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+# Fast Federated Learning in the Presence of Arbitrary Device Unavailability
+
+Xinran Gu∗ IIIS Tsinghua University gxr21@mails.tsinghua.edu.cn
+
+Kaixuan Huang∗ ECE Princeton University kaixuanh@princeton.edu
+
+Jingzhao Zhang EECS Massachusetts Institute of Technology jzhzhang@mit.edu
+
+Longbo Huang† IIIS Tsinghua University longbohuang@tsinghua.edu.cn
+
+# Abstract
+
+Federated Learning (FL) coordinates with numerous heterogeneous devices to collaboratively train a shared model while preserving user privacy. Despite its multiple advantages, FL faces new challenges. One challenge arises when devices drop out of the training process beyond the control of the central server. In this case, the convergence of popular FL algorithms such as FedAvg is severely influenced by the straggling devices. To tackle this challenge, we study federated learning algorithms under arbitrary device unavailability and propose an algorithm named Memory-augmented Impatient Federated Averaging (MIFA). Our algorithm efficiently avoids excessive latency induced by inactive devices, and corrects the gradient bias using the memorized latest updates from the devices. We prove that MIFA achieves minimax optimal convergence rates on non-i.i.d. data for both strongly convex and non-convex smooth functions. We also provide an explicit characterization of the improvement over baseline algorithms through a case study, and validate the results by numerical experiments on real-world datasets.
+
+# 1 Introduction
+
+Federated learning is a machine learning setting in which a central server coordinates with a large number of devices to collectively train a shared model [28, 34, 20, 33, 25, 26]. Practical advantages of this training scheme are mainly twofold. First, each device keeps the private data locally and hence preserves its data privacy. Second, federated learning can make use of idle computing resources and lower computation costs. Although federated learning successfully scales up with data sizes and accelerates training via more affordable computing power [43, 38, 36], the collaborative setup leads to new challenges due to large variations among individual computing devices. Our work aims to formulate and investigate the impact of device variations on FL from an optimization perspective.
+
+In FL, a device can differ from its peers in multiple aspects [17, 25]. First, the data distribution and local task can be different among devices. To address the data variation, non-i.i.d. objective models were proposed and analyzed by [26, 18, 19, 42, 25]. We follow this line of work and formulate our optimization objective as a sum of stochastic functions on individual devices (See Eqn. (1)).
+
+A second variation among devices is caused by different computing and communication speeds. One natural way to formulate the variation in computation speeds is to allow asynchronous updates and model the updates as delayed responses. Lots of novel research has studied the problem with different delay models, e.g., [35, 4, 27, 11, 45, 1, 5, 13]. However, the delayed setup assumes that all devices make roughly the same number of (delayed) responses in the end. This behavior may deviate largely from the FL practice, where each device, e.g., personal cell phones, can have very different active duration when participating in the FL training, and hence make different numbers of responses. For this reason, our work aims to address this third discrepancy among devices caused by individual availability patterns.
+
+The third device heterogeneity caused by different availability patterns is less studied in optimization for federated learning problems. In this model, instead of making a delayed response, devices can abort the training halfway, e.g., due to battery level, incoming calls, etc, and fail to return their responses upon the central server’s requests [28, 7, 17]. To handle missing responses, researchers propose algorithms where the central server may collect responses from only a fraction of the devices and make updates [18, 28, 42, 25, 26, 17, 30, 14].
+
+Previous works on collecting responses from a fraction of devices can be divided into two categories. When the response distribution is known, one could collect only the fastest responses and re-weight according to their response probability [17, 26, 30]. This model can be restrictive, as in practice, the exact distribution may not be available and may evolve. Another line of work assumes that the server can arbitrarily decide and sample a set of devices to collect responses accordingly in every communication round [18, 28, 42, 25, 14]. This model does not require knowing the response possibility. However, the response time can be very long if the selected subset contains unavailable devices.
+
+In this work, we address the above limitations by studying federated learning in the presence of arbitrary device unavailability. Within this practical setup, we propose an algorithm that automatically accommodates for the underlying unavailability and allows patterns of the device unavailability to be non-stationary and even adversarial. Furthermore, our algorithm can achieve optimal convergence rates in the presence of device inactivity and automatically reduce to best-known rates if all devices are active. Our contributions are summarized as follows.
+
+• We investigate the federated learning problem with a practical formulation of device participation, which does not require each device to be online according to an (either known or unknown) distribution. • We propose the Memory-augmented Impatient Federated Averaging (MIFA) algorithm that is agnostic to the availability pattern. It efficiently avoids excessive latency induced by inactive devices, successfully exploits the information about the descent direction in stale and noisy gradients, and corrects the gradient bias using the memorized latest updates. • We prove that MIFA achieves minimax optimal convergence rates $\begin{array} { r } { \mathcal { O } \left( \frac { \bar { \tau } _ { T } + 1 } { N K T } \right) } \end{array}$ for smooth, strongly convex functions, and $\mathcal { O } \left( \sqrt { \frac { \bar { \nu } + 1 } { N K T } } \right)$ for smooth, non-convex functions, and establish matching lower bounds. Here, $N , K$ and $T$ stand for the number of devices, local updates and communication rounds respectively. $\bar { \tau } _ { T }$ and $\bar { \nu }$ characterize how actively devices participate in training (see formal definitions in Sections 3, 5 and 6). MIFA also achieves optimal convergence rates in the ideal case when all devices are active. • We provide an explicit characterization of the improvement over baseline algorithms through a case study and empirically verify our results on real-world datasets.
+
+# 2 Related work
+
+Federated learning. Federated Averaging (FedAvg) was first proposed in [28]. [26, 19, 18, 42] provided convergence analysis for FedAvg on non-i.i.d. data and quantified how data heterogeneity degrades the convergence rate. Several variants of FedAvg were designed to deal with data heterogeneity. FedProx [25] adds a proximal term to local objective functions, while FSVRG [21] and SCAFFOLD [18] employ variance reduction techniques.
+
+One line of work focused on variations in computation capabilities among devices [39, 31, 40]. These models assume that responses are delayed but not missing. To address the missing response, some work assumes that the server can actively sample a subset of devices to respond [18, 28, 42, 25, 14] or that the pattern of device availability is known [26, 17, 10, 30]. These results do not generalize to adversarial inactive patterns. [32] discussed the impact of device inactivity on convergence but their proposed algorithm diverges if there exists an inactive device in each round of communication. However, our setup allows adversarial patterns under certain non-distributional assumptions (see Section 5) while our proposed algorithm still achieves convergence.
+
+Asynchronous distributed optimization. Our work is related to literature in the field of traditional asynchronous distributed optimization in that our proposed algorithm uses stale gradients. The problem setup for asynchronous distributed algorithms can be divided into two categories [13]. One is the shared-data (i.i.d.) setting, where all workers can access the whole dataset. In this setting, the local gradient is an unbiased estimator of the global gradient [35, 4, 27, 11, 45, 1]. In contrast, we assume each worker has non-i.i.d. data, and hence the local stochastic gradient can not be viewed as an unbiased estimator of the global gradient.
+
+The other less studied setting in distributed optimization is the distributed-data setting (non-i.i.d.), where data are partitioned among workers. Specifically, [5] proposed an asynchronous incremental aggregated gradient algorithm that uses buffered gradients to update the global model. Unlike our setup, this algorithm evaluates full local gradients, performs only one local step, and was analyzed under the bounded delay assumption. [13] models the delay as stochastic and assumes that the server has knowledge of the distribution, but our formulation is distribution-free. [6] allows workers to perform multiple local steps and communicate with the server at different times, but the authors assume that all workers are available and compute at the same rate.
+
+Comparison with an independent work. While preparing the manuscript, we were unaware of an independent work [41] that investigated the same setup and proposed a similar algorithm called FedLaAvg. Their main theorem established the convergence rate of $\mathcal { O } \left( \sqrt { \frac { \nu _ { \mathrm { m a x } } } { N ^ { 0 . 5 } T } } ( \mathbf { \bar { G } } ^ { 2 } + \sigma ^ { 2 } ) \right)$ for smooth and non-convex problems, where $G ^ { 2 }$ is the uniform upper bound for the squared norm of stochastic gradients and $\nu _ { m a x }$ is the maximum number of inactive rounds. In comparison, we prove the minimax optimal rate of $\mathcal { O } ( \sqrt { \frac { \bar { \nu } } { N K T } } \sigma ^ { 2 } )$ without the bounded gradient assumption, also improving $\nu _ { \mathrm { m a x } }$ to $\bar { \nu }$ . Furthermore, our result achieves a linear speedup in $N$ and $K$ .
+
+Apart from non-convex functions, we also derive the minimax optimal rates for strongly convex smooth functions under the mild assumption that allows for arbitrary and unbounded number of inactive rounds. Both of our results achieve linear speedups in terms of $N$ and $K$ , and automatically recover the best-known rates of FedAvg when all devices are active. We also show that our proposed algorithm achieves acceleration over unbiased baseline algorithms in the presence of stragglers.
+
+# 3 Problem Setup
+
+We consider optimizing the following problem in a Federated Learning setting:
+
+$$
+\begin{array} { r } { \underset { w \in \mathbb { R } ^ { d } } { \operatorname* { m i n } } f ( w ) : = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } f _ { i } ( w ) : = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \mathbb { E } _ { \xi _ { i } } [ f _ { i } ( w , \xi _ { i } ) ] , } \end{array}
+$$
+
+where $w$ is the optimization variable, e.g., parameters of a machine learning model, $N$ is the number of participating devices, $f _ { i }$ is the local loss function on device $i$ , and $\xi _ { i }$ describes the randomness in local data distribution.
+
+In the ideal federated learning setup (see Figure 1 (a)), all devices return responses within similar time, and hence the central server collects all the local updates. In this case, the computation cost is usually measured by the number of local stochastic oracle evaluations, which is proportional to the number of rounds. In a delayed FL setup (see Figure 1 (b)), devices are always active upon the central server’s request but may return responses with a delay. Here, all devices return almost the same number of responses in the long term.
+
+As we discussed, the above setups do not depict a real-world scenario in which a device can have a longer inactive duration than active duration. In such cases, the communication interval is much longer than the local computation time required for each update, and each device generates an unequal number of responses [28, 17]. This motivates our setup in Figure 1 (c).
+
+![](images/dbec5db87e22ee0dc55043253dc815eff5fd652c994f67ee8a04489f10ef06e1.jpg)  
+Figure 1: An illustration of setup. (a) Ideal setup: all devices return their responses within similar time. (b) Delayed setup: all devices are available, but may return responses with a delay. (c) Our setup: devices can be unavailable arbitrarily, and the communication interval is long enough for active devices to return responses.
+
+In our proposed setup, we use $t$ to index the global communication rounds. We say a device participates or is active at round $t$ if it can complete the computation task and send back the update at the end of round $t$ . We define $\boldsymbol { \mathcal { A } } ( t )$ as the set of all active devices at round $t$ . Notice that we make no assumptions on the distribution of the participation patterns of devices and allow them to be arbitrary.
+
+Directly applying FedAvg to the proposed setup can be problematic due to the existence of inactive devices. To accommodate for inactive devices, we discuss three natural variants of FedAvg and their limitations. The detailed algorithms can be found in Appendix A.
+
+• Biased FedAvg. At each communication round, the global model is updated with a direct average of local updates from the active devices. This naive approach induces bias when data distribution and response patterns vary among devices.   
+• FedAvg with device sampling. The server selects a subset of $S$ devices randomly without replacement, and then waits until all devices in the subset $s$ respond. This is how original FedAvg [28] addresses device unavailability. Note that over $T$ communication rounds, the global model is updated less than $T$ times due to waiting. This approach is prone to stragglers and we refer the readers to Section 5.1 for a detailed discussion.   
+• FedAvg with importance sampling [26, 13, 17]. The local updates from the active devices are weighted by the reciprocal of the participation probabilities to avoid bias. This approach is only applicable when the response of each device is i.i.d. over rounds and it requires the knowledge of participation probabilities.
+
+# 4 Memory-augmented Impatient Federated Averaging (MIFA)
+
+In this section, we introduce our algorithm — Memory-augmented Impatient Federated Averaging (MIFA). MIFA maintains an update-array $\{ G ^ { i } \}$ in the memory that stores the latest updates for all devices. As the name suggests, MIFA has two components. First, the algorithm is impatient and avoids waiting for any specific device when facing heterogeneous devices with arbitrary availability. Second, the algorithm augments the received updates of the active devices with the stored updates of the inactive devices to perform averaging.
+
+Specifically, at the beginning of round $t$ , the server broadcasts the latest model parameter $w _ { t }$ to all active devices $\boldsymbol { \mathcal { A } } ( t )$ . After receiving $w _ { t }$ , each active device, say, the $i$ -th device, sets $w _ { t , 0 } ^ { i } = w _ { t }$ and performs $K$ steps of SGD with respect to the local objective function to get $w _ { t , K } ^ { i }$ :
+
+$$
+\boldsymbol w _ { t , k + 1 } ^ { i } = \boldsymbol w _ { t , k } ^ { i } - \eta _ { t } \tilde { \nabla } f _ { i } ( \boldsymbol w _ { t , k } ^ { i } ) , \boldsymbol k = 0 , \cdots , \boldsymbol K - 1 ,
+$$
+
+where $\eta _ { t }$ is the learning rate and $\tilde { \nabla } f _ { i } ( w _ { t , k } ^ { i } )$ is the stochastic gradient evaluated on device $i$ . Next, the server stores the received update $\begin{array} { r } { \frac { 1 } { \eta _ { t } } \big ( w _ { t } - w _ { t , K } ^ { i } \big ) } \end{array}$ in $G ^ { i }$ . Denote by $\{ G _ { t } ^ { i } \}$ the update-array after round $t$ , then we have
+
+$$
+\begin{array} { r } { G _ { t } ^ { i } = \left\{ \begin{array} { l l } { G _ { t - 1 } ^ { i } , } & { \mathrm { ~ i f ~ } i \notin \mathcal { A } ( t ) , } \\ { \frac { 1 } { \eta _ { t } } ( w _ { t } - w _ { t , K } ^ { i } ) , } & { \mathrm { ~ i f ~ } i \in \mathcal { A } ( t ) . } \end{array} \right. } \end{array}
+$$
+
+At the end of round $t$ , the server updates the global model with the average of $\{ G _ { t } ^ { i } \}$ (line 9). In other words, our algorithm MIFA updates the model with the latest available accumulated gradients for all devices.
+
+Algorithm 1 Memory-augmented Impatient Federated Averaging (MIFA)   
+
+<table><tr><td colspan="2">1: Input: initial wi,learning rate {nt }</td></tr><tr><td>2: Server executes:</td><td>1: DeviceUpdate(i, Wt , nt):</td></tr><tr><td>3: initialize Gi ←O,i∈[N]</td><td>2:w,←Wt</td></tr><tr><td>4: fort=1,.,T-1do</td><td>3: for local step k = O,.,K -1 do</td></tr><tr><td>5: broadcast wt to all active devices i ∈ A(t)</td><td> 4:compute stochastic gradient Vfi(wt,)</td></tr><tr><td>6: for each active device ido</td><td>Wk+1←W,k-ntvfi(w，） 5:</td></tr><tr><td>7: G ←DeviceUpdate(i,Wt, nt) 8: end for</td><td>6: end for</td></tr><tr><td>Wt+1←wt-∑1 N 9: Gi</td><td>7:Return (wt - wi,k) to the server</td></tr></table>
+
+MIFA efficiently progresses without waiting for inactive devices and re-uses their latest updates as the surrogate for missing responses. Being impatient accelerates convergence, whereas memory augmentation corrects the update bias. Our algorithm differs from asynchronous algorithms in traditional distributed optimization [35, 4, 27, 11, 45, 1, 13, 6] in that we utilize the noisy updates of inactive devices more than once to avoid biasing against stragglers. In the following part of the paper, we show that MIFA successfully exploits information about the descent direction contained in the stale and noisy gradients.
+
+Discussion on implementation. In practice, to implement MIFA, the server needs to maintain a huge array to store the latest update for each device, which scales with the model size and the total number of devices. To avoid exhausting the server’s memory, one strategy is to distribute the memory consumption among devices. Specifically, each device, say the $i$ -th, stores its previous update $G _ { t _ { i } ^ { \prime } } ^ { \ i }$ computed at round $t _ { i } ^ { \prime }$ in its local memory. When it becomes active and computes $G _ { t } ^ { i }$ , the device sends $G _ { t } ^ { i } - G _ { t _ { i } ^ { \prime } } ^ { i }$ to the server, which is the difference between the current update and the previous one. In this case, the server only needs to maintain the average $\bar { G }$ in the memory and updates it by $\begin{array} { r } { \bar { G } _ { t } = \bar { G } _ { t - 1 } + \frac { 1 } { N } \sum _ { i \in \mathcal { A } ( t ) } ( G _ { t } ^ { \bar { i } } - G _ { t _ { i } ^ { \prime } } ^ { i } ) } \end{array}$ at round $t$ . Then the server updates the global model by $w _ { t + 1 } = w _ { t } - \eta _ { t } \bar { G } _ { t }$ .
+
+# 5 Convergence Analysis for strongly convex objective functions
+
+In this section, we present the convergence results for MIFA on $\mu$ -strongly convex $L$ -smooth functions. Typical examples for the strongly convex case are $\ell _ { 2 }$ regularized logistic regression and linear regression problems.
+
+In order to capture how the unavailability of devices affects algorithm performance, we introduce the following notion to quantify the dynamics of devices in our setting.
+
+Definition 5.1 (Number of inactive rounds). We define the number of inactive rounds of device $i$ at round $t$ as $\tau ( t , i ) = t - \operatorname* { m a x } \{ t ^ { \prime } \mid t ^ { \prime } \leq t , i \in A ( t ^ { \prime } ) \}$ , which is the difference between current round $t$ and the latest round when device $i$ is active.
+
+It can be seen that $\tau ( t , i ) = 0$ if device $i$ is active at round $t$ and $\tau ( t , i ) = \tau ( t - 1 , i ) + 1$ otherwise. Also, $t - \tau ( t , i )$ is the latest round when the device $i$ is active. Next, we present the assumptions made for establishing our convergence theorem.
+
+Assumption 1. $f _ { 1 } , \cdots , f _ { N }$ are all $L$ -smooth, i.e., for all $w$ and $v _ { \scriptscriptstyle { i } }$ , $\begin{array} { r c l } { f _ { i } ( v ) } & { \leq } & { f _ { i } ( w ) \ + } \end{array}$ $\begin{array} { r } { \left. \nabla f _ { i } ( w ) , v - w \right. + \frac { L } { 2 } \left\| w - v \right\| ^ { 2 } } \end{array}$ .
+
+Assumption 2. $\tilde { \nabla } f _ { i } ( \boldsymbol { w } )$ is an unbiased estimator of $\nabla f _ { i }$ with variance bounded by $\sigma ^ { 2 }$ , i.e., $\mathbb { E } _ { \boldsymbol { \xi } } \left[ \tilde { \nabla } f _ { i } ( \boldsymbol { w } ) \right] = \nabla f _ { i } ( \boldsymbol { w } ) , \mathbb { E } _ { \boldsymbol { \xi } } \left[ \left\| \tilde { \nabla } f _ { i } ( \boldsymbol { w } ) - \nabla f _ { i } ( \boldsymbol { w } ) \right\| ^ { 2 } \right] \leq \sigma ^ { 2 } .$ .
+
+Assumption 3. $f _ { 1 } , \cdots , f _ { N }$ are all $\mu$ -strongly convex: for all $w$ and $v _ { \scriptscriptstyle { i } }$ , $f _ { i } ( v ) \geq f _ { i } ( w ) +$ $\begin{array} { r } { \left. \nabla f _ { i } ( w ) , v - w \right. + \frac { \mu } { 2 } \left\| w - v \right\| ^ { 2 } } \end{array}$ .
+
+Assumption 4. There exists a constant $t _ { 0 } > 0$ , such that for all $t \geq 1$ and $i \in [ N ]$ , the number of inactive rounds of device $i$ at communication round $t$ satisfies $\begin{array} { r } { \tau ( t , i ) \leq t _ { 0 } + \frac { 1 } { b } \dot { t } } \end{array}$ , where $b =$ $4 0 \left( L / \mu \right) ^ { 1 . 5 }$ .
+
+Assumptions 1, 2, and 3 are standard and common in the FL literature, e.g., [26, 18, 19, 42, 36]. In Assumption 2, we relax the bounded gradient assumption that is often required in prior work, e.g., [6, 26, 40, 1]. Lastly, Assumption 4 is a very mild assumption on device availability, since it allows the number of inactive rounds to grow as ${ \mathcal { O } } ( t )$ . In contrast, existing results on asynchronous updates mostly assume a bounded or fixed latency, e.g., [6, 1, 5, 40, 35, 4].
+
+We are now ready to present our first convergence result. Define $\begin{array} { r l } { D = } & { { } \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \left. \nabla f _ { i } ( w _ { * } ) \right. ^ { 2 } } \end{array}$ to measure data dissimilarity, where $w _ { * } = \arg \operatorname* { m i n } f ( w )$ is the global optimum. Also, define $\bar { \tau } _ { T }$ and $\tau _ { \operatorname* { m a x } , T }$ to be the average and maximum numbers of inactive rounds $\tau ( t , i )$ across all devices and rounds, respectively. That is,
+
+$$
+\bar { \tau } _ { T } = \frac { 1 } { N ( T - 1 ) } \sum _ { t = 1 } ^ { T - 1 } \sum _ { i = 1 } ^ { N } \tau ( t , i ) , \tau _ { \operatorname* { m a x } , T } = \operatorname* { m a x } _ { i \in [ N ] } \operatorname* { m a x } _ { 1 \leq t \leq T - 1 } \tau ( t , i ) .
+$$
+
+The following theorem summarizes the performance of MIFA in this case.
+
+Theorem 5.1. Assume that Assumptions $^ { l }$ to 3 hold. Further assume that the device availability sequence $\tau ( t , i )$ satisfies Assumption $^ { 4 }$ and $\tau ( 1 , i ) = 0$ for all $i \in [ N ]$ . By setting the learning rate $\begin{array} { r } { \eta _ { t } = \frac { 4 } { \mu K ( t + a ) } } \end{array}$ with $a = \operatorname* { m a x } \{ 1 0 0 , 4 0 t _ { 0 } \} ( L / \mu ) ^ { 1 . 5 }$ , after $T - 1$ communication rounds, MIFA satisfies:
+
+$$
+\mathbb { E } _ { \xi } \left[ f ( \overline { { w } } _ { T } ) \right] - f ( w _ { * } ) = \mathcal { O } \left( \frac { \bar { \tau } _ { T } + 1 } { \mu N K T } \sigma ^ { 2 } + \frac { \tau _ { \operatorname* { m a x } , T } ^ { 2 } A _ { 1 } + ( K - 1 ) ^ { 2 } A _ { 2 } + A _ { 3 } } { \mu ^ { 2 } T ^ { 2 } } \right) ,
+$$
+
+where $\overline { { w } } _ { T }$ is a weighted average of $w _ { t }$ defined as:
+
+$$
+\overline { { w } } _ { T } = \frac { 1 } { W _ { T } } \sum _ { t = 1 } ^ { T } ( t + a - 1 ) ( t + a - 2 ) w _ { t } , W _ { T } = \sum _ { t = 1 } ^ { T } ( t + a - 1 ) ( t + a - 2 ) ,
+$$
+
+Our results hold under Assumption 4, which allows for arbitrary device availability sequences with $\tau _ { \mathrm { m a x } , T } = \mathcal { O } ( T )$ . However, for MIFA to converge, we require $\tau _ { \operatorname* { m a x } , T } = o ( T )$ and $\dot { t _ { 0 } } = \bar { o } ( T )$ . When ( NK(τ2max,T +t20)τ¯+1 ), the first term dominates and the impact of the second O(1/T 2) term is negligible. In fact the first term in Theorem 5.1 is minimax optimal by our information-theoretic lower bound for the problem in the next proposition.
+
+Proposition 5.1. Let $c _ { 0 } > 0$ be a universal constant. For any potentially randomized algorithm, there exists a stochastic strongly convex problem satisfying Assumptions $^ { l }$ to 3, such that the output $w _ { T }$ after $T$ rounds of communication has expected sub-optimality lower bounded by
+
+$$
+\mathbb { E } [ f ( w _ { T } ) - f ( w ^ { * } ) ] \geq c _ { 0 } \frac { \bar { \tau } _ { T } \sigma ^ { 2 } } { \mu N K T } .
+$$
+
+The proof is based on the observation that the number of gradient evaluation can scale inversely with $\bar { \tau } _ { T }$ and that the oracle complexity is tight even for centralized stochastic optimization problems. The optimality of the first term in Theorem 5.1 is independent of the distributed or the FL setup.
+
+The second term in Theorem 5.1 converges at the rate $\mathcal { O } ( 1 / T ^ { 2 } )$ and consists of three parts, where the first part reflects the slowdown caused by device unavailability through $\tau _ { \operatorname* { m a x } , T }$ , the second part shows the effect of multiple $( K > 1 )$ ) local steps, and the third part tells how the initial error decreases.
+
+Remark 5.1. When $\tau ( t , i ) ~ = ~ 0$ for all $i$ and $t _ { : }$ , our setup reduces to FedAvg with full device pbound $\bar { \tau } _ { T } ~ = ~ 0$ d  , $\tau _ { \mathrm { m a x } , T } ~ = ~ 0$ . Iate $\begin{array} { r } { \mathcal { O } \left( \frac { \sigma ^ { 2 } } { \mu K N T } + \frac { L ( \sigma ^ { 2 } / K + D + L ^ { 2 } \| w _ { 1 } - w _ { * } \| ^ { 2 } ) } { \mu ^ { 2 } T ^ { 2 } } \right) } \end{array}$ $\begin{array} { r } { \mathcal { O } \bigg ( \frac { \sigma ^ { 2 } \log T } { \mu K N T } + \frac { L ( \sigma ^ { 2 } / K + D ) ( \log T ) ^ { 2 } } { \mu ^ { 2 } T ^ { 2 } } + } \end{array}$ $\begin{array} { r } { \mu \left. w _ { 1 } - w _ { * } \right. ^ { 2 } \exp ( - \frac { \mu } { 4 8 L } T ) \Bigr ) } \end{array}$ in [18] (Thm. V. $B ^ { 2 } = 2 , \eta _ { g } = 1 ,$ ) up to logarithmic terms. Besides, in the general case, our $\mathcal { O } ( \tau _ { \mathrm { m a x } , T } ^ { 2 } A _ { 1 } / T ^ { 2 } )$ term matches the last term in $I 6 J \left( C o r . 5 \right)$ .
+
+Remark 5.2. Our analysis relies on the technical assumption that all devices respond in the first round. Intuitively, this is because we need at least one valid stochastic gradient evaluation for each device to get a complete picture of the global objective, or otherwise any update would be biased. In practice, this can be achieved by waiting for the updates from all devices on $w _ { 1 }$ at the very beginning.
+
+# 5.1 Case Study: i.i.d. Bernoulli participation
+
+Though our algorithm can be applied to non-stationary and non-independent response patterns, we show in this subsection that even in the simple i.i.d. Bernoulli participation scenario our algorithm can achieve considerable improvement compared to known algorithms. In particular, we consider a setup where each device becomes active independently with a fixed probability $p _ { i }$ . It serves as the first motivating example towards modeling the participation patterns of devices, and provides a clean view of how the heterogeneity of the device participation influences the Federated optimization algorithms.
+
+We will show that in this scenario, Assumption 4 holds with high probability, and the terms involving the inactive rounds $\tau ( t , i )$ in Theorem 5.1 can also be bounded. Furthermore, we theoretically demonstrate that algorithms such as FedAvg [28] and SCAFFOLD [18], which sample $S$ devices for each global update, are more prone to stragglers than our algorithm.
+
+Definition 5.2. Assume that for all $i \in [ N ]$ , the i-th device is assigned with a probability $p _ { i }$ . We say the participation of the devices follows i.i.d. Bernoulli participation model with participation probabilities $\{ p _ { i } \}$ , if (1). at the first round, all devices are active, and (2). at round $t > 1$ , device $i$ is active with probability $p _ { i }$ , which is independent of the history and other devices.
+
+Next theorem shows that under i.i.d. Bernoulli participation scenario, with high probability, $\tau ( t , i )$ only grows logarithmically in $t$ . Also Assumption 4 holds for a mild choice of $t _ { 0 }$ .
+
+Theorem 5.2. For i.i.d. Bernoulli participation model defined in Definition 5.2, given any $\delta > 0$ , with probability at least $1 - \delta$ , we have the following holds for all $t \geq 1$ and $i \in [ N ]$ simultaneously,
+
+$$
+\tau ( t , i ) \leq \mathcal { O } \Big ( \frac { 1 } { p _ { i } } ( \log ( N t / \delta ) + 1 ) \Big ) .
+$$
+
+Furthermore, $( l )$ . Assumption $^ { 4 }$ holds true $\begin{array} { r } { \mathrm { \ddot { \it f } } t _ { 0 } = \Omega \bigg ( \frac { 1 } { p _ { m i n } } \log \frac { b N } { p _ { m i n } \delta } \bigg ) } \end{array}$ , where $p _ { m i n } = \operatorname* { m i n } \{ p _ { i } \}$ , and $b = 4 0 ( L / \mu ) ^ { 1 . 5 }$ ;(2). $\tau _ { \operatorname* { m a x } , T }$ can be upper bounded as
+
+$$
+\tau _ { \operatorname* { m a x } , T } \leq \mathcal { O } \Big ( \frac { 1 } { p _ { m i n } } \cdot \big ( \log ( T N / \delta ) + 1 \big ) \Big ) .
+$$
+
+The next theorem provides a high probability upper bound for $\bar { \tau } _ { T }$ .
+
+Theorem 5.3. For i.i.d. Bernoulli participation model defined in Definition 5.2, given any $\delta > 0$ and $T > 1$ , with probability at least $1 - \delta$ , we have
+
+$$
+\bar { \tau } _ { T } \leq \Big ( \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \frac { 1 } { p _ { i } } \Big ) \cdot \mathcal { O } \Big ( 1 + \log \frac { 1 } { \delta } \Big ) .
+$$
+
+By Theorem 5.2 and Theorem 5.3, we conclude that the dominant term of our convergence bound is $\begin{array} { r } { \stackrel { \triangledown } { \mathcal { O } } \left( \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \frac { 1 } { p _ { i } } \cdot \frac { \sigma ^ { 2 } } { \mu N K T } \right) } \end{array}$ . Therefore, to achieve $\epsilon$ accuracy, the dominant term of the number of the required rounds is
+
+$$
+T _ { \epsilon } ^ { \left( \mathrm { M I F A } \right) } = \widetilde { \mathcal { O } } \Bigl ( \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \frac { 1 } { p _ { i } } \cdot \frac { \sigma ^ { 2 } } { \mu N K \epsilon } \Bigr ) .
+$$
+
+For both FedAvg and SCAFFOLD that sample $S$ devices uniformly at random, [18] (Thm I. & III.) showed that the dominant term of the number of global updates needed to achieve $\epsilon$ accuracy is $\begin{array} { r } { R _ { \epsilon } = \widetilde { \mathcal { O } } \left( \frac { \sigma ^ { 2 } } { \mu S K \epsilon } \right) } \end{array}$ . Notice that in our setting, to accomplish each global update, the server needs to wait for a few rounds for the $S$ devices to respond. Let $T ( S )$ be the expected rounds for which the server needs to wait for the selected devices $\boldsymbol { S }$ to be active. Then the expected total rounds to achieve $\epsilon$ accuracy is $R _ { \epsilon } \cdot \mathbb { E } _ { S } \left[ T ( S ) \right]$ . For i.i.d. Bernoulli participation model, we have $\begin{array} { r } { T ( S ) \geq \frac { 1 } { \operatorname* { m i n } \{ p _ { i } | i \in S \} } } \end{array}$ , and we can further show that $\begin{array} { r } { \mathbb { E } _ { S } \left[ T ( S ) \right] \ge \frac { 1 } { p _ { m i n } } \frac { S } { N } } \end{array}$ ≥ 1p min SN ( see Appendix D.3 for details). Therefore,
+
+$$
+\mathbb { E } \left[ T _ { \epsilon } ^ { ( \mathsf { F e d A v g , S C A F F O L D } ) } \right] \geq \frac { S } { N } \frac { 1 } { p _ { m i n } } \widetilde { \mathcal { O } } \Big ( \frac { \sigma ^ { 2 } } { \mu S K \epsilon } \Big ) = \widetilde { \mathcal { O } } \Big ( \frac { 1 } { p _ { m i n } } \cdot \frac { \sigma ^ { 2 } } { \mu N K \epsilon } \Big ) .
+$$
+
+By comparing Eqn. 2 and Eqn. 3, we see that both FedAvg and SCAFFOLD are more vulnerable to stragglers, that is, the devices with very small participation probabilities; on the contrary, the convergence rate of MIFA only depends on the average of $1 / p _ { i }$ instead of $1 / p _ { m i n }$ . We also provide empirical experiments showing that MIFA converges faster than FedAvg in Section 7.
+
+# 6 Convergence result for non-convex objective functions
+
+In this section, we present the convergence guarantee of MIFA for the non-convex case. First we list the additional assumptions as below.
+
+Assumption 5 (Hessian Lipschitz). $f _ { 1 } , \cdots , f _ { N }$ are all $\rho$ -Hessian Lipschitz: for all $w$ and $v$ $\bigl \| \nabla ^ { 2 } f _ { i } \bigl ( \tilde { \boldsymbol { w } } \bigr ) - \nabla ^ { 2 } f _ { i } ( \boldsymbol { v } ) \bigr \| \leq \dot { \rho } \bigl \| \boldsymbol { w } - \boldsymbol { \tilde { v } } \bigr \|$ .
+
+Assumption 6 (Bounded noise). The noise of the local stochastic gradients is upper bounded by $a$ constant $\delta$ almost surely: $\begin{array} { r } { \left\| \tilde { \nabla } f _ { i } ( w ) - \nabla f _ { i } ( w ) \right\| \le \delta a . s . , \forall i \in [ N ] } \end{array}$ .
+
+Assand y). There exist . Furthermore, $\alpha > 0$ anfine $\beta _ { i } > 0$ at for all . $w$ $\in [ N ] \colon \left\| \nabla f _ { i } ( w ) \right\| ^ { 2 } \leq \alpha \left\| \nabla f ( w ) \right\| ^ { 2 } + \beta _ { i }$ $\begin{array} { r } { \beta = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \beta _ { i } } \end{array}$
+
+Assumption 8. There exists a constant $\nu _ { i }$ such that $\tau ( t , i ) \ \leq \ \nu _ { i }$ , for all $i \in [ N ]$ and $t \geq 1$ Furthermore, define ν¯ = 1N PNi=1 νi and νmax = maxi∈[N] νi.
+
+The analysis of non-convex functions is much more technically involved, and our results rely on strong assumptions that provide a finer control of the gradient difference (Assumption 5), gradient noise (Assumption 6), gradient dissimilarity among devices (Assumption 7), and device unavailability (Assumption 8). We remark that Assumption 5 is also made in [9, 16], and Assumption 7 is also made in [18, 39]. We leave it as future work to study whether and how MIFA converges for non-convex functions with weaker assumptions.
+
+Theorem 6.1. Assume that Assumptions $I , ~ 2 , ~$ , and $^ { 5 }$ to 7 hold. Further assume that the device availability sequence $\tau ( t , i )$ satisfies Assumption 8 and $\tau ( 1 , i ) = 0$ for all $i \in [ N ]$ . By using $a$ learning rate η = q $\begin{array} { r } { \eta = \sqrt { \frac { N } { K T L \left( 1 + \bar { \nu } \right) } } } \end{array}$ NKT L(1+¯ν), for T ≥ max{32αLNK, 16LNK , 8KNν2max(L2+ρδ) }, after T − 1 communication rounds, MIFA satisfies:
+
+$$
+\operatorname* { m i n } _ { 1 \leq t \leq T } \mathbb { E } _ { \xi } \left[ \left\| \nabla f ( w _ { t } ) \right\| ^ { 2 } \right] = \mathcal { O } \left( \sqrt { \frac { ( 1 + \bar { \nu } ) L } { T K N } } ( f ( w _ { 1 } ) - f ^ { * } + \sigma ^ { 2 } ) + \frac { A _ { 4 } + A _ { 5 } } { T } \right) ,
+$$
+
+where $f ^ { * }$ is the optimal value, and:
+
+$$
+\begin{array} { l } { { A _ { 4 } = N K L \left( \alpha \sigma ^ { 2 } \hat { \nu } + \frac { \sigma ^ { 2 } \nu _ { \mathrm { m a x } } } { \sqrt { K N } } + \sigma \nu _ { \mathrm { m a x } } \sqrt { \beta } \right) + \frac { { ( L ^ { 2 } + \rho \delta ) \sigma ^ { 2 } \nu _ { \mathrm { m a x } } } } { L } , } } \\ { { A _ { 5 } = \frac { { ( K - 1 ) N L ( \beta + \sigma ^ { 2 } / K ) } } { \hat { \nu } + 1 } . } } \end{array}
+$$
+
+Next, we show that the leading $\mathcal { O } ( 1 / \sqrt { T } )$ term is theoretically optimal for zero-respecting algorithms.
+
+Proposition 6.1. Let $c _ { 0 } > 0$ be a universal constant. For any randomized zero-respecting algorithm, there exists a stochastic non-convex problem satisfying Assumption1, 2, 5 and 7, such that the output $w _ { T }$ after $T$ rounds of communication has expected sub-optimality lower bounded by
+
+$$
+\mathbb { E } [ \| \nabla f ( w _ { T } ) \| ^ { 2 } ] \ge \mathbb { E } [ \| \nabla f ( w _ { T } ) \| ] ^ { 2 } \ge c _ { 0 } \sqrt { \frac { \bar { \nu } L \sigma ^ { 2 } ( f ( w _ { 0 } ) - f ^ { * } ) } { N K T } } .
+$$
+
+The above proposition show that when $\sigma \sqrt { ( f ( w _ { 0 } ) - f ^ { * } ) } \sim \sigma ^ { 2 } + ( f ( w _ { 0 } ) - f ^ { * } )$ , the result in Theorem 6.1 is tight. However, note that the counter example we used requires the quantity $\delta$ in Assumption 6 to scale with $T$ , hence requiring $\delta$ to be large enough. This does not change the optimality of the first term as the first term is independent of $\delta$ . Whether this requirement can be relaxed is left as an open problem.
+
+Remark 6.1. When all $\nu _ { i } = 0$ (i.e. all the devices are active), our convergence bound reduces to $\begin{array} { r } { \mathcal { O } \Big ( \sqrt { \frac { L } { T K N } } ( f ( w _ { 1 } ) - f ^ { * } + \sigma ^ { 2 } ) + \frac { ( K - 1 ) N L ( \beta + \sigma ^ { 2 } / K ) } { T } \Big ) } \end{array}$ . This matches the result in [42] (Thm. 1, $\begin{array} { r } { \eta = 1 , \eta _ { L } = \sqrt { \frac { N } { K T L } } ) } \end{array}$ .
+
+# 7 Numerical Experiments
+
+In this section, we conduct numerical experiments3. to verify our theoretical results and investigate how the heterogeneity of the device availability influences the Federated optimization algorithms. We compare the performance of the following four algorithms: FedAvg with importance sampling (FedAvg-IS), Biased FedAvg, FedAvg with device sampling, and our proposed MIFA. For the detailed discussions of the algorithms, we refer the readers to Sections 3 and 4. We remark that for a fair comparison, we deliberately include the first few rounds that MIFA needs to wait to receive responses from all devices for initializing the update-array $\{ G ^ { i } \}$ .
+
+Following [26, 25], we construct non-i.i.d. datasets from two commonly used computer vision datasets — MNIST [23] and CIFAR-10 [22] . Specifically, we divide the data into $N = 1 0 0$ devices with each device holding samples of only two classes, which creates a high level of data heterogeneity. For simplicity, we ensure that each device holds the same number of samples. We do not use any data augmentation. We use multinomial logistic regression as the convex model and LeNet-5 [24] with ReLU activations as the non-convex model. For all experiments, we use weight decay in the training process, which corresponds to adding $\ell _ { 2 }$ penalty. We use logistic models for MNIST dataset, while we use LeNet-5 for CIFAR-10. Our code is adapted from [26], which is under MIT License.
+
+We model the availability of the devices as independent Bernoulli random trials. The $i \cdot$ -th device is assigned with a probability $p _ { i }$ , where at each time step, the device becomes active with probability $p _ { i }$ . In our experiments, the $p _ { i }$ ’s are chosen such that devices holding data of smaller labels participate less frequently. Specifically, if the $i$ -th device holds the data of label $j$ and $k$ , we set $\bar { p } _ { i } = \bar { p _ { \operatorname* { m i n } } } \operatorname* { m i n } ( j , k ) / 9 + \bar { ( 1 - p _ { \operatorname* { m i n } } ) }$ , where $p _ { \mathrm { m i n } }$ controls the lower bound of the participation probabilities. The correlation between the participation patterns and local datasets increases the difficulty of the problem [17]. To investigate this phenomenon, we repeat the experiments for $p _ { \mathrm { m i n } } = 0 . 1$ and 0.2. We control the randomness of device participation when testing different algorithms.
+
+In all the experiments, we set the initial learning rate to be $\eta _ { 0 } = 0 . 1$ and decay the learning rate as $\begin{array} { r } { \eta _ { t } = \eta _ { 0 } \cdot \frac { 1 } { t } } \end{array}$ . We set the weight decay to be 0.001. The local batch size is 100 and each local update consists of 2 epochs. Therefore, the actual number of local steps $K$ depends on the size of the dataset. We run all the experiments with 4 GPUs of type GeForce RTX 2080 Ti. We repeat the experiments for 5 different random seeds, and all of the experiments exhibit similar training curves. We report the averaged training loss and test accuracy with error bars in Figure 2.
+
+We observe that FedAvg with device sampling (FedAvg ( $S = 5 0$ ) and FedAvg $S = 1 0 0$ ) in Figure 2) is severely influenced by the straggling devices and makes progress relatively slowly compared to the other algorithms. Although biased FedAvg converges fast at the beginning, this simple algorithm is biased, and the optimality gaps are prominent for the harder CIFAR-10 dataset and when $p _ { \mathrm { m i n } }$ is small. On the contrary, our proposed MIFA avoids waiting for stragglers, converges fast without bias, and is competitive with FedAvg with importance sampling, which requires knowledge of the participation probabilities. We refer the readers to Appendix G for additional experiments on CIFAR-10.
+
+# 8 Conclusions and Discussions
+
+In this paper, we study FL algorithms in the presence of arbitrary device unavailability and propose MIFA, which avoids waiting for straggling devices and re-uses the memorized latest updates as the surrogate when the device is unavailable. We theoretically analyze MIFA without any structural assumptions on the device availability and prove the convergence for strongly convex and non-convex smooth functions. Different from the literature that studies oracle complexity in terms of stochastic gradient evaluations, we argue that in federated learning system, the bottleneck lies in the nonstationary and possibly adversarial pattern of device participation. Therefore, it is important to study how the number of inactive rounds influences the convergence rate. In Theorem 5.1, the dependency upon $\tau _ { \operatorname* { m a x } , T }$ might be an artifact of our analysis, and a future direction is to study whether we can remove this dependency. Another important direction is to analyze algorithms for non-convex functions under weaker assumptions.
+
+![](images/c1142e9ecbf7a3052e86380ac3f4c818747910b17c29f238aaf9134bafeb112a.jpg)  
+Figure 2: Training losses and test accuracies. Fig. 2(a)–2(d): logistic models on non-iid MNIST. Fig. 2(e)–2(h): LeNet-5 on non-iid CIFAR-10. FedAvg $S = 5 0$ ) and FedAvg $S = 1 0 0$ ) refer to FedAvg with device sampling that samples $S$ devices for each global update. FedAvg-IS is short for FedAvg with importance sampling, which requires knowledge of the participation probabilities.
+
+# Acknowledgments and Disclosure of Funding
+
+The work of Xinran Gu and Longbo Huang is supported in part by the Technology and Innovation Major Project of the Ministry of Science and Technology of China under Grant 2020AAA0108400 and2020AAA0108403. Xinran Gu would like to thank Kaifeng Lyu for the discussion on the convergence analysis. The authors would like to thank Sai Praneeth Karimireddy for the discussion on the problem settings.
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See Section 8.   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] Our work is mainly theoretical and targets at providing theoretical analysis of the FL algorithms, which has almost no negative societal impacts.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] The assumptions are rigorously stated in Section 5 and Section 6.   
+(b) Did you include complete proofs of all theoretical results? [Yes] All the proofs are deferred to the appendix.
+
+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] Our code will be included in the supplemental material. The datasets we use are public. The instructions needed to reproduce the main experimental results are clearly stated in Section 7.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 7.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Section 7.   
+(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] See Section 7.
+
+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] See Section 7.   
+(b) Did you mention the license of the assets? [Yes] See Section 7.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We include our code for reproducing the experiments in the supplemental material.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] We choose the commonly-used MNIST and CIFAR-10 datasets.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] We choose the commonly-used MNIST and CIFAR-10 datasets.
+
+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]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
\ No newline at end of file
diff --git a/parse/train/1_gaHBaRYt/1_gaHBaRYt_content_list.json b/parse/train/1_gaHBaRYt/1_gaHBaRYt_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..7961824bbd98037a484d1c55bc1322e9ea7ee6a7
--- /dev/null
+++ b/parse/train/1_gaHBaRYt/1_gaHBaRYt_content_list.json
@@ -0,0 +1,1665 @@
+[
+    {
+        "type": "text",
+        "text": "Fast Federated Learning in the Presence of Arbitrary Device Unavailability ",
+        "text_level": 1,
+        "bbox": [
+            238,
+            122,
+            761,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Xinran Gu∗ IIIS Tsinghua University gxr21@mails.tsinghua.edu.cn ",
+        "bbox": [
+            245,
+            226,
+            480,
+            282
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Kaixuan Huang∗ ECE Princeton University kaixuanh@princeton.edu ",
+        "bbox": [
+            558,
+            227,
+            751,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jingzhao Zhang EECS Massachusetts Institute of Technology jzhzhang@mit.edu ",
+        "bbox": [
+            225,
+            303,
+            478,
+            358
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Longbo Huang† IIIS Tsinghua University longbohuang@tsinghua.edu.cn ",
+        "bbox": [
+            537,
+            303,
+            771,
+            358
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            395,
+            535,
+            411
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Federated Learning (FL) coordinates with numerous heterogeneous devices to collaboratively train a shared model while preserving user privacy. Despite its multiple advantages, FL faces new challenges. One challenge arises when devices drop out of the training process beyond the control of the central server. In this case, the convergence of popular FL algorithms such as FedAvg is severely influenced by the straggling devices. To tackle this challenge, we study federated learning algorithms under arbitrary device unavailability and propose an algorithm named Memory-augmented Impatient Federated Averaging (MIFA). Our algorithm efficiently avoids excessive latency induced by inactive devices, and corrects the gradient bias using the memorized latest updates from the devices. We prove that MIFA achieves minimax optimal convergence rates on non-i.i.d. data for both strongly convex and non-convex smooth functions. We also provide an explicit characterization of the improvement over baseline algorithms through a case study, and validate the results by numerical experiments on real-world datasets. ",
+        "bbox": [
+            233,
+            428,
+            766,
+            621
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            650,
+            310,
+            667
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Federated learning is a machine learning setting in which a central server coordinates with a large number of devices to collectively train a shared model [28, 34, 20, 33, 25, 26]. Practical advantages of this training scheme are mainly twofold. First, each device keeps the private data locally and hence preserves its data privacy. Second, federated learning can make use of idle computing resources and lower computation costs. Although federated learning successfully scales up with data sizes and accelerates training via more affordable computing power [43, 38, 36], the collaborative setup leads to new challenges due to large variations among individual computing devices. Our work aims to formulate and investigate the impact of device variations on FL from an optimization perspective. ",
+        "bbox": [
+            174,
+            684,
+            823,
+            795
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In FL, a device can differ from its peers in multiple aspects [17, 25]. First, the data distribution and local task can be different among devices. To address the data variation, non-i.i.d. objective models were proposed and analyzed by [26, 18, 19, 42, 25]. We follow this line of work and formulate our optimization objective as a sum of stochastic functions on individual devices (See Eqn. (1)). ",
+        "bbox": [
+            174,
+            801,
+            823,
+            857
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A second variation among devices is caused by different computing and communication speeds. One natural way to formulate the variation in computation speeds is to allow asynchronous updates and model the updates as delayed responses. Lots of novel research has studied the problem with different delay models, e.g., [35, 4, 27, 11, 45, 1, 5, 13]. However, the delayed setup assumes that all devices make roughly the same number of (delayed) responses in the end. This behavior may deviate largely from the FL practice, where each device, e.g., personal cell phones, can have very different active duration when participating in the FL training, and hence make different numbers of responses. For this reason, our work aims to address this third discrepancy among devices caused by individual availability patterns. ",
+        "bbox": [
+            174,
+            90,
+            825,
+            215
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The third device heterogeneity caused by different availability patterns is less studied in optimization for federated learning problems. In this model, instead of making a delayed response, devices can abort the training halfway, e.g., due to battery level, incoming calls, etc, and fail to return their responses upon the central server’s requests [28, 7, 17]. To handle missing responses, researchers propose algorithms where the central server may collect responses from only a fraction of the devices and make updates [18, 28, 42, 25, 26, 17, 30, 14]. ",
+        "bbox": [
+            174,
+            222,
+            825,
+            305
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Previous works on collecting responses from a fraction of devices can be divided into two categories. When the response distribution is known, one could collect only the fastest responses and re-weight according to their response probability [17, 26, 30]. This model can be restrictive, as in practice, the exact distribution may not be available and may evolve. Another line of work assumes that the server can arbitrarily decide and sample a set of devices to collect responses accordingly in every communication round [18, 28, 42, 25, 14]. This model does not require knowing the response possibility. However, the response time can be very long if the selected subset contains unavailable devices. ",
+        "bbox": [
+            173,
+            311,
+            825,
+            422
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we address the above limitations by studying federated learning in the presence of arbitrary device unavailability. Within this practical setup, we propose an algorithm that automatically accommodates for the underlying unavailability and allows patterns of the device unavailability to be non-stationary and even adversarial. Furthermore, our algorithm can achieve optimal convergence rates in the presence of device inactivity and automatically reduce to best-known rates if all devices are active. Our contributions are summarized as follows. ",
+        "bbox": [
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+            428,
+            825,
+            512
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We investigate the federated learning problem with a practical formulation of device participation, which does not require each device to be online according to an (either known or unknown) distribution. • We propose the Memory-augmented Impatient Federated Averaging (MIFA) algorithm that is agnostic to the availability pattern. It efficiently avoids excessive latency induced by inactive devices, successfully exploits the information about the descent direction in stale and noisy gradients, and corrects the gradient bias using the memorized latest updates. • We prove that MIFA achieves minimax optimal convergence rates $\\begin{array} { r } { \\mathcal { O } \\left( \\frac { \\bar { \\tau } _ { T } + 1 } { N K T } \\right) } \\end{array}$ for smooth, strongly convex functions, and $\\mathcal { O } \\left( \\sqrt { \\frac { \\bar { \\nu } + 1 } { N K T } } \\right)$ for smooth, non-convex functions, and establish matching lower bounds. Here, $N , K$ and $T$ stand for the number of devices, local updates and communication rounds respectively. $\\bar { \\tau } _ { T }$ and $\\bar { \\nu }$ characterize how actively devices participate in training (see formal definitions in Sections 3, 5 and 6). MIFA also achieves optimal convergence rates in the ideal case when all devices are active. • We provide an explicit characterization of the improvement over baseline algorithms through a case study and empirically verify our results on real-world datasets. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            775,
+            318,
+            792
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Federated learning. Federated Averaging (FedAvg) was first proposed in [28]. [26, 19, 18, 42] provided convergence analysis for FedAvg on non-i.i.d. data and quantified how data heterogeneity degrades the convergence rate. Several variants of FedAvg were designed to deal with data heterogeneity. FedProx [25] adds a proximal term to local objective functions, while FSVRG [21] and SCAFFOLD [18] employ variance reduction techniques. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "One line of work focused on variations in computation capabilities among devices [39, 31, 40]. These models assume that responses are delayed but not missing. To address the missing response, some work assumes that the server can actively sample a subset of devices to respond [18, 28, 42, 25, 14] or that the pattern of device availability is known [26, 17, 10, 30]. These results do not generalize to adversarial inactive patterns. [32] discussed the impact of device inactivity on convergence but their proposed algorithm diverges if there exists an inactive device in each round of communication. However, our setup allows adversarial patterns under certain non-distributional assumptions (see Section 5) while our proposed algorithm still achieves convergence. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+            90,
+            825,
+            174
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Asynchronous distributed optimization. Our work is related to literature in the field of traditional asynchronous distributed optimization in that our proposed algorithm uses stale gradients. The problem setup for asynchronous distributed algorithms can be divided into two categories [13]. One is the shared-data (i.i.d.) setting, where all workers can access the whole dataset. In this setting, the local gradient is an unbiased estimator of the global gradient [35, 4, 27, 11, 45, 1]. In contrast, we assume each worker has non-i.i.d. data, and hence the local stochastic gradient can not be viewed as an unbiased estimator of the global gradient. ",
+        "bbox": [
+            174,
+            190,
+            825,
+            289
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The other less studied setting in distributed optimization is the distributed-data setting (non-i.i.d.), where data are partitioned among workers. Specifically, [5] proposed an asynchronous incremental aggregated gradient algorithm that uses buffered gradients to update the global model. Unlike our setup, this algorithm evaluates full local gradients, performs only one local step, and was analyzed under the bounded delay assumption. [13] models the delay as stochastic and assumes that the server has knowledge of the distribution, but our formulation is distribution-free. [6] allows workers to perform multiple local steps and communicate with the server at different times, but the authors assume that all workers are available and compute at the same rate. ",
+        "bbox": [
+            173,
+            294,
+            825,
+            405
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Comparison with an independent work. While preparing the manuscript, we were unaware of an independent work [41] that investigated the same setup and proposed a similar algorithm called FedLaAvg. Their main theorem established the convergence rate of $\\mathcal { O } \\left( \\sqrt { \\frac { \\nu _ { \\mathrm { m a x } } } { N ^ { 0 . 5 } T } } ( \\mathbf { \\bar { G } } ^ { 2 } + \\sigma ^ { 2 } ) \\right)$ for smooth and non-convex problems, where $G ^ { 2 }$ is the uniform upper bound for the squared norm of stochastic gradients and $\\nu _ { m a x }$ is the maximum number of inactive rounds. In comparison, we prove the minimax optimal rate of $\\mathcal { O } ( \\sqrt { \\frac { \\bar { \\nu } } { N K T } } \\sigma ^ { 2 } )$ without the bounded gradient assumption, also improving $\\nu _ { \\mathrm { m a x } }$ to $\\bar { \\nu }$ . Furthermore, our result achieves a linear speedup in $N$ and $K$ . ",
+        "bbox": [
+            173,
+            421,
+            825,
+            531
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Apart from non-convex functions, we also derive the minimax optimal rates for strongly convex smooth functions under the mild assumption that allows for arbitrary and unbounded number of inactive rounds. Both of our results achieve linear speedups in terms of $N$ and $K$ , and automatically recover the best-known rates of FedAvg when all devices are active. We also show that our proposed algorithm achieves acceleration over unbiased baseline algorithms in the presence of stragglers. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Problem Setup ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            627,
+            330,
+            645
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We consider optimizing the following problem in a Federated Learning setting: ",
+        "bbox": [
+            174,
+            660,
+            691,
+            675
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/22f078ec690bdea95bad0a3810ee899d3c05899a7210597c3a542826721eabc9.jpg",
+        "text": "$$\n\\begin{array} { r } { \\underset { w \\in \\mathbb { R } ^ { d } } { \\operatorname* { m i n } } f ( w ) : = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } f _ { i } ( w ) : = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\mathbb { E } _ { \\xi _ { i } } [ f _ { i } ( w , \\xi _ { i } ) ] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            308,
+            683,
+            687,
+            710
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $w$ is the optimization variable, e.g., parameters of a machine learning model, $N$ is the number of participating devices, $f _ { i }$ is the local loss function on device $i$ , and $\\xi _ { i }$ describes the randomness in local data distribution. ",
+        "bbox": [
+            176,
+            717,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the ideal federated learning setup (see Figure 1 (a)), all devices return responses within similar time, and hence the central server collects all the local updates. In this case, the computation cost is usually measured by the number of local stochastic oracle evaluations, which is proportional to the number of rounds. In a delayed FL setup (see Figure 1 (b)), devices are always active upon the central server’s request but may return responses with a delay. Here, all devices return almost the same number of responses in the long term. ",
+        "bbox": [
+            173,
+            765,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As we discussed, the above setups do not depict a real-world scenario in which a device can have a longer inactive duration than active duration. In such cases, the communication interval is much longer than the local computation time required for each update, and each device generates an unequal number of responses [28, 17]. This motivates our setup in Figure 1 (c). ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/dbec5db87e22ee0dc55043253dc815eff5fd652c994f67ee8a04489f10ef06e1.jpg",
+        "image_caption": [
+            "Figure 1: An illustration of setup. (a) Ideal setup: all devices return their responses within similar time. (b) Delayed setup: all devices are available, but may return responses with a delay. (c) Our setup: devices can be unavailable arbitrarily, and the communication interval is long enough for active devices to return responses. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            82,
+            823,
+            233
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In our proposed setup, we use $t$ to index the global communication rounds. We say a device participates or is active at round $t$ if it can complete the computation task and send back the update at the end of round $t$ . We define $\\boldsymbol { \\mathcal { A } } ( t )$ as the set of all active devices at round $t$ . Notice that we make no assumptions on the distribution of the participation patterns of devices and allow them to be arbitrary. ",
+        "bbox": [
+            173,
+            299,
+            825,
+            356
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Directly applying FedAvg to the proposed setup can be problematic due to the existence of inactive devices. To accommodate for inactive devices, we discuss three natural variants of FedAvg and their limitations. The detailed algorithms can be found in Appendix A. ",
+        "bbox": [
+            176,
+            361,
+            825,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• Biased FedAvg. At each communication round, the global model is updated with a direct average of local updates from the active devices. This naive approach induces bias when data distribution and response patterns vary among devices.   \n• FedAvg with device sampling. The server selects a subset of $S$ devices randomly without replacement, and then waits until all devices in the subset $s$ respond. This is how original FedAvg [28] addresses device unavailability. Note that over $T$ communication rounds, the global model is updated less than $T$ times due to waiting. This approach is prone to stragglers and we refer the readers to Section 5.1 for a detailed discussion.   \n• FedAvg with importance sampling [26, 13, 17]. The local updates from the active devices are weighted by the reciprocal of the participation probabilities to avoid bias. This approach is only applicable when the response of each device is i.i.d. over rounds and it requires the knowledge of participation probabilities. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 Memory-augmented Impatient Federated Averaging (MIFA) ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            621,
+            709,
+            638
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we introduce our algorithm — Memory-augmented Impatient Federated Averaging (MIFA). MIFA maintains an update-array $\\{ G ^ { i } \\}$ in the memory that stores the latest updates for all devices. As the name suggests, MIFA has two components. First, the algorithm is impatient and avoids waiting for any specific device when facing heterogeneous devices with arbitrary availability. Second, the algorithm augments the received updates of the active devices with the stored updates of the inactive devices to perform averaging. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Specifically, at the beginning of round $t$ , the server broadcasts the latest model parameter $w _ { t }$ to all active devices $\\boldsymbol { \\mathcal { A } } ( t )$ . After receiving $w _ { t }$ , each active device, say, the $i$ -th device, sets $w _ { t , 0 } ^ { i } = w _ { t }$ and performs $K$ steps of SGD with respect to the local objective function to get $w _ { t , K } ^ { i }$ : ",
+        "bbox": [
+            174,
+            743,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2c0127a5706cf7944f95df9680a24f67da0062cc0709e170f05907f9d6d3a845.jpg",
+        "text": "$$\n\\boldsymbol w _ { t , k + 1 } ^ { i } = \\boldsymbol w _ { t , k } ^ { i } - \\eta _ { t } \\tilde { \\nabla } f _ { i } ( \\boldsymbol w _ { t , k } ^ { i } ) , \\boldsymbol k = 0 , \\cdots , \\boldsymbol K - 1 ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            799,
+            666,
+            820
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\eta _ { t }$ is the learning rate and $\\tilde { \\nabla } f _ { i } ( w _ { t , k } ^ { i } )$ is the stochastic gradient evaluated on device $i$ . Next, the server stores the received update $\\begin{array} { r } { \\frac { 1 } { \\eta _ { t } } \\big ( w _ { t } - w _ { t , K } ^ { i } \\big ) } \\end{array}$ in $G ^ { i }$ . Denote by $\\{ G _ { t } ^ { i } \\}$ the update-array after round $t$ , then we have ",
+        "bbox": [
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+            830,
+            826,
+            878
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/72c964c9acb692ffa258397e5523d41c405bf9354c410c37349066a92c8e4d6d.jpg",
+        "text": "$$\n\\begin{array} { r } { G _ { t } ^ { i } = \\left\\{ \\begin{array} { l l } { G _ { t - 1 } ^ { i } , } & { \\mathrm { ~ i f ~ } i \\notin \\mathcal { A } ( t ) , } \\\\ { \\frac { 1 } { \\eta _ { t } } ( w _ { t } - w _ { t , K } ^ { i } ) , } & { \\mathrm { ~ i f ~ } i \\in \\mathcal { A } ( t ) . } \\end{array} \\right. } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            362,
+            880,
+            627,
+            916
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "At the end of round $t$ , the server updates the global model with the average of $\\{ G _ { t } ^ { i } \\}$ (line 9). In other words, our algorithm MIFA updates the model with the latest available accumulated gradients for all devices. ",
+        "bbox": [
+            174,
+            90,
+            823,
+            132
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/5471198ff0b654b45b8a140b5c46c5f6e75ff5794c1529023b59736618813d80.jpg",
+        "table_caption": [
+            "Algorithm 1 Memory-augmented Impatient Federated Averaging (MIFA) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">1: Input: initial wi,learning rate {nt }</td></tr><tr><td>2: Server executes:</td><td>1: DeviceUpdate(i, Wt , nt):</td></tr><tr><td>3: initialize Gi ←O,i∈[N]</td><td>2:w,←Wt</td></tr><tr><td>4: fort=1,.,T-1do</td><td>3: for local step k = O,.,K -1 do</td></tr><tr><td>5: broadcast wt to all active devices i ∈ A(t)</td><td> 4:compute stochastic gradient Vfi(wt,)</td></tr><tr><td>6: for each active device ido</td><td>Wk+1←W,k-ntvfi(w，） 5:</td></tr><tr><td>7: G ←DeviceUpdate(i,Wt, nt) 8: end for</td><td>6: end for</td></tr><tr><td>Wt+1←wt-∑1 N 9: Gi</td><td>7:Return (wt - wi,k) to the server</td></tr></table>",
+        "bbox": [
+            178,
+            164,
+            810,
+            304
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "MIFA efficiently progresses without waiting for inactive devices and re-uses their latest updates as the surrogate for missing responses. Being impatient accelerates convergence, whereas memory augmentation corrects the update bias. Our algorithm differs from asynchronous algorithms in traditional distributed optimization [35, 4, 27, 11, 45, 1, 13, 6] in that we utilize the noisy updates of inactive devices more than once to avoid biasing against stragglers. In the following part of the paper, we show that MIFA successfully exploits information about the descent direction contained in the stale and noisy gradients. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Discussion on implementation. In practice, to implement MIFA, the server needs to maintain a huge array to store the latest update for each device, which scales with the model size and the total number of devices. To avoid exhausting the server’s memory, one strategy is to distribute the memory consumption among devices. Specifically, each device, say the $i$ -th, stores its previous update $G _ { t _ { i } ^ { \\prime } } ^ { \\ i }$ computed at round $t _ { i } ^ { \\prime }$ in its local memory. When it becomes active and computes $G _ { t } ^ { i }$ , the device sends $G _ { t } ^ { i } - G _ { t _ { i } ^ { \\prime } } ^ { i }$ to the server, which is the difference between the current update and the previous one. In this case, the server only needs to maintain the average $\\bar { G }$ in the memory and updates it by $\\begin{array} { r } { \\bar { G } _ { t } = \\bar { G } _ { t - 1 } + \\frac { 1 } { N } \\sum _ { i \\in \\mathcal { A } ( t ) } ( G _ { t } ^ { \\bar { i } } - G _ { t _ { i } ^ { \\prime } } ^ { i } ) } \\end{array}$ at round $t$ . Then the server updates the global model by $w _ { t + 1 } = w _ { t } - \\eta _ { t } \\bar { G } _ { t }$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5 Convergence Analysis for strongly convex objective functions ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present the convergence results for MIFA on $\\mu$ -strongly convex $L$ -smooth functions. Typical examples for the strongly convex case are $\\ell _ { 2 }$ regularized logistic regression and linear regression problems. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In order to capture how the unavailability of devices affects algorithm performance, we introduce the following notion to quantify the dynamics of devices in our setting. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 5.1 (Number of inactive rounds). We define the number of inactive rounds of device $i$ at round $t$ as $\\tau ( t , i ) = t - \\operatorname* { m a x } \\{ t ^ { \\prime } \\mid t ^ { \\prime } \\leq t , i \\in A ( t ^ { \\prime } ) \\}$ , which is the difference between current round $t$ and the latest round when device $i$ is active. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "It can be seen that $\\tau ( t , i ) = 0$ if device $i$ is active at round $t$ and $\\tau ( t , i ) = \\tau ( t - 1 , i ) + 1$ otherwise. Also, $t - \\tau ( t , i )$ is the latest round when the device $i$ is active. Next, we present the assumptions made for establishing our convergence theorem. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Assumption 1. $f _ { 1 } , \\cdots , f _ { N }$ are all $L$ -smooth, i.e., for all $w$ and $v _ { \\scriptscriptstyle { i } }$ , $\\begin{array} { r c l } { f _ { i } ( v ) } & { \\leq } & { f _ { i } ( w ) \\ + } \\end{array}$ $\\begin{array} { r } { \\left. \\nabla f _ { i } ( w ) , v - w \\right. + \\frac { L } { 2 } \\left\\| w - v \\right\\| ^ { 2 } } \\end{array}$ . ",
+        "bbox": [
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+            791,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Assumption 2. $\\tilde { \\nabla } f _ { i } ( \\boldsymbol { w } )$ is an unbiased estimator of $\\nabla f _ { i }$ with variance bounded by $\\sigma ^ { 2 }$ , i.e., $\\mathbb { E } _ { \\boldsymbol { \\xi } } \\left[ \\tilde { \\nabla } f _ { i } ( \\boldsymbol { w } ) \\right] = \\nabla f _ { i } ( \\boldsymbol { w } ) , \\mathbb { E } _ { \\boldsymbol { \\xi } } \\left[ \\left\\| \\tilde { \\nabla } f _ { i } ( \\boldsymbol { w } ) - \\nabla f _ { i } ( \\boldsymbol { w } ) \\right\\| ^ { 2 } \\right] \\leq \\sigma ^ { 2 } .$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Assumption 3. $f _ { 1 } , \\cdots , f _ { N }$ are all $\\mu$ -strongly convex: for all $w$ and $v _ { \\scriptscriptstyle { i } }$ , $f _ { i } ( v ) \\geq f _ { i } ( w ) +$ $\\begin{array} { r } { \\left. \\nabla f _ { i } ( w ) , v - w \\right. + \\frac { \\mu } { 2 } \\left\\| w - v \\right\\| ^ { 2 } } \\end{array}$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Assumption 4. There exists a constant $t _ { 0 } > 0$ , such that for all $t \\geq 1$ and $i \\in [ N ]$ , the number of inactive rounds of device $i$ at communication round $t$ satisfies $\\begin{array} { r } { \\tau ( t , i ) \\leq t _ { 0 } + \\frac { 1 } { b } \\dot { t } } \\end{array}$ , where $b =$ $4 0 \\left( L / \\mu \\right) ^ { 1 . 5 }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assumptions 1, 2, and 3 are standard and common in the FL literature, e.g., [26, 18, 19, 42, 36]. In Assumption 2, we relax the bounded gradient assumption that is often required in prior work, e.g., [6, 26, 40, 1]. Lastly, Assumption 4 is a very mild assumption on device availability, since it allows the number of inactive rounds to grow as ${ \\mathcal { O } } ( t )$ . In contrast, existing results on asynchronous updates mostly assume a bounded or fixed latency, e.g., [6, 1, 5, 40, 35, 4]. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We are now ready to present our first convergence result. Define $\\begin{array} { r l } { D = } & { { } \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\left. \\nabla f _ { i } ( w _ { * } ) \\right. ^ { 2 } } \\end{array}$ to measure data dissimilarity, where $w _ { * } = \\arg \\operatorname* { m i n } f ( w )$ is the global optimum. Also, define $\\bar { \\tau } _ { T }$ and $\\tau _ { \\operatorname* { m a x } , T }$ to be the average and maximum numbers of inactive rounds $\\tau ( t , i )$ across all devices and rounds, respectively. That is, ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/733c5805d202f72a1df49e8a211f10a98cbaf3e0246f5f2ede823112da4e143c.jpg",
+        "text": "$$\n\\bar { \\tau } _ { T } = \\frac { 1 } { N ( T - 1 ) } \\sum _ { t = 1 } ^ { T - 1 } \\sum _ { i = 1 } ^ { N } \\tau ( t , i ) , \\tau _ { \\operatorname* { m a x } , T } = \\operatorname* { m a x } _ { i \\in [ N ] } \\operatorname* { m a x } _ { 1 \\leq t \\leq T - 1 } \\tau ( t , i ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The following theorem summarizes the performance of MIFA in this case. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 5.1. Assume that Assumptions $^ { l }$ to 3 hold. Further assume that the device availability sequence $\\tau ( t , i )$ satisfies Assumption $^ { 4 }$ and $\\tau ( 1 , i ) = 0$ for all $i \\in [ N ]$ . By setting the learning rate $\\begin{array} { r } { \\eta _ { t } = \\frac { 4 } { \\mu K ( t + a ) } } \\end{array}$ with $a = \\operatorname* { m a x } \\{ 1 0 0 , 4 0 t _ { 0 } \\} ( L / \\mu ) ^ { 1 . 5 }$ , after $T - 1$ communication rounds, MIFA satisfies: ",
+        "bbox": [
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+            352,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a586eb10027a5e2bde1e4b791106324315628fa6d12e02e2e3e3c1ae021cee54.jpg",
+        "text": "$$\n\\mathbb { E } _ { \\xi } \\left[ f ( \\overline { { w } } _ { T } ) \\right] - f ( w _ { * } ) = \\mathcal { O } \\left( \\frac { \\bar { \\tau } _ { T } + 1 } { \\mu N K T } \\sigma ^ { 2 } + \\frac { \\tau _ { \\operatorname* { m a x } , T } ^ { 2 } A _ { 1 } + ( K - 1 ) ^ { 2 } A _ { 2 } + A _ { 3 } } { \\mu ^ { 2 } T ^ { 2 } } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            240,
+            412,
+            754,
+            455
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\overline { { w } } _ { T }$ is a weighted average of $w _ { t }$ defined as: ",
+        "bbox": [
+            173,
+            463,
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+            478
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/467ef08efd364fd1d849ca3dc16aa8bd25b2ad4ef2f4136b25ee850e32438dab.jpg",
+        "text": "$$\n\\overline { { w } } _ { T } = \\frac { 1 } { W _ { T } } \\sum _ { t = 1 } ^ { T } ( t + a - 1 ) ( t + a - 2 ) w _ { t } , W _ { T } = \\sum _ { t = 1 } ^ { T } ( t + a - 1 ) ( t + a - 2 ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            243,
+            481,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our results hold under Assumption 4, which allows for arbitrary device availability sequences with $\\tau _ { \\mathrm { m a x } , T } = \\mathcal { O } ( T )$ . However, for MIFA to converge, we require $\\tau _ { \\operatorname* { m a x } , T } = o ( T )$ and $\\dot { t _ { 0 } } = \\bar { o } ( T )$ . When ( NK(τ2max,T +t20)τ¯+1 ), the first term dominates and the impact of the second O(1/T 2) term is negligible. In fact the first term in Theorem 5.1 is minimax optimal by our information-theoretic lower bound for the problem in the next proposition. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proposition 5.1. Let $c _ { 0 } > 0$ be a universal constant. For any potentially randomized algorithm, there exists a stochastic strongly convex problem satisfying Assumptions $^ { l }$ to 3, such that the output $w _ { T }$ after $T$ rounds of communication has expected sub-optimality lower bounded by ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/15944ea925ca625bc0c3fb0cf5ed413d2deb8aa7ced28f61f9ba449d3ce641aa.jpg",
+        "text": "$$\n\\mathbb { E } [ f ( w _ { T } ) - f ( w ^ { * } ) ] \\geq c _ { 0 } \\frac { \\bar { \\tau } _ { T } \\sigma ^ { 2 } } { \\mu N K T } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            385,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The proof is based on the observation that the number of gradient evaluation can scale inversely with $\\bar { \\tau } _ { T }$ and that the oracle complexity is tight even for centralized stochastic optimization problems. The optimality of the first term in Theorem 5.1 is independent of the distributed or the FL setup. ",
+        "bbox": [
+            174,
+            727,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The second term in Theorem 5.1 converges at the rate $\\mathcal { O } ( 1 / T ^ { 2 } )$ and consists of three parts, where the first part reflects the slowdown caused by device unavailability through $\\tau _ { \\operatorname* { m a x } , T }$ , the second part shows the effect of multiple $( K > 1 )$ ) local steps, and the third part tells how the initial error decreases. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Remark 5.1. When $\\tau ( t , i ) ~ = ~ 0$ for all $i$ and $t _ { : }$ , our setup reduces to FedAvg with full device pbound $\\bar { \\tau } _ { T } ~ = ~ 0$ d  , $\\tau _ { \\mathrm { m a x } , T } ~ = ~ 0$ . Iate $\\begin{array} { r } { \\mathcal { O } \\left( \\frac { \\sigma ^ { 2 } } { \\mu K N T } + \\frac { L ( \\sigma ^ { 2 } / K + D + L ^ { 2 } \\| w _ { 1 } - w _ { * } \\| ^ { 2 } ) } { \\mu ^ { 2 } T ^ { 2 } } \\right) } \\end{array}$ $\\begin{array} { r } { \\mathcal { O } \\bigg ( \\frac { \\sigma ^ { 2 } \\log T } { \\mu K N T } + \\frac { L ( \\sigma ^ { 2 } / K + D ) ( \\log T ) ^ { 2 } } { \\mu ^ { 2 } T ^ { 2 } } + } \\end{array}$ $\\begin{array} { r } { \\mu \\left. w _ { 1 } - w _ { * } \\right. ^ { 2 } \\exp ( - \\frac { \\mu } { 4 8 L } T ) \\Bigr ) } \\end{array}$ in [18] (Thm. V. $B ^ { 2 } = 2 , \\eta _ { g } = 1 ,$ ) up to logarithmic terms. Besides, in the general case, our $\\mathcal { O } ( \\tau _ { \\mathrm { m a x } , T } ^ { 2 } A _ { 1 } / T ^ { 2 } )$ term matches the last term in $I 6 J \\left( C o r . 5 \\right)$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Remark 5.2. Our analysis relies on the technical assumption that all devices respond in the first round. Intuitively, this is because we need at least one valid stochastic gradient evaluation for each device to get a complete picture of the global objective, or otherwise any update would be biased. In practice, this can be achieved by waiting for the updates from all devices on $w _ { 1 }$ at the very beginning. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 Case Study: i.i.d. Bernoulli participation ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Though our algorithm can be applied to non-stationary and non-independent response patterns, we show in this subsection that even in the simple i.i.d. Bernoulli participation scenario our algorithm can achieve considerable improvement compared to known algorithms. In particular, we consider a setup where each device becomes active independently with a fixed probability $p _ { i }$ . It serves as the first motivating example towards modeling the participation patterns of devices, and provides a clean view of how the heterogeneity of the device participation influences the Federated optimization algorithms. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We will show that in this scenario, Assumption 4 holds with high probability, and the terms involving the inactive rounds $\\tau ( t , i )$ in Theorem 5.1 can also be bounded. Furthermore, we theoretically demonstrate that algorithms such as FedAvg [28] and SCAFFOLD [18], which sample $S$ devices for each global update, are more prone to stragglers than our algorithm. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Definition 5.2. Assume that for all $i \\in [ N ]$ , the i-th device is assigned with a probability $p _ { i }$ . We say the participation of the devices follows i.i.d. Bernoulli participation model with participation probabilities $\\{ p _ { i } \\}$ , if (1). at the first round, all devices are active, and (2). at round $t > 1$ , device $i$ is active with probability $p _ { i }$ , which is independent of the history and other devices. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Next theorem shows that under i.i.d. Bernoulli participation scenario, with high probability, $\\tau ( t , i )$ only grows logarithmically in $t$ . Also Assumption 4 holds for a mild choice of $t _ { 0 }$ . ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 5.2. For i.i.d. Bernoulli participation model defined in Definition 5.2, given any $\\delta > 0$ , with probability at least $1 - \\delta$ , we have the following holds for all $t \\geq 1$ and $i \\in [ N ]$ simultaneously, ",
+        "bbox": [
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+            454,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/39aa0a52372df38b6d9e71e222db332e84efa7cdb81235ae00a0fb9752de5e1b.jpg",
+        "text": "$$\n\\tau ( t , i ) \\leq \\mathcal { O } \\Big ( \\frac { 1 } { p _ { i } } ( \\log ( N t / \\delta ) + 1 ) \\Big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, $( l )$ . Assumption $^ { 4 }$ holds true $\\begin{array} { r } { \\mathrm { \\ddot { \\it f } } t _ { 0 } = \\Omega \\bigg ( \\frac { 1 } { p _ { m i n } } \\log \\frac { b N } { p _ { m i n } \\delta } \\bigg ) } \\end{array}$ , where $p _ { m i n } = \\operatorname* { m i n } \\{ p _ { i } \\}$ , and $b = 4 0 ( L / \\mu ) ^ { 1 . 5 }$ ;(2). $\\tau _ { \\operatorname* { m a x } , T }$ can be upper bounded as ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/922e99211cb7c2d1fd2b59ad614767b49e2270e68185e6f77ea7f51093a5ac80.jpg",
+        "text": "$$\n\\tau _ { \\operatorname* { m a x } , T } \\leq \\mathcal { O } \\Big ( \\frac { 1 } { p _ { m i n } } \\cdot \\big ( \\log ( T N / \\delta ) + 1 \\big ) \\Big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            577,
+            637,
+            609
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The next theorem provides a high probability upper bound for $\\bar { \\tau } _ { T }$ . ",
+        "bbox": [
+            173,
+            622,
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+            638
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 5.3. For i.i.d. Bernoulli participation model defined in Definition 5.2, given any $\\delta > 0$ and $T > 1$ , with probability at least $1 - \\delta$ , we have ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/79e3a5996cb3f7b4c46ca91911907e624b79420e705c6142b9338d5386fc6e5b.jpg",
+        "text": "$$\n\\bar { \\tau } _ { T } \\leq \\Big ( \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\frac { 1 } { p _ { i } } \\Big ) \\cdot \\mathcal { O } \\Big ( 1 + \\log \\frac { 1 } { \\delta } \\Big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            678,
+            619,
+            722
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "By Theorem 5.2 and Theorem 5.3, we conclude that the dominant term of our convergence bound is $\\begin{array} { r } { \\stackrel { \\triangledown } { \\mathcal { O } } \\left( \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\frac { 1 } { p _ { i } } \\cdot \\frac { \\sigma ^ { 2 } } { \\mu N K T } \\right) } \\end{array}$ . Therefore, to achieve $\\epsilon$ accuracy, the dominant term of the number of the required rounds is ",
+        "bbox": [
+            173,
+            734,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d6f389bee97e0f3555558010eb2588b9917d901810a517b05b4f4f415bf814cb.jpg",
+        "text": "$$\nT _ { \\epsilon } ^ { \\left( \\mathrm { M I F A } \\right) } = \\widetilde { \\mathcal { O } } \\Bigl ( \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\frac { 1 } { p _ { i } } \\cdot \\frac { \\sigma ^ { 2 } } { \\mu N K \\epsilon } \\Bigr ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            785,
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+            828
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For both FedAvg and SCAFFOLD that sample $S$ devices uniformly at random, [18] (Thm I. & III.) showed that the dominant term of the number of global updates needed to achieve $\\epsilon$ accuracy is $\\begin{array} { r } { R _ { \\epsilon } = \\widetilde { \\mathcal { O } } \\left( \\frac { \\sigma ^ { 2 } } { \\mu S K \\epsilon } \\right) } \\end{array}$ . Notice that in our setting, to accomplish each global update, the server needs to wait for a few rounds for the $S$ devices to respond. Let $T ( S )$ be the expected rounds for which the server needs to wait for the selected devices $\\boldsymbol { S }$ to be active. Then the expected total rounds to achieve $\\epsilon$ accuracy is $R _ { \\epsilon } \\cdot \\mathbb { E } _ { S } \\left[ T ( S ) \\right]$ . For i.i.d. Bernoulli participation model, we have $\\begin{array} { r } { T ( S ) \\geq \\frac { 1 } { \\operatorname* { m i n } \\{ p _ { i } | i \\in S \\} } } \\end{array}$ , and we can further show that $\\begin{array} { r } { \\mathbb { E } _ { S } \\left[ T ( S ) \\right] \\ge \\frac { 1 } { p _ { m i n } } \\frac { S } { N } } \\end{array}$ ≥ 1p min SN ( see Appendix D.3 for details). Therefore, ",
+        "bbox": [
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+        "text": "$$\n\\mathbb { E } \\left[ T _ { \\epsilon } ^ { ( \\mathsf { F e d A v g , S C A F F O L D } ) } \\right] \\geq \\frac { S } { N } \\frac { 1 } { p _ { m i n } } \\widetilde { \\mathcal { O } } \\Big ( \\frac { \\sigma ^ { 2 } } { \\mu S K \\epsilon } \\Big ) = \\widetilde { \\mathcal { O } } \\Big ( \\frac { 1 } { p _ { m i n } } \\cdot \\frac { \\sigma ^ { 2 } } { \\mu N K \\epsilon } \\Big ) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "By comparing Eqn. 2 and Eqn. 3, we see that both FedAvg and SCAFFOLD are more vulnerable to stragglers, that is, the devices with very small participation probabilities; on the contrary, the convergence rate of MIFA only depends on the average of $1 / p _ { i }$ instead of $1 / p _ { m i n }$ . We also provide empirical experiments showing that MIFA converges faster than FedAvg in Section 7. ",
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+        "type": "text",
+        "text": "6 Convergence result for non-convex objective functions ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we present the convergence guarantee of MIFA for the non-convex case. First we list the additional assumptions as below. ",
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+        "type": "text",
+        "text": "Assumption 5 (Hessian Lipschitz). $f _ { 1 } , \\cdots , f _ { N }$ are all $\\rho$ -Hessian Lipschitz: for all $w$ and $v$ $\\bigl \\| \\nabla ^ { 2 } f _ { i } \\bigl ( \\tilde { \\boldsymbol { w } } \\bigr ) - \\nabla ^ { 2 } f _ { i } ( \\boldsymbol { v } ) \\bigr \\| \\leq \\dot { \\rho } \\bigl \\| \\boldsymbol { w } - \\boldsymbol { \\tilde { v } } \\bigr \\|$ . ",
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+    {
+        "type": "text",
+        "text": "Assumption 6 (Bounded noise). The noise of the local stochastic gradients is upper bounded by $a$ constant $\\delta$ almost surely: $\\begin{array} { r } { \\left\\| \\tilde { \\nabla } f _ { i } ( w ) - \\nabla f _ { i } ( w ) \\right\\| \\le \\delta a . s . , \\forall i \\in [ N ] } \\end{array}$ . ",
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+        "type": "text",
+        "text": "Assand y). There exist . Furthermore, $\\alpha > 0$ anfine $\\beta _ { i } > 0$ at for all . $w$ $\\in [ N ] \\colon \\left\\| \\nabla f _ { i } ( w ) \\right\\| ^ { 2 } \\leq \\alpha \\left\\| \\nabla f ( w ) \\right\\| ^ { 2 } + \\beta _ { i }$ $\\begin{array} { r } { \\beta = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\beta _ { i } } \\end{array}$ ",
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+        "type": "text",
+        "text": "Assumption 8. There exists a constant $\\nu _ { i }$ such that $\\tau ( t , i ) \\ \\leq \\ \\nu _ { i }$ , for all $i \\in [ N ]$ and $t \\geq 1$ Furthermore, define ν¯ = 1N PNi=1 νi and νmax = maxi∈[N] νi. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The analysis of non-convex functions is much more technically involved, and our results rely on strong assumptions that provide a finer control of the gradient difference (Assumption 5), gradient noise (Assumption 6), gradient dissimilarity among devices (Assumption 7), and device unavailability (Assumption 8). We remark that Assumption 5 is also made in [9, 16], and Assumption 7 is also made in [18, 39]. We leave it as future work to study whether and how MIFA converges for non-convex functions with weaker assumptions. ",
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+        "type": "text",
+        "text": "Theorem 6.1. Assume that Assumptions $I , ~ 2 , ~$ , and $^ { 5 }$ to 7 hold. Further assume that the device availability sequence $\\tau ( t , i )$ satisfies Assumption 8 and $\\tau ( 1 , i ) = 0$ for all $i \\in [ N ]$ . By using $a$ learning rate η = q $\\begin{array} { r } { \\eta = \\sqrt { \\frac { N } { K T L \\left( 1 + \\bar { \\nu } \\right) } } } \\end{array}$ NKT L(1+¯ν), for T ≥ max{32αLNK, 16LNK , 8KNν2max(L2+ρδ) }, after T − 1 communication rounds, MIFA satisfies: ",
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+        "text": "$$\n\\operatorname* { m i n } _ { 1 \\leq t \\leq T } \\mathbb { E } _ { \\xi } \\left[ \\left\\| \\nabla f ( w _ { t } ) \\right\\| ^ { 2 } \\right] = \\mathcal { O } \\left( \\sqrt { \\frac { ( 1 + \\bar { \\nu } ) L } { T K N } } ( f ( w _ { 1 } ) - f ^ { * } + \\sigma ^ { 2 } ) + \\frac { A _ { 4 } + A _ { 5 } } { T } \\right) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $f ^ { * }$ is the optimal value, and: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/068374b3ecfcb8e112e697b4b2961eec732c2ae914494131485a106df94ed394.jpg",
+        "text": "$$\n\\begin{array} { l } { { A _ { 4 } = N K L \\left( \\alpha \\sigma ^ { 2 } \\hat { \\nu } + \\frac { \\sigma ^ { 2 } \\nu _ { \\mathrm { m a x } } } { \\sqrt { K N } } + \\sigma \\nu _ { \\mathrm { m a x } } \\sqrt { \\beta } \\right) + \\frac { { ( L ^ { 2 } + \\rho \\delta ) \\sigma ^ { 2 } \\nu _ { \\mathrm { m a x } } } } { L } , } } \\\\ { { A _ { 5 } = \\frac { { ( K - 1 ) N L ( \\beta + \\sigma ^ { 2 } / K ) } } { \\hat { \\nu } + 1 } . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Next, we show that the leading $\\mathcal { O } ( 1 / \\sqrt { T } )$ term is theoretically optimal for zero-respecting algorithms. ",
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+    {
+        "type": "text",
+        "text": "Proposition 6.1. Let $c _ { 0 } > 0$ be a universal constant. For any randomized zero-respecting algorithm, there exists a stochastic non-convex problem satisfying Assumption1, 2, 5 and 7, such that the output $w _ { T }$ after $T$ rounds of communication has expected sub-optimality lower bounded by ",
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+        "type": "equation",
+        "img_path": "images/ca9299cdc027e71b6e299a9e1e31a7cd5c2809df369d439bba1b7aba2e00a1ef.jpg",
+        "text": "$$\n\\mathbb { E } [ \\| \\nabla f ( w _ { T } ) \\| ^ { 2 } ] \\ge \\mathbb { E } [ \\| \\nabla f ( w _ { T } ) \\| ] ^ { 2 } \\ge c _ { 0 } \\sqrt { \\frac { \\bar { \\nu } L \\sigma ^ { 2 } ( f ( w _ { 0 } ) - f ^ { * } ) } { N K T } } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The above proposition show that when $\\sigma \\sqrt { ( f ( w _ { 0 } ) - f ^ { * } ) } \\sim \\sigma ^ { 2 } + ( f ( w _ { 0 } ) - f ^ { * } )$ , the result in Theorem 6.1 is tight. However, note that the counter example we used requires the quantity $\\delta$ in Assumption 6 to scale with $T$ , hence requiring $\\delta$ to be large enough. This does not change the optimality of the first term as the first term is independent of $\\delta$ . Whether this requirement can be relaxed is left as an open problem. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Remark 6.1. When all $\\nu _ { i } = 0$ (i.e. all the devices are active), our convergence bound reduces to $\\begin{array} { r } { \\mathcal { O } \\Big ( \\sqrt { \\frac { L } { T K N } } ( f ( w _ { 1 } ) - f ^ { * } + \\sigma ^ { 2 } ) + \\frac { ( K - 1 ) N L ( \\beta + \\sigma ^ { 2 } / K ) } { T } \\Big ) } \\end{array}$ . This matches the result in [42] (Thm. 1, $\\begin{array} { r } { \\eta = 1 , \\eta _ { L } = \\sqrt { \\frac { N } { K T L } } ) } \\end{array}$ . ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "7 Numerical Experiments ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In this section, we conduct numerical experiments3. to verify our theoretical results and investigate how the heterogeneity of the device availability influences the Federated optimization algorithms. We compare the performance of the following four algorithms: FedAvg with importance sampling (FedAvg-IS), Biased FedAvg, FedAvg with device sampling, and our proposed MIFA. For the detailed discussions of the algorithms, we refer the readers to Sections 3 and 4. We remark that for a fair comparison, we deliberately include the first few rounds that MIFA needs to wait to receive responses from all devices for initializing the update-array $\\{ G ^ { i } \\}$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Following [26, 25], we construct non-i.i.d. datasets from two commonly used computer vision datasets — MNIST [23] and CIFAR-10 [22] . Specifically, we divide the data into $N = 1 0 0$ devices with each device holding samples of only two classes, which creates a high level of data heterogeneity. For simplicity, we ensure that each device holds the same number of samples. We do not use any data augmentation. We use multinomial logistic regression as the convex model and LeNet-5 [24] with ReLU activations as the non-convex model. For all experiments, we use weight decay in the training process, which corresponds to adding $\\ell _ { 2 }$ penalty. We use logistic models for MNIST dataset, while we use LeNet-5 for CIFAR-10. Our code is adapted from [26], which is under MIT License. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We model the availability of the devices as independent Bernoulli random trials. The $i \\cdot$ -th device is assigned with a probability $p _ { i }$ , where at each time step, the device becomes active with probability $p _ { i }$ . In our experiments, the $p _ { i }$ ’s are chosen such that devices holding data of smaller labels participate less frequently. Specifically, if the $i$ -th device holds the data of label $j$ and $k$ , we set $\\bar { p } _ { i } = \\bar { p _ { \\operatorname* { m i n } } } \\operatorname* { m i n } ( j , k ) / 9 + \\bar { ( 1 - p _ { \\operatorname* { m i n } } ) }$ , where $p _ { \\mathrm { m i n } }$ controls the lower bound of the participation probabilities. The correlation between the participation patterns and local datasets increases the difficulty of the problem [17]. To investigate this phenomenon, we repeat the experiments for $p _ { \\mathrm { m i n } } = 0 . 1$ and 0.2. We control the randomness of device participation when testing different algorithms. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In all the experiments, we set the initial learning rate to be $\\eta _ { 0 } = 0 . 1$ and decay the learning rate as $\\begin{array} { r } { \\eta _ { t } = \\eta _ { 0 } \\cdot \\frac { 1 } { t } } \\end{array}$ . We set the weight decay to be 0.001. The local batch size is 100 and each local update consists of 2 epochs. Therefore, the actual number of local steps $K$ depends on the size of the dataset. We run all the experiments with 4 GPUs of type GeForce RTX 2080 Ti. We repeat the experiments for 5 different random seeds, and all of the experiments exhibit similar training curves. We report the averaged training loss and test accuracy with error bars in Figure 2. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We observe that FedAvg with device sampling (FedAvg ( $S = 5 0$ ) and FedAvg $S = 1 0 0$ ) in Figure 2) is severely influenced by the straggling devices and makes progress relatively slowly compared to the other algorithms. Although biased FedAvg converges fast at the beginning, this simple algorithm is biased, and the optimality gaps are prominent for the harder CIFAR-10 dataset and when $p _ { \\mathrm { m i n } }$ is small. On the contrary, our proposed MIFA avoids waiting for stragglers, converges fast without bias, and is competitive with FedAvg with importance sampling, which requires knowledge of the participation probabilities. We refer the readers to Appendix G for additional experiments on CIFAR-10. ",
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+    {
+        "type": "text",
+        "text": "8 Conclusions and Discussions ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "In this paper, we study FL algorithms in the presence of arbitrary device unavailability and propose MIFA, which avoids waiting for straggling devices and re-uses the memorized latest updates as the surrogate when the device is unavailable. We theoretically analyze MIFA without any structural assumptions on the device availability and prove the convergence for strongly convex and non-convex smooth functions. Different from the literature that studies oracle complexity in terms of stochastic gradient evaluations, we argue that in federated learning system, the bottleneck lies in the nonstationary and possibly adversarial pattern of device participation. Therefore, it is important to study how the number of inactive rounds influences the convergence rate. In Theorem 5.1, the dependency upon $\\tau _ { \\operatorname* { m a x } , T }$ might be an artifact of our analysis, and a future direction is to study whether we can remove this dependency. Another important direction is to analyze algorithms for non-convex functions under weaker assumptions. ",
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+        "img_path": "images/c1142e9ecbf7a3052e86380ac3f4c818747910b17c29f238aaf9134bafeb112a.jpg",
+        "image_caption": [
+            "Figure 2: Training losses and test accuracies. Fig. 2(a)–2(d): logistic models on non-iid MNIST. Fig. 2(e)–2(h): LeNet-5 on non-iid CIFAR-10. FedAvg $S = 5 0$ ) and FedAvg $S = 1 0 0$ ) refer to FedAvg with device sampling that samples $S$ devices for each global update. FedAvg-IS is short for FedAvg with importance sampling, which requires knowledge of the participation probabilities. "
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The work of Xinran Gu and Longbo Huang is supported in part by the Technology and Innovation Major Project of the Ministry of Science and Technology of China under Grant 2020AAA0108400 and2020AAA0108403. Xinran Gu would like to thank Kaifeng Lyu for the discussion on the convergence analysis. The authors would like to thank Sai Praneeth Karimireddy for the discussion on the problem settings. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "References ",
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+        "bbox": [
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+        "page_idx": 9
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+                                    "content": "multiple advantages, FL faces new challenges. One challenge arises when devices",
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+                                    "content": "this case, the convergence of popular FL algorithms such as FedAvg is severely",
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+                                    "score": 1.0,
+                                    "content": "named Memory-augmented Impatient Federated Averaging (MIFA). Our algorithm",
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+                                    "content": "gradient bias using the memorized latest updates from the devices. We prove that",
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+                                    "content": "characterization of the improvement over baseline algorithms through a case study,",
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+                                    "content": "and validate the results by numerical experiments on real-world datasets.",
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+                                    "content": "preserves its data privacy. Second, federated learning can make use of idle computing resources and",
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+                                    "score": 1.0,
+                                    "content": "to new challenges due to large variations among individual computing devices. Our work aims to",
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+                                    "content": "were proposed and analyzed by [26, 18, 19, 42, 25]. We follow this line of work and formulate our",
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+                                    "score": 1.0,
+                                    "content": "this case, the convergence of popular FL algorithms such as FedAvg is severely",
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+                                    "content": "influenced by the straggling devices. To tackle this challenge, we study federated",
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+                                    "score": 1.0,
+                                    "content": "learning algorithms under arbitrary device unavailability and propose an algorithm",
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+                                    "score": 1.0,
+                                    "content": "named Memory-augmented Impatient Federated Averaging (MIFA). Our algorithm",
+                                    "type": "text"
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+                            ],
+                            "index": 26
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "gradient bias using the memorized latest updates from the devices. We prove that",
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+                            "index": 28
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+                                    "score": 1.0,
+                                    "content": "strongly convex and non-convex smooth functions. We also provide an explicit",
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+                            "index": 30
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+                                    "score": 1.0,
+                                    "content": "characterization of the improvement over baseline algorithms through a case study,",
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+                                    "score": 1.0,
+                                    "content": "and validate the results by numerical experiments on real-world datasets.",
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "of this training scheme are mainly twofold. First, each device keeps the private data locally and hence",
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+                                    "score": 1.0,
+                                    "content": "lower computation costs. Although federated learning successfully scales up with data sizes and",
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+                                    "content": "to new challenges due to large variations among individual computing devices. Our work aims to",
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+                                    "score": 1.0,
+                                    "content": "were proposed and analyzed by [26, 18, 19, 42, 25]. We follow this line of work and formulate our",
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+                                    "score": 1.0,
+                                    "content": "model the updates as delayed responses. Lots of novel research has studied the problem with different",
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+                                    "content": "delay models, e.g., [35, 4, 27, 11, 45, 1, 5, 13]. However, the delayed setup assumes that all devices",
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+                            "index": 3
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+                                    "content": "make roughly the same number of (delayed) responses in the end. This behavior may deviate largely",
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+                                    "score": 1.0,
+                                    "content": "from the FL practice, where each device, e.g., personal cell phones, can have very different active",
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+                                    "score": 1.0,
+                                    "content": "duration when participating in the FL training, and hence make different numbers of responses. For",
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+                                    "score": 1.0,
+                                    "content": "this reason, our work aims to address this third discrepancy among devices caused by individual",
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+                                    "score": 1.0,
+                                    "content": "participate in training (see formal definitions in Sections 3, 5 and 6). MIFA also achieves",
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+                                    "content": "a case study and empirically verify our results on real-world datasets.",
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+                                            "content": "time. (b) Delayed setup: all devices are available, but may return responses with a delay. (c) Our",
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+                                            "type": "text"
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+                                    "score": 1.0,
+                                    "content": "if it can complete the computation task and send back the update at",
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+                        {
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+                                    "content": "the end of round",
+                                    "type": "text"
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+                                    "score": 0.92,
+                                    "content": "\\boldsymbol { \\mathcal { A } } ( t )",
+                                    "type": "inline_equation"
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+                                {
+                                    "bbox": [
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+                                    "content": "as the set of all active devices at round",
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+                                    "score": 1.0,
+                                    "content": ". Notice that we make no",
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+                        },
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+                                    "content": "assumptions on the distribution of the participation patterns of devices and allow them to be arbitrary.",
+                                    "type": "text"
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+                            ],
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+                    ],
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+                    "type": "text",
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+                                    "content": "Directly applying FedAvg to the proposed setup can be problematic due to the existence of inactive",
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+                            ],
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+                                    "content": "devices. To accommodate for inactive devices, we discuss three natural variants of FedAvg and their",
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "limitations. The detailed algorithms can be found in Appendix A.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 13
+                        }
+                    ],
+                    "index": 12
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "• Biased FedAvg. At each communication round, the global model is updated with a direct",
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+                            ],
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+                        },
+                        {
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+                                }
+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "• FedAvg with device sampling. The server selects a subset of",
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+                                    "score": 0.82,
+                                    "content": "S",
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+                                    "score": 1.0,
+                                    "content": "replacement, and then waits until all devices in the subset",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.79,
+                                    "content": "s",
+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": "respond. This is how original",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 18
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "FedAvg [28] addresses device unavailability. Note that over",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                        402
+                                    ],
+                                    "score": 0.78,
+                                    "content": "T",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "communication rounds,",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
+                            "bbox": [
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+                                505,
+                                415
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        402,
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+                                        415
+                                    ],
+                                    "score": 1.0,
+                                    "content": "the global model is updated less than",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 0.75,
+                                    "content": "T",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        307,
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+                                        415
+                                    ],
+                                    "score": 1.0,
+                                    "content": "times due to waiting. This approach is prone to",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 20
+                        },
+                        {
+                            "bbox": [
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+                                426
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        414,
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+                                        426
+                                    ],
+                                    "score": 1.0,
+                                    "content": "stragglers and we refer the readers to Section 5.1 for a detailed discussion.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 21
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "• FedAvg with importance sampling [26, 13, 17]. The local updates from the active devices",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 22
+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "are weighted by the reciprocal of the participation probabilities to avoid bias. This approach",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 23
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "is only applicable when the response of each device is i.i.d. over rounds and it requires the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 24
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "knowledge of participation probabilities.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 25
+                        }
+                    ],
+                    "index": 19.5
+                },
+                {
+                    "type": "title",
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+                                    "score": 1.0,
+                                    "content": "4 Memory-augmented Impatient Federated Averaging (MIFA)",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 26
+                        }
+                    ],
+                    "index": 26
+                },
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+                    "type": "text",
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+                                    "score": 1.0,
+                                    "content": "In this section, we introduce our algorithm — Memory-augmented Impatient Federated Averaging",
+                                    "type": "text"
+                                }
+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "(MIFA). MIFA maintains an update-array",
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+                                    "score": 0.92,
+                                    "content": "\\{ G ^ { i } \\}",
+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": "in the memory that stores the latest updates for all",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 28
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "devices. As the name suggests, MIFA has two components. First, the algorithm is impatient and",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 29
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "avoids waiting for any specific device when facing heterogeneous devices with arbitrary availability.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 30
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Second, the algorithm augments the received updates of the active devices with the stored updates of",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 31
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "the inactive devices to perform averaging.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 32
+                        }
+                    ],
+                    "index": 29.5
+                },
+                {
+                    "type": "text",
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+                                    "score": 1.0,
+                                    "content": "Specifically, at the beginning of round",
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+                                    "score": 1.0,
+                                    "content": ", the server broadcasts the latest model parameter",
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+                                    "score": 0.84,
+                                    "content": "w _ { t }",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "to all",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 33
+                        },
+                        {
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+                                    "content": "devices. To accommodate for inactive devices, we discuss three natural variants of FedAvg and their",
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+                                    "score": 1.0,
+                                    "content": "limitations. The detailed algorithms can be found in Appendix A.",
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+                    ],
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+                                    "content": "• Biased FedAvg. At each communication round, the global model is updated with a direct",
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+                                    "score": 1.0,
+                                    "content": "respond. This is how original",
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+                                    "score": 1.0,
+                                    "content": "FedAvg [28] addresses device unavailability. Note that over",
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+                                    "content": "communication rounds,",
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "the global model is updated less than",
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+                                    "score": 0.75,
+                                    "content": "T",
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+                                    "score": 1.0,
+                                    "content": "times due to waiting. This approach is prone to",
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+                            ],
+                            "index": 20
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+                        {
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+                                    "score": 1.0,
+                                    "content": "stragglers and we refer the readers to Section 5.1 for a detailed discussion.",
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+                            ],
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+                            "is_list_end_line": true
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+                                    "content": "• FedAvg with importance sampling [26, 13, 17]. The local updates from the active devices",
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "are weighted by the reciprocal of the participation probabilities to avoid bias. This approach",
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+                                    "score": 1.0,
+                                    "content": "is only applicable when the response of each device is i.i.d. over rounds and it requires the",
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+                                    "content": "In order to capture how the unavailability of devices affects algorithm performance, we introduce the",
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+                                    "content": "[6, 26, 40, 1]. Lastly, Assumption 4 is a very mild assumption on device availability, since it allows",
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+                                    "content": "{ \\mathcal { O } } ( t )",
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+                                    "score": 1.0,
+                                    "content": ". In contrast, existing results on asynchronous updates",
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+                                    "score": 1.0,
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+                                    "content": "Assumptions 1, 2, and 3 are standard and common in the FL literature, e.g., [26, 18, 19, 42, 36]. In",
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+                                    "content": "Assumption 2, we relax the bounded gradient assumption that is often required in prior work, e.g.,",
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+                                    "content": "[6, 26, 40, 1]. Lastly, Assumption 4 is a very mild assumption on device availability, since it allows",
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+                        },
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+                                    "content": "{ \\mathcal { O } } ( t )",
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+                                {
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+                                    "score": 1.0,
+                                    "content": ". In contrast, existing results on asynchronous updates",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 6
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "mostly assume a bounded or fixed latency, e.g., [6, 1, 5, 40, 35, 4].",
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+                                    "score": 1.0,
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+                                    "content": "Furthermore, define ν¯ = 1N PNi=1 νi and νmax = maxi∈[N] νi.",
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+                                    "content": "The analysis of non-convex functions is much more technically involved, and our results rely on",
+                                    "type": "text"
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+                                    "content": "strong assumptions that provide a finer control of the gradient difference (Assumption 5), gradient",
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+                                    "content": "noise (Assumption 6), gradient dissimilarity among devices (Assumption 7), and device unavailability",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "(Assumption 8). We remark that Assumption 5 is also made in [9, 16], and Assumption 7 is also made",
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+                                    "score": 1.0,
+                                    "content": "in [18, 39]. We leave it as future work to study whether and how MIFA converges for non-convex",
+                                    "type": "text"
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+                                    "content": "functions with weaker assumptions.",
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+                                    "content": "Theorem 6.1. Assume that Assumptions",
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+                                    "score": 1.0,
+                                    "content": "to 7 hold. Further assume that the device",
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+                                    "content": "for all",
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+                                    "content": "NKT L(1+¯ν), for T ≥ max{32αLNK, 16LNK",
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+                                    "content": ", 8KNν2max(L2+ρδ) }, after T − 1",
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+                                    "content": "be a universal constant. For any randomized zero-respecting algorithm,",
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+                                    "content": "there exists a stochastic non-convex problem satisfying Assumption1, 2, 5 and 7, such that the output",
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+                                    "content": "rounds of communication has expected sub-optimality lower bounded by",
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+                                    "content": "Theorem 6.1 is tight. However, note that the counter example we used requires the quantity",
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+                                    "content": "Assumption 6 to scale with",
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+                                    "content": "Furthermore, define ν¯ = 1N PNi=1 νi and νmax = maxi∈[N] νi.",
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+                                    "content": "The analysis of non-convex functions is much more technically involved, and our results rely on",
+                                    "type": "text"
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+                                    "content": "strong assumptions that provide a finer control of the gradient difference (Assumption 5), gradient",
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+                                    "content": "noise (Assumption 6), gradient dissimilarity among devices (Assumption 7), and device unavailability",
+                                    "type": "text"
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+                                    "content": "(Assumption 8). We remark that Assumption 5 is also made in [9, 16], and Assumption 7 is also made",
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+                                    "score": 1.0,
+                                    "content": "in [18, 39]. We leave it as future work to study whether and how MIFA converges for non-convex",
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+                                    "content": "functions with weaker assumptions.",
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+                                    "score": 1.0,
+                                    "content": "to be large enough. This does not change the",
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+                                    "content": ". Whether this requirement can be",
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+                                    "content": ". This matches the result in [42] (Thm. 1,",
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+                                    "type": "text"
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+                            "index": 3
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+                    ],
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+                                    "content": "In this section, we conduct numerical experiments3. to verify our theoretical results and investigate",
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+                                    "content": "how the heterogeneity of the device availability influences the Federated optimization algorithms.",
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+                                    "score": 1.0,
+                                    "content": "We compare the performance of the following four algorithms: FedAvg with importance sampling",
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+                                    "score": 1.0,
+                                    "content": "(FedAvg-IS), Biased FedAvg, FedAvg with device sampling, and our proposed MIFA. For the detailed",
+                                    "type": "text"
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+                            ],
+                            "index": 7
+                        },
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+                                    "score": 1.0,
+                                    "content": "discussions of the algorithms, we refer the readers to Sections 3 and 4. We remark that for a fair",
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+                                    "content": "Following [26, 25], we construct non-i.i.d. datasets from two commonly used computer vision",
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+                                    "content": "with each device holding samples of only two classes, which creates a high level of data heterogeneity.",
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+                                    "content": "For simplicity, we ensure that each device holds the same number of samples. We do not use any data",
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+                                    "score": 1.0,
+                                    "content": "penalty. We use logistic models for MNIST dataset, while",
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+                                    "score": 1.0,
+                                    "content": "we use LeNet-5 for CIFAR-10. Our code is adapted from [26], which is under MIT License.",
+                                    "type": "text"
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+                                    "content": ", where at each time step, the device becomes active with proba-",
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+                                    "content": ". In our experiments, the",
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+                                    "score": 1.0,
+                                    "content": "’s are chosen such that devices holding data of smaller labels",
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+                                    "score": 1.0,
+                                    "content": "participate less frequently. Specifically, if the",
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+                                    "score": 0.93,
+                                    "content": "\\bar { p } _ { i } = \\bar { p _ { \\operatorname* { m i n } } } \\operatorname* { m i n } ( j , k ) / 9 + \\bar { ( 1 - p _ { \\operatorname* { m i n } } ) }",
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+                                    "score": 0.89,
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+                                    "type": "inline_equation"
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "controls the lower bound of the participation prob-",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "abilities. The correlation between the participation patterns and local datasets increases the difficulty",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "of the problem [17]. To investigate this phenomenon, we repeat the experiments for",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.91,
+                                    "content": "p _ { \\mathrm { m i n } } = 0 . 1",
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+                        },
+                        {
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "0.2. We control the randomness of device participation when testing different algorithms.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 26
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+                    ],
+                    "index": 22.5
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+                    "lines": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "In all the experiments, we set the initial learning rate to be",
+                                    "type": "text"
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+                                {
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+                                    "score": 0.91,
+                                    "content": "\\eta _ { 0 } = 0 . 1",
+                                    "type": "inline_equation"
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "and decay the learning rate as",
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+                            ],
+                            "index": 27
+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 0.92,
+                                    "content": "\\begin{array} { r } { \\eta _ { t } = \\eta _ { 0 } \\cdot \\frac { 1 } { t } } \\end{array}",
+                                    "type": "inline_equation"
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ". We set the weight decay to be 0.001. The local batch size is 100 and each local update",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 28
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
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+                                        360,
+                                        462
+                                    ],
+                                    "score": 1.0,
+                                    "content": "consists of 2 epochs. Therefore, the actual number of local steps",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.79,
+                                    "content": "K",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "depends on the size of the dataset.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 29
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "We run all the experiments with 4 GPUs of type GeForce RTX 2080 Ti. We repeat the experiments",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 30
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "for 5 different random seeds, and all of the experiments exhibit similar training curves. We report the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 31
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "averaged training loss and test accuracy with error bars in Figure 2.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 32
+                        }
+                    ],
+                    "index": 29.5
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                    ],
+                    "lines": [
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "We observe that FedAvg with device sampling (FedAvg (",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.86,
+                                    "content": "S = 5 0",
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+                                },
+                                {
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+                                    "score": 1.0,
+                                    "content": ") and FedAvg",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.85,
+                                    "content": "S = 1 0 0",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": ") in Figure 2)",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 33
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
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+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "is severely influenced by the straggling devices and makes progress relatively slowly compared to the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 34
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "other algorithms. Although biased FedAvg converges fast at the beginning, this simple algorithm is",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 35
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        545
+                                    ],
+                                    "score": 1.0,
+                                    "content": "biased, and the optimality gaps are prominent for the harder CIFAR-10 dataset and when",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        451,
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+                                        543
+                                    ],
+                                    "score": 0.88,
+                                    "content": "p _ { \\mathrm { m i n } }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "is small.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 36
+                        },
+                        {
+                            "bbox": [
+                                105,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        556
+                                    ],
+                                    "score": 1.0,
+                                    "content": "On the contrary, our proposed MIFA avoids waiting for stragglers, converges fast without bias, and is",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 37
+                        },
+                        {
+                            "bbox": [
+                                105,
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+                                567
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        553,
+                                        506,
+                                        567
+                                    ],
+                                    "score": 1.0,
+                                    "content": "competitive with FedAvg with importance sampling, which requires knowledge of the participation",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 38
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        577
+                                    ],
+                                    "score": 1.0,
+                                    "content": "probabilities. We refer the readers to Appendix G for additional experiments on CIFAR-10.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 39
+                        }
+                    ],
+                    "index": 36
+                },
+                {
+                    "type": "title",
+                    "bbox": [
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+                    ],
+                    "lines": [
+                        {
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+                                    "score": 1.0,
+                                    "content": "8 Conclusions and Discussions",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 40
+                        }
+                    ],
+                    "index": 40
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                    ],
+                    "lines": [
+                        {
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "In this paper, we study FL algorithms in the presence of arbitrary device unavailability and propose",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 41
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "MIFA, which avoids waiting for straggling devices and re-uses the memorized latest updates as the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 42
+                        },
+                        {
+                            "bbox": [
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+                                650
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        638,
+                                        505,
+                                        650
+                                    ],
+                                    "score": 1.0,
+                                    "content": "surrogate when the device is unavailable. We theoretically analyze MIFA without any structural",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 43
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "assumptions on the device availability and prove the convergence for strongly convex and non-convex",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 44
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        660,
+                                        505,
+                                        671
+                                    ],
+                                    "score": 1.0,
+                                    "content": "smooth functions. Different from the literature that studies oracle complexity in terms of stochastic",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 45
+                        },
+                        {
+                            "bbox": [
+                                105,
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+                                682
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        670,
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+                                        682
+                                    ],
+                                    "score": 1.0,
+                                    "content": "gradient evaluations, we argue that in federated learning system, the bottleneck lies in the non-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 46
+                        },
+                        {
+                            "bbox": [
+                                106,
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+                                505,
+                                694
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        682,
+                                        505,
+                                        694
+                                    ],
+                                    "score": 1.0,
+                                    "content": "stationary and possibly adversarial pattern of device participation. Therefore, it is important to study",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 47
+                        },
+                        {
+                            "bbox": [
+                                105,
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+                                705
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        691,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "how the number of inactive rounds influences the convergence rate. In Theorem 5.1, the dependency",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 48
+                        }
+                    ],
+                    "index": 44.5
+                }
+            ],
+            "page_idx": 8,
+            "page_size": [
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+                792
+            ],
+            "discarded_blocks": [
+                {
+                    "type": "discarded",
+                    "bbox": [
+                        118,
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+                    ],
+                    "lines": [
+                        {
+                            "bbox": [
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+                            ],
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+                                {
+                                    "bbox": [
+                                        118,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "3Our code is available at https://github.com/hmgxr128/MIFA_code/",
+                                    "type": "text"
+                                }
+                            ]
+                        }
+                    ]
+                },
+                {
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+                    ],
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "9",
+                                    "type": "text"
+                                }
+                            ]
+                        }
+                    ]
+                }
+            ],
+            "para_blocks": [
+                {
+                    "type": "text",
+                    "bbox": [
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+                    ],
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Remark 6.1. When all",
+                                    "type": "text"
+                                },
+                                {
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+                                    ],
+                                    "score": 0.9,
+                                    "content": "\\nu _ { i } = 0",
+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "(i.e. all the devices are active), our convergence bound reduces",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 0
+                        },
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+                            ],
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+                                    ],
+                                    "score": 1.0,
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+                                    "content": "\\begin{array} { r } { \\mathcal { O } \\Big ( \\sqrt { \\frac { L } { T K N } } ( f ( w _ { 1 } ) - f ^ { * } + \\sigma ^ { 2 } ) + \\frac { ( K - 1 ) N L ( \\beta + \\sigma ^ { 2 } / K ) } { T } \\Big ) } \\end{array}",
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+                                    "score": 1.0,
+                                    "content": ". This matches the result in [42] (Thm. 1,",
+                                    "type": "text"
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+                                    "content": "7 Numerical Experiments",
+                                    "type": "text"
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+                                    "content": "In this section, we conduct numerical experiments3. to verify our theoretical results and investigate",
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+                                    "content": "We compare the performance of the following four algorithms: FedAvg with importance sampling",
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+                                    "score": 1.0,
+                                    "content": "(FedAvg-IS), Biased FedAvg, FedAvg with device sampling, and our proposed MIFA. For the detailed",
+                                    "type": "text"
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+                            ],
+                            "index": 7
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+                                    "score": 1.0,
+                                    "content": "discussions of the algorithms, we refer the readers to Sections 3 and 4. We remark that for a fair",
+                                    "type": "text"
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+                                    "content": "comparison, we deliberately include the first few rounds that MIFA needs to wait to receive responses",
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+                                    "content": "Following [26, 25], we construct non-i.i.d. datasets from two commonly used computer vision",
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+                                    "content": "we use LeNet-5 for CIFAR-10. Our code is adapted from [26], which is under MIT License.",
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+                                    "content": ", where at each time step, the device becomes active with proba-",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "of the problem [17]. To investigate this phenomenon, we repeat the experiments for",
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+                                    "score": 1.0,
+                                    "content": "0.2. We control the randomness of device participation when testing different algorithms.",
+                                    "type": "text"
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+                                    "content": "\\eta _ { 0 } = 0 . 1",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "and decay the learning rate as",
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+                        {
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+                                    "content": "\\begin{array} { r } { \\eta _ { t } = \\eta _ { 0 } \\cdot \\frac { 1 } { t } } \\end{array}",
+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    "score": 1.0,
+                                    "content": ". We set the weight decay to be 0.001. The local batch size is 100 and each local update",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 28
+                        },
+                        {
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "consists of 2 epochs. Therefore, the actual number of local steps",
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+                                },
+                                {
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+                                    "score": 0.79,
+                                    "content": "K",
+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": "depends on the size of the dataset.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 29
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "We run all the experiments with 4 GPUs of type GeForce RTX 2080 Ti. We repeat the experiments",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 30
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "for 5 different random seeds, and all of the experiments exhibit similar training curves. We report the",
+                                    "type": "text"
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+                            ],
+                            "index": 31
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "averaged training loss and test accuracy with error bars in Figure 2.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 32
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+                    ],
+                    "index": 29.5,
+                    "bbox_fs": [
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+                    "type": "text",
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+                                    "score": 1.0,
+                                    "content": "We observe that FedAvg with device sampling (FedAvg (",
+                                    "type": "text"
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": ") in Figure 2)",
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+                                    "score": 1.0,
+                                    "content": "is severely influenced by the straggling devices and makes progress relatively slowly compared to the",
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "other algorithms. Although biased FedAvg converges fast at the beginning, this simple algorithm is",
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "biased, and the optimality gaps are prominent for the harder CIFAR-10 dataset and when",
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+                                    "score": 0.88,
+                                    "content": "p _ { \\mathrm { m i n } }",
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+                                    "score": 1.0,
+                                    "content": "is small.",
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+                            "index": 36
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+                        {
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+                                    "score": 1.0,
+                                    "content": "On the contrary, our proposed MIFA avoids waiting for stragglers, converges fast without bias, and is",
+                                    "type": "text"
+                                }
+                            ],
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+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "competitive with FedAvg with importance sampling, which requires knowledge of the participation",
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+                            "index": 38
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+                        {
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+                                    "score": 1.0,
+                                    "content": "probabilities. We refer the readers to Appendix G for additional experiments on CIFAR-10.",
+                                    "type": "text"
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+                            ],
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+                    ],
+                    "index": 36,
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+                                    "content": "In this paper, we study FL algorithms in the presence of arbitrary device unavailability and propose",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "surrogate when the device is unavailable. We theoretically analyze MIFA without any structural",
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+                                    "score": 1.0,
+                                    "content": "smooth functions. Different from the literature that studies oracle complexity in terms of stochastic",
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diff --git a/parse/train/7R7fAoUygoa/7R7fAoUygoa.md b/parse/train/7R7fAoUygoa/7R7fAoUygoa.md
new file mode 100644
index 0000000000000000000000000000000000000000..f6da37ba570b36447843db821b4581b531f761b0
--- /dev/null
+++ b/parse/train/7R7fAoUygoa/7R7fAoUygoa.md
@@ -0,0 +1,698 @@
+# OPTIMAL REGULARIZATION CAN MITIGATE DOUBLE DESCENT
+
+Preetum Nakkiran   
+Harvard University   
+preetum@cs.harvard.edu   
+Prayaag Venkat   
+Harvard University   
+pvenkat@g.harvard.edu
+
+# Sham Kakade
+
+Microsoft Research & University of Washington sham@cs.washington.edu
+
+Tengyu Ma   
+Stanford University   
+tengyuma@stanford.edu
+
+# ABSTRACT
+
+Recent empirical and theoretical studies have shown that many learning algorithms – from linear regression to neural networks – can have test performance that is non-monotonic in quantities such the sample size and model size. This striking phenomenon, often referred to as “double descent”, has raised questions of if we need to re-think our current understanding of generalization. In this work, we study whether the double-descent phenomenon can be avoided by using optimal regularization. Theoretically, we prove that for certain linear regression models with isotropic data distribution, optimally-tuned $\ell _ { 2 }$ regularization achieves monotonic test performance as we grow either the sample size or the model size. We also demonstrate empirically that optimally-tuned $\ell _ { 2 }$ regularization can mitigate double descent for more general models, including neural networks. Our results suggest that it may also be informative to study the test risk scalings of various algorithms in the context of appropriately tuned regularization.
+
+# 1 INTRODUCTION
+
+Recent works have demonstrated a ubiquitous “double descent” phenomenon present in a range of machine learning models, including decision trees, random features, linear regression, and deep neural networks (Opper, 1995; 2001; Advani & Saxe, 2017; Spigler et al., 2018; Belkin et al., 2018; Geiger et al., 2019b; Nakkiran et al., 2020; Belkin et al., 2019; Hastie et al., 2019; Bartlett et al., 2019; Muthukumar et al., 2019; Bibas et al., 2019; Mitra, 2019; Mei & Montanari, 2019; Liang & Rakhlin, 2018; Liang et al., 2019; Xu & Hsu, 2019; Derezinski et al., 2019; Lampinen & Ganguli, ´ 2018; Deng et al., 2019; Nakkiran, 2019). The phenomenon is that models exhibit a peak of high test risk when they are just barely able to fit the train set, that is, to interpolate. For example, as we increase the size of models, test risk first decreases, then increases to a peak around when effective model size is close to the training data size, and then decreases again in the overparameterized regime. Also surprising is that Nakkiran et al. (2020) observe a double descent as we increase sample size, i.e. for a fixed model, training the model with more data can hurt test performance.
+
+These striking observations highlight a potential gap in our understanding of generalization and an opportunity for improved methods. Ideally, we seek to use learning algorithms which robustly improve performance as the data or model size grow and do not exhibit such unexpected nonmonotonic behaviors. In other words, we aim to improve the test performance in situations which would otherwise exhibit high test risk due to double descent. Here, a natural strategy would be to use a regularizer and tune its strength on a validation set. This motivates the central question of this work:
+
+When does optimally tuned regularization mitigate or remove the double-descent phenomenon?
+
+Another motivation is the fact that double descent is largely observed for unregularized or under regularized models in practice. As an example, Figure 1 shows a simple linear ridge regression setting in which the unregularized estimator exhibits double descent, but an optimally-tuned regularizer has monotonic test performance.
+
+![](images/b5edb5b47ed8f04264d90917cb098c4913b1a417a4d6db231609364ffb35c0c9.jpg)  
+Figure 1: Test Risk vs. Num. Samples for Isotropic Ridge Regression in $d \_ =$ 500 dimensions. Unregularized regression is non-monotonic in samples, but optimallyregularized regression $( \lambda = \lambda _ { o p t } )$ is monotonic. In this setting, the optimal regularizer $\lambda _ { o p t }$ does not depend on number of samples $n$ (Lemma 2), but this is not always true – see Figure 2.
+
+Our Contributions: We study this question from both a theoretical and empirical perspective. Theoretically, we start with the setting of high-dimensional linear regression. Linear regression is a sensible starting point to study these questions, since it already exhibits many of the qualitative features of double descent in more complex models (e.g. Belkin et al. (2019); Hastie et al. (2019) and further related works in Section 1.1). Our work shows that optimally-tuned ridge regression can achieve both sample-wise monotonicity and model-size-wise monotonicity under certain assumptions. Concretely, we show
+
+1. Sample-wise monotonicity: In the setting of well-specified linear regression with isotropic features/covariates (Figure 1), we prove that optimally-tuned ridge regression yields monotonic test performance with increasing samples. That is, more data never hurts for optimally-tuned ridge regression. (See Theorem 1).
+
+2. Model-wise monotonicity: We consider a setting where the input/covariate lives in a highdimensional ambient space with isotropic covariance. Given a fixed model size $d$ (which might be much smaller than ambient dimension), we consider the family of models which first project the input to a random $d$ -dimensional subspace, and then compute a linear function in this projected “feature space.” (This is nearly identical to models of double-descent considered in Hastie et al. (2019, Section 5.1)). We prove that in this setting, as we grow the model-size, optimally-tuned ridge regression over the projected features has monotone test performance. That is, with optimal regularization, bigger models are always better or the same. (See Theorem 3).
+
+3. Monotonicity in the real-world: We also demonstrate several richer empirical settings where optimal $\ell _ { 2 }$ regularization induces monotonicity, including random feature classifiers and convolutional neural networks. This suggests that the mitigating effect of optimal regularization may hold more generally in broad machine learning contexts. (See Section 5).
+
+A few remarks are in order:
+
+Problem-specific vs Minimax and Bayesian. It is worth noting that our results hold for all linear ground-truths, rather than holding for only the worst-case ground-truth or a random ground-truth. Indeed, the minimax optimal estimator or the Bayes optimal estimator are both trivially sample-wise and model-wise monotonic with respect to the minimax risk or the Bayes risk. However, they do not guarantee monotonicity of the risk itself for a given fixed problem. In particular, there exist minimax optimal estimators which are not sample-monotonic in the sense we desire.
+
+Universal vs Asymptotic. We also remark that our analysis is not only non-asymptotic but also works for all possible input dimensions, model sizes, and sample sizes. To our knowledge, the results herein are the first non-asymptotic sample-wise and model-wise monotonicity results for linear regression. (See discussion of related works Hastie et al. (2019); Mei & Montanari (2019) for related results in the asymptotic setting). Our work reveals aspects of the problem that were not present in prior asymptotic works. For example, we empirically show that optimal regularization can eliminate even “triple descent” in ridge regression (Figure 2). Moreover, we show that for nonGaussian covariates, optimally-tuned ridge regression is not always sample-monotonic: we give a counterexample in Section 4.
+
+Towards a more general characterization. Our theoretical results crucially rely on the covariance of the data being isotropic. A natural next question is if and when the same results can hold more generally. A full answer to this question is beyond the scope of this paper, though we give the following results:
+
+1. Optimally-tuned ridge regression is not always sample-monotonic: we show a counterexample for a certain non-Gaussian data distribution and heteroscedastic noise. We are not aware of prior work pointing out this fact. (See Section 4 for the counterexample and intuitions.)   
+2. For non-isotropic Gaussian covariates, we can achieve sample-wise monotonicity with a regularizer that depends on the population covariance matrix of data. This suggests unlabeled data might also help mitigate double descent in some settings, because the population covariance can be estimated from unlabeled data. (See Appendix B).   
+3. For non-isotropic Gaussian covariates, we conjecture that optimally-tuned ridge regression is sample-monotonic even with a standard $\ell _ { 2 }$ regularizer (as in Figure 2). We derive a sufficient condition for this conjecture. Due to that current random matrix theory may be insufficient to verify this conjecture, we verify it numerically on a wide variety of cases. (See Appendix B for details).
+
+The last two results above highlight the importance of the form of the regularizer, which leads to the open question: “How do we design good regularizers which mitigate or remove double descent?” We hope that our results can motivate future work on mitigating the double descent phenomenon, and allow us to train high performance models which do not exhibit nonmonotonic behaviors.
+
+# 1.1 RELATED WORKS
+
+The study of nonmonotonicity in learning algorithms existed prior to double descent and has a long history going back to (at least) Trunk (1979) and LeCun et al. (1991); Le Cun et al. (1991), where the former was largely empirical observations and the latter studied the sample non-nonmonotonicity of unregularized linear regression in terms of the eigenspectrum of the covariance matrix; the difference to our works is that we study this in the context of optimal regularization. In fact, Duin (1995; 2000); Opper (2001); Loog & Duin (2012). Loog et al. (2019) introduces the same notion of risk monotonicity which we consider, and studies several examples of monotonic and non-monotonic procedures.
+
+Double descent of test risk as a function of model size was considered recently in more generality by Belkin et al. (2018). Similar behavior was observed empirically in earlier work in somewhat more restricted settings Trunk (1979); Opper (1995; 2001); Skurichina & Duin (2002); Le Cun et al. (1991); LeCun et al. (1991) and more recently in Advani & Saxe (2017); Geiger et al. (2019a); Spigler et al. (2018); Neal et al. (2018). Recently Nakkiran et al. (2020) demonstrated a generalized double descent phenomenon on modern deep networks, and highlighted “sample non-monotonicity” as an aspect of double descent.
+
+A recent stream of theoretical works consider model-wise double descent in simplified settings— often via linear models for regression or classification. This also connects to works on highdimentional regression in the statistics literature. A partial list of works in these areas include Belkin et al. (2019); Hastie et al. (2019); Bartlett et al. (2019); Muthukumar et al. (2019); Bibas et al. (2019); Mitra (2019); Mei & Montanari (2019); Liang & Rakhlin (2018); Liang et al. (2019); Xu & Hsu (2019); Derezinski et al. (2019); Lampinen & Ganguli (2018); Deng et al. (2019); Nakki- ´ ran (2019); Mahdaviyeh & Naulet (2019); Dobriban et al. (2018); Dobriban & Sheng (2019); Kobak et al. (2018). Of these, most closely related to our work are Hastie et al. (2019); Dobriban et al. (2018); Mei & Montanari (2019). Specifically, Hastie et al. (2019) considers the risk of unregularized and regularized linear regression in an asymptotic regime, where dimension $d$ and number of samples $n$ scale to infinity together, at a constant ratio $d / n$ . In contrast, we show non-asymptotic results, and are able to consider increasing the number of samples for a fixed model, without scaling both together. Mei & Montanari (2019) derive similar results for unregularized and regularized random features, also in an asymptotic limit. The non-asymptotic versions of the settings considered in Hastie et al. (2019) are almost identical to ours— for example, our projection model in Section 3 is nearly identical to the model in Hastie et al. (2019, Section 5.1). Finally, subsequent to our work, d’Ascoli et al. (2020) identified triple descent in an asymptotic setting.
+
+# 2 SAMPLE MONOTONICITY IN RIDGE RIDGRESSION
+
+In this section, we prove that optimally-regularized ridge regression has test risk that is monotonic in samples, for isotropic gaussian covariates and linear response. This confirms the behavior empirically observed in Figure 1. We also show that this monotonicity is not “fragile”, and using larger than larger regularization is still sample-monotonic (consistent with Figure 1).
+
+Formally, we consider the following linear regression problem in $d$ dimensions. The input/covariate $x \in \mathbb { R } ^ { d }$ is generated from $\mathcal { N } ( 0 , I _ { d } )$ , and the output/response is generated by $y = \langle x , \beta ^ { * } \rangle + \varepsilon$ with $\varepsilon \sim \mathcal { N } ( 0 , \bar { \sigma } ^ { 2 } )$ for some unknown parameter $\beta ^ { * } \in \mathbb { R } ^ { d }$ . We denote the joint distribution of $( x , y )$ by $\mathcal { D }$ . We are given $n$ training examples $\{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n }$ i.i.d sampled from $\mathcal { D }$ . We aim to learn a linear model $f _ { \beta } ( x ) = \langle x , \beta \rangle$ with small population risk $\bar { R } ( \beta ) : = \mathbb { E } _ { ( x , y ) \sim \mathcal { D } } [ ( \langle x , \beta \rangle - y ) ^ { 2 } ]$ . For simplicity, let $\ b { X } \in \mathbb { R } ^ { n \times d }$ be the data matrix that contains $x _ { i } ^ { \top }$ ’s as rows and let $\vec { y } \in \mathbb R ^ { n }$ be column vector that contains the responses $y _ { i }$ ’s as entries. For any estimator $\hat { \beta } _ { n } ( X , \vec { y } )$ as a function of $n$ samples, define the expected risk of the estimator as:
+
+$$
+{ \overline { { R } } } ( { \hat { \beta } } _ { n } ) : = \operatorname * { \mathbb { E } } _ { X , y \sim { \mathcal { D } } ^ { n } } [ R ( { \hat { \beta } } _ { n } ( X , { \vec { y } } ) ) ]
+$$
+
+We consider the regularized least-squares estimator, also known as the ridge regression estimator. For a given $\lambda > 0$ , define
+
+$$
+\hat { \beta } _ { n , \lambda } : = \underset { \beta } { \operatorname { a r g m i n } } | | X \beta - \vec { y } | | _ { 2 } ^ { 2 } + \lambda | | \beta | | _ { 2 } ^ { 2 } = ( X ^ { T } X + \lambda I _ { d } ) ^ { - 1 } X ^ { T } \vec { y }
+$$
+
+Here $I _ { d }$ denotes the $d$ dimensional identity matrix. Let $\lambda _ { n } ^ { \mathrm { o p t } }$ be the optimal ridge parameter (that achieves the minimum expected risk) given $n$ samples: $\begin{array} { r } { \lambda _ { n } ^ { \mathrm { o p t } } : = \operatorname * { a r g m i n } _ { \lambda : \lambda \geq 0 } \overline { { R } } ( \hat { \beta } _ { n , \lambda } ) ) } \end{array}$ . Let $\hat { \beta } _ { n } ^ { \mathrm { { o p t } } }$ be the estimator that corresponds to the $\lambda _ { n } ^ { \mathrm { o p t } }$ . That is, $\begin{array} { r } { \hat { \beta } _ { n } ^ { \mathrm { o p t } } : = \operatorname * { a r g m i n } _ { \beta } | | X \beta - \vec { y } | | _ { 2 } ^ { 2 } + \lambda _ { n } ^ { \mathrm { o p t } } | | \beta | | _ { 2 } ^ { 2 } } \end{array}$ . Our main theorem in this section shows that the expected risk of $\hat { \beta } _ { n } ^ { \mathrm { { o p t } } }$ monotonically decreases as $n$ increases.
+
+Theorem 1. In the setting above, the expected test risk of optimally-regularized well-specified isotropic linear regression is monotonic in samples. That is, for all $\beta ^ { * } \in \mathbb { R } ^ { d }$ and all $d \in \mathbb { N } , n \in$ $\mathbb { N } , \sigma > 0$ ,
+
+$$
+\overline { { R } } ( \hat { \beta } _ { n + 1 } ^ { \mathrm { o p t } } ) \leq \overline { { R } } ( \hat { \beta } _ { n } ^ { \mathrm { o p t } } )
+$$
+
+The above theorem shows a strong form of monotonicity, since it holds for every fixed groundtruth $\beta ^ { * }$ , and does not require averaging over any prior on ground-truths. Moreover, it holds nonasymptotically, for every fixed $n , d \in \mathbb { N }$ . Obtaining such non-asymptotic results is nontrivial, since we cannot rely on concentration properties of the involved random variables.
+
+In particular, evaluating $\overline { { R } } ( \hat { \beta } _ { n } ^ { \mathrm { { o p t } } } )$ as a function of the problem parameters $( n , \sigma , \beta ^ { * }$ , and $d$ ) is technically challenging. In fact, we suspect that a simple closed form expression does not exist. The key idea towards proving the theorem is to derive a “partial evaluation” — the following lemmas shows that we can write $\overline { { R } } ( \hat { \beta } _ { n } ^ { \mathrm { { o p t } } } )$ in the form of $\mathbb { E } [ g ( \gamma , \sigma , n , d , \beta ^ { * } ) ]$ where $\gamma \in \mathbb { R } ^ { d }$ contains the singular values of $X$ . We will then couple the randomness of data matrices obtained by adding a single sample, and use singular value interlacing to compare their singular values.
+
+Lemma 1. In the setting of Theorem $^ { l }$ , let $\gamma = ( \gamma _ { 1 } , \ldots , \gamma _ { d } )$ be the singular values of the data matrix $\ b { X } \in \mathbb { R } ^ { n \times d }$ . (If $n < d$ , we pad the $\gamma _ { i } = 0$ for $i > n .$ .) Let $\Gamma _ { n }$ be the distribution of $\gamma .$ . Then, the expected test risk is
+
+$$
+\overline { { R } } ( \hat { \beta } _ { n , \lambda } ) = \operatorname * { \mathbb { E } } _ { ( \gamma _ { 1 } , \dots \gamma _ { d } ) \sim \Gamma _ { n } } \left[ \sum _ { i = 1 } ^ { d } \frac { | | \beta ^ { * } | | _ { 2 } ^ { 2 } \lambda ^ { 2 } / d + \sigma ^ { 2 } \gamma _ { i } ^ { 2 } } { ( \gamma _ { i } ^ { 2 } + \lambda ) ^ { 2 } } \right] + \sigma ^ { 2 }
+$$
+
+From Lemma 1, the below lemma follows directly by taking derivatives to find the optimal $\lambda$ .
+
+Lemma 2. In the setting of Theorem 1, the optimal ridge parameter is constant for all $n$ : $\lambda _ { n } ^ { \mathrm { o p t } } =$ $\frac { d \sigma ^ { 2 } } { | | \beta ^ { * } | | _ { 2 } ^ { 2 } }$ . Moreover, the optimal expected test risk can be written as
+
+$$
+\overline { { R } } ( \hat { \beta } _ { n } ^ { \mathrm { o p t } } ) = \operatorname * { l } _ { ( \gamma _ { 1 } , \ldots \gamma _ { d } ) \sim \Gamma _ { n } } \left[ \sum _ { i = 1 } ^ { d } \frac { \sigma ^ { 2 } } { \gamma _ { i } ^ { 2 } + d \sigma ^ { 2 } / | | \beta ^ { * } | | _ { 2 } ^ { 2 } } \right] + \sigma ^ { 2 }
+$$
+
+Proofs of Lemma 1 and 2 are deferred to the Appendix, Section A.1. Now we are ready to prove Theorem 1.
+
+Proof of Theorem $^ { l }$ . Let $\widetilde { X } \ \in \ \mathbb { R } ^ { ( n + 1 ) \times d }$ and $\boldsymbol { X } \in \mathbb { R } ^ { n \times d }$ be any two matrices which differ by only the last row of $\widetilde { X }$ . By the Cauchy interlacing theorem Theorem 4.3.4 of Horn et al. (1990) (c.f.,Lemma 3.4 of Marcus et al. (2014)), the singular values of $X$ and $\widetilde { X }$ are interlaced: $\forall i$ : $\gamma _ { i - 1 } ( X ) \geq \gamma _ { i } ( { \widetilde { X } } ) \geq \gamma _ { i } ( X )$ where $\gamma _ { i } ( \cdot )$ is the $i$ -th singular value.
+
+If we couple $\widetilde { X }$ and $X$ , it will induce a coupling $\Pi$ between the distributions $\Gamma _ { n + 1 }$ and $\Gamma _ { n }$ , of the singular values of the data matrix for $n + 1$ and $n$ samples. This coupling satisfies that $\widetilde { \gamma } _ { i } \geq \gamma _ { i }$ with probability 1 for $( \{ \widetilde { \gamma } _ { i } \} , \{ \gamma _ { i } \} ) \sim \Pi$ e. Now, expand the test risk using Lemma 2, and observe that each eterm in the sum of Equation (4) below is monotone decreasing with $\gamma _ { i }$ . Thus:
+
+$$
+\begin{array} { r l } & { \overline { { R } } ( \hat { \beta } _ { n } ^ { \mathrm { o p t } } ) = \underset { ( \gamma _ { 1 } , \ldots , \gamma _ { d } ) \sim \Gamma _ { n } } { \mathbb { E } } \left[ \displaystyle \sum _ { i = 1 } ^ { d } \frac { \sigma ^ { 2 } } { \gamma _ { i } ^ { 2 } + d \sigma ^ { 2 } / | | \beta ^ { * } | | _ { 2 } ^ { 2 } } \right] + \sigma ^ { 2 } } \\ & { \qquad \quad \overset { \mathbb { E } } { \ge } \underset { ( \widetilde { \gamma } _ { 1 } , \ldots , \widetilde { \gamma } _ { d } ) \sim \Gamma _ { n + 1 } } { \mathbb { E } } \left[ \displaystyle \sum _ { i = 1 } ^ { d } \frac { \sigma ^ { 2 } } { \widetilde { \gamma } _ { i } ^ { 2 } + d \sigma ^ { 2 } / | | \beta ^ { * } | | _ { 2 } ^ { 2 } } \right] + \sigma ^ { 2 } } \\ & { \qquad = \overline { { R } } ( \hat { \beta } _ { n + 1 } ^ { \mathrm { o p t } } ) } \end{array}
+$$
+
+By similar techniques, we can also prove that overregularization —that is, using ridge parameters $\lambda$ larger than the optimal value— is still monotonic. This proves the behavior empirically observed in Figure 1.
+
+Theorem 2. In the same setting as Theorem $I$ , over-regularized regression is also monotonic in samples. That is, for all $d \in \mathbb { N } , n \in \mathbb { N } , \sigma > 0 , \beta ^ { * } \in \mathbb { R } ^ { d }$ , the following holds
+
+$$
+\forall \lambda \geq \lambda ^ { * } : \quad \overline { { { R } } } ( \widehat { \beta } _ { n + 1 , \lambda } ) \leq \overline { { { R } } } ( \widehat { \beta } _ { n , \lambda } )
+$$
+
+where $\begin{array} { r } { \lambda ^ { * } = \frac { d \sigma ^ { 2 } } { | | \beta ^ { * } | | _ { 2 } ^ { 2 } } } \end{array}$
+
+Proof. In Section A.1.
+
+# 3 MODEL-WISE MONOTONICITY IN RIDGE REGRESSION
+
+In this section, we show that for a certain family of linear models, optimal regularization prevents model-wise double descent. That is, for a fixed number of samples, larger models are not worse than smaller models.
+
+We consider the following learning problem. Informally, covariates live in a $p$ -dimensional ambient space, and we consider models which first linearly project down to a random $d$ -dimensional subspace, then perform ridge regression in that subspace for some $d \leq p$ . Formally, the covariate $x \in \mathbb { R } ^ { p }$ is generated from $\bar { \mathcal { N } } ( 0 , I _ { p } )$ , and the response is generated by $y = \langle x , \theta \rangle + \varepsilon$ with $\varepsilon \sim \mathcal { N } ( 0 , \sigma ^ { 2 } )$ and for some unknown parameter $\theta \in \mathbb { R } ^ { p }$ . Next, $n$ examples $\{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n }$ are sampled i.i.d from this distribution. For a given model size $d \leq p$ , we first sample a random orthonormal matrix $P \in \mathbb { R } ^ { d \times p }$ which specifies our model. We then consider models which operate on $( \widetilde { x } _ { i } , y _ { i } ) \in \mathbb { R } ^ { d } \times \mathbb { R }$ , where $\widetilde { x _ { i } } = P x _ { i }$ . We denote the joint distribution of $( \widetilde { x } , y )$ by $\mathcal { D }$ . Here, we eemphasize that $p$ eis some large ambient dimension and $d \leq p$ eis the size of the model we learn.
+
+For a fixed $P$ , we want to learn a linear model $f _ { \hat { \beta } } ( \tilde { x } ) = \langle \tilde { x } , \hat { \beta } \rangle$ for estimating $y$ , with small mean squared error on distribution: $R _ { P } ( \hat { \beta } ) : = \mathbb { E } _ { ( \tilde { x } , y ) \sim \mathcal { D } } [ ( \langle \tilde { x } , \hat { \beta } \rangle - y ) ^ { 2 } ]$ . For $n$ samples $( x _ { i } , y _ { i } )$ , let $X \in$ $\mathbb { R } ^ { n \times p }$ be the data matrix, $\widetilde { X } \ = X P ^ { T } \ \in \ \mathbb { R } ^ { n \times d }$ be the projected data matrix and $\vec { y } \in \mathbb R ^ { n }$ be the responses. For any estimator $\hat { \beta } ( \widetilde { X } , \vec { y } )$ as a function of the observed samples, define the expected risk of the estimator as:
+
+$$
+\overline { { R } } ( \hat { \beta } ) : = \operatorname * { \mathbb { E } } _ { P } \operatorname * { \mathbb { E } } _ { \tilde { X } , \vec { y } \sim \mathcal { D } ^ { n } } [ R _ { P } ( \hat { \beta } ( \tilde { X } , \vec { y } ) ]
+$$
+
+We consider the regularized least-squares estimator. For a given $\lambda > 0$ , define
+
+$$
+\hat { \beta } _ { d , \lambda } : = \underset { \beta } { \mathrm { a r g m i n } } | | \widetilde { X } \beta - \vec { y } | | _ { 2 } ^ { 2 } + \lambda | | \beta | | _ { 2 } ^ { 2 } = ( \widetilde { X } ^ { T } \widetilde { X } + \lambda I _ { d } ) ^ { - 1 } \widetilde { X } ^ { T } \vec { y }
+$$
+
+Let $\lambda _ { d } ^ { \mathrm { o p t } }$ be the optimal ridge parameter (that achieves the minimum expected risk) for a model of size $d$ d, with $n$ samples: $\lambda _ { d } ^ { \mathrm { o p t } } : = \mathrm { a r g m i n } _ { \lambda \ge 0 } \overline { { R } } ( \hat { \beta } _ { d , \lambda } ) )$ . Let $\hat { \beta } _ { d } ^ { \mathrm { { o p t } } }$ be the estimator that corresponds to the $\lambda _ { d } ^ { \mathrm { o p t } }$ , that is $\begin{array} { r } { \hat { \beta } _ { d } ^ { \mathrm { o p t } } : = \operatorname * { a r g m i n } _ { \beta } | | \widetilde { X } \beta - \vec { y } | | _ { 2 } ^ { 2 } + \lambda _ { d } ^ { \mathrm { o p t } } | | \beta | | _ { 2 } ^ { 2 } } \end{array}$ . Now, our main theorem in this setting shows that with optimal $\ell _ { 2 }$ regularization, test performance is monotonic in model size.
+
+Theorem 3. In the setting above, the expected test risk of the optimally-regularized model is monotonic in the model size $d .$ . That is, for all $p \in \mathbb { N } , \theta \in \mathbb { R } ^ { p } , d \leq p , n \in \mathbb { N } , \sigma > 0$ , we have
+
+$$
+\overline { { R } } ( \hat { \beta } _ { d + 1 } ^ { \mathrm { o p t } } ) \leq \overline { { R } } ( \hat { \beta } _ { d } ^ { \mathrm { o p t } } )
+$$
+
+The proof of Theorem 3 is in Appendix A.2, and follows closely the proof of Theorem 1.
+
+# 4 COUNTEREXAMPLES TO MONOTONICITY
+
+In this section, we show that optimally-regularized ridge regression is not always monotonic in samples. We give a numeric counterexample in $d = 2$ dimensions, with non-gaussian covariates and heteroscedastic noise. This does not contradict our main theorem in Section 2, since this distribution is not jointly Gaussian with isotropic marginals.
+
+Counterexample. Here we give an example of a distribution $( x , y )$ for which the expected error of optimally-regularized ridge regression with $n = 2$ samples is worse than with $n = 1$ samples. This counterexample is most intuitive to understand when the ridge parameter $\lambda$ is allowed to depend on the specific sample instance $( X , { \vec { y } } )$ as well as $n ^ { 1 }$ . We sketch the intuition for this below. Consider the following distribution on $( x , y )$ in $d = 2$ dimensions. This distribution has one “clean” coordinate and one “noisy” coordinate. The distribution is: $( x , y ) = ( { \vec { e } } _ { 1 } , 1 )$ with probability $1 / 2$ , and $( x , y ) =$ $( \vec { e } _ { 2 } , \pm 1 0 )$ w.p. 1/2. Where $\pm 1 0$ is uniformly random independent noise. This distribution is “wellspecified” in that the optimal predictor is linear in $x$ $: \mathbb { E } [ y | x ] = \langle \beta ^ { * } , x \rangle$ for $\beta ^ { * } = [ 1 , 0 ]$ . However, the noise is heteroscedastic.
+
+For $n = 1$ samples, the estimator can decide whether to use small $\lambda$ or large $\lambda$ depending on if the sampled coordinate is the “clean” or “noisy” one. Specifically, for the sample $( x , y )$ : If $x = \vec { e } _ { 1 }$ , then the optimal ridge parameter is $\lambda = 0$ . If $x = \vec { e } _ { 2 }$ , then the optimal parameter is $\lambda = \infty$ .
+
+For $n = 2$ samples, with probability $1 / 2$ the two samples will hit both coordinates. In this case, the estimator must chose a single value of $\lambda$ uniformly for both coordinates. This yields to a suboptimal tradeoff, since the “noisy” coordinate demands large regularization, but this hurts estimation on the “clean” coordinate.
+
+It turns out that a slight modification to the above also serves as a counterexample to monotonicity when the regularization parameter $\lambda$ is chosen only depending on $n$ (and not on the instance $X , y )$ . The distribution is: $( x , y ) = ( \vec { e } _ { 1 } , 1 )$ w.p. 0.98 and $( x , y ) = ( \vec { e } _ { 2 } , \pm 2 0 )$ w.p. 0.02. This distribution has the following property.
+
+Theorem 4. There exists a distribution $\mathcal { D }$ over $( x , y )$ for $x \in \mathbb { R } ^ { 2 } , y \in \mathbb { R }$ with the following properties. Let $\hat { \beta } _ { n } ^ { \mathrm { { o p t } } }$ be the optimally-regularized ridge regression solution for $n$ samples $( X , { \vec { y } } )$ from $\mathcal { D }$ . Then:
+
+1. D is “well-specified” in that $\mathbb { E } p [ y | x ]$ is a linear function of $x$ ,
+
+2. The expected test risk increases as a function of $n _ { \ast }$ , between $n = 1$ and $n = 2$ . Specifically
+
+$$
+\overline { { R } } ( \hat { \beta } _ { n = 1 } ^ { \mathrm { o p t } } ) < \overline { { R } } ( \hat { \beta } _ { n = 2 } ^ { \mathrm { o p t } } )
+$$
+
+Proof. For $n = 1$ samples, it can be confirmed analytically that the expected risk $\overline { { { R } } } ( \hat { \beta } _ { n = 1 } ^ { \mathrm { o p t } } ) ~ <$ 8.157. This is achieved with $\lambda = 4 0 0 / 2 4 0 1 \approx 0 . 1 6 6 5 9 7 .$ For $n = 2$ samples, it can be confirmed numerically (via Mathematica) that the expected risk $\overline { { R } } ( \hat { \beta } _ { n = 2 } ^ { \mathrm { o p t } } ) ~ > ~ 8 . 1 7 9$ . This is achieved with $\lambda = 0 . 6 4 2 5 2 5$ . □
+
+# 5 EXPERIMENTS
+
+We now experimentally demonstrate that optimal $\ell _ { 2 }$ regularization can mitigate double descent, in more general settings than Theorems 1 and 3.
+
+# 5.1 SAMPLE MONOTONICITY
+
+Here we show various settings where optimal $\ell _ { 2 }$ regularization empirically induces samplemonotonic performance.
+
+Nonisotropic Regression. We first consider the setting of Theorem 1, but with non-isotropic covariantes $x$ . That is, we perform ridge regression on samples $( x , y )$ , where the covariate $\boldsymbol { x } \in \mathbb { R } ^ { d }$ is generated from $\mathcal { N } ( 0 , \Sigma )$ for $\Sigma \neq I _ { d }$ . As before, the response is generated by $y = \langle x , \beta ^ { * } \rangle + \varepsilon$ with $\varepsilon \sim \mathcal { N } ( 0 , \sigma ^ { 2 } )$ for some unknown parameter $\beta ^ { * } \in \mathbb { R } ^ { d }$ . We consider the same ridge regression estimator, $\begin{array} { r } { \hat { \beta } _ { n , \lambda } : = \operatorname * { a r g m i n } _ { \boldsymbol { \beta } } \| X { \boldsymbol { \beta } } - \vec { y } \| _ { 2 } ^ { 2 } + \lambda \| { \boldsymbol { \beta } } \| _ { 2 } ^ { 2 } } \end{array}$ .
+
+![](images/6bbe45f3d0113a0cbb86e816d67e9834573c85c456540349b12e6e02c25001ba.jpg)  
+Figure 2: Test Risk vs. Num. Samples for Non-Isotropic Ridge Regression in $d = 3 0$ dimensions. Unregularized regression is non-monotonic in samples, but optimally-regularized regression is monotonic. Note the optimal regularization $\lambda$ depends on the number of samples $n$ .   
+Figure 2 shows one instance of this, for a particular choice of $\Sigma$ and $\beta ^ { * }$ . The covariance $\Sigma$ is diagonal, with $\Sigma _ { i , i } = 1 0$ for $i \leq 1 5$ and $\Sigma _ { i , i } = 1$ for $i > 1 5$ . That is, the covariance has one “large” eigenspace and one “small” eigenspace. The ground-truth $\beta ^ { * } = 0 . 1 \vec { e _ { 1 } } + e _ { 3 0 } ^ {  }$ , which lies almost entirely within the “small” eigenspace of $\Sigma$ . The noise parameter is $\sigma = 0 . 5$ .
+
+We see that unregularized regression $\lambda = 0$ ) actually undergoes “triple descent”2 in this setting, with the first peak around $n = 1 5$ samples due to the 15-dimensional large eigenspace, and the second peak at $n = d$ . In this setting, optimally-regularized ridge regression is empirically monotonic in samples (Figure 2). Unlike the isotropic setting of Section 2, the optimal ridge parameter $\lambda _ { n }$ is no longer a constant, but varies with number of samples $n$ .
+
+Random ReLU Features. We consider random ReLU features, in the random features framework of Rahimi & Recht (2008). For a given number of features $D$ , and number of samples $n$ , the random feature classifier is obtained by performing regularized linear regression on the embedding√ $\tilde { x } : = \mathrm { R e L U } ( W x )$ , where $W \in \mathbb { R } ^ { D \times d }$ is a matrix with each entry sampled i.i.d $\mathcal { N } ( 0 , 1 / \sqrt { d } )$ and ReLU applies pointwise. This is equivalent to a 2-layer fully-connected neural network with a frozen (randomly-initialized) first layer, trained with $\ell _ { 2 }$ loss and weight decay. In Appendix A.4, we apply random features to Fashion-MNIST Xiao et al. (2017). From Appendix Figure 4a, we see that underregularized models are non-monotonic, but optimal $\ell _ { 2 }$ regularization is monotonic in samples. Moreover, the optimal ridge parameter $\lambda$ appears to be constant for all $n$ , similar to our results from the isotropic setting in Theorem 1.
+
+# 5.2 MODEL-SIZE MONOTONICITY
+
+Here we empirically show that optimal $\ell _ { 2 }$ regularization can mitigate model-wise double descent.
+
+Random ReLU Features. We consider the same experimental setup as in Section 5.1, but now fix the number of samples $n$ , and vary the number of random features $D$ . This corresponds to varying the width of the corresponding 2-layer neural network. Figure 4b in Appendix A.4 shows the test error of the random features classifier, for $n = 5 0 0$ train samples and varying number of random features. We see that underregularized models undergo model-wise double descent, but optimal $\ell _ { 2 }$ regularization prevents double descent.
+
+Convolutional Neural Networks. We follow the experimental setup of Nakkiran et al. (2020) for modelwise double descent, and add varying amounts of $\ell _ { 2 }$ regularization (weight decay). We chose the following setting from Nakkiran et al. (2020), because it exhibits double descent even with no added label noise. We consider the same family of 5-layer convolutional neural networks (CNNs) from Nakkiran et al. (2020), consisting of 4 convolutional layers of widths $[ k , 2 k , 4 k , 8 k ]$ for varying $k \in$ $\mathbb { N }$ . We train and test on CIFAR100 (Krizhevsky et al., 2009), an image classification problem with 100 classes. Inputs are normalized to
+
+![](images/b59ffa519889838486fdb1bea9435eca85527fcc4c5ff780a6f8610321941a4b.jpg)  
+Figure 3: Test Error vs. Model Size for 5-layer CNNs on CIFAR-100, with $\ell _ { 2 }$ regularization (weight decay). Note that the optimal regularization $\lambda$ varies with $n$ .
+
+$[ - 1 , 1 ] ^ { d }$ , and we use standard data-augmentation of random horizontal flip and random crop with 4- pixel padding. All models are trained using Stochastic Gradient Descent (SGD) on the cross-entropy loss, with step size $0 . 1 / \sqrt { \lfloor { T / 5 1 2 } \rfloor + 1 }$ at step $T$ . We train for 1e6 gradient steps, and use weight decay $\lambda$ for varying $\lambda$ . Due to optimization instabilities for large $\lambda$ , we use the model with the minimum train loss among the last 5K gradient steps. Figure 3 shows the test error of these models on CIFAR-100. Although unregularized and under-reguarized models exhibit double descent, the test error of optimally-regularized models is largely monotonic. Note that the optimal regularization $\lambda$ varies with the model size — no single regularization value is optimal for all models.
+
+# 6 DISCUSSION AND CONCLUSION
+
+In this work, we study the double descent phenomenon in the context of optimal regularization. We show that, while unregularized or under-regularized models often have non-monotonic behavior, appropriate regularization can eliminate this effect.
+
+Theoretically, we prove that for certain linear regression models with isotropic covariates, optimallytuned $\ell _ { 2 }$ regularization achieves monotonic test performance as we grow either the sample size or the model size. These are the first non-asymptotic monotonicity results we are aware of in linear regression. We also demonstrate empirically that optimally-tuned $\ell _ { 2 }$ regularization can mitigate double descent for more general models, including neural networks. We hope that our results can motivate future work on mitigating the double descent phenomenon, and allow us to train high performance models which do not exhibit unexpected nonmonotonic behaviors.
+
+Open Questions. Our work suggests a number of natural open questions. First, it is open to prove (or disprove) that optimal ridge regression is sample-monotonic for non-isotropic Gaussian covariates. We conjecture that it is, and outline a potential route to proving this (via Conjectures 1 and 2 in the Appendix). Second, more broadly, it is open to prove sample-wise or model-wise monotonicity for more general (non-linear) models with appropriate regularizers. Finally, it is open to understand why large neural networks in practice are often sample-monotonic in realistic regimes of sample sizes, even without careful choice of regularization.
+
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+
+# A APPENDIX
+
+In Section A.1 and A.2 we provide the proofs for sample-monotonicity and model-size monotonicity. In Section A.4 we include additional and omitted plots. In Section B we investigate whether monotonicity provably holds in more general models, and present a monotonicity conjecture for non-isotropic covariates.
+
+# A.1 SAMPLE MONOTONICITY PROOFS
+
+First we prove Lemma 1.
+
+Proof of Lemma $^ { l }$ . For isotropic $x$ , the test risk is related to the parameter error as:
+
+$$
+\begin{array} { r l } & { R ( \hat { \beta } ) : = \underset { ( x , y ) \sim \mathcal { D } } { \mathbb { E } } [ ( \langle x , \hat { \beta } \rangle - y ) ^ { 2 } ] } \\ & { \quad \quad = \underset { x \sim \mathcal { N } ( 0 , I _ { d } ) , \eta \sim \mathcal { N } ( 0 , \sigma ^ { 2 } ) } { \mathbb { E } } [ ( \langle x , \hat { \beta } - \beta ^ { * } \rangle + \eta ) ^ { 2 } ] } \\ & { \quad \quad = | | \hat { \beta } - \beta ^ { * } | | _ { 2 } ^ { 2 } + \sigma ^ { 2 } } \end{array}
+$$
+
+Plugging in the form of $\hat { \beta } _ { n , \lambda }$ and expanding:
+
+$$
+\begin{array} { r l } & { \bar { R } ( \hat { \beta } _ { n , \lambda } ) : = \underset { X , y \sim \mathcal { D } ^ { n } } { \mathbb { E } } [ R ( \hat { \beta } _ { n , \lambda } ) ] } \\ & { = \mathbb { E } [ \| \hat { \beta } _ { n , \lambda } - \beta ^ { * } \| _ { 2 } ^ { 2 } ] + \sigma ^ { 2 } } \\ & { = \underset { X , y } { \mathbb { E } } [ \| ( X ^ { T } X + \lambda I ) ^ { - 1 } X ^ { T } y - \beta \| _ { 2 } ^ { 2 } ] + \sigma ^ { 2 } } \\ & { = \underset { X , \eta \sim \mathcal { N } ( 0 , \sigma ^ { 2 } I _ { n } ) } { \mathbb { E } } [ \| ( X ^ { T } X + \lambda I ) ^ { - 1 } X ^ { T } ( X \beta ^ { * } + \eta ) - \beta ^ { * } \| _ { 2 } ^ { 2 } ] + \sigma ^ { 2 } } \\ & { = \underset { X } { \mathbb { E } } [ \| ( X ^ { T } X + \lambda I ) ^ { - 1 } X ^ { T } X \beta ^ { * } - \beta ^ { * } \| _ { 2 } ^ { 2 } ] + \underset { X , \eta } { \mathbb { E } } [ \| ( X ^ { T } X + \lambda I ) ^ { - 1 } X ^ { T } \eta \| _ { 2 } ^ { 2 } ] + \sigma ^ { 2 } } \\ & { = \underset { X } { \mathbb { E } } [ \| ( X ^ { T } X + \lambda I ) ^ { - 1 } X ^ { T } X \beta ^ { * } - \beta ^ { * } \| _ { 2 } ^ { 2 } ] + \sigma ^ { 2 } \underset { X } { \mathbb { E } } [ \| ( X ^ { T } X + \lambda I ) ^ { - 1 } X ^ { T } \| _ { F } ^ { 2 } ] + \sigma ^ { 2 } } \end{array}
+$$
+
+Now let $X \ = \ U \Sigma V ^ { T }$ be the full singular value decomposition of $X$ , with $U \ \in \ \mathbb { R } ^ { n \times n } , \Sigma \ \in$ $\mathbb { R } ^ { n \times d } , V \in \mathbb { R } ^ { d \times d }$ . Let $( \gamma _ { 1 } , \ldots \gamma _ { d } )$ denote the singular values, defining $\gamma _ { i } = 0$ for $i > \operatorname* { m i n } ( n , d )$ . Then, continuing:
+
+$$
+\begin{array} { r l } & { \overline { { R } } ( \hat { \beta } _ { n , \lambda } ) = \underset { V , \Sigma } { \mathbb { E } } [ | | \mathrm { d i a g } ( \{ \frac { - \lambda } { \gamma _ { i } ^ { 2 } + \lambda } \} ) V ^ { T } \beta ^ { * } | | _ { 2 } ^ { 2 } ] + \sigma ^ { 2 } \underset { \Sigma } { \mathbb { E } } [ \sum _ { i } \frac { \gamma _ { i } ^ { 2 } } { ( \gamma _ { i } ^ { 2 } + \lambda ) ^ { 2 } } ] + \sigma ^ { 2 } } \\ & { \quad \quad = \underset { z \sim \mathrm { U n i f } ( 1 ) \frac { \mathbb { E } } { \delta } } { \underbrace { \mathrm { U } \mathrm { E } } } [ | | \mathrm { d i a g } ( \{ \frac { - \lambda } { \gamma _ { i } ^ { 2 } + \lambda } \} ) z | | _ { 2 } ^ { 2 } ] + \sigma ^ { 2 } \underset { \Sigma } { \mathbb { E } } [ \sum _ { i } \frac { \gamma _ { i } ^ { 2 } } { ( \gamma _ { i } ^ { 2 } + \lambda ) ^ { 2 } } ] + \sigma ^ { 2 } } \\ & { \quad \quad = \frac { | | \beta ^ { * } | | _ { 2 } ^ { 2 } } { d } \underset { \Sigma } { \mathbb { E } } [ \sum _ { i } \frac { \lambda ^ { 2 } } { ( \gamma _ { i } ^ { 2 } + \lambda ) ^ { 2 } } ] + \sigma ^ { 2 } \underset { \Sigma } { \mathbb { E } } [ \sum _ { i } \frac { \gamma _ { i } ^ { 2 } } { ( \gamma _ { i } ^ { 2 } + \lambda ) ^ { 2 } } ] + \sigma ^ { 2 } } \\ & { \quad \quad = \underset { \Sigma } { \mathbb { E } } [ \sum _ { i } \frac { | | \beta ^ { * } | | _ { 2 } ^ { 2 } \lambda ^ { 2 } / d + \sigma ^ { 2 } \gamma _ { i } ^ { 2 } } { ( \gamma _ { i } ^ { 2 } + \lambda ) ^ { 2 } } ] + \sigma ^ { 2 } } \end{array}
+$$
+
+In Line (10) follows because by symmetry, the distribution of $V$ is a uniformly random orthonormal matrix, and $\Sigma$ is independent of $V$ . Thus, $z : = V ^ { T } \beta ^ { * }$ is distributed as a uniformly random point on the unit sphere of radius $| | \beta ^ { * } | | _ { 2 }$ .
+
+Next we prove Lemma 2.
+
+Proof of Lemma 2. First, we determine the optimal ridge parameter. Using Lemma 1, we have
+
+$$
+\begin{array} { r l } & { \frac { \partial } { \partial \lambda } \overline { { R } } ( \hat { \beta } _ { n , \lambda } ) = \frac { \partial } { \partial \lambda } _ { ( \gamma _ { 1 } , \dots \gamma _ { d } ) \sim \Gamma } \left[ \displaystyle \sum _ { i } \frac { | | \beta | | _ { 2 } ^ { 2 } \lambda ^ { 2 } / d + \sigma ^ { 2 } \gamma _ { i } ^ { 2 } } { ( \gamma _ { i } ^ { 2 } + \lambda ) ^ { 2 } } \right] } \\ & { = 2 ( | | \beta ^ { * } | | _ { 2 } ^ { 2 } \lambda / d - \sigma ^ { 2 } ) _ { ( \underbrace { ( \gamma _ { 1 } , \dots \gamma _ { d } ) \sim \Gamma } _ { > 0 } } \left[ \displaystyle \sum _ { i } \frac { \gamma _ { i } ^ { 2 } } { ( \gamma _ { i } ^ { 2 } + \lambda ) ^ { 3 } } \right] } \end{array}
+$$
+
+Thus, $\begin{array} { r } { \frac { \partial } { \partial \lambda } \overline { { R } } ( \widehat { \beta } _ { n , \lambda } ) = 0 \implies \lambda = \frac { d \sigma ^ { 2 } } { | | \beta ^ { * } | | _ { 2 } ^ { 2 } } } \end{array}$ and we conclude that $\begin{array} { r } { \lambda _ { n } ^ { \mathrm { o p t } } = \frac { d \sigma ^ { 2 } } { | | \beta ^ { * } | | _ { 2 } ^ { 2 } } } \end{array}$
+
+For this optimal parameter, the test risk follows from Lemma 1 as
+
+$$
+\begin{array} { l } { \displaystyle \overline { { R } } ( \hat { \beta } _ { n } ^ { \mathrm { o p t } } ) = \overline { { R } } ( \hat { \beta } _ { n , \lambda _ { n } ^ { \mathrm { o p t } } } ) } \\ { \displaystyle = \frac { \mathbb { E } } { ( \gamma _ { 1 } , \dots \gamma _ { d } ) \sim \Gamma _ { n } } \left[ \sum _ { i = 1 } ^ { d } \frac { \sigma ^ { 2 } } { \gamma _ { i } ^ { 2 } + d \sigma ^ { 2 } / | | \beta ^ { * } | | _ { 2 } ^ { 2 } } \right] + \sigma ^ { 2 } } \end{array}
+$$
+
+Proof of Theorem 2. We follow a similar proof strategy as in Theorem 1: we invoke singular value interlacing $( \widetilde { \gamma } _ { i } \geq \gamma _ { i } )$ for the data matrix when adding a single sample. We then apply Lemma 1 to eargue that the test risk varies monotonically with the singular values.
+
+We have
+
+$$
+{ \overline { { R } } } ( \hat { \beta } _ { n , \lambda } ) = \operatorname * { \mathbb { E } } _ { ( \gamma _ { 1 } , \dots \gamma _ { d } ) \sim \Gamma } [ \sum _ { i } \underbrace { | | \beta ^ { * } | | _ { 2 } ^ { 2 } \lambda ^ { 2 } / d + \sigma ^ { 2 } \gamma _ { i } ^ { 2 } } _ { ( \gamma _ { i } ^ { 2 } + \lambda ) ^ { 2 } } ]
+$$
+
+and we compute how each term in the sum varies with $\gamma _ { i }$ :
+
+$$
+\begin{array} { l } { \displaystyle { \frac { \partial } { \partial \gamma _ { i } } \sum _ { i } S ( \gamma _ { i } ) = \frac { \partial } { \partial \gamma _ { i } } S ( \gamma _ { i } ) } } \\ { \displaystyle { \quad = ( \frac { - 2 \gamma _ { i } } { d } ) \frac { 2 | | \beta ^ { * } | | _ { 2 } ^ { 2 } \lambda ^ { 2 } + d \sigma ^ { 2 } ( \gamma _ { i } ^ { 2 } - \lambda ) } { ( \gamma _ { i } ^ { 2 } + \lambda ) ^ { 3 } } } } \end{array}
+$$
+
+Thus we have
+
+$$
+\lambda \geq \frac { d \sigma ^ { 2 } } { 2 \vert \vert \beta ^ { * } \vert \vert ^ { 2 } } \implies \frac { \partial } { \partial \gamma _ { i } } S ( \gamma _ { i } ) \leq 0
+$$
+
+By the coupling argument in Theorem 1, this implies that the test risk is monotonic:
+
+$$
+\begin{array} { r l } & { \overline { { R } } ( \hat { \beta } _ { n + 1 , \lambda } ) - \overline { { R } } ( \hat { \beta } _ { n , \lambda } ) } \\ & { \quad = \underset { ( \widetilde { \gamma } _ { 1 } , \dots , \widetilde { \gamma } _ { d } ) \sim \Gamma _ { n + 1 } } { \mathbb { E } } \left[ \displaystyle \sum _ { i = 1 } ^ { d } S ( \widetilde { \gamma } _ { i } ) \right] - \underset { ( \gamma _ { 1 } , \dots , \gamma _ { d } ) \sim \Gamma _ { n } } { \mathbb { E } } \left[ \displaystyle \sum _ { i = 1 } ^ { d } S ( \gamma _ { i } ) \right] } \\ & { \quad = \underset { ( \{ \widetilde { \gamma } _ { i } \} , \{ \gamma _ { i } \} ) \sim \Pi } { \mathbb { E } } \left[ \displaystyle \sum _ { i = 1 } ^ { d } S ( \widetilde { \gamma } _ { i } ) - S ( \gamma _ { i } ) \right] } \\ & { \quad \le 0 } \end{array}
+$$
+
+where $\Pi$ is the coupling. Line (17) follows from Equation (15), and the fact that the coupling obeys $\widetilde { \gamma } _ { i } \geq \gamma _ { i }$ . □
+
+# A.2 PROJECTION MODEL PROOFS
+
+Lemma 3. For all $\theta \in \mathbb { R } ^ { p }$ , $d , n \in \mathbb { N } ,$ , and $\lambda > 0 ;$ , let $\ b { X } \in \mathbb { R } ^ { n \times p }$ be a matrix with i.i.d. $\mathcal { N } ( 0 , 1 )$ entries. Let $P \in \mathbb { R } ^ { d \times p }$ be a random orthonormal matrix. Define $\widetilde { X } : = X P ^ { T }$ . Let $\left( \gamma _ { 1 } , \dots , \gamma _ { m } \right)$ be the singular values of the data matrix $\tilde { X } \in \mathbb { R } ^ { n \times d }$ , for $m : = \operatorname* { m a x } ( n , d )$ (with $\gamma _ { i } ~ = ~ 0$ for $i > \operatorname* { m i n } ( n , d ) )$ . Let $\Gamma _ { d }$ be the distribution of singular values $\left( \gamma _ { 1 } , \ldots , \gamma _ { m } \right)$ .
+
+Then, the optimal ridge parameter is constant for all $d$ : $\begin{array} { r } { \lambda _ { d } ^ { \mathrm { o p t } } = \frac { p ^ { 2 } \widetilde \sigma ^ { 2 } } { d | | \theta | | _ { 2 } ^ { 2 } } } \end{array}$ . where we define $\widetilde { \sigma } ^ { 2 } : = \ :$ $\begin{array} { r } { \sigma ^ { 2 } + \frac { p - d } { p } | | \boldsymbol { \theta } | | _ { 2 } ^ { 2 } } \end{array}$ . Moreover, the optimal expected test risk can be written as
+
+$$
+\overline { { R } } ( \hat { \beta } _ { d } ^ { \mathrm { o p t } } ) = \widetilde { \sigma } ^ { 2 } + \underset { ( \gamma _ { 1 } , \ldots , \gamma _ { m } ) \sim \Gamma _ { d } } { \mathbb { E } } \left[ \sum _ { i = 1 } ^ { p } \frac { \widetilde { \sigma } ^ { 2 } } { \gamma _ { i } ^ { 2 } + \frac { \widetilde { \sigma } ^ { 2 } p ^ { 2 } } { d | | \theta | | _ { 2 } ^ { 2 } } } \right]
+$$
+
+Proof. This proof follows exactly analogously as the proof of Lemma 2 from Lemma 1, in Section A.1. □
+
+Lemma 4. For all $\theta \in \mathbb { R } ^ { p }$ , $d , n \in \mathbb { N } ,$ , and $\lambda > 0$ , let $\ b { X } \in \mathbb { R } ^ { n \times p }$ be a matrix with i.i.d. $\mathcal { N } ( 0 , 1 )$   
+entries. Let $P \in \mathbb { R } ^ { d \times p }$ be a random orthonormal matrix. Define $\widetilde X : = X P ^ { T }$ and $\beta ^ { * } : = P \theta$ .
+
+Let $\left( \gamma _ { 1 } , \dots , \gamma _ { m } \right)$ be the singular values of the data matrix $\tilde { X } \in \mathbb { R } ^ { n \times d }$ , for $m : = \operatorname* { m a x } ( n , d )$ (with $\gamma _ { i } = 0$ for $i > \operatorname* { m i n } ( n , d ) )$ . Let $\Gamma _ { d }$ be the distribution of singular values $\left( \gamma _ { 1 } , \dots , \gamma _ { m } \right)$ .
+
+Then, the expected test risk is
+
+$$
+\begin{array} { r l } & { \overline { { R } } ( \hat { \beta } _ { d , \lambda } ) : = \underset { P } { \mathbb { E } } \underset { \widetilde { X } , \widetilde { y } \sim \mathcal { D } ^ { n } } { \mathbb { E } } [ R _ { P } ( \hat { \beta } _ { d , \lambda } ( \widetilde { X } , \overrightarrow { y } ) ] } \\ & { = \sigma ^ { 2 } + ( 1 - \frac { d } { p } ) | | \theta | | _ { 2 } ^ { 2 } } \\ & { + \underset { ( \gamma _ { 1 } , \dotsc , \gamma _ { m } ) \sim \Gamma _ { d } } { \mathbb { E } } \left[ \underset { i = 1 } { \overset { p } { \sum } } \frac { ( \sigma ^ { 2 } + \frac { p - d } { p } | | \theta | | _ { 2 } ^ { 2 } ) \gamma _ { i } ^ { 2 } + \frac { d } { p ^ { 2 } } | | \theta | | _ { 2 } ^ { 2 } \lambda ^ { 2 } } { ( \gamma _ { i } ^ { 2 } + \lambda ) ^ { 2 } } \right] } \end{array}
+$$
+
+Proof of Lemma 4. We first define the parameter that minimizes the population risk. It follows directly that:
+
+$$
+\beta _ { P } ^ { * } : = \underset { \beta \in \mathbb { R } ^ { d } } { \mathrm { a r g m i n } } R _ { P } ( \beta ) = P \theta
+$$
+
+First, we can expand the risk as
+
+$$
+\begin{array} { r l } & { R ( \hat { \beta } ) = \underset { ( \bar { x } , y ) \sim \mathcal { D } } { \mathbb { E } } [ ( \langle P x , \hat { \beta } \rangle - y ) ^ { 2 } ] } \\ & { \quad = \underset { ( \bar { x } , y ) \sim \mathcal { D } , \eta \sim \mathcal { N } ( 0 , \sigma ^ { 2 } ) } { \mathbb { E } } [ ( \langle x , P ^ { T } \hat { \beta } - \theta \rangle + \eta ) ^ { 2 } ] } \\ & { \quad = \sigma ^ { 2 } + \vert \vert \theta - P ^ { T } \hat { \beta } \vert \vert _ { 2 } ^ { 2 } } \\ & { \quad = \sigma ^ { 2 } + \vert \vert \theta - P ^ { T } \beta ^ { * } \vert \vert _ { 2 } ^ { 2 } + \vert \vert P ^ { T } \beta ^ { * } - P ^ { T } \hat { \beta } \vert \vert _ { 2 } ^ { 2 } } \\ & { \quad + 2 \langle ( \theta - P ^ { T } \beta ^ { * } ) , P ^ { T } \beta ^ { * } - P ^ { T } \hat { \beta } \rangle } \\ & { \quad = \sigma ^ { 2 } + \vert \vert \theta - P ^ { T } \beta ^ { * } \vert \vert _ { 2 } ^ { 2 } + \vert \vert P ^ { T } \beta ^ { * } - P ^ { T } \hat { \beta } \vert \vert _ { 2 } ^ { 2 } } \\ & { \quad = \sigma ^ { 2 } + \vert \vert \vert \theta - P ^ { T } P \theta \vert \vert _ { 2 } ^ { 2 } + \vert \vert \beta ^ { * } - \hat { \beta } \vert \vert _ { 2 } ^ { 2 } } \end{array}
+$$
+
+The cross terms in Line (22) vanish because the first-order optimality condition for $\beta ^ { * }$ implies that $\beta ^ { * }$ satisfies $P ( \theta ^ { * } - P ^ { T } \beta ^ { * } ) = 0$ . We now simplify each of the two remaining terms.
+
+First, we have that:
+
+$$
+\underset { P } { \mathbb { E } } | | \theta - P ^ { T } P \theta | | _ { 2 } ^ { 2 } = ( 1 - \frac { d } { p } ) | | \theta | | _ { 2 } ^ { 2 }
+$$
+
+since $P ^ { T } P$ is an orthogonal projection onto a random $d$ -dimensional subspace.
+
+Now, recall we have ${ \vec { y } } = X \theta + \eta$ where $\eta \sim \mathcal { N } ( 0 , \sigma ^ { 2 } I _ { n } )$ . Expand this as:
+
+$$
+\begin{array} { l } { { \vec { y } = X \theta + \eta } } \\ { { \ = X P ^ { T } P \theta + X ( 1 - P ^ { T } P ) \theta + \eta } } \\ { { \ = \widetilde { X } \beta ^ { * } + \varepsilon + \eta } } \end{array}
+$$
+
+where $\varepsilon : = X ( 1 - P ^ { T } P ) \theta$ . Note that conditioned on $P$ , the three terms $\widetilde { X } , \varepsilon$ and $\eta$ are conditionally independent, since $P ^ { T } P$ and $( I - P ^ { T } P )$ project $X$ onto orthogonal subspaces. And further, $\varepsilon \sim$ $\mathcal { N } ( \dot { 0 _ { \cdot } } | | ( 1 - P ^ { T } P ) \theta | | ^ { 2 } I _ { n } )$ .
+
+$$
+\begin{array} { r l } & { \frac { \mathbb { E } } { P } \underset { \tilde { X } , y } { \mathbb { E } } | | \hat { \boldsymbol { \beta } } - \boldsymbol { \beta } ^ { * } | | _ { 2 } ^ { 2 } } \\ & { = \underset { P } { \mathbb { E } } \underset { \tilde { X } , y } { \mathbb { E } } | | ( \widetilde { X } ^ { T } \widetilde { X } + \lambda I ) ^ { - 1 } \widetilde { X } ^ { T } \boldsymbol { y } - \boldsymbol { \beta } ^ { * } | | _ { 2 } ^ { 2 } } \\ & { = \underset { P } { \mathbb { E } } \underset { \tilde { X } , y , \varepsilon , \eta } { \mathbb { E } } | | ( \widetilde { X } ^ { T } \widetilde { X } + \lambda I ) ^ { - 1 } \widetilde { X } ^ { T } ( \widetilde { X } \boldsymbol { \beta } ^ { * } + \varepsilon + \eta ) - \boldsymbol { \beta } ^ { * } | | _ { 2 } ^ { 2 } } \\ & { = \underset { P } { \overset { \mathbb { E } } { \mathbb { X } } } \underset { y , \varepsilon , \eta } { \mathbb { E } } | | ( \widetilde { X } ^ { T } \widetilde { X } + \lambda I ) ^ { - 1 } \widetilde { X } ^ { T } \widetilde { X } \boldsymbol { \beta } ^ { * } - \boldsymbol { \beta } ^ { * } | | _ { 2 } ^ { 2 } } \\ & { = \underset { P } { \overset { \mathbb { E } } { \mathbb { X } } } \underset { y , \varepsilon , \eta } { \mathbb { E } } | | ( \widetilde { X } ^ { T } \widetilde { X } + \lambda I ) ^ { - 1 } \widetilde { X } ^ { T } \widetilde { X } \boldsymbol { \beta } ^ { * } - \boldsymbol { \beta } ^ { * } | | _ { 2 } ^ { 2 } } \\ & { + \| ( \widetilde { X } ^ { T } \widetilde { X } + \lambda I ) ^ { - 1 } \widetilde { X } ^ { T } \varepsilon | | _ { 2 } ^ { 2 } + | | ( \widetilde { X } ^ { T } \widetilde { X } + \lambda I ) ^ { - 1 } \widetilde { X } ^ { T } \eta | | _ { 2 } ^ { 2 } ] } \end{array}
+$$
+
+Now, since $\widetilde { X }$ is conditionally independent of $\varepsilon$ conditioned on $P$ ,
+
+$$
+\begin{array} { r l } & { \frac { \mathbb { E } } { F } \underset { \tilde { X } , y , \varepsilon | P } { \mathbb { E } } \ \lVert ( \tilde { X } ^ { T } \tilde { X } + \lambda I ) ^ { - 1 } \tilde { X } ^ { T } \varepsilon \rVert _ { 2 } ^ { 2 } } \\ & { = \mathbb { E } \ \underset { \tilde { Y } } { \mathbb { E } } \ [ \lVert ( \tilde { X } ^ { T } \tilde { X } + \lambda I ) ^ { - 1 } \tilde { X } ^ { T } \rVert _ { F } ^ { 2 } \ \underset { \varepsilon | P } { \mathbb { E } } [ \lVert \varepsilon \rVert _ { 2 } ^ { 2 } ] } \\ & { = \frac { \mathbb { E } } { \frac { \mathbb { E } } { \tilde { X } } } \lVert ( \tilde { X } ^ { T } \tilde { X } + \lambda I ) ^ { - 1 } \tilde { X } ^ { T } \rVert _ { F } ^ { 2 } \ \underset { \varepsilon , \varepsilon } { \mathbb { E } } [ \lVert \varepsilon \rVert _ { 2 } ^ { 2 } ] } \\ & { = \underset { \tilde { X } } { \mathbb { E } } [ \lVert ( \tilde { X } ^ { T } \tilde { X } + \lambda I ) ^ { - 1 } \tilde { X } ^ { T } \rVert _ { F } ^ { 2 } ] \cdot \underset { { P } , X } { \mathbb { E } } [ \lVert X ( 1 - P ^ { T } P ) \theta \rVert _ { 2 } ^ { 2 } ] } \\ & { = \underset { \tilde { X } } { \mathbb { E } } [ \lVert ( \tilde { X } ^ { T } \tilde { X } + \lambda I ) ^ { - 1 } \tilde { X } ^ { T } \rVert _ { F } ^ { 2 } ] ( \frac { p - d } { p } | \theta \rVert _ { 2 } ^ { 2 } ) } \end{array}
+$$
+
+where Line (37) holds because the marginal distribution of $\widetilde { X }$ does not depend on $P$ .
+
+Similarly,
+
+$$
+\begin{array} { r l } & { \frac { \mathbb { E } } { P } \underset { \widetilde { X } , y , \eta | P } { \mathbb { E } } | | ( \widetilde { X } ^ { T } \widetilde { X } + \lambda I ) ^ { - 1 } \widetilde { X } ^ { T } \eta | | _ { 2 } ^ { 2 } } \\ & { = \sigma ^ { 2 } \underset { \widetilde { X } } { \mathbb { E } } [ | | ( \widetilde { X } ^ { T } \widetilde { X } + \lambda I ) ^ { - 1 } \widetilde { X } ^ { T } | | _ { F } ^ { 2 } ] } \end{array}
+$$
+
+Now let $\tilde { \cal X } \ = \ { \cal U } \Sigma { \cal V } ^ { T }$ be the full singular value decomposition of $\widetilde { X }$ , with $U \ \in \ \mathbb { R } ^ { n \times n } , \Sigma \ \in$ $\mathbb { R } ^ { n \times d } , V \in \mathbb { R } ^ { d \times d }$ . Let $( \gamma _ { 1 } , \dots \gamma _ { m } )$ denote the singular values, where $m = \operatorname* { m a x } ( n , d )$ and defining $\gamma _ { i } = 0$ for $i > \operatorname* { m i n } ( n , d )$ .
+
+Observe that by symmetry, $\widetilde { X } = X P ^ { T }$ and $P$ are independent, because the joint distribution $( \widetilde { X } , P )$ is equivalent to the distribution $( \widetilde { X } Q , P Q )$ for a random orthonormal $Q \in \mathbb { R } ^ { p \times p }$ . Thus $\widetilde { X }$ and
+
+$\beta ^ { * } = P \theta$ are also independent, and we have:
+
+$$
+\begin{array} { r l } & { \mathbb { E } \mathbb { E } \| \{ \hat { X } ^ { \prime } \} \hat { X } + \lambda \mathcal { I } \} ^ { - 1 } \hat { X } ^ { \prime } \hat { X } ^ { \prime } - \delta ^ { * } \} \| _ { 2 } ^ { 2 } } \\ & { = \frac { 1 } { b } \mathbb { E } \hat { X } ^ { \prime } \| \mathrm { H i s s } ( \hat { \mathcal { Z } } _ { \frac { e } { 2 } } ^ { - 1 } \lambda ) b ^ { \mathcal { N } / 2 } \| _ { 2 } ^ { 2 } } \\ & { = \mathbb { E } \mathbb { E } \mathbb { E } \mathbb { E } \mathbb { E } \bigg [ \| \mathrm { H i s s } ( \hat { \mathcal { Z } } _ { \frac { e } { 2 } } ^ { - 1 } \lambda ) \| ^ { \mathcal { N } / 2 } b ^ { \mathcal { N } / 2 } \| _ { 2 } ^ { 2 } } \\ & { - \mathbb { E } \mathbb { E } \mathbb { E } \mathbb { E } \bigg [ \| \mathrm { H i s s } ( \hat { \mathcal { Z } } _ { \frac { e } { 2 } } ^ { - 1 } \lambda ) \| ^ { \mathcal { N } / 2 } \bigg ] \| _ { 2 } ^ { 2 } } \\ & { = \mathbb { E } _ { \hat { X } ^ { \prime } \sim \mathrm { N } \times \mathrm { W } ( \hat { \mathcal { Z } } ) } \mathbb { E } \| \mathrm { H i s s } ( \hat { \mathcal { Z } } _ { \frac { e } { 2 } } ^ { - 1 } \lambda ) \| ^ { \mathcal { N } / 2 } } \\ & { = \mathbb { E } \mathbb { E } \| \hat { X } ^ { \prime } \| ^ { 2 } \mathbb { E } \| \mathrm { H } ^ { 2 } \mathbb { E } \bigg [ \mathrm { H } _ { \frac { e } { 2 } } ^ { - 1 } \lambda \Big ] ^ { \| \mathcal { N } / 2 } } \\ & { = \frac { 1 } { b } \mathbb { E } \hat { X } ^ { \prime } \| ^ { 2 } \mathbb { E } \bigg [ \mathcal { P } _ { \frac { e } { 2 } } ^ { - 1 } \frac { \lambda ^ { 2 } } { b } \Big ] ^ { 2 } } \\ &  = \frac { 1 } { b } \mathbb { E } \frac { \hat { X } ^ { \prime } } { \| \mathcal { P } \| \mathcal { P } \| ^ { 2 } \| _ { 2 } ^ { 2 } } \mathbb { E } \bigg [ \int _ { 0 } ^ { \mathcal { N } / 2 } \frac { \lambda ^ { 2 } }  \| \mathcal { P } \| _ { 2 } ^ { 2 } \end{array}
+$$
+
+Finally, continuing from Line (33), we can use Lines (38), (40), and (47) to write:
+
+$$
+\begin{array} { r l } & { \frac { \mathbb { E } } { \rho } \frac { \| \tilde { \rho } - \tilde { \rho } ^ { - 1 } \| ^ { 2 } } { \mathcal { F } _ { x , y } } } \\ & { = ( \sigma ^ { 2 } + \frac { \rho - \tilde { \rho } - d } { p } \| \theta \| _ { 2 } ^ { 2 } \frac { \mathbb { E } \| ( \tilde { X } ^ { \mathcal { T } } \tilde { X } + \lambda I ) ^ { - 1 } \tilde { X } ^ { \mathcal { T } } \| _ { \mathcal { F } _ { x } ^ { 1 } } ^ { 2 } } { \mathcal { F } _ { x } ^ { 1 } } } \\ & { + \frac { d } { \gamma } \| \theta \| _ { 2 } ^ { 2 } \frac { \mathbb { E } } { \varepsilon } \| \sum _ { i } ^ { 2 } \frac { \lambda ^ { 2 } } { ( \sqrt { d } ^ { 2 } + \lambda I ) ^ { 2 } } | } \\ & { = ( \sigma ^ { 2 } + \frac { p - d } { p } \| \theta \| _ { 2 } ^ { 2 } ) \frac { \mathbb { E } } { \Sigma } \frac { \gamma ^ { 2 } } { \varepsilon } \langle \frac { \eta _ { i } ^ { 2 } } { ( \sqrt { d } ^ { 2 } + \lambda I ) ^ { 2 } } | } \\ & { + \frac { d } { \gamma ^ { 2 } } \| \theta \| _ { 2 } ^ { 2 } \frac { \mathbb { E } } { \Sigma } \frac { \lambda ^ { 2 } } { \langle \frac { \eta _ { i } ^ { 2 } + \lambda I \rangle ^ { 2 } } { \varepsilon } } | } \\ &  = \frac { \mathbb { E } } { \nu } \langle \frac { ( \sigma ^ { 2 } + \frac { p - d - \tilde { \rho } - \tilde { \rho } - \tilde { \rho } - \tilde { \rho } } { i } ) | \theta \rangle _ { 2 } ^ { 2 } \sqrt { \frac { \lambda ^ { 2 } } { \varepsilon } + \frac { d } { \gamma ^ { 2 } } | \theta | _ { 2 } ^ { 2 } \lambda ^ { 2 } } } \end{array}
+$$
+
+Now, we can continue from Line (24), and apply lines (25), to conclude:
+
+$$
+\begin{array} { r l } & { \mathbb { E } \big [ R ( \hat { \beta } ) \big ] = \sigma ^ { 2 } + \mathbb { E } [ | | \theta - P ^ { T } P \theta | | _ { 2 } ^ { 2 } ] + \mathbb { E } [ | | \beta ^ { * } - \hat { \beta } | | _ { 2 } ^ { 2 } ] } \\ & { \qquad = \sigma ^ { 2 } + ( 1 - \frac { d } { p } ) | | \theta | | _ { 2 } ^ { 2 } } \\ & { \qquad + \mathbb { E } \left[ \displaystyle \sum _ { i = 1 } ^ { p } \frac { \big ( \sigma ^ { 2 } + \frac { p - d } { p } | | \theta | | _ { 2 } ^ { 2 } \big ) \gamma _ { i } ^ { 2 } + \frac { d } { p ^ { 2 } } | | \theta | | _ { 2 } ^ { 2 } \lambda ^ { 2 } } { \big ( \gamma _ { i } ^ { 2 } + \lambda \big ) ^ { 2 } } \right] } \end{array}
+$$
+
+Proof of Theorem 3. This follows analogously to the proof of Theorem 1, Let $\widetilde { X } _ { d }$ and $\widetilde { X } _ { d + 1 }$ be the observed data matrices for $d$ and $d + 1$ e emodel size. As in Theorem 1, there exists a coupling $\Pi$ between the distributions $\Gamma _ { d }$ and $\Gamma _ { d + 1 }$ of the singular values of $\widetilde { X } _ { d }$ and $\widetilde { X } _ { d + 1 }$ such that these singular values are interlaced.
+
+Thus by Lemma 3,
+
+$$
+\begin{array} { l } { \overline { { R } } ( \hat { \beta } _ { d } ^ { \mathrm { o p t } } ) = \widetilde { \sigma } ^ { 2 } + \underset { ( \gamma _ { 1 } , \ldots , \gamma _ { m } ) \sim \Gamma _ { d + 1 } } { \mathbb { E } } \left[ \overset { p } { \underset { i = 1 } { \sum } } \frac { \widetilde { \sigma } ^ { 2 } } { \gamma _ { i } ^ { 2 } + \frac { \widetilde { \sigma } ^ { 2 } p ^ { 2 } } { d | \boldsymbol { \theta } | | _ { 2 } ^ { 2 } } } \right] } \\ { \geq \widetilde { \sigma } ^ { 2 } + \underset { ( \widetilde { \gamma } _ { 1 } , \ldots , \widetilde { \gamma } _ { m } ) \sim \Gamma _ { d + 1 } } { \mathbb { E } } \left[ \overset { p } { \underset { i = 1 } { \sum } } \frac { \widetilde { \sigma } ^ { 2 } } { \widetilde { \gamma } _ { i } ^ { 2 } + \frac { \widetilde { \sigma } ^ { 2 } p ^ { 2 } } { d | \boldsymbol { \theta } | | _ { 2 } ^ { 2 } } } \right] } \\ { = \overline { { R } } ( \hat { \beta } _ { d + 1 } ^ { \mathrm { o p t } } ) } \end{array}
+$$
+
+# A.3 NONISOTROPIC REDUCTION
+
+Here we observe that results on isotropic regression in Section 2 also imply that ridge regression can be made sample-monotonic even for non-isotropic covariates, if an appropriate regularzier is applied. Specifically, the regularizer depends on the covariance on the inputs. This follows from a general equivalence between the non-isotropic and isotropic problems.
+
+Lemma 5. For all $n \in \mathbb { N } , d \in \mathbb { N } , \lambda \in \mathbb { R } , \sigma \in \mathbb { R } ,$ covariance $\Sigma \in \mathbb { R } ^ { d \times d }$ , PSD matrix $M \in \mathbb { R } ^ { d \times d }$ and ground-truth $\beta ^ { * } \in \mathbb { R } ^ { d }$ , the following holds.
+
+Consider the following two problems:
+
+1. Regularized regression on isotropic covariates, and an $M$ -regularizer. That is, suppose $n$ samples $( x , y )$ are drawn with covariates $x \sim \mathcal { N } ( 0 , I _ { d } )$ and response $y = \langle \beta ^ { * } , x \rangle +$ $\mathcal { N } ( 0 , \bar { \sigma } ^ { 2 } )$ . Let $\ b { \hat { X } } \in \mathbb { R } ^ { n \times d }$ be the matrix of covariates, and $\vec { y }$ the vector of responses. Consider
+
+$$
+{ \hat { \beta } } _ { \lambda } : = \operatorname * { a r g m i n } _ { \beta } | | X \beta - { \vec { y } } | | _ { 2 } ^ { 2 } + \lambda | | \beta | | _ { M } ^ { 2 }
+$$
+
+Let $\overline { { R } } = \mathbb { E } _ { X , y } [ | | \hat { \beta } - \beta ^ { * } | | _ { 2 } ^ { 2 } ] + \sigma ^ { 2 }$ be the expected test risk of the above estimator.
+
+2. Regularized regression with covariance $\Sigma$ , and an $( \Sigma ^ { 1 / 2 } M \Sigma ^ { 1 / 2 } )$ -regularizer. That is, suppose $n$ samples $( \widetilde { x } , y )$ are drawn with covariates $\widetilde { x } \sim \mathcal { N } ( 0 , \Sigma )$ and response $y =$ $\langle z ^ { \ast } , \widetilde { x } \rangle + \mathcal { N } ( 0 , \widetilde { \sigma } ^ { 2 } )$ e, for
+
+$$
+z ^ { * } = \Sigma ^ { - 1 / 2 } \beta ^ { * }
+$$
+
+Let $\widetilde { X } \in \mathbb { R } ^ { n \times d }$ be the matrix of covariates, and $\vec { y }$ the vector of responses. Consider
+
+$$
+\hat { z } _ { \lambda } : = \operatorname * { a r g m i n } _ { z } | | \widetilde { X } z - \vec { y } | | _ { 2 } ^ { 2 } + \lambda | | z | | _ { \Sigma ^ { 1 / 2 } M \Sigma ^ { 1 / 2 } } ^ { 2 }
+$$
+
+Let $\widetilde { R } = \mathbb { E } _ { \widetilde { X } , y } [ | | \hat { z } - z ^ { * } | | _ { \Sigma } ^ { 2 } ] + \sigma ^ { 2 }$ be the expected test risk of the above estimator.
+
+Then, the expected test risks of the above two problems are identical:
+
+$$
+\overline { { R } } = \widetilde { R }
+$$
+
+Proof of Lemma 5. The distribution of $\widetilde { X }$ in the Problem 2 is equivalent to $X \Sigma ^ { 1 / 2 }$ , where $X$ is as in Problem 1. Thus, the two settings are equivalent by the change-of-variable $\beta = \Sigma ^ { 1 / 2 } z$ . Specifically,
+
+$$
+\begin{array} { r l } & { \hat { z } _ { \lambda } : = \underset { z } { \mathrm { a r g m i n } } \left| \lvert \tilde { X } z - \bar { y } \rvert \right| _ { 2 } ^ { 2 } + \lambda \lvert | z \rvert | _ { \Sigma ^ { 1 / 2 } M \Sigma ^ { 1 / 2 } } ^ { 2 } } \\ & { = \underset { z } { \mathrm { a r g m i n } } \left| \lvert X \Sigma ^ { 1 / 2 } z - \bar { y } \rvert \right| _ { 2 } ^ { 2 } + \lambda z ^ { T } \Sigma ^ { 1 / 2 } M \Sigma ^ { 1 / 2 } z } \\ & { = \underset { z } { \mathrm { a r g m i n } } \left| \lvert X \Sigma ^ { 1 / 2 } z - \bar { y } \rvert \right| _ { 2 } ^ { 2 } + \lambda z ^ { T } \Sigma ^ { 1 / 2 } M \Sigma ^ { 1 / 2 } z } \\ & { = \Sigma ^ { - 1 / 2 } \underset { \beta = \Sigma ^ { 1 / 2 } z } { \mathrm { a r g m i n } } \left| \lvert X \beta - \bar { y } \rvert \right| _ { 2 } ^ { 2 } + \lambda \beta ^ { T } M \beta } \end{array}
+$$
+
+Further, the response $\langle z ^ { * } , { \widetilde { x } } \rangle = \langle \beta , x \rangle$ , and the test risk transforms identically:
+
+$$
+\begin{array} { r } { \widetilde { R } = \underset { \widetilde { X } , y } { \mathbb { E } } \left[ \vert \vert \hat { z } - z ^ { * } \vert \vert _ { \Sigma } ^ { 2 } \right] + \sigma ^ { 2 } } \\ { = \underset { X , y } { \mathbb { E } } \left[ \vert \vert \hat { \beta } - \beta ^ { * } \vert \vert _ { 2 } ^ { 2 } \right] + \sigma ^ { 2 } } \\ { = \overline { { R } } } \end{array}
+$$
+
+This implies that if the covariance $\Sigma$ is known, then ridge regression with a $\Sigma ^ { - 1 }$ regularizer is sample-monotonic.
+
+Theorem 5. For all $n \in \mathbb { N } , d \in \mathbb { N } , \sigma \in \mathbb { R } ,$ , covariance $\Sigma \in \mathbb { R } ^ { d \times d }$ , and ground-truths $\beta ^ { * } \in \mathbb { R } ^ { d }$ , th e following holds.
+
+Suppose $n$ samples $( x , y )$ are drawn with covariates $x \sim \mathcal { N } ( 0 , \Sigma )$ and response $y = \langle \beta ^ { * } , x \rangle +$ ${ \mathcal { N } } ( 0 , \sigma ^ { 2 } )$ . Let $\ b { X } \in \mathbb { R } ^ { n \times d }$ be the matrix of covariates, and $\vec { y }$ the vector of responses. For $\lambda > 0$ , consider the ridge regression estimator with $\Sigma ^ { - 1 }$ -regularizer:
+
+$$
+\hat { \beta } _ { n , \lambda } : = \operatorname * { a r g m i n } _ { \boldsymbol { \beta } } \| X \boldsymbol { \beta } - \vec { y } \| _ { 2 } ^ { 2 } + \lambda \| \boldsymbol { \beta } \| _ { \Sigma ^ { - 1 } } ^ { 2 }
+$$
+
+Let $\overline { { R } } ( \hat { \beta } _ { n , \lambda } ) : = \mathbb { E } _ { \hat { \beta } } \| \hat { \beta } - \beta ^ { * } \| _ { \Sigma } + \sigma ^ { 2 }$ be the expected test risk of the above estimator. Let $\lambda _ { n } ^ { \mathrm { o p t } }$ be the optimal ridge parameter (that achieves the minimum expected risk) given $n$ samples:
+
+$$
+\lambda _ { n } ^ { \mathrm { o p t } } : = \operatorname * { a r g m i n } _ { \lambda } \overline { { R } } ( \hat { \beta } _ { n , \lambda } ) )
+$$
+
+And let $\hat { \beta } _ { n } ^ { \mathrm { { o p t } } }$ be the estimator that corresponds to the $\lambda _ { n } ^ { \mathrm { o p t } }$ . Then, the expected test risk of optimallyregularized linear regression is monotonic in samples:
+
+$$
+\overline { { R } } ( \hat { \beta } _ { n + 1 } ^ { \mathrm { o p t } } ) \leq \overline { { R } } ( \hat { \beta } _ { n } ^ { \mathrm { o p t } } )
+$$
+
+Proof. This follows directly by applying the reduction in Lemma 5 for $M = I _ { d }$ to reduce to the isotropic case, and then applying the monotonicity of isotropic regression from Theorem 1. □
+
+# A.4 ADDITIONAL PLOTS
+
+We apply random features to Fashion-MNIST Xiao et al. (2017), an image classification problem with 10 classes. Input images $x \in \mathbb { R } ^ { d }$ are normalized and flattened to $[ - \bar { 1 } , 1 ] ^ { d }$ for $d = 7 8 4$ . Class labels are encoded as one-hot vectors $y \in \{ \vec { e _ { 1 } } , \dots e _ { 1 0 } ^ {  } \} \subset \mathbb { R } ^ { 1 0 }$ .
+
+# B TOWARDS MONOTONICITY WITH GENERAL COVARIATES
+
+Here we investigate whether monotonicity provably holds in more general models, inspired by the experimental results. As a first step, we consider Gaussian (but not isotropic) covariances and homeostatic noise. That is, we consider ridge regression in the setting of Section 2, but with $x \sim \mathcal { N } ( 0 , \Sigma )$ , and $y \sim \langle x , \beta ^ { * } \rangle + N ( 0 , \sigma ^ { 2 } )$ . In this section, we observe that ridge regression can be made sample-monotonic with a modified regularizer. We also conjecture that ridge regression is sample-monotonic without modifying the regularizer, and we outline a potential proof strategy along with numerical evidence.
+
+# B.1 ADAPTIVE REGULARIZATION
+
+The results on isotropic regression in Section 2 imply that ridge regression can be made samplemonotonic even for non-isotropic covariates, if an appropriate regularizer is applied. Specifically, the appropriate regularizer depends on the covariance of the inputs. For $\boldsymbol { x } \sim \mathcal { N } ( 0 , \Sigma )$ , the following estimator is sample-monotonic for optimally-tuned $\lambda$ : $\begin{array} { r } { \hat { \beta } _ { n , \lambda } : = \operatorname * { a r g m i n } _ { \boldsymbol { \beta } } \| X \boldsymbol { \beta } - \vec { y } \| _ { 2 } ^ { 2 } + \lambda \| \boldsymbol { \beta } \| _ { \Sigma ^ { - 1 } } ^ { 2 } } \end{array}$ . This follows directly from Theorem 1 by applying a change-of-variable; full details of this equivalence are in Section A.3. Note that if the population covariance $\Sigma$ is not known, it can potentially be estimated from unlabeled data.
+
+![](images/6d706f197addd390fb2defce66fd1cd5af413a8242fbc58ab89186d03af9186f.jpg)  
+Test Error for Regularized Random Features
+
+(a) Test Classification Error vs. Number of Training Samples.
+
+![](images/ce3f799d5bd5415e26ca617c4e47e654875a8b6c2557b5bf3b3c7007b6a14a4f.jpg)  
+Test Errorfor Regularized Random Features   
+(b) Test Classification Error vs. Model Size (Number of Random Features).
+
+Figure 4: Double-descent for Random ReLU Features. Test classification error as a function of model size and sample size for Random ReLU Features on Fashion-MNIST. Left: with $D = 5 0 0$ features. Right: with $n = 5 0 0$ samples. See Figures 7, 8 for the corresponding test Mean Squared Error. See Appendix D of Nakkiran et al. (2020) for the performance of these unregularized models plotted across Num. Samples $\times$ Model Size simultaneously.
+
+![](images/05df2a46592e323dc1205edaa5fa11d2355d7bba6df96212a7b9f700cd4a121b.jpg)  
+Figure 5: Train Error vs. Model Size for 5-layer CNNs on CIFAR-100, with $\ell _ { 2 }$ regularization (weight decay).
+
+# B.2 TOWARDS PROVING MONOTONICITY
+
+We conjecture that optimally-regularized ridge regression is sample-monotonic for non-isotropic covariates, even without modifying the regularizer (as suggested by the experiment in Figure 2). We derive a sufficient condition for monotonicity, which we have numerically verified in a variety of instances. Specifically, we conjecture the following.
+
+Conjecture 1. For all $d \in \mathbb { N }$ , and all PSD covariances $\Sigma \in \mathbb { R } ^ { d \times d }$ , consider the distribution on $( x , y )$ where $x \sim \mathcal { N } ( 0 , \Sigma )$ , and $y \sim \langle x , \beta ^ { * } \rangle + \mathcal { N } ( 0 , \sigma ^ { 2 } )$ . Then, we conjecture that the expected test risk of the ridge regression estimator: $\begin{array} { r } { \hat { \beta } _ { n , \lambda } : = \operatorname * { a r g m i n } _ { \boldsymbol { \beta } } \| X { \boldsymbol { \beta } } - \vec { y } \| _ { 2 } ^ { 2 } + \lambda \| { \boldsymbol { \beta } } \| _ { 2 } ^ { 2 } } \end{array}$ for optimally-tuned $\lambda \geq 0$ , is monotone non-increasing in number of samples $n$ . That is, for all $n \in \mathbb { N }$ ,
+
+$$
+\operatorname * { i n f } _ { \lambda \geq 0 } \overline { { { R } } } ( \hat { \beta } _ { n + 1 , \lambda } ) \leq \operatorname * { i n f } _ { \lambda \geq 0 } \overline { { { R } } } ( \hat { \beta } _ { n , \lambda } )
+$$
+
+where we define $\begin{array} { r } { \hat { \beta } _ { n , 0 } : = \operatorname* { l i m } _ { \lambda \to 0 + } \hat { \beta } _ { n , \lambda } = X ^ { \dagger } y } \end{array}$ .
+
+![](images/d4048491df9a41611fd80fea0aeac569328ee316a77886c71b4c9d4ba9170d40.jpg)  
+Figure 6: Train MSE vs. Num. Samples for Non-Isotropic Ridge Regression in $d = 3 0$ dimensions, in the setting of Figure 2. Plotting train MSE: $\begin{array} { r } { \frac { 1 } { n } | | X \hat { \boldsymbol { \beta } } - \vec { y } | | _ { 2 } ^ { 2 } } \end{array}$ .
+
+![](images/aa84a8ffbd2beefb1d745ab29a4ccc613c56c1dfa1eef8d51380a0483dcffecb.jpg)  
+Figure 7: Test Mean Squared Error vs. Num Train Samples for Random ReLU Features on FashionMNIST, with $D = 5 0 0$ features.
+
+In Appendix B.3 we present a technical conjecture in random matrix theory (Conjecture 2) which suffices to prove Conjecture 1. Proving this Conjecture 2 presents a number of technical challenges, but we have numerically verified it in a variety of cases. It can also be shown that Conjecture 2 is true when $Q = I$ , corresponding to isotropic covariates. We prove the reduction between Conjecture 2 and 1 in Appendix B.3.
+
+# B.3 MONOTONICITY CONJECTURE PROOFS
+
+In order to establish Conjecture 1, it is sufficient to prove the following technical conjecture.
+
+![](images/8eda0db61a5bd63cb07df9565b6981c5a80a7fd865e013f9ce281e419515418e.jpg)  
+Figure 8: Test Mean Squared Error vs. Num Features for Random ReLU Features on FashionMNIST, with $n = 5 0 0$ samples.
+
+Conjecture 2. For all $n \in \mathbb { N } , d \geq n , \lambda > 0$ , symmetric positive definite matrix $Q \in \mathbb { R } ^ { d \times d }$ , the following holds.
+
+Define
+
+$$
+G _ { \lambda } ^ { n } : = \lambda ^ { 2 } \underset { X } { \mathbb { E } } [ ( X ^ { T } X + \lambda Q ) ^ { - 2 } ]
+$$
+
+where $\ b { X } \in \mathbb { R } ^ { n \times d }$ is sampled with each entry i.i.d. $\mathcal { N } ( 0 , 1 )$ . Similarly, define
+
+$$
+H _ { \lambda } ^ { n } : = \mathbb { E } [ | | X ^ { T } X + \lambda Q ) ^ { - 1 } X ^ { T } | | _ { F } ^ { 2 } ]
+$$
+
+The expected test risk for $n$ samples can be expressed as:
+
+$$
+\overline { { { R } } } ( \hat { \beta } _ { n , \lambda } ) = ( \beta ^ { * } ) ^ { T } G _ { \lambda } ^ { n } \beta ^ { * } + \sigma ^ { 2 } H _ { \lambda } ^ { n } + \sigma ^ { 2 }
+$$
+
+Then, we conjecture that the following two conditions hold.
+
+1.
+
+$$
+G _ { \lambda } ^ { n } \succeq G _ { \lambda } ^ { n + 1 }
+$$
+
+2.
+
+$$
+( G _ { \lambda } ^ { n } - G _ { \lambda } ^ { n + 1 } ) - ( H _ { \lambda } ^ { n } - H _ { \lambda } ^ { n + 1 } ) \frac { d G _ { \lambda } ^ { n } / d \lambda } { d H _ { \lambda } ^ { n } / d \lambda } \succeq 0
+$$
+
+Lemma 6. Conjecture 2 implies Conjecture $^ { l }$
+
+Proof. By the reduction in Section A.3, showing monotonicity for non-isotropic regression with an isotropic regularizer is equivalent to showing monotonicity for isotropic regression with a nonisotropic regularizer. Thus, we consider the latter. Specifically, Conjecture 1 is equivalent to showing monotonicity for the estimator
+
+$$
+\begin{array} { r l } {  { \hat { \beta } _ { n , \lambda } : = \operatorname * { a r g m i n } _ { \beta } \| X \beta - \vec { y } \| _ { 2 } ^ { 2 } + \lambda \| \beta \| _ { \Sigma ^ { - 1 } } ^ { 2 } } } \\ & { = ( X ^ { T } X + \lambda \Sigma ^ { - 1 } ) ^ { - 1 } X ^ { T } y } \end{array}
+$$
+
+where $x \sim \mathcal { N } ( 0 , I )$ is isotropic, and $y \sim \langle x , \beta ^ { * } \rangle + \mathcal { N } ( 0 , \sigma ^ { 2 } )$ .
+
+Now, letting $\boldsymbol { Q } : = \boldsymbol { \Sigma } ^ { - 1 }$ , the expected test risk of this estimator for $n$ samples is:
+
+$$
+\begin{array} { r l } & { \mathbb { E } [ \left. \hat { \beta } _ { n , \lambda } \right. = \underset { X , y } { \mathbb { E } } [ \left. \hat { \beta } _ { n , \lambda } - \beta ^ { * } \right. _ { 2 } ^ { 2 } ] + \sigma ^ { 2 } } \\ & { \quad = \underset { X , y } { \mathbb { E } } [ \left. ( X ^ { T } X + \lambda Q ) ^ { - 1 } X ^ { T } y - \beta ^ { * } \right. _ { 2 } ^ { 2 } ] + \sigma ^ { 2 } } \\ & { \quad = \underset { X , \eta = N ; 0 } { \mathbb { E } } [ \left. ( X ^ { T } X + \lambda Q ) ^ { - 1 } X ^ { T } ( X \beta ^ { * } + \eta ) - \beta ^ { * } \right. _ { 2 } ^ { 2 } ] + \sigma ^ { 2 } } \\ & { \quad = \underset { X , \eta = N ; 0 } { \mathbb { E } } [ ( 0 , \lambda ^ { 2 } Y ^ { T } \eta _ { n } ^ { 2 } ) ] \left. ( X ^ { T } X + \lambda Q ) ^ { - 1 } X ^ { T } ( X \beta ^ { * } + \eta ) - \beta ^ { * } \right. _ { 2 } ^ { 2 } ] + \sigma ^ { 2 } } \\ & { \quad = \underset { X } { \mathbb { E } } [ \left. ( X ^ { T } X + \lambda Q ) ^ { - 1 } X ^ { T } X \beta ^ { * } - \beta ^ { * } ) \right. _ { 2 } ^ { 2 } ] + \sigma ^ { 2 } \underset { X } { \mathbb { E } } [ \left. ( X ^ { T } X + \lambda Q ) ^ { - 1 } X ^ { T } \right. _ { F } ^ { 2 } ] + \sigma ^ { 2 } } \\ & { \quad = \underset { X } { \mathbb { E } } [ \left. ( X ^ { T } X + \lambda Q ) ^ { - 1 } ( X ^ { T } X + \lambda Q - \lambda Q ) \beta ^ { * } - \beta ^ { * } ) \right. _ { 2 } ^ { 2 } ] + \sigma ^ { 2 } \underset { X } { \mathbb { E } } [ \left. ( X ^ { T } X + \lambda Q ) ^ { - 1 } X ^ { T } \right. _ { F } ^ { 2 } ] } \\ &  \quad = \lambda ^ { 2 } \underset { X } { \mathbb { E } } [ \left. ( X ^ { T } X + \lambda Q ) ^ { - 1 } \right. _ { 2 } ^ { 2 } ] + \sigma ^ { 2 } \underset { X } { \mathbb { E } } [ \left. ( X ^ { T } X + \lambda Q ) ^ { - 1 } X ^ { T } \right. _ { F } ^ { 2 } ] + \sigma \end{array}
+$$
+
+Consider the infimum
+
+$$
+\operatorname* { i n f } _ { \lambda \geq 0 } { \overline { { R } } } ( { \hat { \beta } } _ { n , \lambda } )
+$$
+
+We consider several cases below.
+
+Case (1). Suppose the infimum in Equation 71 is achieved in the limit $\lambda \to + \infty$ . In this case, monotonicity trivially holds, since
+
+$$
+\operatorname* { l i m } _ { \lambda \to \infty } \overline { { { R } } } ( \hat { \beta } _ { n , \lambda } ) = \overline { { { R } } } ( \vec { 0 } ) = \operatorname* { l i m } _ { \lambda \to \infty } \overline { { { R } } } ( \hat { \beta } _ { n + 1 , \lambda } )
+$$
+
+Case (2). Suppose the infimum in Equation 71 is achieved by some $\lambda = \lambda _ { n } ^ { \mathrm { o p t } }$ in the interior of the set $( 0 , \infty )$ .
+
+Because $\overline { { R } } ( \hat { \beta } _ { n , \lambda } )$ is continuous and differentiable in $\lambda$ for all $\lambda \in ( 0 , \infty )$ , we have that $\lambda _ { n } ^ { \mathrm { o p t } }$ must satisfy the following first-order optimality condition:
+
+$$
+\begin{array} { r } { \frac { d \overline { { R } } ( \hat { \beta } _ { n , \lambda } ) } { d \lambda } \Big | _ { \lambda = \lambda _ { n } ^ { \mathrm { o p t } } } = 0 } \\ { \implies ( \beta ^ { * } ) ^ { T } \frac { d G _ { \lambda } ^ { n } } { d \lambda } \beta ^ { * } + \sigma ^ { 2 } \frac { d H _ { \lambda } ^ { n } } { d \lambda } \Big | _ { \lambda = \lambda _ { n } ^ { \mathrm { o p t } } } = 0 } \end{array}
+$$
+
+We will later use this condition to show monotonicity.
+
+Case (3). Suppose the infimum in Equation 71 is achieved at $\lambda _ { n } ^ { \mathrm { o p t } } = 0$ . Recall, we define $\hat { \beta } _ { n , 0 } : =$ $\mathrm { l i m } _ { \lambda \to 0 + } \hat { \beta } _ { n , \lambda }$ . This means that,
+
+$$
+\frac { d \overline { { R } } ( \hat { \beta } _ { n , \lambda } ) } { d \lambda } \Big | _ { \lambda = 0 } = ( \beta ^ { * } ) ^ { T } \frac { d G _ { \lambda } ^ { n } } { d \lambda } \beta ^ { * } + \sigma ^ { 2 } \frac { d H _ { \lambda } ^ { n } } { d \lambda } \Big | _ { \lambda = 0 } \geq 0
+$$
+
+Note that since $\begin{array} { r } { \frac { d H _ { \lambda } ^ { n } } { d \lambda } \leq 0 } \end{array}$ , both Equations (73) and (74) in Case (2) and Case (3) respectively imply that
+
+$$
+\sigma ^ { 2 } \leq - \frac { ( \beta ^ { * } ) ^ { T } ( \frac { d G _ { \lambda } ^ { n } } { d \lambda } ) \beta ^ { * } } { d H _ { \lambda } ^ { n } / d \lambda } \Big \vert _ { \lambda = \lambda _ { n } ^ { \mathrm { o p t } } }
+$$
+
+Now, assuming Conjecture 2, we will show that the choice of $\lambda _ { n } ^ { \mathrm { o p t } }$ in Cases (2) and (3) has nonincreasing test risk for $( n + 1 )$ samples. That is,
+
+$$
+\overline { { R } } ( \widehat { \beta } _ { n , \lambda _ { n } ^ { \mathrm { o p t } } } ) \geq \overline { { R } } ( \widehat { \beta } _ { n + 1 , \lambda _ { n } ^ { \mathrm { o p t } } } )
+$$
+
+This implies the desired monotonicity, since R(βˆn+1,λoptn ) ≥ R(βˆn+1,λoptn+1 ).
+
+We first consider the case when $H _ { \lambda } ^ { n } - H _ { \lambda } ^ { n + 1 } | _ { \lambda = \lambda _ { n } ^ { \mathrm { o p t } } } \geq 0$ . In this case, because $G _ { \lambda } ^ { n } - G _ { \lambda } ^ { n + 1 } \succeq 0$ by assumption, we have
+
+$$
+\begin{array} { r l r } {  { \overline { { R } } ( \widehat { \beta } _ { n , \lambda _ { n } ^ { \mathrm { o p t } } } ) - \overline { { R } } ( \widehat { \beta } _ { n + 1 , \lambda _ { n } ^ { \mathrm { o p t } } } ) = ( \beta ^ { * } ) ^ { T } ( G _ { \lambda } ^ { n } - G _ { \lambda } ^ { n + 1 } ) \beta ^ { * } + \sigma ^ { 2 } ( H _ { \lambda } ^ { n } - H _ { \lambda } ^ { n + 1 } ) \Big | _ { \lambda = \lambda _ { n } ^ { \mathrm { o p t } } } } } \\ & { } & { \geq 0 \quad \quad } \end{array}
+$$
+
+Otherwise, assume. $H _ { \lambda } ^ { n } - H _ { \lambda } ^ { n + 1 } | _ { \lambda = \lambda _ { n } ^ { \mathrm { o p t } } } \leq 0$ . Then we have:
+
+$$
+\begin{array} { r l } { \overline { { R } } ( \hat { \beta } _ { n , \lambda _ { n } ^ { \smash { \mathrm { o u t } } } } ) - \overline { { R } } ( \hat { \beta } _ { n + 1 , \lambda _ { n } ^ { \smash { \mathrm { o u t } } } } ) = ( \beta ^ { * } ) ^ { T } ( G _ { \lambda } ^ { m } - G _ { \lambda } ^ { m + 1 } ) \beta ^ { * } + \sigma ^ { 2 } ( H _ { \lambda } ^ { n } - H _ { \lambda } ^ { n + 1 } ) \Big | _ { \lambda = \lambda _ { n } ^ { \smash { \mathrm { o u t } } } } } & { \qquad \mathrm { ( } } \\ & { \qquad \geq ( \beta ^ { * } ) ^ { T } ( G _ { \lambda } ^ { m } - G _ { \lambda } ^ { m + 1 } ) \beta ^ { * } - ( \beta ^ { * } ) ^ { T } ( \frac { d G _ { \lambda } ^ { n } } { d \lambda } ) \beta ^ { * } \frac { ( H _ { \lambda } ^ { n } - H _ { \lambda } ^ { n + 1 } ) } { d H _ { \lambda } ^ { \lambda } / d \lambda } \Big | _ { \lambda = \lambda _ { n } ^ { \smash { \mathrm { o u t } } } } } \\ & { \qquad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { = ( \beta ^ { * } ) ^ { T } \underbrace { \Bigg ( ( G _ { \lambda } ^ { m } - G _ { \lambda } ^ { n + 1 } ) - ( H _ { \lambda } ^ { n } - H _ { \lambda } ^ { n + 1 } ) \frac { d G _ { \lambda } ^ { m } / d \lambda } { d H _ { \lambda } ^ { n } / d \lambda } \Big ) \Big | _ { \lambda = \lambda _ { n } ^ { \mathrm { o u t } } } \beta ^ { * } } _ { \geq 0 \mathrm { b y c o n j e c u r e 2 } } } \\ & { \qquad \quad \quad \quad \quad \quad \quad \quad \quad \geq 0 } \end{array}
+$$
+
+as desired.
\ No newline at end of file
diff --git a/parse/train/7R7fAoUygoa/7R7fAoUygoa_content_list.json b/parse/train/7R7fAoUygoa/7R7fAoUygoa_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..0deaeb37c666af79c1b5e1ae5b69dedfe0dbf22a
--- /dev/null
+++ b/parse/train/7R7fAoUygoa/7R7fAoUygoa_content_list.json
@@ -0,0 +1,3247 @@
+[
+    {
+        "type": "text",
+        "text": "OPTIMAL REGULARIZATION CAN MITIGATE DOUBLE DESCENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
+            823,
+            145
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Preetum Nakkiran   \nHarvard University   \npreetum@cs.harvard.edu   \nPrayaag Venkat   \nHarvard University   \npvenkat@g.harvard.edu ",
+        "bbox": [
+            183,
+            170,
+            401,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            606,
+            171,
+            813,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sham Kakade ",
+        "text_level": 1,
+        "bbox": [
+            184,
+            234,
+            284,
+            247
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Microsoft Research & University of Washington sham@cs.washington.edu ",
+        "bbox": [
+            184,
+            248,
+            501,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Tengyu Ma   \nStanford University   \ntengyuma@stanford.edu ",
+        "bbox": [
+            563,
+            234,
+            771,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            313,
+            544,
+            327
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent empirical and theoretical studies have shown that many learning algorithms – from linear regression to neural networks – can have test performance that is non-monotonic in quantities such the sample size and model size. This striking phenomenon, often referred to as “double descent”, has raised questions of if we need to re-think our current understanding of generalization. In this work, we study whether the double-descent phenomenon can be avoided by using optimal regularization. Theoretically, we prove that for certain linear regression models with isotropic data distribution, optimally-tuned $\\ell _ { 2 }$ regularization achieves monotonic test performance as we grow either the sample size or the model size. We also demonstrate empirically that optimally-tuned $\\ell _ { 2 }$ regularization can mitigate double descent for more general models, including neural networks. Our results suggest that it may also be informative to study the test risk scalings of various algorithms in the context of appropriately tuned regularization. ",
+        "bbox": [
+            233,
+            344,
+            764,
+            523
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            551,
+            336,
+            568
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent works have demonstrated a ubiquitous “double descent” phenomenon present in a range of machine learning models, including decision trees, random features, linear regression, and deep neural networks (Opper, 1995; 2001; Advani & Saxe, 2017; Spigler et al., 2018; Belkin et al., 2018; Geiger et al., 2019b; Nakkiran et al., 2020; Belkin et al., 2019; Hastie et al., 2019; Bartlett et al., 2019; Muthukumar et al., 2019; Bibas et al., 2019; Mitra, 2019; Mei & Montanari, 2019; Liang & Rakhlin, 2018; Liang et al., 2019; Xu & Hsu, 2019; Derezinski et al., 2019; Lampinen & Ganguli, ´ 2018; Deng et al., 2019; Nakkiran, 2019). The phenomenon is that models exhibit a peak of high test risk when they are just barely able to fit the train set, that is, to interpolate. For example, as we increase the size of models, test risk first decreases, then increases to a peak around when effective model size is close to the training data size, and then decreases again in the overparameterized regime. Also surprising is that Nakkiran et al. (2020) observe a double descent as we increase sample size, i.e. for a fixed model, training the model with more data can hurt test performance. ",
+        "bbox": [
+            174,
+            583,
+            825,
+            750
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "These striking observations highlight a potential gap in our understanding of generalization and an opportunity for improved methods. Ideally, we seek to use learning algorithms which robustly improve performance as the data or model size grow and do not exhibit such unexpected nonmonotonic behaviors. In other words, we aim to improve the test performance in situations which would otherwise exhibit high test risk due to double descent. Here, a natural strategy would be to use a regularizer and tune its strength on a validation set. This motivates the central question of this work: ",
+        "bbox": [
+            174,
+            757,
+            823,
+            853
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "When does optimally tuned regularization mitigate or remove the double-descent phenomenon? ",
+        "bbox": [
+            186,
+            867,
+            810,
+            882
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Another motivation is the fact that double descent is largely observed for unregularized or under regularized models in practice. As an example, Figure 1 shows a simple linear ridge regression setting in which the unregularized estimator exhibits double descent, but an optimally-tuned regularizer has monotonic test performance. ",
+        "bbox": [
+            178,
+            895,
+            818,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/b5edb5b47ed8f04264d90917cb098c4913b1a417a4d6db231609364ffb35c0c9.jpg",
+        "image_caption": [
+            "Figure 1: Test Risk vs. Num. Samples for Isotropic Ridge Regression in $d \\_ =$ 500 dimensions. Unregularized regression is non-monotonic in samples, but optimallyregularized regression $( \\lambda = \\lambda _ { o p t } )$ is monotonic. In this setting, the optimal regularizer $\\lambda _ { o p t }$ does not depend on number of samples $n$ (Lemma 2), but this is not always true – see Figure 2. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            204,
+            106,
+            483,
+            321
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            352,
+            823,
+            381
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our Contributions: We study this question from both a theoretical and empirical perspective. Theoretically, we start with the setting of high-dimensional linear regression. Linear regression is a sensible starting point to study these questions, since it already exhibits many of the qualitative features of double descent in more complex models (e.g. Belkin et al. (2019); Hastie et al. (2019) and further related works in Section 1.1). Our work shows that optimally-tuned ridge regression can achieve both sample-wise monotonicity and model-size-wise monotonicity under certain assumptions. Concretely, we show ",
+        "bbox": [
+            173,
+            395,
+            825,
+            492
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. Sample-wise monotonicity: In the setting of well-specified linear regression with isotropic features/covariates (Figure 1), we prove that optimally-tuned ridge regression yields monotonic test performance with increasing samples. That is, more data never hurts for optimally-tuned ridge regression. (See Theorem 1). ",
+        "bbox": [
+            173,
+            500,
+            825,
+            555
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2. Model-wise monotonicity: We consider a setting where the input/covariate lives in a highdimensional ambient space with isotropic covariance. Given a fixed model size $d$ (which might be much smaller than ambient dimension), we consider the family of models which first project the input to a random $d$ -dimensional subspace, and then compute a linear function in this projected “feature space.” (This is nearly identical to models of double-descent considered in Hastie et al. (2019, Section 5.1)). We prove that in this setting, as we grow the model-size, optimally-tuned ridge regression over the projected features has monotone test performance. That is, with optimal regularization, bigger models are always better or the same. (See Theorem 3). ",
+        "bbox": [
+            173,
+            563,
+            825,
+            674
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3. Monotonicity in the real-world: We also demonstrate several richer empirical settings where optimal $\\ell _ { 2 }$ regularization induces monotonicity, including random feature classifiers and convolutional neural networks. This suggests that the mitigating effect of optimal regularization may hold more generally in broad machine learning contexts. (See Section 5). ",
+        "bbox": [
+            174,
+            680,
+            825,
+            736
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A few remarks are in order: ",
+        "bbox": [
+            176,
+            743,
+            354,
+            757
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Problem-specific vs Minimax and Bayesian. It is worth noting that our results hold for all linear ground-truths, rather than holding for only the worst-case ground-truth or a random ground-truth. Indeed, the minimax optimal estimator or the Bayes optimal estimator are both trivially sample-wise and model-wise monotonic with respect to the minimax risk or the Bayes risk. However, they do not guarantee monotonicity of the risk itself for a given fixed problem. In particular, there exist minimax optimal estimators which are not sample-monotonic in the sense we desire. ",
+        "bbox": [
+            174,
+            763,
+            825,
+            847
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Universal vs Asymptotic. We also remark that our analysis is not only non-asymptotic but also works for all possible input dimensions, model sizes, and sample sizes. To our knowledge, the results herein are the first non-asymptotic sample-wise and model-wise monotonicity results for linear regression. (See discussion of related works Hastie et al. (2019); Mei & Montanari (2019) for related results in the asymptotic setting). Our work reveals aspects of the problem that were not present in prior asymptotic works. For example, we empirically show that optimal regularization can eliminate even “triple descent” in ridge regression (Figure 2). Moreover, we show that for nonGaussian covariates, optimally-tuned ridge regression is not always sample-monotonic: we give a counterexample in Section 4. ",
+        "bbox": [
+            174,
+            854,
+            823,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            823,
+            159
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Towards a more general characterization. Our theoretical results crucially rely on the covariance of the data being isotropic. A natural next question is if and when the same results can hold more generally. A full answer to this question is beyond the scope of this paper, though we give the following results: ",
+        "bbox": [
+            176,
+            166,
+            825,
+            222
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "1. Optimally-tuned ridge regression is not always sample-monotonic: we show a counterexample for a certain non-Gaussian data distribution and heteroscedastic noise. We are not aware of prior work pointing out this fact. (See Section 4 for the counterexample and intuitions.)   \n2. For non-isotropic Gaussian covariates, we can achieve sample-wise monotonicity with a regularizer that depends on the population covariance matrix of data. This suggests unlabeled data might also help mitigate double descent in some settings, because the population covariance can be estimated from unlabeled data. (See Appendix B).   \n3. For non-isotropic Gaussian covariates, we conjecture that optimally-tuned ridge regression is sample-monotonic even with a standard $\\ell _ { 2 }$ regularizer (as in Figure 2). We derive a sufficient condition for this conjecture. Due to that current random matrix theory may be insufficient to verify this conjecture, we verify it numerically on a wide variety of cases. (See Appendix B for details). ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "The last two results above highlight the importance of the form of the regularizer, which leads to the open question: “How do we design good regularizers which mitigate or remove double descent?” We hope that our results can motivate future work on mitigating the double descent phenomenon, and allow us to train high performance models which do not exhibit nonmonotonic behaviors. ",
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+    {
+        "type": "text",
+        "text": "1.1 RELATED WORKS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The study of nonmonotonicity in learning algorithms existed prior to double descent and has a long history going back to (at least) Trunk (1979) and LeCun et al. (1991); Le Cun et al. (1991), where the former was largely empirical observations and the latter studied the sample non-nonmonotonicity of unregularized linear regression in terms of the eigenspectrum of the covariance matrix; the difference to our works is that we study this in the context of optimal regularization. In fact, Duin (1995; 2000); Opper (2001); Loog & Duin (2012). Loog et al. (2019) introduces the same notion of risk monotonicity which we consider, and studies several examples of monotonic and non-monotonic procedures. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Double descent of test risk as a function of model size was considered recently in more generality by Belkin et al. (2018). Similar behavior was observed empirically in earlier work in somewhat more restricted settings Trunk (1979); Opper (1995; 2001); Skurichina & Duin (2002); Le Cun et al. (1991); LeCun et al. (1991) and more recently in Advani & Saxe (2017); Geiger et al. (2019a); Spigler et al. (2018); Neal et al. (2018). Recently Nakkiran et al. (2020) demonstrated a generalized double descent phenomenon on modern deep networks, and highlighted “sample non-monotonicity” as an aspect of double descent. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A recent stream of theoretical works consider model-wise double descent in simplified settings— often via linear models for regression or classification. This also connects to works on highdimentional regression in the statistics literature. A partial list of works in these areas include Belkin et al. (2019); Hastie et al. (2019); Bartlett et al. (2019); Muthukumar et al. (2019); Bibas et al. (2019); Mitra (2019); Mei & Montanari (2019); Liang & Rakhlin (2018); Liang et al. (2019); Xu & Hsu (2019); Derezinski et al. (2019); Lampinen & Ganguli (2018); Deng et al. (2019); Nakki- ´ ran (2019); Mahdaviyeh & Naulet (2019); Dobriban et al. (2018); Dobriban & Sheng (2019); Kobak et al. (2018). Of these, most closely related to our work are Hastie et al. (2019); Dobriban et al. (2018); Mei & Montanari (2019). Specifically, Hastie et al. (2019) considers the risk of unregularized and regularized linear regression in an asymptotic regime, where dimension $d$ and number of samples $n$ scale to infinity together, at a constant ratio $d / n$ . In contrast, we show non-asymptotic results, and are able to consider increasing the number of samples for a fixed model, without scaling both together. Mei & Montanari (2019) derive similar results for unregularized and regularized random features, also in an asymptotic limit. The non-asymptotic versions of the settings considered in Hastie et al. (2019) are almost identical to ours— for example, our projection model in Section 3 is nearly identical to the model in Hastie et al. (2019, Section 5.1). Finally, subsequent to our work, d’Ascoli et al. (2020) identified triple descent in an asymptotic setting. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "2 SAMPLE MONOTONICITY IN RIDGE RIDGRESSION ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "In this section, we prove that optimally-regularized ridge regression has test risk that is monotonic in samples, for isotropic gaussian covariates and linear response. This confirms the behavior empirically observed in Figure 1. We also show that this monotonicity is not “fragile”, and using larger than larger regularization is still sample-monotonic (consistent with Figure 1). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Formally, we consider the following linear regression problem in $d$ dimensions. The input/covariate $x \\in \\mathbb { R } ^ { d }$ is generated from $\\mathcal { N } ( 0 , I _ { d } )$ , and the output/response is generated by $y = \\langle x , \\beta ^ { * } \\rangle + \\varepsilon$ with $\\varepsilon \\sim \\mathcal { N } ( 0 , \\bar { \\sigma } ^ { 2 } )$ for some unknown parameter $\\beta ^ { * } \\in \\mathbb { R } ^ { d }$ . We denote the joint distribution of $( x , y )$ by $\\mathcal { D }$ . We are given $n$ training examples $\\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { n }$ i.i.d sampled from $\\mathcal { D }$ . We aim to learn a linear model $f _ { \\beta } ( x ) = \\langle x , \\beta \\rangle$ with small population risk $\\bar { R } ( \\beta ) : = \\mathbb { E } _ { ( x , y ) \\sim \\mathcal { D } } [ ( \\langle x , \\beta \\rangle - y ) ^ { 2 } ]$ . For simplicity, let $\\ b { X } \\in \\mathbb { R } ^ { n \\times d }$ be the data matrix that contains $x _ { i } ^ { \\top }$ ’s as rows and let $\\vec { y } \\in \\mathbb R ^ { n }$ be column vector that contains the responses $y _ { i }$ ’s as entries. For any estimator $\\hat { \\beta } _ { n } ( X , \\vec { y } )$ as a function of $n$ samples, define the expected risk of the estimator as: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/ca8f1f38d0fbad23b37af04d06db0fdee27ea648b14d9cc7188acd0d5ba2d5d0.jpg",
+        "text": "$$\n{ \\overline { { R } } } ( { \\hat { \\beta } } _ { n } ) : = \\operatorname * { \\mathbb { E } } _ { X , y \\sim { \\mathcal { D } } ^ { n } } [ R ( { \\hat { \\beta } } _ { n } ( X , { \\vec { y } } ) ) ]\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "We consider the regularized least-squares estimator, also known as the ridge regression estimator. For a given $\\lambda > 0$ , define ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/d1e09e8985a269d77fd3cb37a62099e25386d5abe73ff5141e77c5f63a87f69b.jpg",
+        "text": "$$\n\\hat { \\beta } _ { n , \\lambda } : = \\underset { \\beta } { \\operatorname { a r g m i n } } | | X \\beta - \\vec { y } | | _ { 2 } ^ { 2 } + \\lambda | | \\beta | | _ { 2 } ^ { 2 } = ( X ^ { T } X + \\lambda I _ { d } ) ^ { - 1 } X ^ { T } \\vec { y }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here $I _ { d }$ denotes the $d$ dimensional identity matrix. Let $\\lambda _ { n } ^ { \\mathrm { o p t } }$ be the optimal ridge parameter (that achieves the minimum expected risk) given $n$ samples: $\\begin{array} { r } { \\lambda _ { n } ^ { \\mathrm { o p t } } : = \\operatorname * { a r g m i n } _ { \\lambda : \\lambda \\geq 0 } \\overline { { R } } ( \\hat { \\beta } _ { n , \\lambda } ) ) } \\end{array}$ . Let $\\hat { \\beta } _ { n } ^ { \\mathrm { { o p t } } }$ be the estimator that corresponds to the $\\lambda _ { n } ^ { \\mathrm { o p t } }$ . That is, $\\begin{array} { r } { \\hat { \\beta } _ { n } ^ { \\mathrm { o p t } } : = \\operatorname * { a r g m i n } _ { \\beta } | | X \\beta - \\vec { y } | | _ { 2 } ^ { 2 } + \\lambda _ { n } ^ { \\mathrm { o p t } } | | \\beta | | _ { 2 } ^ { 2 } } \\end{array}$ . Our main theorem in this section shows that the expected risk of $\\hat { \\beta } _ { n } ^ { \\mathrm { { o p t } } }$ monotonically decreases as $n$ increases. ",
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+        "type": "text",
+        "text": "Theorem 1. In the setting above, the expected test risk of optimally-regularized well-specified isotropic linear regression is monotonic in samples. That is, for all $\\beta ^ { * } \\in \\mathbb { R } ^ { d }$ and all $d \\in \\mathbb { N } , n \\in$ $\\mathbb { N } , \\sigma > 0$ , ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/02c89e1530f055b86ccd8d1b6fb13f02720f9d0d99c45df164c8d8dfcd675713.jpg",
+        "text": "$$\n\\overline { { R } } ( \\hat { \\beta } _ { n + 1 } ^ { \\mathrm { o p t } } ) \\leq \\overline { { R } } ( \\hat { \\beta } _ { n } ^ { \\mathrm { o p t } } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The above theorem shows a strong form of monotonicity, since it holds for every fixed groundtruth $\\beta ^ { * }$ , and does not require averaging over any prior on ground-truths. Moreover, it holds nonasymptotically, for every fixed $n , d \\in \\mathbb { N }$ . Obtaining such non-asymptotic results is nontrivial, since we cannot rely on concentration properties of the involved random variables. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In particular, evaluating $\\overline { { R } } ( \\hat { \\beta } _ { n } ^ { \\mathrm { { o p t } } } )$ as a function of the problem parameters $( n , \\sigma , \\beta ^ { * }$ , and $d$ ) is technically challenging. In fact, we suspect that a simple closed form expression does not exist. The key idea towards proving the theorem is to derive a “partial evaluation” — the following lemmas shows that we can write $\\overline { { R } } ( \\hat { \\beta } _ { n } ^ { \\mathrm { { o p t } } } )$ in the form of $\\mathbb { E } [ g ( \\gamma , \\sigma , n , d , \\beta ^ { * } ) ]$ where $\\gamma \\in \\mathbb { R } ^ { d }$ contains the singular values of $X$ . We will then couple the randomness of data matrices obtained by adding a single sample, and use singular value interlacing to compare their singular values. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1. In the setting of Theorem $^ { l }$ , let $\\gamma = ( \\gamma _ { 1 } , \\ldots , \\gamma _ { d } )$ be the singular values of the data matrix $\\ b { X } \\in \\mathbb { R } ^ { n \\times d }$ . (If $n < d$ , we pad the $\\gamma _ { i } = 0$ for $i > n .$ .) Let $\\Gamma _ { n }$ be the distribution of $\\gamma .$ . Then, the expected test risk is ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b40dba43a33ce50127dbb819e04e578cfe43a13d87ee3e1ab4cc4770fe468e0d.jpg",
+        "text": "$$\n\\overline { { R } } ( \\hat { \\beta } _ { n , \\lambda } ) = \\operatorname * { \\mathbb { E } } _ { ( \\gamma _ { 1 } , \\dots \\gamma _ { d } ) \\sim \\Gamma _ { n } } \\left[ \\sum _ { i = 1 } ^ { d } \\frac { | | \\beta ^ { * } | | _ { 2 } ^ { 2 } \\lambda ^ { 2 } / d + \\sigma ^ { 2 } \\gamma _ { i } ^ { 2 } } { ( \\gamma _ { i } ^ { 2 } + \\lambda ) ^ { 2 } } \\right] + \\sigma ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "From Lemma 1, the below lemma follows directly by taking derivatives to find the optimal $\\lambda$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Lemma 2. In the setting of Theorem 1, the optimal ridge parameter is constant for all $n$ : $\\lambda _ { n } ^ { \\mathrm { o p t } } =$ $\\frac { d \\sigma ^ { 2 } } { | | \\beta ^ { * } | | _ { 2 } ^ { 2 } }$ . Moreover, the optimal expected test risk can be written as ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e5430d69b46fe619155e1f8482e9e60dbcbbee1ff0353cd69134b248c0fec2e1.jpg",
+        "text": "$$\n\\overline { { R } } ( \\hat { \\beta } _ { n } ^ { \\mathrm { o p t } } ) = \\operatorname * { l } _ { ( \\gamma _ { 1 } , \\ldots \\gamma _ { d } ) \\sim \\Gamma _ { n } } \\left[ \\sum _ { i = 1 } ^ { d } \\frac { \\sigma ^ { 2 } } { \\gamma _ { i } ^ { 2 } + d \\sigma ^ { 2 } / | | \\beta ^ { * } | | _ { 2 } ^ { 2 } } \\right] + \\sigma ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proofs of Lemma 1 and 2 are deferred to the Appendix, Section A.1. Now we are ready to prove Theorem 1. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proof of Theorem $^ { l }$ . Let $\\widetilde { X } \\ \\in \\ \\mathbb { R } ^ { ( n + 1 ) \\times d }$ and $\\boldsymbol { X } \\in \\mathbb { R } ^ { n \\times d }$ be any two matrices which differ by only the last row of $\\widetilde { X }$ . By the Cauchy interlacing theorem Theorem 4.3.4 of Horn et al. (1990) (c.f.,Lemma 3.4 of Marcus et al. (2014)), the singular values of $X$ and $\\widetilde { X }$ are interlaced: $\\forall i$ : $\\gamma _ { i - 1 } ( X ) \\geq \\gamma _ { i } ( { \\widetilde { X } } ) \\geq \\gamma _ { i } ( X )$ where $\\gamma _ { i } ( \\cdot )$ is the $i$ -th singular value. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "If we couple $\\widetilde { X }$ and $X$ , it will induce a coupling $\\Pi$ between the distributions $\\Gamma _ { n + 1 }$ and $\\Gamma _ { n }$ , of the singular values of the data matrix for $n + 1$ and $n$ samples. This coupling satisfies that $\\widetilde { \\gamma } _ { i } \\geq \\gamma _ { i }$ with probability 1 for $( \\{ \\widetilde { \\gamma } _ { i } \\} , \\{ \\gamma _ { i } \\} ) \\sim \\Pi$ e. Now, expand the test risk using Lemma 2, and observe that each eterm in the sum of Equation (4) below is monotone decreasing with $\\gamma _ { i }$ . Thus: ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/394fb265e7acb3aca9e90fbd87720cebac655f2a76eaa31a335aed3b7bf29338.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\overline { { R } } ( \\hat { \\beta } _ { n } ^ { \\mathrm { o p t } } ) = \\underset { ( \\gamma _ { 1 } , \\ldots , \\gamma _ { d } ) \\sim \\Gamma _ { n } } { \\mathbb { E } } \\left[ \\displaystyle \\sum _ { i = 1 } ^ { d } \\frac { \\sigma ^ { 2 } } { \\gamma _ { i } ^ { 2 } + d \\sigma ^ { 2 } / | | \\beta ^ { * } | | _ { 2 } ^ { 2 } } \\right] + \\sigma ^ { 2 } } \\\\ & { \\qquad \\quad \\overset { \\mathbb { E } } { \\ge } \\underset { ( \\widetilde { \\gamma } _ { 1 } , \\ldots , \\widetilde { \\gamma } _ { d } ) \\sim \\Gamma _ { n + 1 } } { \\mathbb { E } } \\left[ \\displaystyle \\sum _ { i = 1 } ^ { d } \\frac { \\sigma ^ { 2 } } { \\widetilde { \\gamma } _ { i } ^ { 2 } + d \\sigma ^ { 2 } / | | \\beta ^ { * } | | _ { 2 } ^ { 2 } } \\right] + \\sigma ^ { 2 } } \\\\ & { \\qquad = \\overline { { R } } ( \\hat { \\beta } _ { n + 1 } ^ { \\mathrm { o p t } } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "By similar techniques, we can also prove that overregularization —that is, using ridge parameters $\\lambda$ larger than the optimal value— is still monotonic. This proves the behavior empirically observed in Figure 1. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2. In the same setting as Theorem $I$ , over-regularized regression is also monotonic in samples. That is, for all $d \\in \\mathbb { N } , n \\in \\mathbb { N } , \\sigma > 0 , \\beta ^ { * } \\in \\mathbb { R } ^ { d }$ , the following holds ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a578df214e9bb7e84f8394d7f15346d3c032904414a25d33ee34685761e5d638.jpg",
+        "text": "$$\n\\forall \\lambda \\geq \\lambda ^ { * } : \\quad \\overline { { { R } } } ( \\widehat { \\beta } _ { n + 1 , \\lambda } ) \\leq \\overline { { { R } } } ( \\widehat { \\beta } _ { n , \\lambda } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            621,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { \\lambda ^ { * } = \\frac { d \\sigma ^ { 2 } } { | | \\beta ^ { * } | | _ { 2 } ^ { 2 } } } \\end{array}$ ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proof. In Section A.1. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 MODEL-WISE MONOTONICITY IN RIDGE REGRESSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we show that for a certain family of linear models, optimal regularization prevents model-wise double descent. That is, for a fixed number of samples, larger models are not worse than smaller models. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We consider the following learning problem. Informally, covariates live in a $p$ -dimensional ambient space, and we consider models which first linearly project down to a random $d$ -dimensional subspace, then perform ridge regression in that subspace for some $d \\leq p$ . Formally, the covariate $x \\in \\mathbb { R } ^ { p }$ is generated from $\\bar { \\mathcal { N } } ( 0 , I _ { p } )$ , and the response is generated by $y = \\langle x , \\theta \\rangle + \\varepsilon$ with $\\varepsilon \\sim \\mathcal { N } ( 0 , \\sigma ^ { 2 } )$ and for some unknown parameter $\\theta \\in \\mathbb { R } ^ { p }$ . Next, $n$ examples $\\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { n }$ are sampled i.i.d from this distribution. For a given model size $d \\leq p$ , we first sample a random orthonormal matrix $P \\in \\mathbb { R } ^ { d \\times p }$ which specifies our model. We then consider models which operate on $( \\widetilde { x } _ { i } , y _ { i } ) \\in \\mathbb { R } ^ { d } \\times \\mathbb { R }$ , where $\\widetilde { x _ { i } } = P x _ { i }$ . We denote the joint distribution of $( \\widetilde { x } , y )$ by $\\mathcal { D }$ . Here, we eemphasize that $p$ eis some large ambient dimension and $d \\leq p$ eis the size of the model we learn. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For a fixed $P$ , we want to learn a linear model $f _ { \\hat { \\beta } } ( \\tilde { x } ) = \\langle \\tilde { x } , \\hat { \\beta } \\rangle$ for estimating $y$ , with small mean squared error on distribution: $R _ { P } ( \\hat { \\beta } ) : = \\mathbb { E } _ { ( \\tilde { x } , y ) \\sim \\mathcal { D } } [ ( \\langle \\tilde { x } , \\hat { \\beta } \\rangle - y ) ^ { 2 } ]$ . For $n$ samples $( x _ { i } , y _ { i } )$ , let $X \\in$ $\\mathbb { R } ^ { n \\times p }$ be the data matrix, $\\widetilde { X } \\ = X P ^ { T } \\ \\in \\ \\mathbb { R } ^ { n \\times d }$ be the projected data matrix and $\\vec { y } \\in \\mathbb R ^ { n }$ be the responses. For any estimator $\\hat { \\beta } ( \\widetilde { X } , \\vec { y } )$ as a function of the observed samples, define the expected risk of the estimator as: ",
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/471b0b050ee0b25d6039c368a191eda06e0d7ece3a4fd4107b6fcd61419bdb12.jpg",
+        "text": "$$\n\\overline { { R } } ( \\hat { \\beta } ) : = \\operatorname * { \\mathbb { E } } _ { P } \\operatorname * { \\mathbb { E } } _ { \\tilde { X } , \\vec { y } \\sim \\mathcal { D } ^ { n } } [ R _ { P } ( \\hat { \\beta } ( \\tilde { X } , \\vec { y } ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We consider the regularized least-squares estimator. For a given $\\lambda > 0$ , define ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "equation",
+        "img_path": "images/d84149205d19f5e433d4ef05d807c1baeb63bf431d05a7edb609aeb394e83063.jpg",
+        "text": "$$\n\\hat { \\beta } _ { d , \\lambda } : = \\underset { \\beta } { \\mathrm { a r g m i n } } | | \\widetilde { X } \\beta - \\vec { y } | | _ { 2 } ^ { 2 } + \\lambda | | \\beta | | _ { 2 } ^ { 2 } = ( \\widetilde { X } ^ { T } \\widetilde { X } + \\lambda I _ { d } ) ^ { - 1 } \\widetilde { X } ^ { T } \\vec { y }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Let $\\lambda _ { d } ^ { \\mathrm { o p t } }$ be the optimal ridge parameter (that achieves the minimum expected risk) for a model of size $d$ d, with $n$ samples: $\\lambda _ { d } ^ { \\mathrm { o p t } } : = \\mathrm { a r g m i n } _ { \\lambda \\ge 0 } \\overline { { R } } ( \\hat { \\beta } _ { d , \\lambda } ) )$ . Let $\\hat { \\beta } _ { d } ^ { \\mathrm { { o p t } } }$ be the estimator that corresponds to the $\\lambda _ { d } ^ { \\mathrm { o p t } }$ , that is $\\begin{array} { r } { \\hat { \\beta } _ { d } ^ { \\mathrm { o p t } } : = \\operatorname * { a r g m i n } _ { \\beta } | | \\widetilde { X } \\beta - \\vec { y } | | _ { 2 } ^ { 2 } + \\lambda _ { d } ^ { \\mathrm { o p t } } | | \\beta | | _ { 2 } ^ { 2 } } \\end{array}$ . Now, our main theorem in this setting shows that with optimal $\\ell _ { 2 }$ regularization, test performance is monotonic in model size. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Theorem 3. In the setting above, the expected test risk of the optimally-regularized model is monotonic in the model size $d .$ . That is, for all $p \\in \\mathbb { N } , \\theta \\in \\mathbb { R } ^ { p } , d \\leq p , n \\in \\mathbb { N } , \\sigma > 0$ , we have ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/220172f9a1811307d4805215b213850851fc08625c6f7fcb7e7c4f9092ce607d.jpg",
+        "text": "$$\n\\overline { { R } } ( \\hat { \\beta } _ { d + 1 } ^ { \\mathrm { o p t } } ) \\leq \\overline { { R } } ( \\hat { \\beta } _ { d } ^ { \\mathrm { o p t } } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "The proof of Theorem 3 is in Appendix A.2, and follows closely the proof of Theorem 1. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 COUNTEREXAMPLES TO MONOTONICITY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we show that optimally-regularized ridge regression is not always monotonic in samples. We give a numeric counterexample in $d = 2$ dimensions, with non-gaussian covariates and heteroscedastic noise. This does not contradict our main theorem in Section 2, since this distribution is not jointly Gaussian with isotropic marginals. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Counterexample. Here we give an example of a distribution $( x , y )$ for which the expected error of optimally-regularized ridge regression with $n = 2$ samples is worse than with $n = 1$ samples. This counterexample is most intuitive to understand when the ridge parameter $\\lambda$ is allowed to depend on the specific sample instance $( X , { \\vec { y } } )$ as well as $n ^ { 1 }$ . We sketch the intuition for this below. Consider the following distribution on $( x , y )$ in $d = 2$ dimensions. This distribution has one “clean” coordinate and one “noisy” coordinate. The distribution is: $( x , y ) = ( { \\vec { e } } _ { 1 } , 1 )$ with probability $1 / 2$ , and $( x , y ) =$ $( \\vec { e } _ { 2 } , \\pm 1 0 )$ w.p. 1/2. Where $\\pm 1 0$ is uniformly random independent noise. This distribution is “wellspecified” in that the optimal predictor is linear in $x$ $: \\mathbb { E } [ y | x ] = \\langle \\beta ^ { * } , x \\rangle$ for $\\beta ^ { * } = [ 1 , 0 ]$ . However, the noise is heteroscedastic. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For $n = 1$ samples, the estimator can decide whether to use small $\\lambda$ or large $\\lambda$ depending on if the sampled coordinate is the “clean” or “noisy” one. Specifically, for the sample $( x , y )$ : If $x = \\vec { e } _ { 1 }$ , then the optimal ridge parameter is $\\lambda = 0$ . If $x = \\vec { e } _ { 2 }$ , then the optimal parameter is $\\lambda = \\infty$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For $n = 2$ samples, with probability $1 / 2$ the two samples will hit both coordinates. In this case, the estimator must chose a single value of $\\lambda$ uniformly for both coordinates. This yields to a suboptimal tradeoff, since the “noisy” coordinate demands large regularization, but this hurts estimation on the “clean” coordinate. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "It turns out that a slight modification to the above also serves as a counterexample to monotonicity when the regularization parameter $\\lambda$ is chosen only depending on $n$ (and not on the instance $X , y )$ . The distribution is: $( x , y ) = ( \\vec { e } _ { 1 } , 1 )$ w.p. 0.98 and $( x , y ) = ( \\vec { e } _ { 2 } , \\pm 2 0 )$ w.p. 0.02. This distribution has the following property. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Theorem 4. There exists a distribution $\\mathcal { D }$ over $( x , y )$ for $x \\in \\mathbb { R } ^ { 2 } , y \\in \\mathbb { R }$ with the following properties. Let $\\hat { \\beta } _ { n } ^ { \\mathrm { { o p t } } }$ be the optimally-regularized ridge regression solution for $n$ samples $( X , { \\vec { y } } )$ from $\\mathcal { D }$ . Then: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "1. D is “well-specified” in that $\\mathbb { E } p [ y | x ]$ is a linear function of $x$ , ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "2. The expected test risk increases as a function of $n _ { \\ast }$ , between $n = 1$ and $n = 2$ . Specifically ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5c348faa011f7bcdbd72bd75ff6cd0e528571f73042b72f9974ff9f74bbe14c5.jpg",
+        "text": "$$\n\\overline { { R } } ( \\hat { \\beta } _ { n = 1 } ^ { \\mathrm { o p t } } ) < \\overline { { R } } ( \\hat { \\beta } _ { n = 2 } ^ { \\mathrm { o p t } } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Proof. For $n = 1$ samples, it can be confirmed analytically that the expected risk $\\overline { { { R } } } ( \\hat { \\beta } _ { n = 1 } ^ { \\mathrm { o p t } } ) ~ <$ 8.157. This is achieved with $\\lambda = 4 0 0 / 2 4 0 1 \\approx 0 . 1 6 6 5 9 7 .$ For $n = 2$ samples, it can be confirmed numerically (via Mathematica) that the expected risk $\\overline { { R } } ( \\hat { \\beta } _ { n = 2 } ^ { \\mathrm { o p t } } ) ~ > ~ 8 . 1 7 9$ . This is achieved with $\\lambda = 0 . 6 4 2 5 2 5$ . □ ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "We now experimentally demonstrate that optimal $\\ell _ { 2 }$ regularization can mitigate double descent, in more general settings than Theorems 1 and 3. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 SAMPLE MONOTONICITY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Here we show various settings where optimal $\\ell _ { 2 }$ regularization empirically induces samplemonotonic performance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Nonisotropic Regression. We first consider the setting of Theorem 1, but with non-isotropic covariantes $x$ . That is, we perform ridge regression on samples $( x , y )$ , where the covariate $\\boldsymbol { x } \\in \\mathbb { R } ^ { d }$ is generated from $\\mathcal { N } ( 0 , \\Sigma )$ for $\\Sigma \\neq I _ { d }$ . As before, the response is generated by $y = \\langle x , \\beta ^ { * } \\rangle + \\varepsilon$ with $\\varepsilon \\sim \\mathcal { N } ( 0 , \\sigma ^ { 2 } )$ for some unknown parameter $\\beta ^ { * } \\in \\mathbb { R } ^ { d }$ . We consider the same ridge regression estimator, $\\begin{array} { r } { \\hat { \\beta } _ { n , \\lambda } : = \\operatorname * { a r g m i n } _ { \\boldsymbol { \\beta } } \\| X { \\boldsymbol { \\beta } } - \\vec { y } \\| _ { 2 } ^ { 2 } + \\lambda \\| { \\boldsymbol { \\beta } } \\| _ { 2 } ^ { 2 } } \\end{array}$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/6bbe45f3d0113a0cbb86e816d67e9834573c85c456540349b12e6e02c25001ba.jpg",
+        "image_caption": [
+            "Figure 2: Test Risk vs. Num. Samples for Non-Isotropic Ridge Regression in $d = 3 0$ dimensions. Unregularized regression is non-monotonic in samples, but optimally-regularized regression is monotonic. Note the optimal regularization $\\lambda$ depends on the number of samples $n$ . ",
+            "Figure 2 shows one instance of this, for a particular choice of $\\Sigma$ and $\\beta ^ { * }$ . The covariance $\\Sigma$ is diagonal, with $\\Sigma _ { i , i } = 1 0$ for $i \\leq 1 5$ and $\\Sigma _ { i , i } = 1$ for $i > 1 5$ . That is, the covariance has one “large” eigenspace and one “small” eigenspace. The ground-truth $\\beta ^ { * } = 0 . 1 \\vec { e _ { 1 } } + e _ { 3 0 } ^ {  }$ , which lies almost entirely within the “small” eigenspace of $\\Sigma$ . The noise parameter is $\\sigma = 0 . 5$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We see that unregularized regression $\\lambda = 0$ ) actually undergoes “triple descent”2 in this setting, with the first peak around $n = 1 5$ samples due to the 15-dimensional large eigenspace, and the second peak at $n = d$ . In this setting, optimally-regularized ridge regression is empirically monotonic in samples (Figure 2). Unlike the isotropic setting of Section 2, the optimal ridge parameter $\\lambda _ { n }$ is no longer a constant, but varies with number of samples $n$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Random ReLU Features. We consider random ReLU features, in the random features framework of Rahimi & Recht (2008). For a given number of features $D$ , and number of samples $n$ , the random feature classifier is obtained by performing regularized linear regression on the embedding√ $\\tilde { x } : = \\mathrm { R e L U } ( W x )$ , where $W \\in \\mathbb { R } ^ { D \\times d }$ is a matrix with each entry sampled i.i.d $\\mathcal { N } ( 0 , 1 / \\sqrt { d } )$ and ReLU applies pointwise. This is equivalent to a 2-layer fully-connected neural network with a frozen (randomly-initialized) first layer, trained with $\\ell _ { 2 }$ loss and weight decay. In Appendix A.4, we apply random features to Fashion-MNIST Xiao et al. (2017). From Appendix Figure 4a, we see that underregularized models are non-monotonic, but optimal $\\ell _ { 2 }$ regularization is monotonic in samples. Moreover, the optimal ridge parameter $\\lambda$ appears to be constant for all $n$ , similar to our results from the isotropic setting in Theorem 1. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.2 MODEL-SIZE MONOTONICITY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            262,
+            421,
+            276
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Here we empirically show that optimal $\\ell _ { 2 }$ regularization can mitigate model-wise double descent. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Random ReLU Features. We consider the same experimental setup as in Section 5.1, but now fix the number of samples $n$ , and vary the number of random features $D$ . This corresponds to varying the width of the corresponding 2-layer neural network. Figure 4b in Appendix A.4 shows the test error of the random features classifier, for $n = 5 0 0$ train samples and varying number of random features. We see that underregularized models undergo model-wise double descent, but optimal $\\ell _ { 2 }$ regularization prevents double descent. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Convolutional Neural Networks. We follow the experimental setup of Nakkiran et al. (2020) for modelwise double descent, and add varying amounts of $\\ell _ { 2 }$ regularization (weight decay). We chose the following setting from Nakkiran et al. (2020), because it exhibits double descent even with no added label noise. We consider the same family of 5-layer convolutional neural networks (CNNs) from Nakkiran et al. (2020), consisting of 4 convolutional layers of widths $[ k , 2 k , 4 k , 8 k ]$ for varying $k \\in$ $\\mathbb { N }$ . We train and test on CIFAR100 (Krizhevsky et al., 2009), an image classification problem with 100 classes. Inputs are normalized to ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/b59ffa519889838486fdb1bea9435eca85527fcc4c5ff780a6f8610321941a4b.jpg",
+        "image_caption": [
+            "Figure 3: Test Error vs. Model Size for 5-layer CNNs on CIFAR-100, with $\\ell _ { 2 }$ regularization (weight decay). Note that the optimal regularization $\\lambda$ varies with $n$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "$[ - 1 , 1 ] ^ { d }$ , and we use standard data-augmentation of random horizontal flip and random crop with 4- pixel padding. All models are trained using Stochastic Gradient Descent (SGD) on the cross-entropy loss, with step size $0 . 1 / \\sqrt { \\lfloor { T / 5 1 2 } \\rfloor + 1 }$ at step $T$ . We train for 1e6 gradient steps, and use weight decay $\\lambda$ for varying $\\lambda$ . Due to optimization instabilities for large $\\lambda$ , we use the model with the minimum train loss among the last 5K gradient steps. Figure 3 shows the test error of these models on CIFAR-100. Although unregularized and under-reguarized models exhibit double descent, the test error of optimally-regularized models is largely monotonic. Note that the optimal regularization $\\lambda$ varies with the model size — no single regularization value is optimal for all models. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 DISCUSSION AND CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            801,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this work, we study the double descent phenomenon in the context of optimal regularization. We show that, while unregularized or under-regularized models often have non-monotonic behavior, appropriate regularization can eliminate this effect. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Theoretically, we prove that for certain linear regression models with isotropic covariates, optimallytuned $\\ell _ { 2 }$ regularization achieves monotonic test performance as we grow either the sample size or the model size. These are the first non-asymptotic monotonicity results we are aware of in linear regression. We also demonstrate empirically that optimally-tuned $\\ell _ { 2 }$ regularization can mitigate double descent for more general models, including neural networks. We hope that our results can motivate future work on mitigating the double descent phenomenon, and allow us to train high performance models which do not exhibit unexpected nonmonotonic behaviors. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Open Questions. Our work suggests a number of natural open questions. First, it is open to prove (or disprove) that optimal ridge regression is sample-monotonic for non-isotropic Gaussian covariates. We conjecture that it is, and outline a potential route to proving this (via Conjectures 1 and 2 in the Appendix). Second, more broadly, it is open to prove sample-wise or model-wise monotonicity for more general (non-linear) models with appropriate regularizers. Finally, it is open to understand why large neural networks in practice are often sample-monotonic in realistic regimes of sample sizes, even without careful choice of regularization. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+        "bbox": [
+            174,
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+    {
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            297,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Section A.1 and A.2 we provide the proofs for sample-monotonicity and model-size monotonicity. In Section A.4 we include additional and omitted plots. In Section B we investigate whether monotonicity provably holds in more general models, and present a monotonicity conjecture for non-isotropic covariates. ",
+        "bbox": [
+            174,
+            137,
+            825,
+            194
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 SAMPLE MONOTONICITY PROOFS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            215,
+            452,
+            229
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "First we prove Lemma 1. ",
+        "bbox": [
+            174,
+            244,
+            338,
+            260
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proof of Lemma $^ { l }$ . For isotropic $x$ , the test risk is related to the parameter error as: ",
+        "bbox": [
+            173,
+            284,
+            715,
+            300
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c97d13ce47a347c33a9d571dff6f8d2d6723326494388228cf1b8a50d62e7ccc.jpg",
+        "text": "$$\n\\begin{array} { r l } & { R ( \\hat { \\beta } ) : = \\underset { ( x , y ) \\sim \\mathcal { D } } { \\mathbb { E } } [ ( \\langle x , \\hat { \\beta } \\rangle - y ) ^ { 2 } ] } \\\\ & { \\quad \\quad = \\underset { x \\sim \\mathcal { N } ( 0 , I _ { d } ) , \\eta \\sim \\mathcal { N } ( 0 , \\sigma ^ { 2 } ) } { \\mathbb { E } } [ ( \\langle x , \\hat { \\beta } - \\beta ^ { * } \\rangle + \\eta ) ^ { 2 } ] } \\\\ & { \\quad \\quad = | | \\hat { \\beta } - \\beta ^ { * } | | _ { 2 } ^ { 2 } + \\sigma ^ { 2 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            334,
+            308,
+            663,
+            392
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Plugging in the form of $\\hat { \\beta } _ { n , \\lambda }$ and expanding: ",
+        "bbox": [
+            176,
+            400,
+            467,
+            416
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c911b8b3710c18864e6f659d837cf52911f7d4f019ed2f3dd867f5c0d66b6396.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\bar { R } ( \\hat { \\beta } _ { n , \\lambda } ) : = \\underset { X , y \\sim \\mathcal { D } ^ { n } } { \\mathbb { E } } [ R ( \\hat { \\beta } _ { n , \\lambda } ) ] } \\\\ & { = \\mathbb { E } [ \\| \\hat { \\beta } _ { n , \\lambda } - \\beta ^ { * } \\| _ { 2 } ^ { 2 } ] + \\sigma ^ { 2 } } \\\\ & { = \\underset { X , y } { \\mathbb { E } } [ \\| ( X ^ { T } X + \\lambda I ) ^ { - 1 } X ^ { T } y - \\beta \\| _ { 2 } ^ { 2 } ] + \\sigma ^ { 2 } } \\\\ & { = \\underset { X , \\eta \\sim \\mathcal { N } ( 0 , \\sigma ^ { 2 } I _ { n } ) } { \\mathbb { E } } [ \\| ( X ^ { T } X + \\lambda I ) ^ { - 1 } X ^ { T } ( X \\beta ^ { * } + \\eta ) - \\beta ^ { * } \\| _ { 2 } ^ { 2 } ] + \\sigma ^ { 2 } } \\\\ & { = \\underset { X } { \\mathbb { E } } [ \\| ( X ^ { T } X + \\lambda I ) ^ { - 1 } X ^ { T } X \\beta ^ { * } - \\beta ^ { * } \\| _ { 2 } ^ { 2 } ] + \\underset { X , \\eta } { \\mathbb { E } } [ \\| ( X ^ { T } X + \\lambda I ) ^ { - 1 } X ^ { T } \\eta \\| _ { 2 } ^ { 2 } ] + \\sigma ^ { 2 } } \\\\ & { = \\underset { X } { \\mathbb { E } } [ \\| ( X ^ { T } X + \\lambda I ) ^ { - 1 } X ^ { T } X \\beta ^ { * } - \\beta ^ { * } \\| _ { 2 } ^ { 2 } ] + \\sigma ^ { 2 } \\underset { X } { \\mathbb { E } } [ \\| ( X ^ { T } X + \\lambda I ) ^ { - 1 } X ^ { T } \\| _ { F } ^ { 2 } ] + \\sigma ^ { 2 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            230,
+            424,
+            764,
+            590
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Now let $X \\ = \\ U \\Sigma V ^ { T }$ be the full singular value decomposition of $X$ , with $U \\ \\in \\ \\mathbb { R } ^ { n \\times n } , \\Sigma \\ \\in$ $\\mathbb { R } ^ { n \\times d } , V \\in \\mathbb { R } ^ { d \\times d }$ . Let $( \\gamma _ { 1 } , \\ldots \\gamma _ { d } )$ denote the singular values, defining $\\gamma _ { i } = 0$ for $i > \\operatorname* { m i n } ( n , d )$ . Then, continuing: ",
+        "bbox": [
+            174,
+            598,
+            825,
+            642
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/040b59fb6e2e081b3c86b7cc23e606f3f2af2702a88eacaab6aa38b59506e573.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\overline { { R } } ( \\hat { \\beta } _ { n , \\lambda } ) = \\underset { V , \\Sigma } { \\mathbb { E } } [ | | \\mathrm { d i a g } ( \\{ \\frac { - \\lambda } { \\gamma _ { i } ^ { 2 } + \\lambda } \\} ) V ^ { T } \\beta ^ { * } | | _ { 2 } ^ { 2 } ] + \\sigma ^ { 2 } \\underset { \\Sigma } { \\mathbb { E } } [ \\sum _ { i } \\frac { \\gamma _ { i } ^ { 2 } } { ( \\gamma _ { i } ^ { 2 } + \\lambda ) ^ { 2 } } ] + \\sigma ^ { 2 } } \\\\ & { \\quad \\quad = \\underset { z \\sim \\mathrm { U n i f } ( 1 ) \\frac { \\mathbb { E } } { \\delta } } { \\underbrace { \\mathrm { U } \\mathrm { E } } } [ | | \\mathrm { d i a g } ( \\{ \\frac { - \\lambda } { \\gamma _ { i } ^ { 2 } + \\lambda } \\} ) z | | _ { 2 } ^ { 2 } ] + \\sigma ^ { 2 } \\underset { \\Sigma } { \\mathbb { E } } [ \\sum _ { i } \\frac { \\gamma _ { i } ^ { 2 } } { ( \\gamma _ { i } ^ { 2 } + \\lambda ) ^ { 2 } } ] + \\sigma ^ { 2 } } \\\\ & { \\quad \\quad = \\frac { | | \\beta ^ { * } | | _ { 2 } ^ { 2 } } { d } \\underset { \\Sigma } { \\mathbb { E } } [ \\sum _ { i } \\frac { \\lambda ^ { 2 } } { ( \\gamma _ { i } ^ { 2 } + \\lambda ) ^ { 2 } } ] + \\sigma ^ { 2 } \\underset { \\Sigma } { \\mathbb { E } } [ \\sum _ { i } \\frac { \\gamma _ { i } ^ { 2 } } { ( \\gamma _ { i } ^ { 2 } + \\lambda ) ^ { 2 } } ] + \\sigma ^ { 2 } } \\\\ & { \\quad \\quad = \\underset { \\Sigma } { \\mathbb { E } } [ \\sum _ { i } \\frac { | | \\beta ^ { * } | | _ { 2 } ^ { 2 } \\lambda ^ { 2 } / d + \\sigma ^ { 2 } \\gamma _ { i } ^ { 2 } } { ( \\gamma _ { i } ^ { 2 } + \\lambda ) ^ { 2 } } ] + \\sigma ^ { 2 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            205,
+            650,
+            764,
+            810
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Line (10) follows because by symmetry, the distribution of $V$ is a uniformly random orthonormal matrix, and $\\Sigma$ is independent of $V$ . Thus, $z : = V ^ { T } \\beta ^ { * }$ is distributed as a uniformly random point on the unit sphere of radius $| | \\beta ^ { * } | | _ { 2 }$ . ",
+        "bbox": [
+            174,
+            818,
+            823,
+            861
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Next we prove Lemma 2. ",
+        "bbox": [
+            174,
+            909,
+            341,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proof of Lemma 2. First, we determine the optimal ridge parameter. Using Lemma 1, we have ",
+        "bbox": [
+            169,
+            102,
+            792,
+            119
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a623c8e79a24fdd865da11d4fdff60646c3d940ef3f190067fd57aa38bac12da.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\frac { \\partial } { \\partial \\lambda } \\overline { { R } } ( \\hat { \\beta } _ { n , \\lambda } ) = \\frac { \\partial } { \\partial \\lambda } _ { ( \\gamma _ { 1 } , \\dots \\gamma _ { d } ) \\sim \\Gamma } \\left[ \\displaystyle \\sum _ { i } \\frac { | | \\beta | | _ { 2 } ^ { 2 } \\lambda ^ { 2 } / d + \\sigma ^ { 2 } \\gamma _ { i } ^ { 2 } } { ( \\gamma _ { i } ^ { 2 } + \\lambda ) ^ { 2 } } \\right] } \\\\ & { = 2 ( | | \\beta ^ { * } | | _ { 2 } ^ { 2 } \\lambda / d - \\sigma ^ { 2 } ) _ { ( \\underbrace { ( \\gamma _ { 1 } , \\dots \\gamma _ { d } ) \\sim \\Gamma } _ { > 0 } } \\left[ \\displaystyle \\sum _ { i } \\frac { \\gamma _ { i } ^ { 2 } } { ( \\gamma _ { i } ^ { 2 } + \\lambda ) ^ { 3 } } \\right] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            127,
+            686,
+            233
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Thus, $\\begin{array} { r } { \\frac { \\partial } { \\partial \\lambda } \\overline { { R } } ( \\widehat { \\beta } _ { n , \\lambda } ) = 0 \\implies \\lambda = \\frac { d \\sigma ^ { 2 } } { | | \\beta ^ { * } | | _ { 2 } ^ { 2 } } } \\end{array}$ and we conclude that $\\begin{array} { r } { \\lambda _ { n } ^ { \\mathrm { o p t } } = \\frac { d \\sigma ^ { 2 } } { | | \\beta ^ { * } | | _ { 2 } ^ { 2 } } } \\end{array}$ ",
+        "bbox": [
+            174,
+            241,
+            673,
+            265
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For this optimal parameter, the test risk follows from Lemma 1 as ",
+        "bbox": [
+            173,
+            272,
+            606,
+            287
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/48d72fd4dc42f61ccda5d47fc0360cd5f8963fbe8f3f90da572f6323560f904d.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\overline { { R } } ( \\hat { \\beta } _ { n } ^ { \\mathrm { o p t } } ) = \\overline { { R } } ( \\hat { \\beta } _ { n , \\lambda _ { n } ^ { \\mathrm { o p t } } } ) } \\\\ { \\displaystyle = \\frac { \\mathbb { E } } { ( \\gamma _ { 1 } , \\dots \\gamma _ { d } ) \\sim \\Gamma _ { n } } \\left[ \\sum _ { i = 1 } ^ { d } \\frac { \\sigma ^ { 2 } } { \\gamma _ { i } ^ { 2 } + d \\sigma ^ { 2 } / | | \\beta ^ { * } | | _ { 2 } ^ { 2 } } \\right] + \\sigma ^ { 2 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            295,
+            653,
+            364
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Proof of Theorem 2. We follow a similar proof strategy as in Theorem 1: we invoke singular value interlacing $( \\widetilde { \\gamma } _ { i } \\geq \\gamma _ { i } )$ for the data matrix when adding a single sample. We then apply Lemma 1 to eargue that the test risk varies monotonically with the singular values. ",
+        "bbox": [
+            173,
+            412,
+            825,
+            455
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We have ",
+        "bbox": [
+            174,
+            465,
+            233,
+            479
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/de3244b98b022f93b1ca9519f7631fa81e4e9af584f4d8319f85bbc231b1557d.jpg",
+        "text": "$$\n{ \\overline { { R } } } ( \\hat { \\beta } _ { n , \\lambda } ) = \\operatorname * { \\mathbb { E } } _ { ( \\gamma _ { 1 } , \\dots \\gamma _ { d } ) \\sim \\Gamma } [ \\sum _ { i } \\underbrace { | | \\beta ^ { * } | | _ { 2 } ^ { 2 } \\lambda ^ { 2 } / d + \\sigma ^ { 2 } \\gamma _ { i } ^ { 2 } } _ { ( \\gamma _ { i } ^ { 2 } + \\lambda ) ^ { 2 } } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
+            478,
+            660,
+            535
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "and we compute how each term in the sum varies with $\\gamma _ { i }$ : ",
+        "bbox": [
+            173,
+            541,
+            553,
+            556
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/083dc399d19565db2a21c827567743ee4f98ee2e1b3dd3f08e16f076c93b48f9.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle { \\frac { \\partial } { \\partial \\gamma _ { i } } \\sum _ { i } S ( \\gamma _ { i } ) = \\frac { \\partial } { \\partial \\gamma _ { i } } S ( \\gamma _ { i } ) } } \\\\ { \\displaystyle { \\quad = ( \\frac { - 2 \\gamma _ { i } } { d } ) \\frac { 2 | | \\beta ^ { * } | | _ { 2 } ^ { 2 } \\lambda ^ { 2 } + d \\sigma ^ { 2 } ( \\gamma _ { i } ^ { 2 } - \\lambda ) } { ( \\gamma _ { i } ^ { 2 } + \\lambda ) ^ { 3 } } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            564,
+            671,
+            640
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Thus we have ",
+        "bbox": [
+            173,
+            648,
+            267,
+            662
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/56481a4012179ea03ab8bd2ae7aed121ef63b3b0de4415a7172237c75670635e.jpg",
+        "text": "$$\n\\lambda \\geq \\frac { d \\sigma ^ { 2 } } { 2 \\vert \\vert \\beta ^ { * } \\vert \\vert ^ { 2 } } \\implies \\frac { \\partial } { \\partial \\gamma _ { i } } S ( \\gamma _ { i } ) \\leq 0\n$$",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            669,
+            614,
+            705
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "By the coupling argument in Theorem 1, this implies that the test risk is monotonic: ",
+        "bbox": [
+            174,
+            713,
+            722,
+            728
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e72e37a8203a6e75f3017eb409e9edb1d55d482ed226ed8392499a42f1c091c7.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\overline { { R } } ( \\hat { \\beta } _ { n + 1 , \\lambda } ) - \\overline { { R } } ( \\hat { \\beta } _ { n , \\lambda } ) } \\\\ & { \\quad = \\underset { ( \\widetilde { \\gamma } _ { 1 } , \\dots , \\widetilde { \\gamma } _ { d } ) \\sim \\Gamma _ { n + 1 } } { \\mathbb { E } } \\left[ \\displaystyle \\sum _ { i = 1 } ^ { d } S ( \\widetilde { \\gamma } _ { i } ) \\right] - \\underset { ( \\gamma _ { 1 } , \\dots , \\gamma _ { d } ) \\sim \\Gamma _ { n } } { \\mathbb { E } } \\left[ \\displaystyle \\sum _ { i = 1 } ^ { d } S ( \\gamma _ { i } ) \\right] } \\\\ & { \\quad = \\underset { ( \\{ \\widetilde { \\gamma } _ { i } \\} , \\{ \\gamma _ { i } \\} ) \\sim \\Pi } { \\mathbb { E } } \\left[ \\displaystyle \\sum _ { i = 1 } ^ { d } S ( \\widetilde { \\gamma } _ { i } ) - S ( \\gamma _ { i } ) \\right] } \\\\ & { \\quad \\le 0 } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            757,
+            692,
+            887
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where $\\Pi$ is the coupling. Line (17) follows from Equation (15), and the fact that the coupling obeys $\\widetilde { \\gamma } _ { i } \\geq \\gamma _ { i }$ . □ ",
+        "bbox": [
+            173,
+            895,
+            826,
+            925
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 PROJECTION MODEL PROOFS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            423,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3. For all $\\theta \\in \\mathbb { R } ^ { p }$ , $d , n \\in \\mathbb { N } ,$ , and $\\lambda > 0 ;$ , let $\\ b { X } \\in \\mathbb { R } ^ { n \\times p }$ be a matrix with i.i.d. $\\mathcal { N } ( 0 , 1 )$ entries. Let $P \\in \\mathbb { R } ^ { d \\times p }$ be a random orthonormal matrix. Define $\\widetilde { X } : = X P ^ { T }$ . Let $\\left( \\gamma _ { 1 } , \\dots , \\gamma _ { m } \\right)$ be the singular values of the data matrix $\\tilde { X } \\in \\mathbb { R } ^ { n \\times d }$ , for $m : = \\operatorname* { m a x } ( n , d )$ (with $\\gamma _ { i } ~ = ~ 0$ for $i > \\operatorname* { m i n } ( n , d ) )$ . Let $\\Gamma _ { d }$ be the distribution of singular values $\\left( \\gamma _ { 1 } , \\ldots , \\gamma _ { m } \\right)$ . ",
+        "bbox": [
+            173,
+            128,
+            825,
+            193
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Then, the optimal ridge parameter is constant for all $d$ : $\\begin{array} { r } { \\lambda _ { d } ^ { \\mathrm { o p t } } = \\frac { p ^ { 2 } \\widetilde \\sigma ^ { 2 } } { d | | \\theta | | _ { 2 } ^ { 2 } } } \\end{array}$ . where we define $\\widetilde { \\sigma } ^ { 2 } : = \\ :$ $\\begin{array} { r } { \\sigma ^ { 2 } + \\frac { p - d } { p } | | \\boldsymbol { \\theta } | | _ { 2 } ^ { 2 } } \\end{array}$ . Moreover, the optimal expected test risk can be written as ",
+        "bbox": [
+            173,
+            196,
+            826,
+            239
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/468fe85c9587c5a5c7cbe90d3a3b9b112d10447b03fa0a4f7669fc3630f11d15.jpg",
+        "text": "$$\n\\overline { { R } } ( \\hat { \\beta } _ { d } ^ { \\mathrm { o p t } } ) = \\widetilde { \\sigma } ^ { 2 } + \\underset { ( \\gamma _ { 1 } , \\ldots , \\gamma _ { m } ) \\sim \\Gamma _ { d } } { \\mathbb { E } } \\left[ \\sum _ { i = 1 } ^ { p } \\frac { \\widetilde { \\sigma } ^ { 2 } } { \\gamma _ { i } ^ { 2 } + \\frac { \\widetilde { \\sigma } ^ { 2 } p ^ { 2 } } { d | | \\theta | | _ { 2 } ^ { 2 } } } \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            246,
+            663,
+            297
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Proof. This proof follows exactly analogously as the proof of Lemma 2 from Lemma 1, in Section A.1. □ ",
+        "bbox": [
+            173,
+            310,
+            821,
+            339
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Lemma 4. For all $\\theta \\in \\mathbb { R } ^ { p }$ , $d , n \\in \\mathbb { N } ,$ , and $\\lambda > 0$ , let $\\ b { X } \\in \\mathbb { R } ^ { n \\times p }$ be a matrix with i.i.d. $\\mathcal { N } ( 0 , 1 )$   \nentries. Let $P \\in \\mathbb { R } ^ { d \\times p }$ be a random orthonormal matrix. Define $\\widetilde X : = X P ^ { T }$ and $\\beta ^ { * } : = P \\theta$ . ",
+        "bbox": [
+            171,
+            347,
+            823,
+            380
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Let $\\left( \\gamma _ { 1 } , \\dots , \\gamma _ { m } \\right)$ be the singular values of the data matrix $\\tilde { X } \\in \\mathbb { R } ^ { n \\times d }$ , for $m : = \\operatorname* { m a x } ( n , d )$ (with $\\gamma _ { i } = 0$ for $i > \\operatorname* { m i n } ( n , d ) )$ . Let $\\Gamma _ { d }$ be the distribution of singular values $\\left( \\gamma _ { 1 } , \\dots , \\gamma _ { m } \\right)$ . ",
+        "bbox": [
+            173,
+            386,
+            825,
+            417
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Then, the expected test risk is ",
+        "bbox": [
+            174,
+            422,
+            369,
+            438
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/741d2372fad09c0cbce14bcf5839b8447076a4cbcd48b57657173a4429b1c348.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\overline { { R } } ( \\hat { \\beta } _ { d , \\lambda } ) : = \\underset { P } { \\mathbb { E } } \\underset { \\widetilde { X } , \\widetilde { y } \\sim \\mathcal { D } ^ { n } } { \\mathbb { E } } [ R _ { P } ( \\hat { \\beta } _ { d , \\lambda } ( \\widetilde { X } , \\overrightarrow { y } ) ] } \\\\ & { = \\sigma ^ { 2 } + ( 1 - \\frac { d } { p } ) | | \\theta | | _ { 2 } ^ { 2 } } \\\\ & { + \\underset { ( \\gamma _ { 1 } , \\dotsc , \\gamma _ { m } ) \\sim \\Gamma _ { d } } { \\mathbb { E } } \\left[ \\underset { i = 1 } { \\overset { p } { \\sum } } \\frac { ( \\sigma ^ { 2 } + \\frac { p - d } { p } | | \\theta | | _ { 2 } ^ { 2 } ) \\gamma _ { i } ^ { 2 } + \\frac { d } { p ^ { 2 } } | | \\theta | | _ { 2 } ^ { 2 } \\lambda ^ { 2 } } { ( \\gamma _ { i } ^ { 2 } + \\lambda ) ^ { 2 } } \\right] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            444,
+            684,
+            554
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Proof of Lemma 4. We first define the parameter that minimizes the population risk. It follows directly that: ",
+        "bbox": [
+            176,
+            566,
+            823,
+            594
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/35bea86cddbd631fb6b4870684bfe035967b6d8211cf49b4cb25e93f8ad7fe5c.jpg",
+        "text": "$$\n\\beta _ { P } ^ { * } : = \\underset { \\beta \\in \\mathbb { R } ^ { d } } { \\mathrm { a r g m i n } } R _ { P } ( \\beta ) = P \\theta\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            593,
+            596,
+            621
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "First, we can expand the risk as ",
+        "bbox": [
+            173,
+            632,
+            382,
+            647
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/03d187ef1b8071dac932f45747b3294d7774fd0b3adde9b08928e633c8afcb6b.jpg",
+        "text": "$$\n\\begin{array} { r l } & { R ( \\hat { \\beta } ) = \\underset { ( \\bar { x } , y ) \\sim \\mathcal { D } } { \\mathbb { E } } [ ( \\langle P x , \\hat { \\beta } \\rangle - y ) ^ { 2 } ] } \\\\ & { \\quad = \\underset { ( \\bar { x } , y ) \\sim \\mathcal { D } , \\eta \\sim \\mathcal { N } ( 0 , \\sigma ^ { 2 } ) } { \\mathbb { E } } [ ( \\langle x , P ^ { T } \\hat { \\beta } - \\theta \\rangle + \\eta ) ^ { 2 } ] } \\\\ & { \\quad = \\sigma ^ { 2 } + \\vert \\vert \\theta - P ^ { T } \\hat { \\beta } \\vert \\vert _ { 2 } ^ { 2 } } \\\\ & { \\quad = \\sigma ^ { 2 } + \\vert \\vert \\theta - P ^ { T } \\beta ^ { * } \\vert \\vert _ { 2 } ^ { 2 } + \\vert \\vert P ^ { T } \\beta ^ { * } - P ^ { T } \\hat { \\beta } \\vert \\vert _ { 2 } ^ { 2 } } \\\\ & { \\quad + 2 \\langle ( \\theta - P ^ { T } \\beta ^ { * } ) , P ^ { T } \\beta ^ { * } - P ^ { T } \\hat { \\beta } \\rangle } \\\\ & { \\quad = \\sigma ^ { 2 } + \\vert \\vert \\theta - P ^ { T } \\beta ^ { * } \\vert \\vert _ { 2 } ^ { 2 } + \\vert \\vert P ^ { T } \\beta ^ { * } - P ^ { T } \\hat { \\beta } \\vert \\vert _ { 2 } ^ { 2 } } \\\\ & { \\quad = \\sigma ^ { 2 } + \\vert \\vert \\vert \\theta - P ^ { T } P \\theta \\vert \\vert _ { 2 } ^ { 2 } + \\vert \\vert \\beta ^ { * } - \\hat { \\beta } \\vert \\vert _ { 2 } ^ { 2 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            652,
+            665,
+            823
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The cross terms in Line (22) vanish because the first-order optimality condition for $\\beta ^ { * }$ implies that $\\beta ^ { * }$ satisfies $P ( \\theta ^ { * } - P ^ { T } \\beta ^ { * } ) = 0$ . We now simplify each of the two remaining terms. ",
+        "bbox": [
+            173,
+            833,
+            825,
+            863
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "First, we have that: ",
+        "bbox": [
+            173,
+            869,
+            300,
+            883
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4669a600f5dddc038e7c628ec71b91a5531e739c9c42893ae9b17ddadb44a8bd.jpg",
+        "text": "$$\n\\underset { P } { \\mathbb { E } } | | \\theta - P ^ { T } P \\theta | | _ { 2 } ^ { 2 } = ( 1 - \\frac { d } { p } ) | | \\theta | | _ { 2 } ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            888,
+            611,
+            922
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "since $P ^ { T } P$ is an orthogonal projection onto a random $d$ -dimensional subspace. ",
+        "bbox": [
+            173,
+            102,
+            692,
+            119
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Now, recall we have ${ \\vec { y } } = X \\theta + \\eta$ where $\\eta \\sim \\mathcal { N } ( 0 , \\sigma ^ { 2 } I _ { n } )$ . Expand this as: ",
+        "bbox": [
+            174,
+            126,
+            660,
+            143
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c1a9b8472650c02f7455387333ca5dfa7f0349357d7a1ca1091251cf1e45a972.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\vec { y } = X \\theta + \\eta } } \\\\ { { \\ = X P ^ { T } P \\theta + X ( 1 - P ^ { T } P ) \\theta + \\eta } } \\\\ { { \\ = \\widetilde { X } \\beta ^ { * } + \\varepsilon + \\eta } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            174,
+            620,
+            236
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "where $\\varepsilon : = X ( 1 - P ^ { T } P ) \\theta$ . Note that conditioned on $P$ , the three terms $\\widetilde { X } , \\varepsilon$ and $\\eta$ are conditionally independent, since $P ^ { T } P$ and $( I - P ^ { T } P )$ project $X$ onto orthogonal subspaces. And further, $\\varepsilon \\sim$ $\\mathcal { N } ( \\dot { 0 _ { \\cdot } } | | ( 1 - P ^ { T } P ) \\theta | | ^ { 2 } I _ { n } )$ . ",
+        "bbox": [
+            173,
+            244,
+            825,
+            291
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/60e0378b64b76b4eab2103bf6418c25dfbf2648197e5f3b7ca1f73f584e033df.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\frac { \\mathbb { E } } { P } \\underset { \\tilde { X } , y } { \\mathbb { E } } | | \\hat { \\boldsymbol { \\beta } } - \\boldsymbol { \\beta } ^ { * } | | _ { 2 } ^ { 2 } } \\\\ & { = \\underset { P } { \\mathbb { E } } \\underset { \\tilde { X } , y } { \\mathbb { E } } | | ( \\widetilde { X } ^ { T } \\widetilde { X } + \\lambda I ) ^ { - 1 } \\widetilde { X } ^ { T } \\boldsymbol { y } - \\boldsymbol { \\beta } ^ { * } | | _ { 2 } ^ { 2 } } \\\\ & { = \\underset { P } { \\mathbb { E } } \\underset { \\tilde { X } , y , \\varepsilon , \\eta } { \\mathbb { E } } | | ( \\widetilde { X } ^ { T } \\widetilde { X } + \\lambda I ) ^ { - 1 } \\widetilde { X } ^ { T } ( \\widetilde { X } \\boldsymbol { \\beta } ^ { * } + \\varepsilon + \\eta ) - \\boldsymbol { \\beta } ^ { * } | | _ { 2 } ^ { 2 } } \\\\ & { = \\underset { P } { \\overset { \\mathbb { E } } { \\mathbb { X } } } \\underset { y , \\varepsilon , \\eta } { \\mathbb { E } } | | ( \\widetilde { X } ^ { T } \\widetilde { X } + \\lambda I ) ^ { - 1 } \\widetilde { X } ^ { T } \\widetilde { X } \\boldsymbol { \\beta } ^ { * } - \\boldsymbol { \\beta } ^ { * } | | _ { 2 } ^ { 2 } } \\\\ & { = \\underset { P } { \\overset { \\mathbb { E } } { \\mathbb { X } } } \\underset { y , \\varepsilon , \\eta } { \\mathbb { E } } | | ( \\widetilde { X } ^ { T } \\widetilde { X } + \\lambda I ) ^ { - 1 } \\widetilde { X } ^ { T } \\widetilde { X } \\boldsymbol { \\beta } ^ { * } - \\boldsymbol { \\beta } ^ { * } | | _ { 2 } ^ { 2 } } \\\\ & { + \\| ( \\widetilde { X } ^ { T } \\widetilde { X } + \\lambda I ) ^ { - 1 } \\widetilde { X } ^ { T } \\varepsilon | | _ { 2 } ^ { 2 } + | | ( \\widetilde { X } ^ { T } \\widetilde { X } + \\lambda I ) ^ { - 1 } \\widetilde { X } ^ { T } \\eta | | _ { 2 } ^ { 2 } ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            308,
+            319,
+            691,
+            472
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Now, since $\\widetilde { X }$ is conditionally independent of $\\varepsilon$ conditioned on $P$ , ",
+        "bbox": [
+            173,
+            507,
+            607,
+            525
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/15570c750f46fbfdd10667eca45b3e9a24a57fe46a295a8570217d0ece4d3a0e.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\frac { \\mathbb { E } } { F } \\underset { \\tilde { X } , y , \\varepsilon | P } { \\mathbb { E } } \\ \\lVert ( \\tilde { X } ^ { T } \\tilde { X } + \\lambda I ) ^ { - 1 } \\tilde { X } ^ { T } \\varepsilon \\rVert _ { 2 } ^ { 2 } } \\\\ & { = \\mathbb { E } \\ \\underset { \\tilde { Y } } { \\mathbb { E } } \\ [ \\lVert ( \\tilde { X } ^ { T } \\tilde { X } + \\lambda I ) ^ { - 1 } \\tilde { X } ^ { T } \\rVert _ { F } ^ { 2 } \\ \\underset { \\varepsilon | P } { \\mathbb { E } } [ \\lVert \\varepsilon \\rVert _ { 2 } ^ { 2 } ] } \\\\ & { = \\frac { \\mathbb { E } } { \\frac { \\mathbb { E } } { \\tilde { X } } } \\lVert ( \\tilde { X } ^ { T } \\tilde { X } + \\lambda I ) ^ { - 1 } \\tilde { X } ^ { T } \\rVert _ { F } ^ { 2 } \\ \\underset { \\varepsilon , \\varepsilon } { \\mathbb { E } } [ \\lVert \\varepsilon \\rVert _ { 2 } ^ { 2 } ] } \\\\ & { = \\underset { \\tilde { X } } { \\mathbb { E } } [ \\lVert ( \\tilde { X } ^ { T } \\tilde { X } + \\lambda I ) ^ { - 1 } \\tilde { X } ^ { T } \\rVert _ { F } ^ { 2 } ] \\cdot \\underset { { P } , X } { \\mathbb { E } } [ \\lVert X ( 1 - P ^ { T } P ) \\theta \\rVert _ { 2 } ^ { 2 } ] } \\\\ & { = \\underset { \\tilde { X } } { \\mathbb { E } } [ \\lVert ( \\tilde { X } ^ { T } \\tilde { X } + \\lambda I ) ^ { - 1 } \\tilde { X } ^ { T } \\rVert _ { F } ^ { 2 } ] ( \\frac { p - d } { p } | \\theta \\rVert _ { 2 } ^ { 2 } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            250,
+            531,
+            627,
+            694
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "where Line (37) holds because the marginal distribution of $\\widetilde { X }$ does not depend on $P$ . ",
+        "bbox": [
+            173,
+            703,
+            728,
+            722
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Similarly, ",
+        "bbox": [
+            173,
+            729,
+            240,
+            744
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/27105b3c05a2ab3257c0e5570b44af98c9a9c874e94dd0667e9c9edeff10bd14.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\frac { \\mathbb { E } } { P } \\underset { \\widetilde { X } , y , \\eta | P } { \\mathbb { E } } | | ( \\widetilde { X } ^ { T } \\widetilde { X } + \\lambda I ) ^ { - 1 } \\widetilde { X } ^ { T } \\eta | | _ { 2 } ^ { 2 } } \\\\ & { = \\sigma ^ { 2 } \\underset { \\widetilde { X } } { \\mathbb { E } } [ | | ( \\widetilde { X } ^ { T } \\widetilde { X } + \\lambda I ) ^ { - 1 } \\widetilde { X } ^ { T } | | _ { F } ^ { 2 } ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            380,
+            752,
+            619,
+            815
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Now let $\\tilde { \\cal X } \\ = \\ { \\cal U } \\Sigma { \\cal V } ^ { T }$ be the full singular value decomposition of $\\widetilde { X }$ , with $U \\ \\in \\ \\mathbb { R } ^ { n \\times n } , \\Sigma \\ \\in$ $\\mathbb { R } ^ { n \\times d } , V \\in \\mathbb { R } ^ { d \\times d }$ . Let $( \\gamma _ { 1 } , \\dots \\gamma _ { m } )$ denote the singular values, where $m = \\operatorname* { m a x } ( n , d )$ and defining $\\gamma _ { i } = 0$ for $i > \\operatorname* { m i n } ( n , d )$ . ",
+        "bbox": [
+            173,
+            835,
+            825,
+            881
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Observe that by symmetry, $\\widetilde { X } = X P ^ { T }$ and $P$ are independent, because the joint distribution $( \\widetilde { X } , P )$ is equivalent to the distribution $( \\widetilde { X } Q , P Q )$ for a random orthonormal $Q \\in \\mathbb { R } ^ { p \\times p }$ . Thus $\\widetilde { X }$ and ",
+        "bbox": [
+            173,
+            890,
+            825,
+            925
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "$\\beta ^ { * } = P \\theta$ are also independent, and we have: ",
+        "bbox": [
+            174,
+            103,
+            468,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2c5cf2c29a1457b5e11f9237a8780aec3ba4175cc252997049b6d706d1666b43.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } \\mathbb { E } \\| \\{ \\hat { X } ^ { \\prime } \\} \\hat { X } + \\lambda \\mathcal { I } \\} ^ { - 1 } \\hat { X } ^ { \\prime } \\hat { X } ^ { \\prime } - \\delta ^ { * } \\} \\| _ { 2 } ^ { 2 } } \\\\ & { = \\frac { 1 } { b } \\mathbb { E } \\hat { X } ^ { \\prime } \\| \\mathrm { H i s s } ( \\hat { \\mathcal { Z } } _ { \\frac { e } { 2 } } ^ { - 1 } \\lambda ) b ^ { \\mathcal { N } / 2 } \\| _ { 2 } ^ { 2 } } \\\\ & { = \\mathbb { E } \\mathbb { E } \\mathbb { E } \\mathbb { E } \\mathbb { E } \\bigg [ \\| \\mathrm { H i s s } ( \\hat { \\mathcal { Z } } _ { \\frac { e } { 2 } } ^ { - 1 } \\lambda ) \\| ^ { \\mathcal { N } / 2 } b ^ { \\mathcal { N } / 2 } \\| _ { 2 } ^ { 2 } } \\\\ & { - \\mathbb { E } \\mathbb { E } \\mathbb { E } \\mathbb { E } \\bigg [ \\| \\mathrm { H i s s } ( \\hat { \\mathcal { Z } } _ { \\frac { e } { 2 } } ^ { - 1 } \\lambda ) \\| ^ { \\mathcal { N } / 2 } \\bigg ] \\| _ { 2 } ^ { 2 } } \\\\ & { = \\mathbb { E } _ { \\hat { X } ^ { \\prime } \\sim \\mathrm { N } \\times \\mathrm { W } ( \\hat { \\mathcal { Z } } ) } \\mathbb { E } \\| \\mathrm { H i s s } ( \\hat { \\mathcal { Z } } _ { \\frac { e } { 2 } } ^ { - 1 } \\lambda ) \\| ^ { \\mathcal { N } / 2 } } \\\\ & { = \\mathbb { E } \\mathbb { E } \\| \\hat { X } ^ { \\prime } \\| ^ { 2 } \\mathbb { E } \\| \\mathrm { H } ^ { 2 } \\mathbb { E } \\bigg [ \\mathrm { H } _ { \\frac { e } { 2 } } ^ { - 1 } \\lambda \\Big ] ^ { \\| \\mathcal { N } / 2 } } \\\\ & { = \\frac { 1 } { b } \\mathbb { E } \\hat { X } ^ { \\prime } \\| ^ { 2 } \\mathbb { E } \\bigg [ \\mathcal { P } _ { \\frac { e } { 2 } } ^ { - 1 } \\frac { \\lambda ^ { 2 } } { b } \\Big ] ^ { 2 } } \\\\ &  = \\frac { 1 } { b } \\mathbb { E } \\frac { \\hat { X } ^ { \\prime } } { \\| \\mathcal { P } \\| \\mathcal { P } \\| ^ { 2 } \\| _ { 2 } ^ { 2 } } \\mathbb { E } \\bigg [ \\int _ { 0 } ^ { \\mathcal { N } / 2 } \\frac { \\lambda ^ { 2 } }  \\| \\mathcal { P } \\| _ { 2 } ^ { 2 } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
+            125,
+            663,
+            383
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Finally, continuing from Line (33), we can use Lines (38), (40), and (47) to write: ",
+        "bbox": [
+            173,
+            400,
+            712,
+            416
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/afc96577bb1030ba4580be2296ccfb837633a4009485fbd8ae0a3eea348ab376.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\frac { \\mathbb { E } } { \\rho } \\frac { \\| \\tilde { \\rho } - \\tilde { \\rho } ^ { - 1 } \\| ^ { 2 } } { \\mathcal { F } _ { x , y } } } \\\\ & { = ( \\sigma ^ { 2 } + \\frac { \\rho - \\tilde { \\rho } - d } { p } \\| \\theta \\| _ { 2 } ^ { 2 } \\frac { \\mathbb { E } \\| ( \\tilde { X } ^ { \\mathcal { T } } \\tilde { X } + \\lambda I ) ^ { - 1 } \\tilde { X } ^ { \\mathcal { T } } \\| _ { \\mathcal { F } _ { x } ^ { 1 } } ^ { 2 } } { \\mathcal { F } _ { x } ^ { 1 } } } \\\\ & { + \\frac { d } { \\gamma } \\| \\theta \\| _ { 2 } ^ { 2 } \\frac { \\mathbb { E } } { \\varepsilon } \\| \\sum _ { i } ^ { 2 } \\frac { \\lambda ^ { 2 } } { ( \\sqrt { d } ^ { 2 } + \\lambda I ) ^ { 2 } } | } \\\\ & { = ( \\sigma ^ { 2 } + \\frac { p - d } { p } \\| \\theta \\| _ { 2 } ^ { 2 } ) \\frac { \\mathbb { E } } { \\Sigma } \\frac { \\gamma ^ { 2 } } { \\varepsilon } \\langle \\frac { \\eta _ { i } ^ { 2 } } { ( \\sqrt { d } ^ { 2 } + \\lambda I ) ^ { 2 } } | } \\\\ & { + \\frac { d } { \\gamma ^ { 2 } } \\| \\theta \\| _ { 2 } ^ { 2 } \\frac { \\mathbb { E } } { \\Sigma } \\frac { \\lambda ^ { 2 } } { \\langle \\frac { \\eta _ { i } ^ { 2 } + \\lambda I \\rangle ^ { 2 } } { \\varepsilon } } | } \\\\ &  = \\frac { \\mathbb { E } } { \\nu } \\langle \\frac { ( \\sigma ^ { 2 } + \\frac { p - d - \\tilde { \\rho } - \\tilde { \\rho } - \\tilde { \\rho } - \\tilde { \\rho } } { i } ) | \\theta \\rangle _ { 2 } ^ { 2 } \\sqrt { \\frac { \\lambda ^ { 2 } } { \\varepsilon } + \\frac { d } { \\gamma ^ { 2 } } | \\theta | _ { 2 } ^ { 2 } \\lambda ^ { 2 } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            330,
+            424,
+            666,
+            656
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Now, we can continue from Line (24), and apply lines (25), to conclude: ",
+        "bbox": [
+            173,
+            672,
+            647,
+            688
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/de85f54a0f8d77f7a08c992950801522d81c9da4a94877208449fe5367012368.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } \\big [ R ( \\hat { \\beta } ) \\big ] = \\sigma ^ { 2 } + \\mathbb { E } [ | | \\theta - P ^ { T } P \\theta | | _ { 2 } ^ { 2 } ] + \\mathbb { E } [ | | \\beta ^ { * } - \\hat { \\beta } | | _ { 2 } ^ { 2 } ] } \\\\ & { \\qquad = \\sigma ^ { 2 } + ( 1 - \\frac { d } { p } ) | | \\theta | | _ { 2 } ^ { 2 } } \\\\ & { \\qquad + \\mathbb { E } \\left[ \\displaystyle \\sum _ { i = 1 } ^ { p } \\frac { \\big ( \\sigma ^ { 2 } + \\frac { p - d } { p } | | \\theta | | _ { 2 } ^ { 2 } \\big ) \\gamma _ { i } ^ { 2 } + \\frac { d } { p ^ { 2 } } | | \\theta | | _ { 2 } ^ { 2 } \\lambda ^ { 2 } } { \\big ( \\gamma _ { i } ^ { 2 } + \\lambda \\big ) ^ { 2 } } \\right] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            717,
+            674,
+            818
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Proof of Theorem 3. This follows analogously to the proof of Theorem 1, Let $\\widetilde { X } _ { d }$ and $\\widetilde { X } _ { d + 1 }$ be the observed data matrices for $d$ and $d + 1$ e emodel size. As in Theorem 1, there exists a coupling $\\Pi$ between the distributions $\\Gamma _ { d }$ and $\\Gamma _ { d + 1 }$ of the singular values of $\\widetilde { X } _ { d }$ and $\\widetilde { X } _ { d + 1 }$ such that these singular values are interlaced. ",
+        "bbox": [
+            174,
+            863,
+            825,
+            924
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Thus by Lemma 3, ",
+        "bbox": [
+            174,
+            103,
+            299,
+            118
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1ea674240f9e02dc5bd479332892e9eca74d4cd7c9535d5e38da57b023ff0bac.jpg",
+        "text": "$$\n\\begin{array} { l } { \\overline { { R } } ( \\hat { \\beta } _ { d } ^ { \\mathrm { o p t } } ) = \\widetilde { \\sigma } ^ { 2 } + \\underset { ( \\gamma _ { 1 } , \\ldots , \\gamma _ { m } ) \\sim \\Gamma _ { d + 1 } } { \\mathbb { E } } \\left[ \\overset { p } { \\underset { i = 1 } { \\sum } } \\frac { \\widetilde { \\sigma } ^ { 2 } } { \\gamma _ { i } ^ { 2 } + \\frac { \\widetilde { \\sigma } ^ { 2 } p ^ { 2 } } { d | \\boldsymbol { \\theta } | | _ { 2 } ^ { 2 } } } \\right] } \\\\ { \\geq \\widetilde { \\sigma } ^ { 2 } + \\underset { ( \\widetilde { \\gamma } _ { 1 } , \\ldots , \\widetilde { \\gamma } _ { m } ) \\sim \\Gamma _ { d + 1 } } { \\mathbb { E } } \\left[ \\overset { p } { \\underset { i = 1 } { \\sum } } \\frac { \\widetilde { \\sigma } ^ { 2 } } { \\widetilde { \\gamma } _ { i } ^ { 2 } + \\frac { \\widetilde { \\sigma } ^ { 2 } p ^ { 2 } } { d | \\boldsymbol { \\theta } | | _ { 2 } ^ { 2 } } } \\right] } \\\\ { = \\overline { { R } } ( \\hat { \\beta } _ { d + 1 } ^ { \\mathrm { o p t } } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            125,
+            668,
+            248
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "A.3 NONISOTROPIC REDUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            284,
+            415,
+            299
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Here we observe that results on isotropic regression in Section 2 also imply that ridge regression can be made sample-monotonic even for non-isotropic covariates, if an appropriate regularzier is applied. Specifically, the regularizer depends on the covariance on the inputs. This follows from a general equivalence between the non-isotropic and isotropic problems. ",
+        "bbox": [
+            173,
+            309,
+            825,
+            367
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Lemma 5. For all $n \\in \\mathbb { N } , d \\in \\mathbb { N } , \\lambda \\in \\mathbb { R } , \\sigma \\in \\mathbb { R } ,$ covariance $\\Sigma \\in \\mathbb { R } ^ { d \\times d }$ , PSD matrix $M \\in \\mathbb { R } ^ { d \\times d }$ and ground-truth $\\beta ^ { * } \\in \\mathbb { R } ^ { d }$ , the following holds. ",
+        "bbox": [
+            173,
+            369,
+            821,
+            400
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Consider the following two problems: ",
+        "bbox": [
+            176,
+            406,
+            421,
+            421
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "1. Regularized regression on isotropic covariates, and an $M$ -regularizer. That is, suppose $n$ samples $( x , y )$ are drawn with covariates $x \\sim \\mathcal { N } ( 0 , I _ { d } )$ and response $y = \\langle \\beta ^ { * } , x \\rangle +$ $\\mathcal { N } ( 0 , \\bar { \\sigma } ^ { 2 } )$ . Let $\\ b { \\hat { X } } \\in \\mathbb { R } ^ { n \\times d }$ be the matrix of covariates, and $\\vec { y }$ the vector of responses. Consider ",
+        "bbox": [
+            212,
+            433,
+            825,
+            489
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "img_path": "images/02363f18ece46693d693587db157e2dcdb085e1e18082f637c3f89a762f72274.jpg",
+        "text": "$$\n{ \\hat { \\beta } } _ { \\lambda } : = \\operatorname * { a r g m i n } _ { \\beta } | | X \\beta - { \\vec { y } } | | _ { 2 } ^ { 2 } + \\lambda | | \\beta | | _ { M } ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            493,
+            653,
+            523
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Let $\\overline { { R } } = \\mathbb { E } _ { X , y } [ | | \\hat { \\beta } - \\beta ^ { * } | | _ { 2 } ^ { 2 } ] + \\sigma ^ { 2 }$ be the expected test risk of the above estimator. ",
+        "bbox": [
+            232,
+            531,
+            756,
+            550
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "2. Regularized regression with covariance $\\Sigma$ , and an $( \\Sigma ^ { 1 / 2 } M \\Sigma ^ { 1 / 2 } )$ -regularizer. That is, suppose $n$ samples $( \\widetilde { x } , y )$ are drawn with covariates $\\widetilde { x } \\sim \\mathcal { N } ( 0 , \\Sigma )$ and response $y =$ $\\langle z ^ { \\ast } , \\widetilde { x } \\rangle + \\mathcal { N } ( 0 , \\widetilde { \\sigma } ^ { 2 } )$ e, for ",
+        "bbox": [
+            214,
+            558,
+            823,
+            602
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "img_path": "images/df06493abeb238642e8c8f74a3139cd40fd2e81f06e4936a7d7cad291c1fde94.jpg",
+        "text": "$$\nz ^ { * } = \\Sigma ^ { - 1 / 2 } \\beta ^ { * }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            477,
+            601,
+            578,
+            619
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Let $\\widetilde { X } \\in \\mathbb { R } ^ { n \\times d }$ be the matrix of covariates, and $\\vec { y }$ the vector of responses. Consider ",
+        "bbox": [
+            232,
+            625,
+            779,
+            641
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cda5be67a395c20ef4b5af3f81051ab51976eb309c81bc6b7d01c1f24c2eb8c2.jpg",
+        "text": "$$\n\\hat { z } _ { \\lambda } : = \\operatorname * { a r g m i n } _ { z } | | \\widetilde { X } z - \\vec { y } | | _ { 2 } ^ { 2 } + \\lambda | | z | | _ { \\Sigma ^ { 1 / 2 } M \\Sigma ^ { 1 / 2 } } ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            646,
+            678,
+            674
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Let $\\widetilde { R } = \\mathbb { E } _ { \\widetilde { X } , y } [ | | \\hat { z } - z ^ { * } | | _ { \\Sigma } ^ { 2 } ] + \\sigma ^ { 2 }$ be the expected test risk of the above estimator. ",
+        "bbox": [
+            230,
+            683,
+            756,
+            702
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Then, the expected test risks of the above two problems are identical: ",
+        "bbox": [
+            174,
+            713,
+            625,
+            728
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "img_path": "images/db5f0c07ae954bc35d619d405280e49178abec00d0dba3e56fe2951dd749e615.jpg",
+        "text": "$$\n\\overline { { R } } = \\widetilde { R }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            472,
+            734,
+            526,
+            752
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Proof of Lemma 5. The distribution of $\\widetilde { X }$ in the Problem 2 is equivalent to $X \\Sigma ^ { 1 / 2 }$ , where $X$ is as in Problem 1. Thus, the two settings are equivalent by the change-of-variable $\\beta = \\Sigma ^ { 1 / 2 } z$ . Specifically, ",
+        "bbox": [
+            173,
+            768,
+            825,
+            800
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f0beeb018cf7057c374d082f97552de1cc35ea24d3586872370a3e8651b97efd.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\hat { z } _ { \\lambda } : = \\underset { z } { \\mathrm { a r g m i n } } \\left| \\lvert \\tilde { X } z - \\bar { y } \\rvert \\right| _ { 2 } ^ { 2 } + \\lambda \\lvert | z \\rvert | _ { \\Sigma ^ { 1 / 2 } M \\Sigma ^ { 1 / 2 } } ^ { 2 } } \\\\ & { = \\underset { z } { \\mathrm { a r g m i n } } \\left| \\lvert X \\Sigma ^ { 1 / 2 } z - \\bar { y } \\rvert \\right| _ { 2 } ^ { 2 } + \\lambda z ^ { T } \\Sigma ^ { 1 / 2 } M \\Sigma ^ { 1 / 2 } z } \\\\ & { = \\underset { z } { \\mathrm { a r g m i n } } \\left| \\lvert X \\Sigma ^ { 1 / 2 } z - \\bar { y } \\rvert \\right| _ { 2 } ^ { 2 } + \\lambda z ^ { T } \\Sigma ^ { 1 / 2 } M \\Sigma ^ { 1 / 2 } z } \\\\ & { = \\Sigma ^ { - 1 / 2 } \\underset { \\beta = \\Sigma ^ { 1 / 2 } z } { \\mathrm { a r g m i n } } \\left| \\lvert X \\beta - \\bar { y } \\rvert \\right| _ { 2 } ^ { 2 } + \\lambda \\beta ^ { T } M \\beta } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            805,
+            663,
+            922
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Further, the response $\\langle z ^ { * } , { \\widetilde { x } } \\rangle = \\langle \\beta , x \\rangle$ , and the test risk transforms identically: ",
+        "bbox": [
+            171,
+            102,
+            686,
+            119
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ba8c337424aa66b3e400992cc04938b8a52ee3d159056665cd8655fcb3352401.jpg",
+        "text": "$$\n\\begin{array} { r } { \\widetilde { R } = \\underset { \\widetilde { X } , y } { \\mathbb { E } } \\left[ \\vert \\vert \\hat { z } - z ^ { * } \\vert \\vert _ { \\Sigma } ^ { 2 } \\right] + \\sigma ^ { 2 } } \\\\ { = \\underset { X , y } { \\mathbb { E } } \\left[ \\vert \\vert \\hat { \\beta } - \\beta ^ { * } \\vert \\vert _ { 2 } ^ { 2 } \\right] + \\sigma ^ { 2 } } \\\\ { = \\overline { { R } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            408,
+            122,
+            589,
+            203
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "This implies that if the covariance $\\Sigma$ is known, then ridge regression with a $\\Sigma ^ { - 1 }$ regularizer is sample-monotonic. ",
+        "bbox": [
+            174,
+            234,
+            823,
+            263
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Theorem 5. For all $n \\in \\mathbb { N } , d \\in \\mathbb { N } , \\sigma \\in \\mathbb { R } ,$ , covariance $\\Sigma \\in \\mathbb { R } ^ { d \\times d }$ , and ground-truths $\\beta ^ { * } \\in \\mathbb { R } ^ { d }$ , th e following holds. ",
+        "bbox": [
+            174,
+            265,
+            823,
+            295
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Suppose $n$ samples $( x , y )$ are drawn with covariates $x \\sim \\mathcal { N } ( 0 , \\Sigma )$ and response $y = \\langle \\beta ^ { * } , x \\rangle +$ ${ \\mathcal { N } } ( 0 , \\sigma ^ { 2 } )$ . Let $\\ b { X } \\in \\mathbb { R } ^ { n \\times d }$ be the matrix of covariates, and $\\vec { y }$ the vector of responses. For $\\lambda > 0$ , consider the ridge regression estimator with $\\Sigma ^ { - 1 }$ -regularizer: ",
+        "bbox": [
+            173,
+            301,
+            825,
+            343
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/44c9a87571ee9323acf796aef81530d801ad3a2c6383f1a76da675ec70121a13.jpg",
+        "text": "$$\n\\hat { \\beta } _ { n , \\lambda } : = \\operatorname * { a r g m i n } _ { \\boldsymbol { \\beta } } \\| X \\boldsymbol { \\beta } - \\vec { y } \\| _ { 2 } ^ { 2 } + \\lambda \\| \\boldsymbol { \\beta } \\| _ { \\Sigma ^ { - 1 } } ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            361,
+            348,
+            635,
+            376
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Let $\\overline { { R } } ( \\hat { \\beta } _ { n , \\lambda } ) : = \\mathbb { E } _ { \\hat { \\beta } } \\| \\hat { \\beta } - \\beta ^ { * } \\| _ { \\Sigma } + \\sigma ^ { 2 }$ be the expected test risk of the above estimator. Let $\\lambda _ { n } ^ { \\mathrm { o p t } }$ be the optimal ridge parameter (that achieves the minimum expected risk) given $n$ samples: ",
+        "bbox": [
+            169,
+            388,
+            823,
+            421
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6a255aa91999f671497ccad33844b9a826f3e210b626c3c0aa2f0b93fc75385e.jpg",
+        "text": "$$\n\\lambda _ { n } ^ { \\mathrm { o p t } } : = \\operatorname * { a r g m i n } _ { \\lambda } \\overline { { R } } ( \\hat { \\beta } _ { n , \\lambda } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            415,
+            424,
+            583,
+            452
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "And let $\\hat { \\beta } _ { n } ^ { \\mathrm { { o p t } } }$ be the estimator that corresponds to the $\\lambda _ { n } ^ { \\mathrm { o p t } }$ . Then, the expected test risk of optimallyregularized linear regression is monotonic in samples: ",
+        "bbox": [
+            173,
+            457,
+            823,
+            487
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e4739674d976e6c2ef07dabe2d407a650cf2cf0edb0ee50ad6200a5333c108d0.jpg",
+        "text": "$$\n\\overline { { R } } ( \\hat { \\beta } _ { n + 1 } ^ { \\mathrm { o p t } } ) \\leq \\overline { { R } } ( \\hat { \\beta } _ { n } ^ { \\mathrm { o p t } } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            431,
+            491,
+            566,
+            512
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Proof. This follows directly by applying the reduction in Lemma 5 for $M = I _ { d }$ to reduce to the isotropic case, and then applying the monotonicity of isotropic regression from Theorem 1. □ ",
+        "bbox": [
+            174,
+            523,
+            825,
+            553
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "A.4 ADDITIONAL PLOTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            568,
+            357,
+            582
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "We apply random features to Fashion-MNIST Xiao et al. (2017), an image classification problem with 10 classes. Input images $x \\in \\mathbb { R } ^ { d }$ are normalized and flattened to $[ - \\bar { 1 } , 1 ] ^ { d }$ for $d = 7 8 4$ . Class labels are encoded as one-hot vectors $y \\in \\{ \\vec { e _ { 1 } } , \\dots e _ { 1 0 } ^ {  } \\} \\subset \\mathbb { R } ^ { 1 0 }$ . ",
+        "bbox": [
+            174,
+            593,
+            825,
+            636
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "B TOWARDS MONOTONICITY WITH GENERAL COVARIATES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            654,
+            681,
+            671
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Here we investigate whether monotonicity provably holds in more general models, inspired by the experimental results. As a first step, we consider Gaussian (but not isotropic) covariances and homeostatic noise. That is, we consider ridge regression in the setting of Section 2, but with $x \\sim \\mathcal { N } ( 0 , \\Sigma )$ , and $y \\sim \\langle x , \\beta ^ { * } \\rangle + N ( 0 , \\sigma ^ { 2 } )$ . In this section, we observe that ridge regression can be made sample-monotonic with a modified regularizer. We also conjecture that ridge regression is sample-monotonic without modifying the regularizer, and we outline a potential proof strategy along with numerical evidence. ",
+        "bbox": [
+            173,
+            684,
+            825,
+            782
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "B.1 ADAPTIVE REGULARIZATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            797,
+            421,
+            813
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "The results on isotropic regression in Section 2 imply that ridge regression can be made samplemonotonic even for non-isotropic covariates, if an appropriate regularizer is applied. Specifically, the appropriate regularizer depends on the covariance of the inputs. For $\\boldsymbol { x } \\sim \\mathcal { N } ( 0 , \\Sigma )$ , the following estimator is sample-monotonic for optimally-tuned $\\lambda$ : $\\begin{array} { r } { \\hat { \\beta } _ { n , \\lambda } : = \\operatorname * { a r g m i n } _ { \\boldsymbol { \\beta } } \\| X \\boldsymbol { \\beta } - \\vec { y } \\| _ { 2 } ^ { 2 } + \\lambda \\| \\boldsymbol { \\beta } \\| _ { \\Sigma ^ { - 1 } } ^ { 2 } } \\end{array}$ . This follows directly from Theorem 1 by applying a change-of-variable; full details of this equivalence are in Section A.3. Note that if the population covariance $\\Sigma$ is not known, it can potentially be estimated from unlabeled data. ",
+        "bbox": [
+            173,
+            823,
+            825,
+            924
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/6d706f197addd390fb2defce66fd1cd5af413a8242fbc58ab89186d03af9186f.jpg",
+        "image_caption": [
+            "Test Error for Regularized Random Features "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            109,
+            460,
+            321
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "(a) Test Classification Error vs. Number of Training Samples. ",
+        "bbox": [
+            173,
+            333,
+            467,
+            359
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/ce3f799d5bd5415e26ca617c4e47e654875a8b6c2557b5bf3b3c7007b6a14a4f.jpg",
+        "image_caption": [
+            "Test Errorfor Regularized Random Features ",
+            "(b) Test Classification Error vs. Model Size (Number of Random Features). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            535,
+            111,
+            816,
+            321
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Figure 4: Double-descent for Random ReLU Features. Test classification error as a function of model size and sample size for Random ReLU Features on Fashion-MNIST. Left: with $D = 5 0 0$ features. Right: with $n = 5 0 0$ samples. See Figures 7, 8 for the corresponding test Mean Squared Error. See Appendix D of Nakkiran et al. (2020) for the performance of these unregularized models plotted across Num. Samples $\\times$ Model Size simultaneously. ",
+        "bbox": [
+            173,
+            371,
+            826,
+            440
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/05df2a46592e323dc1205edaa5fa11d2355d7bba6df96212a7b9f700cd4a121b.jpg",
+        "image_caption": [
+            "Figure 5: Train Error vs. Model Size for 5-layer CNNs on CIFAR-100, with $\\ell _ { 2 }$ regularization (weight decay). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            279,
+            462,
+            717,
+            654
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "B.2 TOWARDS PROVING MONOTONICITY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            724,
+            472,
+            739
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "We conjecture that optimally-regularized ridge regression is sample-monotonic for non-isotropic covariates, even without modifying the regularizer (as suggested by the experiment in Figure 2). We derive a sufficient condition for monotonicity, which we have numerically verified in a variety of instances. Specifically, we conjecture the following. ",
+        "bbox": [
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+            751,
+            825,
+            808
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Conjecture 1. For all $d \\in \\mathbb { N }$ , and all PSD covariances $\\Sigma \\in \\mathbb { R } ^ { d \\times d }$ , consider the distribution on $( x , y )$ where $x \\sim \\mathcal { N } ( 0 , \\Sigma )$ , and $y \\sim \\langle x , \\beta ^ { * } \\rangle + \\mathcal { N } ( 0 , \\sigma ^ { 2 } )$ . Then, we conjecture that the expected test risk of the ridge regression estimator: $\\begin{array} { r } { \\hat { \\beta } _ { n , \\lambda } : = \\operatorname * { a r g m i n } _ { \\boldsymbol { \\beta } } \\| X { \\boldsymbol { \\beta } } - \\vec { y } \\| _ { 2 } ^ { 2 } + \\lambda \\| { \\boldsymbol { \\beta } } \\| _ { 2 } ^ { 2 } } \\end{array}$ for optimally-tuned $\\lambda \\geq 0$ , is monotone non-increasing in number of samples $n$ . That is, for all $n \\in \\mathbb { N }$ , ",
+        "bbox": [
+            173,
+            810,
+            825,
+            869
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c9c57ee5be88035eb586dc57d2d1e0b886158489d7bd35b146d1aa191662b2a5.jpg",
+        "text": "$$\n\\operatorname * { i n f } _ { \\lambda \\geq 0 } \\overline { { { R } } } ( \\hat { \\beta } _ { n + 1 , \\lambda } ) \\leq \\operatorname * { i n f } _ { \\lambda \\geq 0 } \\overline { { { R } } } ( \\hat { \\beta } _ { n , \\lambda } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            875,
+            607,
+            901
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "where we define $\\begin{array} { r } { \\hat { \\beta } _ { n , 0 } : = \\operatorname* { l i m } _ { \\lambda \\to 0 + } \\hat { \\beta } _ { n , \\lambda } = X ^ { \\dagger } y } \\end{array}$ . ",
+        "bbox": [
+            173,
+            906,
+            488,
+            926
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/d4048491df9a41611fd80fea0aeac569328ee316a77886c71b4c9d4ba9170d40.jpg",
+        "image_caption": [
+            "Figure 6: Train MSE vs. Num. Samples for Non-Isotropic Ridge Regression in $d = 3 0$ dimensions, in the setting of Figure 2. Plotting train MSE: $\\begin{array} { r } { \\frac { 1 } { n } | | X \\hat { \\boldsymbol { \\beta } } - \\vec { y } | | _ { 2 } ^ { 2 } } \\end{array}$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            326,
+            107,
+            671,
+            371
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "image",
+        "img_path": "images/aa84a8ffbd2beefb1d745ab29a4ccc613c56c1dfa1eef8d51380a0483dcffecb.jpg",
+        "image_caption": [
+            "Figure 7: Test Mean Squared Error vs. Num Train Samples for Random ReLU Features on FashionMNIST, with $D = 5 0 0$ features. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            326,
+            443,
+            671,
+            707
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "In Appendix B.3 we present a technical conjecture in random matrix theory (Conjecture 2) which suffices to prove Conjecture 1. Proving this Conjecture 2 presents a number of technical challenges, but we have numerically verified it in a variety of cases. It can also be shown that Conjecture 2 is true when $Q = I$ , corresponding to isotropic covariates. We prove the reduction between Conjecture 2 and 1 in Appendix B.3. ",
+        "bbox": [
+            174,
+            784,
+            825,
+            854
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "B.3 MONOTONICITY CONJECTURE PROOFS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            877,
+            486,
+            892
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "In order to establish Conjecture 1, it is sufficient to prove the following technical conjecture. ",
+        "bbox": [
+            168,
+            909,
+            776,
+            924
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "image",
+        "img_path": "images/8eda0db61a5bd63cb07df9565b6981c5a80a7fd865e013f9ce281e419515418e.jpg",
+        "image_caption": [
+            "Figure 8: Test Mean Squared Error vs. Num Features for Random ReLU Features on FashionMNIST, with $n = 5 0 0$ samples. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            326,
+            106,
+            671,
+            372
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Conjecture 2. For all $n \\in \\mathbb { N } , d \\geq n , \\lambda > 0$ , symmetric positive definite matrix $Q \\in \\mathbb { R } ^ { d \\times d }$ , the following holds. ",
+        "bbox": [
+            174,
+            443,
+            823,
+            473
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Define ",
+        "bbox": [
+            173,
+            479,
+            220,
+            494
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ad76a8c087b005ddee977955e15dff2088f817370c0193f16b774104a3fb7cc1.jpg",
+        "text": "$$\nG _ { \\lambda } ^ { n } : = \\lambda ^ { 2 } \\underset { X } { \\mathbb { E } } [ ( X ^ { T } X + \\lambda Q ) ^ { - 2 } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            491,
+            601,
+            516
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "where $\\ b { X } \\in \\mathbb { R } ^ { n \\times d }$ is sampled with each entry i.i.d. $\\mathcal { N } ( 0 , 1 )$ . Similarly, define ",
+        "bbox": [
+            173,
+            520,
+            676,
+            536
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "img_path": "images/231a0f4ef6fbdc38727280f01461eda169b50c285dd0fde6c57d63e46a831858.jpg",
+        "text": "$$\nH _ { \\lambda } ^ { n } : = \\mathbb { E } [ | | X ^ { T } X + \\lambda Q ) ^ { - 1 } X ^ { T } | | _ { F } ^ { 2 } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            542,
+            616,
+            568
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "The expected test risk for $n$ samples can be expressed as: ",
+        "bbox": [
+            173,
+            575,
+            549,
+            590
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c89974b87c986d9d15a75f2dd3fb1e0db209a7da6355aa40f2d30e736ed328c1.jpg",
+        "text": "$$\n\\overline { { { R } } } ( \\hat { \\beta } _ { n , \\lambda } ) = ( \\beta ^ { * } ) ^ { T } G _ { \\lambda } ^ { n } \\beta ^ { * } + \\sigma ^ { 2 } H _ { \\lambda } ^ { n } + \\sigma ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            597,
+            629,
+            616
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Then, we conjecture that the following two conditions hold. ",
+        "bbox": [
+            173,
+            630,
+            562,
+            645
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "1. ",
+        "bbox": [
+            212,
+            657,
+            225,
+            671
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "img_path": "images/336184f3133bbe3606ffc5cf5d47e56e1ddb0b17a540f79cdd78752af7fc88b0.jpg",
+        "text": "$$\nG _ { \\lambda } ^ { n } \\succeq G _ { \\lambda } ^ { n + 1 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            486,
+            669,
+            570,
+            688
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "2. ",
+        "bbox": [
+            212,
+            699,
+            225,
+            712
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5c3c2fe8f8069e1ad2ae74cc0f35a3b3a716f86e6780793a14502d60454d7f41.jpg",
+        "text": "$$\n( G _ { \\lambda } ^ { n } - G _ { \\lambda } ^ { n + 1 } ) - ( H _ { \\lambda } ^ { n } - H _ { \\lambda } ^ { n + 1 } ) \\frac { d G _ { \\lambda } ^ { n } / d \\lambda } { d H _ { \\lambda } ^ { n } / d \\lambda } \\succeq 0\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            717,
+            681,
+            751
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Lemma 6. Conjecture 2 implies Conjecture $^ { l }$ ",
+        "bbox": [
+            173,
+            757,
+            480,
+            773
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Proof. By the reduction in Section A.3, showing monotonicity for non-isotropic regression with an isotropic regularizer is equivalent to showing monotonicity for isotropic regression with a nonisotropic regularizer. Thus, we consider the latter. Specifically, Conjecture 1 is equivalent to showing monotonicity for the estimator ",
+        "bbox": [
+            173,
+            787,
+            825,
+            844
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6c41e816d4f887f9bde12f642fb19207753ffa9bfb323e88287605e629300b7c.jpg",
+        "text": "$$\n\\begin{array} { r l } {  { \\hat { \\beta } _ { n , \\lambda } : = \\operatorname * { a r g m i n } _ { \\beta } \\| X \\beta - \\vec { y } \\| _ { 2 } ^ { 2 } + \\lambda \\| \\beta \\| _ { \\Sigma ^ { - 1 } } ^ { 2 } } } \\\\ & { = ( X ^ { T } X + \\lambda \\Sigma ^ { - 1 } ) ^ { - 1 } X ^ { T } y } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            361,
+            851,
+            633,
+            902
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "where $x \\sim \\mathcal { N } ( 0 , I )$ is isotropic, and $y \\sim \\langle x , \\beta ^ { * } \\rangle + \\mathcal { N } ( 0 , \\sigma ^ { 2 } )$ . ",
+        "bbox": [
+            173,
+            907,
+            578,
+            925
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Now, letting $\\boldsymbol { Q } : = \\boldsymbol { \\Sigma } ^ { - 1 }$ , the expected test risk of this estimator for $n$ samples is: ",
+        "bbox": [
+            173,
+            102,
+            696,
+            119
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b0c0704953e261076194e5ff818a464b49094b93133c27b3b95b7aa2feff2dd1.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } [ \\left. \\hat { \\beta } _ { n , \\lambda } \\right. = \\underset { X , y } { \\mathbb { E } } [ \\left. \\hat { \\beta } _ { n , \\lambda } - \\beta ^ { * } \\right. _ { 2 } ^ { 2 } ] + \\sigma ^ { 2 } } \\\\ & { \\quad = \\underset { X , y } { \\mathbb { E } } [ \\left. ( X ^ { T } X + \\lambda Q ) ^ { - 1 } X ^ { T } y - \\beta ^ { * } \\right. _ { 2 } ^ { 2 } ] + \\sigma ^ { 2 } } \\\\ & { \\quad = \\underset { X , \\eta = N ; 0 } { \\mathbb { E } } [ \\left. ( X ^ { T } X + \\lambda Q ) ^ { - 1 } X ^ { T } ( X \\beta ^ { * } + \\eta ) - \\beta ^ { * } \\right. _ { 2 } ^ { 2 } ] + \\sigma ^ { 2 } } \\\\ & { \\quad = \\underset { X , \\eta = N ; 0 } { \\mathbb { E } } [ ( 0 , \\lambda ^ { 2 } Y ^ { T } \\eta _ { n } ^ { 2 } ) ] \\left. ( X ^ { T } X + \\lambda Q ) ^ { - 1 } X ^ { T } ( X \\beta ^ { * } + \\eta ) - \\beta ^ { * } \\right. _ { 2 } ^ { 2 } ] + \\sigma ^ { 2 } } \\\\ & { \\quad = \\underset { X } { \\mathbb { E } } [ \\left. ( X ^ { T } X + \\lambda Q ) ^ { - 1 } X ^ { T } X \\beta ^ { * } - \\beta ^ { * } ) \\right. _ { 2 } ^ { 2 } ] + \\sigma ^ { 2 } \\underset { X } { \\mathbb { E } } [ \\left. ( X ^ { T } X + \\lambda Q ) ^ { - 1 } X ^ { T } \\right. _ { F } ^ { 2 } ] + \\sigma ^ { 2 } } \\\\ & { \\quad = \\underset { X } { \\mathbb { E } } [ \\left. ( X ^ { T } X + \\lambda Q ) ^ { - 1 } ( X ^ { T } X + \\lambda Q - \\lambda Q ) \\beta ^ { * } - \\beta ^ { * } ) \\right. _ { 2 } ^ { 2 } ] + \\sigma ^ { 2 } \\underset { X } { \\mathbb { E } } [ \\left. ( X ^ { T } X + \\lambda Q ) ^ { - 1 } X ^ { T } \\right. _ { F } ^ { 2 } ] } \\\\ &  \\quad = \\lambda ^ { 2 } \\underset { X } { \\mathbb { E } } [ \\left. ( X ^ { T } X + \\lambda Q ) ^ { - 1 } \\right. _ { 2 } ^ { 2 } ] + \\sigma ^ { 2 } \\underset { X } { \\mathbb { E } } [ \\left. ( X ^ { T } X + \\lambda Q ) ^ { - 1 } X ^ { T } \\right. _ { F } ^ { 2 } ] + \\sigma \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            123,
+            839,
+            315
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Consider the infimum ",
+        "bbox": [
+            173,
+            328,
+            320,
+            343
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "img_path": "images/13c55acf6258506f84d51c2ef9587f1fae07d58b644a432c0ea44cbe3d20e3df.jpg",
+        "text": "$$\n\\operatorname* { i n f } _ { \\lambda \\geq 0 } { \\overline { { R } } } ( { \\hat { \\beta } } _ { n , \\lambda } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            454,
+            348,
+            542,
+            375
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "We consider several cases below. ",
+        "bbox": [
+            174,
+            382,
+            392,
+            397
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Case (1). Suppose the infimum in Equation 71 is achieved in the limit $\\lambda \\to + \\infty$ . In this case, monotonicity trivially holds, since ",
+        "bbox": [
+            173,
+            404,
+            823,
+            433
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6ae7607f4c9c2ddad443112fd1792e84f875d21cc7082bbeefac25b604d59f7f.jpg",
+        "text": "$$\n\\operatorname* { l i m } _ { \\lambda \\to \\infty } \\overline { { { R } } } ( \\hat { \\beta } _ { n , \\lambda } ) = \\overline { { { R } } } ( \\vec { 0 } ) = \\operatorname* { l i m } _ { \\lambda \\to \\infty } \\overline { { { R } } } ( \\hat { \\beta } _ { n + 1 , \\lambda } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            439,
+            640,
+            465
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Case (2). Suppose the infimum in Equation 71 is achieved by some $\\lambda = \\lambda _ { n } ^ { \\mathrm { o p t } }$ in the interior of the set $( 0 , \\infty )$ . ",
+        "bbox": [
+            173,
+            483,
+            823,
+            513
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Because $\\overline { { R } } ( \\hat { \\beta } _ { n , \\lambda } )$ is continuous and differentiable in $\\lambda$ for all $\\lambda \\in ( 0 , \\infty )$ , we have that $\\lambda _ { n } ^ { \\mathrm { o p t } }$ must satisfy the following first-order optimality condition: ",
+        "bbox": [
+            174,
+            521,
+            825,
+            551
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bd5812ddb8e5b3a3003763dbc4df1c1083a51b8491f6169af12151e16bdf1fe2.jpg",
+        "text": "$$\n\\begin{array} { r } { \\frac { d \\overline { { R } } ( \\hat { \\beta } _ { n , \\lambda } ) } { d \\lambda } \\Big | _ { \\lambda = \\lambda _ { n } ^ { \\mathrm { o p t } } } = 0 } \\\\ { \\implies ( \\beta ^ { * } ) ^ { T } \\frac { d G _ { \\lambda } ^ { n } } { d \\lambda } \\beta ^ { * } + \\sigma ^ { 2 } \\frac { d H _ { \\lambda } ^ { n } } { d \\lambda } \\Big | _ { \\lambda = \\lambda _ { n } ^ { \\mathrm { o p t } } } = 0 } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            362,
+            558,
+            640,
+            628
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "We will later use this condition to show monotonicity. ",
+        "bbox": [
+            173,
+            632,
+            529,
+            647
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Case (3). Suppose the infimum in Equation 71 is achieved at $\\lambda _ { n } ^ { \\mathrm { o p t } } = 0$ . Recall, we define $\\hat { \\beta } _ { n , 0 } : =$ $\\mathrm { l i m } _ { \\lambda \\to 0 + } \\hat { \\beta } _ { n , \\lambda }$ . This means that, ",
+        "bbox": [
+            174,
+            655,
+            823,
+            689
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c95a4e0017eb4b8708bb725a904978d94b5de54f2554ec63647774db25cd0a72.jpg",
+        "text": "$$\n\\frac { d \\overline { { R } } ( \\hat { \\beta } _ { n , \\lambda } ) } { d \\lambda } \\Big | _ { \\lambda = 0 } = ( \\beta ^ { * } ) ^ { T } \\frac { d G _ { \\lambda } ^ { n } } { d \\lambda } \\beta ^ { * } + \\sigma ^ { 2 } \\frac { d H _ { \\lambda } ^ { n } } { d \\lambda } \\Big | _ { \\lambda = 0 } \\geq 0\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            696,
+            674,
+            732
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Note that since $\\begin{array} { r } { \\frac { d H _ { \\lambda } ^ { n } } { d \\lambda } \\leq 0 } \\end{array}$ , both Equations (73) and (74) in Case (2) and Case (3) respectively imply that ",
+        "bbox": [
+            173,
+            746,
+            825,
+            779
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6e211998885ace57711bcff68cdbeb605671fe5b714974d9087bbcd54d6a9332.jpg",
+        "text": "$$\n\\sigma ^ { 2 } \\leq - \\frac { ( \\beta ^ { * } ) ^ { T } ( \\frac { d G _ { \\lambda } ^ { n } } { d \\lambda } ) \\beta ^ { * } } { d H _ { \\lambda } ^ { n } / d \\lambda } \\Big \\vert _ { \\lambda = \\lambda _ { n } ^ { \\mathrm { o p t } } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            782,
+            599,
+            821
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "Now, assuming Conjecture 2, we will show that the choice of $\\lambda _ { n } ^ { \\mathrm { o p t } }$ in Cases (2) and (3) has nonincreasing test risk for $( n + 1 )$ samples. That is, ",
+        "bbox": [
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+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1ef487276cca9054802b0d0ae7cb3f1d48e7c75f3f3742cad65981740d6a61b4.jpg",
+        "text": "$$\n\\overline { { R } } ( \\widehat { \\beta } _ { n , \\lambda _ { n } ^ { \\mathrm { o p t } } } ) \\geq \\overline { { R } } ( \\widehat { \\beta } _ { n + 1 , \\lambda _ { n } ^ { \\mathrm { o p t } } } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            876,
+            586,
+            897
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "This implies the desired monotonicity, since R(βˆn+1,λoptn ) ≥ R(βˆn+1,λoptn+1 ). ",
+        "bbox": [
+            174,
+            905,
+            669,
+            926
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "We first consider the case when $H _ { \\lambda } ^ { n } - H _ { \\lambda } ^ { n + 1 } | _ { \\lambda = \\lambda _ { n } ^ { \\mathrm { o p t } } } \\geq 0$ . In this case, because $G _ { \\lambda } ^ { n } - G _ { \\lambda } ^ { n + 1 } \\succeq 0$ by assumption, we have ",
+        "bbox": [
+            173,
+            102,
+            825,
+            133
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "img_path": "images/07aa3c990d95a8ac383860c5d76349205b0b166f3e65d75762e8e6f9ec27e44e.jpg",
+        "text": "$$\n\\begin{array} { r l r } {  { \\overline { { R } } ( \\widehat { \\beta } _ { n , \\lambda _ { n } ^ { \\mathrm { o p t } } } ) - \\overline { { R } } ( \\widehat { \\beta } _ { n + 1 , \\lambda _ { n } ^ { \\mathrm { o p t } } } ) = ( \\beta ^ { * } ) ^ { T } ( G _ { \\lambda } ^ { n } - G _ { \\lambda } ^ { n + 1 } ) \\beta ^ { * } + \\sigma ^ { 2 } ( H _ { \\lambda } ^ { n } - H _ { \\lambda } ^ { n + 1 } ) \\Big | _ { \\lambda = \\lambda _ { n } ^ { \\mathrm { o p t } } } } } \\\\ & { } & { \\geq 0 \\quad \\quad } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            236,
+            138,
+            759,
+            188
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "Otherwise, assume. $H _ { \\lambda } ^ { n } - H _ { \\lambda } ^ { n + 1 } | _ { \\lambda = \\lambda _ { n } ^ { \\mathrm { o p t } } } \\leq 0$ . Then we have: ",
+        "bbox": [
+            173,
+            193,
+            570,
+            212
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b15faae1a9fffa8999e8e06d3794f4e39afa6731de30d9facd85224487832d0a.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\overline { { R } } ( \\hat { \\beta } _ { n , \\lambda _ { n } ^ { \\smash { \\mathrm { o u t } } } } ) - \\overline { { R } } ( \\hat { \\beta } _ { n + 1 , \\lambda _ { n } ^ { \\smash { \\mathrm { o u t } } } } ) = ( \\beta ^ { * } ) ^ { T } ( G _ { \\lambda } ^ { m } - G _ { \\lambda } ^ { m + 1 } ) \\beta ^ { * } + \\sigma ^ { 2 } ( H _ { \\lambda } ^ { n } - H _ { \\lambda } ^ { n + 1 } ) \\Big | _ { \\lambda = \\lambda _ { n } ^ { \\smash { \\mathrm { o u t } } } } } & { \\qquad \\mathrm { ( } } \\\\ & { \\qquad \\geq ( \\beta ^ { * } ) ^ { T } ( G _ { \\lambda } ^ { m } - G _ { \\lambda } ^ { m + 1 } ) \\beta ^ { * } - ( \\beta ^ { * } ) ^ { T } ( \\frac { d G _ { \\lambda } ^ { n } } { d \\lambda } ) \\beta ^ { * } \\frac { ( H _ { \\lambda } ^ { n } - H _ { \\lambda } ^ { n + 1 } ) } { d H _ { \\lambda } ^ { \\lambda } / d \\lambda } \\Big | _ { \\lambda = \\lambda _ { n } ^ { \\smash { \\mathrm { o u t } } } } } \\\\ & { \\qquad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { = ( \\beta ^ { * } ) ^ { T } \\underbrace { \\Bigg ( ( G _ { \\lambda } ^ { m } - G _ { \\lambda } ^ { n + 1 } ) - ( H _ { \\lambda } ^ { n } - H _ { \\lambda } ^ { n + 1 } ) \\frac { d G _ { \\lambda } ^ { m } / d \\lambda } { d H _ { \\lambda } ^ { n } / d \\lambda } \\Big ) \\Big | _ { \\lambda = \\lambda _ { n } ^ { \\mathrm { o u t } } } \\beta ^ { * } } _ { \\geq 0 \\mathrm { b y c o n j e c u r e 2 } } } \\\\ & { \\qquad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\geq 0 } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            186,
+            219,
+            799,
+            378
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "text",
+        "text": "as desired. ",
+        "bbox": [
+            173,
+            382,
+            245,
+            396
+        ],
+        "page_idx": 22
+    }
+]
\ No newline at end of file
diff --git a/parse/train/7R7fAoUygoa/7R7fAoUygoa_middle.json b/parse/train/7R7fAoUygoa/7R7fAoUygoa_middle.json
new file mode 100644
index 0000000000000000000000000000000000000000..863c1ec06805f6812a2b21c193f8fefd607557e0
--- /dev/null
+++ b/parse/train/7R7fAoUygoa/7R7fAoUygoa_middle.json
@@ -0,0 +1,65877 @@
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "(which might be",
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+                                    "score": 1.0,
+                                    "content": "much smaller than ambient dimension), we consider the family of models which first project the",
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+                                    "content": "-dimensional subspace, and then compute a linear function in this projected",
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+                                    "score": 1.0,
+                                    "content": "“feature space.” (This is nearly identical to models of double-descent considered in Hastie et al.",
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+                                    "score": 1.0,
+                                    "content": "(2019, Section 5.1)). We prove that in this setting, as we grow the model-size, optimally-tuned",
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+                                    "score": 1.0,
+                                    "content": "ridge regression over the projected features has monotone test performance. That is, with optimal",
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+                            "index": 41
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+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "regularization, bigger models are always better or the same. (See Theorem 3).",
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+                    ],
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+                                    "content": "3. Monotonicity in the real-world: We also demonstrate several richer empirical settings where",
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+                                    "score": 1.0,
+                                    "content": "tional neural networks. This suggests that the mitigating effect of optimal regularization may hold",
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+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "more generally in broad machine learning contexts. (See Section 5).",
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+                                    "content": "Indeed, the minimax optimal estimator or the Bayes optimal estimator are both trivially sample-wise",
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+                                    "score": 1.0,
+                                    "content": "and model-wise monotonic with respect to the minimax risk or the Bayes risk. However, they do not",
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+                                    "score": 1.0,
+                                    "content": "guarantee monotonicity of the risk itself for a given fixed problem. In particular, there exist minimax",
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+                                    "content": "optimal estimators which are not sample-monotonic in the sense we desire.",
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+                                    "content": "Universal vs Asymptotic. We also remark that our analysis is not only non-asymptotic but also",
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+                                    "content": "for related results in the asymptotic setting). Our work reveals aspects of the problem that were not",
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+                                            "score": 0.64,
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+                                            "score": 1.0,
+                                            "content": "(Lemma 2), but this is not always true – see",
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+                                    ],
+                                    "index": 20
+                                },
+                                {
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+                                            "score": 1.0,
+                                            "content": "Figure 2.",
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+                                }
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+                        }
+                    ],
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+                                    "content": "Our Contributions: We study this question from both a theoretical and empirical perspective.",
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+                            "index": 24
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Theoretically, we start with the setting of high-dimensional linear regression. Linear regression is",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "a sensible starting point to study these questions, since it already exhibits many of the qualitative",
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+                            ],
+                            "index": 26
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "features of double descent in more complex models (e.g. Belkin et al. (2019); Hastie et al. (2019)",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 27
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "and further related works in Section 1.1). Our work shows that optimally-tuned ridge regression can",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 28
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "achieve both sample-wise monotonicity and model-size-wise monotonicity under certain assump-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 29
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "tions. Concretely, we show",
+                                    "type": "text"
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+                            ],
+                            "index": 30
+                        }
+                    ],
+                    "index": 27,
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+                                    "score": 1.0,
+                                    "content": "1. Sample-wise monotonicity: In the setting of well-specified linear regression with isotropic",
+                                    "type": "text"
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+                            "index": 31
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "features/covariates (Figure 1), we prove that optimally-tuned ridge regression yields monotonic test",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 32
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "performance with increasing samples. That is, more data never hurts for optimally-tuned ridge",
+                                    "type": "text"
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+                            ],
+                            "index": 33
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "regression. (See Theorem 1).",
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+                            "index": 34
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+                    ],
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+                    "bbox_fs": [
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+                                    "score": 1.0,
+                                    "content": "2. Model-wise monotonicity: We consider a setting where the input/covariate lives in a high-",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "dimensional ambient space with isotropic covariance. Given a fixed model size",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "(which might be",
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+                            ],
+                            "index": 36
+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "much smaller than ambient dimension), we consider the family of models which first project the",
+                                    "type": "text"
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+                            "index": 37
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "input to a random",
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+                                    "score": 1.0,
+                                    "content": "-dimensional subspace, and then compute a linear function in this projected",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 38
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "“feature space.” (This is nearly identical to models of double-descent considered in Hastie et al.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 39
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "(2019, Section 5.1)). We prove that in this setting, as we grow the model-size, optimally-tuned",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 40
+                        },
+                        {
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "ridge regression over the projected features has monotone test performance. That is, with optimal",
+                                    "type": "text"
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+                            ],
+                            "index": 41
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    "score": 1.0,
+                                    "content": "regularization, bigger models are always better or the same. (See Theorem 3).",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 42
+                        }
+                    ],
+                    "index": 38.5,
+                    "bbox_fs": [
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+                {
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+                    "lines": [
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+                                    "score": 1.0,
+                                    "content": "3. Monotonicity in the real-world: We also demonstrate several richer empirical settings where",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "type": "text"
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+                                    "score": 0.85,
+                                    "content": "\\ell _ { 2 }",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "regularization induces monotonicity, including random feature classifiers and convolu-",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "tional neural networks. This suggests that the mitigating effect of optimal regularization may hold",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 45
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "more generally in broad machine learning contexts. (See Section 5).",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 46
+                        }
+                    ],
+                    "index": 44.5,
+                    "bbox_fs": [
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+                    "type": "text",
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+                                    "score": 1.0,
+                                    "content": "A few remarks are in order:",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "Problem-specific vs Minimax and Bayesian. It is worth noting that our results hold for all linear",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 48
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "ground-truths, rather than holding for only the worst-case ground-truth or a random ground-truth.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 49
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Indeed, the minimax optimal estimator or the Bayes optimal estimator are both trivially sample-wise",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 50
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "and model-wise monotonic with respect to the minimax risk or the Bayes risk. However, they do not",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 51
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    "score": 1.0,
+                                    "content": "guarantee monotonicity of the risk itself for a given fixed problem. In particular, there exist minimax",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 52
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "optimal estimators which are not sample-monotonic in the sense we desire.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 53
+                        }
+                    ],
+                    "index": 50.5,
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+                                    "score": 1.0,
+                                    "content": "regularizer that depends on the population covariance matrix of data. This suggests unla-",
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+                                    "score": 1.0,
+                                    "content": "beled data might also help mitigate double descent in some settings, because the population",
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+                                    "score": 1.0,
+                                    "content": "covariance can be estimated from unlabeled data. (See Appendix B).",
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+                            "index": 15
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+                                    "score": 1.0,
+                                    "content": "3. For non-isotropic Gaussian covariates, we conjecture that optimally-tuned ridge regression",
+                                    "type": "text"
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+                                    "content": "\\ell _ { 2 }",
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+                                    "score": 1.0,
+                                    "content": "regularizer (as in Figure 2). We derive a",
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+                                    "score": 1.0,
+                                    "content": "sufficient condition for this conjecture. Due to that current random matrix theory may be",
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+                            "index": 18
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+                                    "score": 1.0,
+                                    "content": "insufficient to verify this conjecture, we verify it numerically on a wide variety of cases.",
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+                                    "score": 1.0,
+                                    "content": "(See Appendix B for details).",
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+                    ],
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+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "The last two results above highlight the importance of the form of the regularizer, which leads to the",
+                                    "type": "text"
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+                            "index": 22
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "We hope that our results can motivate future work on mitigating the double descent phenomenon,",
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+                            "index": 23
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "and allow us to train high performance models which do not exhibit nonmonotonic behaviors.",
+                                    "type": "text"
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+                            "index": 24
+                        }
+                    ],
+                    "index": 22.5
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+                {
+                    "type": "title",
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+                                    "score": 1.0,
+                                    "content": "1.1 RELATED WORKS",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 25
+                        }
+                    ],
+                    "index": 25
+                },
+                {
+                    "type": "text",
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "The study of nonmonotonicity in learning algorithms existed prior to double descent and has a long",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 26
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "history going back to (at least) Trunk (1979) and LeCun et al. (1991); Le Cun et al. (1991), where the",
+                                    "type": "text"
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+                            ],
+                            "index": 27
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+                                    "score": 1.0,
+                                    "content": "former was largely empirical observations and the latter studied the sample non-nonmonotonicity of",
+                                    "type": "text"
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+                            ],
+                            "index": 28
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "unregularized linear regression in terms of the eigenspectrum of the covariance matrix; the difference",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 29
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "to our works is that we study this in the context of optimal regularization. In fact, Duin (1995;",
+                                    "type": "text"
+                                }
+                            ],
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+                                    "content": "2000); Opper (2001); Loog & Duin (2012). Loog et al. (2019) introduces the same notion of risk",
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+                                    "content": "more restricted settings Trunk (1979); Opper (1995; 2001); Skurichina & Duin (2002); Le Cun et al.",
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+                                    "score": 1.0,
+                                    "content": "Spigler et al. (2018); Neal et al. (2018). Recently Nakkiran et al. (2020) demonstrated a generalized",
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+                            "index": 38
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "double descent phenomenon on modern deep networks, and highlighted “sample non-monotonicity”",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "as an aspect of double descent.",
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+                            ],
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+                        }
+                    ],
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+                {
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+                    "bbox": [
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+                                    "content": "A recent stream of theoretical works consider model-wise double descent in simplified settings—",
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+                                    "content": "Belkin et al. (2019); Hastie et al. (2019); Bartlett et al. (2019); Muthukumar et al. (2019); Bibas",
+                                    "type": "text"
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+                            ],
+                            "index": 44
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+                        {
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+                                    "score": 1.0,
+                                    "content": "et al. (2019); Mitra (2019); Mei & Montanari (2019); Liang & Rakhlin (2018); Liang et al. (2019);",
+                                    "type": "text"
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+                            ],
+                            "index": 45
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+                                    "score": 1.0,
+                                    "content": "Xu & Hsu (2019); Derezinski et al. (2019); Lampinen & Ganguli (2018); Deng et al. (2019); Nakki- ´",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 46
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "ran (2019); Mahdaviyeh & Naulet (2019); Dobriban et al. (2018); Dobriban & Sheng (2019); Kobak",
+                                    "type": "text"
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+                            ],
+                            "index": 47
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+                                    "score": 1.0,
+                                    "content": "et al. (2018). Of these, most closely related to our work are Hastie et al. (2019); Dobriban et al.",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": ". In contrast, we show non-asymptotic",
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+                                    "type": "text"
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "intuitions.)",
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "regularizer that depends on the population covariance matrix of data. This suggests unla-",
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+                                    "score": 1.0,
+                                    "content": "beled data might also help mitigate double descent in some settings, because the population",
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+                                    "score": 1.0,
+                                    "content": "covariance can be estimated from unlabeled data. (See Appendix B).",
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+                                    "content": "3. For non-isotropic Gaussian covariates, we conjecture that optimally-tuned ridge regression",
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+                                    "score": 1.0,
+                                    "content": "regularizer (as in Figure 2). We derive a",
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+                                    "score": 1.0,
+                                    "content": "sufficient condition for this conjecture. Due to that current random matrix theory may be",
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+                                    "score": 1.0,
+                                    "content": "insufficient to verify this conjecture, we verify it numerically on a wide variety of cases.",
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "(See Appendix B for details).",
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+                            ],
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+                        }
+                    ],
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+                                    "score": 1.0,
+                                    "content": "The last two results above highlight the importance of the form of the regularizer, which leads to the",
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "and allow us to train high performance models which do not exhibit nonmonotonic behaviors.",
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+                        }
+                    ],
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+                                    "content": "1.1 RELATED WORKS",
+                                    "type": "text"
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+                                    "content": "The study of nonmonotonicity in learning algorithms existed prior to double descent and has a long",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "content": "history going back to (at least) Trunk (1979) and LeCun et al. (1991); Le Cun et al. (1991), where the",
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "former was largely empirical observations and the latter studied the sample non-nonmonotonicity of",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 28
+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "unregularized linear regression in terms of the eigenspectrum of the covariance matrix; the difference",
+                                    "type": "text"
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+                            ],
+                            "index": 29
+                        },
+                        {
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "to our works is that we study this in the context of optimal regularization. In fact, Duin (1995;",
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+                        },
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+                                    "score": 1.0,
+                                    "content": "2000); Opper (2001); Loog & Duin (2012). Loog et al. (2019) introduces the same notion of risk",
+                                    "type": "text"
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+                            ],
+                            "index": 31
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+                                    "score": 1.0,
+                                    "content": "monotonicity which we consider, and studies several examples of monotonic and non-monotonic",
+                                    "type": "text"
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+                            ],
+                            "index": 32
+                        },
+                        {
+                            "bbox": [
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+                                    "content": "procedures.",
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+                        }
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+                                    "content": "Double descent of test risk as a function of model size was considered recently in more generality",
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+                                    "score": 1.0,
+                                    "content": "by Belkin et al. (2018). Similar behavior was observed empirically in earlier work in somewhat",
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "more restricted settings Trunk (1979); Opper (1995; 2001); Skurichina & Duin (2002); Le Cun et al.",
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+                            "index": 36
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+                                    "content": "(1991); LeCun et al. (1991) and more recently in Advani & Saxe (2017); Geiger et al. (2019a);",
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+                            "index": 37
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+                                    "score": 1.0,
+                                    "content": "Spigler et al. (2018); Neal et al. (2018). Recently Nakkiran et al. (2020) demonstrated a generalized",
+                                    "type": "text"
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+                            ],
+                            "index": 38
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "double descent phenomenon on modern deep networks, and highlighted “sample non-monotonicity”",
+                                    "type": "text"
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+                            ],
+                            "index": 39
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "as an aspect of double descent.",
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+                {
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+                                    "score": 1.0,
+                                    "content": "A recent stream of theoretical works consider model-wise double descent in simplified settings—",
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+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "often via linear models for regression or classification. This also connects to works on high-",
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+                            ],
+                            "index": 42
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "dimentional regression in the statistics literature. A partial list of works in these areas include",
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+                            ],
+                            "index": 43
+                        },
+                        {
+                            "bbox": [
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+                                    "content": "Belkin et al. (2019); Hastie et al. (2019); Bartlett et al. (2019); Muthukumar et al. (2019); Bibas",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 44
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        104,
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+                                    "score": 1.0,
+                                    "content": "et al. (2019); Mitra (2019); Mei & Montanari (2019); Liang & Rakhlin (2018); Liang et al. (2019);",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 45
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    "score": 1.0,
+                                    "content": "Xu & Hsu (2019); Derezinski et al. (2019); Lampinen & Ganguli (2018); Deng et al. (2019); Nakki- ´",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 46
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        678
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+                                    "score": 1.0,
+                                    "content": "ran (2019); Mahdaviyeh & Naulet (2019); Dobriban et al. (2018); Dobriban & Sheng (2019); Kobak",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 47
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "et al. (2018). Of these, most closely related to our work are Hastie et al. (2019); Dobriban et al.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 48
+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "(2018); Mei & Montanari (2019). Specifically, Hastie et al. (2019) considers the risk of unregular-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 49
+                        },
+                        {
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+                                    "bbox": [
+                                        105,
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+                                    "score": 1.0,
+                                    "content": "ized and regularized linear regression in an asymptotic regime, where dimension",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.75,
+                                    "content": "d",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "and number of",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 50
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "samples",
+                                    "type": "text"
+                                },
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+                                    "score": 0.66,
+                                    "content": "n",
+                                    "type": "inline_equation"
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+                                {
+                                    "bbox": [
+                                        150,
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+                                    "score": 1.0,
+                                    "content": "scale to infinity together, at a constant ratio",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                        722
+                                    ],
+                                    "score": 0.9,
+                                    "content": "d / n",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        348,
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+                                        722
+                                    ],
+                                    "score": 1.0,
+                                    "content": ". In contrast, we show non-asymptotic",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 51
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "results, and are able to consider increasing the number of samples for a fixed model, without scaling",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 52
+                        },
+                        {
+                            "bbox": [
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+                                    "content": "\\begin{array} { r } { \\hat { \\beta } _ { d } ^ { \\mathrm { o p t } } : = \\operatorname * { a r g m i n } _ { \\beta } | | \\widetilde { X } \\beta - \\vec { y } | | _ { 2 } ^ { 2 } + \\lambda _ { d } ^ { \\mathrm { o p t } } | | \\beta | | _ { 2 } ^ { 2 } } \\end{array}",
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+                                    "score": 1.0,
+                                    "content": ". Now, our main theorem in this setting",
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+                                    "content": "Theorem 3. In the setting above, the expected test risk of the optimally-regularized model is mono-",
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+                                    "content": "d .",
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+                                    "content": ". That is, for all",
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+                                    "score": 0.89,
+                                    "content": "p \\in \\mathbb { N } , \\theta \\in \\mathbb { R } ^ { p } , d \\leq p , n \\in \\mathbb { N } , \\sigma > 0",
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+                                    "score": 1.0,
+                                    "content": ", we have",
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+                                    "score": 0.9,
+                                    "content": "\\overline { { R } } ( \\hat { \\beta } _ { d + 1 } ^ { \\mathrm { o p t } } ) \\leq \\overline { { R } } ( \\hat { \\beta } _ { d } ^ { \\mathrm { o p t } } )",
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+                                    "content": "The proof of Theorem 3 is in Appendix A.2, and follows closely the proof of Theorem 1.",
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+                                    "content": "4 COUNTEREXAMPLES TO MONOTONICITY",
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+                                    "content": "In this section, we show that optimally-regularized ridge regression is not always monotonic in",
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+                                    "score": 1.0,
+                                    "content": "samples. We give a numeric counterexample in",
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+                                    "content": "d = 2",
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+                                    "score": 1.0,
+                                    "content": "dimensions, with non-gaussian covariates and",
+                                    "type": "text"
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+                            "index": 18
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+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "heteroscedastic noise. This does not contradict our main theorem in Section 2, since this distribution",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "is not jointly Gaussian with isotropic marginals.",
+                                    "type": "text"
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+                            "index": 20
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+                                    "score": 1.0,
+                                    "content": "Counterexample. Here we give an example of a distribution",
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+                                    "content": "( x , y )",
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+                                    "score": 1.0,
+                                    "content": "for which the expected error of",
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+                                    "content": "n = 1",
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+                                    "content": "samples. This",
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+                                    "score": 1.0,
+                                    "content": "counterexample is most intuitive to understand when the ridge parameter",
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+                                    "score": 0.79,
+                                    "content": "\\lambda",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "is allowed to depend on",
+                                    "type": "text"
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+                            "index": 23
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+                        {
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+                                    "score": 1.0,
+                                    "content": "the specific sample instance",
+                                    "type": "text"
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+                                    "content": "( X , { \\vec { y } } )",
+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    "score": 1.0,
+                                    "content": "as well as",
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+                                    "content": "n ^ { 1 }",
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+                                    "score": 1.0,
+                                    "content": ". We sketch the intuition for this below. Consider the",
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+                                    "content": "d = 2",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "dimensions. This distribution has one “clean” coordinate",
+                                    "type": "text"
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+                            ],
+                            "index": 25
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "and one “noisy” coordinate. The distribution is:",
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+                                    "content": "( x , y ) = ( { \\vec { e } } _ { 1 } , 1 )",
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+                                    "content": "with probability",
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+                                    "content": "( x , y ) =",
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+                                    "content": "( \\vec { e } _ { 2 } , \\pm 1 0 )",
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+                                    "content": "w.p. 1/2. Where",
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+                                    "score": 1.0,
+                                    "content": "is uniformly random independent noise. This distribution is “well-",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "specified” in that the optimal predictor is linear in",
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+                                    "content": "x",
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+                                    "content": ": \\mathbb { E } [ y | x ] = \\langle \\beta ^ { * } , x \\rangle",
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+                                },
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+                                    "content": "\\beta ^ { * } = [ 1 , 0 ]",
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+                                    "score": 1.0,
+                                    "content": ". However, the",
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+                                    "content": "The proof of Theorem 3 is in Appendix A.2, and follows closely the proof of Theorem 1.",
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+                                    "content": "4 COUNTEREXAMPLES TO MONOTONICITY",
+                                    "type": "text"
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+                                    "content": "In this section, we show that optimally-regularized ridge regression is not always monotonic in",
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+                                    "content": "d = 2",
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+                                    "score": 1.0,
+                                    "content": "dimensions, with non-gaussian covariates and",
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+                                    "score": 1.0,
+                                    "content": "heteroscedastic noise. This does not contradict our main theorem in Section 2, since this distribution",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "is not jointly Gaussian with isotropic marginals.",
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+                                    "score": 1.0,
+                                    "content": "Counterexample. Here we give an example of a distribution",
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+                                    "content": "( x , y )",
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+                                    "score": 1.0,
+                                    "content": "for which the expected error of",
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+                                    "content": "n = 2",
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+                                    "content": "n = 1",
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+                                    "score": 1.0,
+                                    "content": "samples. This",
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+                                    "score": 1.0,
+                                    "content": "counterexample is most intuitive to understand when the ridge parameter",
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+                                    "score": 0.79,
+                                    "content": "\\lambda",
+                                    "type": "inline_equation"
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "is allowed to depend on",
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+                                    "score": 1.0,
+                                    "content": "the specific sample instance",
+                                    "type": "text"
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+                                    "content": "( X , { \\vec { y } } )",
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+                                {
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+                                    "score": 1.0,
+                                    "content": "as well as",
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+                                    "content": "n ^ { 1 }",
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+                                    "score": 1.0,
+                                    "content": ". We sketch the intuition for this below. Consider the",
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+                                    "content": "d = 2",
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+                                    "score": 1.0,
+                                    "content": "dimensions. This distribution has one “clean” coordinate",
+                                    "type": "text"
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+                            ],
+                            "index": 25
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "and one “noisy” coordinate. The distribution is:",
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+                                    "content": "( x , y ) = ( { \\vec { e } } _ { 1 } , 1 )",
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+                                    "content": "with probability",
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+                                    "content": "( x , y ) =",
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+                                    "score": 0.91,
+                                    "content": "( \\vec { e } _ { 2 } , \\pm 1 0 )",
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+                                    "score": 1.0,
+                                    "content": "w.p. 1/2. Where",
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+                                    "score": 1.0,
+                                    "content": "is uniformly random independent noise. This distribution is “well-",
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+                        },
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+                                    "score": 1.0,
+                                    "content": "specified” in that the optimal predictor is linear in",
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+                                    "content": "x",
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+                                    "content": ": \\mathbb { E } [ y | x ] = \\langle \\beta ^ { * } , x \\rangle",
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+                                },
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+                                    "content": "\\beta ^ { * } = [ 1 , 0 ]",
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+                                    "score": 1.0,
+                                    "content": ". However, the",
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+                                    "score": 1.0,
+                                    "content": "or large",
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+                                    "score": 1.0,
+                                    "content": "depending on if the",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "sampled coordinate is the “clean” or “noisy” one. Specifically, for the sample",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.9,
+                                    "content": "( x , y )",
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+                                    "content": ": If",
+                                    "type": "text"
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+                                    "bbox": [
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+                                    "score": 0.91,
+                                    "content": "x = \\vec { e } _ { 1 }",
+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    "score": 1.0,
+                                    "content": ", then",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "the optimal ridge parameter is",
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+                                    "content": "\\lambda = 0",
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+                                    "score": 1.0,
+                                    "content": ". If",
+                                    "type": "text"
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+                                {
+                                    "bbox": [
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+                                    "score": 0.92,
+                                    "content": "x = \\vec { e } _ { 2 }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ", then the optimal parameter is",
+                                    "type": "text"
+                                },
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+                                    "score": 1.0,
+                                    "content": "numerically (via Mathematica) that the expected risk",
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+                                    "score": 0.91,
+                                    "content": "\\overline { { R } } ( \\hat { \\beta } _ { n = 2 } ^ { \\mathrm { o p t } } ) ~ > ~ 8 . 1 7 9",
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+                                    "score": 1.0,
+                                    "content": ". This is achieved with",
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+                                    "content": "\\lambda = 0 . 6 4 2 5 2 5",
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+                                    "score": 1.0,
+                                    "content": ".",
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+                                    "content": "□",
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+                                    "content": "5 EXPERIMENTS",
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+                                    "score": 1.0,
+                                    "content": "We now experimentally demonstrate that optimal",
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+                                    "content": "regularization can mitigate double descent, in",
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+                                    "score": 1.0,
+                                    "content": "more general settings than Theorems 1 and 3.",
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+                                    "content": "5.1 SAMPLE MONOTONICITY",
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+                                    "score": 1.0,
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+                                    "content": "Nonisotropic Regression. We first consider the setting of Theorem 1, but with non-isotropic co-",
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+                                    "score": 1.0,
+                                    "content": ". That is, we perform ridge regression on samples",
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+                                    "content": "\\boldsymbol { x } \\in \\mathbb { R } ^ { d }",
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+                                    "score": 1.0,
+                                    "content": ". As before, the response is generated by",
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+                                    "score": 1.0,
+                                    "content": ". We consider the same ridge regression",
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+                                    "score": 0.91,
+                                    "content": "\\overline { { R } } ( \\hat { \\beta } _ { n = 2 } ^ { \\mathrm { o p t } } ) ~ > ~ 8 . 1 7 9",
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+                                    "content": ". This is achieved with",
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+                                    "content": "□",
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+                                    "content": "5 EXPERIMENTS",
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+                                    "content": "more general settings than Theorems 1 and 3.",
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+                                    "content": ". That is, we perform ridge regression on samples",
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+                                    "score": 1.0,
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+                                    "content": ". In this setting, optimally-regularized ridge regression is empirically monotonic",
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+                                    "score": 1.0,
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+                            ],
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+                                    "score": 1.0,
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+                                    "content": "\\tilde { x } : = \\mathrm { R e L U } ( W x )",
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+                                    "content": "and",
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "loss and weight decay. In Appendix A.4, we",
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+                                    "score": 1.0,
+                                    "content": "apply random features to Fashion-MNIST Xiao et al. (2017). From Appendix Figure 4a, we see that",
+                                    "type": "text"
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+                                    "type": "text"
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+                                    "score": 1.0,
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "appears to be constant for all",
+                                    "type": "text"
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+                                {
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+                                    "score": 1.0,
+                                    "content": ", similar to our results from",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "the isotropic setting in Theorem 1.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 9
+                        }
+                    ],
+                    "index": 4.5
+                },
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+                    "type": "title",
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+                                    "content": "5.2 MODEL-SIZE MONOTONICITY",
+                                    "type": "text"
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+                                    "content": ", and we use standard data-augmentation of random horizontal flip and random crop with 4-",
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+                                    "score": 1.0,
+                                    "content": "pixel padding. All models are trained using Stochastic Gradient Descent (SGD) on the cross-entropy",
+                                    "type": "text"
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+                                    "content": "0 . 1 / \\sqrt { \\lfloor { T / 5 1 2 } \\rfloor + 1 }",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": ". We train for 1e6 gradient steps, and use weight",
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+                                    "score": 1.0,
+                                    "content": ". Due to optimization instabilities for large",
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+                                    "score": 1.0,
+                                    "content": ", we use the model with the min-",
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+                                    "score": 1.0,
+                                    "content": "imum train loss among the last 5K gradient steps. Figure 3 shows the test error of these models on",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "CIFAR-100. Although unregularized and under-reguarized models exhibit double descent, the test",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 53
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "error of optimally-regularized models is largely monotonic. Note that the optimal regularization",
+                                    "type": "text"
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+                                {
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+                                    "content": "\\lambda",
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+                            ],
+                            "index": 54
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+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "varies with the model size — no single regularization value is optimal for all models.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 55
+                        }
+                    ],
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+                                    "content": "6 DISCUSSION AND CONCLUSION",
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+                        {
+                            "bbox": [
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+                                    "content": "show that, while unregularized or under-regularized models often have non-monotonic behavior,",
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+                                    "content": "regularization achieves monotonic test performance as we grow either the sample size or",
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+                                    "score": 1.0,
+                                    "content": "the model size. These are the first non-asymptotic monotonicity results we are aware of in linear",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 62
+                        }
+                    ],
+                    "index": 61
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+            ],
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+                                    "score": 1.0,
+                                    "content": "Published as a conference paper at ICLR 2021",
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+                            ],
+                            "index": 0
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+                                    "score": 1.0,
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+                            "index": 1
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+                                    "score": 1.0,
+                                    "content": "random feature classifier is obtained by performing regularized linear regression on the embedding√",
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+                            "index": 2
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+                            "bbox": [
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+                                    "score": 0.9,
+                                    "content": "\\tilde { x } : = \\mathrm { R e L U } ( W x )",
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+                                    "score": 0.93,
+                                    "content": "W \\in \\mathbb { R } ^ { D \\times d }",
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "is a matrix with each entry sampled i.i.d",
+                                    "type": "text"
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+                                {
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+                                    "score": 0.93,
+                                    "content": "\\mathcal { N } ( 0 , 1 / \\sqrt { d } )",
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+                                {
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+                                    "score": 1.0,
+                                    "content": "and",
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+                                    "type": "text"
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "loss and weight decay. In Appendix A.4, we",
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+                                    "score": 1.0,
+                                    "content": "apply random features to Fashion-MNIST Xiao et al. (2017). From Appendix Figure 4a, we see that",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "appears to be constant for all",
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+                                    "score": 1.0,
+                                    "content": ", similar to our results from",
+                                    "type": "text"
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+                                    "content": "0 . 1 / \\sqrt { \\lfloor { T / 5 1 2 } \\rfloor + 1 }",
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+                                    "score": 1.0,
+                                    "content": ". We train for 1e6 gradient steps, and use weight",
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+                                    "score": 1.0,
+                                    "content": ". Due to optimization instabilities for large",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": ", we use the model with the min-",
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+                                    "score": 1.0,
+                                    "content": "imum train loss among the last 5K gradient steps. Figure 3 shows the test error of these models on",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 52
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "CIFAR-100. Although unregularized and under-reguarized models exhibit double descent, the test",
+                                    "type": "text"
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+                            ],
+                            "index": 53
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "error of optimally-regularized models is largely monotonic. Note that the optimal regularization",
+                                    "type": "text"
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+                        {
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+                                    "score": 1.0,
+                                    "content": "varies with the model size — no single regularization value is optimal for all models.",
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+                                    "score": 1.0,
+                                    "content": "6 DISCUSSION AND CONCLUSION",
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+                                    "content": "regularization achieves monotonic test performance as we grow either the sample size or",
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+                                    "score": 1.0,
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+                            "index": 1
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+                                    "content": "performance models which do not exhibit unexpected nonmonotonic behaviors.",
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+                                    "score": 1.0,
+                                    "content": "Open Questions. Our work suggests a number of natural open questions. First, it is open to",
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+                            ],
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+                        },
+                        {
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+                                    "content": "prove (or disprove) that optimal ridge regression is sample-monotonic for non-isotropic Gaussian",
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+                                    "score": 1.0,
+                                    "content": "covariates. We conjecture that it is, and outline a potential route to proving this (via Conjectures 1",
+                                    "type": "text"
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+                        },
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+                                    "score": 1.0,
+                                    "content": "and 2 in the Appendix). Second, more broadly, it is open to prove sample-wise or model-wise",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "monotonicity for more general (non-linear) models with appropriate regularizers. Finally, it is open",
+                                    "type": "text"
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+                                    "content": "and 2 in the Appendix). Second, more broadly, it is open to prove sample-wise or model-wise",
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+                                    "content": "Proof. This proof follows exactly analogously as the proof of Lemma 2 from Lemma 1, in Sec-",
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+                                    "content": "Proof of Theorem 3. This follows analogously to the proof of Theorem 1, Let",
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+                                    "content": "B.2 TOWARDS PROVING MONOTONICITY",
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diff --git a/parse/train/7R7fAoUygoa/7R7fAoUygoa_model.json b/parse/train/7R7fAoUygoa/7R7fAoUygoa_model.json
new file mode 100644
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diff --git a/parse/train/ByRWCqvT-/ByRWCqvT-.md b/parse/train/ByRWCqvT-/ByRWCqvT-.md
new file mode 100644
index 0000000000000000000000000000000000000000..3217906cfb4e299ad2a7a9215d5c3f8bf404912d
--- /dev/null
+++ b/parse/train/ByRWCqvT-/ByRWCqvT-.md
@@ -0,0 +1,384 @@
+# LEARNING TO CLUSTER IN ORDER TO TRANSFER ACROSS DOMAINS AND TASKS
+
+Yen-Chang Hsu, Zhaoyang Lv   
+Georgia Institute of Technology   
+Atlanta, GA 30332, USA   
+{yenchang.hsu, zhaoyang.lv}@gatech.edu
+
+Zsolt Kira Georgia Tech Research Institute Atlanta, GA 30318, USA zkira@gatech.edu
+
+# ABSTRACT
+
+This paper introduces a novel method to perform transfer learning across domains and tasks, formulating it as a problem of learning to cluster. The key insight is that, in addition to features, we can transfer similarity information and this is sufficient to learn a similarity function and clustering network to perform both domain adaptation and cross-task transfer learning. We begin by reducing categorical information to pairwise constraints, which only considers whether two instances belong to the same class or not (pairwise semantic similarity). This similarity is category-agnostic and can be learned from data in the source domain using a similarity network. We then present two novel approaches for performing transfer learning using this similarity function. First, for unsupervised domain adaptation, we design a new loss function to regularize classification with a constrained clustering loss, hence learning a clustering network with the transferred similarity metric generating the training inputs. Second, for cross-task learning (i.e., unsupervised clustering with unseen categories), we propose a framework to reconstruct and estimate the number of semantic clusters, again using the clustering network. Since the similarity network is noisy, the key is to use a robust clustering algorithm, and we show that our formulation is more robust than the alternative constrained and unconstrained clustering approaches. Using this method, we first show state of the art results for the challenging cross-task problem, applied on Omniglot and ImageNet. Our results show that we can reconstruct semantic clusters with high accuracy. We then evaluate the performance of cross-domain transfer using images from the Office-31 and SVHN-MNIST tasks and present top accuracy on both datasets. Our approach doesn’t explicitly deal with domain discrepancy. If we combine with a domain adaptation loss, it shows further improvement.
+
+# 1 INTRODUCTION
+
+Supervised learning has made significant strides in the past decade, with substantial advancements arising from the use of deep neural networks. However, a large part of this success has come from the existence of extensive labeled datasets. In many situations, it is not practical to obtain such data due to the amount of effort required or when the task or data distributions change dynamically. To deal with these situations, the fields of transfer learning and domain adaptation have explored how to transfer learned knowledge across tasks or domains. Many approaches have focused on cases where the distributions of the features and labels have changed, but the task is the same (e.g., classification across datasets with the same categories). Cross-task transfer learning strategies, on the other hand, have been widely adopted especially in the computer vision community where features learned by a deep neural network on a large classification task have been applied to a wide variety of other tasks (Donahue et al., 2014).
+
+Most of the prior cross-task transfer learning works, however, require labeled target data to learn classifiers for the new task. If labels of the target data are absent, there is little choice other than to apply unsupervised approaches such as clustering on the target data with pre-trained feature representations. In this paper, we focus on the question of what can be transferred (besides features) to support both cross-domain and cross-task transfer learning. We address it with a learned similarity function as the fundamental component of clustering. Clustering can then be realized using a neural network trained using the output of the similarity function, which can be successfully used to achieve both cross-task and cross-domain transfer.
+
+The key idea is to formulate the clustering objective to use a learnable (and transferable) term, which in our proposed work is a similarity prediction function. Our proposed objective function can be easily combined with deep neural networks and optimized end-to-end. The features and clustering are optimized jointly, hence taking advantage of such side information in a robust way. Using this method, we show that unsupervised learning can benefit from learning performed on a distinct task, and demonstrate the flexibility of further combining it with a classification loss and domain discrepancy loss.
+
+In summary, we make several contributions. First, we propose to use predictive pairwise similarity as the knowledge that is transferred and formulate a learnable objective function to utilize the pairwise information in a fashion similar to constrained clustering. We then provide the methodologies to deploy the objective function in both cross-task and cross-domain scenarios with deep neural networks. The experimental results for cross-task learning on Omniglot and ImageNet show that we can achieve state of the art clustering results with predicted similarities. On the standard domain adaptation benchmark Office-31 dataset, we demonstrate improvements over state-of-art even when not performing any explicit domain adaptation, and further improvements if we do. Finally, on another domain adaptation task, SVHN-to-MNIST, our approach using Omniglot as the auxiliary dataset achieves top performance with a large margin.
+
+# 2 RELATED WORK
+
+Transfer Learning: Transfer learning aims to leverage knowledge from the source domain to help learn in the target domain, while only focusing on the performance on the target domain. The type of transferred knowledge includes training instances, features, model parameters and relational knowledge (Pan & Yang, 2010). Pairwise similarity is the meta-knowledge we propose to transfer, which falls in between the last two types. The similarity prediction function is a neural network with learned parameters, while the output is the most simplified form of relational knowledge, i.e., only considering pairwise semantic similarity.
+
+Cross-task Transfer Learning: Features learned when trained for ImageNet classification (Russakovsky et al., 2015) have boosted the performance of a variety of vision tasks in a supervised setting. For example, new classification tasks (Donahue et al., 2014; Yosinski et al., 2014), object detection (Girshick et al., 2014), semantic segmentation (Long et al., 2015a), and image captioning (Vinyals et al., 2015). Translated Learning (Dai et al., 2009) has an unsupervised setting similar to ours, but it again focuses only on transferring features across tasks. Our work explores how learning could benefit from transferring pairwise similarity in an unsupervised setting.
+
+Cross-domain Transfer Learning: Also known as domain adaptation (Pan & Yang, 2010), there has recently been a large body of work dealing with domain shift between image datasets by minimizing domain discrepancy (Tzeng et al., 2014; Long et al., 2015b; Ganin et al., 2016; Sun et al., 2016; Long et al., 2016; Sener et al., 2016; Carlucci et al., 2017; Long et al., 2017; Zellinger et al., 2017; Bousmalis et al., 2017). We address the problem in a complementary way that transfers extra information from the auxiliary dataset and show a larger performance boost with further gains using an additional domain discrepancy loss.
+
+Constrained Clustering: Constrained clustering algorithms can be categorized by how they utilize constraints (e.g. pairwise information). The first set of work use the constraints to learn a distance metric. For example, DML (Xing et al., 2003), ITML (Davis et al., 2007), SKMS (Anand et al., 2014), SKKm (Anand et al., 2014; Amid et al., 2016), and SKLR (Amid et al., 2016). This group of approaches closely relates to metric learning and needs a clustering algorithm such as K-means in a separate stage to obtain cluster assignments. The second group of work use constraints to formulate the clustering loss. For example, CSP (Wang et al., 2014) and COSC (Rangapuram & Hein, 2012). The third group uses constraints for both metric learning and the clustering objective, such as MPCKMeans (Bilenko et al., 2004) and CECM (Antoine et al., 2012). The fourth group does not use constraints at all. A generic clustering algorithms such as K-means (MacQueen et al., 1967), LSC (Chen & Cai, 2011), and LPNMF (Cai et al., 2009) all belong to this category. There is a long list of associated works and they are summarized in survey papers, e.g. Davidson & Basu (2007) and Dinler & Tural (2016). Our proposed clustering strategy belongs to the third group. The constrained clustering methods above are applied to the semi-supervised setting where the groundtruth constraints are sparsely available. In our unsupervised setting, the ground-truth is unavailable but predicted constraints are densely available. We include all four groups of algorithms in our comparison and show the advantages of the third group.
+
+![](images/e45febb1495b4ec74c335fc4134b2e9707a9552a0aad1ab6b240e7e63479843e.jpg)  
+Figure 1: Overview of the transfer scheme with a learnable clustering objective (LCO). The LCO and pairwise similarity are the two key components of our approach and are described in section 4. The dashed rectangles and light gray arrows are only available in cross-domain transfer. Details are described in section 3.
+
+# 3 THE TRANSFER LEARNING TASKS
+
+To define the transfer learning problem addressed in this work, we follow the notations used by Pan & Yang (2010). The goal is to transfer knowledge from source data $\boldsymbol { S } = ( X _ { S } , Y _ { S } )$ , where $X _ { S }$ is the set of data instances and $Y _ { S }$ is the corresponding categorical labels, to a target data noted as $T = \left( X _ { T } , Y _ { T } \right)$ . The learning is unsupervised since $Y _ { T }$ is unknown. The scenario is divided into two cases. One is $\{ Y _ { T } \} \ne \{ { \bar { Y } } _ { S } \}$ , which means the set of categories are not the same and hence the transfer is across tasks. The second case is $\{ Y _ { T } \} = \{ Y _ { S } \}$ , but with a domain shift. In other words, the marginal probability distributions of the input data are different, i.e., $P ( X _ { T } ) \neq P ( X _ { S } )$ . The latter is a cross-domain learning problem also called transductive learning. The domain adaptation approaches which have gained significant attention recently belong to the second scenario.
+
+To align with common benchmarks for evaluating transfer learning performance, we further use the notion of an auxiliary dataset and split the source data into $S = { \bar { S } } ^ { \prime } \cup A$ . $S ^ { \prime } = ( X _ { S } ^ { \prime } , Y _ { S } ^ { \prime } )$ which is only present in the cross-domain transfer scheme and has $\{ Y _ { S } ^ { \prime } \} = \{ Y _ { T } \}$ . $A = ( X _ { A } , Y _ { A } )$ is the auxiliary dataset which has a large amount of labeled data and potentially categories as well, and may or may not contain the categories of $Y _ { T }$ . For the cross task scenario, only $A$ and unlabeled $T$ are included, while cross-domain transfer involves $A$ , $S ^ { \prime }$ , and $T$ . In the following sections we use the above notations and describe the two transfer learning tasks in detail. Figure 1 illustrates how our approach relates to both tasks.
+
+Transfer across tasks: If the target task has different categories, we cannot directly transfer the classifier from source to target, and there is no labeled target data to use for fine-tuning transferred features. Here we propose to first reduce the categorization problem to a surrogate same-task problem. We can directly apply transductive transfer learning (Pan & Yang, 2010) to the transformed task. The cluster structure in the target data is then reconstructed using the predictions in the transformed task. See figure 2 for an illustration.
+
+The source involves a labeled auxiliary dataset $A$ and an unlabeled target dataset $T$ . $Y _ { T }$ is the target that must be inferred. In this scenario the set of categories $\{ Y _ { A } \} \ne \{ Y _ { T } \}$ , and $Y _ { T }$ is unknown. We first transform the problem of categorization into a pairwise similarity prediction problem. In other words, we specify a transformation function $R$ such that $R ( A ) = ( X _ { A } ^ { R ^ { \prime } } , \bar { Y } ^ { R } )$ , and $\dot { X } _ { A } ^ { R } = \{ ( x _ { A , i } , x _ { A , j } ) \} _ { \forall i , j }$ contains all pairs of data, where $\{ Y ^ { R } \} = \{ d i s s i m i l a r , s i m i l a r \}$ . The transformation on the labeled auxiliary data is straightforward. It will be similar if two data instances are from the same category, and vice versa. We then use it to learn a pairwise similarity prediction function $G ( x _ { i } , x _ { j } ) = \mathbf { \bar { \Gamma } } y _ { i , j }$
+
+![](images/2b04f9cd1aa8b76244e957c10c6c523af984b0b1e4ea8542e4f1f432440b0e65.jpg)  
+Figure 2: The concept for reconstructing clusters of unseen categories. The proposed approach follows the arrows in the counter-clockwise direction, which converts cross-task transfer learning to cross-domain transfer learning. The colors of dots represent data of different categories. The hollow circle and cross symbol represent similar and dissimilar data pairs. The $G$ function and the cluster reconstruction (via constrained clustering) are the two key components in the diagram.
+
+By applying $G$ on $T$ , we can obtain $G ( x _ { T , i } , x _ { T , j } ) = y _ { T , i , j }$ . The last step is to infer $Y _ { T }$ from $Y _ { T } ^ { R } = \{ y _ { T , i , j } \} _ { \forall i , j }$ , which can be solved using constrained clustering algorithms. Note that since the actual $Y _ { T }$ is unknown, the algorithm only infers the indices of categories, which could be in arbitrary order. The resulting clusters are expected to contain coherent semantic categories.
+
+Transfer across domains: The problem setting we consider here is the same as unsupervised domain adaptation. Following the standard evaluation procedure (Long et al., 2017; Ganin et al., 2016), the labeled datasets $A$ is ImageNet and $S ^ { \prime }$ is one domain in the Office-31 dataset. The unlabeled $T$ is another domain in Office-31. The goal is to enhance classification performance on $T$ by utilizing $A$ , $S ^ { \prime }$ , and $T$ together.
+
+# 4 THE LEARNABLE CLUSTERING OBJECTIVE (LCO)
+
+The key to our approach is the design of a learning objective that can use (noisy) predicted pairwise similarities, and is inspired from constrained clustering which involves using pairwise information in the loss function. The pairwise information is called must-link/cannot-link constraints or similar/dissimilar pairs (we use the latter). Note that the information is binarized to one and zero for similar and dissimilar pairs, accordingly.
+
+Although many constrained clustering algorithms have been developed, few of them are scalable with respect to the number of pairwise relationships. Further, none of them can be easily integrated into deep neural networks. Inspired by the work of Hsu & Kira (2016), we construct a contrastive loss for clustering on the probability outputs of a softmax classifier. However, each output node does not have to map to a fixed category but instead each output node represents a probabilistic assignment of a data point to a cluster. The assignment between output nodes and clusters are formed stochastically during the optimization and is guided by the pairwise similarity. If there is a similar pair, their output distribution should be similar, and vice-versa.
+
+Specifically, we use the pair-wise KL-divergence to evaluate the distance between $k$ cluster assignment distributions of two data instances, and use predicted similarity to construct the contrastive loss. Given a pair of data $x _ { p } , x _ { q }$ , their corresponding output distributions are defined as $\mathcal { P } = f ( x _ { p } )$ and $\mathcal { Q } = f ( x _ { q } )$ , while $f$ is the neural network. The cost of a similar pair is described as :
+
+$$
+\mathcal { L } ( \boldsymbol { x } _ { p } , \boldsymbol { x } _ { q } ) ^ { + } = \mathcal { D } _ { K L } ( \mathcal { P } ^ { \star } | | \boldsymbol { \mathcal { Q } } ) + \mathcal { D } _ { K L } ( \mathcal { Q } ^ { \star } | | \mathcal { P } )
+$$
+
+$$
+\mathcal { D } _ { K L } ( \mathcal { P } ^ { \star } | | \mathcal { Q } ) = \sum _ { c = 1 } ^ { k } p _ { c } l o g ( \frac { p _ { c } } { q _ { c } } )
+$$
+
+![](images/b7321399ab1bca7f870eb89cd648d445662c407854095766daa46df0319beb58.jpg)  
+Figure 3: The similarity prediction network (the $G$ function).
+
+The cost $\mathcal { L } ( x _ { p } , x _ { q } ) ^ { + }$ is symmetric w.r.t. $x _ { p } , x _ { q }$ , in which ${ \mathcal { P } } ^ { \star }$ and $\mathcal { Q } ^ { \star }$ are alternatively assumed to be constant. Each KL-divergence factor $\hat { \mathcal { D } } _ { K L } ^ { \star } ( \hat { \mathcal { P } } ^ { \star } | | \mathcal { Q } )$ becomes a unary function whose gradient is simply $\partial { \cal D } _ { K L } ( { \mathcal { P } } ^ { \star } | | { \mathcal { Q } } ) / \partial { \mathcal { Q } }$ .
+
+If $x _ { p } , x _ { q }$ comes from a pair which is dissimilar, their output distributions are expected to be different, which can be defined as a hinge-loss function:
+
+$$
+\mathcal { L } ( \boldsymbol { x } _ { p } , \boldsymbol { x } _ { q } ) ^ { - } = L _ { h } ( \mathcal { D } _ { K L } ( \mathcal { P } ^ { \star } | | \boldsymbol { \mathcal { Q } } ) , \boldsymbol { \sigma } ) + L _ { h } ( \mathcal { D } _ { K L } ( \mathcal { Q } ^ { \star } | | \mathcal { P } ) , \boldsymbol { \sigma } )
+$$
+
+$$
+L _ { h } ( e , \sigma ) = m a x ( 0 , \sigma - e )
+$$
+
+Given a pair of data with similarity prediction function $G ( x _ { p } , x _ { q } ) \in \{ 0 , 1 \}$ , which is introduced in section 3, the total loss can be defined as a contrastive loss (where we use integer 1 to represent a similar pair):
+
+$$
+\mathcal { L } ( x _ { p } , x _ { q } ) = G ( x _ { p } , x _ { q } ) \mathcal { L } ( x _ { p } , x _ { q } ) ^ { + } + ( 1 - G ( x _ { p } , x _ { q } ) ) \mathcal { L } ( x _ { p } , x _ { q } ) ^ { - }
+$$
+
+We refer to equation 5 as LCO. Function $G$ is the learnable part that utilizes prior knowledge and is trained with auxiliary dataset $A$ before optimizing LCO. Two particular characteristics of the clustering criterion are worth mentioning: (1) There is no need to define cluster centers. (2) There is no predefined metric applied on the feature representation. Instead, the divergence is calculated directly on the cluster assignment; therefore, both feature representation and clustering are jointly optimized using back-propagation through the deep neural networks.
+
+# 4.1 THE PAIRWISE SIMILARITY PREDICTION NETWORK
+
+Although there is no restriction of how $G$ should be constructed, we choose deep convolution neural networks due to their efficiency in vision tasks. We design a network architecture inspired from Zagoruyko & Komodakis (2015). While they use it to predict image patch similarity, we use it to predict image-level semantic similarity. However, the Siamese architecture used in Zagoruyko & Komodakis (2015) is not efficient in both training and inference, especially when pairwise information is dense. Therefore, instead of using Siamese architecture, we keep the single column backbone but add a pair-enumeration layer on top of the feature extraction network. The pair-enumeration layer enumerates all pairs of feature vectors within a mini-batch and concatenates the features. Suppose the input of the layer is $1 0 \times 5 1 2$ with the mini-batch size 10 and the feature dimension 512; then the output of the pair-enumeration layer will be $1 0 0 \times 1 0 2 4$ (self-pairs included).
+
+The architecture is illustrated in figure 3. We add one hidden fully connected layer on top of the enumeration layer and a binary classifier at the end. We use the standard cross-entropy loss and train it end-to-end. The supervision for training was obtained by converting the ground-truth category labels into binary similarity, i.e., if two samples are from the same class then their label will be similar, otherwise dissimilar. The inference is also end-to-end, and it outputs predictions among all similarity pairs in a mini-batch. The output probability $g \in [ 0 , 1 ]$ with 1 means more similar. We binarize $g$ at 0.5 to obtain discrete similarity predictions. In the following sections, we simplified the notation of the pairwise similarity prediction network as $G$ . Once $G$ is learned, it then works as a static function in our experiments.
+
+# 4.2 THE OBJECTIVE FUNCTION WITH DENSE SIMILARITY PREDICTION
+
+Since the pairwise prediction of a mini-batch can be densely obtained from $G$ , to efficiently utilize the pair-wise information without forwarding each data multiple times we also combine the pairenumeration layer described in section 4.1 with equation 5. In this case, the outputs of softmax are
+
+![](images/6ee14a32f3d38107f344e558ad27ba8aa4c6dd2a6632d85ddac1a4bb2284c5fa.jpg)  
+Figure 4: The constrained clustering network (CCN) for transfer learning across tasks. The input is unlabeled target data $T$ . The cluster assignment block contains two fully connected layers and has the number of output nodes equal to $k$ . The $f$ described in section 4 is the backbone network plus the cluster assignment block. To optimize LCO, the full pipeline in the diagram is used. After the optimization, it uses another forward propagation with only $f$ to obtain the final cluster assignment.
+
+![](images/480735bbbbbc31b0524612dd6b214d318a05c8bfdb6bc3ec5a4e395d164bbae6.jpg)  
+Figure 5: The network for transfer learning across domains. The input is the mix of $S ^ { \prime }$ and $T$ . The architecture is a direct extension of CCN. We use $\mathrm { C C N ^ { + } }$ to represent the mandatory parts (upper branch) which implements eq. (7). $\mathrm { C C N ^ { + + } }$ includes the domain adaptation method (optional branch).
+
+enumerated in pairs. Let $D$ be the set of all tuples $( p , q )$ , while $p$ and $q$ are the indices of a sample in a mini-batch. The dense clustering loss $\mathcal { L } _ { d }$ for a mini-batch is calculated by:
+
+$$
+\mathcal { L } _ { d } = \sum _ { \forall ( p , q ) \in D } \mathcal { L } ( x _ { p } , x _ { q } ) .
+$$
+
+We use $\mathcal { L } _ { d }$ standalone with deep neural networks to reconstruct semantic clusters and for transfer learning across tasks (figure 4). We call the architecture the constrained clustering network (CCN).
+
+# 4.3 COMBINING WITH OTHER OBJECTIVES
+
+In the cross-domain case, we additionally have labeled source data. This enables us to use LCO for $T$ while using classification loss for $S ^ { \prime }$ . The overall training procedure is similar to previous domain adaptation approaches in that both source and target data are mixed in a mini-batch, but different losses are applied. We denote the source domain images in a mini-batch $b$ as $X _ { b ^ { S } }$ and the target domain images $X _ { b ^ { T } }$ with its set of dense pairs $D _ { b ^ { T } }$ . The loss function $\mathcal { L } _ { c d }$ for cross-domain transfer in a mini-batch can then be formulated as:
+
+$$
+\mathcal { L } _ { c d } = \mathcal { L } _ { c l s } + \mathcal { L } _ { c l u s t e r }
+$$
+
+$$
+\mathcal { L } _ { c l s } = \frac { 1 } { | b ^ { S } | } \sum _ { \forall x _ { i } \in X _ { b } s } C r o s s E n t r o p y L o s s ( f ( x _ { i } ) ) .
+$$
+
+$$
+\mathcal { L } _ { c l u s t e r } = \frac { 1 } { | b ^ { T } | ^ { 2 } } \sum _ { \forall ( i , j ) \in D _ { b ^ { T } } } \mathcal { L } ( x _ { i } , x _ { j } ) .
+$$
+
+The $\mathcal { L } _ { c l u s t e r }$ and $\mathcal { L } _ { c l s }$ share the same outputs from network $f$ . Although $\mathcal { L } _ { c l u s t e r }$ does not force the mapping between clusters and categories, $\mathcal { L } _ { c l s }$ does. Therefore the learned classifier can be applied on target data directly and picks the maximum probability for the predicted class. Note that in our loss function, there is no term to explicitly match the feature distribution between source and target; it merely transfers more knowledge in the form of constraints to regularize the learning of the classifier. There is also no requirement for the architecture to utilize hidden layers. Therefore our approach has the large flexibility to be combined with other domain adaptation strategies. Figure 5 illustrates the architectures $\mathrm { C C N ^ { + } }$ and $\mathrm { C C N ^ { + + } }$ used in our cross-domain experiments.
+
+# 5 EXPERIMENTS
+
+This section contains evaluations with four image datasets and covers both cross-task and crossdomain schemes. The details are described below, and the differences between experimental settings are illustrated in appendix A.
+
+# 5.1 UNSUPERVISED CROSS-TASK TRANSFER LEARNING
+
+# 5.1.1 SETUP
+
+The Omniglot dataset (Lake et al., 2015) contains 1623 different handwritten characters and each of them has 20 images drawn by different people. The characters are from 50 different alphabets and were separated to 30 background sets $O m n i g l o t _ { b g }$ and 20 evaluation sets $O m n i g l o t _ { e v a l }$ by the author. We use the $O m n i g l o t _ { b g }$ as the auxiliary dataset $( A )$ and the $O m n i g l o t _ { e v a l }$ as the target data $( T )$ . The total number of characters in $O m n g l o t _ { b g }$ is 964, which can be regarded as the number of categories available to learn the semantic similarity. The goal is to cluster Omniglot $_ { e v a l }$ to reconstruct its semantic categories, without ever having any labels.
+
+The $G$ function has a backbone neural network with four $3 \mathrm { x } 3$ convolution layers followed by $2 \mathbf { x } 2$ max-pooling with stride 2. All hidden layers are followed by batch normalization (Ioffe & Szegedy, 2015) and rectified linear unit. To prepare the inputs for training, the images from $O m n i g l o t _ { b g }$ were resized to $3 2 \mathrm { x } 3 2$ and normalized to zero mean with unit standard deviation. Each mini-batch has a size of 100 and is sampled from a random 20 characters to make sure the amount of similar pairs is reasonable. After pair enumeration, there are 10000 pairs subject to the loss function, which is a two-class cross entropy loss. The ground-truth similarity is obtained by converting the categorical labels. The loss of $G$ is optimized by stochastic gradient descent and is the only part trained in a supervised manner in this section.
+
+The Constrained Clustering Network (CCN) is used to reconstruct the semantic clusters in the target dataset using outputs from $G$ . The network has four 3x3 convolution layers followed by $2 \mathbf { x } 2$ max-pooling with stride 2, one hidden fully-connected layer, and one cluster assignment layer which is also fully connected. The number of output nodes in the last layer is equal to the number of potential clusters in the target data. The output distribution is enumerated in pairs before sending to LCO. The network is randomly initialized, trained end-to-end, and optimized by stochastic gradient descent with randomly sampled 100 images per mini-batch. Note the $G$ function used by LCO is fixed during the optimization. The input data preparation is the same as above, except now the data is from $O m n i g l o t _ { e v a l }$ . Specifically, during the training, the same mini-batch is given to both $G$ and CCN. The dense pairwise similarity predictions from $G$ are sent to LCO and are then fully utilized. The only hyper-parameter in LCO is $\sigma$ , and we set it to 2 for all our experiments.
+
+Omniglot $e v a l$ contains 20 alphabets and each one can be used as a standalone dataset. The 20 target datasets contain a varied number (20 to 47) of characters. Therefore we can evaluate the reconstruction performance under a varied number of ground-truth clusters. We tested the situations when the number of character $( K )$ in an alphabet is known and unknown. When $K$ is known, we set the target number of clusters in the clustering algorithm equal to the true number of characters. If $K$ is unknown, the common practice is to set it to a large number so that the data from different categories will not be forced to be in the same cluster. In the experiment, we merely set $K$ to 100, which is much larger than the largest dataset (47).
+
+All constrained clustering algorithms can be used to reconstruct the semantic clusters for our problem. Since there are mis-predictions in $G$ , robustness to noise is the most important factor. Here we include all four types of constrained clustering algorithms introduced in section 2 as the baselines. We provide the full set of pairwise constraints to all algorithms including ours. In other words, given an alphabet of 20 characters, it contains 400 images and 160000 predicted similarities from $G$ (note that $G$ makes predictions for both orders of a pair and for self-pairs). The full pairwise similarities were presented to all algorithms in random order, while we empirically found it has no noticeable effect on results. We pick the baseline approaches based on code availability and scalability concerning the number of pairwise constraints. Therefore we have shown results for K-means (MacQueen et al., 1967), LPNMF (Cai et al., 2009), LSC (Chen & Cai, 2011), ITML (Davis et al., 2007), SKKm (Anand et al., 2014), SKLR (Amid et al., 2016), SKMS (Anand et al., 2014), CSP (Wang et al., 2014) , and MPCK-means (Bilenko et al., 2004) as our baselines. We use the default parameters for each algorithm provided by the original author except for $K$ , the number of clusters. We use the same normalized images used in CCN for all algorithms.
+
+Table 1: Unsupervised cross-task transfer from Omniglot $b g$ to $O m n i g l o t _ { e v a l }$ . The performance is averaged across 20 alphabets which have 20 to 47 letters. The ACC and NMI without brackets have the number of clusters equal to ground-truth. The "(100)" means the algorithms use $K = 1 0 0$ . The characteristics of how each algorithm utilizes the pairwise constraints are marked in the "Constraints in" column, where metric stands for the metric learning of feature representation.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Constraints in</td><td rowspan="2">ACC</td><td rowspan="2">ACC (100)</td><td rowspan="2">NMI</td><td rowspan="2">NMI (100)</td></tr><tr><td>Metric</td><td>Clustering</td></tr><tr><td>K-means</td><td></td><td></td><td>21.7%</td><td>18.9%</td><td>0.353</td><td>0.464</td></tr><tr><td>LPNMF</td><td></td><td></td><td>22.2%</td><td>16.3%</td><td>0.372</td><td>0.498</td></tr><tr><td>LSC</td><td></td><td></td><td>23.6%</td><td>18.0%</td><td>0.376</td><td>0.500</td></tr><tr><td>ITML</td><td>0</td><td></td><td>56.7%</td><td>47.2%</td><td>0.674</td><td>0.727</td></tr><tr><td>SKMS</td><td>0</td><td></td><td>=</td><td>45.5%</td><td>1</td><td>0.693</td></tr><tr><td>SKKm</td><td>0</td><td></td><td>62.4%</td><td>46.9%</td><td>0.770</td><td>0.781</td></tr><tr><td>SKLR</td><td>0</td><td></td><td>66.9%</td><td>46.8%</td><td>0.791</td><td>0.760</td></tr><tr><td>CSP</td><td></td><td>0</td><td>62.5%</td><td>65.4%</td><td>0.812</td><td>0.812</td></tr><tr><td>MPCK-means</td><td>0</td><td>0</td><td>81.9%</td><td>53.9%</td><td>0.871</td><td>0.816</td></tr><tr><td>CCN (Ours)</td><td>0</td><td>0</td><td>82.4%</td><td>78.1%</td><td>0.889</td><td>0.874</td></tr></table>
+
+The evaluation uses two clustering metrics. The first is normalized-mutual information (NMI) (Strehl & Ghosh, 2002) which is widely used for clustering. The second one is the clustering accuracy (ACC) (Yang et al., 2010). The ACC metric first finds the one-to-one matching between predicted clusters and ground-truth labels, and then calculates the classification accuracy based on the mapping. All data outside the matched clusters will be regarded as mis-predictions. To get high ACC, the algorithm has to generate coherent clusters where each cluster includes most of the data in a category; otherwise the score drops quickly. Therefore ACC provides better discrimination to evaluate whether the semantic clusters have been reconstructed well.
+
+# 5.1.2 RESULTS AND DISCUSSIONS
+
+We report the average performance over the 20 alphabets in table 1. Our approach achieved the top performance on both metrics. The CCN demonstrates strong robustness on the challenging scenario of unknown $K$ . It achieved $7 8 . 1 \%$ average accuracy. Compared with $8 2 . 4 \%$ when $K$ is known, CCN has a relatively small drop. Compared to the second best algorithm, CSP, which is $6 5 . 4 \%$ , CCN outperforms it with a large gap. The classical approach MPCK-means works surprisingly well when the number of clusters is known, but its performance dropped dramatically from $8 1 . 9 \%$ to $5 3 . 9 \%$ when $K = 1 0 0$ . In the performance breakdown for the 20 individual alphabets, CCN achieved $94 \%$ clustering accuracy on Old Church Slavonic Cyrillic, which has 45 characters (appendix table 5). Therefore the results show the feasibility of reconstructing semantic clusters using only noisy similarity predictions.
+
+When to use the semantic similarity? The experiments in table 1 show a clear trend that utilizing the pairwise constraints jointly for both metric learning and minimizing the clustering loss achieves the best performance, including both MPCK-means and CCN. In the case of unknown number of clusters, where we set $K = 1 0 0$ , the algorithms that use constraints to optimize clustering loss have better robustness, for example, CSP and CCN. The group that only use constraints for metric learning (ITML, SKMS, SKKm, and SKLR) significantly outperform the group that does not use it (K-means, LPNMF, LSC). However, their performance are still far behind CCN. Our results confirm the importance of jointly optimizing the metric and clustering.
+
+The robustness against noisy similarity prediction is the key factor to enable the cross-task transfer framework. To the best of our knowledge, table 1 is the first comprehensive robustness comparisons using predicted constraints learned from real data instead of converting from ground-truth labels. The accuracy of $G$ in our experiment is shown in appendix table 7 and demonstrates the reasonable performance of $G$ which is on par with Matching-Net (Vinyals et al., 2016). After binarizing the prediction at 0.5 probability, the similar pair precision, similar pair recall, dissimilar pair precision, and dissimilar pair recall among the 659 characters are (0.392, 0.927, 0.999, 0.995), accordingly. The binarized predictions are better than uniform random guess (0.002, 0.500, 0.998, 0.500), but are still noisy. Therefore it is very challenging for constrained clustering. The visualization of the robustness range of CCN are provided in appendix D, and shows that the robustness is related to the density of pairs involved in a mini-batch. We hypothesize that during the optimization, the gradients from wrongly predicted pairs are canceled out by each other or by the correctly predicted pairs. Therefore the overall gradient still moves the solution towards a better clustering result.
+
+How to predict $K ?$ Inferring the number of clusters $( N C )$ is a hard problem, but with the pairwise similarity information, it becomes feasible. For evaluation, we compute the difference between the number of dominant clusters $( N D C )$ and the true number of categories $( N C ^ { g t } )$ in a dataset. We use a naive definition for $N D C$ , which is the number of clusters that have a size larger than expected size when data is sampled uniformly. In other words, $\begin{array} { r } { N D C = \sum _ { i = 1 } ^ { K } \left[ C _ { i } > = \bar { E } [ C _ { i } ] \right] } \end{array}$ where $[ \cdot ]$ is an Iverson Bracket and $C _ { i }$ is the size of cluster $i$ . For example, $\bar { E } [ \bar { C } _ { i } ]$ will be 10 if the alphabet has 1000 images and $K = 1 0 0$ . Then the average difference $( A D i f )$ is calculated by $\begin{array} { r } { A D i f = \frac { 1 } { | D | } \sum _ { d \in D } { | N D C _ { d } - N C _ { d } ^ { g t } | } } \end{array}$ , where $d$ (i.e., alphabet) is a dataset in $D$ . A smaller $A D i f$ indicates a better estimate of $K$ . CCN achieves a score of 6.35 (appendix table 6). We compare this with the baseline approach SKMS (Anand et al., 2014), which does not require a given $K$ and supports a pipeline to estimate $K$ automatically (therefore we only put it into the column $K = 1 0 0$ in table 1.). SKMS gets 16.3. Furthermore, 10 out of 20 datasets from CCN’s prediction have a difference between $N D C _ { d }$ and $N C _ { d } ^ { g t }$ smaller or equal to 3, which shows the feasibility of estimating $K$ with predicted similarity.
+
+# 5.1.3 EXPERIMENTS USING THE IMAGENET DATASET
+
+To demonstrate the scalability of our approach, we applied the same scheme on the ImageNet dataset. The 1000-class dataset is separated into 882-class (ImageNet882) and 118-class (ImageNet118) subsets as the random split in Vinyals et al. (2016). We use ImageN et882 for $A$ and 30 classes $\mathrm { \sim } 3 9 \mathrm { k }$ images) are randomly sampled from $I m a g e N e t _ { 1 1 8 }$ for $T$ . The difference from section 5.1.1 is that here we use Resnet-18 for both $G$ and CCN, and the weights of the backbone are pre-trained with ImageN et882. Since the number of pairs is high and it is not feasible to feed them into other constrained clustering algorithms, we compare CCN with K-means, LSC(Chen & Cai, 2011), and LPNMF (Cai et al., 2009). We use the output from the average pooling layer of Resnet-18 as the input to these clustering algorithms. CCN gives the top performance with average ACC $7 3 . 8 \%$ when $K$ is known, and $6 5 . 2 \%$ when the number of clusters is unknown, which outperforms the second $( 3 4 . 5 \%$ by K-means) with a large margin. The full comparison is in appendix table 8. And the performance of $G$ is provided in appendix table 9.
+
+# 5.2 UNSUPERVISED CROSS-DOMAIN TRANSFER LEARNING
+
+# 5.2.1 SETUP
+
+Office-31 (Saenko et al., 2010) has images from 31 categories of office objects. The 4652 images are obtained from three domains: Amazon $( a )$ , DSLR $( d )$ , and Webcam $( w )$ . The dataset is the standard benchmark for evaluating domain adaptation performance in computer vision. We experiment with all six combinations (source $S ^ { \prime } \to \mathrm { t a r g e t } T$ ): $a  w$ , $a \to d$ , $d \to a$ , $d \to w$ , $w  a$ , $w \to d$ , and report the average accuracy based on five random experiments for each setting.
+
+The $G$ function learns the semantic similarity function from the auxiliary dataset $A$ , which is ImageNet with all 1000 categories. The backbone network of $G$ is Resnet-18 and has the weights initialized by ImageNet classification. The training process is the same as section 5.1.1 except the images are resized to 224.
+
+We follow the standard protocols using deep neural networks (Long et al., 2017; Ganin et al., 2016) for unsupervised domain adaptation. The backbone network of $\mathrm { C C N ^ { + } }$ is pre-trained with ImageNet. Appendix figure 7 illustrates the scheme. During the training, all source data and target data are used. Each mini-batch is constructed by 32 labeled samples from source and 96 unlabeled samples from target. Since the target dataset has no labels and could only be randomly sampled, it is crucial to have sufficient mini-batch size to ensure that similar pairs are sampled. The loss function used in our approach is equation (7) and is optimized by stochastic gradient descent. The $\mathrm { C C N ^ { + / + + } }$ and DANN (RevGrad) with ResNet backbone are implemented with Torch. We use the code from original author for JAN. Both DANN and JAN use a 256-dimension bottleneck feature layer.
+
+Table 2: Unsupervised cross-domain transfer (domain adaptation) on the Office-31 dataset. The backbone network used here is Resnet-18 (He et al., 2016) pre-trained with ImageNet.   
+
+<table><tr><td></td><td>A→W</td><td>D→W</td><td>W→D</td><td>A→D</td><td>D→A</td><td>W→A</td><td>Avg</td></tr><tr><td>Source-Only</td><td>66.8</td><td>92.8</td><td>96.8</td><td>67.1</td><td>51.4</td><td>53.0</td><td>71.3</td></tr><tr><td>DANN (Ganin et al., 2016)</td><td>73.2</td><td>97.0</td><td>99.0</td><td>69.3</td><td>58.0</td><td>57.8</td><td>75.7</td></tr><tr><td>JAN (Long et al., 2017)</td><td>74.5</td><td>94.1</td><td>99.6</td><td>75.9</td><td>58.7</td><td>59.0</td><td>76.9</td></tr><tr><td>CCN+ (ours)</td><td>76.7</td><td>97.3</td><td>98.2</td><td>71.2</td><td>61.0</td><td>60.5</td><td>77.5</td></tr><tr><td>CCN++ (with DANN)</td><td>78.2</td><td>97.4</td><td>98.6</td><td>73.5</td><td>62.8</td><td>60.6</td><td>78.5</td></tr></table>
+
+# 5.2.2 RESULTS AND DISCUSSIONS
+
+The results are summarized in table 2. Our approach $\mathrm { ( C C N ^ { + } ) }$ demonstrates a strong performance boost for the unsupervised cross-domain transfer problem. It reaches $7 7 . 5 \%$ average accuracy which gained 6.2 points from the $7 1 . 3 \%$ source-only baseline. Although our approach merely transfers more information from the auxiliary dataset, it outperforms the strong approach DANN $( 7 5 . 7 \% )$ , and state-of-the-art JAN $( 7 6 . 9 \% )$ . When combining ours with DANN $\mathrm { ( C C N ^ { + + } ) }$ , the performance is further boosted. This indicates that LCO helps mitigate the transfer problem in a certain way that is orthogonal to minimizing the domain discrepancy. We observe the same trend when using a deeper backbone network, i.e., ResNet-34. In such a case the average accuracy achieved is $7 7 . 9 \%$ , $8 1 . 1 \%$ and $82 \%$ for source-only, $\mathrm { C C N ^ { + } }$ and $\mathrm { C C N ^ { + + } }$ , respectively, though we used exactly the same $G$ as before (with ResNet-18 backbone for $G$ ). This indicates that the information carried in the similarity predictions is not equivalent to transferring features with deeper networks. More discussions are in appendix C and the performance of $G$ is provided in appendix table 11 to show that although the prediction has low precision for similar pairs $( \sim 0 . 2 )$ , our approach still benefits from the dense similarity predictions.
+
+# 5.2.3 EXPERIMENTS USING SVHN-TO-MNIST
+
+We also evaluated the $\mathrm { C C N ^ { + } }$ on another widely compared scenario, which uses color Street View House Numbers images (SVHN) (Netzer et al., 2011) as $S ^ { \prime }$ , the gray-scale hand-written digits (MNIST) (LeCun, 1998) as $T$ . To learn $G$ , we use the $O m n g l o t _ { b g }$ as $A$ . We train all the networks in this section from scratch. Our experimental setting is similar to Sener et al. (2016). We achieve the top performance with $8 9 . 1 \%$ accuracy. The performance gain from source-only in our approach is $+ 3 7 . 1 \%$ , which wins by a large margin compared to the $+ 2 3 . 9 \%$ of LTR (Sener et al., 2016). The full comparison is presented in appendix table 12.
+
+# 6 CONCLUSION AND OUTLOOK
+
+In this paper, we demonstrated the usefulness of transferring information in the form of pairwise similarity predictions. Such information can be transferred as a function and utilized by a loss formulation inspired from constrained clustering, but implemented more robustly within a neural network that can jointly optimize both features and clustering outputs based on these noisy predictions. The experiments for both cross-task and cross-domain transfer learning show strong benefits of using the semantic similarity predictions resulting in new state of the art results across several datasets. This is true even without explicit domain adaptation for the cross-domain task, and if we add a domain discrepancy loss the benefits increase further.
+
+There are two key factors that determine the performance of the proposed framework. The first is the robustness of the constrained clustering and second is the performance of the similarity prediction function. We show robustness of CCN empirically, but we do not explore situations where learning the similarity function is harder. For example, such cases arise when there are a small number of categories in source or a large domain discrepancy between source and target. One idea to deal with such a situation is learning G with domain adaptation strategies. We leave these aspects for future work.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by the National Science Foundation and National Robotics Initiative (grant # IIS-1426998).
+
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+# APPENDICES
+
+# A COMPARISON OF EXPERIMENTAL SETTINGS
+
+Table 3: The list of datasets involved in each experiment. $G$ learns the similarity function from dataset A. The $\mathbf { C C N } ^ { * }$ is optimized with dataset T or T∪S’, while $\mathbf { C C N } ^ { * }$ means CCN for cross-task transfer and $\mathrm { C C N ^ { + / + + } }$ for cross-domain transfer. The rows for network initialization indicate whether the network has weights initialized by training a classification task with the specified dataset. The weights are randomly initialized if not specified.   
+
+<table><tr><td colspan="2">Scheme</td><td colspan="2">Cross-Task Transfer</td><td colspan="2">Cross-Domain Transfer</td></tr><tr><td colspan="2">Experiment Section</td><td>Omniglot 5.1.1</td><td>ImageNet 5.1.3</td><td>Office-31 5.2.1</td><td>SVHN-MNIST 5.2.3</td></tr><tr><td rowspan="3">Datasets</td><td>A</td><td>Omniglotbg</td><td>ImageNet882</td><td>ImageNet1000</td><td>Omniglotbg</td></tr><tr><td>S’</td><td></td><td></td><td>Office-31 {a,d,w}</td><td>SVHN</td></tr><tr><td>T</td><td>Omnigloteval</td><td>ImageNet118</td><td>Office-31 {a,d,w)</td><td>MNIST</td></tr><tr><td rowspan="2">Network Initialization</td><td>G</td><td></td><td>ImageNet882</td><td>ImageNet1000</td><td>1</td></tr><tr><td>CCN*</td><td></td><td>ImageNet882</td><td>ImageNet1000</td><td>1</td></tr></table>
+
+Table 4: The list of loss functions used for training networks. The similarity prediction function (network) $G$ uses the cross-entropy (CE) loss with two classes (similar/dissimilar). The training of constrained clustering network $( \mathbf { C C N } ^ { * } )$ involves the combinations of the learnable clustering objective (LCO), cross-entropy, and domain adaptation loss (DA).   
+
+<table><tr><td></td><td>LCO</td><td>CE</td><td>DA</td></tr><tr><td>G</td><td></td><td>√</td><td></td></tr><tr><td>CCN</td><td></td><td></td><td></td></tr><tr><td>CCN+</td><td>广</td><td>√</td><td></td></tr><tr><td>CCN++</td><td>√</td><td></td><td>√</td></tr></table>
+
+![](images/6077c2618b017816b79d4cf6b62b30b24d7ebf9ecfe529286747a494f9f155c8.jpg)  
+Figure 6: A diagram depicting the cross-task transfer experiment. Two experiments follow this flow: Omniglot $b g$ to $O m n i g l o t _ { e v a l }$ , and $I m a g e N e t _ { 8 8 2 }$ to ImageNet118. Both have exclusive classes between source and target domain. In the ImageNet experiment, the backbone network is initialized with the weights pre-trained with ImageN et882.
+
+![](images/62fef6aa1814051042be0542b4068cc2863de90390604a41a4bf68f744e5059c.jpg)  
+Figure 7: Comparison between domain adaptation approaches (a) Transferring semantic similarity from auxiliary data (our method), and (b) Minimizing the domain discrepancy. The diagram uses office-31 benchmark as the scenario of transferring.
+
+# B SUPPLEMENTARY FOR UNSUPERVISED CROSS TASK TRANSFER LEARNING
+
+Table 5: A breakdown of the results for each alphabet in Omnigloteval. The unsupervised cross task transfer experiment is described in section 5.1. This table shows the clustering accuracy with $K = 1 0 0$ to simulate the situation of unknown number of clusters.   
+
+<table><tr><td>Alphabet</td><td>K-means</td><td>LPNMF</td><td>LSC</td><td>ITML</td><td>SKMS</td><td>SKKm</td><td>SKLR</td><td>CSP</td><td>MPCK-means</td><td>CCN</td></tr><tr><td>Angelic</td><td>24%</td><td>26%</td><td>27%</td><td>48%</td><td>78%</td><td>40%</td><td>38%</td><td>72%</td><td>50%</td><td>82%</td></tr><tr><td>Atemayar_Qelisayer</td><td>19%</td><td>14%</td><td>17%</td><td>51%</td><td>61%</td><td>51%</td><td>44%</td><td>61%</td><td>50%</td><td>82%</td></tr><tr><td>Atlantean</td><td>19%</td><td>15%</td><td>19%</td><td>57%</td><td>72%</td><td>55%</td><td>51%</td><td>77%</td><td>47%</td><td>72%</td></tr><tr><td>Aurek_Besh</td><td>23%</td><td>18%</td><td>21%</td><td>45%</td><td>49%</td><td>44%</td><td>45%</td><td>76%</td><td>71%</td><td>90%</td></tr><tr><td>Avesta</td><td>22%</td><td>18%</td><td>19%</td><td>47%</td><td>29%</td><td>39%</td><td>38%</td><td>65%</td><td>52%</td><td>76%</td></tr><tr><td>Ge_ez</td><td>19%</td><td>17%</td><td>17%</td><td>56%</td><td>58%</td><td>47%</td><td>42%</td><td>72%</td><td>55%</td><td>82%</td></tr><tr><td>Glagolitic</td><td>22%</td><td>19%</td><td>21%</td><td>42%</td><td>38%</td><td>58%</td><td>62%</td><td>66%</td><td>61%</td><td>85%</td></tr><tr><td>Gurmukhi</td><td>15%</td><td>13%</td><td>15%</td><td>44%</td><td>26%</td><td>54%</td><td>48%</td><td>60%</td><td>56%</td><td>80%</td></tr><tr><td>Kannada</td><td>18%</td><td>14%</td><td>16%</td><td>48%</td><td>37%</td><td>46%</td><td>53%</td><td>60%</td><td>48%</td><td>62%</td></tr><tr><td>Keble</td><td>20%</td><td>14%</td><td>18%</td><td>44%</td><td>60%</td><td>46%</td><td>44%</td><td>75%</td><td>68%</td><td>90%</td></tr><tr><td>Malayalam</td><td>18%</td><td>15%</td><td>16%</td><td>36%</td><td>24%</td><td>49%</td><td>52%</td><td>50%</td><td>54%</td><td>72%</td></tr><tr><td>Manipuri</td><td>17%</td><td>15%</td><td>16%</td><td>49%</td><td>40%</td><td>53%</td><td>54%</td><td>66%</td><td>62%</td><td>85%</td></tr><tr><td>Mongolian</td><td>18%</td><td>18%</td><td>19%</td><td>50%</td><td>40%</td><td>37%</td><td>48%</td><td>75%</td><td>57%</td><td>86%</td></tr><tr><td>Old_Church_Slavonic_Cyrilic</td><td>19%</td><td>16%</td><td>19%</td><td>42%</td><td>41%</td><td>52%</td><td>67%</td><td>71%</td><td>67%</td><td>94%</td></tr><tr><td>Oriya</td><td>15%</td><td>13%</td><td>14%</td><td>46%</td><td>40%</td><td>54%</td><td>45%</td><td>57%</td><td>52%</td><td>67%</td></tr><tr><td>Sylheti</td><td>14%</td><td>13%</td><td>14%</td><td>49%</td><td>32%</td><td>35%</td><td>37%</td><td>59%</td><td>43%</td><td>64%</td></tr><tr><td>Syriac_Serto</td><td>20%</td><td>18%</td><td>19%</td><td>55%</td><td>70%</td><td>42%</td><td>35%</td><td>70%</td><td>47%</td><td>73%</td></tr><tr><td>Tengwar</td><td>18%</td><td>18%</td><td>18%</td><td>51%</td><td>44%</td><td>49%</td><td>40%</td><td>61%</td><td>44%</td><td>66%</td></tr><tr><td>Tibetan</td><td>18%</td><td>14%</td><td>16%</td><td>44%</td><td>31%</td><td>47%</td><td>54%</td><td>59%</td><td>55%</td><td>84%</td></tr><tr><td>ULOG</td><td>20%</td><td>18%</td><td>18%</td><td>41%</td><td>41%</td><td>40%</td><td>41%</td><td>56%</td><td>38%</td><td>70%</td></tr><tr><td>Average</td><td>19%</td><td>16%</td><td>18%</td><td>47%</td><td>46%</td><td>47%</td><td>47%</td><td>65%</td><td>54%</td><td>78%</td></tr></table>
+
+Table 6: Estimates for the number of characters across the 20 datasets in Omniglot $e v a l$ . The bold number means the prediction has error smaller or equal to 3. The $A D i f$ is defined in section 5.1.2.   
+
+<table><tr><td>Alphabet</td><td>True #class</td><td>SKMS</td><td>CCN (K=100)</td></tr><tr><td>Angelic</td><td>20</td><td>16</td><td>26</td></tr><tr><td>Atemayar_Qelisayer</td><td>26</td><td>17</td><td>34</td></tr><tr><td>Atlantean</td><td>26</td><td>21</td><td>41</td></tr><tr><td>Aurek_Besh</td><td>26</td><td>14</td><td>28</td></tr><tr><td>Avesta</td><td>26</td><td>8</td><td>32</td></tr><tr><td>Ge_ez</td><td>26</td><td>18</td><td>32</td></tr><tr><td>Glagolitic</td><td>45</td><td>18</td><td>45</td></tr><tr><td>Gurmukhi</td><td>45</td><td>12</td><td>43</td></tr><tr><td>Kannada</td><td>41</td><td>19</td><td>44</td></tr><tr><td>Keble</td><td>26</td><td>16</td><td>28</td></tr><tr><td>Malayalam</td><td>47</td><td>12</td><td>47</td></tr><tr><td>Manipuri</td><td>40</td><td>17</td><td>41</td></tr><tr><td>Mongolian</td><td>30</td><td>28</td><td>36</td></tr><tr><td>Old_Church_Slavonic_Cyrillic</td><td>45</td><td>23</td><td>45</td></tr><tr><td>Oriya</td><td>46</td><td>22</td><td>49</td></tr><tr><td>Sylheti</td><td>28</td><td>11</td><td>50</td></tr><tr><td>Syriac_Serto</td><td>23</td><td>19</td><td>38</td></tr><tr><td>Tengwar</td><td>25</td><td>12</td><td>41</td></tr><tr><td>Tibetan</td><td>42</td><td>15</td><td>42</td></tr><tr><td>ULOG</td><td>26</td><td>15</td><td>40</td></tr><tr><td>ADif</td><td></td><td>16.3</td><td>6.35</td></tr></table>
+
+Table 7: The performance of the similarity prediction function used in section 5.1. We leverage the N-way test which is commonly used in one-shot learning evaluation. The similarity is learned with Omniglot $b g$ and has N-way test with $O m n i g l o t _ { e v a l }$ and MNIST. The experimental settings follow Vinyals et al. (2016). The raw probability output (without binarization) from our $G$ is used to find the nearest exemplar in the N-way test.   
+
+<table><tr><td>Method</td><td colspan="2">Omniglot-eval 5-way</td><td>MNIST 10-way</td></tr><tr><td>Siamese-Nets (Koch et al., 2015)</td><td>0.967</td><td>20-way 0.880</td><td>0.703</td></tr><tr><td>Match-Net (Vinyals et al., 2016)</td><td>0.981</td><td>0.938</td><td>0.720</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>Ours</td><td>0.979</td><td>0.935</td><td>0.720</td></tr></table>
+
+Table 8: Unsupervised cross-task transfer learning on ImageNet. The values are the average of three random subsets in $I m a g e N e t _ { 1 1 8 }$ . Each subset has 30 classes. The "ACC" has $K = 3 0$ while the "ACC (100)" sets $K = 1 0 0$ . All methods use the features (outputs of average pooling) from Resnet-18 pre-trained with ImageNet882 classification.   
+
+<table><tr><td>Method</td><td>ACC</td><td>ACC(100)</td><td>NMI</td><td>NMI(100)</td></tr><tr><td>K-means</td><td>71.9%</td><td>34.5%</td><td>0.713</td><td>0.671</td></tr><tr><td>LSC</td><td>73.3%</td><td>33.5%</td><td>0.733</td><td>0.655</td></tr><tr><td>LPNMF</td><td>43.0%</td><td>21.8%</td><td>0.526</td><td>0.500</td></tr><tr><td>CCN (ours)</td><td>73.8%</td><td>65.2%</td><td>0.750</td><td>0.715</td></tr></table>
+
+Table 9: Performance of the similarity prediction function ( $G$ , trained with ImageN et882) applied to three subsets of $I m a g e N e t _ { 1 1 8 }$ . Each subset contains 30 random classes of $I m a g e N e t _ { 1 1 8 }$ . The predictions are binarized at 0.5 to calculate the precision and recall. Random\*: The expected performance when classes are uniformly distributed and make uniform random guess for similarity. It is an approximation since the number of images in each class is only roughly equal in ImageNet. We sampled 12M pairs for each set to collect the statistics. Sampling more pairs has no noticeable change to the values.   
+
+<table><tr><td>Set</td><td>Similar Precision</td><td> Similar Recall</td><td>Dissimilar Precision</td><td>Dissimilar Recall</td></tr><tr><td>Random*</td><td>0.033</td><td>0.500</td><td>0.967</td><td>0.500</td></tr><tr><td>Set A</td><td>0.840</td><td>0.664</td><td>0.983</td><td>0.994</td></tr><tr><td>SetB</td><td>0.825</td><td>0.631</td><td>0.981</td><td>0.993</td></tr><tr><td>Set C</td><td>0.771</td><td>0.671</td><td>0.983</td><td>0.990</td></tr><tr><td>Average</td><td>0.812</td><td>0.655</td><td>0.982</td><td>0.992</td></tr></table>
+
+# C SUPPLEMENTARY FOR UNSUPERVISED CROSS DOMAIN TRANSFER LEARNING
+
+# C.1 MORE DISCUSSION FOR THE OFFICE-31 EXPERIMENTS
+
+Our experiment (table 10) shows that simply using deeper pre-trained networks produces a significant performance boost. Specifically, using Resnet-50 increases the performance 8.4 points from Alexnet, which surpasses the 6.2 points gained by using one of the state-of-the-art domain adaptation algorithms (JAN (Long et al., 2017)). If we regard pre-training with deeper networks as transferring more information, our approach is similar in this aspect since both transfer more information from the auxiliary dataset. In this case, memory limitations precluded the application of LCO to such models, and multi-GPU implementations for this problem is an area of future work.
+
+Table 10: The performance of unsupervised transfer across domains on Office-31 dataset. The backbone networks in the comparison have different numbers of convolutional layers. AlexNet has 5 layers and the ResNets have $1 8 { \sim } 5 0$ layers. SO is the abbreviation for source-only, which simply trains on $S ^ { \prime }$ and directly applies the classifier on $T$ . The first two rows are directly copied from Long et al. (2017). The features learned with deeper networks generalize better across domains.   
+
+<table><tr><td colspan="8">A→W D→W W→D A→D D→A W→A</td></tr><tr><td>AlexNet SO</td><td>61.6</td><td>95.4</td><td>99.0</td><td>63.8</td><td>51.1</td><td>49.8</td><td>Avg 70.1</td><td>1</td></tr><tr><td>AlexNet (JAN-A)</td><td>75.2</td><td>96.6</td><td>99.6</td><td>72.8</td><td>57.5</td><td>56.3</td><td>76.3</td><td>+6.2</td></tr><tr><td>Resnet-18 SO</td><td>66.8</td><td>92.8</td><td>96.8</td><td>67.1</td><td>51.4</td><td>53.0</td><td>71.3</td><td>+1.2</td></tr><tr><td>Resnet-34 SO</td><td>73.1</td><td>96.4</td><td>98.8</td><td>73.8</td><td>63.2</td><td>62.1</td><td>77.9</td><td>+7.8</td></tr><tr><td>Resnet-50 SO</td><td>74.3</td><td>96.8</td><td>98.8</td><td>79.5</td><td>61.2</td><td>60.6</td><td>78.5</td><td>+8.4</td></tr></table>
+
+Table 11: Performance of the similarity prediction function ( $G$ , trained with ImageNet882) applied on three domains of the Office-31 dataset. In total, 1.4M pairs are examined to calculate the table.   
+
+<table><tr><td>Domain</td><td>Similar Precision</td><td>Similar Recall</td><td>Dissimilar Precision</td><td>Dissimilar Recall</td></tr><tr><td>Amazon</td><td>0.194</td><td>0.696</td><td>0.988</td><td>0.894</td></tr><tr><td>DSLR</td><td>0.196</td><td>0.882</td><td>0.995</td><td>0.858</td></tr><tr><td>Webcam</td><td>0.213</td><td>0.865</td><td>0.994</td><td>0.876</td></tr></table>
+
+Table 12: Unsupervised transferring across domains (S’: SVHN, T: MNIST A: Omniglotbg) without pre-trained backbone network weights. Our setup is similar to Sener et al. (2016) and Ganin et al. (2016) which therefore has a similar source-only performance.   
+
+<table><tr><td colspan="2">Method</td><td>SVHN-&gt;MNIST</td><td>Gain</td></tr><tr><td>Ganin et al. (2016)</td><td>Source-Only DANN</td><td>54.9 73.9</td><td>1 +19.0</td></tr><tr><td>Sener et al. (2016)</td><td>Source-Only LTR</td><td>54.9 78.8</td><td>1 +23.9</td></tr><tr><td>Saito et al. (2017)</td><td>Source-Only ATDA</td><td>70.1 86.2</td><td>- +16.1</td></tr><tr><td>Tzeng et al. (2017)</td><td>Source-Only ADDA</td><td>60.1 76.0</td><td>- +15.9</td></tr><tr><td>Ours</td><td>Source-Only CCN+</td><td>52.0 89.1</td><td>1 +37.1</td></tr></table>
+
+Table 13: Performance of the similarity prediction function ( $G$ , trained with Omniglotbg) applied on the MNIST dataset. In total, 5M pairs are sampled to calculate the table.   
+
+<table><tr><td></td><td>Similar Precision</td><td>Similar Recall</td><td>Dissimilar Precision</td><td>Dissimilar Recall</td></tr><tr><td>Random</td><td>0.100</td><td>0.500</td><td>0.900</td><td>0.500</td></tr><tr><td>MNIST</td><td>0.782</td><td>0.509</td><td>0.946</td><td>0.984</td></tr></table>
+
+# D ROBUSTNESS ANALYSIS OF CONSTRAINED CLUSTERING NETWORK
+
+# D.1 SETUP
+
+To quickly explore the large combination of factors that may affect the clustering, we use a small dataset (MNIST) and a small network which has two convolution layers followed by two fully connected layers. The MNIST dataset is a dataset of handwritten digits that contains 60k training and 10k testing images with size $2 8 \mathbf { x } 2 8$ . Only the training set is used in this section and the raw pixels, which were normalized to zero mean and unit standard deviation, were fed into networks directly.
+
+The networks were randomly initialized and the clustering training was run five times under each combination of factors; we show the best final results, as is usual in the random restart regime. The mini-batch size was set to 256, thus up to 65536 pairs were presented to the LCO per mini-batch if using full density $\mathrm { ( D } { = } 1 \mathrm { ) }$ ). There were 235 mini-batches in an epoch and the optimization proceeded for 15 epochs. The clustering loss was minimized by stochastic gradient descent with learning rate 0.1 and momentum 0.9. The predicted cluster was assigned at the end by forwarding samples through the clustering networks. The best result in the five runs was reported.
+
+To simulate different performance of the similarity prediction, the label of pairs were flipped according to the designated recall. For example, to simulate a $90 \%$ recall of similar pair, $10 \%$ of the ground truth similar pair in a mini-batch were flipped. The precision of similar/dissimilar pairs is a function of the recall of both type of pairs, thus controlling the recall is sufficient for the evaluation. The recalls for both similar and dissimilar pairs were gradually reduced from one to zero at intervals of 0.1.
+
+# D.2 DISCUSSION
+
+The resulting performance w.r.t different values of recall, density, and number of clusters is visualized in Figure 8. A bright color means high NMI score and is desired. The larger the bright region, the more robust the clustering is against the noise of similarity prediction. The ACC score shows almost the same trend and is thus not shown here.
+
+How does similarity prediction affect clustering? Looking at the top-left heat map in figure 8, which has $D = 1$ and 10 clusters, it can be observed that the NMI score is very robust to low similar pair recall, even lower than 0.5. For recall of dissimilar pairs, the effect of recall is divided at the 0.5 value: the clustering performance can be very robust to noise in dissimilar pairs if the recall is greater than 0.5; however, it can completely fail if recall is below 0.5. For similar pairs, the clustering works on a wide range of recalls when the recall of dissimilar pairs is high.
+
+In practical terms, robustness to the recall of similar pairs is desirable because it is much easier to predict dissimilar pairs than similar pairs in real scenarios. In a dataset with 10 categories e.g. Cifar-10, we can easily get $90 \%$ recall for dissimilar pairs with purely random guess if the number of classes is known, while the recall for similar pairs will be $10 \%$ .
+
+How does the density of the constraints affect clustering? We argue that the density of pairwise relationships is the key factor to improving the robustness of clustering. The density $D = 1$ means that every pair in a mini-batch is utilized by the clustering loss. For density $D = 0 . 1$ , it means only 1 out of 10 possible constraints is used. We could regard the higher density as better utilization of the pairwise information in a mini-batch, thus more learning instances contribute to the gradients at once. Consider a scenario where there is one sample associated with 5 true similar pairs and 3 false similar pairs. In such a case, the gradients introduced by the false similar pairs have a higher chance to be overridden by true similar pairs within the mini-batch, thus the loss can converge faster and is less affected by errors. In Figure 8, we could see when density decreases, the size of the bright region shrinks significantly.
+
+![](images/12cec765ecdee0ae03b658c3f3798b3d511e16695116130fd85ecccd06d459c9.jpg)  
+Figure 8: The clustering performance with different pairwise density and number of clusters. A bright color means that the NMI score is close to 1 while black corresponds to 0. The density is defined as a ratio compared to the total number of pair-wise combinations in a mini-batch. The number of clusters defines the final softmax output dimensionality. In each sub-figure, we show how the scores change w.r.t. the similar pair recall and dissimiliar pair recall.
+
+In our implementation, enumerating the full pairwise relationships introduces negligible overhead in computation time using GPU. Although there is overhead for memory consumption, it is limited because only the vector of predicted distributions has to be enumerated for calculating the clustering loss.
+
+The effect of varying the number of Clusters In the MNIST experiments, the number of categories is 10. We augment the softmax output number up to 100. The rows of figure 8 show that even when the number of output categories is significant larger than the number of true object categories, e.g. $1 0 0 > 1 0$ , the clustering performance NMI score only degrades slightly.
\ No newline at end of file
diff --git a/parse/train/ByRWCqvT-/ByRWCqvT-_content_list.json b/parse/train/ByRWCqvT-/ByRWCqvT-_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..04257fe34b037f33792c6bb937c4f8a87eecbca4
--- /dev/null
+++ b/parse/train/ByRWCqvT-/ByRWCqvT-_content_list.json
@@ -0,0 +1,2038 @@
+[
+    {
+        "type": "text",
+        "text": "LEARNING TO CLUSTER IN ORDER TO TRANSFER ACROSS DOMAINS AND TASKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            101,
+            820,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yen-Chang Hsu, Zhaoyang Lv   \nGeorgia Institute of Technology   \nAtlanta, GA 30332, USA   \n{yenchang.hsu, zhaoyang.lv}@gatech.edu ",
+        "bbox": [
+            184,
+            170,
+            557,
+            227
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Zsolt Kira Georgia Tech Research Institute Atlanta, GA 30318, USA zkira@gatech.edu ",
+        "bbox": [
+            588,
+            170,
+            800,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            262,
+            544,
+            277
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This paper introduces a novel method to perform transfer learning across domains and tasks, formulating it as a problem of learning to cluster. The key insight is that, in addition to features, we can transfer similarity information and this is sufficient to learn a similarity function and clustering network to perform both domain adaptation and cross-task transfer learning. We begin by reducing categorical information to pairwise constraints, which only considers whether two instances belong to the same class or not (pairwise semantic similarity). This similarity is category-agnostic and can be learned from data in the source domain using a similarity network. We then present two novel approaches for performing transfer learning using this similarity function. First, for unsupervised domain adaptation, we design a new loss function to regularize classification with a constrained clustering loss, hence learning a clustering network with the transferred similarity metric generating the training inputs. Second, for cross-task learning (i.e., unsupervised clustering with unseen categories), we propose a framework to reconstruct and estimate the number of semantic clusters, again using the clustering network. Since the similarity network is noisy, the key is to use a robust clustering algorithm, and we show that our formulation is more robust than the alternative constrained and unconstrained clustering approaches. Using this method, we first show state of the art results for the challenging cross-task problem, applied on Omniglot and ImageNet. Our results show that we can reconstruct semantic clusters with high accuracy. We then evaluate the performance of cross-domain transfer using images from the Office-31 and SVHN-MNIST tasks and present top accuracy on both datasets. Our approach doesn’t explicitly deal with domain discrepancy. If we combine with a domain adaptation loss, it shows further improvement. ",
+        "bbox": [
+            232,
+            294,
+            766,
+            625
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            648,
+            336,
+            665
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Supervised learning has made significant strides in the past decade, with substantial advancements arising from the use of deep neural networks. However, a large part of this success has come from the existence of extensive labeled datasets. In many situations, it is not practical to obtain such data due to the amount of effort required or when the task or data distributions change dynamically. To deal with these situations, the fields of transfer learning and domain adaptation have explored how to transfer learned knowledge across tasks or domains. Many approaches have focused on cases where the distributions of the features and labels have changed, but the task is the same (e.g., classification across datasets with the same categories). Cross-task transfer learning strategies, on the other hand, have been widely adopted especially in the computer vision community where features learned by a deep neural network on a large classification task have been applied to a wide variety of other tasks (Donahue et al., 2014). ",
+        "bbox": [
+            174,
+            680,
+            825,
+            833
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Most of the prior cross-task transfer learning works, however, require labeled target data to learn classifiers for the new task. If labels of the target data are absent, there is little choice other than to apply unsupervised approaches such as clustering on the target data with pre-trained feature representations. In this paper, we focus on the question of what can be transferred (besides features) to support both cross-domain and cross-task transfer learning. We address it with a learned similarity function as the fundamental component of clustering. Clustering can then be realized using a neural network trained using the output of the similarity function, which can be successfully used to achieve both cross-task and cross-domain transfer. ",
+        "bbox": [
+            174,
+            840,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The key idea is to formulate the clustering objective to use a learnable (and transferable) term, which in our proposed work is a similarity prediction function. Our proposed objective function can be easily combined with deep neural networks and optimized end-to-end. The features and clustering are optimized jointly, hence taking advantage of such side information in a robust way. Using this method, we show that unsupervised learning can benefit from learning performed on a distinct task, and demonstrate the flexibility of further combining it with a classification loss and domain discrepancy loss. ",
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+        "type": "text",
+        "text": "In summary, we make several contributions. First, we propose to use predictive pairwise similarity as the knowledge that is transferred and formulate a learnable objective function to utilize the pairwise information in a fashion similar to constrained clustering. We then provide the methodologies to deploy the objective function in both cross-task and cross-domain scenarios with deep neural networks. The experimental results for cross-task learning on Omniglot and ImageNet show that we can achieve state of the art clustering results with predicted similarities. On the standard domain adaptation benchmark Office-31 dataset, we demonstrate improvements over state-of-art even when not performing any explicit domain adaptation, and further improvements if we do. Finally, on another domain adaptation task, SVHN-to-MNIST, our approach using Omniglot as the auxiliary dataset achieves top performance with a large margin. ",
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+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Transfer Learning: Transfer learning aims to leverage knowledge from the source domain to help learn in the target domain, while only focusing on the performance on the target domain. The type of transferred knowledge includes training instances, features, model parameters and relational knowledge (Pan & Yang, 2010). Pairwise similarity is the meta-knowledge we propose to transfer, which falls in between the last two types. The similarity prediction function is a neural network with learned parameters, while the output is the most simplified form of relational knowledge, i.e., only considering pairwise semantic similarity. ",
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+    {
+        "type": "text",
+        "text": "Cross-task Transfer Learning: Features learned when trained for ImageNet classification (Russakovsky et al., 2015) have boosted the performance of a variety of vision tasks in a supervised setting. For example, new classification tasks (Donahue et al., 2014; Yosinski et al., 2014), object detection (Girshick et al., 2014), semantic segmentation (Long et al., 2015a), and image captioning (Vinyals et al., 2015). Translated Learning (Dai et al., 2009) has an unsupervised setting similar to ours, but it again focuses only on transferring features across tasks. Our work explores how learning could benefit from transferring pairwise similarity in an unsupervised setting. ",
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+        "type": "text",
+        "text": "Cross-domain Transfer Learning: Also known as domain adaptation (Pan & Yang, 2010), there has recently been a large body of work dealing with domain shift between image datasets by minimizing domain discrepancy (Tzeng et al., 2014; Long et al., 2015b; Ganin et al., 2016; Sun et al., 2016; Long et al., 2016; Sener et al., 2016; Carlucci et al., 2017; Long et al., 2017; Zellinger et al., 2017; Bousmalis et al., 2017). We address the problem in a complementary way that transfers extra information from the auxiliary dataset and show a larger performance boost with further gains using an additional domain discrepancy loss. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Constrained Clustering: Constrained clustering algorithms can be categorized by how they utilize constraints (e.g. pairwise information). The first set of work use the constraints to learn a distance metric. For example, DML (Xing et al., 2003), ITML (Davis et al., 2007), SKMS (Anand et al., 2014), SKKm (Anand et al., 2014; Amid et al., 2016), and SKLR (Amid et al., 2016). This group of approaches closely relates to metric learning and needs a clustering algorithm such as K-means in a separate stage to obtain cluster assignments. The second group of work use constraints to formulate the clustering loss. For example, CSP (Wang et al., 2014) and COSC (Rangapuram & Hein, 2012). The third group uses constraints for both metric learning and the clustering objective, such as MPCKMeans (Bilenko et al., 2004) and CECM (Antoine et al., 2012). The fourth group does not use constraints at all. A generic clustering algorithms such as K-means (MacQueen et al., 1967), LSC (Chen & Cai, 2011), and LPNMF (Cai et al., 2009) all belong to this category. There is a long list of associated works and they are summarized in survey papers, e.g. Davidson & Basu (2007) and Dinler & Tural (2016). Our proposed clustering strategy belongs to the third group. The constrained clustering methods above are applied to the semi-supervised setting where the groundtruth constraints are sparsely available. In our unsupervised setting, the ground-truth is unavailable but predicted constraints are densely available. We include all four groups of algorithms in our comparison and show the advantages of the third group. ",
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+            "Figure 1: Overview of the transfer scheme with a learnable clustering objective (LCO). The LCO and pairwise similarity are the two key components of our approach and are described in section 4. The dashed rectangles and light gray arrows are only available in cross-domain transfer. Details are described in section 3. "
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+        "type": "text",
+        "text": "3 THE TRANSFER LEARNING TASKS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To define the transfer learning problem addressed in this work, we follow the notations used by Pan & Yang (2010). The goal is to transfer knowledge from source data $\\boldsymbol { S } = ( X _ { S } , Y _ { S } )$ , where $X _ { S }$ is the set of data instances and $Y _ { S }$ is the corresponding categorical labels, to a target data noted as $T = \\left( X _ { T } , Y _ { T } \\right)$ . The learning is unsupervised since $Y _ { T }$ is unknown. The scenario is divided into two cases. One is $\\{ Y _ { T } \\} \\ne \\{ { \\bar { Y } } _ { S } \\}$ , which means the set of categories are not the same and hence the transfer is across tasks. The second case is $\\{ Y _ { T } \\} = \\{ Y _ { S } \\}$ , but with a domain shift. In other words, the marginal probability distributions of the input data are different, i.e., $P ( X _ { T } ) \\neq P ( X _ { S } )$ . The latter is a cross-domain learning problem also called transductive learning. The domain adaptation approaches which have gained significant attention recently belong to the second scenario. ",
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+        "type": "text",
+        "text": "To align with common benchmarks for evaluating transfer learning performance, we further use the notion of an auxiliary dataset and split the source data into $S = { \\bar { S } } ^ { \\prime } \\cup A$ . $S ^ { \\prime } = ( X _ { S } ^ { \\prime } , Y _ { S } ^ { \\prime } )$ which is only present in the cross-domain transfer scheme and has $\\{ Y _ { S } ^ { \\prime } \\} = \\{ Y _ { T } \\}$ . $A = ( X _ { A } , Y _ { A } )$ is the auxiliary dataset which has a large amount of labeled data and potentially categories as well, and may or may not contain the categories of $Y _ { T }$ . For the cross task scenario, only $A$ and unlabeled $T$ are included, while cross-domain transfer involves $A$ , $S ^ { \\prime }$ , and $T$ . In the following sections we use the above notations and describe the two transfer learning tasks in detail. Figure 1 illustrates how our approach relates to both tasks. ",
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+        "type": "text",
+        "text": "Transfer across tasks: If the target task has different categories, we cannot directly transfer the classifier from source to target, and there is no labeled target data to use for fine-tuning transferred features. Here we propose to first reduce the categorization problem to a surrogate same-task problem. We can directly apply transductive transfer learning (Pan & Yang, 2010) to the transformed task. The cluster structure in the target data is then reconstructed using the predictions in the transformed task. See figure 2 for an illustration. ",
+        "bbox": [
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+        "type": "text",
+        "text": "The source involves a labeled auxiliary dataset $A$ and an unlabeled target dataset $T$ . $Y _ { T }$ is the target that must be inferred. In this scenario the set of categories $\\{ Y _ { A } \\} \\ne \\{ Y _ { T } \\}$ , and $Y _ { T }$ is unknown. We first transform the problem of categorization into a pairwise similarity prediction problem. In other words, we specify a transformation function $R$ such that $R ( A ) = ( X _ { A } ^ { R ^ { \\prime } } , \\bar { Y } ^ { R } )$ , and $\\dot { X } _ { A } ^ { R } = \\{ ( x _ { A , i } , x _ { A , j } ) \\} _ { \\forall i , j }$ contains all pairs of data, where $\\{ Y ^ { R } \\} = \\{ d i s s i m i l a r , s i m i l a r \\}$ . The transformation on the labeled auxiliary data is straightforward. It will be similar if two data instances are from the same category, and vice versa. We then use it to learn a pairwise similarity prediction function $G ( x _ { i } , x _ { j } ) = \\mathbf { \\bar { \\Gamma } } y _ { i , j }$ ",
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+    {
+        "type": "image",
+        "img_path": "images/2b04f9cd1aa8b76244e957c10c6c523af984b0b1e4ea8542e4f1f432440b0e65.jpg",
+        "image_caption": [
+            "Figure 2: The concept for reconstructing clusters of unseen categories. The proposed approach follows the arrows in the counter-clockwise direction, which converts cross-task transfer learning to cross-domain transfer learning. The colors of dots represent data of different categories. The hollow circle and cross symbol represent similar and dissimilar data pairs. The $G$ function and the cluster reconstruction (via constrained clustering) are the two key components in the diagram. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "By applying $G$ on $T$ , we can obtain $G ( x _ { T , i } , x _ { T , j } ) = y _ { T , i , j }$ . The last step is to infer $Y _ { T }$ from $Y _ { T } ^ { R } = \\{ y _ { T , i , j } \\} _ { \\forall i , j }$ , which can be solved using constrained clustering algorithms. Note that since the actual $Y _ { T }$ is unknown, the algorithm only infers the indices of categories, which could be in arbitrary order. The resulting clusters are expected to contain coherent semantic categories. ",
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+        "type": "text",
+        "text": "Transfer across domains: The problem setting we consider here is the same as unsupervised domain adaptation. Following the standard evaluation procedure (Long et al., 2017; Ganin et al., 2016), the labeled datasets $A$ is ImageNet and $S ^ { \\prime }$ is one domain in the Office-31 dataset. The unlabeled $T$ is another domain in Office-31. The goal is to enhance classification performance on $T$ by utilizing $A$ , $S ^ { \\prime }$ , and $T$ together. ",
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+        "type": "text",
+        "text": "4 THE LEARNABLE CLUSTERING OBJECTIVE (LCO) ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The key to our approach is the design of a learning objective that can use (noisy) predicted pairwise similarities, and is inspired from constrained clustering which involves using pairwise information in the loss function. The pairwise information is called must-link/cannot-link constraints or similar/dissimilar pairs (we use the latter). Note that the information is binarized to one and zero for similar and dissimilar pairs, accordingly. ",
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+    {
+        "type": "text",
+        "text": "Although many constrained clustering algorithms have been developed, few of them are scalable with respect to the number of pairwise relationships. Further, none of them can be easily integrated into deep neural networks. Inspired by the work of Hsu & Kira (2016), we construct a contrastive loss for clustering on the probability outputs of a softmax classifier. However, each output node does not have to map to a fixed category but instead each output node represents a probabilistic assignment of a data point to a cluster. The assignment between output nodes and clusters are formed stochastically during the optimization and is guided by the pairwise similarity. If there is a similar pair, their output distribution should be similar, and vice-versa. ",
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+        "type": "text",
+        "text": "Specifically, we use the pair-wise KL-divergence to evaluate the distance between $k$ cluster assignment distributions of two data instances, and use predicted similarity to construct the contrastive loss. Given a pair of data $x _ { p } , x _ { q }$ , their corresponding output distributions are defined as $\\mathcal { P } = f ( x _ { p } )$ and $\\mathcal { Q } = f ( x _ { q } )$ , while $f$ is the neural network. The cost of a similar pair is described as : ",
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+    {
+        "type": "equation",
+        "img_path": "images/43126e3207a3614b82cb0757e647c76f3175b51f2528f5306705302aa203b6ea.jpg",
+        "text": "$$\n\\mathcal { L } ( \\boldsymbol { x } _ { p } , \\boldsymbol { x } _ { q } ) ^ { + } = \\mathcal { D } _ { K L } ( \\mathcal { P } ^ { \\star } | | \\boldsymbol { \\mathcal { Q } } ) + \\mathcal { D } _ { K L } ( \\mathcal { Q } ^ { \\star } | | \\mathcal { P } )\n$$",
+        "text_format": "latex",
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+        "type": "equation",
+        "img_path": "images/e92760371a7f6c535154e12438d4d96a95c16ec07580a674a153986b2223c479.jpg",
+        "text": "$$\n\\mathcal { D } _ { K L } ( \\mathcal { P } ^ { \\star } | | \\mathcal { Q } ) = \\sum _ { c = 1 } ^ { k } p _ { c } l o g ( \\frac { p _ { c } } { q _ { c } } )\n$$",
+        "text_format": "latex",
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+        "img_path": "images/b7321399ab1bca7f870eb89cd648d445662c407854095766daa46df0319beb58.jpg",
+        "image_caption": [
+            "Figure 3: The similarity prediction network (the $G$ function). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The cost $\\mathcal { L } ( x _ { p } , x _ { q } ) ^ { + }$ is symmetric w.r.t. $x _ { p } , x _ { q }$ , in which ${ \\mathcal { P } } ^ { \\star }$ and $\\mathcal { Q } ^ { \\star }$ are alternatively assumed to be constant. Each KL-divergence factor $\\hat { \\mathcal { D } } _ { K L } ^ { \\star } ( \\hat { \\mathcal { P } } ^ { \\star } | | \\mathcal { Q } )$ becomes a unary function whose gradient is simply $\\partial { \\cal D } _ { K L } ( { \\mathcal { P } } ^ { \\star } | | { \\mathcal { Q } } ) / \\partial { \\mathcal { Q } }$ . ",
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+    {
+        "type": "text",
+        "text": "If $x _ { p } , x _ { q }$ comes from a pair which is dissimilar, their output distributions are expected to be different, which can be defined as a hinge-loss function: ",
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+        "type": "equation",
+        "img_path": "images/51ccf19abcaebb370f91fd57f44f84142543d559f1a6e585eb5f7e9a8a2b8135.jpg",
+        "text": "$$\n\\mathcal { L } ( \\boldsymbol { x } _ { p } , \\boldsymbol { x } _ { q } ) ^ { - } = L _ { h } ( \\mathcal { D } _ { K L } ( \\mathcal { P } ^ { \\star } | | \\boldsymbol { \\mathcal { Q } } ) , \\boldsymbol { \\sigma } ) + L _ { h } ( \\mathcal { D } _ { K L } ( \\mathcal { Q } ^ { \\star } | | \\mathcal { P } ) , \\boldsymbol { \\sigma } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "equation",
+        "img_path": "images/932eafed8b20b7e34e4d9da7e6abb747c2e87c7130ec49c672fa75935d5f6b54.jpg",
+        "text": "$$\nL _ { h } ( e , \\sigma ) = m a x ( 0 , \\sigma - e )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Given a pair of data with similarity prediction function $G ( x _ { p } , x _ { q } ) \\in \\{ 0 , 1 \\}$ , which is introduced in section 3, the total loss can be defined as a contrastive loss (where we use integer 1 to represent a similar pair): ",
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+    {
+        "type": "equation",
+        "img_path": "images/557947567cc8ebcd6b5062b9e15b936183c860874dd1e623234b5a2b028ebed7.jpg",
+        "text": "$$\n\\mathcal { L } ( x _ { p } , x _ { q } ) = G ( x _ { p } , x _ { q } ) \\mathcal { L } ( x _ { p } , x _ { q } ) ^ { + } + ( 1 - G ( x _ { p } , x _ { q } ) ) \\mathcal { L } ( x _ { p } , x _ { q } ) ^ { - }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "We refer to equation 5 as LCO. Function $G$ is the learnable part that utilizes prior knowledge and is trained with auxiliary dataset $A$ before optimizing LCO. Two particular characteristics of the clustering criterion are worth mentioning: (1) There is no need to define cluster centers. (2) There is no predefined metric applied on the feature representation. Instead, the divergence is calculated directly on the cluster assignment; therefore, both feature representation and clustering are jointly optimized using back-propagation through the deep neural networks. ",
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+    {
+        "type": "text",
+        "text": "4.1 THE PAIRWISE SIMILARITY PREDICTION NETWORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Although there is no restriction of how $G$ should be constructed, we choose deep convolution neural networks due to their efficiency in vision tasks. We design a network architecture inspired from Zagoruyko & Komodakis (2015). While they use it to predict image patch similarity, we use it to predict image-level semantic similarity. However, the Siamese architecture used in Zagoruyko & Komodakis (2015) is not efficient in both training and inference, especially when pairwise information is dense. Therefore, instead of using Siamese architecture, we keep the single column backbone but add a pair-enumeration layer on top of the feature extraction network. The pair-enumeration layer enumerates all pairs of feature vectors within a mini-batch and concatenates the features. Suppose the input of the layer is $1 0 \\times 5 1 2$ with the mini-batch size 10 and the feature dimension 512; then the output of the pair-enumeration layer will be $1 0 0 \\times 1 0 2 4$ (self-pairs included). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The architecture is illustrated in figure 3. We add one hidden fully connected layer on top of the enumeration layer and a binary classifier at the end. We use the standard cross-entropy loss and train it end-to-end. The supervision for training was obtained by converting the ground-truth category labels into binary similarity, i.e., if two samples are from the same class then their label will be similar, otherwise dissimilar. The inference is also end-to-end, and it outputs predictions among all similarity pairs in a mini-batch. The output probability $g \\in [ 0 , 1 ]$ with 1 means more similar. We binarize $g$ at 0.5 to obtain discrete similarity predictions. In the following sections, we simplified the notation of the pairwise similarity prediction network as $G$ . Once $G$ is learned, it then works as a static function in our experiments. ",
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+    {
+        "type": "text",
+        "text": "4.2 THE OBJECTIVE FUNCTION WITH DENSE SIMILARITY PREDICTION ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Since the pairwise prediction of a mini-batch can be densely obtained from $G$ , to efficiently utilize the pair-wise information without forwarding each data multiple times we also combine the pairenumeration layer described in section 4.1 with equation 5. In this case, the outputs of softmax are ",
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+    {
+        "type": "image",
+        "img_path": "images/6ee14a32f3d38107f344e558ad27ba8aa4c6dd2a6632d85ddac1a4bb2284c5fa.jpg",
+        "image_caption": [
+            "Figure 4: The constrained clustering network (CCN) for transfer learning across tasks. The input is unlabeled target data $T$ . The cluster assignment block contains two fully connected layers and has the number of output nodes equal to $k$ . The $f$ described in section 4 is the backbone network plus the cluster assignment block. To optimize LCO, the full pipeline in the diagram is used. After the optimization, it uses another forward propagation with only $f$ to obtain the final cluster assignment. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/480735bbbbbc31b0524612dd6b214d318a05c8bfdb6bc3ec5a4e395d164bbae6.jpg",
+        "image_caption": [
+            "Figure 5: The network for transfer learning across domains. The input is the mix of $S ^ { \\prime }$ and $T$ . The architecture is a direct extension of CCN. We use $\\mathrm { C C N ^ { + } }$ to represent the mandatory parts (upper branch) which implements eq. (7). $\\mathrm { C C N ^ { + + } }$ includes the domain adaptation method (optional branch). "
+        ],
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+    {
+        "type": "text",
+        "text": "enumerated in pairs. Let $D$ be the set of all tuples $( p , q )$ , while $p$ and $q$ are the indices of a sample in a mini-batch. The dense clustering loss $\\mathcal { L } _ { d }$ for a mini-batch is calculated by: ",
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+        "type": "equation",
+        "img_path": "images/2536ed3e74d2ee43afd3d960df5668474cec18206ed2d1958279712458d3e6d4.jpg",
+        "text": "$$\n\\mathcal { L } _ { d } = \\sum _ { \\forall ( p , q ) \\in D } \\mathcal { L } ( x _ { p } , x _ { q } ) .\n$$",
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+        "type": "text",
+        "text": "We use $\\mathcal { L } _ { d }$ standalone with deep neural networks to reconstruct semantic clusters and for transfer learning across tasks (figure 4). We call the architecture the constrained clustering network (CCN). ",
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+    {
+        "type": "text",
+        "text": "4.3 COMBINING WITH OTHER OBJECTIVES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In the cross-domain case, we additionally have labeled source data. This enables us to use LCO for $T$ while using classification loss for $S ^ { \\prime }$ . The overall training procedure is similar to previous domain adaptation approaches in that both source and target data are mixed in a mini-batch, but different losses are applied. We denote the source domain images in a mini-batch $b$ as $X _ { b ^ { S } }$ and the target domain images $X _ { b ^ { T } }$ with its set of dense pairs $D _ { b ^ { T } }$ . The loss function $\\mathcal { L } _ { c d }$ for cross-domain transfer in a mini-batch can then be formulated as: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/70260d3727ed67e5a59fd974700b2ab49d04ad5b5e5cdc3ecf8874b45be70bd8.jpg",
+        "text": "$$\n\\mathcal { L } _ { c d } = \\mathcal { L } _ { c l s } + \\mathcal { L } _ { c l u s t e r }\n$$",
+        "text_format": "latex",
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+        "type": "equation",
+        "img_path": "images/c8faa5cd2cfb258d9ee286453aeb92b063fa9022dc543a5611e434e7dbd32356.jpg",
+        "text": "$$\n\\mathcal { L } _ { c l s } = \\frac { 1 } { | b ^ { S } | } \\sum _ { \\forall x _ { i } \\in X _ { b } s } C r o s s E n t r o p y L o s s ( f ( x _ { i } ) ) .\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/56ffd6cf3c83633f9f743ca9c45f42e3446398e208b61a557a090c6441db0b55.jpg",
+        "text": "$$\n\\mathcal { L } _ { c l u s t e r } = \\frac { 1 } { | b ^ { T } | ^ { 2 } } \\sum _ { \\forall ( i , j ) \\in D _ { b ^ { T } } } \\mathcal { L } ( x _ { i } , x _ { j } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The $\\mathcal { L } _ { c l u s t e r }$ and $\\mathcal { L } _ { c l s }$ share the same outputs from network $f$ . Although $\\mathcal { L } _ { c l u s t e r }$ does not force the mapping between clusters and categories, $\\mathcal { L } _ { c l s }$ does. Therefore the learned classifier can be applied on target data directly and picks the maximum probability for the predicted class. Note that in our loss function, there is no term to explicitly match the feature distribution between source and target; it merely transfers more knowledge in the form of constraints to regularize the learning of the classifier. There is also no requirement for the architecture to utilize hidden layers. Therefore our approach has the large flexibility to be combined with other domain adaptation strategies. Figure 5 illustrates the architectures $\\mathrm { C C N ^ { + } }$ and $\\mathrm { C C N ^ { + + } }$ used in our cross-domain experiments. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "This section contains evaluations with four image datasets and covers both cross-task and crossdomain schemes. The details are described below, and the differences between experimental settings are illustrated in appendix A. ",
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+    {
+        "type": "text",
+        "text": "5.1 UNSUPERVISED CROSS-TASK TRANSFER LEARNING ",
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+    {
+        "type": "text",
+        "text": "5.1.1 SETUP ",
+        "text_level": 1,
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+        "text": "The Omniglot dataset (Lake et al., 2015) contains 1623 different handwritten characters and each of them has 20 images drawn by different people. The characters are from 50 different alphabets and were separated to 30 background sets $O m n i g l o t _ { b g }$ and 20 evaluation sets $O m n i g l o t _ { e v a l }$ by the author. We use the $O m n i g l o t _ { b g }$ as the auxiliary dataset $( A )$ and the $O m n i g l o t _ { e v a l }$ as the target data $( T )$ . The total number of characters in $O m n g l o t _ { b g }$ is 964, which can be regarded as the number of categories available to learn the semantic similarity. The goal is to cluster Omniglot $_ { e v a l }$ to reconstruct its semantic categories, without ever having any labels. ",
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+        "type": "text",
+        "text": "The $G$ function has a backbone neural network with four $3 \\mathrm { x } 3$ convolution layers followed by $2 \\mathbf { x } 2$ max-pooling with stride 2. All hidden layers are followed by batch normalization (Ioffe & Szegedy, 2015) and rectified linear unit. To prepare the inputs for training, the images from $O m n i g l o t _ { b g }$ were resized to $3 2 \\mathrm { x } 3 2$ and normalized to zero mean with unit standard deviation. Each mini-batch has a size of 100 and is sampled from a random 20 characters to make sure the amount of similar pairs is reasonable. After pair enumeration, there are 10000 pairs subject to the loss function, which is a two-class cross entropy loss. The ground-truth similarity is obtained by converting the categorical labels. The loss of $G$ is optimized by stochastic gradient descent and is the only part trained in a supervised manner in this section. ",
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+        "type": "text",
+        "text": "The Constrained Clustering Network (CCN) is used to reconstruct the semantic clusters in the target dataset using outputs from $G$ . The network has four 3x3 convolution layers followed by $2 \\mathbf { x } 2$ max-pooling with stride 2, one hidden fully-connected layer, and one cluster assignment layer which is also fully connected. The number of output nodes in the last layer is equal to the number of potential clusters in the target data. The output distribution is enumerated in pairs before sending to LCO. The network is randomly initialized, trained end-to-end, and optimized by stochastic gradient descent with randomly sampled 100 images per mini-batch. Note the $G$ function used by LCO is fixed during the optimization. The input data preparation is the same as above, except now the data is from $O m n i g l o t _ { e v a l }$ . Specifically, during the training, the same mini-batch is given to both $G$ and CCN. The dense pairwise similarity predictions from $G$ are sent to LCO and are then fully utilized. The only hyper-parameter in LCO is $\\sigma$ , and we set it to 2 for all our experiments. ",
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+        "type": "text",
+        "text": "Omniglot $e v a l$ contains 20 alphabets and each one can be used as a standalone dataset. The 20 target datasets contain a varied number (20 to 47) of characters. Therefore we can evaluate the reconstruction performance under a varied number of ground-truth clusters. We tested the situations when the number of character $( K )$ in an alphabet is known and unknown. When $K$ is known, we set the target number of clusters in the clustering algorithm equal to the true number of characters. If $K$ is unknown, the common practice is to set it to a large number so that the data from different categories will not be forced to be in the same cluster. In the experiment, we merely set $K$ to 100, which is much larger than the largest dataset (47). ",
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+        "text": "All constrained clustering algorithms can be used to reconstruct the semantic clusters for our problem. Since there are mis-predictions in $G$ , robustness to noise is the most important factor. Here we include all four types of constrained clustering algorithms introduced in section 2 as the baselines. We provide the full set of pairwise constraints to all algorithms including ours. In other words, given an alphabet of 20 characters, it contains 400 images and 160000 predicted similarities from $G$ (note that $G$ makes predictions for both orders of a pair and for self-pairs). The full pairwise similarities were presented to all algorithms in random order, while we empirically found it has no noticeable effect on results. We pick the baseline approaches based on code availability and scalability concerning the number of pairwise constraints. Therefore we have shown results for K-means (MacQueen et al., 1967), LPNMF (Cai et al., 2009), LSC (Chen & Cai, 2011), ITML (Davis et al., 2007), SKKm (Anand et al., 2014), SKLR (Amid et al., 2016), SKMS (Anand et al., 2014), CSP (Wang et al., 2014) , and MPCK-means (Bilenko et al., 2004) as our baselines. We use the default parameters for each algorithm provided by the original author except for $K$ , the number of clusters. We use the same normalized images used in CCN for all algorithms. ",
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+    {
+        "type": "table",
+        "img_path": "images/2ae0c7337bcbc45fa8408cbf5514b9f871c789e8f5603902752e52690914bc95.jpg",
+        "table_caption": [
+            "Table 1: Unsupervised cross-task transfer from Omniglot $b g$ to $O m n i g l o t _ { e v a l }$ . The performance is averaged across 20 alphabets which have 20 to 47 letters. The ACC and NMI without brackets have the number of clusters equal to ground-truth. The \"(100)\" means the algorithms use $K = 1 0 0$ . The characteristics of how each algorithm utilizes the pairwise constraints are marked in the \"Constraints in\" column, where metric stands for the metric learning of feature representation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">Constraints in</td><td rowspan=\"2\">ACC</td><td rowspan=\"2\">ACC (100)</td><td rowspan=\"2\">NMI</td><td rowspan=\"2\">NMI (100)</td></tr><tr><td>Metric</td><td>Clustering</td></tr><tr><td>K-means</td><td></td><td></td><td>21.7%</td><td>18.9%</td><td>0.353</td><td>0.464</td></tr><tr><td>LPNMF</td><td></td><td></td><td>22.2%</td><td>16.3%</td><td>0.372</td><td>0.498</td></tr><tr><td>LSC</td><td></td><td></td><td>23.6%</td><td>18.0%</td><td>0.376</td><td>0.500</td></tr><tr><td>ITML</td><td>0</td><td></td><td>56.7%</td><td>47.2%</td><td>0.674</td><td>0.727</td></tr><tr><td>SKMS</td><td>0</td><td></td><td>=</td><td>45.5%</td><td>1</td><td>0.693</td></tr><tr><td>SKKm</td><td>0</td><td></td><td>62.4%</td><td>46.9%</td><td>0.770</td><td>0.781</td></tr><tr><td>SKLR</td><td>0</td><td></td><td>66.9%</td><td>46.8%</td><td>0.791</td><td>0.760</td></tr><tr><td>CSP</td><td></td><td>0</td><td>62.5%</td><td>65.4%</td><td>0.812</td><td>0.812</td></tr><tr><td>MPCK-means</td><td>0</td><td>0</td><td>81.9%</td><td>53.9%</td><td>0.871</td><td>0.816</td></tr><tr><td>CCN (Ours)</td><td>0</td><td>0</td><td>82.4%</td><td>78.1%</td><td>0.889</td><td>0.874</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "The evaluation uses two clustering metrics. The first is normalized-mutual information (NMI) (Strehl & Ghosh, 2002) which is widely used for clustering. The second one is the clustering accuracy (ACC) (Yang et al., 2010). The ACC metric first finds the one-to-one matching between predicted clusters and ground-truth labels, and then calculates the classification accuracy based on the mapping. All data outside the matched clusters will be regarded as mis-predictions. To get high ACC, the algorithm has to generate coherent clusters where each cluster includes most of the data in a category; otherwise the score drops quickly. Therefore ACC provides better discrimination to evaluate whether the semantic clusters have been reconstructed well. ",
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+    {
+        "type": "text",
+        "text": "5.1.2 RESULTS AND DISCUSSIONS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We report the average performance over the 20 alphabets in table 1. Our approach achieved the top performance on both metrics. The CCN demonstrates strong robustness on the challenging scenario of unknown $K$ . It achieved $7 8 . 1 \\%$ average accuracy. Compared with $8 2 . 4 \\%$ when $K$ is known, CCN has a relatively small drop. Compared to the second best algorithm, CSP, which is $6 5 . 4 \\%$ , CCN outperforms it with a large gap. The classical approach MPCK-means works surprisingly well when the number of clusters is known, but its performance dropped dramatically from $8 1 . 9 \\%$ to $5 3 . 9 \\%$ when $K = 1 0 0$ . In the performance breakdown for the 20 individual alphabets, CCN achieved $94 \\%$ clustering accuracy on Old Church Slavonic Cyrillic, which has 45 characters (appendix table 5). Therefore the results show the feasibility of reconstructing semantic clusters using only noisy similarity predictions. ",
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+    {
+        "type": "text",
+        "text": "When to use the semantic similarity? The experiments in table 1 show a clear trend that utilizing the pairwise constraints jointly for both metric learning and minimizing the clustering loss achieves the best performance, including both MPCK-means and CCN. In the case of unknown number of clusters, where we set $K = 1 0 0$ , the algorithms that use constraints to optimize clustering loss have better robustness, for example, CSP and CCN. The group that only use constraints for metric learning (ITML, SKMS, SKKm, and SKLR) significantly outperform the group that does not use it (K-means, LPNMF, LSC). However, their performance are still far behind CCN. Our results confirm the importance of jointly optimizing the metric and clustering. ",
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+        "type": "text",
+        "text": "The robustness against noisy similarity prediction is the key factor to enable the cross-task transfer framework. To the best of our knowledge, table 1 is the first comprehensive robustness comparisons using predicted constraints learned from real data instead of converting from ground-truth labels. The accuracy of $G$ in our experiment is shown in appendix table 7 and demonstrates the reasonable performance of $G$ which is on par with Matching-Net (Vinyals et al., 2016). After binarizing the prediction at 0.5 probability, the similar pair precision, similar pair recall, dissimilar pair precision, and dissimilar pair recall among the 659 characters are (0.392, 0.927, 0.999, 0.995), accordingly. The binarized predictions are better than uniform random guess (0.002, 0.500, 0.998, 0.500), but are still noisy. Therefore it is very challenging for constrained clustering. The visualization of the robustness range of CCN are provided in appendix D, and shows that the robustness is related to the density of pairs involved in a mini-batch. We hypothesize that during the optimization, the gradients from wrongly predicted pairs are canceled out by each other or by the correctly predicted pairs. Therefore the overall gradient still moves the solution towards a better clustering result. ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "How to predict $K ?$ Inferring the number of clusters $( N C )$ is a hard problem, but with the pairwise similarity information, it becomes feasible. For evaluation, we compute the difference between the number of dominant clusters $( N D C )$ and the true number of categories $( N C ^ { g t } )$ in a dataset. We use a naive definition for $N D C$ , which is the number of clusters that have a size larger than expected size when data is sampled uniformly. In other words, $\\begin{array} { r } { N D C = \\sum _ { i = 1 } ^ { K } \\left[ C _ { i } > = \\bar { E } [ C _ { i } ] \\right] } \\end{array}$ where $[ \\cdot ]$ is an Iverson Bracket and $C _ { i }$ is the size of cluster $i$ . For example, $\\bar { E } [ \\bar { C } _ { i } ]$ will be 10 if the alphabet has 1000 images and $K = 1 0 0$ . Then the average difference $( A D i f )$ is calculated by $\\begin{array} { r } { A D i f = \\frac { 1 } { | D | } \\sum _ { d \\in D } { | N D C _ { d } - N C _ { d } ^ { g t } | } } \\end{array}$ , where $d$ (i.e., alphabet) is a dataset in $D$ . A smaller $A D i f$ indicates a better estimate of $K$ . CCN achieves a score of 6.35 (appendix table 6). We compare this with the baseline approach SKMS (Anand et al., 2014), which does not require a given $K$ and supports a pipeline to estimate $K$ automatically (therefore we only put it into the column $K = 1 0 0$ in table 1.). SKMS gets 16.3. Furthermore, 10 out of 20 datasets from CCN’s prediction have a difference between $N D C _ { d }$ and $N C _ { d } ^ { g t }$ smaller or equal to 3, which shows the feasibility of estimating $K$ with predicted similarity. ",
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+    {
+        "type": "text",
+        "text": "5.1.3 EXPERIMENTS USING THE IMAGENET DATASET ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To demonstrate the scalability of our approach, we applied the same scheme on the ImageNet dataset. The 1000-class dataset is separated into 882-class (ImageNet882) and 118-class (ImageNet118) subsets as the random split in Vinyals et al. (2016). We use ImageN et882 for $A$ and 30 classes $\\mathrm { \\sim } 3 9 \\mathrm { k }$ images) are randomly sampled from $I m a g e N e t _ { 1 1 8 }$ for $T$ . The difference from section 5.1.1 is that here we use Resnet-18 for both $G$ and CCN, and the weights of the backbone are pre-trained with ImageN et882. Since the number of pairs is high and it is not feasible to feed them into other constrained clustering algorithms, we compare CCN with K-means, LSC(Chen & Cai, 2011), and LPNMF (Cai et al., 2009). We use the output from the average pooling layer of Resnet-18 as the input to these clustering algorithms. CCN gives the top performance with average ACC $7 3 . 8 \\%$ when $K$ is known, and $6 5 . 2 \\%$ when the number of clusters is unknown, which outperforms the second $( 3 4 . 5 \\%$ by K-means) with a large margin. The full comparison is in appendix table 8. And the performance of $G$ is provided in appendix table 9. ",
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+    {
+        "type": "text",
+        "text": "5.2 UNSUPERVISED CROSS-DOMAIN TRANSFER LEARNING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.2.1 SETUP ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Office-31 (Saenko et al., 2010) has images from 31 categories of office objects. The 4652 images are obtained from three domains: Amazon $( a )$ , DSLR $( d )$ , and Webcam $( w )$ . The dataset is the standard benchmark for evaluating domain adaptation performance in computer vision. We experiment with all six combinations (source $S ^ { \\prime } \\to \\mathrm { t a r g e t } T$ ): $a  w$ , $a \\to d$ , $d \\to a$ , $d \\to w$ , $w  a$ , $w \\to d$ , and report the average accuracy based on five random experiments for each setting. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "The $G$ function learns the semantic similarity function from the auxiliary dataset $A$ , which is ImageNet with all 1000 categories. The backbone network of $G$ is Resnet-18 and has the weights initialized by ImageNet classification. The training process is the same as section 5.1.1 except the images are resized to 224. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We follow the standard protocols using deep neural networks (Long et al., 2017; Ganin et al., 2016) for unsupervised domain adaptation. The backbone network of $\\mathrm { C C N ^ { + } }$ is pre-trained with ImageNet. Appendix figure 7 illustrates the scheme. During the training, all source data and target data are used. Each mini-batch is constructed by 32 labeled samples from source and 96 unlabeled samples from target. Since the target dataset has no labels and could only be randomly sampled, it is crucial to have sufficient mini-batch size to ensure that similar pairs are sampled. The loss function used in our approach is equation (7) and is optimized by stochastic gradient descent. The $\\mathrm { C C N ^ { + / + + } }$ and DANN (RevGrad) with ResNet backbone are implemented with Torch. We use the code from original author for JAN. Both DANN and JAN use a 256-dimension bottleneck feature layer. ",
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+    {
+        "type": "table",
+        "img_path": "images/f5936f0f0feec0a34a07606096c3ce59fc69a3f0e85a74616b90a6b7ecc8c3b3.jpg",
+        "table_caption": [
+            "Table 2: Unsupervised cross-domain transfer (domain adaptation) on the Office-31 dataset. The backbone network used here is Resnet-18 (He et al., 2016) pre-trained with ImageNet. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>A→W</td><td>D→W</td><td>W→D</td><td>A→D</td><td>D→A</td><td>W→A</td><td>Avg</td></tr><tr><td>Source-Only</td><td>66.8</td><td>92.8</td><td>96.8</td><td>67.1</td><td>51.4</td><td>53.0</td><td>71.3</td></tr><tr><td>DANN (Ganin et al., 2016)</td><td>73.2</td><td>97.0</td><td>99.0</td><td>69.3</td><td>58.0</td><td>57.8</td><td>75.7</td></tr><tr><td>JAN (Long et al., 2017)</td><td>74.5</td><td>94.1</td><td>99.6</td><td>75.9</td><td>58.7</td><td>59.0</td><td>76.9</td></tr><tr><td>CCN+ (ours)</td><td>76.7</td><td>97.3</td><td>98.2</td><td>71.2</td><td>61.0</td><td>60.5</td><td>77.5</td></tr><tr><td>CCN++ (with DANN)</td><td>78.2</td><td>97.4</td><td>98.6</td><td>73.5</td><td>62.8</td><td>60.6</td><td>78.5</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5.2.2 RESULTS AND DISCUSSIONS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The results are summarized in table 2. Our approach $\\mathrm { ( C C N ^ { + } ) }$ demonstrates a strong performance boost for the unsupervised cross-domain transfer problem. It reaches $7 7 . 5 \\%$ average accuracy which gained 6.2 points from the $7 1 . 3 \\%$ source-only baseline. Although our approach merely transfers more information from the auxiliary dataset, it outperforms the strong approach DANN $( 7 5 . 7 \\% )$ , and state-of-the-art JAN $( 7 6 . 9 \\% )$ . When combining ours with DANN $\\mathrm { ( C C N ^ { + + } ) }$ , the performance is further boosted. This indicates that LCO helps mitigate the transfer problem in a certain way that is orthogonal to minimizing the domain discrepancy. We observe the same trend when using a deeper backbone network, i.e., ResNet-34. In such a case the average accuracy achieved is $7 7 . 9 \\%$ , $8 1 . 1 \\%$ and $82 \\%$ for source-only, $\\mathrm { C C N ^ { + } }$ and $\\mathrm { C C N ^ { + + } }$ , respectively, though we used exactly the same $G$ as before (with ResNet-18 backbone for $G$ ). This indicates that the information carried in the similarity predictions is not equivalent to transferring features with deeper networks. More discussions are in appendix C and the performance of $G$ is provided in appendix table 11 to show that although the prediction has low precision for similar pairs $( \\sim 0 . 2 )$ , our approach still benefits from the dense similarity predictions. ",
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+    {
+        "type": "text",
+        "text": "5.2.3 EXPERIMENTS USING SVHN-TO-MNIST ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We also evaluated the $\\mathrm { C C N ^ { + } }$ on another widely compared scenario, which uses color Street View House Numbers images (SVHN) (Netzer et al., 2011) as $S ^ { \\prime }$ , the gray-scale hand-written digits (MNIST) (LeCun, 1998) as $T$ . To learn $G$ , we use the $O m n g l o t _ { b g }$ as $A$ . We train all the networks in this section from scratch. Our experimental setting is similar to Sener et al. (2016). We achieve the top performance with $8 9 . 1 \\%$ accuracy. The performance gain from source-only in our approach is $+ 3 7 . 1 \\%$ , which wins by a large margin compared to the $+ 2 3 . 9 \\%$ of LTR (Sener et al., 2016). The full comparison is presented in appendix table 12. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION AND OUTLOOK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper, we demonstrated the usefulness of transferring information in the form of pairwise similarity predictions. Such information can be transferred as a function and utilized by a loss formulation inspired from constrained clustering, but implemented more robustly within a neural network that can jointly optimize both features and clustering outputs based on these noisy predictions. The experiments for both cross-task and cross-domain transfer learning show strong benefits of using the semantic similarity predictions resulting in new state of the art results across several datasets. This is true even without explicit domain adaptation for the cross-domain task, and if we add a domain discrepancy loss the benefits increase further. ",
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+    {
+        "type": "text",
+        "text": "There are two key factors that determine the performance of the proposed framework. The first is the robustness of the constrained clustering and second is the performance of the similarity prediction function. We show robustness of CCN empirically, but we do not explore situations where learning the similarity function is harder. For example, such cases arise when there are a small number of categories in source or a large domain discrepancy between source and target. One idea to deal with such a situation is learning G with domain adaptation strategies. We leave these aspects for future work. ",
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+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
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+        "text": "This work was supported by the National Science Foundation and National Robotics Initiative (grant # IIS-1426998). ",
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+    {
+        "type": "table",
+        "img_path": "images/669369c47c732375ec24da585d0c99869e157e4a7cf270d59df4477454679e65.jpg",
+        "table_caption": [
+            "Table 3: The list of datasets involved in each experiment. $G$ learns the similarity function from dataset A. The $\\mathbf { C C N } ^ { * }$ is optimized with dataset T or T∪S’, while $\\mathbf { C C N } ^ { * }$ means CCN for cross-task transfer and $\\mathrm { C C N ^ { + / + + } }$ for cross-domain transfer. The rows for network initialization indicate whether the network has weights initialized by training a classification task with the specified dataset. The weights are randomly initialized if not specified. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Scheme</td><td colspan=\"2\">Cross-Task Transfer</td><td colspan=\"2\">Cross-Domain Transfer</td></tr><tr><td colspan=\"2\">Experiment Section</td><td>Omniglot 5.1.1</td><td>ImageNet 5.1.3</td><td>Office-31 5.2.1</td><td>SVHN-MNIST 5.2.3</td></tr><tr><td rowspan=\"3\">Datasets</td><td>A</td><td>Omniglotbg</td><td>ImageNet882</td><td>ImageNet1000</td><td>Omniglotbg</td></tr><tr><td>S’</td><td></td><td></td><td>Office-31 {a,d,w}</td><td>SVHN</td></tr><tr><td>T</td><td>Omnigloteval</td><td>ImageNet118</td><td>Office-31 {a,d,w)</td><td>MNIST</td></tr><tr><td rowspan=\"2\">Network Initialization</td><td>G</td><td></td><td>ImageNet882</td><td>ImageNet1000</td><td>1</td></tr><tr><td>CCN*</td><td></td><td>ImageNet882</td><td>ImageNet1000</td><td>1</td></tr></table>",
+        "bbox": [
+            194,
+            252,
+            803,
+            411
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/2385e43b5137c9f8e365706d4a2896d3faaa824f5c9736b5ead9792edd2cfe5c.jpg",
+        "table_caption": [
+            "Table 4: The list of loss functions used for training networks. The similarity prediction function (network) $G$ uses the cross-entropy (CE) loss with two classes (similar/dissimilar). The training of constrained clustering network $( \\mathbf { C C N } ^ { * } )$ involves the combinations of the learnable clustering objective (LCO), cross-entropy, and domain adaptation loss (DA). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>LCO</td><td>CE</td><td>DA</td></tr><tr><td>G</td><td></td><td>√</td><td></td></tr><tr><td>CCN</td><td></td><td></td><td></td></tr><tr><td>CCN+</td><td>广</td><td>√</td><td></td></tr><tr><td>CCN++</td><td>√</td><td></td><td>√</td></tr></table>",
+        "bbox": [
+            395,
+            498,
+            601,
+            583
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/6077c2618b017816b79d4cf6b62b30b24d7ebf9ecfe529286747a494f9f155c8.jpg",
+        "image_caption": [
+            "Figure 6: A diagram depicting the cross-task transfer experiment. Two experiments follow this flow: Omniglot $b g$ to $O m n i g l o t _ { e v a l }$ , and $I m a g e N e t _ { 8 8 2 }$ to ImageNet118. Both have exclusive classes between source and target domain. In the ImageNet experiment, the backbone network is initialized with the weights pre-trained with ImageN et882. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            101,
+            826,
+            236
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/62fef6aa1814051042be0542b4068cc2863de90390604a41a4bf68f744e5059c.jpg",
+        "image_caption": [
+            "Figure 7: Comparison between domain adaptation approaches (a) Transferring semantic similarity from auxiliary data (our method), and (b) Minimizing the domain discrepancy. The diagram uses office-31 benchmark as the scenario of transferring. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            339,
+            825,
+            598
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B SUPPLEMENTARY FOR UNSUPERVISED CROSS TASK TRANSFER LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
+            821,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/edd5c8527438fb985fa3aae8e157b710127e887e112b22304a27c43541aa280a.jpg",
+        "table_caption": [
+            "Table 5: A breakdown of the results for each alphabet in Omnigloteval. The unsupervised cross task transfer experiment is described in section 5.1. This table shows the clustering accuracy with $K = 1 0 0$ to simulate the situation of unknown number of clusters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Alphabet</td><td>K-means</td><td>LPNMF</td><td>LSC</td><td>ITML</td><td>SKMS</td><td>SKKm</td><td>SKLR</td><td>CSP</td><td>MPCK-means</td><td>CCN</td></tr><tr><td>Angelic</td><td>24%</td><td>26%</td><td>27%</td><td>48%</td><td>78%</td><td>40%</td><td>38%</td><td>72%</td><td>50%</td><td>82%</td></tr><tr><td>Atemayar_Qelisayer</td><td>19%</td><td>14%</td><td>17%</td><td>51%</td><td>61%</td><td>51%</td><td>44%</td><td>61%</td><td>50%</td><td>82%</td></tr><tr><td>Atlantean</td><td>19%</td><td>15%</td><td>19%</td><td>57%</td><td>72%</td><td>55%</td><td>51%</td><td>77%</td><td>47%</td><td>72%</td></tr><tr><td>Aurek_Besh</td><td>23%</td><td>18%</td><td>21%</td><td>45%</td><td>49%</td><td>44%</td><td>45%</td><td>76%</td><td>71%</td><td>90%</td></tr><tr><td>Avesta</td><td>22%</td><td>18%</td><td>19%</td><td>47%</td><td>29%</td><td>39%</td><td>38%</td><td>65%</td><td>52%</td><td>76%</td></tr><tr><td>Ge_ez</td><td>19%</td><td>17%</td><td>17%</td><td>56%</td><td>58%</td><td>47%</td><td>42%</td><td>72%</td><td>55%</td><td>82%</td></tr><tr><td>Glagolitic</td><td>22%</td><td>19%</td><td>21%</td><td>42%</td><td>38%</td><td>58%</td><td>62%</td><td>66%</td><td>61%</td><td>85%</td></tr><tr><td>Gurmukhi</td><td>15%</td><td>13%</td><td>15%</td><td>44%</td><td>26%</td><td>54%</td><td>48%</td><td>60%</td><td>56%</td><td>80%</td></tr><tr><td>Kannada</td><td>18%</td><td>14%</td><td>16%</td><td>48%</td><td>37%</td><td>46%</td><td>53%</td><td>60%</td><td>48%</td><td>62%</td></tr><tr><td>Keble</td><td>20%</td><td>14%</td><td>18%</td><td>44%</td><td>60%</td><td>46%</td><td>44%</td><td>75%</td><td>68%</td><td>90%</td></tr><tr><td>Malayalam</td><td>18%</td><td>15%</td><td>16%</td><td>36%</td><td>24%</td><td>49%</td><td>52%</td><td>50%</td><td>54%</td><td>72%</td></tr><tr><td>Manipuri</td><td>17%</td><td>15%</td><td>16%</td><td>49%</td><td>40%</td><td>53%</td><td>54%</td><td>66%</td><td>62%</td><td>85%</td></tr><tr><td>Mongolian</td><td>18%</td><td>18%</td><td>19%</td><td>50%</td><td>40%</td><td>37%</td><td>48%</td><td>75%</td><td>57%</td><td>86%</td></tr><tr><td>Old_Church_Slavonic_Cyrilic</td><td>19%</td><td>16%</td><td>19%</td><td>42%</td><td>41%</td><td>52%</td><td>67%</td><td>71%</td><td>67%</td><td>94%</td></tr><tr><td>Oriya</td><td>15%</td><td>13%</td><td>14%</td><td>46%</td><td>40%</td><td>54%</td><td>45%</td><td>57%</td><td>52%</td><td>67%</td></tr><tr><td>Sylheti</td><td>14%</td><td>13%</td><td>14%</td><td>49%</td><td>32%</td><td>35%</td><td>37%</td><td>59%</td><td>43%</td><td>64%</td></tr><tr><td>Syriac_Serto</td><td>20%</td><td>18%</td><td>19%</td><td>55%</td><td>70%</td><td>42%</td><td>35%</td><td>70%</td><td>47%</td><td>73%</td></tr><tr><td>Tengwar</td><td>18%</td><td>18%</td><td>18%</td><td>51%</td><td>44%</td><td>49%</td><td>40%</td><td>61%</td><td>44%</td><td>66%</td></tr><tr><td>Tibetan</td><td>18%</td><td>14%</td><td>16%</td><td>44%</td><td>31%</td><td>47%</td><td>54%</td><td>59%</td><td>55%</td><td>84%</td></tr><tr><td>ULOG</td><td>20%</td><td>18%</td><td>18%</td><td>41%</td><td>41%</td><td>40%</td><td>41%</td><td>56%</td><td>38%</td><td>70%</td></tr><tr><td>Average</td><td>19%</td><td>16%</td><td>18%</td><td>47%</td><td>46%</td><td>47%</td><td>47%</td><td>65%</td><td>54%</td><td>78%</td></tr></table>",
+        "bbox": [
+            173,
+            189,
+            826,
+            425
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/a3aa1654884a293f29e57c2b20dac4b2622b4b5a81d2fa80fab3a1c67ce91da7.jpg",
+        "table_caption": [
+            "Table 6: Estimates for the number of characters across the 20 datasets in Omniglot $e v a l$ . The bold number means the prediction has error smaller or equal to 3. The $A D i f$ is defined in section 5.1.2. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Alphabet</td><td>True #class</td><td>SKMS</td><td>CCN (K=100)</td></tr><tr><td>Angelic</td><td>20</td><td>16</td><td>26</td></tr><tr><td>Atemayar_Qelisayer</td><td>26</td><td>17</td><td>34</td></tr><tr><td>Atlantean</td><td>26</td><td>21</td><td>41</td></tr><tr><td>Aurek_Besh</td><td>26</td><td>14</td><td>28</td></tr><tr><td>Avesta</td><td>26</td><td>8</td><td>32</td></tr><tr><td>Ge_ez</td><td>26</td><td>18</td><td>32</td></tr><tr><td>Glagolitic</td><td>45</td><td>18</td><td>45</td></tr><tr><td>Gurmukhi</td><td>45</td><td>12</td><td>43</td></tr><tr><td>Kannada</td><td>41</td><td>19</td><td>44</td></tr><tr><td>Keble</td><td>26</td><td>16</td><td>28</td></tr><tr><td>Malayalam</td><td>47</td><td>12</td><td>47</td></tr><tr><td>Manipuri</td><td>40</td><td>17</td><td>41</td></tr><tr><td>Mongolian</td><td>30</td><td>28</td><td>36</td></tr><tr><td>Old_Church_Slavonic_Cyrillic</td><td>45</td><td>23</td><td>45</td></tr><tr><td>Oriya</td><td>46</td><td>22</td><td>49</td></tr><tr><td>Sylheti</td><td>28</td><td>11</td><td>50</td></tr><tr><td>Syriac_Serto</td><td>23</td><td>19</td><td>38</td></tr><tr><td>Tengwar</td><td>25</td><td>12</td><td>41</td></tr><tr><td>Tibetan</td><td>42</td><td>15</td><td>42</td></tr><tr><td>ULOG</td><td>26</td><td>15</td><td>40</td></tr><tr><td>ADif</td><td></td><td>16.3</td><td>6.35</td></tr></table>",
+        "bbox": [
+            259,
+            486,
+            740,
+            810
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/b3acd2713316e53455d67e85e1a9e12c7140f850c530ca53e0204f69f9c4d0f0.jpg",
+        "table_caption": [
+            "Table 7: The performance of the similarity prediction function used in section 5.1. We leverage the N-way test which is commonly used in one-shot learning evaluation. The similarity is learned with Omniglot $b g$ and has N-way test with $O m n i g l o t _ { e v a l }$ and MNIST. The experimental settings follow Vinyals et al. (2016). The raw probability output (without binarization) from our $G$ is used to find the nearest exemplar in the N-way test. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td colspan=\"2\">Omniglot-eval 5-way</td><td>MNIST 10-way</td></tr><tr><td>Siamese-Nets (Koch et al., 2015)</td><td>0.967</td><td>20-way 0.880</td><td>0.703</td></tr><tr><td>Match-Net (Vinyals et al., 2016)</td><td>0.981</td><td>0.938</td><td>0.720</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>Ours</td><td>0.979</td><td>0.935</td><td>0.720</td></tr></table>",
+        "bbox": [
+            284,
+            224,
+            714,
+            315
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/ed7b58dab6ef5fe60a2c9d95bf27a1b4443d28d6a8838d1fd037c05b989d02e4.jpg",
+        "table_caption": [
+            "Table 8: Unsupervised cross-task transfer learning on ImageNet. The values are the average of three random subsets in $I m a g e N e t _ { 1 1 8 }$ . Each subset has 30 classes. The \"ACC\" has $K = 3 0$ while the \"ACC (100)\" sets $K = 1 0 0$ . All methods use the features (outputs of average pooling) from Resnet-18 pre-trained with ImageNet882 classification. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>ACC</td><td>ACC(100)</td><td>NMI</td><td>NMI(100)</td></tr><tr><td>K-means</td><td>71.9%</td><td>34.5%</td><td>0.713</td><td>0.671</td></tr><tr><td>LSC</td><td>73.3%</td><td>33.5%</td><td>0.733</td><td>0.655</td></tr><tr><td>LPNMF</td><td>43.0%</td><td>21.8%</td><td>0.526</td><td>0.500</td></tr><tr><td>CCN (ours)</td><td>73.8%</td><td>65.2%</td><td>0.750</td><td>0.715</td></tr></table>",
+        "bbox": [
+            300,
+            476,
+            696,
+            566
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/2fd973ba9b8e5a822c55ca7487481ec288ef5988660c0a3a7768ad39ede6159e.jpg",
+        "table_caption": [
+            "Table 9: Performance of the similarity prediction function ( $G$ , trained with ImageN et882) applied to three subsets of $I m a g e N e t _ { 1 1 8 }$ . Each subset contains 30 random classes of $I m a g e N e t _ { 1 1 8 }$ . The predictions are binarized at 0.5 to calculate the precision and recall. Random\\*: The expected performance when classes are uniformly distributed and make uniform random guess for similarity. It is an approximation since the number of images in each class is only roughly equal in ImageNet. We sampled 12M pairs for each set to collect the statistics. Sampling more pairs has no noticeable change to the values. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Set</td><td>Similar Precision</td><td> Similar Recall</td><td>Dissimilar Precision</td><td>Dissimilar Recall</td></tr><tr><td>Random*</td><td>0.033</td><td>0.500</td><td>0.967</td><td>0.500</td></tr><tr><td>Set A</td><td>0.840</td><td>0.664</td><td>0.983</td><td>0.994</td></tr><tr><td>SetB</td><td>0.825</td><td>0.631</td><td>0.981</td><td>0.993</td></tr><tr><td>Set C</td><td>0.771</td><td>0.671</td><td>0.983</td><td>0.990</td></tr><tr><td>Average</td><td>0.812</td><td>0.655</td><td>0.982</td><td>0.992</td></tr></table>",
+        "bbox": [
+            189,
+            767,
+            808,
+            880
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "C SUPPLEMENTARY FOR UNSUPERVISED CROSS DOMAIN TRANSFER LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            761,
+            136
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "C.1 MORE DISCUSSION FOR THE OFFICE-31 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            151,
+            596,
+            166
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Our experiment (table 10) shows that simply using deeper pre-trained networks produces a significant performance boost. Specifically, using Resnet-50 increases the performance 8.4 points from Alexnet, which surpasses the 6.2 points gained by using one of the state-of-the-art domain adaptation algorithms (JAN (Long et al., 2017)). If we regard pre-training with deeper networks as transferring more information, our approach is similar in this aspect since both transfer more information from the auxiliary dataset. In this case, memory limitations precluded the application of LCO to such models, and multi-GPU implementations for this problem is an area of future work. ",
+        "bbox": [
+            173,
+            176,
+            825,
+            275
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "table",
+        "img_path": "images/8b547758927a8194ad9bb2d90cb1ef3c9d64fc483b82dadb28405678cd15be07.jpg",
+        "table_caption": [
+            "Table 10: The performance of unsupervised transfer across domains on Office-31 dataset. The backbone networks in the comparison have different numbers of convolutional layers. AlexNet has 5 layers and the ResNets have $1 8 { \\sim } 5 0$ layers. SO is the abbreviation for source-only, which simply trains on $S ^ { \\prime }$ and directly applies the classifier on $T$ . The first two rows are directly copied from Long et al. (2017). The features learned with deeper networks generalize better across domains. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"8\">A→W D→W W→D A→D D→A W→A</td></tr><tr><td>AlexNet SO</td><td>61.6</td><td>95.4</td><td>99.0</td><td>63.8</td><td>51.1</td><td>49.8</td><td>Avg 70.1</td><td>1</td></tr><tr><td>AlexNet (JAN-A)</td><td>75.2</td><td>96.6</td><td>99.6</td><td>72.8</td><td>57.5</td><td>56.3</td><td>76.3</td><td>+6.2</td></tr><tr><td>Resnet-18 SO</td><td>66.8</td><td>92.8</td><td>96.8</td><td>67.1</td><td>51.4</td><td>53.0</td><td>71.3</td><td>+1.2</td></tr><tr><td>Resnet-34 SO</td><td>73.1</td><td>96.4</td><td>98.8</td><td>73.8</td><td>63.2</td><td>62.1</td><td>77.9</td><td>+7.8</td></tr><tr><td>Resnet-50 SO</td><td>74.3</td><td>96.8</td><td>98.8</td><td>79.5</td><td>61.2</td><td>60.6</td><td>78.5</td><td>+8.4</td></tr></table>",
+        "bbox": [
+            178,
+            369,
+            820,
+            468
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "table",
+        "img_path": "images/6710c86e7661394e049d39d0714ba25355b67f32155cc9ea97e9c59e48599d1d.jpg",
+        "table_caption": [
+            "Table 11: Performance of the similarity prediction function ( $G$ , trained with ImageNet882) applied on three domains of the Office-31 dataset. In total, 1.4M pairs are examined to calculate the table. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Domain</td><td>Similar Precision</td><td>Similar Recall</td><td>Dissimilar Precision</td><td>Dissimilar Recall</td></tr><tr><td>Amazon</td><td>0.194</td><td>0.696</td><td>0.988</td><td>0.894</td></tr><tr><td>DSLR</td><td>0.196</td><td>0.882</td><td>0.995</td><td>0.858</td></tr><tr><td>Webcam</td><td>0.213</td><td>0.865</td><td>0.994</td><td>0.876</td></tr></table>",
+        "bbox": [
+            191,
+            529,
+            805,
+            598
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "table",
+        "img_path": "images/25c5d5e868a3d729788c298939bb3733d14f6e33ffebc69fbaad1b8d49e8a041.jpg",
+        "table_caption": [
+            "Table 12: Unsupervised transferring across domains (S’: SVHN, T: MNIST A: Omniglotbg) without pre-trained backbone network weights. Our setup is similar to Sener et al. (2016) and Ganin et al. (2016) which therefore has a similar source-only performance. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Method</td><td>SVHN-&gt;MNIST</td><td>Gain</td></tr><tr><td>Ganin et al. (2016)</td><td>Source-Only DANN</td><td>54.9 73.9</td><td>1 +19.0</td></tr><tr><td>Sener et al. (2016)</td><td>Source-Only LTR</td><td>54.9 78.8</td><td>1 +23.9</td></tr><tr><td>Saito et al. (2017)</td><td>Source-Only ATDA</td><td>70.1 86.2</td><td>- +16.1</td></tr><tr><td>Tzeng et al. (2017)</td><td>Source-Only ADDA</td><td>60.1 76.0</td><td>- +15.9</td></tr><tr><td>Ours</td><td>Source-Only CCN+</td><td>52.0 89.1</td><td>1 +37.1</td></tr></table>",
+        "bbox": [
+            277,
+            672,
+            720,
+            866
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "table",
+        "img_path": "images/de85c2a20514b9e73a5603c61cc374d3b3140630ae5017bceac12a520b099289.jpg",
+        "table_caption": [
+            "Table 13: Performance of the similarity prediction function ( $G$ , trained with Omniglotbg) applied on the MNIST dataset. In total, 5M pairs are sampled to calculate the table. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Similar Precision</td><td>Similar Recall</td><td>Dissimilar Precision</td><td>Dissimilar Recall</td></tr><tr><td>Random</td><td>0.100</td><td>0.500</td><td>0.900</td><td>0.500</td></tr><tr><td>MNIST</td><td>0.782</td><td>0.509</td><td>0.946</td><td>0.984</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "D ROBUSTNESS ANALYSIS OF CONSTRAINED CLUSTERING NETWORK ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            231,
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+            246
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "D.1 SETUP ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            261,
+            264,
+            275
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "To quickly explore the large combination of factors that may affect the clustering, we use a small dataset (MNIST) and a small network which has two convolution layers followed by two fully connected layers. The MNIST dataset is a dataset of handwritten digits that contains 60k training and 10k testing images with size $2 8 \\mathbf { x } 2 8$ . Only the training set is used in this section and the raw pixels, which were normalized to zero mean and unit standard deviation, were fed into networks directly. ",
+        "bbox": [
+            174,
+            287,
+            825,
+            357
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "The networks were randomly initialized and the clustering training was run five times under each combination of factors; we show the best final results, as is usual in the random restart regime. The mini-batch size was set to 256, thus up to 65536 pairs were presented to the LCO per mini-batch if using full density $\\mathrm { ( D } { = } 1 \\mathrm { ) }$ ). There were 235 mini-batches in an epoch and the optimization proceeded for 15 epochs. The clustering loss was minimized by stochastic gradient descent with learning rate 0.1 and momentum 0.9. The predicted cluster was assigned at the end by forwarding samples through the clustering networks. The best result in the five runs was reported. ",
+        "bbox": [
+            174,
+            363,
+            825,
+            462
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "To simulate different performance of the similarity prediction, the label of pairs were flipped according to the designated recall. For example, to simulate a $90 \\%$ recall of similar pair, $10 \\%$ of the ground truth similar pair in a mini-batch were flipped. The precision of similar/dissimilar pairs is a function of the recall of both type of pairs, thus controlling the recall is sufficient for the evaluation. The recalls for both similar and dissimilar pairs were gradually reduced from one to zero at intervals of 0.1. ",
+        "bbox": [
+            174,
+            468,
+            825,
+            551
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "D.2 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            569,
+            305,
+            584
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "The resulting performance w.r.t different values of recall, density, and number of clusters is visualized in Figure 8. A bright color means high NMI score and is desired. The larger the bright region, the more robust the clustering is against the noise of similarity prediction. The ACC score shows almost the same trend and is thus not shown here. ",
+        "bbox": [
+            174,
+            595,
+            825,
+            651
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "How does similarity prediction affect clustering? Looking at the top-left heat map in figure 8, which has $D = 1$ and 10 clusters, it can be observed that the NMI score is very robust to low similar pair recall, even lower than 0.5. For recall of dissimilar pairs, the effect of recall is divided at the 0.5 value: the clustering performance can be very robust to noise in dissimilar pairs if the recall is greater than 0.5; however, it can completely fail if recall is below 0.5. For similar pairs, the clustering works on a wide range of recalls when the recall of dissimilar pairs is high. ",
+        "bbox": [
+            174,
+            659,
+            825,
+            742
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "In practical terms, robustness to the recall of similar pairs is desirable because it is much easier to predict dissimilar pairs than similar pairs in real scenarios. In a dataset with 10 categories e.g. Cifar-10, we can easily get $90 \\%$ recall for dissimilar pairs with purely random guess if the number of classes is known, while the recall for similar pairs will be $10 \\%$ . ",
+        "bbox": [
+            174,
+            750,
+            825,
+            805
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "How does the density of the constraints affect clustering? We argue that the density of pairwise relationships is the key factor to improving the robustness of clustering. The density $D = 1$ means that every pair in a mini-batch is utilized by the clustering loss. For density $D = 0 . 1$ , it means only 1 out of 10 possible constraints is used. We could regard the higher density as better utilization of the pairwise information in a mini-batch, thus more learning instances contribute to the gradients at once. Consider a scenario where there is one sample associated with 5 true similar pairs and 3 false similar pairs. In such a case, the gradients introduced by the false similar pairs have a higher chance to be overridden by true similar pairs within the mini-batch, thus the loss can converge faster and is less affected by errors. In Figure 8, we could see when density decreases, the size of the bright region shrinks significantly. ",
+        "bbox": [
+            174,
+            811,
+            825,
+            924
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/12cec765ecdee0ae03b658c3f3798b3d511e16695116130fd85ecccd06d459c9.jpg",
+        "image_caption": [
+            "Figure 8: The clustering performance with different pairwise density and number of clusters. A bright color means that the NMI score is close to 1 while black corresponds to 0. The density is defined as a ratio compared to the total number of pair-wise combinations in a mini-batch. The number of clusters defines the final softmax output dimensionality. In each sub-figure, we show how the scores change w.r.t. the similar pair recall and dissimiliar pair recall. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            282,
+            106,
+            743,
+            378
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            494,
+            821,
+            522
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "In our implementation, enumerating the full pairwise relationships introduces negligible overhead in computation time using GPU. Although there is overhead for memory consumption, it is limited because only the vector of predicted distributions has to be enumerated for calculating the clustering loss. ",
+        "bbox": [
+            174,
+            530,
+            825,
+            585
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "The effect of varying the number of Clusters In the MNIST experiments, the number of categories is 10. We augment the softmax output number up to 100. The rows of figure 8 show that even when the number of output categories is significant larger than the number of true object categories, e.g. $1 0 0 > 1 0$ , the clustering performance NMI score only degrades slightly. ",
+        "bbox": [
+            173,
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+        "page_idx": 19
+    }
+]
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diff --git a/parse/train/ByRWCqvT-/ByRWCqvT-_middle.json b/parse/train/ByRWCqvT-/ByRWCqvT-_middle.json
new file mode 100644
index 0000000000000000000000000000000000000000..eec95a1de84f25d61e2c10fd63c32635db42cb8a
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+                                    "score": 1.0,
+                                    "content": "estimate the number of semantic clusters, again using the clustering network. Since",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "the similarity network is noisy, the key is to use a robust clustering algorithm, and",
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+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "we show that our formulation is more robust than the alternative constrained and",
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+                                    "content": "See figure 2 for an illustration.",
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+                                    "content": "Zagoruyko & Komodakis (2015). While they use it to predict image patch similarity, we use it to",
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+                                    "score": 1.0,
+                                    "content": "is dense. Therefore, instead of using Siamese architecture, we keep the single column backbone but",
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+                                    "content": "add a pair-enumeration layer on top of the feature extraction network. The pair-enumeration layer",
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+                                    "content": "which can be defined as a hinge-loss function:",
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+                                    "content": "\\mathcal { L } ( \\boldsymbol { x } _ { p } , \\boldsymbol { x } _ { q } ) ^ { - } = L _ { h } ( \\mathcal { D } _ { K L } ( \\mathcal { P } ^ { \\star } | | \\boldsymbol { \\mathcal { Q } } ) , \\boldsymbol { \\sigma } ) + L _ { h } ( \\mathcal { D } _ { K L } ( \\mathcal { Q } ^ { \\star } | | \\mathcal { P } ) , \\boldsymbol { \\sigma } )",
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+                                    "content": "section 3, the total loss can be defined as a contrastive loss (where we use integer 1 to represent a",
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+                                    "score": 1.0,
+                                    "content": "clustering criterion are worth mentioning: (1) There is no need to define cluster centers. (2) There",
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+                            ],
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+                                    "score": 1.0,
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+                                    "content": "4.1 THE PAIRWISE SIMILARITY PREDICTION NETWORK",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "Zagoruyko & Komodakis (2015). While they use it to predict image patch similarity, we use it to",
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+                        },
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+                                    "content": "predict image-level semantic similarity. However, the Siamese architecture used in Zagoruyko &",
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+                                    "content": "Komodakis (2015) is not efficient in both training and inference, especially when pairwise information",
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+                                    "score": 1.0,
+                                    "content": "is dense. Therefore, instead of using Siamese architecture, we keep the single column backbone but",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "add a pair-enumeration layer on top of the feature extraction network. The pair-enumeration layer",
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+                            ],
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+                                    "content": "1 0 \\times 5 1 2",
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+                                    "score": 1.0,
+                                    "content": "with the mini-batch size 10 and the feature dimension 512; then the",
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+                                    "score": 1.0,
+                                    "content": "output of the pair-enumeration layer will be",
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+                                    "score": 0.9,
+                                    "content": "1 0 0 \\times 1 0 2 4",
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+                                    "content": "The architecture is illustrated in figure 3. We add one hidden fully connected layer on top of the",
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+                                    "content": "enumeration layer and a binary classifier at the end. We use the standard cross-entropy loss and train",
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+                                    "score": 1.0,
+                                    "content": "it end-to-end. The supervision for training was obtained by converting the ground-truth category",
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+                                    "score": 1.0,
+                                    "content": "labels into binary similarity, i.e., if two samples are from the same class then their label will be",
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+                                    "score": 1.0,
+                                    "content": "similar, otherwise dissimilar. The inference is also end-to-end, and it outputs predictions among all",
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+                                    "score": 1.0,
+                                    "content": "to all algorithms in random order, while we empirically found it has no noticeable effect on results.",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "(Bilenko et al., 2004) as our baselines. We use the default parameters for each algorithm provided by",
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+                                    "score": 1.0,
+                                    "content": "predictions for both orders of a pair and for self-pairs). The full pairwise similarities were presented",
+                                    "type": "text"
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+                        {
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+                                    "score": 1.0,
+                                    "content": "to all algorithms in random order, while we empirically found it has no noticeable effect on results.",
+                                    "type": "text"
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+                            ],
+                            "index": 47
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "We pick the baseline approaches based on code availability and scalability concerning the number of",
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "pairwise constraints. Therefore we have shown results for K-means (MacQueen et al., 1967), LPNMF",
+                                    "type": "text"
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+                            ],
+                            "index": 49
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+                                    "score": 1.0,
+                                    "content": "(Cai et al., 2009), LSC (Chen & Cai, 2011), ITML (Davis et al., 2007), SKKm (Anand et al., 2014),",
+                                    "type": "text"
+                                }
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+                                    "content": "SKLR (Amid et al., 2016), SKMS (Anand et al., 2014), CSP (Wang et al., 2014) , and MPCK-means",
+                                    "type": "text"
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "(Bilenko et al., 2004) as our baselines. We use the default parameters for each algorithm provided by",
+                                    "type": "text"
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+                                    "content": ", the number of clusters. We use the same normalized images used in",
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+                                    "cross_page": true
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+                                            "score": 1.0,
+                                            "content": "averaged across 20 alphabets which have 20 to 47 letters. The ACC and NMI without brackets have",
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+                                            "score": 1.0,
+                                            "content": "characteristics of how each algorithm utilizes the pairwise constraints are marked in the \"Constraints",
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+                                            "score": 1.0,
+                                            "content": "in\" column, where metric stands for the metric learning of feature representation.",
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+                                    "content": "& Ghosh, 2002) which is widely used for clustering. The second one is the clustering accuracy",
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+                                    "content": "(ACC) (Yang et al., 2010). The ACC metric first finds the one-to-one matching between predicted",
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+                                    "score": 1.0,
+                                    "content": "otherwise the score drops quickly. Therefore ACC provides better discrimination to evaluate whether",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "range of CCN are provided in appendix D, and shows that the robustness is related to the density",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "of pairs involved in a mini-batch. We hypothesize that during the optimization, the gradients from",
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+                                    "score": 1.0,
+                                    "content": "wrongly predicted pairs are canceled out by each other or by the correctly predicted pairs. Therefore",
+                                    "type": "text"
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+                        {
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+                                    "score": 1.0,
+                                    "content": "the overall gradient still moves the solution towards a better clustering result.",
+                                    "type": "text"
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+                                    "score": 1.0,
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+                                    "content": "\\begin{array} { r } { N D C = \\sum _ { i = 1 } ^ { K } \\left[ C _ { i } > = \\bar { E } [ C _ { i } ] \\right] } \\end{array}",
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+                                    "score": 1.0,
+                                    "content": "is an Iverson Bracket and",
+                                    "type": "text"
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+                                    "content": "\\bar { E } [ \\bar { C } _ { i } ]",
+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": "will be 10 if",
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+                        {
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+                                    "score": 1.0,
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+                                    "content": "K = 1 0 0",
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+                                    "score": 1.0,
+                                    "content": ". Then the average difference",
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+                                    "content": "( A D i f )",
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+                                    "score": 1.0,
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+                                    "score": 0.91,
+                                    "content": "\\begin{array} { r } { A D i f = \\frac { 1 } { | D | } \\sum _ { d \\in D } { | N D C _ { d } - N C _ { d } ^ { g t } | } } \\end{array}",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
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+                                    "content": "d",
+                                    "type": "inline_equation"
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "(i.e., alphabet) is a dataset in",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.78,
+                                    "content": "D",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        430,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": ". A smaller",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 0.84,
+                                    "content": "A D i f",
+                                    "type": "inline_equation"
+                                }
+                            ],
+                            "index": 15
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "indicates a better estimate of",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.79,
+                                    "content": "K",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ". CCN achieves a score of 6.35 (appendix table 6). We compare",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 16
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "this with the baseline approach SKMS (Anand et al., 2014), which does not require a given",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.8,
+                                    "content": "K",
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+                                {
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+                                    "score": 1.0,
+                                    "content": "and",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 17
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "supports a pipeline to estimate",
+                                    "type": "text"
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+                                    "score": 0.82,
+                                    "content": "K",
+                                    "type": "inline_equation"
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "automatically (therefore we only put it into the column",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.89,
+                                    "content": "K = 1 0 0",
+                                    "type": "inline_equation"
+                                }
+                            ],
+                            "index": 18
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "in table 1.). SKMS gets 16.3. Furthermore, 10 out of 20 datasets from CCN’s prediction have a",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "difference between",
+                                    "type": "text"
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+                                    "score": 0.91,
+                                    "content": "N D C _ { d }",
+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    "score": 1.0,
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+                                    "score": 0.92,
+                                    "content": "N C _ { d } ^ { g t }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "smaller or equal to 3, which shows the feasibility of estimating",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 20
+                        },
+                        {
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+                                    "score": 0.79,
+                                    "content": "K",
+                                    "type": "inline_equation"
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "with predicted similarity.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 21
+                        }
+                    ],
+                    "index": 14.5
+                },
+                {
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+                                    "content": "5.1.3 EXPERIMENTS USING THE IMAGENET DATASET",
+                                    "type": "text"
+                                }
+                            ],
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+                    ],
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+                                    "content": "To demonstrate the scalability of our approach, we applied the same scheme on the ImageNet dataset.",
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+                                    "content": "The 1000-class dataset is separated into 882-class (ImageNet882) and 118-class (ImageNet118)",
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+                                    "score": 1.0,
+                                    "content": "subsets as the random split in Vinyals et al. (2016). We use ImageN et882 for",
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+                                    "score": 1.0,
+                                    "content": "in table 1.). SKMS gets 16.3. Furthermore, 10 out of 20 datasets from CCN’s prediction have a",
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+                                    "score": 1.0,
+                                    "content": "in this section from scratch. Our experimental setting is similar to Sener et al. (2016). We achieve the",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": ". Only the training set is used in this section and the raw pixels,",
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+                            "index": 10
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+                        {
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+                                    "score": 1.0,
+                                    "content": "which were normalized to zero mean and unit standard deviation, were fed into networks directly.",
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+                    ],
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+                                    "score": 1.0,
+                                    "content": "combination of factors; we show the best final results, as is usual in the random restart regime. The",
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+                                    "score": 1.0,
+                                    "content": "mini-batch size was set to 256, thus up to 65536 pairs were presented to the LCO per mini-batch if",
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+                                    "content": "using full density",
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+                                    "score": 0.77,
+                                    "content": "\\mathrm { ( D } { = } 1 \\mathrm { ) }",
+                                    "type": "inline_equation"
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "). There were 235 mini-batches in an epoch and the optimization proceeded",
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "for 15 epochs. The clustering loss was minimized by stochastic gradient descent with learning rate",
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+                            ],
+                            "index": 16
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "0.1 and momentum 0.9. The predicted cluster was assigned at the end by forwarding samples through",
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "the clustering networks. The best result in the five runs was reported.",
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+                                }
+                            ],
+                            "index": 18
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+                    ],
+                    "index": 15
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+                                    "score": 1.0,
+                                    "content": "To simulate different performance of the similarity prediction, the label of pairs were flipped according",
+                                    "type": "text"
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+                            ],
+                            "index": 19
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "of the ground",
+                                    "type": "text"
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+                            ],
+                            "index": 20
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "truth similar pair in a mini-batch were flipped. The precision of similar/dissimilar pairs is a function",
+                                    "type": "text"
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+                            ],
+                            "index": 21
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "of the recall of both type of pairs, thus controlling the recall is sufficient for the evaluation. The",
+                                    "type": "text"
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+                            ],
+                            "index": 22
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "recalls for both similar and dissimilar pairs were gradually reduced from one to zero at intervals of",
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+                            ],
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+                        },
+                        {
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+                                    "content": "0.1.",
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+                    ],
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+                    "type": "title",
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+                                    "score": 1.0,
+                                    "content": "D.2 DISCUSSION",
+                                    "type": "text"
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+                            "index": 25
+                        }
+                    ],
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+                },
+                {
+                    "type": "text",
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+                                    "content": "The resulting performance w.r.t different values of recall, density, and number of clusters is visualized",
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+                                    "score": 1.0,
+                                    "content": "in Figure 8. A bright color means high NMI score and is desired. The larger the bright region, the",
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+                            ],
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+                        },
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+                                    "score": 1.0,
+                                    "content": "more robust the clustering is against the noise of similarity prediction. The ACC score shows almost",
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "the same trend and is thus not shown here.",
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+                    ],
+                    "index": 27.5
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+                                    "score": 1.0,
+                                    "content": "How does similarity prediction affect clustering? Looking at the top-left heat map in figure 8,",
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+                                    "content": "D = 1",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "and 10 clusters, it can be observed that the NMI score is very robust to low similar",
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+                                    "content": "pair recall, even lower than 0.5. For recall of dissimilar pairs, the effect of recall is divided at the 0.5",
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+                            "index": 32
+                        },
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+                                    "score": 1.0,
+                                    "content": "value: the clustering performance can be very robust to noise in dissimilar pairs if the recall is greater",
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+                            ],
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+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "than 0.5; however, it can completely fail if recall is below 0.5. For similar pairs, the clustering works",
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+                            "index": 34
+                        },
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+                                    "score": 1.0,
+                                    "content": "on a wide range of recalls when the recall of dissimilar pairs is high.",
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+                                    "score": 1.0,
+                                    "content": "In practical terms, robustness to the recall of similar pairs is desirable because it is much easier",
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+                                    "score": 1.0,
+                                    "content": "Cifar-10, we can easily get",
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+                                    "content": "90 \\%",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "recall for dissimilar pairs with purely random guess if the number of",
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+                                    "score": 1.0,
+                                    "content": "classes is known, while the recall for similar pairs will be",
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+                                    "score": 0.87,
+                                    "content": "10 \\%",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
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+                    ],
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+                                    "score": 1.0,
+                                    "content": "How does the density of the constraints affect clustering? We argue that the density of pairwise",
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+                                    "content": "D = 1",
+                                    "type": "inline_equation"
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+                                {
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+                                    "content": "means",
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+                                    "score": 1.0,
+                                    "content": "that every pair in a mini-batch is utilized by the clustering loss. For density",
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+                                    "score": 0.91,
+                                    "content": "D = 0 . 1",
+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": ", it means only 1",
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+                                    "score": 1.0,
+                                    "content": "which were normalized to zero mean and unit standard deviation, were fed into networks directly.",
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+                                    "content": "). There were 235 mini-batches in an epoch and the optimization proceeded",
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+                                    "score": 1.0,
+                                    "content": "of the recall of both type of pairs, thus controlling the recall is sufficient for the evaluation. The",
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+}
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diff --git a/parse/train/ByRWCqvT-/ByRWCqvT-_model.json b/parse/train/ByRWCqvT-/ByRWCqvT-_model.json
new file mode 100644
index 0000000000000000000000000000000000000000..83bf6f433b5a7de718ec239533ab106dd0f98ff4
--- /dev/null
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\ No newline at end of file
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+# FREELB: ENHANCED ADVERSARIAL TRAINING FOR NATURAL LANGUAGE UNDERSTANDING
+
+Chen $\mathbf { Z } \mathbf { h } \mathbf { u } ^ { \mathbf { 1 } }$ , Yu Cheng2, Zhe $\mathbf { G a n } ^ { 2 }$ , Siqi $\mathbf { S u n } ^ { 2 }$ , Tom Goldstein1, Jingjing Liu2 1University of Maryland, College Park 2Microsoft Dynamics 365 AI Research {chenzhu,tomg}@cs.umd.edu, {yu.cheng,zhe.gan,siqi.sun,jingjl}@microsoft.com
+
+# ABSTRACT
+
+Adversarial training, which minimizes the maximal risk for label-preserving input perturbations, has proved to be effective for improving the generalization of language models. In this work, we propose a novel adversarial training algorithm, FreeLB, that promotes higher invariance in the embedding space, by adding adversarial perturbations to word embeddings and minimizing the resultant adversarial risk inside different regions around input samples. To validate the effectiveness of the proposed approach, we apply it to Transformer-based models for natural language understanding and commonsense reasoning tasks. Experiments on the GLUE benchmark show that when applied only to the finetuning stage, it is able to improve the overall test scores of BERT-base model from 78.3 to 79.4, and RoBERTa-large model from 88.5 to 88.8. In addition, the proposed approach achieves state-of-the-art single-model test accuracies of $8 5 . 4 4 \%$ and $6 7 . 7 5 \%$ on ARC-Easy and ARC-Challenge. Experiments on CommonsenseQA benchmark further demonstrate that FreeLB can be generalized and boost the performance of RoBERTa-large model on other tasks as well. 1
+
+# 1 INTRODUCTION
+
+Adversarial training is a method for creating robust neural networks. During adversarial training, mini-batches of training samples are contaminated with adversarial perturbations (alterations that are small and yet cause misclassification), and then used to update network parameters until the resulting model learns to resist such attacks. Adversarial training was originally proposed as a means to enhance the security of machine learning systems (Goodfellow et al., 2015), especially for safety-critical systems like self-driving cars (Xiao et al., 2018) and copyright detection (Saadatpanah et al., 2019).
+
+In this paper, we turn our focus away from the security benefits of adversarial training, and instead study its effects on generalization. While adversarial training boosts the robustness, it is widely accepted by computer vision researchers that it is at odds with generalization, with classification accuracy on non-corrupted images dropping as much as $1 0 \%$ on CIFAR-10, and $1 5 \%$ on Imagenet (Madry et al., 2018; Xie et al., 2019). Surprisingly, people observe the opposite result for language models (Miyato et al., 2017; Cheng et al., 2019), showing that adversarial training can improve both generalization and robustness.
+
+We will show that adversarial training significantly improves performance of state-of-the-art models for many language understanding tasks. In particular, we propose a novel adversarial training algorithm, called FreeLB (Free Large-Batch), which adds adversarial perturbations to word embeddings and minimizes the resultant adversarial loss around input samples. The method leverages recently proposed “free” training strategies (Shafahi et al., 2019; Zhang et al., 2019) to enrich the training data with diversified adversarial samples under different norm constraints at no extra cost than PGD-based (Projected Gradient Descent) adversarial training (Madry et al., 2018), which enables us to perform such diversified adversarial training on large-scale state-of-the-art models. We observe improved invariance in the embedding space for models trained with FreeLB, which is positively correlated with generalization.
+
+We perform comprehensive experiments to evaluate the performance of a variety of adversarial training algorithms on state-of-the-art language understanding models and tasks. In the comparisons with standard PGD (Madry et al., 2018), FreeAT (Shafahi et al., 2019) and YOPO (Zhang et al., 2019), FreeLB stands out to be the best for the datasets and models we evaluated. With FreeLB, we achieve state-of-the-art results on several important language understanding benchmarks. On the GLUE benchmark, FreeLB pushes the performance of the BERT-base model from 78.3 to 79.4. The overall score of the RoBERTa-large models on the GLUE benchmark is also lifted from 88.5 to 88.8, achieving best results on most of its sub-tasks. Experiments also show that FreeLB can boost the performance of RoBERTa-large on question answering tasks, such as the ARC and CommonsenseQA benchmarks. We also provide a comprehensive ablation study and analysis to demonstrate the effectiveness of our training process.
+
+# 2 RELATED WORK
+
+# 2.1 ADVERSARIAL TRAINING
+
+To improve the robustness of neural networks against adversarial examples, many defense strategies and models have been proposed, in which PGD-based adversarial training (Madry et al., 2018) is widely considered to be the most effective, since it largely avoids the the obfuscated gradient problem (Athalye et al., 2018). It formulates a class of adversarial training algorithms (Kurakin et al., 2017) into solving a minimax problem on the cross-entropy loss, which can be achieved reliably through multiple projected gradient ascent steps followed by a SGD (Stochastic Gradient Descent) step.
+
+Despite being verified by Athalye et al. (2018) to avoid obfuscated gradients, Qin et al. (2019) shows that PGD-based adversarial training still leads to highly convolved and non-linear loss surfaces when $K$ is small, which could be readily broken under stronger adversaries. Thus, to be effective, the cost of PGD-based adversarial training is much higher than conventional training. To mitigate this cost, Shafahi et al. (2019) proposed a “free” adversarial training algorithm that simultaneously updates both model parameters and adversarial perturbations on a single backward pass. Using a similar formulation, Zhang et al. (2019) effectively reduce the total number of full forward and backward propagations for obtaining adversarial examples by restricting most of its adversarial updates in the first layer.
+
+# 2.2 ADVERSARIAL EXAMPLES IN NATURAL LANGUAGES
+
+Adversarial examples have been explored primarily in the image domain, and received many attention in text domain recently. Previous works on text adversaries have focused on heuristics for creating adversarial examples in the black-box setting, or on specific tasks. Jia & Liang (2017) propose to add distracting sentences to the input document in order to induce mis-classification. Zhao et al. (2018) generate text adversaries by projecting the input data to a latent space using GANs, and searching for adversaries close to the original instance. Belinkov & Bisk (2018) manipulate every word in a sentence with synthetic or natural noise in machine translation systems. Iyyer et al. (2018) propose a neural paraphrase model based on back-translated data to produce paraphrases that have different sentence structures. Different from previous work, ours is not to produce actual adversarial examples, but only take the benefit of adversarial training for natural language understanding.
+
+We are not the first to observe that robust language models may perform better on clean test data. Miyato et al. (2017) extend adversarial and virtual adversarial training (Miyato et al., 2019) to the text domain to improve the performance on semi-supervised classification tasks. Ebrahimi et al. (2018) propose a character/word replacement for crafting attacks, and show employing adversarial examples in training renders the models more robust. Ribeiro et al. (2018) show that adversarial attacks can be used as a valuable tool for debugging NLP models. Cheng et al. (2019) also find that crafting adversarial examples can help neural machine translation significantly. Notably, these studies have focused on simple models or text generation tasks. Our work explores how to efficiently use the gradients obtained in adversarial training to boost the performance of state-of-the-art transformer-based models.
+
+# 3 ADVERSARIAL TRAINING FOR LANGUAGE UNDERSTANDING
+
+Pre-trained large-scale language models, such as BERT (Devlin et al., 2019), RoBERTa (Liu et al., 2019b), ALBERT (Lan et al., 2020) and T5 (Raffel et al., 2019), have proven to be highly effective for downstream tasks. We aim to further improve the generalization of these pre-trained language models on the downstream language understanding tasks by enhancing their robustness in the embedding space during finetuning on these tasks. We achieve this goal by creating “virtual” adversarial examples in the embedding space, and then perform parameter updates on these adversarial embeddings. Creating actual adversarial examples for language is difficult; even with state-of-theart language models as guidance (e.g., (Cheng et al., 2019)), it remains unclear how to construct label-preserving adversarial examples via word/character replacement without human evaluations, because the meaning of each word/character depends on the context (Ribeiro et al., 2018). Since we are only interested in the effects of adversarial training, rather than producing actual adversarial examples, we add norm-bounded adversarial perturbations to the embeddings of the input sentences using a gradient-based method. Note that our embedding-based adversary is strictly stronger than a more conventional text-based adversary, as our adversary can make manipulations on word embeddings that are not possible in the text domain.
+
+For models that incorporate various input representations, including word or subword embeddings, segment embeddings and position embeddings, our adversaries only modify the concatenated word or sub-word embeddings, leaving other components of the sentence representation unchanged. 2 Denote the sequence of one-hot representations of the input subwords as $\pmb { Z } = [ z _ { 1 } , z _ { 2 } , . . . , z _ { n } ]$ , the embedding matrix as $V$ , and the language model (encoder) as a function ${ \pmb y } = f _ { \pmb \theta } ( { \pmb X } )$ , where $\dot { \boldsymbol { X } } = \boldsymbol { V } \boldsymbol { Z }$ is the subword embeddings, $\textbf {  { y } }$ is the output of the model (e.g., class probabilities for classification models), and $\pmb \theta$ denotes all the learnable parameters including the embedding matrix $V$ . We add adversarial perturbations $\delta$ to the embeddings such that the prediction becomes $\pmb { y } ^ { \prime } = f _ { \pmb { \theta } } ( \pmb { X } + \pmb { \delta } )$ . To preserve the semantics, we constrain the norm of $\pmb { \delta }$ to be small, and assume the model’s prediction should not change after the perturbation. This formulation is analogous to Miyato et al. (2017), with the difference that we do not require $\boldsymbol { X }$ to be normalized.
+
+# 3.1 PGD FOR ADVERSARIAL TRAINING
+
+Standard adversarial training seeks to find optimal parameters $\pmb { \theta } ^ { * }$ to minimize the maximum risk for any $\pmb { \delta }$ within a norm ball as:
+
+$$
+\operatorname* { m i n } _ { \theta } \mathbb { E } _ { ( Z , y ) \sim \mathcal { D } } \left[ \operatorname* { m a x } _ { \| \delta \| \leq \epsilon } L ( f _ { \theta } ( X + \delta ) , y ) \right] ,
+$$
+
+where $\mathcal { D }$ is the data distribution, $y$ is the label, and $L$ is some loss function. We use the Frobenius norm to constrain $\delta$ . For neural networks, the outer “min” is non-convex, and the inner “max” is non-concave. Nonetheless, Madry et al. (2018) demonstrated that this saddle-point problem can be solved reliably with SGD for the outer minimization and PGD (a standard method for large-scale constrained optimization, see (Combettes & Pesquet, 2011) and (Goldstein et al., 2014)), for the inner maximization. In particular, for the constraint $\| \delta \| _ { F } \le \epsilon$ , with an additional assumption that the loss function is locally linear, PGD takes the following step (with step size $\alpha$ ) in each iteration:
+
+$$
+\delta _ { t + 1 } = \Pi _ { \| \pmb { \delta } \| _ { F } \leq \epsilon } \left( \pmb { \delta } _ { t } + \alpha g ( \pmb { \delta } _ { t } ) / \| g ( \pmb { \delta } _ { t } ) \| _ { F } \right) ,
+$$
+
+where $g ( \delta _ { t } ) = \nabla _ { \delta } L ( f _ { \theta } ( X + \delta _ { t } ) , y )$ is the gradient of the loss with respect to $\delta$ , and $\Pi _ { \| \delta \| _ { F } \leq \epsilon }$ performs a projection onto the $\epsilon$ -ball. To achieve high-level robustness, multi-step adversarial examples are needed during training, which is computationally expensive. The $K$ -step PGD ( $K$ -PGD) requires $K$ forward-backward passes through the network, while the standard SGD update requires only one. As a result, the adversary generation step in adversarial training increases run-time by an order of magnitude—a catastrophic amount when training large state-of-the-art language models.
+
+# 3.2 LARGE-BATCH ADVERSARIAL TRAINING FOR FREE
+
+In the inner ascent steps of PGD, the gradients of the parameters can be obtained with almost no overhead when computing the gradients of the inputs. From this observation, FreeAT (Shafahi et al.,
+
+# Algorithm 1 “Free” Large-Batch Adversarial Training (FreeLB- $K$ )
+
+<table><tr><td>ascent step size α</td><td>Require: Training samples X = {(Z,y)}, perturbation bound ε, learning rate T, ascent steps K,</td></tr><tr><td>1:Initialize 0</td><td></td></tr><tr><td>2: for epoch =1... Nep do</td><td></td></tr><tr><td>3:</td><td>for minibatchB ℃Xdo</td></tr><tr><td>4:</td><td>U(-∈,∈)</td></tr><tr><td>5:</td><td>90↑0</td></tr><tr><td>6:</td><td></td></tr><tr><td>7:</td><td>fort=1...K do</td></tr><tr><td>8:</td><td>Accumulate gradient of parameters θ</td></tr><tr><td>9:</td><td>gt←gt-1+kE(z,y)∈B[VθL(fe(X+δt-1),y)]</td></tr><tr><td>10:</td><td>Update the perturbation δ via gradient ascend</td></tr><tr><td>11:</td><td>gadu ←VsL(fθ(X+δt-1),y) δt ←II|δ|lr≤e(δt-1+α:gadu/llgadullF)</td></tr><tr><td>12:</td><td></td></tr><tr><td>13:</td><td>end for</td></tr><tr><td>14:</td><td>0←0-TgK</td></tr><tr><td>end for 15: end for</td><td></td></tr></table>
+
+2019) and YOPO (Zhang et al., 2019) have been proposed to accelerate adversarial training. They achieve comparable robustness and generalization as standard PGD-trained models using only the same or a slightly larger number of forward-backward passes as natural training (i.e., SGD on clean samples). FreeAT takes one descent step on the parameters together with each of the $K$ ascent steps on the perturbation. As a result, FreeAT may suffer from the “stale gradient” problem (Dutta et al., 2018), where in every step $t$ , $\delta _ { t }$ does not necessarily maximize the model with parameter $\theta _ { t }$ since its update is based on $\nabla _ { \delta } L \big ( f _ { \pmb { \theta } _ { t - 1 } } ( \pmb { X } + \delta _ { t - 1 } ) , y \big )$ , and vice versa, $\theta _ { t }$ does not necessarily minimize the adversarial risk with adversary $\delta _ { t }$ since its update is based on $\nabla _ { \pmb { \theta } } L \big ( f _ { \pmb { \theta } _ { t - 1 } } ( \pmb { X } + \delta _ { t - 1 } ) , y \big )$ . Such a problem may be more significant when the step size is large.
+
+Different from FreeAT, YOPO accumulates the gradient of the parameters from each of the ascent steps, and updates the parameters only once after the $K$ inner ascent steps. YOPO also advocates that after each back-propagation, one should take the gradient of the first hidden layer as a constant and perform several additional updates on the adversary using the product of this constant and the Jacobian of the first layer of the network to obtain strong adversaries. However, when the first hidden layer is a linear layer as in their implementation, such an operation is equivalent to taking a larger step size on the adversary. The analysis backing the extra update steps also assumes a twice continuously differentiable loss, which does not hold for ReLU-based neural networks they experimented with, and thus the reasons for the success of such an algorithm remains obscure. We give empirical comparisons between YOPO and our approach in Sec. 4.3.
+
+To obtain better solutions for the inner max and avoid fundamental limitations on the function class, we propose FreeLB, which performs multiple PGD iterations to craft adversarial examples, and simultaneously accumulates the “free” parameter gradients $\nabla _ { \boldsymbol { \theta } } L$ in each iteration. After that, it updates the model parameter $\pmb \theta$ all at once with the accumulated gradients. The overall procedure is shown in Algorithm 1, in which $X + \delta _ { t }$ is an approximation to the local maximum within the intersection of two balls $\mathcal { T } _ { t } = \mathcal { B } _ { X + \delta _ { 0 } } ( \alpha t ) \cap \mathcal { B } _ { X } ( \epsilon )$ . By taking a descent step along the averaged gradients at $X + \delta _ { 0 } , . . . , X + \delta _ { K - 1 }$ , we approximately optimize the following objective:
+
+$$
+\underset { \pmb { \theta } } { \operatorname* { m i n } } \mathbb { E } _ { ( Z , y ) \sim \mathcal { D } } \left[ \frac { 1 } { K } \sum _ { t = 0 } ^ { K - 1 } \underset { \delta _ { t } \in \mathcal { Z } _ { t } } { \operatorname* { m a x } } L \big ( f _ { \pmb { \theta } } \big ( \pmb { X } + \delta _ { t } \big ) , y \big ) \right] ,
+$$
+
+which is equivalent to replacing the original batch $\boldsymbol { X }$ with a $K$ -times larger virtual batch, consisting of samples whose embeddings are $X + \delta _ { 0 } , . . . , X + \delta _ { K - 1 }$ . Compared with PGD-based adversarial training (Eq. 1), which minimizes the maximum risk at a single estimated point in the vicinity of each training sample, FreeLB minimizes the maximum risk at each ascent step at almost no overhead.
+
+Intuitively, FreeLB could be a learning method with lower generalization error than PGD. Sokolic et al. (2017) have proved that the generalization error of a learning method invariant to a set of√ $T$ transformations may be up to $\sqrt { T }$ smaller than a non-invariant learning method. According to their theory, FreeLB could have a more significant improvement over natural training, since FreeLB enforces the invariance to $K$ adversaries from a set of up to $K$ different norm constraints,3 while PGD only enforces invariance to a single norm constraint $\epsilon$ .
+
+Empirically, FreeLB does lead to higher robustness and invariance than PGD in the embedding space, in the sense that the maximum increase of loss in the vicinity of $\boldsymbol { X }$ for models trained with FreeLB is smaller than that with PGD. See Sec. 4.3 for details. In theory, such improved robustness can lead to better generalization (Xu & Mannor, 2012), which is consistent with our experiments. Qin et al. (2019) also demonstrated that PGD-based method leads to highly convolved and non-linear loss surfaces in the vicinity of input samples when $K$ is small, indicating a lack of robustness.
+
+# 3.3 WHEN ADVERSARIAL TRAINING MEETS DROPOUT
+
+Usually, adversarial training is not used together with dropout (Srivastava et al., 2014). However, for some language models like RoBERTa (Liu et al., 2019b), dropout is used during the finetuning stage. In practice, when dropout is turned on, each ascent step of Algorithm 1 is optimizing $\delta$ for a different network. Specifically, denote the dropout mask as $_ { m }$ with each entry $m _ { i } \sim \mathrm { B e r n o u l l i } ( p )$ . Similar to our analysis for FreeAT, the ascent step from $\delta _ { t - 1 }$ to $\delta _ { t }$ is based on $\nabla _ { \delta } L \big ( f _ { \theta ( m _ { t - 1 } ) } \big ( \boldsymbol { X } + \delta _ { t - 1 } \big ) , \boldsymbol { y } \big )$ , so $\delta _ { t }$ is sub-optimal for $L ( f _ { \theta ( m _ { t } ) } ( \boldsymbol { X } + \delta ) , y )$ . Here $\theta ( m )$ is the effective parameters under dropout mask $_ { m }$ .
+
+The more plausible solution is to use the same $_ { \mathbf { \nabla } } \mathbf { m }$ in each step. When applying dropout to any network, the objective for $\pmb \theta$ is to minimize the expectation of loss under different networks determined by the dropout masks, which is achieved by minimizing the Monte Carlo estimation of the expected loss. In our case, the objective becomes:
+
+$$
+\operatorname* { m i n } _ { \theta } \mathbb { E } _ { ( Z , y ) \sim \mathcal { D } , m \sim \mathcal { M } } \left[ \frac { 1 } { K } \sum _ { t = 0 } ^ { K - 1 } \operatorname* { m a x } _ { \delta _ { t } \in \mathcal { T } _ { t } } L \big ( f _ { \theta ( m ) } ( X + \delta _ { t } ) , y \big ) \right] ,
+$$
+
+where the 1-sample Monte Carlo estimation should be $\begin{array} { r } { \frac { 1 } { K } \sum _ { t = 0 } ^ { K - 1 } \operatorname* { m a x } _ { \delta _ { t } \in \mathcal { T } _ { t } } L ( f _ { \theta ( m _ { 0 } ) } ( X + \delta _ { t } ) , y ) } \end{array}$ and can be minimized by using FreeLB with dropout mask $m _ { 0 }$ in each ascent step. This is similar to applying Variational Dropout to RNNs as used in Gal & Ghahramani (2016).
+
+# 4 EXPERIMENTS
+
+In this section, we provide comprehensive analysis on FreeLB through extensive experiments on three Natural Language Understanding benchmarks: GLUE (Wang et al., 2019), ARC (Clark et al., 2018) and CommonsenseQA (Talmor et al., 2019). We also compare the robustness and generalization of FreeLB with other adversarial training algorithms to demonstrate its strength. Additional experimental details are provided in the Appendix.
+
+# 4.1 DATASETS
+
+GLUE Benchmark. The GLUE benchmark is a collection of 9 natural language understanding tasks, namely Corpus of Linguistic Acceptability (CoLA; Warstadt et al. (2018)), Stanford Sentiment Treebank (SST; Socher et al. (2013)), Microsoft Research Paraphrase Corpus (MRPC; Dolan & Brockett (2005)), Semantic Textual Similarity Benchmark (STS; Agirre et al. (2007)), Quora Question Pairs (QQP; Iyer et al. (2017)), Multi-Genre NLI (MNLI; Williams et al. (2018)), Question NLI (QNLI; Rajpurkar et al. (2016)), Recognizing Textual Entailment (RTE; Dagan et al. (2006); Bar Haim et al. (2006); Giampiccolo et al. (2007); Bentivogli et al. (2009)) and Winograd NLI (WNLI; Levesque et al. (2011)). 8 of the tasks are formulated as classification problems and only STS-B is formulated as regression, but FreeLB applies to all of them. For BERT-base, we use the HuggingFace implementation4, and follow the single-task finetuning procedure as in Devlin et al. (2019). For RoBERTa, we use the fairseq implementation5. Same as Liu et al. (2019b), we also use single-task finetuning for all dev set results, and start with MNLI-finetuned models on RTE, MRPC and STS-B for the test submissions.
+
+Table 1: Results (median and variance) on the dev sets of GLUE based on the RoBERTa-large model, from 5 runs with the same hyperparameter but different random seeds. ReImp is our reimplementation of RoBERTalarge. The training process can be very unstable even with the vanilla version. Here, both PGD on STS-B and FreeAT on RTE demonstrates such instability, with one unconverged instance out of five.   
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>MNLI(Acc)</td><td rowspan=1 colspan=1>QNLI(Acc)</td><td rowspan=1 colspan=1>QQP(Acc)</td><td rowspan=1 colspan=1>RTE(Acc)</td><td rowspan=1 colspan=1>SST-2(Acc)</td><td rowspan=1 colspan=1>MRPC(Acc)</td><td rowspan=1 colspan=1>CoLA(Mcc)</td><td rowspan=1 colspan=1>STS-B(Pearson)</td></tr><tr><td rowspan=1 colspan=1>Reported</td><td rowspan=1 colspan=1>90.2</td><td rowspan=1 colspan=1>94.7</td><td rowspan=1 colspan=1>92.2</td><td rowspan=1 colspan=1>86.6</td><td rowspan=1 colspan=1>96.4</td><td rowspan=1 colspan=1>90.9</td><td rowspan=1 colspan=1>68.0</td><td rowspan=1 colspan=1>92.4</td></tr><tr><td rowspan=1 colspan=1>ReImp</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>85.61 (1.7)</td><td rowspan=1 colspan=1>96.56 (.3)</td><td rowspan=1 colspan=1>90.69(.5)</td><td rowspan=1 colspan=1>67.57 (1.3)</td><td rowspan=1 colspan=1>92.20 (.2)</td></tr><tr><td rowspan=1 colspan=1>PGD</td><td rowspan=1 colspan=1>90.53 (.2)</td><td rowspan=1 colspan=1>94.87 (.2)</td><td rowspan=1 colspan=1>92.49 (.07)</td><td rowspan=1 colspan=1>87.41 (.9)</td><td rowspan=1 colspan=1>96.44 (.1)</td><td rowspan=1 colspan=1>90.93 (.2)</td><td rowspan=1 colspan=1>69.67 (1.2)</td><td rowspan=1 colspan=1>92.43 (7.)</td></tr><tr><td rowspan=1 colspan=1>FreeAT</td><td rowspan=1 colspan=1>90.02 (.2)</td><td rowspan=1 colspan=1>94.66 (.2)</td><td rowspan=1 colspan=1>92.48(.08)</td><td rowspan=1 colspan=1>86.69 (15.)</td><td rowspan=1 colspan=1>96.10 (.2)</td><td rowspan=1 colspan=1>90.69 (.4)</td><td rowspan=1 colspan=1>68.80 (1.3)</td><td rowspan=1 colspan=1>92.40 (.3)</td></tr><tr><td rowspan=1 colspan=1>FreeLB</td><td rowspan=1 colspan=1>90.61 (.1)</td><td rowspan=1 colspan=1>94.98 (.2)</td><td rowspan=1 colspan=1>92.60 (.03)</td><td rowspan=1 colspan=1>88.13 (1.2)</td><td rowspan=1 colspan=1>96.79 (.2)</td><td rowspan=1 colspan=1>91.42 (.7)</td><td rowspan=1 colspan=1>71.12 (.9)</td><td rowspan=1 colspan=1>92.67 (.08)</td></tr></table>
+
+Table 2: Results on GLUE from the evaluation server, as of Sep 25, 2019. Metrics are the same as the leaderboard. Number under each task’s name is the size of the training set. FreeLB-BERT is the single-model results of BERT-base finetuned with FreeLB, and FreeLB-RoB is the ensemble of 7 RoBERTa-Large models for each task. References: 1: (Devlin et al., 2019); 2: (Liu et al., 2019a); 3: (Yang et al., 2019); 4: (Liu et al., 2019b).   
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Score</td><td rowspan=1 colspan=1>CoLA8.5k</td><td rowspan=1 colspan=1>SST-267k</td><td rowspan=1 colspan=1>MRPC3.7k</td><td rowspan=1 colspan=1>STS-B7k</td><td rowspan=1 colspan=1>QQP364k</td><td rowspan=1 colspan=1>MNLI-m/mm393k</td><td rowspan=1 colspan=1>QNLI108k</td><td rowspan=1 colspan=1>RTE2.5k</td><td rowspan=1 colspan=1>WNLI634</td><td rowspan=1 colspan=1>AX</td></tr><tr><td rowspan=1 colspan=1>BERT-base1</td><td rowspan=1 colspan=1>78.3</td><td rowspan=1 colspan=1>52.1</td><td rowspan=1 colspan=1>93.5</td><td rowspan=1 colspan=1>88.9/84.8</td><td rowspan=1 colspan=1>87.1/85.8</td><td rowspan=1 colspan=1>71.2/89.2</td><td rowspan=1 colspan=1>84.6/83.4</td><td rowspan=1 colspan=1>90.5</td><td rowspan=1 colspan=1>66.4</td><td rowspan=1 colspan=1>65.1</td><td rowspan=1 colspan=1>34.2</td></tr><tr><td rowspan=1 colspan=1>FreeLB-BERT</td><td rowspan=1 colspan=1>79.4</td><td rowspan=1 colspan=1>54.5</td><td rowspan=1 colspan=1>93.6</td><td rowspan=1 colspan=1>88.1/83.5</td><td rowspan=1 colspan=1>87.7/86.7</td><td rowspan=1 colspan=1>72.7/89.6</td><td rowspan=1 colspan=1>85.7/84.6</td><td rowspan=1 colspan=1>91.8</td><td rowspan=1 colspan=1>70.1</td><td rowspan=1 colspan=1>65.1</td><td rowspan=1 colspan=1>36.9</td></tr><tr><td rowspan=1 colspan=1>MT-DNN2</td><td rowspan=1 colspan=1>87.6</td><td rowspan=1 colspan=1>68.4</td><td rowspan=1 colspan=1>96.5</td><td rowspan=1 colspan=1>92.7/90.3</td><td rowspan=1 colspan=1>91.1/90.7</td><td rowspan=1 colspan=1>73.7789.9</td><td rowspan=1 colspan=1>87.9/87.4</td><td rowspan=1 colspan=1>96.0</td><td rowspan=1 colspan=1>86.3</td><td rowspan=1 colspan=1>89.0</td><td rowspan=1 colspan=1>42.8</td></tr><tr><td rowspan=1 colspan=1>XLNet-Large</td><td rowspan=1 colspan=1>88.4</td><td rowspan=1 colspan=1>67.8</td><td rowspan=1 colspan=1>96.8</td><td rowspan=1 colspan=1>93.0/90.7</td><td rowspan=1 colspan=1>91.6/91.1</td><td rowspan=1 colspan=1>74.2/90.3</td><td rowspan=1 colspan=1>90.2/89.8</td><td rowspan=1 colspan=1>98.6</td><td rowspan=1 colspan=1>86.3</td><td rowspan=1 colspan=1>90.4</td><td rowspan=1 colspan=1>47.5</td></tr><tr><td rowspan=1 colspan=1>RoBERTa4</td><td rowspan=1 colspan=1>88.5</td><td rowspan=1 colspan=1>67.8</td><td rowspan=1 colspan=1>96.7</td><td rowspan=1 colspan=1>92.3/89.8</td><td rowspan=1 colspan=1>92.2/91.9</td><td rowspan=1 colspan=1>74.3/90.2</td><td rowspan=1 colspan=1>90.8/90.2</td><td rowspan=1 colspan=1>98.9</td><td rowspan=1 colspan=1>88.2</td><td rowspan=1 colspan=1>89.0</td><td rowspan=1 colspan=1>48.7</td></tr><tr><td rowspan=1 colspan=1>FreeLB-RoB</td><td rowspan=1 colspan=1>88.8</td><td rowspan=1 colspan=1>68.0</td><td rowspan=1 colspan=1>96.8</td><td rowspan=1 colspan=1>93.1/90.8</td><td rowspan=1 colspan=1>92.4/92.2</td><td rowspan=1 colspan=1>74.8/90.3</td><td rowspan=1 colspan=1>91.1/90.7</td><td rowspan=1 colspan=1>98.8</td><td rowspan=1 colspan=1>88.7</td><td rowspan=1 colspan=1>89.0</td><td rowspan=1 colspan=1>50.1</td></tr><tr><td rowspan=1 colspan=1>Human</td><td rowspan=1 colspan=1>87.1</td><td rowspan=1 colspan=1>66.4</td><td rowspan=1 colspan=1>97.8</td><td rowspan=1 colspan=1>86.3/80.8</td><td rowspan=1 colspan=1>92.7/92.6</td><td rowspan=1 colspan=1>59.5/80.4</td><td rowspan=1 colspan=1>92.0/92.8</td><td rowspan=1 colspan=1>91.2</td><td rowspan=1 colspan=1>93.6</td><td rowspan=1 colspan=1>95.9</td><td rowspan=1 colspan=1>1</td></tr></table>
+
+ARC Benchmark. The ARC dataset (Clark et al., 2018) is a collection of multi-choice science questions from grade-school level exams. It is further divided into ARC-Challenge set with 2,590 question answer (QA) pairs and ARC-Easy set with 5,197 QA pairs. Questions in ARC-Challenge are more difficult and cannot be handled by simply using a retrieval and co-occurence based algorithm (Clark et al., 2018). A typical question is:
+
+Which property of a mineral can be determined just by looking at it?
+
+(A) luster [correct] $( B )$ mass $( C )$ weight $( D )$ hardness.
+
+CommonsenseQA Benchmark. The CommonsenseQA dataset (Talmor et al., 2019) consists of 12,102 natural language questions that require human commonsense reasoning ability to answer. A typical question is :
+
+Where can I stand on a river to see water falling without getting wet?
+
+(A) waterfall, (B) bridge [correct], (C) valley, $( D )$ stream, (E) bottom.
+
+Each question has five candidate answers from ConceptNet (Speer et al., 2017). To make the question more difficult to solve, most answers have the same relation in ConceptNet to the key concept in the question. As shown in the above example, most answers can be connected to “river” by “AtLocation” relation in ConceptNet. For a fair comparison with the reported results in papers and leaderboard6, we use the official random split 1.11.
+
+# 4.2 EXPERIMENTAL RESULTS
+
+GLUE We summarize results on the dev sets of GLUE in Table 1, comparing the proposed FreeLB against other adversatial training algorithms (PGD (Madry et al., 2018) and FreeAT (Shafahi et al., 2019)). We use the same step size $\alpha$ and number of steps $m$ for PGD, FreeAT and FreeLB. FreeLB is consistently better than the two baselines. Comparisons and detailed discussions about
+
+Table 3: Results on ARC and CommonsenseQA (CQA). ARC-Merge is the combination of ARC-Easy and ARC-Challenge, “MTL” stands for multi-task learning and “Ens” stands for ensemble. Results of XLNet $^ +$ RoBERTa $\left( \mathbf { M } \mathbf { T } \mathbf { L } { + } \mathbf { E } \mathbf { n } \mathbf { s } \right)$ and AristoRoBERTaV7 (MTL) are from the ARC leaderboards. Test (E) denotes the test set results with ensembles. For CQA, we report the highest dev and test accuracies among all models. The models with 78.81/72.19 dev/test accuracy (as in the table) have 71.84/78.64 test/dev accuracies respectively.   
+
+<table><tr><td rowspan=2 colspan=1></td><td rowspan=2 colspan=9>ARC-Easy    ARC-Challenge   ARC-Merge            CQADev   Test   Dev    Test    Dev   Test   Dev  Test Test (E)</td></tr><tr><td rowspan=1 colspan=1>Test</td><td rowspan=1 colspan=1>Dev</td><td rowspan=1 colspan=1>Test</td><td rowspan=1 colspan=1>Dev</td><td rowspan=1 colspan=2>Test   Dev</td><td rowspan=1 colspan=1>Test</td></tr><tr><td rowspan=1 colspan=1>RoBERTa(Reported)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>78.43</td><td rowspan=1 colspan=1>72.1</td><td rowspan=1 colspan=1>72.5</td></tr><tr><td rowspan=1 colspan=1>RoBERTa (ReImp)</td><td rowspan=1 colspan=1>84.39</td><td rowspan=1 colspan=1>84.13</td><td rowspan=1 colspan=1>64.54</td><td rowspan=1 colspan=1>64.44</td><td rowspan=1 colspan=1>77.83</td><td rowspan=1 colspan=1>77.62</td><td rowspan=1 colspan=1>77.56</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>FreeLB-RoBERTa</td><td rowspan=1 colspan=1>84.91</td><td rowspan=1 colspan=1>84.81</td><td rowspan=1 colspan=1>65.89</td><td rowspan=1 colspan=1>65.36</td><td rowspan=1 colspan=1>78.37</td><td rowspan=1 colspan=1>78.39</td><td rowspan=1 colspan=1>78.81</td><td rowspan=1 colspan=1>72.2</td><td rowspan=1 colspan=1>73.1</td></tr><tr><td rowspan=1 colspan=1>AristoRoBERTaV7(MTL)</td><td rowspan=1 colspan=1>:</td><td rowspan=1 colspan=1>85.02</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>66.47</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>78.89</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>:</td></tr><tr><td rowspan=1 colspan=1>XLNet +RoBERTa (MTL+Ens)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>67.06</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>二</td></tr><tr><td rowspan=1 colspan=1>FreeLB-RoBERTa(MTL)</td><td rowspan=1 colspan=1>84.91</td><td rowspan=1 colspan=1>85.44</td><td rowspan=1 colspan=1>70.23</td><td rowspan=1 colspan=1>67.75</td><td rowspan=1 colspan=1>79.86</td><td rowspan=1 colspan=1>79.60</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr></table>
+
+YOPO (Zhang et al., 2019) are provided in Sec. 4.3. We have also submitted our results to the evaluation server, results provided in Table 2. FreeLB lifts the performance of the BERT-base model from 78.3 to 79.4, and RoBERTa-large model from 88.5 to 88.8 on overall scores.
+
+ARC For ARC, a corpus of 14 million related science documents (from ARC Corpus, Wikipedia and other sources) is provided. For each QA pair, we first use a retrieval model to select top 10 related documents. Then, given these retrieved documents7, we use RoBERTa-large model to encode $\langle s \rangle$ Retrieved Documents $\langle / s \rangle$ Question $^ +$ Answer $\langle / s \rangle$ , where $\langle \mathbf { s } \rangle$ and $\langle / \mathrm { s } \rangle$ are special tokens for RoBERTa model8. We then apply a fully-connected layer to the representation of the [CLS] token to compute the final logit, and use standard cross-entropy loss for model training.
+
+Results are summarized in Table 3. Following Sun et al. (2018), we first finetune the RoBERTa model on the RACE dataset (Lai et al., 2017). The finetuned RoBERTa model achieves $8 5 . 7 0 \%$ and $8 5 . 2 4 \%$ accuracy on the development and test set of RACE, respectively. Based on this, we further finetune the model on both ARC-Easy and ARC-Challenge datasets with the same hyper-parameter searching strategy (for 5 epochs), which achieves $8 4 . 1 3 \% / 6 4 . 4 4 \%$ test accuracy on ARC-Easy/ARCChallenge. And by adding FreeLB finetuning, we can reach $8 4 . 8 1 \% / 6 5 . 3 6 \%$ , a significant boost on ARC benchmark, demonstrating the effectiveness of FreeLB.
+
+To further improve the results, we apply a multi-task learning (MTL) strategy using additional datasets. We first finetune the model on RACE (Lai et al., 2017), and then finetune on a joint dataset of ARC-Easy, ARC-Challenge, OpenbookQA (Mihaylov et al., 2018) and Regents Living Environment9. Based on this, we further finetune our model on ARC-Easy and ARC-Challenge with FreeLB. After finetuning, our single model achieves $6 7 . 7 5 \%$ test accuracy on ARC-Challenge and $8 5 . 4 4 \%$ on ARC-Easy, both outperforming the best submission on the official leaderboard10.
+
+CommonsenseQA Similar to the training strategy in Liu et al. (2019b), we construct five inputs for each question by concatenating the question and each answer separately, then encode each input with the representation of the [CLS] token. A final score is calculated by applying the representation of [CLS] to a fully-connected layer. Following the fairseq repository11, the input is formatted as: ${ \mathfrak { n } } ( s ) Q$ : Where can I stand on a river to see water falling without getting wet? $\langle / s \rangle A$ : waterfall $\langle \langle s \rangle ^ { \ast } \cdot \mathbf { \vec { \mathbf { \mu } } }$ , where $\cdot _ { Q }$ :’ and $^ { , } A$ :’ are the prefix for question and answer, respectively.
+
+Results are summarized in Table 3. We obtained a dev-set accuracy of $7 7 . 5 6 \%$ with the RoBERTalarge model. When using FreeLB finetuning, we achieved $7 8 . 8 1 \%$ , a $1 . 2 5 \%$ absolute gain. Compared with the results reported from fairseq repository, which obtains $7 8 . 4 3 \%$ accuracy on the devset, FreeLB still achieves better performance. Our submission to the CommonsenseQA leaderboard achieves $7 2 . 2 \%$ single-model test set accuracy, and the result of a 20-model ensemble is $7 3 . 1 \%$ , which achieves No.1 among all the submissions without making use of ConceptNet.
+
+Table 4: The median and standard deviation of the scores on the dev sets of RTE, CoLA and MRPC from the GLUE benchmark, computed from 5 runs with the same hyper-parameters except for the random seeds. We use FreeLB- $\mathbf { \nabla } m$ to denote FreeLB with $m$ ascent steps, and FreeLB- ${ . 3 ^ { * } }$ to denote the version without reusing the dropout mask.   
+
+<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=1>Vanilla</td><td rowspan=1 colspan=1>FreeLB-3*</td><td rowspan=1 colspan=1>FreeLB-3</td><td rowspan=1 colspan=1>YOPO-3-2</td><td rowspan=1 colspan=1>YOPO-3-3</td></tr><tr><td rowspan=1 colspan=1>RTE</td><td rowspan=1 colspan=1>85.61 (1.67)</td><td rowspan=1 colspan=1>87.14 (1.29)</td><td rowspan=1 colspan=1>88.13 (1.21)</td><td rowspan=1 colspan=1>87.05 (1.36)</td><td rowspan=1 colspan=1>87.05 (0.20)</td></tr><tr><td rowspan=1 colspan=1>CoLA</td><td rowspan=1 colspan=1>67.57 (1.30)</td><td rowspan=1 colspan=1>69.31 (1.16)</td><td rowspan=1 colspan=1>71.12 (0.90)</td><td rowspan=1 colspan=1>70.40 (0.91)</td><td rowspan=1 colspan=1>69.91 (1.16)</td></tr><tr><td rowspan=1 colspan=1>MRPC</td><td rowspan=1 colspan=1>90.69 (0.54)</td><td rowspan=1 colspan=1>90.93 (0.66)</td><td rowspan=1 colspan=1>91.42 (0.72)</td><td rowspan=1 colspan=1>90.44 (0.62)</td><td rowspan=1 colspan=1>90.69 (0.37)</td></tr></table>
+
+<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=3>RTE</td><td rowspan=1 colspan=3>CoLA</td><td rowspan=1 colspan=3>MRPC</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>M-Inc(10-4)</td><td rowspan=1 colspan=1>M-Inc (R)(10-4)</td><td rowspan=1 colspan=1>N-Loss(10-4)</td><td rowspan=1 colspan=1>M-Inc(10-4)</td><td rowspan=1 colspan=1>M-Inc (R)(10-4)</td><td rowspan=1 colspan=1>N-Loss(10-4)</td><td rowspan=1 colspan=1>M-Inc(10-3)</td><td rowspan=1 colspan=1>M-Inc (R)(10-3)</td><td rowspan=1 colspan=1>N-Loss(10-3)</td></tr><tr><td rowspan=1 colspan=1>Vanilla</td><td rowspan=1 colspan=1>5.1</td><td rowspan=1 colspan=1>5.3</td><td rowspan=1 colspan=1>4.5</td><td rowspan=1 colspan=1>6.1</td><td rowspan=1 colspan=1>5.7</td><td rowspan=1 colspan=1>5.2</td><td rowspan=1 colspan=1>10.2</td><td rowspan=1 colspan=1>10.2</td><td rowspan=1 colspan=1>1.9</td></tr><tr><td rowspan=1 colspan=1>PGD</td><td rowspan=1 colspan=1>4.7</td><td rowspan=1 colspan=1>4.9</td><td rowspan=1 colspan=1>6.2</td><td rowspan=1 colspan=1>128.2</td><td rowspan=1 colspan=1>130.1</td><td rowspan=1 colspan=1>436.1</td><td rowspan=1 colspan=1>5.7</td><td rowspan=1 colspan=1>5.7</td><td rowspan=1 colspan=1>5.4</td></tr><tr><td rowspan=1 colspan=1>FreeLB</td><td rowspan=1 colspan=1>3.0</td><td rowspan=1 colspan=1>2.6</td><td rowspan=1 colspan=1>4.1</td><td rowspan=1 colspan=1>1.4</td><td rowspan=1 colspan=1>1.3</td><td rowspan=1 colspan=1>7.2</td><td rowspan=1 colspan=1>3.6</td><td rowspan=1 colspan=1>3.6</td><td rowspan=1 colspan=1>2.7</td></tr></table>
+
+Table 5: Median of the maximum increase in loss in the vicinity of the dev set samples for RoBERTa-Large model finetuned with different methods. Vanilla models are naturally trained RoBERTa’s. M-Inc: Max Inc, MInc (R): Max Inc (R). Nat Loss (N-Loss) is the loss value on clean samples. Notice we require all clean samples here to be correctly classified by all models, which results in 227, 850 and 355 samples for RTE, CoLA and MRPC, respectively. We also give the variance in the Appendix.
+
+# 4.3 ABLATION STUDY AND ANALYSIS
+
+In this sub-section, we first show the importance of reusing dropout mask, then conduct a thorough ablation study on FreeLB over the GLUE benchmark to analyze the robustness and generalization strength of different approaches. We observe that it is unnecessary to perform shallow-layer updates on the adversary as YOPO for our case, and FreeLB results in improved robustness and generalization compared with PGD.
+
+Importance of Reusing Mask Table 4 (columns 2 to 4) compares the results of FreeLB with and without reusing the same dropout mask in each ascent step, as proposed in Sec. 3.3. With reusing, FreeLB can achieve a larger improvement over the naturally trained models. Thus, we enable mask reusing for all experiments involving RoBERTa.
+
+Comparing the Robustness Table 5 provides the comparisons of the maximum increment of loss in the vicinity of each sample, defined as:
+
+$$
+\displaystyle \Delta L _ { \mathrm { m a x } } ( \boldsymbol { X } , \epsilon ) = \operatorname* { m a x } _ { \| \delta \| \leq \epsilon } L ( f _ { \boldsymbol { \theta } } ( \boldsymbol { X } + \delta ) , \boldsymbol { y } ) - L ( f _ { \boldsymbol { \theta } } ( \boldsymbol { X } ) , \boldsymbol { y } ) ,
+$$
+
+which reflects the robustness and invariance of the model in the embedding space. In practice, we use PGD steps as in Eq. 2 to find the value of ${ \Delta } L _ { \mathrm { m a x } } ( \boldsymbol { X } , \boldsymbol { \epsilon } )$ . We found that when using a step size of $5 { \cdot } 1 0 ^ { - 3 }$ and $\epsilon = 0 . 0 1 \| X \| _ { F }$ , the PGD iterations converge to almost the same value, starting from 100 different random initializations of $\pmb { \delta }$ for the RoBERTa models, trained with or without FreeLB. This indicates that PGD reliably finds $\Delta L _ { \mathrm { m a x } }$ for these models. Therefore, we compute ${ \Delta } L _ { \mathrm { m a x } } ( X , \epsilon )$ for each $\boldsymbol { X }$ via a 2000-step PGD.
+
+Samples with small margins exist even for models with perfect accuracy, which could give a false sense of vulnerability of the model. To rule out the outlier effect and make ${ \Delta } L _ { \mathrm { m a x } } ( \boldsymbol { X } , \epsilon )$ comparable across different samples, we only consider samples that all the evaluated models can correctly classify, and search for an $\epsilon$ for each sample such that the reference model can correctly classify all samples within the $\epsilon$ ball.12 However, such choice of per-sample $\epsilon$ favors the reference model by design. To make fair comparisons, Table 5 provides the median of ${ \Delta } L _ { \mathrm { m a x } } ( \boldsymbol { X } , \epsilon )$ with per-sample $\epsilon$ from models trained by FreeLB (Max Inc) and PGD (Mac Inc (R)), respectively.
+
+Across all three datasets and different reference models, FreeLB has the smallest median increment even when starting from a larger natural loss than vanilla models. This demonstrates that FreeLB is more robust and invariant in most cases. Such results are also consistent with the models’ dev set performance (the performances for Vanilla/PGD/FreeLB models on RTE, CoLA and MRPC are 86.69/87.41/89.21, 69.91/70.84/71.40, 91.67/91.17/91.17, respectively).
+
+Comparing with YOPO The original implementation of YOPO (Zhang et al., 2019) chooses the first convolutional layer of the ResNets as $f _ { 0 }$ for updating the adversary in the “s-loop”. As a result, each step of the “s-loop” should be using exactly the same value to update the adversary,and YOPO$m { - } n$ degenerates into FreeLB with a $n$ -times large step size. To avoid that, we choose the layers up to the output of the first Transformer block as $f _ { 0 }$ when implementing YOPO. To make the total amount of update on the adversary equal, we take the hyper-parameters for FreeLB- $m$ and only change the step size $\alpha$ into $\alpha / n$ for YOPO- $m { \cdot } n$ . Table 4 shows that FreeLB performs consistently better than YOPO on all three datasets. Accidentally, we also give the results comparing with YOPO- $\mathbf { \nabla } m - n$ without changing the step size $\alpha$ for YOPO in Table 8. The gap between two approaches seem to shrink, which may be caused by using a larger total step size for the YOPO adversaries. We leave exhaustive hyperparameter search for both models as our future work.
+
+# 5 CONCLUSION
+
+In this work, we have developed an adversarial training approach, FreeLB, to improve natural language understanding. The proposed approach adds perturbations to continuous word embeddings using a gradient method, and minimizes the resultant adversarial risk in an efficient way. FreeLB is able to boost Transformer-based model (BERT and RoBERTa) on several datasets and achieve new state of the art on GLUE and ARC benchmarks. Empirical study demonstrates that our method results in both higher robustness in the embedding space than natural training and better generalization ability. Such observation is also consistent with recent findings in Computer Vision. However, adversarial training still takes significant overhead compared with vanilla SGD. How to accelerate this process while improving generalization is an interesting future direction.
+
+Acknowledgements: Goldstein and Zhu were supported in part by the DARPA GARD, DARPA QED for RML, and AFOSR MURI programs.
+
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+
+# A ADDITIONAL EXPERIMENTAL DETAILS
+
+# A.1 PROBLEM FORMULATIONS
+
+For tasks with ranking loss like ARC, CommonsenseQA, WNLI and QNLI, add the perturbation to the concatenation of the embeddings of all question/answer pairs.
+
+Additional tricks are required to achieve high performance on WNLI and QNLI for the GLUE benchmark. We use the same tricks as Liu et al. (2019b). For WNLI, we use the same WSC data provided by Liu et al. (2019b) for training. For testing, Liu et al. (2019b) also provided the test set with span annotations, but the order is different form the GLUE dataset. We re-order their test set by matching. For the QNLI, we follow Liu et al. (2019b) and formulate the problem as pairwise ranking problem, which is the same for CommonsenseQA. We find the matching pairs for both training set and testing set by matching the queries in the dev set. We predict “entailment” if the candidate has the higher score, and “not entailment” otherwise.
+
+# A.2 HYPER-PARAMETERS
+
+Table 6: Additional hyper-parameters on GLUE tasks.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>MNLI</td><td rowspan=1 colspan=1>QNLI</td><td rowspan=1 colspan=1>QQP</td><td rowspan=1 colspan=1>RTE</td><td rowspan=1 colspan=1>SST-2</td><td rowspan=1 colspan=1>MRPC</td><td rowspan=1 colspan=1>CoLA</td><td rowspan=1 colspan=1>STS-B</td></tr><tr><td rowspan=1 colspan=1>E</td><td rowspan=1 colspan=1>2E-1</td><td rowspan=1 colspan=1>1.5E-1</td><td rowspan=1 colspan=1>4.5E-1</td><td rowspan=1 colspan=1>1.5E-1</td><td rowspan=1 colspan=1>6E-1</td><td rowspan=1 colspan=1>4E-1</td><td rowspan=1 colspan=1>2E-1</td><td rowspan=1 colspan=1>3E-1</td></tr><tr><td rowspan=1 colspan=1>a</td><td rowspan=1 colspan=1>1E-1</td><td rowspan=1 colspan=1>1E-1</td><td rowspan=1 colspan=1>1.5E-1</td><td rowspan=1 colspan=1>3E-2</td><td rowspan=1 colspan=1>1E-1</td><td rowspan=1 colspan=1>4E-2</td><td rowspan=1 colspan=1>2.5E-2</td><td rowspan=1 colspan=1>1E-1</td></tr><tr><td rowspan=1 colspan=1>m</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>3</td></tr></table>
+
+As other adversarial training methods, introduces three additional hyper-parameters: step size $\alpha$ , maximum perturbation $\epsilon$ , number of steps $m$ . For all other hyper-parameters such as learning rate and number of iterations, we either search in the same interval as RoBERTa (on CommonsenseQA, ARC, and WNLI), or use exactly the same setting as RoBERTa (except for MRPC, where we find using a learning rate of $5 \times 1 0 ^ { - 6 }$ gives better results).13. We list the best combinations for $\alpha , \epsilon$ and $m$ for each of the GLUE tasks in Table 6. For WSC/WNLI, the best combination is $\epsilon = 1 e - 2 , \alpha =$ $5 e - 3 , m = 2$ . Notice even when $m \alpha < \epsilon$ , the maximum perturbation could still reach $\epsilon$ due to the random initialization.
+
+# B VARIANCE OF MAXIMUM INCREMENT OF LOSS
+
+<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=3>RTE</td><td rowspan=1 colspan=3>CoLA</td><td rowspan=1 colspan=3>MRPC</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>M-Inc(10-4)</td><td rowspan=1 colspan=1>M-Inc (R)(10-4）</td><td rowspan=1 colspan=1>N-Loss(10-4)</td><td rowspan=1 colspan=1>M-Inc(10-4)</td><td rowspan=1 colspan=1>M-Inc (R)(10-4）</td><td rowspan=1 colspan=1>N-Loss(10-4)</td><td rowspan=1 colspan=1>M-Inc(10-3)</td><td rowspan=1 colspan=1>M-Inc (R)(10-3）</td><td rowspan=1 colspan=1>N-Loss(10-3)</td></tr><tr><td rowspan=1 colspan=1>Vanilla</td><td rowspan=1 colspan=1>5.1(15087)</td><td rowspan=1 colspan=1>5.3(14346)</td><td rowspan=1 colspan=1>4.5(920)</td><td rowspan=1 colspan=1>6.1(13118)</td><td rowspan=1 colspan=1>5.7(14122)</td><td rowspan=1 colspan=1>5.2(447)</td><td rowspan=1 colspan=1>10.2(929)</td><td rowspan=1 colspan=1>10.2(955)</td><td rowspan=1 colspan=1>1.9(76)</td></tr><tr><td rowspan=1 colspan=1>PGD</td><td rowspan=1 colspan=1>4.7(10138)</td><td rowspan=1 colspan=1>4.9(1828)</td><td rowspan=1 colspan=1>6.2(752)</td><td rowspan=1 colspan=1>128.2(1300)</td><td rowspan=1 colspan=1>130.1(893)</td><td rowspan=1 colspan=1>436.1(1276)</td><td rowspan=1 colspan=1>5.7(321)</td><td rowspan=1 colspan=1>5.7(155)</td><td rowspan=1 colspan=1>5.4(54)</td></tr><tr><td rowspan=1 colspan=1>FreeLB</td><td rowspan=1 colspan=1>3.0(1614)</td><td rowspan=1 colspan=1>2.6(11114)</td><td rowspan=1 colspan=1>4.1(595)</td><td rowspan=1 colspan=1>1.4(1392)</td><td rowspan=1 colspan=1>1.3(6231)</td><td rowspan=1 colspan=1>7.2(615)</td><td rowspan=1 colspan=1>3.6(167)</td><td rowspan=1 colspan=1>3.6(601)</td><td rowspan=1 colspan=1>2.7(54)</td></tr></table>
+
+Table 7: Median and Standard Deviation of the maximum increase in loss in the vicinity of the dev set samples for RoBERTa-Large model finetuned with different methods. Vanilla models are naturally trained RoBERTa’s. Nat Loss is the loss value on clean samples. Notice we require all clean samples here to be correctly classified by all models, which results in 227, 850 and 355 samples for RTE, CoLA and MRPC.
+
+Table 7 provides the complete results for the increment of loss in the interval, with median and standard deviation.
+
+# C ADDITIONAL RESULTS FOR ABLATION STUDIES
+
+Here we provide some additional results for comparison with YOPO as complementary results to Table 4. We will release the complete results for comparing with YOPO and without variational dropout on each of the GLUE tasks in our next revision. From the current results, there is no need
+
+<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=1>Vanilla</td><td rowspan=1 colspan=1>FreeLB-3</td><td rowspan=1 colspan=1>YOPO-3-2</td><td rowspan=1 colspan=1>YOPO-3-3</td></tr><tr><td rowspan=1 colspan=1>STS-B</td><td rowspan=1 colspan=1>92.20 (.2)</td><td rowspan=1 colspan=1>92.67 (.08)</td><td rowspan=1 colspan=1>92.60 (.17)</td><td rowspan=1 colspan=1>92.60 (0.20)</td></tr><tr><td rowspan=1 colspan=1>SST-2</td><td rowspan=1 colspan=1>96.56 (.3)</td><td rowspan=1 colspan=1>96.79 (.2)</td><td rowspan=1 colspan=1>96.44 (.2)</td><td rowspan=1 colspan=1>96.33 (.1)</td></tr><tr><td rowspan=1 colspan=1>QNLI</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>94.98 (.2)</td><td rowspan=1 colspan=1>94.96 (.1)</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>QQP</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>92.60 (.03)</td><td rowspan=1 colspan=1>92.55 (.05)*</td><td rowspan=1 colspan=1>92.50 (.02)*</td></tr><tr><td rowspan=1 colspan=1>MNLI</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>90.61 (.1)</td><td rowspan=1 colspan=1>90.59 (.2)</td><td rowspan=1 colspan=1>90.45 (.2)</td></tr></table>
+
+Table 8: The median and standard deviation of the scores on the dev sets of STS-B, SST-2, QNLI, QQP and MNLI from the GLUE benchmark, each computed from 5 runs with the same hyper-parameters except for the random seeds (except for the results with YOPO on QQP, which are from 4 runs). Also note here we use a step size of $\alpha$ for the adversary of YOPO-m-n, so YOPO effectively uses a step size of $_ { n \alpha }$ . We use FreeLB- $\mathbf { \nabla } m$ to denote FreeLB with $m$ ascent steps, and YOPO-3- $\mathbf { \nabla } \cdot \mathbf { n }$ to denote YOPO with $_ n$ shallow-layer ascents.
+
+in using extra shallow-layer updates that YOPO advocates, since this consistently deteriorates the performance while introducing extra computations.
\ No newline at end of file
diff --git a/parse/train/BygzbyHFvB/BygzbyHFvB_content_list.json b/parse/train/BygzbyHFvB/BygzbyHFvB_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..1bc68a57b8443d99203e20ee8dfea75b42cb9f97
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@@ -0,0 +1,1710 @@
+[
+    {
+        "type": "text",
+        "text": "FREELB: ENHANCED ADVERSARIAL TRAINING FOR NATURAL LANGUAGE UNDERSTANDING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            820,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Chen $\\mathbf { Z } \\mathbf { h } \\mathbf { u } ^ { \\mathbf { 1 } }$ , Yu Cheng2, Zhe $\\mathbf { G a n } ^ { 2 }$ , Siqi $\\mathbf { S u n } ^ { 2 }$ , Tom Goldstein1, Jingjing Liu2 1University of Maryland, College Park 2Microsoft Dynamics 365 AI Research {chenzhu,tomg}@cs.umd.edu, {yu.cheng,zhe.gan,siqi.sun,jingjl}@microsoft.com ",
+        "bbox": [
+            184,
+            167,
+            714,
+            213
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            250,
+            544,
+            265
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Adversarial training, which minimizes the maximal risk for label-preserving input perturbations, has proved to be effective for improving the generalization of language models. In this work, we propose a novel adversarial training algorithm, FreeLB, that promotes higher invariance in the embedding space, by adding adversarial perturbations to word embeddings and minimizing the resultant adversarial risk inside different regions around input samples. To validate the effectiveness of the proposed approach, we apply it to Transformer-based models for natural language understanding and commonsense reasoning tasks. Experiments on the GLUE benchmark show that when applied only to the finetuning stage, it is able to improve the overall test scores of BERT-base model from 78.3 to 79.4, and RoBERTa-large model from 88.5 to 88.8. In addition, the proposed approach achieves state-of-the-art single-model test accuracies of $8 5 . 4 4 \\%$ and $6 7 . 7 5 \\%$ on ARC-Easy and ARC-Challenge. Experiments on CommonsenseQA benchmark further demonstrate that FreeLB can be generalized and boost the performance of RoBERTa-large model on other tasks as well. 1 ",
+        "bbox": [
+            233,
+            282,
+            764,
+            489
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            518,
+            334,
+            534
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Adversarial training is a method for creating robust neural networks. During adversarial training, mini-batches of training samples are contaminated with adversarial perturbations (alterations that are small and yet cause misclassification), and then used to update network parameters until the resulting model learns to resist such attacks. Adversarial training was originally proposed as a means to enhance the security of machine learning systems (Goodfellow et al., 2015), especially for safety-critical systems like self-driving cars (Xiao et al., 2018) and copyright detection (Saadatpanah et al., 2019). ",
+        "bbox": [
+            174,
+            550,
+            825,
+            647
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we turn our focus away from the security benefits of adversarial training, and instead study its effects on generalization. While adversarial training boosts the robustness, it is widely accepted by computer vision researchers that it is at odds with generalization, with classification accuracy on non-corrupted images dropping as much as $1 0 \\%$ on CIFAR-10, and $1 5 \\%$ on Imagenet (Madry et al., 2018; Xie et al., 2019). Surprisingly, people observe the opposite result for language models (Miyato et al., 2017; Cheng et al., 2019), showing that adversarial training can improve both generalization and robustness. ",
+        "bbox": [
+            174,
+            655,
+            823,
+            752
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We will show that adversarial training significantly improves performance of state-of-the-art models for many language understanding tasks. In particular, we propose a novel adversarial training algorithm, called FreeLB (Free Large-Batch), which adds adversarial perturbations to word embeddings and minimizes the resultant adversarial loss around input samples. The method leverages recently proposed “free” training strategies (Shafahi et al., 2019; Zhang et al., 2019) to enrich the training data with diversified adversarial samples under different norm constraints at no extra cost than PGD-based (Projected Gradient Descent) adversarial training (Madry et al., 2018), which enables us to perform such diversified adversarial training on large-scale state-of-the-art models. We observe improved invariance in the embedding space for models trained with FreeLB, which is positively correlated with generalization. ",
+        "bbox": [
+            174,
+            758,
+            825,
+            897
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We perform comprehensive experiments to evaluate the performance of a variety of adversarial training algorithms on state-of-the-art language understanding models and tasks. In the comparisons with standard PGD (Madry et al., 2018), FreeAT (Shafahi et al., 2019) and YOPO (Zhang et al., 2019), FreeLB stands out to be the best for the datasets and models we evaluated. With FreeLB, we achieve state-of-the-art results on several important language understanding benchmarks. On the GLUE benchmark, FreeLB pushes the performance of the BERT-base model from 78.3 to 79.4. The overall score of the RoBERTa-large models on the GLUE benchmark is also lifted from 88.5 to 88.8, achieving best results on most of its sub-tasks. Experiments also show that FreeLB can boost the performance of RoBERTa-large on question answering tasks, such as the ARC and CommonsenseQA benchmarks. We also provide a comprehensive ablation study and analysis to demonstrate the effectiveness of our training process. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            256
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            291,
+            343,
+            308
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 ADVERSARIAL TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            328,
+            393,
+            342
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To improve the robustness of neural networks against adversarial examples, many defense strategies and models have been proposed, in which PGD-based adversarial training (Madry et al., 2018) is widely considered to be the most effective, since it largely avoids the the obfuscated gradient problem (Athalye et al., 2018). It formulates a class of adversarial training algorithms (Kurakin et al., 2017) into solving a minimax problem on the cross-entropy loss, which can be achieved reliably through multiple projected gradient ascent steps followed by a SGD (Stochastic Gradient Descent) step. ",
+        "bbox": [
+            174,
+            353,
+            825,
+            450
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Despite being verified by Athalye et al. (2018) to avoid obfuscated gradients, Qin et al. (2019) shows that PGD-based adversarial training still leads to highly convolved and non-linear loss surfaces when $K$ is small, which could be readily broken under stronger adversaries. Thus, to be effective, the cost of PGD-based adversarial training is much higher than conventional training. To mitigate this cost, Shafahi et al. (2019) proposed a “free” adversarial training algorithm that simultaneously updates both model parameters and adversarial perturbations on a single backward pass. Using a similar formulation, Zhang et al. (2019) effectively reduce the total number of full forward and backward propagations for obtaining adversarial examples by restricting most of its adversarial updates in the first layer. ",
+        "bbox": [
+            174,
+            457,
+            825,
+            583
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.2 ADVERSARIAL EXAMPLES IN NATURAL LANGUAGES",
+        "text_level": 1,
+        "bbox": [
+            174,
+            614,
+            581,
+            628
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Adversarial examples have been explored primarily in the image domain, and received many attention in text domain recently. Previous works on text adversaries have focused on heuristics for creating adversarial examples in the black-box setting, or on specific tasks. Jia & Liang (2017) propose to add distracting sentences to the input document in order to induce mis-classification. Zhao et al. (2018) generate text adversaries by projecting the input data to a latent space using GANs, and searching for adversaries close to the original instance. Belinkov & Bisk (2018) manipulate every word in a sentence with synthetic or natural noise in machine translation systems. Iyyer et al. (2018) propose a neural paraphrase model based on back-translated data to produce paraphrases that have different sentence structures. Different from previous work, ours is not to produce actual adversarial examples, but only take the benefit of adversarial training for natural language understanding. ",
+        "bbox": [
+            174,
+            638,
+            825,
+            777
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We are not the first to observe that robust language models may perform better on clean test data. Miyato et al. (2017) extend adversarial and virtual adversarial training (Miyato et al., 2019) to the text domain to improve the performance on semi-supervised classification tasks. Ebrahimi et al. (2018) propose a character/word replacement for crafting attacks, and show employing adversarial examples in training renders the models more robust. Ribeiro et al. (2018) show that adversarial attacks can be used as a valuable tool for debugging NLP models. Cheng et al. (2019) also find that crafting adversarial examples can help neural machine translation significantly. Notably, these studies have focused on simple models or text generation tasks. Our work explores how to efficiently use the gradients obtained in adversarial training to boost the performance of state-of-the-art transformer-based models. ",
+        "bbox": [
+            174,
+            785,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 ADVERSARIAL TRAINING FOR LANGUAGE UNDERSTANDING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            709,
+            118
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Pre-trained large-scale language models, such as BERT (Devlin et al., 2019), RoBERTa (Liu et al., 2019b), ALBERT (Lan et al., 2020) and T5 (Raffel et al., 2019), have proven to be highly effective for downstream tasks. We aim to further improve the generalization of these pre-trained language models on the downstream language understanding tasks by enhancing their robustness in the embedding space during finetuning on these tasks. We achieve this goal by creating “virtual” adversarial examples in the embedding space, and then perform parameter updates on these adversarial embeddings. Creating actual adversarial examples for language is difficult; even with state-of-theart language models as guidance (e.g., (Cheng et al., 2019)), it remains unclear how to construct label-preserving adversarial examples via word/character replacement without human evaluations, because the meaning of each word/character depends on the context (Ribeiro et al., 2018). Since we are only interested in the effects of adversarial training, rather than producing actual adversarial examples, we add norm-bounded adversarial perturbations to the embeddings of the input sentences using a gradient-based method. Note that our embedding-based adversary is strictly stronger than a more conventional text-based adversary, as our adversary can make manipulations on word embeddings that are not possible in the text domain. ",
+        "bbox": [
+            173,
+            133,
+            825,
+            342
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For models that incorporate various input representations, including word or subword embeddings, segment embeddings and position embeddings, our adversaries only modify the concatenated word or sub-word embeddings, leaving other components of the sentence representation unchanged. 2 Denote the sequence of one-hot representations of the input subwords as $\\pmb { Z } = [ z _ { 1 } , z _ { 2 } , . . . , z _ { n } ]$ , the embedding matrix as $V$ , and the language model (encoder) as a function ${ \\pmb y } = f _ { \\pmb \\theta } ( { \\pmb X } )$ , where $\\dot { \\boldsymbol { X } } = \\boldsymbol { V } \\boldsymbol { Z }$ is the subword embeddings, $\\textbf {  { y } }$ is the output of the model (e.g., class probabilities for classification models), and $\\pmb \\theta$ denotes all the learnable parameters including the embedding matrix $V$ . We add adversarial perturbations $\\delta$ to the embeddings such that the prediction becomes $\\pmb { y } ^ { \\prime } = f _ { \\pmb { \\theta } } ( \\pmb { X } + \\pmb { \\delta } )$ . To preserve the semantics, we constrain the norm of $\\pmb { \\delta }$ to be small, and assume the model’s prediction should not change after the perturbation. This formulation is analogous to Miyato et al. (2017), with the difference that we do not require $\\boldsymbol { X }$ to be normalized. ",
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+    {
+        "type": "text",
+        "text": "3.1 PGD FOR ADVERSARIAL TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Standard adversarial training seeks to find optimal parameters $\\pmb { \\theta } ^ { * }$ to minimize the maximum risk for any $\\pmb { \\delta }$ within a norm ball as: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/17c85187abb9a6ea71a8c3e97ea4dd66fd478797846cf419f4a2d777a9491c4c.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta } \\mathbb { E } _ { ( Z , y ) \\sim \\mathcal { D } } \\left[ \\operatorname* { m a x } _ { \\| \\delta \\| \\leq \\epsilon } L ( f _ { \\theta } ( X + \\delta ) , y ) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { D }$ is the data distribution, $y$ is the label, and $L$ is some loss function. We use the Frobenius norm to constrain $\\delta$ . For neural networks, the outer “min” is non-convex, and the inner “max” is non-concave. Nonetheless, Madry et al. (2018) demonstrated that this saddle-point problem can be solved reliably with SGD for the outer minimization and PGD (a standard method for large-scale constrained optimization, see (Combettes & Pesquet, 2011) and (Goldstein et al., 2014)), for the inner maximization. In particular, for the constraint $\\| \\delta \\| _ { F } \\le \\epsilon$ , with an additional assumption that the loss function is locally linear, PGD takes the following step (with step size $\\alpha$ ) in each iteration: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/be4aa28dea2f09e48329453f621909356a76b82a9114c7d9674cce7e2f1707f6.jpg",
+        "text": "$$\n\\delta _ { t + 1 } = \\Pi _ { \\| \\pmb { \\delta } \\| _ { F } \\leq \\epsilon } \\left( \\pmb { \\delta } _ { t } + \\alpha g ( \\pmb { \\delta } _ { t } ) / \\| g ( \\pmb { \\delta } _ { t } ) \\| _ { F } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $g ( \\delta _ { t } ) = \\nabla _ { \\delta } L ( f _ { \\theta } ( X + \\delta _ { t } ) , y )$ is the gradient of the loss with respect to $\\delta$ , and $\\Pi _ { \\| \\delta \\| _ { F } \\leq \\epsilon }$ performs a projection onto the $\\epsilon$ -ball. To achieve high-level robustness, multi-step adversarial examples are needed during training, which is computationally expensive. The $K$ -step PGD ( $K$ -PGD) requires $K$ forward-backward passes through the network, while the standard SGD update requires only one. As a result, the adversary generation step in adversarial training increases run-time by an order of magnitude—a catastrophic amount when training large state-of-the-art language models. ",
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+    {
+        "type": "text",
+        "text": "3.2 LARGE-BATCH ADVERSARIAL TRAINING FOR FREE ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In the inner ascent steps of PGD, the gradients of the parameters can be obtained with almost no overhead when computing the gradients of the inputs. From this observation, FreeAT (Shafahi et al., ",
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+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 “Free” Large-Batch Adversarial Training (FreeLB- $K$ ) ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/faa4884339b2dc56b051cb475bba7c56913f2fcee10d81fc04db122bc7add815.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>ascent step size α</td><td>Require: Training samples X = {(Z,y)}, perturbation bound ε, learning rate T, ascent steps K,</td></tr><tr><td>1:Initialize 0</td><td></td></tr><tr><td>2: for epoch =1... Nep do</td><td></td></tr><tr><td>3:</td><td>for minibatchB ℃Xdo</td></tr><tr><td>4:</td><td>U(-∈,∈)</td></tr><tr><td>5:</td><td>90↑0</td></tr><tr><td>6:</td><td></td></tr><tr><td>7:</td><td>fort=1...K do</td></tr><tr><td>8:</td><td>Accumulate gradient of parameters θ</td></tr><tr><td>9:</td><td>gt←gt-1+kE(z,y)∈B[VθL(fe(X+δt-1),y)]</td></tr><tr><td>10:</td><td>Update the perturbation δ via gradient ascend</td></tr><tr><td>11:</td><td>gadu ←VsL(fθ(X+δt-1),y) δt ←II|δ|lr≤e(δt-1+α:gadu/llgadullF)</td></tr><tr><td>12:</td><td></td></tr><tr><td>13:</td><td>end for</td></tr><tr><td>14:</td><td>0←0-TgK</td></tr><tr><td>end for 15: end for</td><td></td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "2019) and YOPO (Zhang et al., 2019) have been proposed to accelerate adversarial training. They achieve comparable robustness and generalization as standard PGD-trained models using only the same or a slightly larger number of forward-backward passes as natural training (i.e., SGD on clean samples). FreeAT takes one descent step on the parameters together with each of the $K$ ascent steps on the perturbation. As a result, FreeAT may suffer from the “stale gradient” problem (Dutta et al., 2018), where in every step $t$ , $\\delta _ { t }$ does not necessarily maximize the model with parameter $\\theta _ { t }$ since its update is based on $\\nabla _ { \\delta } L \\big ( f _ { \\pmb { \\theta } _ { t - 1 } } ( \\pmb { X } + \\delta _ { t - 1 } ) , y \\big )$ , and vice versa, $\\theta _ { t }$ does not necessarily minimize the adversarial risk with adversary $\\delta _ { t }$ since its update is based on $\\nabla _ { \\pmb { \\theta } } L \\big ( f _ { \\pmb { \\theta } _ { t - 1 } } ( \\pmb { X } + \\delta _ { t - 1 } ) , y \\big )$ . Such a problem may be more significant when the step size is large. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Different from FreeAT, YOPO accumulates the gradient of the parameters from each of the ascent steps, and updates the parameters only once after the $K$ inner ascent steps. YOPO also advocates that after each back-propagation, one should take the gradient of the first hidden layer as a constant and perform several additional updates on the adversary using the product of this constant and the Jacobian of the first layer of the network to obtain strong adversaries. However, when the first hidden layer is a linear layer as in their implementation, such an operation is equivalent to taking a larger step size on the adversary. The analysis backing the extra update steps also assumes a twice continuously differentiable loss, which does not hold for ReLU-based neural networks they experimented with, and thus the reasons for the success of such an algorithm remains obscure. We give empirical comparisons between YOPO and our approach in Sec. 4.3. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To obtain better solutions for the inner max and avoid fundamental limitations on the function class, we propose FreeLB, which performs multiple PGD iterations to craft adversarial examples, and simultaneously accumulates the “free” parameter gradients $\\nabla _ { \\boldsymbol { \\theta } } L$ in each iteration. After that, it updates the model parameter $\\pmb \\theta$ all at once with the accumulated gradients. The overall procedure is shown in Algorithm 1, in which $X + \\delta _ { t }$ is an approximation to the local maximum within the intersection of two balls $\\mathcal { T } _ { t } = \\mathcal { B } _ { X + \\delta _ { 0 } } ( \\alpha t ) \\cap \\mathcal { B } _ { X } ( \\epsilon )$ . By taking a descent step along the averaged gradients at $X + \\delta _ { 0 } , . . . , X + \\delta _ { K - 1 }$ , we approximately optimize the following objective: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ccae4d8ef9e13199ab20f9f8a5afaca5aac1d2c3bfff645daf32d9544c002c1d.jpg",
+        "text": "$$\n\\underset { \\pmb { \\theta } } { \\operatorname* { m i n } } \\mathbb { E } _ { ( Z , y ) \\sim \\mathcal { D } } \\left[ \\frac { 1 } { K } \\sum _ { t = 0 } ^ { K - 1 } \\underset { \\delta _ { t } \\in \\mathcal { Z } _ { t } } { \\operatorname* { m a x } } L \\big ( f _ { \\pmb { \\theta } } \\big ( \\pmb { X } + \\delta _ { t } \\big ) , y \\big ) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "which is equivalent to replacing the original batch $\\boldsymbol { X }$ with a $K$ -times larger virtual batch, consisting of samples whose embeddings are $X + \\delta _ { 0 } , . . . , X + \\delta _ { K - 1 }$ . Compared with PGD-based adversarial training (Eq. 1), which minimizes the maximum risk at a single estimated point in the vicinity of each training sample, FreeLB minimizes the maximum risk at each ascent step at almost no overhead. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, FreeLB could be a learning method with lower generalization error than PGD. Sokolic et al. (2017) have proved that the generalization error of a learning method invariant to a set of√ $T$ transformations may be up to $\\sqrt { T }$ smaller than a non-invariant learning method. According to their theory, FreeLB could have a more significant improvement over natural training, since FreeLB enforces the invariance to $K$ adversaries from a set of up to $K$ different norm constraints,3 while PGD only enforces invariance to a single norm constraint $\\epsilon$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Empirically, FreeLB does lead to higher robustness and invariance than PGD in the embedding space, in the sense that the maximum increase of loss in the vicinity of $\\boldsymbol { X }$ for models trained with FreeLB is smaller than that with PGD. See Sec. 4.3 for details. In theory, such improved robustness can lead to better generalization (Xu & Mannor, 2012), which is consistent with our experiments. Qin et al. (2019) also demonstrated that PGD-based method leads to highly convolved and non-linear loss surfaces in the vicinity of input samples when $K$ is small, indicating a lack of robustness. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 WHEN ADVERSARIAL TRAINING MEETS DROPOUT ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Usually, adversarial training is not used together with dropout (Srivastava et al., 2014). However, for some language models like RoBERTa (Liu et al., 2019b), dropout is used during the finetuning stage. In practice, when dropout is turned on, each ascent step of Algorithm 1 is optimizing $\\delta$ for a different network. Specifically, denote the dropout mask as $_ { m }$ with each entry $m _ { i } \\sim \\mathrm { B e r n o u l l i } ( p )$ . Similar to our analysis for FreeAT, the ascent step from $\\delta _ { t - 1 }$ to $\\delta _ { t }$ is based on $\\nabla _ { \\delta } L \\big ( f _ { \\theta ( m _ { t - 1 } ) } \\big ( \\boldsymbol { X } + \\delta _ { t - 1 } \\big ) , \\boldsymbol { y } \\big )$ , so $\\delta _ { t }$ is sub-optimal for $L ( f _ { \\theta ( m _ { t } ) } ( \\boldsymbol { X } + \\delta ) , y )$ . Here $\\theta ( m )$ is the effective parameters under dropout mask $_ { m }$ . ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "The more plausible solution is to use the same $_ { \\mathbf { \\nabla } } \\mathbf { m }$ in each step. When applying dropout to any network, the objective for $\\pmb \\theta$ is to minimize the expectation of loss under different networks determined by the dropout masks, which is achieved by minimizing the Monte Carlo estimation of the expected loss. In our case, the objective becomes: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e662cc84309c58d4d57852d609bdc508f80860aaba1115045b618ae79744dc0a.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta } \\mathbb { E } _ { ( Z , y ) \\sim \\mathcal { D } , m \\sim \\mathcal { M } } \\left[ \\frac { 1 } { K } \\sum _ { t = 0 } ^ { K - 1 } \\operatorname* { m a x } _ { \\delta _ { t } \\in \\mathcal { T } _ { t } } L \\big ( f _ { \\theta ( m ) } ( X + \\delta _ { t } ) , y \\big ) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where the 1-sample Monte Carlo estimation should be $\\begin{array} { r } { \\frac { 1 } { K } \\sum _ { t = 0 } ^ { K - 1 } \\operatorname* { m a x } _ { \\delta _ { t } \\in \\mathcal { T } _ { t } } L ( f _ { \\theta ( m _ { 0 } ) } ( X + \\delta _ { t } ) , y ) } \\end{array}$ and can be minimized by using FreeLB with dropout mask $m _ { 0 }$ in each ascent step. This is similar to applying Variational Dropout to RNNs as used in Gal & Ghahramani (2016). ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide comprehensive analysis on FreeLB through extensive experiments on three Natural Language Understanding benchmarks: GLUE (Wang et al., 2019), ARC (Clark et al., 2018) and CommonsenseQA (Talmor et al., 2019). We also compare the robustness and generalization of FreeLB with other adversarial training algorithms to demonstrate its strength. Additional experimental details are provided in the Appendix. ",
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+    {
+        "type": "text",
+        "text": "4.1 DATASETS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "GLUE Benchmark. The GLUE benchmark is a collection of 9 natural language understanding tasks, namely Corpus of Linguistic Acceptability (CoLA; Warstadt et al. (2018)), Stanford Sentiment Treebank (SST; Socher et al. (2013)), Microsoft Research Paraphrase Corpus (MRPC; Dolan & Brockett (2005)), Semantic Textual Similarity Benchmark (STS; Agirre et al. (2007)), Quora Question Pairs (QQP; Iyer et al. (2017)), Multi-Genre NLI (MNLI; Williams et al. (2018)), Question NLI (QNLI; Rajpurkar et al. (2016)), Recognizing Textual Entailment (RTE; Dagan et al. (2006); Bar Haim et al. (2006); Giampiccolo et al. (2007); Bentivogli et al. (2009)) and Winograd NLI (WNLI; Levesque et al. (2011)). 8 of the tasks are formulated as classification problems and only STS-B is formulated as regression, but FreeLB applies to all of them. For BERT-base, we use the HuggingFace implementation4, and follow the single-task finetuning procedure as in Devlin et al. (2019). For RoBERTa, we use the fairseq implementation5. Same as Liu et al. (2019b), we also use single-task finetuning for all dev set results, and start with MNLI-finetuned models on RTE, MRPC and STS-B for the test submissions. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/e1f0b3873d4f551620e6abc0f84fd1bd6b832d5a4995eee5316a5213471483a2.jpg",
+        "table_caption": [
+            "Table 1: Results (median and variance) on the dev sets of GLUE based on the RoBERTa-large model, from 5 runs with the same hyperparameter but different random seeds. ReImp is our reimplementation of RoBERTalarge. The training process can be very unstable even with the vanilla version. Here, both PGD on STS-B and FreeAT on RTE demonstrates such instability, with one unconverged instance out of five. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>MNLI(Acc)</td><td rowspan=1 colspan=1>QNLI(Acc)</td><td rowspan=1 colspan=1>QQP(Acc)</td><td rowspan=1 colspan=1>RTE(Acc)</td><td rowspan=1 colspan=1>SST-2(Acc)</td><td rowspan=1 colspan=1>MRPC(Acc)</td><td rowspan=1 colspan=1>CoLA(Mcc)</td><td rowspan=1 colspan=1>STS-B(Pearson)</td></tr><tr><td rowspan=1 colspan=1>Reported</td><td rowspan=1 colspan=1>90.2</td><td rowspan=1 colspan=1>94.7</td><td rowspan=1 colspan=1>92.2</td><td rowspan=1 colspan=1>86.6</td><td rowspan=1 colspan=1>96.4</td><td rowspan=1 colspan=1>90.9</td><td rowspan=1 colspan=1>68.0</td><td rowspan=1 colspan=1>92.4</td></tr><tr><td rowspan=1 colspan=1>ReImp</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>85.61 (1.7)</td><td rowspan=1 colspan=1>96.56 (.3)</td><td rowspan=1 colspan=1>90.69(.5)</td><td rowspan=1 colspan=1>67.57 (1.3)</td><td rowspan=1 colspan=1>92.20 (.2)</td></tr><tr><td rowspan=1 colspan=1>PGD</td><td rowspan=1 colspan=1>90.53 (.2)</td><td rowspan=1 colspan=1>94.87 (.2)</td><td rowspan=1 colspan=1>92.49 (.07)</td><td rowspan=1 colspan=1>87.41 (.9)</td><td rowspan=1 colspan=1>96.44 (.1)</td><td rowspan=1 colspan=1>90.93 (.2)</td><td rowspan=1 colspan=1>69.67 (1.2)</td><td rowspan=1 colspan=1>92.43 (7.)</td></tr><tr><td rowspan=1 colspan=1>FreeAT</td><td rowspan=1 colspan=1>90.02 (.2)</td><td rowspan=1 colspan=1>94.66 (.2)</td><td rowspan=1 colspan=1>92.48(.08)</td><td rowspan=1 colspan=1>86.69 (15.)</td><td rowspan=1 colspan=1>96.10 (.2)</td><td rowspan=1 colspan=1>90.69 (.4)</td><td rowspan=1 colspan=1>68.80 (1.3)</td><td rowspan=1 colspan=1>92.40 (.3)</td></tr><tr><td rowspan=1 colspan=1>FreeLB</td><td rowspan=1 colspan=1>90.61 (.1)</td><td rowspan=1 colspan=1>94.98 (.2)</td><td rowspan=1 colspan=1>92.60 (.03)</td><td rowspan=1 colspan=1>88.13 (1.2)</td><td rowspan=1 colspan=1>96.79 (.2)</td><td rowspan=1 colspan=1>91.42 (.7)</td><td rowspan=1 colspan=1>71.12 (.9)</td><td rowspan=1 colspan=1>92.67 (.08)</td></tr></table>",
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/1ffd483862d6f9e2fd41f54a7e4c5ac87fd33bba048718a29e5bb6a776b93922.jpg",
+        "table_caption": [
+            "Table 2: Results on GLUE from the evaluation server, as of Sep 25, 2019. Metrics are the same as the leaderboard. Number under each task’s name is the size of the training set. FreeLB-BERT is the single-model results of BERT-base finetuned with FreeLB, and FreeLB-RoB is the ensemble of 7 RoBERTa-Large models for each task. References: 1: (Devlin et al., 2019); 2: (Liu et al., 2019a); 3: (Yang et al., 2019); 4: (Liu et al., 2019b). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Score</td><td rowspan=1 colspan=1>CoLA8.5k</td><td rowspan=1 colspan=1>SST-267k</td><td rowspan=1 colspan=1>MRPC3.7k</td><td rowspan=1 colspan=1>STS-B7k</td><td rowspan=1 colspan=1>QQP364k</td><td rowspan=1 colspan=1>MNLI-m/mm393k</td><td rowspan=1 colspan=1>QNLI108k</td><td rowspan=1 colspan=1>RTE2.5k</td><td rowspan=1 colspan=1>WNLI634</td><td rowspan=1 colspan=1>AX</td></tr><tr><td rowspan=1 colspan=1>BERT-base1</td><td rowspan=1 colspan=1>78.3</td><td rowspan=1 colspan=1>52.1</td><td rowspan=1 colspan=1>93.5</td><td rowspan=1 colspan=1>88.9/84.8</td><td rowspan=1 colspan=1>87.1/85.8</td><td rowspan=1 colspan=1>71.2/89.2</td><td rowspan=1 colspan=1>84.6/83.4</td><td rowspan=1 colspan=1>90.5</td><td rowspan=1 colspan=1>66.4</td><td rowspan=1 colspan=1>65.1</td><td rowspan=1 colspan=1>34.2</td></tr><tr><td rowspan=1 colspan=1>FreeLB-BERT</td><td rowspan=1 colspan=1>79.4</td><td rowspan=1 colspan=1>54.5</td><td rowspan=1 colspan=1>93.6</td><td rowspan=1 colspan=1>88.1/83.5</td><td rowspan=1 colspan=1>87.7/86.7</td><td rowspan=1 colspan=1>72.7/89.6</td><td rowspan=1 colspan=1>85.7/84.6</td><td rowspan=1 colspan=1>91.8</td><td rowspan=1 colspan=1>70.1</td><td rowspan=1 colspan=1>65.1</td><td rowspan=1 colspan=1>36.9</td></tr><tr><td rowspan=1 colspan=1>MT-DNN2</td><td rowspan=1 colspan=1>87.6</td><td rowspan=1 colspan=1>68.4</td><td rowspan=1 colspan=1>96.5</td><td rowspan=1 colspan=1>92.7/90.3</td><td rowspan=1 colspan=1>91.1/90.7</td><td rowspan=1 colspan=1>73.7789.9</td><td rowspan=1 colspan=1>87.9/87.4</td><td rowspan=1 colspan=1>96.0</td><td rowspan=1 colspan=1>86.3</td><td rowspan=1 colspan=1>89.0</td><td rowspan=1 colspan=1>42.8</td></tr><tr><td rowspan=1 colspan=1>XLNet-Large</td><td rowspan=1 colspan=1>88.4</td><td rowspan=1 colspan=1>67.8</td><td rowspan=1 colspan=1>96.8</td><td rowspan=1 colspan=1>93.0/90.7</td><td rowspan=1 colspan=1>91.6/91.1</td><td rowspan=1 colspan=1>74.2/90.3</td><td rowspan=1 colspan=1>90.2/89.8</td><td rowspan=1 colspan=1>98.6</td><td rowspan=1 colspan=1>86.3</td><td rowspan=1 colspan=1>90.4</td><td rowspan=1 colspan=1>47.5</td></tr><tr><td rowspan=1 colspan=1>RoBERTa4</td><td rowspan=1 colspan=1>88.5</td><td rowspan=1 colspan=1>67.8</td><td rowspan=1 colspan=1>96.7</td><td rowspan=1 colspan=1>92.3/89.8</td><td rowspan=1 colspan=1>92.2/91.9</td><td rowspan=1 colspan=1>74.3/90.2</td><td rowspan=1 colspan=1>90.8/90.2</td><td rowspan=1 colspan=1>98.9</td><td rowspan=1 colspan=1>88.2</td><td rowspan=1 colspan=1>89.0</td><td rowspan=1 colspan=1>48.7</td></tr><tr><td rowspan=1 colspan=1>FreeLB-RoB</td><td rowspan=1 colspan=1>88.8</td><td rowspan=1 colspan=1>68.0</td><td rowspan=1 colspan=1>96.8</td><td rowspan=1 colspan=1>93.1/90.8</td><td rowspan=1 colspan=1>92.4/92.2</td><td rowspan=1 colspan=1>74.8/90.3</td><td rowspan=1 colspan=1>91.1/90.7</td><td rowspan=1 colspan=1>98.8</td><td rowspan=1 colspan=1>88.7</td><td rowspan=1 colspan=1>89.0</td><td rowspan=1 colspan=1>50.1</td></tr><tr><td rowspan=1 colspan=1>Human</td><td rowspan=1 colspan=1>87.1</td><td rowspan=1 colspan=1>66.4</td><td rowspan=1 colspan=1>97.8</td><td rowspan=1 colspan=1>86.3/80.8</td><td rowspan=1 colspan=1>92.7/92.6</td><td rowspan=1 colspan=1>59.5/80.4</td><td rowspan=1 colspan=1>92.0/92.8</td><td rowspan=1 colspan=1>91.2</td><td rowspan=1 colspan=1>93.6</td><td rowspan=1 colspan=1>95.9</td><td rowspan=1 colspan=1>1</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "ARC Benchmark. The ARC dataset (Clark et al., 2018) is a collection of multi-choice science questions from grade-school level exams. It is further divided into ARC-Challenge set with 2,590 question answer (QA) pairs and ARC-Easy set with 5,197 QA pairs. Questions in ARC-Challenge are more difficult and cannot be handled by simply using a retrieval and co-occurence based algorithm (Clark et al., 2018). A typical question is: ",
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+    {
+        "type": "text",
+        "text": "Which property of a mineral can be determined just by looking at it? ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "(A) luster [correct] $( B )$ mass $( C )$ weight $( D )$ hardness. ",
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+    },
+    {
+        "type": "text",
+        "text": "CommonsenseQA Benchmark. The CommonsenseQA dataset (Talmor et al., 2019) consists of 12,102 natural language questions that require human commonsense reasoning ability to answer. A typical question is : ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Where can I stand on a river to see water falling without getting wet? ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "(A) waterfall, (B) bridge [correct], (C) valley, $( D )$ stream, (E) bottom. ",
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+    {
+        "type": "text",
+        "text": "Each question has five candidate answers from ConceptNet (Speer et al., 2017). To make the question more difficult to solve, most answers have the same relation in ConceptNet to the key concept in the question. As shown in the above example, most answers can be connected to “river” by “AtLocation” relation in ConceptNet. For a fair comparison with the reported results in papers and leaderboard6, we use the official random split 1.11. ",
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+    {
+        "type": "text",
+        "text": "4.2 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
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+        "text": "GLUE We summarize results on the dev sets of GLUE in Table 1, comparing the proposed FreeLB against other adversatial training algorithms (PGD (Madry et al., 2018) and FreeAT (Shafahi et al., 2019)). We use the same step size $\\alpha$ and number of steps $m$ for PGD, FreeAT and FreeLB. FreeLB is consistently better than the two baselines. Comparisons and detailed discussions about ",
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+        "img_path": "images/e0294ede408252845a3f452283bba1eeb68e7d5fc0d948c4421b775c75aff2ef.jpg",
+        "table_caption": [
+            "Table 3: Results on ARC and CommonsenseQA (CQA). ARC-Merge is the combination of ARC-Easy and ARC-Challenge, “MTL” stands for multi-task learning and “Ens” stands for ensemble. Results of XLNet $^ +$ RoBERTa $\\left( \\mathbf { M } \\mathbf { T } \\mathbf { L } { + } \\mathbf { E } \\mathbf { n } \\mathbf { s } \\right)$ and AristoRoBERTaV7 (MTL) are from the ARC leaderboards. Test (E) denotes the test set results with ensembles. For CQA, we report the highest dev and test accuracies among all models. The models with 78.81/72.19 dev/test accuracy (as in the table) have 71.84/78.64 test/dev accuracies respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=2 colspan=1></td><td rowspan=2 colspan=9>ARC-Easy    ARC-Challenge   ARC-Merge            CQADev   Test   Dev    Test    Dev   Test   Dev  Test Test (E)</td></tr><tr><td rowspan=1 colspan=1>Test</td><td rowspan=1 colspan=1>Dev</td><td rowspan=1 colspan=1>Test</td><td rowspan=1 colspan=1>Dev</td><td rowspan=1 colspan=2>Test   Dev</td><td rowspan=1 colspan=1>Test</td></tr><tr><td rowspan=1 colspan=1>RoBERTa(Reported)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>78.43</td><td rowspan=1 colspan=1>72.1</td><td rowspan=1 colspan=1>72.5</td></tr><tr><td rowspan=1 colspan=1>RoBERTa (ReImp)</td><td rowspan=1 colspan=1>84.39</td><td rowspan=1 colspan=1>84.13</td><td rowspan=1 colspan=1>64.54</td><td rowspan=1 colspan=1>64.44</td><td rowspan=1 colspan=1>77.83</td><td rowspan=1 colspan=1>77.62</td><td rowspan=1 colspan=1>77.56</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>FreeLB-RoBERTa</td><td rowspan=1 colspan=1>84.91</td><td rowspan=1 colspan=1>84.81</td><td rowspan=1 colspan=1>65.89</td><td rowspan=1 colspan=1>65.36</td><td rowspan=1 colspan=1>78.37</td><td rowspan=1 colspan=1>78.39</td><td rowspan=1 colspan=1>78.81</td><td rowspan=1 colspan=1>72.2</td><td rowspan=1 colspan=1>73.1</td></tr><tr><td rowspan=1 colspan=1>AristoRoBERTaV7(MTL)</td><td rowspan=1 colspan=1>:</td><td rowspan=1 colspan=1>85.02</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>66.47</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>78.89</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>:</td></tr><tr><td rowspan=1 colspan=1>XLNet +RoBERTa (MTL+Ens)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>67.06</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>二</td></tr><tr><td rowspan=1 colspan=1>FreeLB-RoBERTa(MTL)</td><td rowspan=1 colspan=1>84.91</td><td rowspan=1 colspan=1>85.44</td><td rowspan=1 colspan=1>70.23</td><td rowspan=1 colspan=1>67.75</td><td rowspan=1 colspan=1>79.86</td><td rowspan=1 colspan=1>79.60</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "YOPO (Zhang et al., 2019) are provided in Sec. 4.3. We have also submitted our results to the evaluation server, results provided in Table 2. FreeLB lifts the performance of the BERT-base model from 78.3 to 79.4, and RoBERTa-large model from 88.5 to 88.8 on overall scores. ",
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+    {
+        "type": "text",
+        "text": "ARC For ARC, a corpus of 14 million related science documents (from ARC Corpus, Wikipedia and other sources) is provided. For each QA pair, we first use a retrieval model to select top 10 related documents. Then, given these retrieved documents7, we use RoBERTa-large model to encode $\\langle s \\rangle$ Retrieved Documents $\\langle / s \\rangle$ Question $^ +$ Answer $\\langle / s \\rangle$ , where $\\langle \\mathbf { s } \\rangle$ and $\\langle / \\mathrm { s } \\rangle$ are special tokens for RoBERTa model8. We then apply a fully-connected layer to the representation of the [CLS] token to compute the final logit, and use standard cross-entropy loss for model training. ",
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+    {
+        "type": "text",
+        "text": "Results are summarized in Table 3. Following Sun et al. (2018), we first finetune the RoBERTa model on the RACE dataset (Lai et al., 2017). The finetuned RoBERTa model achieves $8 5 . 7 0 \\%$ and $8 5 . 2 4 \\%$ accuracy on the development and test set of RACE, respectively. Based on this, we further finetune the model on both ARC-Easy and ARC-Challenge datasets with the same hyper-parameter searching strategy (for 5 epochs), which achieves $8 4 . 1 3 \\% / 6 4 . 4 4 \\%$ test accuracy on ARC-Easy/ARCChallenge. And by adding FreeLB finetuning, we can reach $8 4 . 8 1 \\% / 6 5 . 3 6 \\%$ , a significant boost on ARC benchmark, demonstrating the effectiveness of FreeLB. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To further improve the results, we apply a multi-task learning (MTL) strategy using additional datasets. We first finetune the model on RACE (Lai et al., 2017), and then finetune on a joint dataset of ARC-Easy, ARC-Challenge, OpenbookQA (Mihaylov et al., 2018) and Regents Living Environment9. Based on this, we further finetune our model on ARC-Easy and ARC-Challenge with FreeLB. After finetuning, our single model achieves $6 7 . 7 5 \\%$ test accuracy on ARC-Challenge and $8 5 . 4 4 \\%$ on ARC-Easy, both outperforming the best submission on the official leaderboard10. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "CommonsenseQA Similar to the training strategy in Liu et al. (2019b), we construct five inputs for each question by concatenating the question and each answer separately, then encode each input with the representation of the [CLS] token. A final score is calculated by applying the representation of [CLS] to a fully-connected layer. Following the fairseq repository11, the input is formatted as: ${ \\mathfrak { n } } ( s ) Q$ : Where can I stand on a river to see water falling without getting wet? $\\langle / s \\rangle A$ : waterfall $\\langle \\langle s \\rangle ^ { \\ast } \\cdot \\mathbf { \\vec { \\mathbf { \\mu } } }$ , where $\\cdot _ { Q }$ :’ and $^ { , } A$ :’ are the prefix for question and answer, respectively. ",
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+    {
+        "type": "text",
+        "text": "Results are summarized in Table 3. We obtained a dev-set accuracy of $7 7 . 5 6 \\%$ with the RoBERTalarge model. When using FreeLB finetuning, we achieved $7 8 . 8 1 \\%$ , a $1 . 2 5 \\%$ absolute gain. Compared with the results reported from fairseq repository, which obtains $7 8 . 4 3 \\%$ accuracy on the devset, FreeLB still achieves better performance. Our submission to the CommonsenseQA leaderboard achieves $7 2 . 2 \\%$ single-model test set accuracy, and the result of a 20-model ensemble is $7 3 . 1 \\%$ , which achieves No.1 among all the submissions without making use of ConceptNet. ",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 4: The median and standard deviation of the scores on the dev sets of RTE, CoLA and MRPC from the GLUE benchmark, computed from 5 runs with the same hyper-parameters except for the random seeds. We use FreeLB- $\\mathbf { \\nabla } m$ to denote FreeLB with $m$ ascent steps, and FreeLB- ${ . 3 ^ { * } }$ to denote the version without reusing the dropout mask. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=1>Vanilla</td><td rowspan=1 colspan=1>FreeLB-3*</td><td rowspan=1 colspan=1>FreeLB-3</td><td rowspan=1 colspan=1>YOPO-3-2</td><td rowspan=1 colspan=1>YOPO-3-3</td></tr><tr><td rowspan=1 colspan=1>RTE</td><td rowspan=1 colspan=1>85.61 (1.67)</td><td rowspan=1 colspan=1>87.14 (1.29)</td><td rowspan=1 colspan=1>88.13 (1.21)</td><td rowspan=1 colspan=1>87.05 (1.36)</td><td rowspan=1 colspan=1>87.05 (0.20)</td></tr><tr><td rowspan=1 colspan=1>CoLA</td><td rowspan=1 colspan=1>67.57 (1.30)</td><td rowspan=1 colspan=1>69.31 (1.16)</td><td rowspan=1 colspan=1>71.12 (0.90)</td><td rowspan=1 colspan=1>70.40 (0.91)</td><td rowspan=1 colspan=1>69.91 (1.16)</td></tr><tr><td rowspan=1 colspan=1>MRPC</td><td rowspan=1 colspan=1>90.69 (0.54)</td><td rowspan=1 colspan=1>90.93 (0.66)</td><td rowspan=1 colspan=1>91.42 (0.72)</td><td rowspan=1 colspan=1>90.44 (0.62)</td><td rowspan=1 colspan=1>90.69 (0.37)</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/123f8ec948de592b6c31088d940779fcf701f6d633979bb44b4a997bb3feb667.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=3>RTE</td><td rowspan=1 colspan=3>CoLA</td><td rowspan=1 colspan=3>MRPC</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>M-Inc(10-4)</td><td rowspan=1 colspan=1>M-Inc (R)(10-4)</td><td rowspan=1 colspan=1>N-Loss(10-4)</td><td rowspan=1 colspan=1>M-Inc(10-4)</td><td rowspan=1 colspan=1>M-Inc (R)(10-4)</td><td rowspan=1 colspan=1>N-Loss(10-4)</td><td rowspan=1 colspan=1>M-Inc(10-3)</td><td rowspan=1 colspan=1>M-Inc (R)(10-3)</td><td rowspan=1 colspan=1>N-Loss(10-3)</td></tr><tr><td rowspan=1 colspan=1>Vanilla</td><td rowspan=1 colspan=1>5.1</td><td rowspan=1 colspan=1>5.3</td><td rowspan=1 colspan=1>4.5</td><td rowspan=1 colspan=1>6.1</td><td rowspan=1 colspan=1>5.7</td><td rowspan=1 colspan=1>5.2</td><td rowspan=1 colspan=1>10.2</td><td rowspan=1 colspan=1>10.2</td><td rowspan=1 colspan=1>1.9</td></tr><tr><td rowspan=1 colspan=1>PGD</td><td rowspan=1 colspan=1>4.7</td><td rowspan=1 colspan=1>4.9</td><td rowspan=1 colspan=1>6.2</td><td rowspan=1 colspan=1>128.2</td><td rowspan=1 colspan=1>130.1</td><td rowspan=1 colspan=1>436.1</td><td rowspan=1 colspan=1>5.7</td><td rowspan=1 colspan=1>5.7</td><td rowspan=1 colspan=1>5.4</td></tr><tr><td rowspan=1 colspan=1>FreeLB</td><td rowspan=1 colspan=1>3.0</td><td rowspan=1 colspan=1>2.6</td><td rowspan=1 colspan=1>4.1</td><td rowspan=1 colspan=1>1.4</td><td rowspan=1 colspan=1>1.3</td><td rowspan=1 colspan=1>7.2</td><td rowspan=1 colspan=1>3.6</td><td rowspan=1 colspan=1>3.6</td><td rowspan=1 colspan=1>2.7</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 5: Median of the maximum increase in loss in the vicinity of the dev set samples for RoBERTa-Large model finetuned with different methods. Vanilla models are naturally trained RoBERTa’s. M-Inc: Max Inc, MInc (R): Max Inc (R). Nat Loss (N-Loss) is the loss value on clean samples. Notice we require all clean samples here to be correctly classified by all models, which results in 227, 850 and 355 samples for RTE, CoLA and MRPC, respectively. We also give the variance in the Appendix. ",
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+    {
+        "type": "text",
+        "text": "4.3 ABLATION STUDY AND ANALYSIS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this sub-section, we first show the importance of reusing dropout mask, then conduct a thorough ablation study on FreeLB over the GLUE benchmark to analyze the robustness and generalization strength of different approaches. We observe that it is unnecessary to perform shallow-layer updates on the adversary as YOPO for our case, and FreeLB results in improved robustness and generalization compared with PGD. ",
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+    {
+        "type": "text",
+        "text": "Importance of Reusing Mask Table 4 (columns 2 to 4) compares the results of FreeLB with and without reusing the same dropout mask in each ascent step, as proposed in Sec. 3.3. With reusing, FreeLB can achieve a larger improvement over the naturally trained models. Thus, we enable mask reusing for all experiments involving RoBERTa. ",
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+    {
+        "type": "text",
+        "text": "Comparing the Robustness Table 5 provides the comparisons of the maximum increment of loss in the vicinity of each sample, defined as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/cf1340b64e7dea43e7ba8813f0bad358c01a42d13b806c2cf4c77f077ec893d9.jpg",
+        "text": "$$\n\\displaystyle \\Delta L _ { \\mathrm { m a x } } ( \\boldsymbol { X } , \\epsilon ) = \\operatorname* { m a x } _ { \\| \\delta \\| \\leq \\epsilon } L ( f _ { \\boldsymbol { \\theta } } ( \\boldsymbol { X } + \\delta ) , \\boldsymbol { y } ) - L ( f _ { \\boldsymbol { \\theta } } ( \\boldsymbol { X } ) , \\boldsymbol { y } ) ,\n$$",
+        "text_format": "latex",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "which reflects the robustness and invariance of the model in the embedding space. In practice, we use PGD steps as in Eq. 2 to find the value of ${ \\Delta } L _ { \\mathrm { m a x } } ( \\boldsymbol { X } , \\boldsymbol { \\epsilon } )$ . We found that when using a step size of $5 { \\cdot } 1 0 ^ { - 3 }$ and $\\epsilon = 0 . 0 1 \\| X \\| _ { F }$ , the PGD iterations converge to almost the same value, starting from 100 different random initializations of $\\pmb { \\delta }$ for the RoBERTa models, trained with or without FreeLB. This indicates that PGD reliably finds $\\Delta L _ { \\mathrm { m a x } }$ for these models. Therefore, we compute ${ \\Delta } L _ { \\mathrm { m a x } } ( X , \\epsilon )$ for each $\\boldsymbol { X }$ via a 2000-step PGD. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Samples with small margins exist even for models with perfect accuracy, which could give a false sense of vulnerability of the model. To rule out the outlier effect and make ${ \\Delta } L _ { \\mathrm { m a x } } ( \\boldsymbol { X } , \\epsilon )$ comparable across different samples, we only consider samples that all the evaluated models can correctly classify, and search for an $\\epsilon$ for each sample such that the reference model can correctly classify all samples within the $\\epsilon$ ball.12 However, such choice of per-sample $\\epsilon$ favors the reference model by design. To make fair comparisons, Table 5 provides the median of ${ \\Delta } L _ { \\mathrm { m a x } } ( \\boldsymbol { X } , \\epsilon )$ with per-sample $\\epsilon$ from models trained by FreeLB (Max Inc) and PGD (Mac Inc (R)), respectively. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Across all three datasets and different reference models, FreeLB has the smallest median increment even when starting from a larger natural loss than vanilla models. This demonstrates that FreeLB is more robust and invariant in most cases. Such results are also consistent with the models’ dev set performance (the performances for Vanilla/PGD/FreeLB models on RTE, CoLA and MRPC are 86.69/87.41/89.21, 69.91/70.84/71.40, 91.67/91.17/91.17, respectively). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Comparing with YOPO The original implementation of YOPO (Zhang et al., 2019) chooses the first convolutional layer of the ResNets as $f _ { 0 }$ for updating the adversary in the “s-loop”. As a result, each step of the “s-loop” should be using exactly the same value to update the adversary,and YOPO$m { - } n$ degenerates into FreeLB with a $n$ -times large step size. To avoid that, we choose the layers up to the output of the first Transformer block as $f _ { 0 }$ when implementing YOPO. To make the total amount of update on the adversary equal, we take the hyper-parameters for FreeLB- $m$ and only change the step size $\\alpha$ into $\\alpha / n$ for YOPO- $m { \\cdot } n$ . Table 4 shows that FreeLB performs consistently better than YOPO on all three datasets. Accidentally, we also give the results comparing with YOPO- $\\mathbf { \\nabla } m - n$ without changing the step size $\\alpha$ for YOPO in Table 8. The gap between two approaches seem to shrink, which may be caused by using a larger total step size for the YOPO adversaries. We leave exhaustive hyperparameter search for both models as our future work. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we have developed an adversarial training approach, FreeLB, to improve natural language understanding. The proposed approach adds perturbations to continuous word embeddings using a gradient method, and minimizes the resultant adversarial risk in an efficient way. FreeLB is able to boost Transformer-based model (BERT and RoBERTa) on several datasets and achieve new state of the art on GLUE and ARC benchmarks. Empirical study demonstrates that our method results in both higher robustness in the embedding space than natural training and better generalization ability. Such observation is also consistent with recent findings in Computer Vision. However, adversarial training still takes significant overhead compared with vanilla SGD. How to accelerate this process while improving generalization is an interesting future direction. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements: Goldstein and Zhu were supported in part by the DARPA GARD, DARPA QED for RML, and AFOSR MURI programs. ",
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+    },
+    {
+        "type": "text",
+        "text": "For tasks with ranking loss like ARC, CommonsenseQA, WNLI and QNLI, add the perturbation to the concatenation of the embeddings of all question/answer pairs. ",
+        "bbox": [
+            174,
+            160,
+            823,
+            188
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Additional tricks are required to achieve high performance on WNLI and QNLI for the GLUE benchmark. We use the same tricks as Liu et al. (2019b). For WNLI, we use the same WSC data provided by Liu et al. (2019b) for training. For testing, Liu et al. (2019b) also provided the test set with span annotations, but the order is different form the GLUE dataset. We re-order their test set by matching. For the QNLI, we follow Liu et al. (2019b) and formulate the problem as pairwise ranking problem, which is the same for CommonsenseQA. We find the matching pairs for both training set and testing set by matching the queries in the dev set. We predict “entailment” if the candidate has the higher score, and “not entailment” otherwise. ",
+        "bbox": [
+            173,
+            194,
+            825,
+            306
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 HYPER-PARAMETERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            324,
+            367,
+            338
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/4a6af4a2ca697bb9e27081b61d517c61fbe771acb7ab6a114f22e5ee01062b9f.jpg",
+        "table_caption": [
+            "Table 6: Additional hyper-parameters on GLUE tasks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>MNLI</td><td rowspan=1 colspan=1>QNLI</td><td rowspan=1 colspan=1>QQP</td><td rowspan=1 colspan=1>RTE</td><td rowspan=1 colspan=1>SST-2</td><td rowspan=1 colspan=1>MRPC</td><td rowspan=1 colspan=1>CoLA</td><td rowspan=1 colspan=1>STS-B</td></tr><tr><td rowspan=1 colspan=1>E</td><td rowspan=1 colspan=1>2E-1</td><td rowspan=1 colspan=1>1.5E-1</td><td rowspan=1 colspan=1>4.5E-1</td><td rowspan=1 colspan=1>1.5E-1</td><td rowspan=1 colspan=1>6E-1</td><td rowspan=1 colspan=1>4E-1</td><td rowspan=1 colspan=1>2E-1</td><td rowspan=1 colspan=1>3E-1</td></tr><tr><td rowspan=1 colspan=1>a</td><td rowspan=1 colspan=1>1E-1</td><td rowspan=1 colspan=1>1E-1</td><td rowspan=1 colspan=1>1.5E-1</td><td rowspan=1 colspan=1>3E-2</td><td rowspan=1 colspan=1>1E-1</td><td rowspan=1 colspan=1>4E-2</td><td rowspan=1 colspan=1>2.5E-2</td><td rowspan=1 colspan=1>1E-1</td></tr><tr><td rowspan=1 colspan=1>m</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>3</td></tr></table>",
+        "bbox": [
+            223,
+            354,
+            774,
+            416
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "As other adversarial training methods, introduces three additional hyper-parameters: step size $\\alpha$ , maximum perturbation $\\epsilon$ , number of steps $m$ . For all other hyper-parameters such as learning rate and number of iterations, we either search in the same interval as RoBERTa (on CommonsenseQA, ARC, and WNLI), or use exactly the same setting as RoBERTa (except for MRPC, where we find using a learning rate of $5 \\times 1 0 ^ { - 6 }$ gives better results).13. We list the best combinations for $\\alpha , \\epsilon$ and $m$ for each of the GLUE tasks in Table 6. For WSC/WNLI, the best combination is $\\epsilon = 1 e - 2 , \\alpha =$ $5 e - 3 , m = 2$ . Notice even when $m \\alpha < \\epsilon$ , the maximum perturbation could still reach $\\epsilon$ due to the random initialization. ",
+        "bbox": [
+            173,
+            459,
+            825,
+            571
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B VARIANCE OF MAXIMUM INCREMENT OF LOSS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            592,
+            604,
+            608
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/dc24cb509b23de841340a676916c10c0a5f0d23eba704bba7d0ecdae595f6dc7.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=3>RTE</td><td rowspan=1 colspan=3>CoLA</td><td rowspan=1 colspan=3>MRPC</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>M-Inc(10-4)</td><td rowspan=1 colspan=1>M-Inc (R)(10-4）</td><td rowspan=1 colspan=1>N-Loss(10-4)</td><td rowspan=1 colspan=1>M-Inc(10-4)</td><td rowspan=1 colspan=1>M-Inc (R)(10-4）</td><td rowspan=1 colspan=1>N-Loss(10-4)</td><td rowspan=1 colspan=1>M-Inc(10-3)</td><td rowspan=1 colspan=1>M-Inc (R)(10-3）</td><td rowspan=1 colspan=1>N-Loss(10-3)</td></tr><tr><td rowspan=1 colspan=1>Vanilla</td><td rowspan=1 colspan=1>5.1(15087)</td><td rowspan=1 colspan=1>5.3(14346)</td><td rowspan=1 colspan=1>4.5(920)</td><td rowspan=1 colspan=1>6.1(13118)</td><td rowspan=1 colspan=1>5.7(14122)</td><td rowspan=1 colspan=1>5.2(447)</td><td rowspan=1 colspan=1>10.2(929)</td><td rowspan=1 colspan=1>10.2(955)</td><td rowspan=1 colspan=1>1.9(76)</td></tr><tr><td rowspan=1 colspan=1>PGD</td><td rowspan=1 colspan=1>4.7(10138)</td><td rowspan=1 colspan=1>4.9(1828)</td><td rowspan=1 colspan=1>6.2(752)</td><td rowspan=1 colspan=1>128.2(1300)</td><td rowspan=1 colspan=1>130.1(893)</td><td rowspan=1 colspan=1>436.1(1276)</td><td rowspan=1 colspan=1>5.7(321)</td><td rowspan=1 colspan=1>5.7(155)</td><td rowspan=1 colspan=1>5.4(54)</td></tr><tr><td rowspan=1 colspan=1>FreeLB</td><td rowspan=1 colspan=1>3.0(1614)</td><td rowspan=1 colspan=1>2.6(11114)</td><td rowspan=1 colspan=1>4.1(595)</td><td rowspan=1 colspan=1>1.4(1392)</td><td rowspan=1 colspan=1>1.3(6231)</td><td rowspan=1 colspan=1>7.2(615)</td><td rowspan=1 colspan=1>3.6(167)</td><td rowspan=1 colspan=1>3.6(601)</td><td rowspan=1 colspan=1>2.7(54)</td></tr></table>",
+        "bbox": [
+            173,
+            627,
+            825,
+            695
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 7: Median and Standard Deviation of the maximum increase in loss in the vicinity of the dev set samples for RoBERTa-Large model finetuned with different methods. Vanilla models are naturally trained RoBERTa’s. Nat Loss is the loss value on clean samples. Notice we require all clean samples here to be correctly classified by all models, which results in 227, 850 and 355 samples for RTE, CoLA and MRPC. ",
+        "bbox": [
+            174,
+            707,
+            825,
+            757
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 7 provides the complete results for the increment of loss in the interval, with median and standard deviation. ",
+        "bbox": [
+            176,
+            776,
+            823,
+            804
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C ADDITIONAL RESULTS FOR ABLATION STUDIES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            825,
+            609,
+            842
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Here we provide some additional results for comparison with YOPO as complementary results to Table 4. We will release the complete results for comparing with YOPO and without variational dropout on each of the GLUE tasks in our next revision. From the current results, there is no need ",
+        "bbox": [
+            176,
+            857,
+            825,
+            898
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/8c04a7228cc8d5061e850a97d495da84f386812487797d0894db7899fc19501f.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=1>Vanilla</td><td rowspan=1 colspan=1>FreeLB-3</td><td rowspan=1 colspan=1>YOPO-3-2</td><td rowspan=1 colspan=1>YOPO-3-3</td></tr><tr><td rowspan=1 colspan=1>STS-B</td><td rowspan=1 colspan=1>92.20 (.2)</td><td rowspan=1 colspan=1>92.67 (.08)</td><td rowspan=1 colspan=1>92.60 (.17)</td><td rowspan=1 colspan=1>92.60 (0.20)</td></tr><tr><td rowspan=1 colspan=1>SST-2</td><td rowspan=1 colspan=1>96.56 (.3)</td><td rowspan=1 colspan=1>96.79 (.2)</td><td rowspan=1 colspan=1>96.44 (.2)</td><td rowspan=1 colspan=1>96.33 (.1)</td></tr><tr><td rowspan=1 colspan=1>QNLI</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>94.98 (.2)</td><td rowspan=1 colspan=1>94.96 (.1)</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>QQP</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>92.60 (.03)</td><td rowspan=1 colspan=1>92.55 (.05)*</td><td rowspan=1 colspan=1>92.50 (.02)*</td></tr><tr><td rowspan=1 colspan=1>MNLI</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>90.61 (.1)</td><td rowspan=1 colspan=1>90.59 (.2)</td><td rowspan=1 colspan=1>90.45 (.2)</td></tr></table>",
+        "bbox": [
+            285,
+            101,
+            709,
+            185
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table 8: The median and standard deviation of the scores on the dev sets of STS-B, SST-2, QNLI, QQP and MNLI from the GLUE benchmark, each computed from 5 runs with the same hyper-parameters except for the random seeds (except for the results with YOPO on QQP, which are from 4 runs). Also note here we use a step size of $\\alpha$ for the adversary of YOPO-m-n, so YOPO effectively uses a step size of $_ { n \\alpha }$ . We use FreeLB- $\\mathbf { \\nabla } m$ to denote FreeLB with $m$ ascent steps, and YOPO-3- $\\mathbf { \\nabla } \\cdot \\mathbf { n }$ to denote YOPO with $_ n$ shallow-layer ascents. ",
+        "bbox": [
+            173,
+            189,
+            826,
+            252
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "in using extra shallow-layer updates that YOPO advocates, since this consistently deteriorates the performance while introducing extra computations. ",
+        "bbox": [
+            173,
+            272,
+            823,
+            301
+        ],
+        "page_idx": 13
+    }
+]
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diff --git a/parse/train/BygzbyHFvB/BygzbyHFvB_middle.json b/parse/train/BygzbyHFvB/BygzbyHFvB_middle.json
new file mode 100644
index 0000000000000000000000000000000000000000..4811bda55f8fc9e74111ed9649403c4cda86eb33
--- /dev/null
+++ b/parse/train/BygzbyHFvB/BygzbyHFvB_middle.json
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+                                    "content": "is some loss function. We use the Frobenius",
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+                                    "score": 1.0,
+                                    "content": ". For neural networks, the outer “min” is non-convex, and the inner “max” is",
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+                                    "content": "is some loss function. We use the Frobenius",
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+                                    "score": 1.0,
+                                    "content": ". For neural networks, the outer “min” is non-convex, and the inner “max” is",
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+                                    "score": 1.0,
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+                                    "content": "since its update is based on",
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+                                    "content": "\\nabla _ { \\pmb { \\theta } } L \\big ( f _ { \\pmb { \\theta } _ { t - 1 } } ( \\pmb { X } + \\delta _ { t - 1 } ) , y \\big )",
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+                                    "content": ". Such",
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+                            "index": 11
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+                        {
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+                                    "score": 1.0,
+                                    "content": "a problem may be more significant when the step size is large.",
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+                                    "content": "Different from FreeAT, YOPO accumulates the gradient of the parameters from each of the ascent",
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+                                    "content": "inner ascent steps. YOPO also advocates",
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+                                    "score": 1.0,
+                                    "content": "that after each back-propagation, one should take the gradient of the first hidden layer as a constant",
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "and perform several additional updates on the adversary using the product of this constant and the",
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+                                    "score": 1.0,
+                                    "content": "Jacobian of the first layer of the network to obtain strong adversaries. However, when the first",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "hidden layer is a linear layer as in their implementation, such an operation is equivalent to taking",
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "a larger step size on the adversary. The analysis backing the extra update steps also assumes a",
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+                                    "score": 1.0,
+                                    "content": "twice continuously differentiable loss, which does not hold for ReLU-based neural networks they",
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "experimented with, and thus the reasons for the success of such an algorithm remains obscure. We",
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+                                    "score": 1.0,
+                                    "content": "give empirical comparisons between YOPO and our approach in Sec. 4.3.",
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+                                    "score": 1.0,
+                                    "content": "training (Eq. 1), which minimizes the maximum risk at a single estimated point in the vicinity of each",
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+                                    "score": 1.0,
+                                    "content": "et al. (2017) have proved that the generalization error of a learning method invariant to a set of√",
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+                                            "content": "{ . 3 ^ { * } }",
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+                                            "score": 1.0,
+                                            "content": "to denote the version without reusing the",
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+                                            "html": "<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=3>RTE</td><td rowspan=1 colspan=3>CoLA</td><td rowspan=1 colspan=3>MRPC</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>M-Inc(10-4)</td><td rowspan=1 colspan=1>M-Inc (R)(10-4)</td><td rowspan=1 colspan=1>N-Loss(10-4)</td><td rowspan=1 colspan=1>M-Inc(10-4)</td><td rowspan=1 colspan=1>M-Inc (R)(10-4)</td><td rowspan=1 colspan=1>N-Loss(10-4)</td><td rowspan=1 colspan=1>M-Inc(10-3)</td><td rowspan=1 colspan=1>M-Inc (R)(10-3)</td><td rowspan=1 colspan=1>N-Loss(10-3)</td></tr><tr><td rowspan=1 colspan=1>Vanilla</td><td rowspan=1 colspan=1>5.1</td><td rowspan=1 colspan=1>5.3</td><td rowspan=1 colspan=1>4.5</td><td rowspan=1 colspan=1>6.1</td><td rowspan=1 colspan=1>5.7</td><td rowspan=1 colspan=1>5.2</td><td rowspan=1 colspan=1>10.2</td><td rowspan=1 colspan=1>10.2</td><td rowspan=1 colspan=1>1.9</td></tr><tr><td rowspan=1 colspan=1>PGD</td><td rowspan=1 colspan=1>4.7</td><td rowspan=1 colspan=1>4.9</td><td rowspan=1 colspan=1>6.2</td><td rowspan=1 colspan=1>128.2</td><td rowspan=1 colspan=1>130.1</td><td rowspan=1 colspan=1>436.1</td><td rowspan=1 colspan=1>5.7</td><td rowspan=1 colspan=1>5.7</td><td rowspan=1 colspan=1>5.4</td></tr><tr><td rowspan=1 colspan=1>FreeLB</td><td rowspan=1 colspan=1>3.0</td><td rowspan=1 colspan=1>2.6</td><td rowspan=1 colspan=1>4.1</td><td rowspan=1 colspan=1>1.4</td><td rowspan=1 colspan=1>1.3</td><td rowspan=1 colspan=1>7.2</td><td rowspan=1 colspan=1>3.6</td><td rowspan=1 colspan=1>3.6</td><td rowspan=1 colspan=1>2.7</td></tr></table>",
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+                                    "content": "Table 5: Median of the maximum increase in loss in the vicinity of the dev set samples for RoBERTa-Large",
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+                                    "content": "on the adversary as YOPO for our case, and FreeLB results in improved robustness and generaliza-",
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+                                    "content": "Importance of Reusing Mask Table 4 (columns 2 to 4) compares the results of FreeLB with and",
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+                                    "content": "\\displaystyle \\Delta L _ { \\mathrm { m a x } } ( \\boldsymbol { X } , \\epsilon ) = \\operatorname* { m a x } _ { \\| \\delta \\| \\leq \\epsilon } L ( f _ { \\boldsymbol { \\theta } } ( \\boldsymbol { X } + \\delta ) , \\boldsymbol { y } ) - L ( f _ { \\boldsymbol { \\theta } } ( \\boldsymbol { X } ) , \\boldsymbol { y } ) ,",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "ball.12 However, such choice of per-sample",
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+                                    "score": 1.0,
+                                    "content": "favors the reference model by",
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+                                    "score": 1.0,
+                                    "content": "design. To make fair comparisons, Table 5 provides the median of",
+                                    "type": "text"
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+                                    "content": "{ \\Delta } L _ { \\mathrm { m a x } } ( \\boldsymbol { X } , \\epsilon )",
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+                            "index": 39
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+                        {
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+                                    "score": 1.0,
+                                    "content": "from models trained by FreeLB (Max Inc) and PGD (Mac Inc (R)), respectively.",
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+                    ],
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+                                    "content": "Across all three datasets and different reference models, FreeLB has the smallest median increment",
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+                                    "score": 1.0,
+                                    "content": "is more robust and invariant in most cases. Such results are also consistent with the models’ dev",
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+                                    "score": 1.0,
+                                    "content": "set performance (the performances for Vanilla/PGD/FreeLB models on RTE, CoLA and MRPC are",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "86.69/87.41/89.21, 69.91/70.84/71.40, 91.67/91.17/91.17, respectively).",
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+                                    "score": 1.0,
+                                    "content": "Comparing with YOPO The original implementation of YOPO (Zhang et al., 2019) chooses the",
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+                                    "score": 1.0,
+                                    "content": "for updating the adversary in the “s-loop”. As a result,",
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+                            ],
+                            "index": 1
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+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "each step of the “s-loop” should be using exactly the same value to update the adversary,and YOPO-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 2
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "degenerates into FreeLB with a",
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+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "-times large step size. To avoid that, we choose the layers up to",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 3
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                        139
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+                                    "score": 1.0,
+                                    "content": "the output of the first Transformer block as",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.88,
+                                    "content": "f _ { 0 }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "when implementing YOPO. To make the total amount",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 4
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "of update on the adversary equal, we take the hyper-parameters for FreeLB-",
+                                    "type": "text"
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+                                {
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+                                    "score": 0.75,
+                                    "content": "m",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "and only change the",
+                                    "type": "text"
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+                            ],
+                            "index": 5
+                        },
+                        {
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+                            "spans": [
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+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "step size",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.77,
+                                    "content": "\\alpha",
+                                    "type": "inline_equation"
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+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "into",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.91,
+                                    "content": "\\alpha / n",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "for YOPO-",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.32,
+                                    "content": "m { \\cdot } n",
+                                    "type": "inline_equation"
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ". Table 4 shows that FreeLB performs consistently better than",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 6
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "YOPO on all three datasets. Accidentally, we also give the results comparing with YOPO-",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.25,
+                                    "content": "\\mathbf { \\nabla } m - n",
+                                    "type": "inline_equation"
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "without changing the step size",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.76,
+                                    "content": "\\alpha",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "for YOPO in Table 8. The gap between two approaches seem to",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 8
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "shrink, which may be caused by using a larger total step size for the YOPO adversaries. We leave",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "exhaustive hyperparameter search for both models as our future work.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 10
+                        }
+                    ],
+                    "index": 5
+                },
+                {
+                    "type": "title",
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+                                    "score": 1.0,
+                                    "content": "5 CONCLUSION",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 11
+                        }
+                    ],
+                    "index": 11
+                },
+                {
+                    "type": "text",
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+                                    "score": 1.0,
+                                    "content": "In this work, we have developed an adversarial training approach, FreeLB, to improve natural lan-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 12
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "guage understanding. The proposed approach adds perturbations to continuous word embeddings",
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+                            ],
+                            "index": 13
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "using a gradient method, and minimizes the resultant adversarial risk in an efficient way. FreeLB",
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "is able to boost Transformer-based model (BERT and RoBERTa) on several datasets and achieve",
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+                                    "score": 1.0,
+                                    "content": "new state of the art on GLUE and ARC benchmarks. Empirical study demonstrates that our method",
+                                    "type": "text"
+                                }
+                            ],
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+                                    "score": 1.0,
+                                    "content": "-times large step size. To avoid that, we choose the layers up to",
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+                                    "score": 1.0,
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+                                    "score": 0.88,
+                                    "content": "f _ { 0 }",
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+                                    "score": 1.0,
+                                    "content": "when implementing YOPO. To make the total amount",
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+                            "index": 4
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+                                    "score": 1.0,
+                                    "content": "of update on the adversary equal, we take the hyper-parameters for FreeLB-",
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+                                    "score": 0.75,
+                                    "content": "m",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "and only change the",
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+                                    "score": 1.0,
+                                    "content": "step size",
+                                    "type": "text"
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+                                    "score": 0.77,
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+                                {
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+                                    "score": 1.0,
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+                                    "score": 0.91,
+                                    "content": "\\alpha / n",
+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": "for YOPO-",
+                                    "type": "text"
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+                                {
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+                                    "score": 0.32,
+                                    "content": "m { \\cdot } n",
+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": ". Table 4 shows that FreeLB performs consistently better than",
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+                                    "score": 1.0,
+                                    "content": "YOPO on all three datasets. Accidentally, we also give the results comparing with YOPO-",
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+                                    "score": 0.25,
+                                    "content": "\\mathbf { \\nabla } m - n",
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+                            "index": 7
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+                                    "score": 1.0,
+                                    "content": "without changing the step size",
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+                                    "score": 1.0,
+                                    "content": "for YOPO in Table 8. The gap between two approaches seem to",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 8
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "shrink, which may be caused by using a larger total step size for the YOPO adversaries. We leave",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 9
+                        },
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+                                    "score": 1.0,
+                                    "content": "exhaustive hyperparameter search for both models as our future work.",
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+                    ],
+                    "index": 5,
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+                                    "content": "5 CONCLUSION",
+                                    "type": "text"
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+                    ],
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+                                    "content": "using a gradient method, and minimizes the resultant adversarial risk in an efficient way. FreeLB",
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+                                    "score": 1.0,
+                                    "content": "new state of the art on GLUE and ARC benchmarks. Empirical study demonstrates that our method",
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+                            ],
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+                        },
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+                                    "content": "results in both higher robustness in the embedding space than natural training and better generaliza-",
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+                                    "score": 1.0,
+                                    "content": "tion ability. Such observation is also consistent with recent findings in Computer Vision. However,",
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+                            ],
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+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "adversarial training still takes significant overhead compared with vanilla SGD. How to accelerate",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
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+                            ],
+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "this process while improving generalization is an interesting future direction.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 20
+                        }
+                    ],
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+                                    "score": 1.0,
+                                    "content": "Acknowledgements: Goldstein and Zhu were supported in part by the DARPA GARD, DARPA",
+                                    "type": "text"
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+                                    "content": "REFERENCES",
+                                    "type": "text"
+                                }
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+                        }
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+                },
+                {
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+                                    "content": "Eneko Agirre, Llu’is M‘arquez, and Richard Wicentowski (eds.). Proceedings of the Fourth In-",
+                                    "type": "text"
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+                                    "content": "Idan Szpektor. The second PASCAL recognising textual entailment challenge. 2006.",
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+                                    "content": "fifth PASCAL recognizing textual entailment challenge. 2009.",
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+                                    "content": "benchmark. We use the same tricks as Liu et al. (2019b). For WNLI, we use the same WSC data",
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+                                    "content": "provided by Liu et al. (2019b) for training. For testing, Liu et al. (2019b) also provided the test set",
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+                                    "score": 1.0,
+                                    "content": "with span annotations, but the order is different form the GLUE dataset. We re-order their test set by",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "matching. For the QNLI, we follow Liu et al. (2019b) and formulate the problem as pairwise ranking",
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+                                    "score": 1.0,
+                                    "content": "problem, which is the same for CommonsenseQA. We find the matching pairs for both training set",
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+                                    "score": 1.0,
+                                    "content": "and testing set by matching the queries in the dev set. We predict “entailment” if the candidate has",
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+                                            "html": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>MNLI</td><td rowspan=1 colspan=1>QNLI</td><td rowspan=1 colspan=1>QQP</td><td rowspan=1 colspan=1>RTE</td><td rowspan=1 colspan=1>SST-2</td><td rowspan=1 colspan=1>MRPC</td><td rowspan=1 colspan=1>CoLA</td><td rowspan=1 colspan=1>STS-B</td></tr><tr><td rowspan=1 colspan=1>E</td><td rowspan=1 colspan=1>2E-1</td><td rowspan=1 colspan=1>1.5E-1</td><td rowspan=1 colspan=1>4.5E-1</td><td rowspan=1 colspan=1>1.5E-1</td><td rowspan=1 colspan=1>6E-1</td><td rowspan=1 colspan=1>4E-1</td><td rowspan=1 colspan=1>2E-1</td><td rowspan=1 colspan=1>3E-1</td></tr><tr><td rowspan=1 colspan=1>a</td><td rowspan=1 colspan=1>1E-1</td><td rowspan=1 colspan=1>1E-1</td><td rowspan=1 colspan=1>1.5E-1</td><td rowspan=1 colspan=1>3E-2</td><td rowspan=1 colspan=1>1E-1</td><td rowspan=1 colspan=1>4E-2</td><td rowspan=1 colspan=1>2.5E-2</td><td rowspan=1 colspan=1>1E-1</td></tr><tr><td rowspan=1 colspan=1>m</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>3</td></tr></table>",
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+                                    "score": 1.0,
+                                    "content": ", number of steps",
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+                                    "score": 1.0,
+                                    "content": ". For all other hyper-parameters such as learning rate",
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+                                    "score": 1.0,
+                                    "content": "and number of iterations, we either search in the same interval as RoBERTa (on CommonsenseQA,",
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+                                    "content": "5 \\times 1 0 ^ { - 6 }",
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+                                    "score": 1.0,
+                                    "content": "gives better results).13. We list the best combinations for",
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+                                    "score": 0.67,
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+                                    "score": 1.0,
+                                    "content": "for each of the GLUE tasks in Table 6. For WSC/WNLI, the best combination is",
+                                    "type": "text"
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+                                    "score": 0.89,
+                                    "content": "\\epsilon = 1 e - 2 , \\alpha =",
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+                                    "score": 0.82,
+                                    "content": "5 e - 3 , m = 2",
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ". Notice even when",
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+                                    "content": "m \\alpha < \\epsilon",
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+                                    "score": 1.0,
+                                    "content": "due to the",
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+                                    "content": "random initialization.",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": ". For all other hyper-parameters such as learning rate",
+                                    "type": "text"
+                                }
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+                            "index": 18
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "and number of iterations, we either search in the same interval as RoBERTa (on CommonsenseQA,",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "ARC, and WNLI), or use exactly the same setting as RoBERTa (except for MRPC, where we find",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "using a learning rate of",
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+                                    "score": 0.92,
+                                    "content": "5 \\times 1 0 ^ { - 6 }",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "gives better results).13. We list the best combinations for",
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+                                    "score": 0.76,
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+                                {
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+                                    "score": 1.0,
+                                    "content": "and",
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 0.67,
+                                    "content": "m",
+                                    "type": "inline_equation"
+                                },
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "for each of the GLUE tasks in Table 6. For WSC/WNLI, the best combination is",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.89,
+                                    "content": "\\epsilon = 1 e - 2 , \\alpha =",
+                                    "type": "inline_equation"
+                                }
+                            ],
+                            "index": 22
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 0.82,
+                                    "content": "5 e - 3 , m = 2",
+                                    "type": "inline_equation"
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ". Notice even when",
+                                    "type": "text"
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+                                    "score": 0.86,
+                                    "content": "m \\alpha < \\epsilon",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": ", the maximum perturbation could still reach",
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+                                    "content": "\\epsilon",
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+                                    "score": 1.0,
+                                    "content": "due to the",
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+                                    "score": 1.0,
+                                    "content": "random initialization.",
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+                                    "content": "B VARIANCE OF MAXIMUM INCREMENT OF LOSS",
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+                                    "content": "Table 7: Median and Standard Deviation of the maximum increase in loss in the vicinity of the dev set samples",
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+                                    "content": "for RoBERTa-Large model finetuned with different methods. Vanilla models are naturally trained RoBERTa’s.",
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+                                    "score": 1.0,
+                                    "content": "Nat Loss is the loss value on clean samples. Notice we require all clean samples here to be correctly classified",
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+                                    "score": 1.0,
+                                    "content": "by all models, which results in 227, 850 and 355 samples for RTE, CoLA and MRPC.",
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+                                    "content": "Table 7 provides the complete results for the increment of loss in the interval, with median and",
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+                                    "content": "C ADDITIONAL RESULTS FOR ABLATION STUDIES",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "Here we provide some additional results for comparison with YOPO as complementary results to",
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+                                    "score": 1.0,
+                                    "content": "Table 4. We will release the complete results for comparing with YOPO and without variational",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "dropout on each of the GLUE tasks in our next revision. From the current results, there is no need",
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+                                            "score": 0.978,
+                                            "html": "<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=1>Vanilla</td><td rowspan=1 colspan=1>FreeLB-3</td><td rowspan=1 colspan=1>YOPO-3-2</td><td rowspan=1 colspan=1>YOPO-3-3</td></tr><tr><td rowspan=1 colspan=1>STS-B</td><td rowspan=1 colspan=1>92.20 (.2)</td><td rowspan=1 colspan=1>92.67 (.08)</td><td rowspan=1 colspan=1>92.60 (.17)</td><td rowspan=1 colspan=1>92.60 (0.20)</td></tr><tr><td rowspan=1 colspan=1>SST-2</td><td rowspan=1 colspan=1>96.56 (.3)</td><td rowspan=1 colspan=1>96.79 (.2)</td><td rowspan=1 colspan=1>96.44 (.2)</td><td rowspan=1 colspan=1>96.33 (.1)</td></tr><tr><td rowspan=1 colspan=1>QNLI</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>94.98 (.2)</td><td rowspan=1 colspan=1>94.96 (.1)</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>QQP</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>92.60 (.03)</td><td rowspan=1 colspan=1>92.55 (.05)*</td><td rowspan=1 colspan=1>92.50 (.02)*</td></tr><tr><td rowspan=1 colspan=1>MNLI</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>90.61 (.1)</td><td rowspan=1 colspan=1>90.59 (.2)</td><td rowspan=1 colspan=1>90.45 (.2)</td></tr></table>",
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+                                    "score": 1.0,
+                                    "content": "random seeds (except for the results with YOPO on QQP, which are from 4 runs). Also note here we use a step",
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index 0000000000000000000000000000000000000000..11cd4fc9ff8e5b6a4b7b9e4e14837d603e7bbcd2
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@@ -0,0 +1,496 @@
+# DEEPCODER: LEARNING TO WRITE PROGRAMS
+
+Matej Balog∗ Department of Engineering University of Cambridge
+
+Alexander L. Gaunt, Marc Brockschmidt, Sebastian Nowozin, Daniel Tarlow Microsoft Research
+
+# ABSTRACT
+
+We develop a first line of attack for solving programming competition-style problems from input-output examples using deep learning. The approach is to train a neural network to predict properties of the program that generated the outputs from the inputs. We use the neural network’s predictions to augment search techniques from the programming languages community, including enumerative search and an SMT-based solver. Empirically, we show that our approach leads to an order of magnitude speedup over the strong non-augmented baselines and a Recurrent Neural Network approach, and that we are able to solve problems of difficulty comparable to the simplest problems on programming competition websites.
+
+# 1 INTRODUCTION
+
+A dream of artificial intelligence is to build systems that can write computer programs. Recently, there has been much interest in program-like neural network models (Graves et al., 2014; Weston et al., 2015; Kurach et al., 2015; Joulin & Mikolov, 2015; Grefenstette et al., 2015; Sukhbaatar et al., 2015; Neelakantan et al., 2016; Kaiser & Sutskever, 2016; Reed & de Freitas, 2016; Zaremba et al., 2016; Graves et al., 2016), but none of these can write programs; that is, they do not generate human-readable source code. Only very recently, Riedel et al. (2016); Bunel et al. (2016); Gaunt et al. (2016) explored the use of gradient descent to induce source code from input-output examples via differentiable interpreters, and Ling et al. (2016) explored the generation of source code from unstructured text descriptions. However, Gaunt et al. (2016) showed that differentiable interpreterbased program induction is inferior to discrete search-based techniques used by the programming languages community. We are then left with the question of how to make progress on program induction using machine learning techniques.
+
+In this work, we propose two main ideas: (1) learn to induce programs; that is, use a corpus of program induction problems to learn strategies that generalize across problems, and (2) integrate neural network architectures with search-based techniques rather than replace them.
+
+In more detail, we can contrast our approach to existing work on differentiable interpreters. In differentiable interpreters, the idea is to define a differentiable mapping from source code and inputs to outputs. After observing inputs and outputs, gradient descent can be used to search for a program that matches the input-output examples. This approach leverages gradient-based optimization, which has proven powerful for training neural networks, but each synthesis problem is still solved independently—solving many synthesis problems does not help to solve the next problem.
+
+We argue that machine learning can provide significant value towards solving Inductive Program Synthesis (IPS) by re-casting the problem as a big data problem. We show that training a neural network on a large number of generated IPS problems to predict cues from the problem description can help a search-based technique. In this work, we focus on predicting an order on the program space and show how to use it to guide search-based techniques that are common in the programming languages community. This approach has three desirable properties: first, we transform a difficult search problem into a supervised learning problem; second, we soften the effect of failures of the neural network by searching over program space rather than relying on a single prediction; and third, the neural network’s predictions are used to guide existing program synthesis systems, allowing us to use and improve on the best solvers from the programming languages community. Empirically, we show orders-of-magnitude improvements over optimized standard search techniques and a Recurrent Neural Network-based approach to the problem.
+
+In summary, we define and instantiate a framework for using deep learning for program synthesis problems like ones appearing on programming competition websites. Our concrete contributions are:
+
+1. defining a programming language that is expressive enough to include real-world programming problems while being high-level enough to be predictable from input-output examples; 2. models for mapping sets of input-output examples to program properties; and 3. experiments that show an order of magnitude speedup over standard program synthesis techniques, which makes this approach feasible for solving problems of similar difficulty as the simplest problems that appear on programming competition websites.
+
+# 2 BACKGROUND ON INDUCTIVE PROGRAM SYNTHESIS
+
+We begin by providing background on Inductive Program Synthesis, including a brief overview of how it is typically formulated and solved in the programming languages community.
+
+The Inductive Program Synthesis (IPS) problem is the following: given input-output examples, produce a program that has behavior consistent with the examples.
+
+Building an IPS system requires solving two problems. First, the search problem: to find consistent programs we need to search over a suitable set of possible programs. We need to define the set (i.e., the program space) and search procedure. Second, the ranking problem: if there are multiple programs consistent with the input-output examples, which one do we return? Both of these problems are dependent on the specifics of the problem formulation. Thus, the first important decision in formulating an approach to program synthesis is the choice of a Domain Specific Language.
+
+Domain Specific Languages (DSLs). DSLs are programming languages that are suitable for a specialized domain but are more restrictive than full-featured programming languages. For example, one might disallow loops or other control flow, and only allow string data types and a small number of primitive operations like concatenation. Most of program synthesis research focuses on synthesizing programs in DSLs, because full-featured languages like $\mathrm { C } { + + }$ enlarge the search space and complicate synthesis. Restricted DSLs can also enable more efficient special-purpose search algorithms. For example, if a DSL only allows concatenations of substrings of an input string, a dynamic programming algorithm can efficiently search over all possible programs (Polozov & Gulwani, 2015). The choice of DSL also affects the difficulty of the ranking problem. For example, in a DSL without if statements, the same algorithm is applied to all inputs, reducing the number of programs consistent with any set of input-output examples, and thus the ranking problem becomes easier. Of course, the restrictiveness of the chosen DSL also determines which problems the system can solve at all.
+
+Search Techniques. There are many techniques for searching for programs consistent with inputoutput examples. Perhaps the simplest approach is to define a grammar and then enumerate all derivations of the grammar, checking each one for consistency with the examples. This approach can be combined with pruning based on types and other logical reasoning (Feser et al., 2015). While simple, these approaches can be implemented efficiently, and they can be surprisingly effective.
+
+In restricted domains such as the concatenation example discussed above, special-purpose algorithms can be used. FlashMeta (Polozov & Gulwani, 2015) describes a framework for DSLs which allow decomposition of the search problem, e.g., where the production of an output string from an input string can be reduced to finding a program for producing the first part of the output and concatenating it with a program for producing the latter part of the output string.
+
+Another class of systems is based on Satisfiability Modulo Theories (SMT) solving. SMT combines SAT-style search with theories like arithmetic and inequalities, with the benefit that theory-dependent subproblems can be handled by special-purpose solvers. For example, a special-purpose solver can easily find integers $x , y$ such that $x < y$ and $y < - 1 0 0$ hold, whereas an enumeration strategy may need to consider many values before satisfying the constraints. Many program synthesis engines based on SMT solvers exist, e.g., Sketch (Solar-Lezama, 2008) and Brahma (Gulwani et al., 2011). They convert the semantics of a DSL into a set of constraints between variables representing the program and the input-output values, and then call an SMT solver to find a satisfying setting of the program variables. This approach shines when special-purpose reasoning can be leveraged, but complex DSLs can lead to very large constraint problems where constructing and manipulating the constraints can be a lot slower than an enumerative approach.
+
+Finally, stochastic local search can be employed to search over program space, and there is a long history of applying genetic algorithms to this problem. One of the most successful recent examples is the STOKE super-optimization system (Schkufza et al., 2016), which uses stochastic local search to find assembly programs that have the same semantics as an input program but execute faster.
+
+Ranking. While we focus on the search problem in this work, we briefly mention the ranking problem here. A popular choice for ranking is to choose the shortest program consistent with inputoutput examples (Gulwani, 2016). A more sophisticated approach is employed by FlashFill (Singh & Gulwani, 2015). It works in a manner similar to max-margin structured prediction, where known ground truth programs are given, and the learning task is to assign scores to programs such that the ground truth programs score higher than other programs that satisfy the input-output specification.
+
+# 3 LEARNING INDUCTIVE PROGRAM SYNTHESIS (LIPS)
+
+In this section we outline the general approach that we follow in this work, which we call Learning Inductive Program Synthesis (LIPS). The details of our instantiation of LIPS appear in Sect. 4. The components of LIPS are (1) a DSL specification, (2) a data-generation procedure, (3) a machine learning model that maps from input-output examples to program attributes, and (4) a search procedure that searches program space in an order guided by the model from (3). The framework is related to the formulation of Menon et al. (2013); the relationship and key differences are discussed in Sect. 6.
+
+(1) DSL and Attributes. The choice of DSL is important in LIPS, just as it is in any program synthesis system. It should be expressive enough to capture the problems that we wish to solve, but restricted as much as possible to limit the difficulty of the search. In LIPS we additionally specify an attribute function $\mathcal { A }$ that maps programs $P$ of the DSL to finite attribute vectors $\mathbf { \boldsymbol { a } } \overset { \cdot } { = } \mathbf { \bar { A } } ( \mathbf { \boldsymbol { P } } )$ . (Attribute vectors of different programs need not have equal length.) Attributes serve as the link between the machine learning and the search component of LIPS: the machine learning model predicts a distribution $q ( \pmb { a } \mid \mathcal { E } )$ , where $\mathcal { E }$ is the set of input-output examples, and the search procedure aims to search over programs $P$ as ordered by $q ( { \bar { \mathcal { A } } } ( { \bar { P } } ) \mid \varepsilon )$ . Thus an attribute is useful if it is both predictable from input-output examples, and if conditioning on its value significantly reduces the effective size of the search space.
+
+Possible attributes are the (perhaps position-dependent) presence or absence of high-level functions (e.g., does the program contain or end in a call to SORT). Other possible attributes include control flow templates (e.g., the number of loops and conditionals). In the extreme case, one may set $\mathcal { A }$ to the identity function, in which case the attribute is equivalent to the program; however, in our experiments we find that performance is improved by choosing a more abstract attribute function.
+
+(2) Data Generation. Step 2 is to generate a dataset $( ( P ^ { ( n ) } , \pmb { a } ^ { ( n ) } , \pmb { \mathcal { E } } ^ { ( n ) } ) ) _ { n = 1 } ^ { N }$ of programs $P ^ { ( n ) }$ in the chosen DSL, their attributes $\mathbf { \pmb { a } } ^ { ( n ) }$ , and accompanying input-output examples ${ \mathcal { E } } ^ { ( n ) }$ . Different approaches are possible, ranging from enumerating valid programs in the DSL and pruning, to training a more sophisticated generative model of programs in the DSL. The key in the LIPS formulation is to ensure that it is feasible to generate a large dataset (ideally millions of programs).
+
+(3) Machine Learning Model. The machine learning problem is to learn a distribution of attributes given input-output examples, $q ( \pmb { a } \mid \mathcal { E } )$ . There is freedom to explore a large space of models, so long as the input component can encode $\mathcal { E }$ , and the output is a proper distribution over attributes (e.g., if attributes are a fixed-size binary vector, then a neural network with independent sigmoid outputs is appropriate; if attributes are variable size, then a recurrent neural network output could be used). Attributes are observed at training time, so training can use a maximum likelihood objective.
+
+(4) Search. The aim of the search component is to interface with an existing solver, using the predicted $q ( \pmb { a } \mid \mathcal { E } )$ to guide the search. We describe specific approaches in the next section.
+
+# 4 DEEPCODER
+
+Here we describe DeepCoder, our instantiation of LIPS including a choice of DSL, a data generation strategy, models for encoding input-output sets, and algorithms for searching over program space.
+
+# 4.1 DOMAIN SPECIFIC LANGUAGE AND ATTRIBUTES
+
+We consider binary attributes indicating the presence or absence of high-level functions in the target program. To make this effective, the chosen DSL needs to contain constructs that are not so low-level that they all appear in the vast majority of programs, but at the same time should be common enough so that predicting their occurrence from input-output examples can be learned successfully.
+
+Following this observation, our DSL is loosely inspired by query languages such as SQL or LINQ, where high-level functions are used in sequence to manipulate data. A program in our DSL is a sequence of function calls, where the result of each call initializes a fresh variable that is either a singleton integer or an integer array. Functions can be applied to any of the inputs or previously computed (intermediate) variables. The output of the program is the return value of the last function call, i.e., the last variable. See Fig. 1 for an example program of length $T = 4$ in our DSL.
+
+![](images/95255e32da9df33db9a517e94978033a462b93ebbbe24b571c4b80d5e2c7f8e6.jpg)  
+Figure 1: An example program in our DSL that takes a single integer array as its input.
+
+Overall, our DSL contains the first-order functions HEAD, LAST, TAKE, DROP, ACCESS, MINIMUM, MAXIMUM, REVERSE, SORT, SUM, and the higher-order functions MAP, FILTER, COUNT, ZIPWITH, SCANL1. Higher-order functions require suitable lambda functions for their behavior to be fully specified: for MAP our DSL provides lambdas $( + 1 )$ , $( - 1 )$ , $( \star 2 )$ , $( / 2 )$ ), $( \star \ ( - 1 ) \ )$ ), $( \star \star 2 )$ , $( \star 3 )$ , $( / 3 )$ , $( \star 4 )$ , $( / 4 )$ ; for FILTER and COUNT there are predicates $( > 0 )$ ), $( < 0 )$ , ( $\scriptstyle { \frac { 0 } { 0 } } 2 = = 0$ ), ( $\scriptstyle { \frac { 0 } { 0 } } 2 = = 1$ ) and for ZIPWITH and SCANL1 the DSL provides lambdas $( + ) , ( - ) , ( \star ) ,$ MIN, MAX. A description of the semantics of all functions is provided in Appendix F.
+
+Note that while the language only allows linear control flow, many of its functions do perform branching and looping internally (e.g., SORT, COUNT, ...). Examples of more sophisticated programs expressible in our DSL, which were inspired by the simplest problems appearing on programming competition websites, are shown in Appendix A.
+
+# 4.2 DATA GENERATION
+
+To generate a dataset, we enumerate programs in the DSL, heuristically pruning away those with easily detectable issues such as a redundant variable whose value does not affect the program output, or, more generally, existence of a shorter equivalent program (equivalence can be overapproximated by identical behavior on randomly or carefully chosen inputs). To generate valid inputs for a program, we enforce a constraint on the output value bounding integers to some predetermined range, and then propagate these constraints backward through the program to obtain a range of valid values for each input. If one of these ranges is empty, we discard the program. Otherwise, input-output pairs can be generated by picking inputs from the pre-computed valid ranges and executing the program to obtain the output values. The binary attribute vectors are easily computed from the program source codes.
+
+# 4.3 MACHINE LEARNING MODEL
+
+Observe how the input-output data in Fig. 1 is informative of the functions appearing in the program: the values in the output are all negative, divisible by 4, they are sorted in decreasing order, and they happen to be multiples of numbers appearing in the input. Our aim is to learn to recognize such patterns in the input-output examples, and to leverage them to predict the presence or absence of individual functions. We employ neural networks to model and learn the mapping from input-output examples to attributes. We can think of these networks as consisting of two parts:
+
+1. an encoder: a differentiable mapping from a set of $M$ input-output examples generated by a single program to a latent real-valued vector, and 2. a decoder: a differentiable mapping from the latent vector representing a set of $M$ inputoutput examples to predictions of the ground truth program’s attributes.
+
+For the encoder we use a simple feed-forward architecture. First, we represent the input and output types (singleton or array) by a one-hot-encoding, and we pad the inputs and outputs to a maximum length $L$ with a special NULL value. Second, each integer in the inputs and in the output is mapped to a learned embedding vector of size $E = 2 0$ . (The range of integers is restricted to a finite range and each embedding is parametrized individually.) Third, for each input-output example separately, we concatenate the embeddings of the input types, the inputs, the output type, and the output into a single (fixed-length) vector, and pass this vector through $H = 3$ hidden layers containing $K = 2 5 6$ sigmoid units each. The third hidden layer thus provides an encoding of each individual input-output example. Finally, for input-output examples in a set generated from the same program, we pool these representations together by simple arithmetic averaging. See Appendix C for more details.
+
+The advantage of this encoder lies in its simplicity, and we found it reasonably easy to train. A disadvantage is that it requires an upper bound $L$ on the length of arrays appearing in the input and output. We confirmed that the chosen encoder architecture is sensible in that it performs empirically at least as well as an RNN encoder, a natural baseline, which may however be more difficult to train.
+
+DeepCoder learns to predict presence or absence of individual functions of the DSL. We shall see this can already be exploited by various search techniques to large computational gains. We use a decoder that pre-multiplies the encoding of input-output examples by a learned $C \times K$ matrix, where $C = 3 4$ is the number of functions in our DSL (higher-order functions and lambdas are predicted independently), and treats the resulting $C$ numbers as log-unnormalized probabilities (logits) of each function appearing in the source code. Fig. 2 shows the predictions a trained neural network made from 5 input-output examples for the program shown in Fig. 1.
+
+![](images/c3c080b2a8813d24f061fe69fe44372e58ba574c230d52185f026a848a7215ca.jpg)  
+Figure 2: Neural network predicts the probability of each function appearing in the source code.
+
+# 4.4 SEARCH
+
+One of the central ideas of this work is to use a neural network to guide the search for a program consistent with a set of input-output examples instead of directly predicting the entire source code. This section briefly describes the search techniques and how they integrate the predicted attributes.
+
+Depth-first search (DFS). We use an optimized version of DFS to search over programs with a given maximum length $T$ (see Appendix D for details). When the search procedure extends a partial program by a new function, it has to try the functions in the DSL in some order. At this point DFS can opt to consider the functions as ordered by their predicted probabilities from the neural network.
+
+“Sort and add” enumeration. A stronger way of utilizing the predicted probabilities of functions in an enumerative search procedure is to use a Sort and add scheme, which maintains a set of active functions and performs DFS with the active function set only. Whenever the search fails, the next most probable function (or several) are added to the active set and the search restarts with this larger active set. Note that this scheme has the deficiency of potentially re-exploring some parts of the search space several times, which could be avoided by a more sophisticated search procedure.
+
+Sketch. Sketch (Solar-Lezama, 2008) is a successful SMT-based program synthesis tool from the programming languages research community. While its main use case is to synthesize programs by filling in “holes” in incomplete source code so as to match specified requirements, it is flexible enough for our use case as well. The function in each step and its arguments can be treated as the “holes”, and the requirement to be satisfied is consistency with the provided set of input-output examples. Sketch can utilize the neural network predictions in a Sort and add scheme as described above, as the possibilities for each function hole can be restricted to the current active set.
+
+$\lambda ^ { 2 }$ . $\lambda ^ { 2 }$ (Feser et al., 2015) is a program synthesis tool from the programming languages community that combines enumerative search with deduction to prune the search space. It is designed to infer small functional programs for data structure manipulation from input-output examples, by combining functions from a provided library. $\lambda ^ { 2 }$ can be used in our framework using a Sort and add scheme as described above by choosing the library of functions according to the neural network predictions.
+
+# 4.5 TRAINING LOSS FUNCTION
+
+We use the negative cross entropy loss to train the neural network described in Sect. 4.3, so that its predictions about each function can be interpreted as marginal probabilities. The LIPS framework dictates learning $q ( \pmb { a } \mid \mathcal { E } )$ , the joint distribution of all attributes $\textbf { \em a }$ given the input-output examples, and it is not clear a priori how much DeepCoder loses by ignoring correlations between functions. However, under the simplifying assumption that the runtime of searching for a program of length $T$ with $C$ functions made available to a search routine is proportional to $\check { C } ^ { T }$ , the following result for Sort and add procedures shows that their runtime can be optimized using marginal probabilities.
+
+Lemma 1. For any fixed program length $T$ , the expected total runtime of a Sort and add search scheme can be upper bounded by a quantity that is minimized by adding the functions in the order of decreasing true marginal probabilities.
+
+Proof. Predicting source code functions from input-output examples can be seen as a multi-label classification problem, where each set of input-output examples is associated with a set of relevant labels (functions appearing in the ground truth source code). Dembczynski et al. (2010) showed that in multi-label classification under a so-called Rank loss, it is Bayes optimal to rank the labels according to their marginal probabilities. If the runtime of search with $C$ functions is proportional to $C ^ { T }$ , the total runtime of a Sort and add procedure can be monotonically transformed so that it is upper bounded by this Rank loss. See Appendix E for more details. □
+
+# 5 EXPERIMENTS
+
+In this section we report results from two categories of experiments. Our main experiments (Sect. 5.1) show that the LIPS framework can lead to significant performance gains in solving IPS by demonstrating such gains with DeepCoder. In Sect. 5.2 we illustrate the robustness of the method by demonstrating a strong kind of generalization ability across programs of different lengths.
+
+# 5.1 DEEPCODER COMPARED TO BASELINES
+
+We trained a neural network as described in Sect. 4.3 to predict used functions from input-output examples and constructed a test set of $P = 5 0 0$ programs, guaranteed to be semantically disjoint from all programs on which the neural network was trained (similarly to the equivalence check described in Sect. 4.2, we have ensured that all test programs behave differently from all programs used during training on at least one input). For each test program we generated $M = 5$ input-output examples involving integers of magnitudes up to 256, passed the examples to the trained neural network, and fed the obtained predictions to the search procedures from Sect. 4.4. We also considered a RNN-based decoder generating programs using beam search (see Sect. 5.3 for details). To evaluate DeepCoder, we then recorded the time the search procedures needed to find a program consistent with the $M$ input-output examples. As a baseline, we also ran all search procedures using a simple prior as function probabilities, computed from their global incidence in the program corpus.
+
+In the first, smaller-scale experiment (program search space size $\sim 2 \times 1 0 ^ { 6 }$ ) we trained the neural network on programs of length $T = 3$ , and the test programs were of the same length. Table 1 shows the per-task timeout required such that a solution could be found for given proportions of the test tasks (in time less than or equal to the timeout). For example, in a hypothetical test set with 4 tasks and runtimes of 3s, 2s, 1s, 4s, the timeout required to solve $50 \%$ of tasks would be 2s. More detailed experimental results are discussed in Appendix B.
+
+Table 1: Search speedups on programs of length $T = 3$ due to using neural network predictions.   
+
+<table><tr><td rowspan="2">Timeout needed to solve</td><td colspan="3">DFS</td><td colspan="3">Enumeration</td><td colspan="3">12</td><td colspan="2">Sketch</td><td>Beam</td></tr><tr><td>20%</td><td>40%</td><td>60%</td><td>20%</td><td>40%</td><td>60%</td><td>20%</td><td>40%</td><td>60%</td><td>20%</td><td>40%</td><td>20%</td></tr><tr><td>Baseline</td><td>41ms</td><td>126ms</td><td>314ms</td><td>80ms</td><td>：335ms861ms</td><td></td><td>18.9s</td><td>49.6s 84.2s</td><td></td><td>&gt;103s</td><td>&gt;103s</td><td>&gt;103s</td></tr><tr><td>DeepCoder</td><td>2.7ms</td><td>33ms</td><td>110ms</td><td>1.3ms</td><td>6.1ms</td><td>27ms</td><td>0.23s</td><td>0.52s 13.5s</td><td></td><td>2.13s</td><td>455s</td><td>292s</td></tr><tr><td>Speedup</td><td>15.2×</td><td>3.9×</td><td>2.9×</td><td>62.2×</td><td>54.6×</td><td>31.5×</td><td>80.4×</td><td>94.6×</td><td>6.2×</td><td>&gt;467×</td><td>&gt;2.2×</td><td>&gt;3.4×</td></tr></table>
+
+In the main experiment, we tackled a large-scale problem of searching for programs consistent with input-output examples generated from programs of length $T = 5$ (search space size on the order of $1 0 ^ { \overline { { 1 0 } } }$ ), supported by a neural network trained with programs of shorter length $T = 4$ . Here, we only consider $P = 1 0 0$ programs for reasons of computational efficiency, after having verified that this does not significantly affect the results in Table 1. The table in Fig. 3a shows significant speedups for DFS, Sort and add enumeration, and $\lambda ^ { 2 }$ with Sort and add enumeration, the search techniques capable of solving the search problem in reasonable time frames. Note that Sort and add enumeration without the neural network (using prior probabilities of functions) exceeded the $1 0 ^ { 4 }$ second timeout in two cases, so the relative speedups shown are crude lower bounds.
+
+![](images/a7cd9438bdca3d6b3e822af385015998ac105636d940628b2f68f443f81b05e3.jpg)  
+Figure 3: Search speedups on programs of length $T = 5$ and influence of length of training programs.
+
+We hypothesize that the substantially larger performance gains on Sort and add schemes as compared to gains on DFS can be explained by the fact that the choice of attribute function (predicting presence of functions anywhere in the program) and learning objective of the neural network are better matched to the Sort and add schemes. Indeed, a more appropriate attribute function for DFS would be one that is more informative of the functions appearing early in the program, since exploring an incorrect first function is costly with DFS. On the other hand, the discussion in Sect. 4.5 provides theoretical indication that ignoring the correlations between functions is not cataclysmic for Sort and add enumeration, since a Rank loss that upper bounds the Sort and add runtime can still be minimized.
+
+In Appendix G we analyse the performance of the neural networks used in these experiments, by investigating which attributes (program instructions) tend to be difficult to distinguish from each other.
+
+# 5.2 GENERALIZATION ACROSS PROGRAM LENGTHS
+
+To investigate the encoder’s generalization ability across programs of different lengths, we trained a network to predict used functions from input-output examples that were generated from programs of length $T _ { \mathrm { t r a i n } } \in \{ 1 , . . . , 4 \}$ . We then used each of these networks to predict functions on 5 test sets containing input-output examples generated from programs of lengths $T _ { \mathrm { t e s t } } \in \{ 1 , . . . , 5 \}$ , respectively. The test programs of a given length $T$ were semantically disjoint from all training programs of the same length $T$ and also from all training and test programs of shorter lengths $T ^ { \prime } < T$ .
+
+For each of the combinations of $T _ { \mathrm { t r a i n } }$ and $T _ { \mathrm { t e s t } }$ , Sort and add enumerative search was run both with and without using the neural network’s predictions (in the latter case using prior probabilities) until it solved $2 0 \%$ of the test set tasks. Fig. 3b shows the relative speedup of the solver having access to predictions from the trained neural networks. These results indicate that the neural networks are able to generalize beyond programs of the same length that they were trained on. This is partly due to the search procedure on top of their predictions, which has the opportunity to correct for the presence of functions that the neural network failed to predict. Note that a sequence-to-sequence model trained on programs of a fixed length could not be expected to exhibit this kind of generalization ability.
+
+# 5.3 ALTERNATIVE MODELS
+
+Encoder We evaluated replacing the feed-forward architecture encoder (Sect. 4.3) with an RNN, a natural baseline. Using a GRU-based RNN we were able to achieve results almost as good as using the feed-forward architecture, but found the RNN encoder more difficult to train.
+
+Decoder We also considered a purely neural network-based approach, where an RNN decoder is trained to predict the entire program token-by-token. We combined this with our feed-forward encoder by initializing the RNN using the pooled final layer of the encoder. We found it substantially more difficult to train an RNN decoder as compared to the independent binary classifiers employed above. Beam search was used to explore likely programs predicted by the RNN, but it only lead to a solution comparable with the other techniques when searching for programs of lengths $T \le 2$ , where the search space size is very small (on the order of $1 0 ^ { 3 }$ ). Note that using an RNN for both the encoder and decoder corresponds to a standard sequence-to-sequence model. However, we do do not rule out that a more sophisticated RNN decoder or training procedure could be possibly more successful.
+
+# 6 RELATED WORK
+
+Machine Learning for Inductive Program Synthesis. There is relatively little work on using machine learning for programming by example. The most closely related work is that of Menon et al. (2013), in which a hand-coded set of features of input-output examples are used as “clues.” When a clue appears in the input-output examples (e.g., the output is a permutation of the input), it reweights the probabilities of productions in a probabilistic context free grammar by a learned amount. This work shares the idea of learning to guide the search over program space conditional on input-output examples. One difference is in the domains. Menon et al. (2013) operate on short string manipulation programs, where it is arguably easier to hand-code features to recognize patterns in the input-output examples (e.g., if the outputs are always permutations or substrings of the input). Our work shows that there are strong cues in patterns in input-output examples in the domain of numbers and lists. However, the main difference is the scale. Menon et al. (2013) learns from a small (280 examples), manually-constructed dataset, which limits the capacity of the machine learning model that can be trained. Thus, it forces the machine learning component to be relatively simple. Indeed, Menon et al. (2013) use a log-linear model and rely on hand-constructed features. LIPS automatically generates training data, which yields datasets with millions of programs and enables high-capacity deep learning models to be brought to bear on the problem.
+
+Learning Representations of Program State. Piech et al. (2015) propose to learn joint embeddings of program states and programs to automatically extend teacher feedback to many similar programs in the MOOC setting. This work is similar in that it considers embedding program states, but the domain is different, and it otherwise specifically focuses on syntactic differences between semantically equivalent programs to provide stylistic feedback. Li et al. (2016) use graph neural networks (GNNs) to predict logical descriptions from program states, focusing on data structure shapes instead of numerical and list data. Such GNNs may be a suitable architecture to encode states appearing when extending our DSL to handle more complex data structures.
+
+Learning to Infer. Very recently, Alemi et al. (2016) used neural sequence models in tandem with an automated theorem prover. Similar to our Sort and Add strategy, a neural network component is trained to select premises that the theorem prover can use to prove a theorem. A recent extension (Loos et al., 2017) is similar to our DFS enumeration strategy and uses a neural network to guide the proof search at intermediate steps. The main differences are in the domains, and that they train on an existing corpus of theorems. More broadly, if we view a DSL as defining a model and search as a form of inference algorithm, then there is a large body of work on using discriminatively-trained models to aid inference in generative models. Examples include Dayan et al. (1995); Kingma & Welling (2014); Shotton et al. (2013); Stuhlmuller et al. (2013); Heess et al. (2013); Jampani et al. ¨ (2015).
+
+# 7 DISCUSSION AND FUTURE WORK
+
+We have presented a framework for improving IPS systems by using neural networks to translate cues in input-output examples to guidance over where to search in program space. Our empirical results show that for many programs, this technique improves the runtime of a wide range of IPS baselines by 1-3 orders. We have found several problems in real online programming challenges that can be solved with a program in our language, which validates the relevance of the class of problems that we have studied in this work. In sum, this suggests that we have made significant progress towards being able to solve programming competition problems, and the machine learning component plays an important role in making it tractable.
+
+There remain some limitations, however. First, the programs we can synthesize are only the simplest problems on programming competition websites and are simpler than most competition problems. Many problems require more complex algorithmic solutions like dynamic programming and search, which are currently beyond our reach. Our chosen DSL currently cannot express solutions to many problems. To do so, it would need to be extended by adding more primitives and allow for more flexibility in program constructs (such as allowing loops). Second, we currently use five input-output examples with relatively large integer values (up to 256 in magnitude), which are probably more informative than typical (smaller) examples. While we remain optimistic about LIPS’s applicability as the DSL becomes more complex and the input-output examples become less informative, it remains to be seen what the magnitude of these effects are as we move towards solving large subsets of programming competition problems.
+
+We foresee many extensions of DeepCoder. We are most interested in better data generation procedures by using generative models of source code, and to incorporate natural language problem descriptions to lessen the information burden required from input-output examples. In sum, DeepCoder represents a promising direction forward, and we are optimistic about the future prospects of using machine learning to synthesize programs.
+
+# ACKNOWLEDGMENTS
+
+The authors would like to express their gratitude to Rishabh Singh and Jack Feser for their valuable guidance and help on using the Sketch and $\lambda ^ { 2 }$ program synthesis systems.
+
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+
+# A EXAMPLE PROGRAMS
+
+This section shows example programs in our Domain Specific Language (DSL), together with inputoutput examples and short descriptions. These programs have been inspired by simple tasks appearing on real programming competition websites, and are meant to illustrate the expressive power of our DSL.
+
+# Program 0:
+
+$\mathrm { k } $ int b ← [int] c ← SORT b d ← TAKE k c e ← SUM d
+
+Input-output example:   
+Input:   
+2, [3 5 4 7 5]   
+Output:   
+[7]
+
+Description:
+
+A new shop near you is selling $n$ paintings. You have $k \ < \ n$ friends and you would like to buy each of your friends a painting from the shop. Return the minimal amount of money you will need to spend.
+
+# Program 1:
+
+$\bar { w } $ [int]   
+t ← [int]   
+c ← MAP (\*3) w   
+d ← ZIPWITH (+) c t   
+e ← MAXIMUM d   
+Input-output example:   
+Input:   
+[6 2 4 7 9],   
+[5 3 6 1 0]   
+Output:   
+27
+
+Description:
+
+In soccer leagues, match winners are awarded 3 points, losers 0 points, and both teams get 1 point in the case of a tie. Compute the number of points awarded to the winner of a league given two arrays $w , t$ of the same length, where $w [ i ]$ (resp. $t [ i ] ,$ ) is the number of times team $i$ won (resp. tied).
+
+# Program 2:
+
+a ← [int]   
+b ← [int]   
+c ← ZIPWITH (-) b a $\mathsf { d } \gets \mathbf { C o u v r } \mathsf { T }$ $( > 0 )$ ) c
+
+# Input-output example:
+
+Input:   
+[6 2 4 7 9],   
+[5 3 2 1 0]   
+Output:   
+4
+
+Description:
+
+Alice and Bob are comparing their results in a recent exam. Given their marks per question as two arrays $a$ and $b$ , count on how many questions Alice got more points than Bob.
+
+# Program 3:
+
+h ← [int]   
+b ← SCANL1 MIN h   
+c ← ZIPWITH (-) h b $\mathrm { ~ d ~ } $ FILTER $( > 0 )$ ) c   
+$\mathrm { ~ e ~ } { \gets } \mathbf { S } \mathbf { U } \mathbf { M } \mathrm { ~ d ~ }$   
+Input-output example:   
+Input:   
+[8 5 7 2 5]   
+Output:   
+5
+
+Description:
+
+Perditia is very peculiar about her garden and wants that the trees standing in a row are all of non-increasing heights. Given the tree heights in centimeters in order of the row as an array h, compute how many centimeters she needs to trim the trees in total.
+
+# Program 4:
+
+x ← [int]   
+y ← [int]   
+$\mathbf { \boldsymbol { \mathrm { c } } } \gets \mathbf { \boldsymbol { S } } \mathbf { \boldsymbol { O } } \mathbf { R } \mathbf { \boldsymbol { T } } \ \mathbf { \boldsymbol { x } }$   
+d ← SORT y   
+e ← REVERSE d   
+f ← ZIPWITH (\*) d e   
+$\mathsf { g } \gets \mathsf { S U M } \mathrm { ~ f ~ }$   
+Input-output example:   
+Input:   
+[7 3 8 2 5],   
+[2 8 9 1 3]   
+Output:   
+79
+
+Description:
+
+Xavier and Yasmine are laying sticks to form non-overlapping rectangles on the ground. They both have fixed sets of pairs of sticks of certain lengths (represented as arrays x and y of numbers). Xavier only lays sticks parallel to the x axis, and Yasmine lays sticks only parallel to y axis. Compute the area their rectangles will cover at least.
+
+Program 5:   
+$\ a \gets$ [int]   
+$\ u \text b \gets$ REVERSE a   
+$\subset $ ZIPWITH MIN a b   
+Input-output example:   
+Input:   
+[3 7 5 2 8]   
+Output:   
+[3 2 5 2 3]
+
+# Description:
+
+A sequence called Billy is looking into the mirror, wondering how much weight it could lose by replacing any of its elements by their mirror images. Given a description of Billy as an array $b$ of length $n$ , return an array $c$ of minimal sum where each element $c [ i ]$ is either $b [ i ]$ or its mirror image $b [ n - i - 1 ]$ .
+
+<table><tr><td colspan="2"></td></tr><tr><td>Program 6: IO example: t←[int] Input: p←[int] [4 8 11 2], C←MAP（-1）t [2341] d←MAP（-1）p Output: e ←ZIPWITH（+）c d 1 f←MINIMUMe</td><td>Description: Umberto has a large collection of ties and match- ing pocket squares—too large, his wife says—and he needs to sell one pair. Given their values as arrays t and p,assuming that he sells the cheapest pair, and selling costs 2,how much will he lose from the sale?</td></tr><tr><td>Program 7: IO example: s←[int] Input: p←[int] [4723], C ← SCANL1（+）p [2131] d ←ZIPWITH（*）S C Output: e←SUMd 62</td><td>Description: Zack always promised his n friends to buy them candy, but never did. Now he won the lottery and counts how often and how much candy he promised to his friends,obtaining arrays p (num- ber of promises) and s (number of promised sweets). He announces that to repay them,he will buy s[1]+s[2]+...+s[n] pieces of candy for the firstp[1]days,thens[2]+s[3]+...+s[n]for p[2] days,and so on,until he has fulfilled all promises. How much candy will he buy in total?</td></tr><tr><td>Program 8: IO example: s←[int] Input: b ←REVERSE S [12457] C ←ZIPWITH（-）b s Output: d ←FILTER（&gt;O）C 9 e←SuMd</td><td>Description: Vivian loves rearranging things.Most of all,when she sees a row of heaps, she wants to make sure that each heap has more items than the one to its left. She is also obsessed with efficiency, so always moves the least possible number of items.Her dad really dislikes if she changes the size of heaps,so she only moves single items between them,making sure that the set of sizes of the heaps is the same as at the start; they are only in a different order. When you come in, you see heaps of sizes (of course,sizes strictly monotonically increasing）s[O]，s[1]，... s[n].What is the maximal number of items that Vivian could have moved?</td></tr></table>
+
+![](images/6d29878119ef08e20bad2e228bb9a4b1140e7a88251e4c13cbe46fb4f2478484.jpg)  
+Fig. 4 shows the predictions made by a neural network trained on programs of length $T = 4$ that were ensured to be semantically disjoint from all 9 example programs shown in this section. For each task, the neural network was provided with 5 input-output examples.   
+Figure 4: Predictions of a neural network on the 9 example programs described in this section. Numbers in squares would ideally be close to 1 (function is present in the ground truth source code), whereas all other numbers should ideally be close to 0 (function is not needed).
+
+# B EXPERIMENTAL RESULTS
+
+Results presented in Sect. 5.1 showcased the computational speedups obtained from the LIPS framework (using DeepCoder), as opposed to solving each program synthesis problem with only the information about global incidence of functions in source code available. For completeness, here we show plots of raw computation times of each search procedure to solve a given number of problems.
+
+Fig. 5 shows the computation times of DFS, of Enumerative search with a Sort and add scheme, of the $\lambda ^ { 2 }$ and Sketch solvers with a Sort and add scheme, and of Beam search, when searching for a program consistent with input-output examples generated from $P = 5 0 0$ different test programs of length $T = 3$ . As discussed in Sect. 5.1, these test programs were ensured to be semantically disjoint from all programs used to train the neural networks, as well as from all programs of shorter length (as discussed in Sect. 4.2).
+
+![](images/69617f5d9c5e8b7b5a7be3c5b88b0456a2865590e84e41413e9067abe2f41f17.jpg)  
+Figure 5: Number of test problems solved versus computation time.
+
+The “steps” in the results for Beam search are due to our search strategy, which doubles the size of the considered beam until reaching the timeout (of 1000 seconds) and thus steps occur whenever the search for a beam of size $2 ^ { k }$ is finished. For $\dot { \lambda ^ { 2 } }$ , we observed that no solution for a given set of allowed functions was ever found after about 5 seconds (on the benchmark machines), but that $\lambda ^ { 2 }$ continued to search. Hence, we introduced a hard timeout after 6 seconds for all but the last iterations of our Sort and add scheme.
+
+Fig. 6 shows the computation times of DFS, Enumerative search with a Sort and add scheme, and $\lambda ^ { 2 }$ with a Sort and add scheme when searching for programs consistent with input-output examples generated from $P = 1 0 0$ different test programs of length $T = 5$ . The neural network was trained on programs of length $T = 4$ .
+
+![](images/8d83b6177f99db604465ccbe078168c96645752256eb9f14da14c4754aa158fd.jpg)  
+Figure 6: Number of test problems solved versus computation time.
+
+# C THE NEURAL NETWORK
+
+As briefly described in Sect. 4.3, we used the following simple feed-forward architecture encoder:
+
+• For each input-output example in the set generated from a single ground truth program:
+
+– Pad arrays appearing in the inputs and in the output to a maximum length $L = 2 0$ with a special NULL value.   
+– Represent the type (singleton integer or integer array) of each input and of the output using a one-hot-encoding vector. Embed each integer in the valid integer range $( - 2 5 6$ to 255) using a learned embedding into $E = 2 0$ dimensional space. Also learn an embedding for the padding NULL value.
+
+– Concatenate the representations of the input types, the embeddings of integers in the inputs, the representation of the output type, and the embeddings of integers in the output into a single (fixed-length) vector. – Pass this vector through $H = 3$ hidden layers containing $K = 2 5 6$ sigmoid units each.
+
+• Pool the last hidden layer encodings of each input-output example together by simple arithmetic averaging.
+
+Fig. 7 shows a schematic drawing of this encoder architecture, together with the decoder that performs independent binary classification for each function in the DSL, indicating whether or not it appears in the ground truth source code.
+
+![](images/1b4ae4043ae84d4ac119b02713a27cb85173d4a2d0a85116dc32199039ab2623.jpg)  
+Figure 7: Schematic representation of our feed-forward encoder, and the decoder.
+
+While DeepCoder learns to embed integers into a $E = 2 0$ dimensional space, we built the system up gradually, starting with a $E = 2$ dimensional space and only training on programs of length $T = 1$ . Such a small scale setting allowed easier investigation of the workings of the neural network, and indeed Fig. 8 below shows a learned embedding of integers in $\mathbb { R } ^ { 2 }$ . The figure demonstrates that the network has learnt the concepts of number magnitude, sign (positive or negative) and evenness, presumably due to FILTER $( > 0 )$ , FILTER $( < 0 )$ , FILTER ( $\scriptstyle { \frac { 0 } { 0 } } 2 = = 0$ ) and FILTER ( $\scriptstyle { \frac { 0 } { 0 } } 2 = = 1$ ) all being among the programs on which the network was trained.
+
+# D DEPTH-FIRST SEARCH
+
+We use an optimized $\mathrm { C } { + } { + }$ implementation of depth-first search (DFS) to search over programs with a given maximum length $T$ . In depth-first search, we start by choosing the first function (and its arguments) of a potential solution program, and then recursively consider all ways of filling in the rest of the program (up to length $T$ ), before moving on to a next choice of first instruction (if a solution has not yet been found).
+
+A program is considered a solution if it is consistent with all $M = 5$ provided input-output examples. Note that this requires evaluating all candidate programs on the $M$ inputs and checking the results for equality with the provided $M$ respective outputs. Our implementation of DFS exploits the sequential structure of programs in our DSL by caching the results of evaluating all prefixes of the currently considered program on the example inputs, thus allowing efficient reuse of computation between candidate programs with common prefixes.
+
+This allows us to explore the search space at roughly the speed of $\sim 3 \times 1 0 ^ { 6 }$ programs per second.
+
+![](images/94148f752106bd924a4f80c9c825128a3efe588caf08184e8765377bb5e169db.jpg)  
+Figure 8: A learned embedding of integers $\{ - 2 5 6 , - 2 5 5 , \ldots , - 1 , 0 , 1 , \ldots , 2 5 5 \}$ in $\mathbb { R } ^ { 2 }$ . The color intensity corresponds to the magnitude of the embedded integer.
+
+When the search procedure extends a partial program by a new function, it has to try the functions in the DSL in some order. At this point DFS can opt to consider the functions as ordered by their predicted probabilities from the neural network. The probability of a function consisting of a higherorder function and a lambda is taken to be the minimum of the probabilities of the two constituent functions.
+
+# E TRAINING LOSS FUNCTION
+
+In Sect. 4.5 we outlined a justification for using marginal probabilities of individual functions as a sensible intermediate representation to provide a solver employing a Sort and add scheme (we considered Enumerative search and the Sketch solver with this scheme). Here we provide a more detailed discussion.
+
+Predicting program components from input-output examples can be cast as a multilabel classification problem, where each instance (set of input-output examples) is associated with a set of relevant labels (functions appearing in the code that generated the examples). We denote the number of labels (functions) by $C$ , and note that throughout this work $C = 3 4$ .
+
+When the task is to predict a subset of labels $\mathbf { y } \in \{ 0 , 1 \} ^ { C }$ , different loss functions can be employed to measure the prediction error of a classifier $\mathbf { h } ( \mathbf { x } )$ or ranking function $\mathbf { f } \left( \mathbf { x } \right)$ . Dembczynski et al. (2010) discuss the following three loss functions:
+
+• Hamming loss counts the number of labels that are predicted incorrectly by a classifier h:
+
+$$
+L _ { H } ( \mathbf { y } , \mathbf { h } ( \mathbf { x } ) ) = \sum _ { c = 1 } ^ { C } \mathbb { 1 } _ { \{ y _ { c } \neq h _ { c } ( \mathbf { x } ) \} }
+$$
+
+• Rank loss counts the number of label pairs violating the condition that relevant labels are ranked higher than irrelevant ones by a scoring function f:
+
+$$
+L _ { r } ( \mathbf { y } , \mathbf { f } ( \mathbf { x } ) ) = \sum _ { ( i , j ) : y _ { i } = 1 , y _ { j } = 0 } ^ { C } \mathbb { 1 } _ { \{ f _ { i } < f _ { j } \} }
+$$
+
+• Subset Zero-One loss indicates whether all labels have been correctly predicted by $\mathbf { h }$ :
+
+$$
+L _ { s } ( \mathbf { y } , \mathbf { h } ( \mathbf { x } ) ) = \mathbb { 1 } _ { \{ \mathbf { y } \neq \mathbf { h } ( \mathbf { x } ) \} }
+$$
+
+Dembczynski et al. (2010) proved that Bayes optimal decisions under the Hamming and Rank loss functions, i.e., decisions minimizing the expected loss under these loss functions, can be computed from marginal probabilities $p _ { c } ( y _ { c } | \mathbf { x } )$ . This suggests that:
+
+• Multilabel classification under these two loss functions may not benefit from considering dependencies between the labels.   
+• ”Instead of minimizing the Rank loss directly, one can simply use any approach for single label prediction that properly estimates the marginal probabilities.” (Dembczynski et al., ´ 2012)
+
+Training the neural network with the negative cross entropy loss function as the training objective is precisely a method for properly estimating the marginal probabilities of labels (functions appearing in source code). It is thus a sensible step in preparation for making predictions under a Rank loss.
+
+It remains to discuss the relationship between the Rank loss and the actual quantity we care about, which is the total runtime of a Sort and add search procedure. Recall the simplifying assumption that the runtime of searching for a program of length $T$ with $C$ functions made available to the search is proportional to $C ^ { T }$ , and consider a Sort and add search for a program of length $T$ , where the size of the active set is increased by 1 whenever the search fails. Starting with an active set of size 1, the total time until a solution is found can be upper bounded by
+
+$$
+1 ^ { T } + 2 ^ { T } + \cdot \cdot \cdot + C _ { A } ^ { T } \leq C _ { A } ^ { T + 1 } \leq C C _ { A } ^ { T }
+$$
+
+where $C _ { A }$ is the size of the active set when the search finally succeeds (i.e., when the active set finally contains all necessary functions for a solution to exist). Hence the total runtime of a Sort and add search can be upper bounded by a quantity that is proportional to $C _ { A } ^ { T }$ .
+
+Now fix a valid program solution $P$ that requires $C _ { P }$ functions, and let $\mathbf { y } _ { P } \in \{ 0 , 1 \} ^ { C }$ be the indicator vector of functions used by $P$ . Let $D : = C _ { A } - C _ { P }$ be the number of redundant operations added into the active set until all operations from $P$ have been added.
+
+Example 1. Suppose the labels, as sorted by decreasing predicted marginal probabilities $\mathbf { f } \left( \mathbf { x } \right)$ , are as follows:
+
+$$
+\begin{array} { r l r l r } { 1 \mathrm { ~ 1 ~ 1 ~ 1 ~ } } & { { } } & { 1 } &  \qquad 1 \mathrm  ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 0 ~ 0 ~ 0 ~ 0 ~ 0 0  \end{array}
+$$
+
+Then the solution $P$ contains $C _ { P } = 6$ functions, but the active set needs to grow to size $C _ { A } = 1 1$ to include all of them, adding $D = 5$ redundant functions along the way. Note that the rank loss of the predictions $\mathbf { f } \left( \mathbf { x } \right)$ is $L _ { r } ( \mathbf { y } _ { P } , \mathbf { f } ( \mathbf { x } ) ) = 2 + 5 = 7$ , as it double counts the two redundant functions which are scored higher than two relevant labels.
+
+Noting that in general $L _ { r } ( \mathbf { y } _ { P } , \mathbf { f } ( \mathbf { x } ) ) \ge D$ , the previous upper bound on the runtime of Sort and add can be further upper bounded as follows:
+
+$$
+C _ { A } ^ { T } = ( C _ { P } + D ) ^ { T } \leq \mathrm { c o n s t } + \mathrm { c o n s t } \times D ^ { T } \leq \mathrm { c o n s t } + \mathrm { c o n s t } \times L _ { r } ( \mathbf { y } _ { P } , \mathbf { f } ( \mathbf { x } ) ) ^ { T }
+$$
+
+Hence we see that for a constant value of $T$ , this upper bound can be minimized by optimizing the Rank loss of the predictions $\mathbf { f } \left( \mathbf { x } \right)$ . Note also that $\bar { L _ { r } } ( { \bf y } _ { P } , { \bf f } ( { \bf x } ) ) = 0$ would imply $D = 0$ , in which case $C _ { A } = C _ { P }$ .
+
+# F DOMAIN SPECIFIC LANGUAGE OF DEEPCODER
+
+Here we provide a description of the semantics of our DSL from Sect. 4.1, both in English and as a Python implementation. Throughout, NULL is a special value that can be set e.g. to an integer outside the working integer range.
+
+First-order functions:
+
+• HEAD :: [int] -> int lambda xs: xs[0] if len $( \mathbf { \nabla } _ { \mathbf { X } } \mathbf { S } \mathbf { \nabla } ) > 0$ else Null Given an array, returns its first element (or NULL if the array is empty).   
+• LAST :: [int] $- >$ int lambda xs: xs[-1] if len $( \mathbf { \nabla } _ { \mathbf { X } } \mathbf { s } ) > 0$ else Null Given an array, returns its last element (or NULL if the array is empty).   
+• TAKE :: int $- >$ [int] -> int lambda n, xs: xs[:n] Given an integer n and array xs, returns the array truncated after the $\boldsymbol { \mathrm { n } }$ -th element. (If the length of xs was no larger than n in the first place, it is returned without modification.)   
+• DROP :: int $- >$ [int] $- >$ int lambda n, xs: xs[n:] Given an integer n and array $_ { \textrm { x S } }$ , returns the array with the first n elements dropped. (If the length of xs was no larger than n in the first place, an empty array is returned.)   
+• ACCESS :: int $- >$ [int] $- >$ int lambda n, xs: xs[n] if $\mathrm { n } > = 0$ and len $( \mathsf { x s } ) > \mathsf { n }$ else Null Given an integer n and array $_ { \textrm { x S } }$ , returns the $( \mathrm { n } { + } 1 )$ -st element of $_ { \textrm { x S } }$ . (If the length of xs was less than or equal to $\boldsymbol { \mathrm { n } }$ , the value NULL is returned instead.)   
+• MINIMUM :: [int] $- >$ int lambda xs: min(xs) if len $( \mathbf { \nabla } _ { \mathbf { X } } \mathbf { S } ) > 0$ else Null Given an array, returns its minimum (or NULL if the array is empty).   
+• MAXIMUM :: [int] $- >$ int lambda xs: max(xs) if len $( \mathbf { \nabla } _ { \mathbf { X } } \mathbf { S } ) > 0$ else Null Given an array, returns its maximum (or NULL if the array is empty). REVERSE :: [int] $- >$ [int] lambda xs: list(reversed(xs)) Given an array, returns its elements in reversed order.   
+• SORT :: [int] -> [int] lambda xs: sorted(xs) Given an array, return its elements in non-decreasing order. SUM :: [int] $- >$ int lambda xs: sum(xs) Given an array, returns the sum of its elements. (The sum of an empty array is 0.)
+
+Higher-order functions:
+
+• MAP :: (int -> int) -> [int] -> [int] lambda f, xs: [f(x) for x in xs] Given a lambda function f mapping from integers to integers, and an array xs, returns the array resulting from applying f to each element of xs.   
+• FILTER :: (int $- >$ bool) $- >$ [int] -> [int] lambda f, xs: [x for x in xs if f(x)] Given a predicate f mapping from integers to truth values, and an array xs, returns the elements of xs satisfying the predicate in their original order.   
+• COUNT :: (int $- >$ bool) $- >$ [int] $- >$ int lambda f, xs: len([x for x in xs if f(x)]) Given a predicate f mapping from integers to truth values, and an array xs, returns the number of elements in xs satisfying the predicate.   
+• ZIPWITH :: (int $- >$ int $- >$ int) $- >$ [int] -> [int] -> [int] lambda f, xs, ys: [f(x, y) for (x, y) in zip(xs, ys)] Given a lambda function f mapping integer pairs to integers, and two arrays xs and ys, returns the array resulting from applying f to corresponding elements of xs and ys. The length of the returned array is the minimum of the lengths of $_ { \textrm { x S } }$ and ys. SCANL1 :: (int $- >$ int $- >$ int) $- >$ [int] -> [int] Given a lambda function f mapping integer pairs to integers, and an array $_ { \textrm { x S } }$ , returns an array ys of the same length as xs and with its content defined by the recurrence $\begin{array} { r l } { \mathrm { y s } \left[ 0 \right] } & { { } = } \end{array}$ $\mathbf { x } \mathbf { s } \left[ 0 \right] , \mathbf { y } \mathbf { s } \left[ \mathrm { n } \right] \mathbf { \Psi } = \mathbf { \Psi } \mathbf { f } \left( \mathbf { y } \mathbf { s } \left[ \mathrm { n - 1 } \right] \right.$ , $\mathtt { x s } [ \mathtt { n } ]$ ) for $n \geq 1$ .
+
+The INT INT lambdas $\left( + 1 \right) , \left( - 1 \right) , \left( \star 2 \right) , \left( / 2 \right) , \left( \star \left( - 1 \right) \right) , \left( \star \star 2 \right) , \left( \star 3 \right) , \left( / 3 \right) , \left( \star 4 \right) , \left( / 4 \right) , \left( \star 3 \right) , \left( \star 4 \right) .$ ) provided by our DSL map integers to integers in a self-explanatory manner. The $\mathrm { I N T } {  } \mathrm { B O O L }$ lambdas $( > 0 )$ , $( < 0 )$ , ( $\scriptstyle { \frac { 0 } { 0 } } 2 = = 0$ ), ( $\scriptstyle { \frac { 0 } { 0 } } 2 = = 1$ ) respectively test positivity, negativity, evenness and oddness of the input integer value. Finally, the $\mathrm { I N T } {  } \mathrm { I N T } {  } \mathrm { I N T }$ lambdas $( + ) , ( - ) , ( \star )$ , MIN, MAX apply a function to a pair of integers and produce a single integer.
+
+As an example, consider the function SCANL1 MAX, consisting of the higher-order function SCANL1 and the $\mathrm { I N T } {  } \mathrm { I N T } {  } \mathrm { I N T }$ lambda MAX. Given an integer array a of length $L$ , this function computes the running maximum of the array a. Specifically, it returns an array $\mathrm { b }$ of the same length $L$ whose $i$ -th element is the maximum of the first $i$ elements in $\exists$ .
+
+![](images/da5620eb6e4f0a76e4439147498cbeb617797fc9a1d092bb07c51b51e15fdcb8.jpg)  
+Figure 9: Conditional confusion matrix for the neural network and test set of $P = 5 0 0$ programs of length $T = 3$ that were used to obtain the results presented in Table 1. Each cell contains the average false positive probability (in larger font) and the number of test programs from which this average was computed (smaller font, in brackets). The color intensity of each cell’s shading coresponds to the magnitude of the average false positive probability.
+
+# G ANALYSIS OF TRAINED NEURAL NETWORKS
+
+We analyzed the performance of trained neural networks by investigating which program instructions tend to get confused by the networks. To this end, we looked at a generalization of confusion matrices to the multilabel classification setting: for each attribute in a ground truth program (rows) measure how likely each other attribute (columns) is predicted as a false positive. More formally, in this matrix the $( i , j )$ -entry is the average predicted probability of attribute $j$ among test programs that do possess attribute $i$ and do not possess attribute $j$ . Intuitively, the $i$ -th row of this matrix shows how the presence of attribute $i$ confuses the network into incorrectly predicting each other attribute $j$ .
+
+Figure 9 shows this conditional confusion matrix for the neural network and $P = 5 0 0$ program test set configuration used to obtain Table 1. We re-ordered the confusion matrix to try to expose block structure in the false positive probabilities, revealing groups of instructions that tend to be difficult to distinguish. Figure 10 show the conditional confusion matrix for the neural network used to obtain the table in Fig. 3a. While the results are somewhat noisy, we observe a few general tendencies:
+
+• There is increased confusion amongst instructions that select out a single element from an array: HEAD, LAST, ACCESS, MINIMUM, MAXIMUM.   
+• Some common attributes get predicted more often regardless of the ground truth program: FILTER, $( > 0 )$ ), $( < 0 )$ , ( $\scriptstyle { \frac { \circ } { \circ } } 2 = = 1$ ), ( $\scriptstyle { \frac { 0 } { \circ } } 2 = = 0$ ), MIN, MAX, (+), (-), ZIPWITH.   
+There are some groups of lambdas that are more difficult for the network to distinguish within: $( + )$ vs $( - ) ~ ; ~ ( + 1 )$ vs $( - 1 ) \ ; \ ( \ / 2 )$ vs (/3) vs (/4).   
+• When a program uses $( \star \star 2 )$ , the network often thinks it’s using $( \star )$ , presumably because both can lead to large values in the output.
+
+![](images/a2f6e1ddd3c6cc3aed7aca84775a0568e1abe730347297f96f96da4cfd262c8d.jpg)
+
+Figure 10: Conditional confusion matrix for the neural network and test set of $P = 5 0 0$ programs of length $T = 5$ . The presentation is the same as in Figure 9.
\ No newline at end of file
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new file mode 100644
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+                                    "content": "The Inductive Program Synthesis (IPS) problem is the following: given input-output examples,",
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+                                    "score": 1.0,
+                                    "content": "formulating an approach to program synthesis is the choice of a Domain Specific Language.",
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+                                    "content": "Domain Specific Languages (DSLs). DSLs are programming languages that are suitable for a",
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+                                    "content": "primitive operations like concatenation. Most of program synthesis research focuses on synthesizing",
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+                                    "score": 1.0,
+                                    "content": "formulating an approach to program synthesis is the choice of a Domain Specific Language.",
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+                                    "content": "ming algorithm can efficiently search over all possible programs (Polozov & Gulwani, 2015). The",
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+                                    "content": "output examples (Gulwani, 2016). A more sophisticated approach is employed by FlashFill (Singh",
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+                                    "score": 1.0,
+                                    "content": "& Gulwani, 2015). It works in a manner similar to max-margin structured prediction, where known",
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+                                    "content": "3 LEARNING INDUCTIVE PROGRAM SYNTHESIS (LIPS)",
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+                                    "content": "Inductive Program Synthesis (LIPS). The details of our instantiation of LIPS appear in Sect. 4. The",
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+                                    "score": 1.0,
+                                    "content": "ing model that maps from input-output examples to program attributes, and (4) a search procedure",
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+                                    "score": 1.0,
+                                    "content": "that searches program space in an order guided by the model from (3). The framework is related to",
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+                                    "content": "the formulation of Menon et al. (2013); the relationship and key differences are discussed in Sect. 6.",
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+                                    "content": "\\mathcal { A }",
+                                    "type": "inline_equation"
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+                                    "content": "\\mathbf { \\boldsymbol { a } } \\overset { \\cdot } { = } \\mathbf { \\bar { A } } ( \\mathbf { \\boldsymbol { P } } )",
+                                    "type": "inline_equation"
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+                                    "content": "(Attribute vectors of different programs need not have equal length.) Attributes serve as the link",
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+                            "index": 25
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+                        {
+                            "bbox": [
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+                                    "score": 0.92,
+                                    "content": "q ( \\pmb { a } \\mid \\mathcal { E } )",
+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": ", where",
+                                    "type": "text"
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+                                    "score": 0.8,
+                                    "content": "\\mathcal { E }",
+                                    "type": "inline_equation"
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "is the set of input-output examples, and the search procedure",
+                                    "type": "text"
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+                        {
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+                                    "score": 1.0,
+                                    "content": ". Thus an attribute is useful if it is both",
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+                                    "content": "Possible attributes are the (perhaps position-dependent) presence or absence of high-level functions",
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+                                    "content": "(e.g., does the program contain or end in a call to SORT). Other possible attributes include control",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "flow templates (e.g., the number of loops and conditionals). In the extreme case, one may set",
+                                    "type": "text"
+                                },
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+                                    "score": 0.74,
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+                        {
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+                                    "score": 1.0,
+                                    "content": "to the identity function, in which case the attribute is equivalent to the program; however, in our",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "experiments we find that performance is improved by choosing a more abstract attribute function.",
+                                    "type": "text"
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+                            ],
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+                    ],
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+                                    "content": "(2) Data Generation. Step 2 is to generate a dataset",
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+                                    "content": "( ( P ^ { ( n ) } , \\pmb { a } ^ { ( n ) } , \\pmb { \\mathcal { E } } ^ { ( n ) } ) ) _ { n = 1 } ^ { N }",
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+                                    "score": 1.0,
+                                    "content": "the chosen DSL, their attributes",
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+                                    "score": 1.0,
+                                    "content": ", and accompanying input-output examples",
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+                                {
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+                                    "score": 0.89,
+                                    "content": "{ \\mathcal { E } } ^ { ( n ) }",
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+                                {
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+                                    "score": 1.0,
+                                    "content": ". Different ap-",
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "proaches are possible, ranging from enumerating valid programs in the DSL and pruning, to training",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "a more sophisticated generative model of programs in the DSL. The key in the LIPS formulation is",
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+                                    "score": 1.0,
+                                    "content": "to ensure that it is feasible to generate a large dataset (ideally millions of programs).",
+                                    "type": "text"
+                                }
+                            ],
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+                        }
+                    ],
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+                                    "content": "(3) Machine Learning Model. The machine learning problem is to learn a distribution of at-",
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+                                    "content": "q ( \\pmb { a } \\mid \\mathcal { E } )",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": ". There is freedom to explore a large space of models,",
+                                    "type": "text"
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "(e.g., if attributes are a fixed-size binary vector, then a neural network with independent sigmoid",
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+                                    "content": "outputs is appropriate; if attributes are variable size, then a recurrent neural network output could be",
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+                            ],
+                            "index": 45
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "used). Attributes are observed at training time, so training can use a maximum likelihood objective.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 46
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+                    ],
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+                                    "score": 1.0,
+                                    "content": "(4) Search. The aim of the search component is to interface with an existing solver, using the",
+                                    "type": "text"
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+                                    "content": "to guide the search. We describe specific approaches in the next section.",
+                                    "type": "text"
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+                                    "content": "Finally, stochastic local search can be employed to search over program space, and there is a long",
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+                                    "content": "is the STOKE super-optimization system (Schkufza et al., 2016), which uses stochastic local search",
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+                                    "score": 1.0,
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+                                    "content": "output examples (Gulwani, 2016). A more sophisticated approach is employed by FlashFill (Singh",
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+                                    "score": 1.0,
+                                    "content": "& Gulwani, 2015). It works in a manner similar to max-margin structured prediction, where known",
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+                            "index": 11
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+                        {
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+                                    "content": "ground truth programs are given, and the learning task is to assign scores to programs such that the",
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+                                    "score": 1.0,
+                                    "content": "ground truth programs score higher than other programs that satisfy the input-output specification.",
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+                                    "score": 1.0,
+                                    "content": "3 LEARNING INDUCTIVE PROGRAM SYNTHESIS (LIPS)",
+                                    "type": "text"
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+                                    "content": "In this section we outline the general approach that we follow in this work, which we call Learning",
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+                                    "content": "Inductive Program Synthesis (LIPS). The details of our instantiation of LIPS appear in Sect. 4. The",
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+                                    "content": "ing model that maps from input-output examples to program attributes, and (4) a search procedure",
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+                                    "score": 1.0,
+                                    "content": "that searches program space in an order guided by the model from (3). The framework is related to",
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+                        {
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+                            "index": 24
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+                                    "content": "(Attribute vectors of different programs need not have equal length.) Attributes serve as the link",
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+                            "index": 25
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+                            "bbox": [
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+                                    "content": "q ( \\pmb { a } \\mid \\mathcal { E } )",
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+                                    "score": 0.8,
+                                    "content": "\\mathcal { E }",
+                                    "type": "inline_equation"
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "is the set of input-output examples, and the search procedure",
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+                        {
+                            "bbox": [
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+                                    "score": 1.0,
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+                                    "score": 0.92,
+                                    "content": "q ( { \\bar { \\mathcal { A } } } ( { \\bar { P } } ) \\mid \\varepsilon )",
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+                                    "score": 1.0,
+                                    "content": ". Thus an attribute is useful if it is both",
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+                                    "content": "to the identity function, in which case the attribute is equivalent to the program; however, in our",
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+                        {
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+                                    "content": "experiments we find that performance is improved by choosing a more abstract attribute function.",
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+                                    "content": "{ \\mathcal { E } } ^ { ( n ) }",
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+                                    "score": 1.0,
+                                    "content": ". Different ap-",
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+                            ],
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+                        },
+                        {
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+                                    "content": "to ensure that it is feasible to generate a large dataset (ideally millions of programs).",
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+                        }
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+                                    "content": "(3) Machine Learning Model. The machine learning problem is to learn a distribution of at-",
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+                                    "content": ". There is freedom to explore a large space of models,",
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+                                    "content": "that they all appear in the vast majority of programs, but at the same time should be common enough",
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+                                    "content": "by identical behavior on randomly or carefully chosen inputs). To generate valid inputs for a program,",
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "with a special NULL value. Second, each integer in the inputs and in the output is mapped",
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+                                    "content": ". (The range of integers is restricted to a finite range",
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+                                    "score": 1.0,
+                                    "content": "and each embedding is parametrized individually.) Third, for each input-output example separately,",
+                                    "type": "text"
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+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "sigmoid units each. The third hidden layer thus provides an encoding of each individual input-output",
+                                    "type": "text"
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+                        {
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+                                    "score": 1.0,
+                                    "content": "example. Finally, for input-output examples in a set generated from the same program, we pool these",
+                                    "type": "text"
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+                        {
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+                                    "score": 1.0,
+                                    "content": "representations together by simple arithmetic averaging. See Appendix C for more details.",
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+                                    "score": 1.0,
+                                    "content": "at least as well as an RNN encoder, a natural baseline, which may however be more difficult to train.",
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+                                    "score": 1.0,
+                                    "content": "to a learned embedding vector of size",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "content": "E = 2 0",
+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    "score": 1.0,
+                                    "content": ". (The range of integers is restricted to a finite range",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "and each embedding is parametrized individually.) Third, for each input-output example separately,",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "we concatenate the embeddings of the input types, the inputs, the output type, and the output into a",
+                                    "type": "text"
+                                }
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+                        },
+                        {
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+                                    "score": 1.0,
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+                                    "type": "text"
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+                                    "bbox": [
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+                                    "score": 0.9,
+                                    "content": "H = 3",
+                                    "type": "inline_equation"
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+                                    "score": 0.87,
+                                    "content": "K = 2 5 6",
+                                    "type": "inline_equation"
+                                }
+                            ],
+                            "index": 12
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "sigmoid units each. The third hidden layer thus provides an encoding of each individual input-output",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 13
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+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "example. Finally, for input-output examples in a set generated from the same program, we pool these",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 14
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "representations together by simple arithmetic averaging. See Appendix C for more details.",
+                                    "type": "text"
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+                            "index": 15
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+                    ],
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+                                    "content": "output. We confirmed that the chosen encoder architecture is sensible in that it performs empirically",
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+                            ],
+                            "index": 18
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "at least as well as an RNN encoder, a natural baseline, which may however be more difficult to train.",
+                                    "type": "text"
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+                            ],
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+                        }
+                    ],
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+                                    "score": 1.0,
+                                    "content": "DeepCoder learns to predict presence or absence of individual functions of the DSL. We shall see",
+                                    "type": "text"
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+                            ],
+                            "index": 20
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "this can already be exploited by various search techniques to large computational gains. We use a",
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+                        },
+                        {
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+                                    "score": 1.0,
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+                                    "score": 0.9,
+                                    "content": "C \\times K",
+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    "type": "text"
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+                            ],
+                            "index": 22
+                        },
+                        {
+                            "bbox": [
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+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": "is the number of functions in our DSL (higher-order functions and lambdas are predicted",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "numbers as log-unnormalized probabilities (logits) of each",
+                                    "type": "text"
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+                                    "score": 1.0,
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "from 5 input-output examples for the program shown in Fig. 1.",
+                                    "type": "text"
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+                                    "type": "text"
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+                                    "content": "One of the central ideas of this work is to use a neural network to guide the search for a program",
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+                                    "content": "consistent with a set of input-output examples instead of directly predicting the entire source code.",
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+                                    "content": "This section briefly describes the search techniques and how they integrate the predicted attributes.",
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+                                    "content": "Depth-first search (DFS). We use an optimized version of DFS to search over programs with a",
+                                    "type": "text"
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+                            ],
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+                        },
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "(see Appendix D for details). When the search procedure extends a partial",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 36
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "program by a new function, it has to try the functions in the DSL in some order. At this point DFS",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 37
+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "can opt to consider the functions as ordered by their predicted probabilities from the neural network.",
+                                    "type": "text"
+                                }
+                            ],
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+                    ],
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+                                    "score": 1.0,
+                                    "content": "“Sort and add” enumeration. A stronger way of utilizing the predicted probabilities of functions",
+                                    "type": "text"
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+                            ],
+                            "index": 39
+                        },
+                        {
+                            "bbox": [
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+                                    "content": "in an enumerative search procedure is to use a Sort and add scheme, which maintains a set of active",
+                                    "type": "text"
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+                                    "content": "functions and performs DFS with the active function set only. Whenever the search fails, the next",
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+                                    "content": "labels (functions appearing in the ground truth source code). Dembczynski et al. (2010) showed",
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+                                    "content": "that in multi-label classification under a so-called Rank loss, it is Bayes optimal to rank the labels",
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+                                    "content": "C ^ { T }",
+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": ", the total runtime of a Sort and add procedure can be monotonically transformed so that it is",
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+                                    "score": 1.0,
+                                    "content": "upper bounded by this Rank loss. See Appendix E for more details.",
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+                                    "content": "5 EXPERIMENTS",
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+                                    "content": "strating such gains with DeepCoder. In Sect. 5.2 we illustrate the robustness of the method by",
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+                                    "content": "and it is not clear a priori how much DeepCoder loses by ignoring correlations between functions.",
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+                                    "score": 1.0,
+                                    "content": ", the following result for",
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+                            ],
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+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "Sort and add procedures shows that their runtime can be optimized using marginal probabilities.",
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+                    ],
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+                                    "score": 1.0,
+                                    "content": ", the expected total runtime of a Sort and add search",
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+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "scheme can be upper bounded by a quantity that is minimized by adding the functions in the order of",
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "Proof. Predicting source code functions from input-output examples can be seen as a multi-label",
+                                    "type": "text"
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+                            ],
+                            "index": 21
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "labels (functions appearing in the ground truth source code). Dembczynski et al. (2010) showed",
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+                                    "score": 1.0,
+                                    "content": "that in multi-label classification under a so-called Rank loss, it is Bayes optimal to rank the labels",
+                                    "type": "text"
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+                            "index": 24
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+                        {
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+                                    "score": 0.88,
+                                    "content": "C ^ { T }",
+                                    "type": "inline_equation"
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+                                {
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+                                    "content": ", the total runtime of a Sort and add procedure can be monotonically transformed so that it is",
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "upper bounded by this Rank loss. See Appendix E for more details.",
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+                                    "content": "□",
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+                                    "content": "5 EXPERIMENTS",
+                                    "type": "text"
+                                }
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+                    ],
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+                },
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+                                    "content": "show that the LIPS framework can lead to significant performance gains in solving IPS by demon-",
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+                                    "score": 1.0,
+                                    "content": "strating such gains with DeepCoder. In Sect. 5.2 we illustrate the robustness of the method by",
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+                                    "content": "P = 5 0 0",
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+                                    "content": "all programs on which the neural network was trained (similarly to the equivalence check described",
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+                        },
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+                                    "score": 1.0,
+                                    "content": "in Sect. 4.2, we have ensured that all test programs behave differently from all programs used during",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "of the test set tasks. Fig. 3b shows the relative speedup of the solver having access to",
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+                                    "content": "Encoder We evaluated replacing the feed-forward architecture encoder (Sect. 4.3) with an RNN, a",
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+                                    "content": "natural baseline. Using a GRU-based RNN we were able to achieve results almost as good as using",
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+                                    "score": 1.0,
+                                    "content": "Decoder We also considered a purely neural network-based approach, where an RNN decoder",
+                                    "type": "text"
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+                            ],
+                            "index": 7
+                        },
+                        {
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+                                    "content": "is trained to predict the entire program token-by-token. We combined this with our feed-forward",
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+                                    "score": 1.0,
+                                    "content": "encoder by initializing the RNN using the pooled final layer of the encoder. We found it substantially",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "more difficult to train an RNN decoder as compared to the independent binary classifiers employed",
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+                            "index": 10
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+                        {
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+                                    "score": 1.0,
+                                    "content": "above. Beam search was used to explore likely programs predicted by the RNN, but it only lead to a",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "solution comparable with the other techniques when searching for programs of lengths",
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+                                    "score": 1.0,
+                                    "content": ", where",
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+                                    "score": 1.0,
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+                                    "content": "1 0 ^ { 3 }",
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+                                    "score": 1.0,
+                                    "content": "). Note that using an RNN for both the encoder",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "and decoder corresponds to a standard sequence-to-sequence model. However, we do do not rule out",
+                                    "type": "text"
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+                            ],
+                            "index": 14
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+                        {
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+                                    "score": 1.0,
+                                    "content": "that a more sophisticated RNN decoder or training procedure could be possibly more successful.",
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+                                    "type": "text"
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+                                    "content": "Decoder We also considered a purely neural network-based approach, where an RNN decoder",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "is trained to predict the entire program token-by-token. We combined this with our feed-forward",
+                                    "type": "text"
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+                        },
+                        {
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "encoder by initializing the RNN using the pooled final layer of the encoder. We found it substantially",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "more difficult to train an RNN decoder as compared to the independent binary classifiers employed",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "above. Beam search was used to explore likely programs predicted by the RNN, but it only lead to a",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 11
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "solution comparable with the other techniques when searching for programs of lengths",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.9,
+                                    "content": "T \\le 2",
+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    "score": 1.0,
+                                    "content": ", where",
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+                            "index": 12
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "the search space size is very small (on the order of",
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+                                    "score": 0.86,
+                                    "content": "1 0 ^ { 3 }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "). Note that using an RNN for both the encoder",
+                                    "type": "text"
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+                            "index": 13
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "and decoder corresponds to a standard sequence-to-sequence model. However, we do do not rule out",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 14
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "that a more sophisticated RNN decoder or training procedure could be possibly more successful.",
+                                    "type": "text"
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+                            "index": 15
+                        }
+                    ],
+                    "index": 11,
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+                {
+                    "type": "title",
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+                                    "score": 1.0,
+                                    "content": "6 RELATED WORK",
+                                    "type": "text"
+                                }
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+                        }
+                    ],
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+                },
+                {
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+                                    "score": 1.0,
+                                    "content": "Machine Learning for Inductive Program Synthesis. There is relatively little work on using",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 17
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+                        {
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+                                    "score": 1.0,
+                                    "content": "machine learning for programming by example. The most closely related work is that of Menon",
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+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "et al. (2013), in which a hand-coded set of features of input-output examples are used as “clues.”",
+                                    "type": "text"
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+                            "index": 19
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "When a clue appears in the input-output examples (e.g., the output is a permutation of the input),",
+                                    "type": "text"
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+                            ],
+                            "index": 20
+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "it reweights the probabilities of productions in a probabilistic context free grammar by a learned",
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "amount. This work shares the idea of learning to guide the search over program space conditional on",
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "input-output examples. One difference is in the domains. Menon et al. (2013) operate on short string",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 23
+                        },
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+                                    "score": 1.0,
+                                    "content": "manipulation programs, where it is arguably easier to hand-code features to recognize patterns in the",
+                                    "type": "text"
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+                        {
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+                                    "score": 1.0,
+                                    "content": "input-output examples (e.g., if the outputs are always permutations or substrings of the input). Our",
+                                    "type": "text"
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+                                    "content": "work shows that there are strong cues in patterns in input-output examples in the domain of numbers",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "and lists. However, the main difference is the scale. Menon et al. (2013) learns from a small (280",
+                                    "type": "text"
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+                            "index": 27
+                        },
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+                                    "score": 1.0,
+                                    "content": "examples), manually-constructed dataset, which limits the capacity of the machine learning model",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "that can be trained. Thus, it forces the machine learning component to be relatively simple. Indeed,",
+                                    "type": "text"
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+                            ],
+                            "index": 29
+                        },
+                        {
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+                                    "type": "text"
+                                }
+                            ],
+                            "index": 30
+                        },
+                        {
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+                    ],
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+                                    "score": 1.0,
+                                    "content": "Learning Representations of Program State. Piech et al. (2015) propose to learn joint embed-",
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+                                    "score": 1.0,
+                                    "content": "appearing when extending our DSL to handle more complex data structures.",
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+                        }
+                    ],
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "is trained to select premises that the theorem prover can use to prove a theorem. A recent exten-",
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+                                    "score": 1.0,
+                                    "content": "sion (Loos et al., 2017) is similar to our DFS enumeration strategy and uses a neural network to guide",
+                                    "type": "text"
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+                        },
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+                                    "content": "the proof search at intermediate steps. The main differences are in the domains, and that they train",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "on an existing corpus of theorems. More broadly, if we view a DSL as defining a model and search",
+                                    "type": "text"
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+                            ],
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+                                    "content": "as a form of inference algorithm, then there is a large body of work on using discriminatively-trained",
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+                                    "score": 1.0,
+                                    "content": "models to aid inference in generative models. Examples include Dayan et al. (1995); Kingma &",
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+                                    "content": "Welling (2014); Shotton et al. (2013); Stuhlmuller et al. (2013); Heess et al. (2013); Jampani et al. ¨",
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+                                    "content": "(2015).",
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+                                    "content": "solved with a program in our language, which validates the relevance of the class of problems that",
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+                                    "score": 1.0,
+                                    "content": "descriptions to lessen the information burden required from input-output examples. In sum, Deep-",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "Coder represents a promising direction forward, and we are optimistic about the future prospects of",
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+                            ],
+                            "index": 23
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "using machine learning to synthesize programs.",
+                                    "type": "text"
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+                        }
+                    ],
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+                },
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+                                    "score": 1.0,
+                                    "content": "ACKNOWLEDGMENTS",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 25
+                        }
+                    ],
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+                },
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+                                    "score": 1.0,
+                                    "content": "The authors would like to express their gratitude to Rishabh Singh and Jack Feser for their valuable",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 26
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+                                    "content": "guidance and help on using the Sketch and",
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+                                    "score": 0.85,
+                                    "content": "\\lambda ^ { 2 }",
+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    "score": 1.0,
+                                    "content": "program synthesis systems.",
+                                    "type": "text"
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+                            ],
+                            "index": 27
+                        }
+                    ],
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+                                    "content": "REFERENCES",
+                                    "type": "text"
+                                }
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+                            "index": 28
+                        }
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+                {
+                    "type": "text",
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Alex A. Alemi, Franc¸ois Chollet, Geoffrey Irving, Christian Szegedy, and Josef Urban. DeepMath -",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 29
+                        },
+                        {
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+                                {
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+                                    "score": 1.0,
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+                                    "type": "text"
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+                        {
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+                                523
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+                            "spans": [
+                                {
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+                                {
+                                    "bbox": [
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+                                    ],
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+                                    "type": "text"
+                                }
+                            ],
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+                            "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "Processing Systems (NIPS), 2016.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 34
+                        },
+                        {
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+                                    "score": 1.0,
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+                                    "type": "text"
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+                                    "content": "Krzysztof Dembczynski, Willem Waegeman, Weiwei Cheng, and Eyke H ´ ullermeier. On label de- ¨",
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+                                    "content": "pendence and loss minimization in multi-label classification. Machine Learning, 88(1):5–45,",
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+                                    "content": "Krzysztof J. Dembczynski, Weiwei Cheng, and Eyke Hllermeier. Bayes optimal multilabel classifi-",
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+                                    "score": 1.0,
+                                    "content": "cation via probabilistic classifier chains. In Proceedings of the 27th International Conference on",
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+                            "index": 41
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+                                    "content": "John K. Feser, Swarat Chaudhuri, and Isil Dillig. Synthesizing data structure transformations from",
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+                                    "content": "input-output examples. In Proceedings of the 36th ACM SIGPLAN Conference on Programming",
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+                                    "content": "Alexander L. Gaunt, Marc Brockschmidt, Rishabh Singh, Nate Kushman, Pushmeet Kohli, Jonathan",
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+                                    "content": "Taylor, and Daniel Tarlow. Terpret: A probabilistic programming language for program induction.",
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+                                    "content": "CoRR, abs/1608.04428, 2016. URL http://arxiv.org/abs/1608.04428.",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "descriptions to lessen the information burden required from input-output examples. In sum, Deep-",
+                                    "type": "text"
+                                }
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+                        },
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+                                    "score": 1.0,
+                                    "content": "Coder represents a promising direction forward, and we are optimistic about the future prospects of",
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "using machine learning to synthesize programs.",
+                                    "type": "text"
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+                        }
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+                                    "content": "ACKNOWLEDGMENTS",
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+                                            "html": "<table><tr><td colspan=\"2\"></td></tr><tr><td>Program 6: IO example: t←[int] Input: p←[int] [4 8 11 2], C←MAP（-1）t [2341] d←MAP（-1）p Output: e ←ZIPWITH（+）c d 1 f←MINIMUMe</td><td>Description: Umberto has a large collection of ties and match- ing pocket squares—too large, his wife says—and he needs to sell one pair. Given their values as arrays t and p,assuming that he sells the cheapest pair, and selling costs 2,how much will he lose from the sale?</td></tr><tr><td>Program 7: IO example: s←[int] Input: p←[int] [4723], C ← SCANL1（+）p [2131] d ←ZIPWITH（*）S C Output: e←SUMd 62</td><td>Description: Zack always promised his n friends to buy them candy, but never did. Now he won the lottery and counts how often and how much candy he promised to his friends,obtaining arrays p (num- ber of promises) and s (number of promised sweets). He announces that to repay them,he will buy s[1]+s[2]+...+s[n] pieces of candy for the firstp[1]days,thens[2]+s[3]+...+s[n]for p[2] days,and so on,until he has fulfilled all promises. How much candy will he buy in total?</td></tr><tr><td>Program 8: IO example: s←[int] Input: b ←REVERSE S [12457] C ←ZIPWITH（-）b s Output: d ←FILTER（&gt;O）C 9 e←SuMd</td><td>Description: Vivian loves rearranging things.Most of all,when she sees a row of heaps, she wants to make sure that each heap has more items than the one to its left. She is also obsessed with efficiency, so always moves the least possible number of items.Her dad really dislikes if she changes the size of heaps,so she only moves single items between them,making sure that the set of sizes of the heaps is the same as at the start; they are only in a different order. When you come in, you see heaps of sizes (of course,sizes strictly monotonically increasing）s[O]，s[1]，... s[n].What is the maximal number of items that Vivian could have moved?</td></tr></table>",
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+                                    "score": 1.0,
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+                                    "content": "E TRAINING LOSS FUNCTION",
+                                    "type": "text"
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+                    ],
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+                                    "content": "In Sect. 4.5 we outlined a justification for using marginal probabilities of individual functions as",
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+                                    "score": 1.0,
+                                    "content": "a sensible intermediate representation to provide a solver employing a Sort and add scheme (we",
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+                                    "score": 1.0,
+                                    "content": "considered Enumerative search and the Sketch solver with this scheme). Here we provide a more",
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+                                    "content": "Predicting program components from input-output examples can be cast as a multilabel classification",
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+                                    "content": "problem, where each instance (set of input-output examples) is associated with a set of relevant",
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+                                    "score": 1.0,
+                                    "content": "labels (functions appearing in the code that generated the examples). We denote the number of labels",
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+                                    "score": 1.0,
+                                    "content": ". This suggests that:",
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+                                    "content": "• Multilabel classification under these two loss functions may not benefit from considering",
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+                                    "content": "• ”Instead of minimizing the Rank loss directly, one can simply use any approach for single",
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+                                    "score": 1.0,
+                                    "content": "label prediction that properly estimates the marginal probabilities.” (Dembczynski et al., ´",
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+                                    "score": 1.0,
+                                    "content": "in source code). It is thus a sensible step in preparation for making predictions under a Rank loss.",
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+                                    "content": "F DOMAIN SPECIFIC LANGUAGE OF DEEPCODER",
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+                                    "content": "Here we provide a description of the semantics of our DSL from Sect. 4.1, both in English and as a",
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+                                    "type": "text"
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+                                {
+                                    "bbox": [
+                                        141,
+                                        115,
+                                        492,
+                                        127
+                                    ],
+                                    "score": 1.0,
+                                    "content": "length of xs was no larger than n in the first place, it is returned without modification.)",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 3
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                131,
+                                306,
+                                142
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        131,
+                                        214,
+                                        142
+                                    ],
+                                    "score": 1.0,
+                                    "content": "• DROP :: int",
+                                    "type": "text",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        214,
+                                        132,
+                                        229,
+                                        141
+                                    ],
+                                    "score": 0.43,
+                                    "content": "- >",
+                                    "type": "inline_equation",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        229,
+                                        131,
+                                        267,
+                                        142
+                                    ],
+                                    "score": 1.0,
+                                    "content": "[int]",
+                                    "type": "text",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        268,
+                                        132,
+                                        282,
+                                        141
+                                    ],
+                                    "score": 0.47,
+                                    "content": "- >",
+                                    "type": "inline_equation",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        283,
+                                        131,
+                                        306,
+                                        142
+                                    ],
+                                    "score": 1.0,
+                                    "content": "int",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 4,
+                            "is_list_start_line": true,
+                            "is_list_end_line": true
+                        },
+                        {
+                            "bbox": [
+                                142,
+                                142,
+                                263,
+                                154
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        142,
+                                        142,
+                                        263,
+                                        154
+                                    ],
+                                    "score": 1.0,
+                                    "content": "lambda n, xs: xs[n:]",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 5,
+                            "is_list_end_line": true
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                152,
+                                505,
+                                165
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        152,
+                                        258,
+                                        165
+                                    ],
+                                    "score": 1.0,
+                                    "content": "Given an integer n and array",
+                                    "type": "text",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        258,
+                                        155,
+                                        271,
+                                        163
+                                    ],
+                                    "score": 0.36,
+                                    "content": "_ { \\textrm { x S } }",
+                                    "type": "inline_equation",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        271,
+                                        152,
+                                        505,
+                                        165
+                                    ],
+                                    "score": 1.0,
+                                    "content": ", returns the array with the first n elements dropped. (If the",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 6
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                163,
+                                461,
+                                176
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        163,
+                                        461,
+                                        176
+                                    ],
+                                    "score": 1.0,
+                                    "content": "length of xs was no larger than n in the first place, an empty array is returned.)",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 7,
+                            "is_list_end_line": true
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                178,
+                                316,
+                                191
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        178,
+                                        223,
+                                        191
+                                    ],
+                                    "score": 1.0,
+                                    "content": "• ACCESS :: int",
+                                    "type": "text",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        224,
+                                        180,
+                                        239,
+                                        190
+                                    ],
+                                    "score": 0.52,
+                                    "content": "- >",
+                                    "type": "inline_equation",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        239,
+                                        178,
+                                        277,
+                                        191
+                                    ],
+                                    "score": 1.0,
+                                    "content": "[int]",
+                                    "type": "text",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        277,
+                                        180,
+                                        292,
+                                        190
+                                    ],
+                                    "score": 0.57,
+                                    "content": "- >",
+                                    "type": "inline_equation",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        293,
+                                        178,
+                                        316,
+                                        191
+                                    ],
+                                    "score": 1.0,
+                                    "content": "int",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 8,
+                            "is_list_start_line": true,
+                            "is_list_end_line": true
+                        },
+                        {
+                            "bbox": [
+                                142,
+                                190,
+                                449,
+                                201
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        142,
+                                        190,
+                                        279,
+                                        201
+                                    ],
+                                    "score": 1.0,
+                                    "content": "lambda n, xs: xs[n] if",
+                                    "type": "text",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        280,
+                                        191,
+                                        306,
+                                        201
+                                    ],
+                                    "score": 0.87,
+                                    "content": "\\mathrm { n } > = 0",
+                                    "type": "inline_equation",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        306,
+                                        190,
+                                        354,
+                                        201
+                                    ],
+                                    "score": 1.0,
+                                    "content": "and len",
+                                    "type": "text",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        355,
+                                        191,
+                                        389,
+                                        201
+                                    ],
+                                    "score": 0.71,
+                                    "content": "( \\mathsf { x s } ) > \\mathsf { n }",
+                                    "type": "inline_equation",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        389,
+                                        190,
+                                        449,
+                                        201
+                                    ],
+                                    "score": 1.0,
+                                    "content": "else Null",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 9,
+                            "is_list_end_line": true
+                        },
+                        {
+                            "bbox": [
+                                142,
+                                201,
+                                505,
+                                213
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        142,
+                                        201,
+                                        261,
+                                        213
+                                    ],
+                                    "score": 1.0,
+                                    "content": "Given an integer n and array",
+                                    "type": "text",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        262,
+                                        203,
+                                        275,
+                                        212
+                                    ],
+                                    "score": 0.53,
+                                    "content": "_ { \\textrm { x S } }",
+                                    "type": "inline_equation",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        275,
+                                        201,
+                                        325,
+                                        213
+                                    ],
+                                    "score": 1.0,
+                                    "content": ", returns the",
+                                    "type": "text",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        325,
+                                        202,
+                                        349,
+                                        212
+                                    ],
+                                    "score": 0.8,
+                                    "content": "( \\mathrm { n } { + } 1 )",
+                                    "type": "inline_equation",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        349,
+                                        201,
+                                        406,
+                                        213
+                                    ],
+                                    "score": 1.0,
+                                    "content": "-st element of",
+                                    "type": "text",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        406,
+                                        203,
+                                        419,
+                                        211
+                                    ],
+                                    "score": 0.63,
+                                    "content": "_ { \\textrm { x S } }",
+                                    "type": "inline_equation",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        420,
+                                        201,
+                                        505,
+                                        213
+                                    ],
+                                    "score": 1.0,
+                                    "content": ". (If the length of xs",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
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+                                213,
+                                400,
+                                223
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        142,
+                                        213,
+                                        242,
+                                        223
+                                    ],
+                                    "score": 1.0,
+                                    "content": "was less than or equal to",
+                                    "type": "text",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        242,
+                                        214,
+                                        249,
+                                        222
+                                    ],
+                                    "score": 0.27,
+                                    "content": "\\boldsymbol { \\mathrm { n } }",
+                                    "type": "inline_equation",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        249,
+                                        213,
+                                        400,
+                                        223
+                                    ],
+                                    "score": 1.0,
+                                    "content": ", the value NULL is returned instead.)",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 11,
+                            "is_list_end_line": true
+                        },
+                        {
+                            "bbox": [
+                                136,
+                                227,
+                                284,
+                                239
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        136,
+                                        227,
+                                        245,
+                                        239
+                                    ],
+                                    "score": 1.0,
+                                    "content": "• MINIMUM :: [int]",
+                                    "type": "text",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        245,
+                                        228,
+                                        260,
+                                        238
+                                    ],
+                                    "score": 0.72,
+                                    "content": "- >",
+                                    "type": "inline_equation",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        260,
+                                        227,
+                                        284,
+                                        239
+                                    ],
+                                    "score": 1.0,
+                                    "content": "int",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 12,
+                            "is_list_start_line": true,
+                            "is_list_end_line": true
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                238,
+                                389,
+                                250
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        238,
+                                        295,
+                                        250
+                                    ],
+                                    "score": 1.0,
+                                    "content": "lambda xs: min(xs) if len",
+                                    "type": "text",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        295,
+                                        239,
+                                        329,
+                                        249
+                                    ],
+                                    "score": 0.52,
+                                    "content": "( \\mathbf { \\nabla } _ { \\mathbf { X } } \\mathbf { S } ) > 0",
+                                    "type": "inline_equation",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        330,
+                                        238,
+                                        389,
+                                        250
+                                    ],
+                                    "score": 1.0,
+                                    "content": "else Null",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 13,
+                            "is_list_end_line": true
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                248,
+                                418,
+                                262
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        248,
+                                        418,
+                                        262
+                                    ],
+                                    "score": 1.0,
+                                    "content": "Given an array, returns its minimum (or NULL if the array is empty).",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 14,
+                            "is_list_end_line": true
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                264,
+                                286,
+                                276
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        264,
+                                        248,
+                                        276
+                                    ],
+                                    "score": 1.0,
+                                    "content": "• MAXIMUM :: [int]",
+                                    "type": "text",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        248,
+                                        266,
+                                        262,
+                                        275
+                                    ],
+                                    "score": 0.37,
+                                    "content": "- >",
+                                    "type": "inline_equation",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        263,
+                                        264,
+                                        286,
+                                        276
+                                    ],
+                                    "score": 1.0,
+                                    "content": "int",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 15,
+                            "is_list_start_line": true,
+                            "is_list_end_line": true
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                276,
+                                389,
+                                287
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        276,
+                                        294,
+                                        287
+                                    ],
+                                    "score": 1.0,
+                                    "content": "lambda xs: max(xs) if len",
+                                    "type": "text",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        295,
+                                        276,
+                                        329,
+                                        286
+                                    ],
+                                    "score": 0.56,
+                                    "content": "( \\mathbf { \\nabla } _ { \\mathbf { X } } \\mathbf { S } ) > 0",
+                                    "type": "inline_equation",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        330,
+                                        276,
+                                        389,
+                                        287
+                                    ],
+                                    "score": 1.0,
+                                    "content": "else Null",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 16,
+                            "is_list_end_line": true
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                286,
+                                419,
+                                299
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        286,
+                                        419,
+                                        299
+                                    ],
+                                    "score": 1.0,
+                                    "content": "Given an array, returns its maximum (or NULL if the array is empty).",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 17,
+                            "is_list_end_line": true
+                        },
+                        {
+                            "bbox": [
+                                140,
+                                302,
+                                291,
+                                313
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        140,
+                                        302,
+                                        242,
+                                        313
+                                    ],
+                                    "score": 1.0,
+                                    "content": "REVERSE :: [int]",
+                                    "type": "text",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        242,
+                                        303,
+                                        257,
+                                        312
+                                    ],
+                                    "score": 0.33,
+                                    "content": "- >",
+                                    "type": "inline_equation",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        257,
+                                        302,
+                                        291,
+                                        313
+                                    ],
+                                    "score": 1.0,
+                                    "content": "[int]",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 18,
+                            "is_list_end_line": true
+                        },
+                        {
+                            "bbox": [
+                                142,
+                                313,
+                                316,
+                                325
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        142,
+                                        313,
+                                        316,
+                                        325
+                                    ],
+                                    "score": 1.0,
+                                    "content": "lambda xs: list(reversed(xs))",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 19,
+                            "is_list_end_line": true
+                        },
+                        {
+                            "bbox": [
+                                142,
+                                324,
+                                357,
+                                336
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        142,
+                                        324,
+                                        357,
+                                        336
+                                    ],
+                                    "score": 1.0,
+                                    "content": "Given an array, returns its elements in reversed order.",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 20,
+                            "is_list_end_line": true
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                339,
+                                275,
+                                352
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        339,
+                                        275,
+                                        352
+                                    ],
+                                    "score": 1.0,
+                                    "content": "• SORT :: [int] -> [int]",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 21,
+                            "is_list_start_line": true,
+                            "is_list_end_line": true
+                        },
+                        {
+                            "bbox": [
+                                142,
+                                350,
+                                268,
+                                362
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        142,
+                                        350,
+                                        268,
+                                        362
+                                    ],
+                                    "score": 1.0,
+                                    "content": "lambda xs: sorted(xs)",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 22,
+                            "is_list_end_line": true
+                        },
+                        {
+                            "bbox": [
+                                142,
+                                361,
+                                380,
+                                373
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        142,
+                                        361,
+                                        380,
+                                        373
+                                    ],
+                                    "score": 1.0,
+                                    "content": "Given an array, return its elements in non-decreasing order.",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 23,
+                            "is_list_end_line": true
+                        },
+                        {
+                            "bbox": [
+                                140,
+                                376,
+                                260,
+                                389
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        140,
+                                        376,
+                                        221,
+                                        389
+                                    ],
+                                    "score": 1.0,
+                                    "content": "SUM :: [int]",
+                                    "type": "text",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        221,
+                                        378,
+                                        236,
+                                        387
+                                    ],
+                                    "score": 0.3,
+                                    "content": "- >",
+                                    "type": "inline_equation",
+                                    "cross_page": true
+                                },
+                                {
+                                    "bbox": [
+                                        237,
+                                        376,
+                                        260,
+                                        389
+                                    ],
+                                    "score": 1.0,
+                                    "content": "int",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 24,
+                            "is_list_end_line": true
+                        },
+                        {
+                            "bbox": [
+                                142,
+                                388,
+                                250,
+                                399
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        142,
+                                        388,
+                                        250,
+                                        399
+                                    ],
+                                    "score": 1.0,
+                                    "content": "lambda xs: sum(xs)",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 25,
+                            "is_list_end_line": true
+                        },
+                        {
+                            "bbox": [
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+                                397,
+                                469,
+                                411
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        397,
+                                        469,
+                                        411
+                                    ],
+                                    "score": 1.0,
+                                    "content": "Given an array, returns the sum of its elements. (The sum of an empty array is 0.)",
+                                    "type": "text",
+                                    "cross_page": true
+                                }
+                            ],
+                            "index": 26,
+                            "is_list_end_line": true
+                        }
+                    ],
+                    "index": 44.5,
+                    "bbox_fs": [
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+                        662,
+                        429,
+                        733
+                    ]
+                }
+            ]
+        },
+        {
+            "preproc_blocks": [
+                {
+                    "type": "text",
+                    "bbox": [
+                        131,
+                        82,
+                        505,
+                        410
+                    ],
+                    "lines": [
+                        {
+                            "bbox": [
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+                                95
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        82,
+                                        213,
+                                        95
+                                    ],
+                                    "score": 1.0,
+                                    "content": "• TAKE :: int",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                        83,
+                                        228,
+                                        93
+                                    ],
+                                    "score": 0.32,
+                                    "content": "- >",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                        82,
+                                        306,
+                                        95
+                                    ],
+                                    "score": 1.0,
+                                    "content": "[int] -> int",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 0
+                        },
+                        {
+                            "bbox": [
+                                142,
+                                93,
+                                263,
+                                107
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        142,
+                                        93,
+                                        263,
+                                        107
+                                    ],
+                                    "score": 1.0,
+                                    "content": "lambda n, xs: xs[:n]",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 1
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                104,
+                                505,
+                                117
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        104,
+                                        421,
+                                        117
+                                    ],
+                                    "score": 1.0,
+                                    "content": "Given an integer n and array xs, returns the array truncated after the",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        421,
+                                        106,
+                                        428,
+                                        115
+                                    ],
+                                    "score": 0.4,
+                                    "content": "\\boldsymbol { \\mathrm { n } }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        429,
+                                        104,
+                                        505,
+                                        117
+                                    ],
+                                    "score": 1.0,
+                                    "content": "-th element. (If the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 2
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                115,
+                                492,
+                                127
+                            ],
+                            "spans": [
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+                                    "bbox": [
+                                        141,
+                                        115,
+                                        492,
+                                        127
+                                    ],
+                                    "score": 1.0,
+                                    "content": "length of xs was no larger than n in the first place, it is returned without modification.)",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 3
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                131,
+                                306,
+                                142
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        131,
+                                        214,
+                                        142
+                                    ],
+                                    "score": 1.0,
+                                    "content": "• DROP :: int",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                        132,
+                                        229,
+                                        141
+                                    ],
+                                    "score": 0.43,
+                                    "content": "- >",
+                                    "type": "inline_equation"
+                                },
+                                {
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+                                        131,
+                                        267,
+                                        142
+                                    ],
+                                    "score": 1.0,
+                                    "content": "[int]",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                        132,
+                                        282,
+                                        141
+                                    ],
+                                    "score": 0.47,
+                                    "content": "- >",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        283,
+                                        131,
+                                        306,
+                                        142
+                                    ],
+                                    "score": 1.0,
+                                    "content": "int",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 4
+                        },
+                        {
+                            "bbox": [
+                                142,
+                                142,
+                                263,
+                                154
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        142,
+                                        142,
+                                        263,
+                                        154
+                                    ],
+                                    "score": 1.0,
+                                    "content": "lambda n, xs: xs[n:]",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                152,
+                                505,
+                                165
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        152,
+                                        258,
+                                        165
+                                    ],
+                                    "score": 1.0,
+                                    "content": "Given an integer n and array",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        258,
+                                        155,
+                                        271,
+                                        163
+                                    ],
+                                    "score": 0.36,
+                                    "content": "_ { \\textrm { x S } }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        271,
+                                        152,
+                                        505,
+                                        165
+                                    ],
+                                    "score": 1.0,
+                                    "content": ", returns the array with the first n elements dropped. (If the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 6
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                163,
+                                461,
+                                176
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        163,
+                                        461,
+                                        176
+                                    ],
+                                    "score": 1.0,
+                                    "content": "length of xs was no larger than n in the first place, an empty array is returned.)",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                178,
+                                316,
+                                191
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        178,
+                                        223,
+                                        191
+                                    ],
+                                    "score": 1.0,
+                                    "content": "• ACCESS :: int",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        224,
+                                        180,
+                                        239,
+                                        190
+                                    ],
+                                    "score": 0.52,
+                                    "content": "- >",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        239,
+                                        178,
+                                        277,
+                                        191
+                                    ],
+                                    "score": 1.0,
+                                    "content": "[int]",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        277,
+                                        180,
+                                        292,
+                                        190
+                                    ],
+                                    "score": 0.57,
+                                    "content": "- >",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        293,
+                                        178,
+                                        316,
+                                        191
+                                    ],
+                                    "score": 1.0,
+                                    "content": "int",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 8
+                        },
+                        {
+                            "bbox": [
+                                142,
+                                190,
+                                449,
+                                201
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        142,
+                                        190,
+                                        279,
+                                        201
+                                    ],
+                                    "score": 1.0,
+                                    "content": "lambda n, xs: xs[n] if",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        280,
+                                        191,
+                                        306,
+                                        201
+                                    ],
+                                    "score": 0.87,
+                                    "content": "\\mathrm { n } > = 0",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        306,
+                                        190,
+                                        354,
+                                        201
+                                    ],
+                                    "score": 1.0,
+                                    "content": "and len",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        355,
+                                        191,
+                                        389,
+                                        201
+                                    ],
+                                    "score": 0.71,
+                                    "content": "( \\mathsf { x s } ) > \\mathsf { n }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        389,
+                                        190,
+                                        449,
+                                        201
+                                    ],
+                                    "score": 1.0,
+                                    "content": "else Null",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
+                                142,
+                                201,
+                                505,
+                                213
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        142,
+                                        201,
+                                        261,
+                                        213
+                                    ],
+                                    "score": 1.0,
+                                    "content": "Given an integer n and array",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        262,
+                                        203,
+                                        275,
+                                        212
+                                    ],
+                                    "score": 0.53,
+                                    "content": "_ { \\textrm { x S } }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        275,
+                                        201,
+                                        325,
+                                        213
+                                    ],
+                                    "score": 1.0,
+                                    "content": ", returns the",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        325,
+                                        202,
+                                        349,
+                                        212
+                                    ],
+                                    "score": 0.8,
+                                    "content": "( \\mathrm { n } { + } 1 )",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        349,
+                                        201,
+                                        406,
+                                        213
+                                    ],
+                                    "score": 1.0,
+                                    "content": "-st element of",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        406,
+                                        203,
+                                        419,
+                                        211
+                                    ],
+                                    "score": 0.63,
+                                    "content": "_ { \\textrm { x S } }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        420,
+                                        201,
+                                        505,
+                                        213
+                                    ],
+                                    "score": 1.0,
+                                    "content": ". (If the length of xs",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
+                                142,
+                                213,
+                                400,
+                                223
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        142,
+                                        213,
+                                        242,
+                                        223
+                                    ],
+                                    "score": 1.0,
+                                    "content": "was less than or equal to",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        242,
+                                        214,
+                                        249,
+                                        222
+                                    ],
+                                    "score": 0.27,
+                                    "content": "\\boldsymbol { \\mathrm { n } }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        249,
+                                        213,
+                                        400,
+                                        223
+                                    ],
+                                    "score": 1.0,
+                                    "content": ", the value NULL is returned instead.)",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 11
+                        },
+                        {
+                            "bbox": [
+                                136,
+                                227,
+                                284,
+                                239
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        136,
+                                        227,
+                                        245,
+                                        239
+                                    ],
+                                    "score": 1.0,
+                                    "content": "• MINIMUM :: [int]",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        245,
+                                        228,
+                                        260,
+                                        238
+                                    ],
+                                    "score": 0.72,
+                                    "content": "- >",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        260,
+                                        227,
+                                        284,
+                                        239
+                                    ],
+                                    "score": 1.0,
+                                    "content": "int",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 12
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                238,
+                                389,
+                                250
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        238,
+                                        295,
+                                        250
+                                    ],
+                                    "score": 1.0,
+                                    "content": "lambda xs: min(xs) if len",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        295,
+                                        239,
+                                        329,
+                                        249
+                                    ],
+                                    "score": 0.52,
+                                    "content": "( \\mathbf { \\nabla } _ { \\mathbf { X } } \\mathbf { S } ) > 0",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        330,
+                                        238,
+                                        389,
+                                        250
+                                    ],
+                                    "score": 1.0,
+                                    "content": "else Null",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 13
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                248,
+                                418,
+                                262
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        248,
+                                        418,
+                                        262
+                                    ],
+                                    "score": 1.0,
+                                    "content": "Given an array, returns its minimum (or NULL if the array is empty).",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 14
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                264,
+                                286,
+                                276
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        264,
+                                        248,
+                                        276
+                                    ],
+                                    "score": 1.0,
+                                    "content": "• MAXIMUM :: [int]",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        248,
+                                        266,
+                                        262,
+                                        275
+                                    ],
+                                    "score": 0.37,
+                                    "content": "- >",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        263,
+                                        264,
+                                        286,
+                                        276
+                                    ],
+                                    "score": 1.0,
+                                    "content": "int",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 15
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                276,
+                                389,
+                                287
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        276,
+                                        294,
+                                        287
+                                    ],
+                                    "score": 1.0,
+                                    "content": "lambda xs: max(xs) if len",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        295,
+                                        276,
+                                        329,
+                                        286
+                                    ],
+                                    "score": 0.56,
+                                    "content": "( \\mathbf { \\nabla } _ { \\mathbf { X } } \\mathbf { S } ) > 0",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        330,
+                                        276,
+                                        389,
+                                        287
+                                    ],
+                                    "score": 1.0,
+                                    "content": "else Null",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 16
+                        },
+                        {
+                            "bbox": [
+                                141,
+                                286,
+                                419,
+                                299
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        141,
+                                        286,
+                                        419,
+                                        299
+                                    ],
+                                    "score": 1.0,
+                                    "content": "Given an array, returns its maximum (or NULL if the array is empty).",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 17
+                        },
+                        {
+                            "bbox": [
+                                140,
+                                302,
+                                291,
+                                313
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        140,
+                                        302,
+                                        242,
+                                        313
+                                    ],
+                                    "score": 1.0,
+                                    "content": "REVERSE :: [int]",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        242,
+                                        303,
+                                        257,
+                                        312
+                                    ],
+                                    "score": 0.33,
+                                    "content": "- >",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        257,
+                                        302,
+                                        291,
+                                        313
+                                    ],
+                                    "score": 1.0,
+                                    "content": "[int]",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 18
+                        },
+                        {
+                            "bbox": [
+                                142,
+                                313,
+                                316,
+                                325
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        142,
+                                        313,
+                                        316,
+                                        325
+                                    ],
+                                    "score": 1.0,
+                                    "content": "lambda xs: list(reversed(xs))",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
+                            "bbox": [
+                                142,
+                                324,
+                                357,
+                                336
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        142,
+                                        324,
+                                        357,
+                                        336
+                                    ],
+                                    "score": 1.0,
+                                    "content": "Given an array, returns its elements in reversed order.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 20
+                        },
+                        {
+                            "bbox": [
+                                132,
+                                339,
+                                275,
+                                352
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        132,
+                                        339,
+                                        275,
+                                        352
+                                    ],
+                                    "score": 1.0,
+                                    "content": "• SORT :: [int] -> [int]",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 21
+                        },
+                        {
+                            "bbox": [
+                                142,
+                                350,
+                                268,
+                                362
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        142,
+                                        350,
+                                        268,
+                                        362
+                                    ],
+                                    "score": 1.0,
+                                    "content": "lambda xs: sorted(xs)",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 22
+                        },
+                        {
+                            "bbox": [
+                                142,
+                                361,
+                                380,
+                                373
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        142,
+                                        361,
+                                        380,
+                                        373
+                                    ],
+                                    "score": 1.0,
+                                    "content": "Given an array, return its elements in non-decreasing order.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 23
+                        },
+                        {
+                            "bbox": [
+                                140,
+                                376,
+                                260,
+                                389
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        140,
+                                        376,
+                                        221,
+                                        389
+                                    ],
+                                    "score": 1.0,
+                                    "content": "SUM :: [int]",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        221,
+                                        378,
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diff --git a/parse/train/ByldLrqlx/ByldLrqlx_model.json b/parse/train/ByldLrqlx/ByldLrqlx_model.json
new file mode 100644
index 0000000000000000000000000000000000000000..25ba4723abde1b9d0a2e51e15c3b1d5453ff0742
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diff --git a/parse/train/ByxXZpVtPB/ByxXZpVtPB.md b/parse/train/ByxXZpVtPB/ByxXZpVtPB.md
new file mode 100644
index 0000000000000000000000000000000000000000..5c9c2ef86ffaa7fce669fe4a7863a1b95de2b550
--- /dev/null
+++ b/parse/train/ByxXZpVtPB/ByxXZpVtPB.md
@@ -0,0 +1,204 @@
+# HOMOGENEOUS LINEAR INEQUALITY CONSTRAINTS FOR NEURAL NETWORK ACTIVATIONS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We propose a method to impose homogeneous linear inequality constraints of the form $A x \le 0$ on neural network activations. The proposed method allows a datadriven training approach to be combined with modeling prior knowledge about the task. One way to achieve this task is by means of a projection step at test time after unconstrained training. However, this is an expensive operation. By directly incorporating the constraints into the architecture, we can significantly speed-up inference at test time; for instance, our experiments show a speed-up of up to two orders of magnitude over a projection method. Our algorithm computes a suitable parameterization of the feasible set at initialization and uses standard variants of stochastic gradient descent to find solutions to the constrained network. Thus, the modeling constraints are always satisfied during training. Crucially, our approach avoids to solve an optimization problem at each training step or to manually trade-off data and constraint fidelity with additional hyperparameters. We consider constrained generative modeling as an important application domain and experimentally demonstrate the proposed method by constraining a variational autoencoder.
+
+# 1 INTRODUCTION
+
+Deep learning models (LeCun et al., 2015) have demonstrated remarkable success in tasks that require exploitation of subtle correlations, such as computer vision (Krizhevsky et al., 2012) and sequence learning (Sutskever et al., 2014). Typically, humans have strong prior knowledge about a task, e.g., based on symmetry, geometry, or physics. Learning such a priori assumptions in a purely data-driven manner is inefficient and, in some situations, may not be feasible at all. While certain prior knowledge was successfully imposed – for example translational symmetry through convolutional architectures (LeCun et al., 1998) – incorporating more general modeling assumptions in the training of deep networks remains an open challenge. Recently, generative neural networks have advanced significantly (Goodfellow et al., 2014; Kingma & Welling, 2014). With such models, controlling the generative process beyond a data-driven, black-box approach is particularly important.
+
+In this paper, we present a method to impose prior knowledge through homogeneous linear inequality constraints of the form $A x \le 0$ on the activations of deep learning models. We directly impose these constraints through a suitable parameterization of the feasible set. This has several advantages:
+
+• The constraints are hard-constraints in the sense that they are satisfied at any point during training and inference.   
+• Inference on the constrained network incurs no overhead compared to unconstrained inference.   
+• There is no manual trade-off between constraint satisfaction and data representation.
+
+In summary, the main contribution of our method is a reparameterization that incorporates homogeneous linear inequality hard-constraints on neural network activations and allows for efficient test time predictions, i.e., our method is faster up to two orders of magnitude. The model can be optimized by standard variants of stochastic gradient descent. As an application in generative modeling, we demonstrate that our method is able to produce authentic samples from a variational autoencoder while satisfying the imposed constraints.
+
+![](images/6ff2de8059afd1546259716b3fa3bcc6f284f19dca82e98544dec5426da694f1.jpg)  
+Figure 1: Samples drawn from a variational autoencoder trained on MNIST without constraints (left) and with a checkerboard constraint on the output domain (right). For a pixel intensity domain $[ - 1 , 1 ]$ , the checkerboard constraint forces the image tiles to have average positive or negative brightness.
+
+# 2 RELATED WORK
+
+Various works have introduced methods to impose some type of hard constraint on neural network activations. This differs from a classical constrained optimization problem (Nocedal & Wright, 2006) in that the constraints are on the image of a parameterized function rather than on the neural network parameters.
+
+Marquez-Neila et al. (2017) formulated generic differentiable equality constraints as soft constraints ´ and employed a Lagrangian approach to train their model. While this is a principled approach to constrained optimization, it does not scale well to practical deep neural network models with their vast number of parameters. To make their method computationally tractable, a subset of the constraints is selected at each training step. In addition, these constraints are locally linearized; thus, there is no guarantee that this subset will be satisfied after a parameter update.
+
+For the specific problem of weakly supervised segmentation, Pathak et al. (2015) proposed an optimization scheme that alternates between optimizing the deep learning model and fitting a constrained distribution to these intermediate models. However, this method involves solving a (convex) optimization problem at each training step. Furthermore, the overall convergence path depends on how the alternating optimization steps are combined, which introduces an additional hyperparameter that must be tuned. Briq et al. (2018) approached the weakly supervised segmentation problem with a layer that implements the orthogonal projection onto a simplex, thereby directly constraining the activations to a probability distribution. This optimization problem can be solved efficiently, but does not generalize to other types of inequality constraints.
+
+OptNet, an approach to solve a generic quadratic program as a differentiable network layer, was proposed by Amos & Kolter (2017). OptNet backpropagates through the first-order optimality conditions of the quadratic program, and linear inequality constraints can be enforced as a special case. The formulation is flexible; however, it scales cubically with the number of variables and constraints. Thus, it becomes prohibitively expensive to train large-scale deep learning models.
+
+Finally, several works have proposed handcrafted solutions for specific applications, such as skeleton prediction (Zhou et al., 2016) and prediction of rigid body motion (Byravan & Fox, 2017). In contrast, to avoid laborious architecture design, we argue for the value of generically modeling constraint classes. In practice, this makes constraint methods more accessible for a broader class of problems.
+
+Contribution In this work, we tackle the problem of imposing homogeneous linear inequality constraints on neural network activations. Rather than solving an optimization problem during training, we split this task into a feasibility step at initialization and an optimality step during training. At initialization, we compute a suitable parameterization of the constraint set (a polyhedral cone) and use the neural network training algorithm to find a good solution within this feasible set. Conceptually, we are trading-off computational cost during initialization to obtain a model that has no overhead at test time. The proposed method is implemented as a neural network layer that is specified by a set of homogeneous linear inequalities and whose output parameterizes the feasible set.
+
+We consider a generic $L$ layer neural network $F _ { \theta }$ with model parameters $\theta$ for inputs $x$ as follows:
+
+$$
+F _ { \theta } ( x ) = f _ { \theta _ { L } } ^ { ( L ) } ( \sigma ( f _ { \theta _ { L - 1 } } ^ { ( L - 1 ) } ( \sigma ( \dots f _ { \theta _ { 1 } } ^ { ( 1 ) } ( x ) \dots ) ) ) ) ,
+$$
+
+where $f _ { \theta _ { l } } ^ { ( l ) }$ are affine functions, e.g., a fully-connected or convolutional layer, and $\sigma$ is an elementwise non-linearity1, e.g., a sigmoid or rectified linear unit (ReLU). In supervised learning, training targets $y$ are known and a loss $\mathcal { L } _ { y } ( F _ { \theta } ( x ) )$ is minimized as a function of the network parameters $\theta$ . A typical loss for a classification task is the cross entropy between the network output and the empirical target distribution, while the mean-squared error is commonly used for a regression task. The proposed method can be applied to constrain any linear activations $z ^ { ( l ) } = f _ { \theta _ { l } } ^ { ( l ) } ( a ^ { ( l - 1 ) } )$ or non-linear activations $a ^ { ( l ) } = \sigma ( z ^ { ( l ) } )$ . In most cases, one would like to constrain the output $F _ { \theta } ( x )$ .
+
+The feasible set for $m$ linear inequality constraints in $d$ dimensions is the convex polyhedron
+
+$$
+\mathcal { C } : = \left\{ z \bigg | A z \leq b , A \in \mathbb { R } ^ { m \times d } , b \in \mathbb { R } ^ { m } \right\} \subseteq \mathbb { R } ^ { d } .
+$$
+
+A suitable description of the convex polyhedron $\mathcal { C }$ is obtained by the decomposition theorem for polyhedra.
+
+Theorem 1 (Decomposition of polyhedra, Minkowski-Weyl). $A$ set $\mathcal { C } \subset \mathbb { R } ^ { d }$ is a convex polyhedron of the form (2) if and only $i f$
+
+$$
+\begin{array} { l } { { \displaystyle { \mathcal C } = \mathrm { c o n v } \big ( \upsilon _ { 1 } , \ldots , \upsilon _ { n } \big ) + \mathrm { c o n e } \big ( r _ { 1 } , \ldots , r _ { s } \big ) } \ ~ } \\ { { \displaystyle ~ = \left\{ \sum _ { i = 1 } ^ { n } \lambda _ { i } \upsilon _ { i } + \sum _ { j = 1 } ^ { s } \mu _ { j } r _ { j } \bigg | \lambda _ { i } , \mu _ { j } \geq 0 , \sum _ { i = 1 } ^ { n } \lambda _ { i } = 1 \right\} } \ } \end{array}
+$$
+
+for finitely many vertices $\{ v _ { 1 } , \ldots , v _ { n } \}$ and rays $\{ r _ { 1 } , \ldots , r _ { s } \}$
+
+Furthermore, $\mathcal { C } = \left\{ z | A z \leq 0 , A \in \mathbb { R } ^ { m \times d } \right\}$ if and only if
+
+$$
+\mathcal { C } = \mathrm { c o n e } ( r _ { 1 } , . . . , r _ { s } )
+$$
+
+for finitely many rays $\{ r _ { 1 } , \ldots , r _ { s } \}$
+
+The theorem states that an intersection of half-spaces (half-space or $\mathrm { H } \cdot$ -representation) can be written as the Minkowski sum of a convex combination of the polyhedron’s vertices and a conical combination of some rays (vertex or $\mathrm { V } .$ -representation). One can switch algorithmically between these two viewpoints via the double description method (Motzkin et al., 1953; Fukuda & Prodon, 1996), which we discuss in the following. Thus, the $\mathrm { H }$ -representation, which is natural when modeling inequality constraints, can be transformed into the V-representation, which can be incorporated into gradient-based neural network training.
+
+In this paper, we focus on homogeneous constraints of the form (4), for which the feasible set is a polyhedral cone. Due to the special structure of this set, we can avoid to work with the convex combination parameters in (3), which is numerically advantageous (Section 3.5), and we can efficiently combine modeling constraints and domain constraints, such as a $[ - 1 , 1 ]$ -pixel domain for images (Section 3.3). Such a polyhedral cone is shown in Figure 2.
+
+# 3.1 DOUBLE DESCRIPTION METHOD
+
+The double description method converts between the half-space and vertex representation of a system of linear inequalities. It was originally proposed by Motzkin et al. (1953) and further refined by Fukuda & Prodon (1996).2 Here, we are only interested in the conversion from $\mathrm { H } \cdot$ -representation to V-representation for homogeneous constraints (4),
+
+$$
+\mathcal { H } \to \mathrm { c o n e } ( r _ { 1 } , \dots , r _ { s } ) \ .
+$$
+
+![](images/e34f7d5efaaf63be07414ce913edd9a6616ef7e1dcd0ac505ab2653737bf016f.jpg)  
+Figure 2: Diagram illustrating an iteration of the double description method. Adding a constraint to the $k$ -constraint set $A ^ { k }$ at iteration $k + 1$ introduces a hyperplane $H$ . The intersection points of $H$ with the boundary of the current polyhedron $R ^ { k }$ (marked by $\circ ^ { \prime }$ ) are added as rays $r _ { 6 }$ and $r _ { 7 }$ to the polyhedral cone. The ray $r _ { 2 }$ is cut-off by the hyperplane $H$ and is removed from $R ^ { k }$ . The result is the next iterate Rk+1.
+
+The core algorithm proceeds as follows. Let the rows of $A$ define a set of homogeneous inequalities and let $R = [ r _ { 1 } , \ldots , r _ { s } ]$ be the matrix whose columns are the rays of the corresponding cone. Here, $( A , R )$ form a double description pair. The algorithm iteratively builds a double description pair $( A ^ { k + 1 } , R ^ { k + 1 } )$ from $( A ^ { k } , R ^ { k } )$ in the following manner. The rows in $A ^ { k }$ represent a $k$ -subset of the rows of $A$ and thus define a convex polyhedron associated with $R ^ { k }$ . Adding a single row to $A ^ { k }$ introduces an additional half-space constraint, which corresponds to a hyperplane. If the vector $r _ { i } - r _ { j }$ for two columns $r _ { i } , r _ { j }$ of $\bar { \boldsymbol { R } } ^ { k }$ intersects with this hyperplane then this intersection point is added to $R ^ { k }$ . Existing rays that are cut-off by the additional hyperplane are removed from $R ^ { k }$ . The result is the double description pair $( A ^ { k + 1 } , \dot { R } ^ { k + 1 } )$ . This procedure is shown in Figure 2.
+
+Adding a hyperplane might drastically increase the number of rays in intermediate representations, which, in turn, contribute combinatorically in the subsequent iteration. In fact, there exist worst case polyhedra for which the algorithm has exponential run time as a function of the number of inequalities and the input dimension, as well as the number of rays (Dyer, 1983; Bremner, 1999). Overall, one can expect the algorithm to be efficient only for problems with a reasonably small number $m$ of inequalities and dimension $d$ .
+
+# 3.2 INTEGRATION IN NEURAL NETWORK ARCHITECTURES
+
+We parameterize the homogeneous form (4) via a neural network layer. This layer takes as input some (latent) representation of the data, which is mapped to activations satisfying the desired hard constraints. The algorithm is provided with the $_ \mathrm { H }$ -representation of linear inequality constraints, i.e., a matrix $A \in \breve { \mathbb { R } } ^ { m \times d }$ for $m$ constraints in $d$ dimensions to specify the feasible set (4). At initialization, we convert this to the $\mathrm { V } .$ -representation via the double description method (Section 3.1). This corresponds to computing the set of rays $\{ r _ { 1 } , \ldots , r _ { s } \}$ to represent the polyhedral cone. During training, the neural network training algorithm is used to optimize within in the feasible set. There are two critical aspects in this procedure. First, as outlined in Section 3.1, the run-time complexity of the double description method may be prohibitive. Conceptually, the proposed approach allows for significant compute time at initialization to obtain an algorithm that is very efficient at training and test time. Second, we must ensure that the mapping from the latent representation to the parameters integrates well with the training algorithm. We assume that the model is trained with gradient-based backpropagation, as is common for current deep learning applications. The constraint layer comprises a batch normalization layer and an affine mapping (fully-connected layer with biases) followed by the element-wise absolute value function that ensures the non-negativity required by the conical combination parameters. In theory, any function $f : \mathbb { R } \to \mathbb { R } _ { \geq 0 }$ would fulfill this requirement; however, care must be taken to not interfere with backpropagated gradients.
+
+# 3.3 COMBINING MODELING AND DOMAIN CONSTRAINTS
+
+Domain constraints are often formulated as unit box constraints, $B : = \{ x \in \mathbb { R } ^ { d } | - 1 \leq x _ { i } \leq 1 \}$ , such as a pixel domain for images. Box constraints are particularly unfit to be converted using the double description method because the number of vertices is exponential in the dimension. Therefore, we distinguish modeling constraints and domain constraints and only convert the former into $\mathrm { V } .$ - representation. Based on this representation, we obtain a point in the modeling constraint set, $x \in { \mathcal { C } }$ . However, this point may not be in the unit box $\boldsymbol { B }$ . To arrive at a point in the intersection $\mathcal { C } \cap \mathcal { B }$ , we normalize $x$ by its infinity norm if $x \notin B$ , $\hat { x } = x / \operatorname* { m a x } \{ \| x \| _ { \infty } , 1 \bar  \}$ . Indeed, $\hat { x } \in \mathcal { C } \cap B$ since scaling by a positive constant remains in the cone, i.e., if $x \in { \mathcal { C } }$ , then $\alpha x \in \mathcal { C } \forall \alpha \geq 0$ .
+
+# 3.4 APPLICATIONS OF HOMOGENEOUS LINEAR INEQUALITY CONSTRAINTS
+
+A natural application of constraints of the form $A x \le 0$ is a parameterization of a set of binary classifiers. If each row $a _ { i }$ of $A$ is such a binary classifier, then the method presented in this paper parameterizes the set $\{ x | a _ { i } ^ { T } x \le 0 \ \forall i \}$ . Consequently, it can be guaranteed that neural network activations satisfy a set of binary criteria. Another domain is to express certain direct relations between neural network activations. Notably, one can guarantee mathematical properties such as monotonicity via $x _ { i + 1 } \geq x _ { i }$ and convexity via $x _ { i + 1 } - 2 x _ { i } + x _ { i - 1 } \geq 0$ .
+
+# 3.5 EXTENSION TO GENERAL LINEAR INEQUALITY CONSTRAINTS
+
+The proposed method takes advantage of the special structure of a polyhedral cone to efficiently combine modeling and domain constraints (Section 3.3). General linear inequality constraints of the form $A x \leq b$ without restrictions on $A$ and $b$ possibly require the conic and convex component of (3) for their $\mathrm { v } .$ -representation. The main approach of this paper may be used in this case, i.e., our layer additionally needs to predict convex combination parameters. However, we observed slow convergence, which we ascribe to the simplex parameterization for the convex combination parameters. We used a softmax function $f ( x ) _ { i } = { \mathrm { { \bar { e x p } } } ( x _ { i } ) } / { \sum _ { j = 1 } ^ { m } \exp ( x _ { j } ) }$ to enforce the constraints $\begin{array} { r } { \lambda _ { i } \ge 0 , \sum _ { i = 1 } ^ { m } \lambda _ { i } = 1 } \end{array}$ of the convex combination parameters in (3). This function has vanishing gradients when one $x _ { i }$ is significantly greater than the other vector entries. Furthermore, this most general setting does not allow for efficient incorporation of domain constraints, as this would require an efficient parameterization of the intersection of a general convex polyhedron and the unit box.
+
+# 4 NUMERICAL RESULTS
+
+We compare the proposed constraint parameterization algorithm with an algorithm that trains without constraints, but requires a projection step at test time. We call this latter algorithm test time projection. We analyze these algorithms in two different settings. In an initial experiment, we learn the orthogonal projection onto a constraint set to demonstrate properties of these algorithms. Here, the result can be compared to the optimal solution of the convex optimization problem. In a second experiment, consistent with our motivation to constrain the output of generative models, we apply these algorithms to a variational autoencoder. Finally, we evaluate the running time of inference for these problems and show that the proposed algorithm is significantly more efficient compared to the test time projection method.
+
+We used the MNIST dataset (LeCun et al.) for both experiments (59000 training, 1000 validation, and 10000 test samples). We chose PyTorch (Paszke et al., 2017) for our implementation3 and all experiments were performed on a single Nvidia Titan X GPU. All networks were optimized with the Adam optimizer and we evaluated learning rates in the range $[ 1 0 ^ { - 5 } , 1 0 ^ { - 3 } ]$ . The initial learning rate was annealed by a factor of $1 / 2$ if progress on the validation loss stagnated for more than 5 epochs. We used OSQP (Stellato et al., 2017) as an efficient solver to compute orthogonal projections.
+
+Both experiments were performed with a checkerboard constraint with 16 tiles, where neighboring tiles are constrained to be on average either below or above pixel domain midpoint. For a $[ - 1 , 1 ]$ -pixel domain, the tiles’ average intensity is positive or negative, respectively. The initial computational cost of converting these constraints into $\mathrm { v } .$ -representation via the double description method is negligible (less than 1s). We observed that it is numerically advantageous to activate unit box scaling after the constraint parameterization model was initially optimized only with modeling constraints for a specified number of epochs.
+
+One might consider OptNet (Amos & Kolter, 2017) and an analogous version of the method introduced by Pathak et al. (2015) as baselines. However, these approaches incur a significant drawback for the setup presented in this paper as they are are computationally expensive at training time. An OptNet layer solves a generic quadratic program as a differentiable network layer, which scales cubically with the number of variables and constraints. The method by Pathak et al. (2015) for the regression problems in this paper alternates between optimization steps in the network parameters via a variant of stochastic gradient descent and projecting the network output onto the constraints, which is computationally expensive.
+
+# 4.1 ORTHOGONAL PROJECTION ONTO A CONSTRAINT SET
+
+We learn an orthogonal projection to demonstrate general properties of both algorithms. For given linear inequalities specified in H-representation, we solve the following problem:
+
+$$
+\operatorname* { m i n } _ { z \in \mathbb { R } ^ { d } } \left\| z - y \right\| _ { 2 } \quad \mathrm { s . t . } A z \leq 0 \qquad ,
+$$
+
+where $y$ is an MNIST image. Here, the problem is convex; therefore, the global optimum can be readily computed and compared to the performance of the learning algorithms. In this setting, we can expect that training an unconstrained network with subsequent projection onto the constraint set at test time yields good results, which can be seen as follows. Let $\begin{array} { r } { \mathcal { P } c ( y ) : = \arg \operatorname* { m i n } _ { z \in \mathcal { C } } \| z - y \| _ { 2 } } \end{array}$ be the orthogonal projection onto the constraint set $\mathcal { C }$ and denote the mean-squared error as $\mathcal { L } _ { y } ( x ) : =$ $\| x - y \| _ { 2 }$ . Both mappings are Lipschitz continuous with Lipschitz constant $L = 1$ . Consequently, for an output $\hat { y }$ of an unconstrained model,
+
+$$
+\begin{array} { r } { \displaystyle \Big | \mathcal { L } _ { y } ( \mathcal { P } _ { \mathcal { C } } ( \hat { y } ) ) - \mathcal { L } _ { y } ( \mathcal { P } _ { \mathcal { C } } ( y ) ) \Big | \le \left\| \mathcal { P } _ { \mathcal { C } } ( \hat { y } ) - \mathcal { P } _ { \mathcal { C } } ( y ) \right\| _ { 2 } \le \left\| \hat { y } - y \right\| _ { 2 } , } \end{array}
+$$
+
+where, by definition, the term $\mathcal { L } _ { y } ( \mathcal { P } c ( y ) )$ is the optimal value of problem (6). The training algorithm fits $\hat { y }$ to $y$ ; therefore, projecting the unconstrained output $\hat { y }$ onto the constraint set will yield an objective value that is close to the optimal value of the constrained optimization problem.
+
+To have a comparable number of parameters for both methods, we use a single fully-connected layer in both cases. For the unconstrained model, we employ an $F C ( 7 8 4 , 7 8 4 )$ layer, and for the constrained model we employ an $F C ( 7 8 4 , n _ { r } )$ layer with $n _ { r } = 1 5 5 2$ many rays to represent the constraint set in V-representation. Additionally, the constraint layer first applies a batch normalization operation (Ioffe & Szegedy, 2015). Both models were optimized with an initial learning rate of $1 0 ^ { - 4 }$ , which was annealed by a factor of 0.1 if progress on the validation loss stagnated for more than 5 epochs. The batch size was chosen to be 256. The unit box constraints were activated after 25 epochs. Additionally, the data for training the model with all constraints being active is shown. This mode eventually results in worse generalization. Figure 3 shows that the mean-squared validation objective for both algorithms converges close to the average optimum. The constraint parameterization method has a larger variance and optimality gap, which hints at the numerical difficulty of training the constrained network. To be precise, the best average validation error during training is within $9 \%$ of the optimum for the constraint parameterization method and within $1 \%$ of the optimum for the test time projection method. Figure 4 shows a test set sample and the respective output of the learned models.
+
+![](images/ccb26edf90e7c914ec8c93ff4de1521e67d3e69261ccf2961a943d7f3e77649f.jpg)  
+Figure 3: Mean-squared validation loss averaged over all pixels for 10 runs; shaded area denotes standard deviation. The objective function (6) is computed on a held-out validation set for the proposed constraint parameterization method and unconstrained optimization with subsequent test time projection. The average optimum over the validation set is obtained as a solution to a convex optimization problem. For the box delay curve, the box constraints are activated after 25 epochs (after $\sim \mathrm { 3 0 s }$ ), which results in better generalization. The best average validation error during training is within $9 \%$ of the optimum for the constraint parameterization method with box constraint delay and within $1 \%$ of the optimum for the test time projection method.
+
+![](images/49fc0106d7c90222b808e52dcf5a411c2b40977fc67fcc1408d6bbddb9492dbf.jpg)  
+Figure 4: Learning to solve the orthogonal projection onto a constraint set as defined in (6). From left to right: MNIST sample from a test set, optimal projection by solving a quadratic program, constraint parameterization model inference, and test time projection model inference.
+
+# 4.2 CONSTRAINED GENERATIVE MODELING WITH VARIATIONAL AUTOENCODERS
+
+Variational autoencoders (VAE) are a class of generative models that are jointly trained to encode observations into latent variables via an encoder or inference network and decode observations from latent variables using a decoder or generative network (Kingma & Welling, 2014). We base our implementation on (Baumgartner, 2018). The model has a fully-connected architecture: ¨
+
+Here, $\mathrm { R e L U } ( x ) = \operatorname* { m a x } ( 0 , x )$ and the sigmoid non-linearity takes the form $\sigma ( x ) = 1 / ( 1 { + } \mathrm { e x p } ( - x ) )$ . In contrast to a standard VAE, we constrain the samples generated by the model to obey a checkerboard constraint. The model was optimized with an initial learning rate of $1 0 ^ { - 4 }$ , which was annealed by a factor of 0.1 if progress on the validation loss stagnated for more than 5 epochs. The batch size was chosen to be 64. The model was trained for 200 epochs while the unit box constraints were activated after 100 epochs. To generate images, we sample the latent space prior $z \sim \mathcal { N } ( 0 , I )$ and evaluate the decoding neural network (Figure 5). The model is able to sample authentic digits while obeying the checkerboard constraint.
+
+# 4.3 FAST INFERENCE WITH CONSTRAINED NEURAL NETWORKS
+
+The main advantage of the proposed method over a simple projection method is a vast speed-up at test time. Since the constraint is incorporated into the neural network architecture, a forward pass has almost no overhead compared to an unconstrained network. On the other hand, for a network that was trained without constraints, a final projection step is necessary; this requires solving a convex optimization problem, which is relatively costly. Table 1 shows inference times for both models for the above numerical experiments. The constraint parameterization approach is up to two orders of magnitude faster at test time compared to the test time projection algorithm.
+
+![](images/02f0cfe4745a3f1d0401396a1e80304dc0a37a9588d7f9d2d50cf658cde58b72.jpg)  
+Figure 5: Samples from a constrained variational autoencoder trained with the test time projection method and our constraint parameterization method. The images represent authentic digits while satisfying the imposed checkerboard constraint. Inference is significantly faster using our method.
+
+Table 1: Inference time for test time projection and constraint parameterization methods. Mean and standard deviation of running times are computed over 100 runs of 59000 samples with a batch size of 256.   
+
+<table><tr><td>METHOD</td><td>PROJECTION</td><td>VAE</td></tr><tr><td>Test time projection</td><td>82±1 s</td><td>40±1s</td></tr><tr><td>Constraint parameterization (ours)</td><td>0.46 ± 0.02 s</td><td>0.75 ± 0.04 s</td></tr></table>
+
+# 5 CONCLUSION
+
+To combine a data-driven task with modeling constraints, we have developed a method to impose homogeneous linear inequality constraints on neural network activations. At initialization, a suitable parameterization is computed and subsequently a standard variant of stochastic gradient descent is used to train the reparameterized network. In this way, we can efficiently guarantee that network activations – in the final or any intermediate layer – satisfy the constraints at any point during training. The main advantage of our method over simply projecting onto the feasible set after unconstrained training is a significant speed-up at test time of up to two orders of magnitude. An important application of the proposed method is generative modeling with prior assumptions. Therefore, we demonstrated experimentally that the proposed method can be used successfully to constrain the output of a variational autoencoder. Our method is implemented as a layer, which is simple to combine with existing and novel neural network architectures in modern deep learning frameworks and is therefore readily available in practice.
+
+# REFERENCES
+
+Brandon Amos and J. Zico Kolter. OptNet: Differentiable optimization as a layer in neural networks. In Proceedings of the 34th International Conference on Machine Learning (ICML 2017), pp. 136– 145, 2017.
+
+Tim Baumgartner.¨ VAE-CVAE-MNIST. https://github.com/timbmg/ VAE-CVAE-MNIST, 2018. commit: e4ba231.
+
+David Bremner. Incremental convex hull algorithms are not output sensitive. Discrete & Computational Geometry, 21(1):57–68, 1999.
+
+Rania Briq, Michael Moeller, and Juergen Gall. Convolutional Simplex Projection Network (CSPN) for Weakly Supervised Semantic Segmentation. BMVC 2018, 2018.
+
+Arunkumar Byravan and Dieter Fox. SE3-nets: Learning rigid body motion using deep neural networks. In 2017 IEEE International Conference on Robotics and Automation (ICRA), 2017.
+
+Martin E. Dyer. The complexity of vertex enumeration methods. Mathematics of Operations Research, 8(3):381–402, 1983.
+
+Komei Fukuda and Alain Prodon. Double description method revisited, pp. 91–111. Combinatorics and Computer Science: 8th Franco-Japanese and 4th Franco-Chinese Conference Brest, France, July 3–5, 1995 Selected Papers. Springer Berlin Heidelberg, 1996.
+
+Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in Neural Information Processing Systems 27 (NIPS 2014), pp. 2672–2680. 2014.
+
+Sergey Ioffe and Christian Szegedy. Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift. In Proceedings of the 32nd International Conference on Machine Learning, volume 37, pp. 448–456. PMLR, 07–09 Jul 2015.
+
+Diederik Kingma and Max Welling. Auto-encoding variational bayes. In International Conference on Learning Representations (ICLR 2014), 2014.
+
+Alex Krizhevsky, Ilya Sutskever, and Geoffrey Hinton. ImageNet classification with deep convolutional neural networks. In Proceedings of the 25th International Conference of Neural Information Processing Systems (NIPS 2012), 2012.
+
+Yann LeCun, Corinna Cortes, and Christopher Burges. The MNIST database of handwritten digits. URL http://yann.lecun.com/exdb/mnist/.
+
+Yann LeCun, Leon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.
+
+Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning. Nature, 521(7553):436–444, 2015.
+
+Pablo Marquez-Neila, Mathieu Salzmann, and Pascal Fua. Imposing hard constraints on deep net- ´ works: Promises and limitations. First Workshop on Negative Results in Computer Vision, CVPR 2017, 2017.
+
+T. S. Motzkin, H. Raiffa, G. L. Thompson, and R. M. Thrall. The double description method. In Contributions to the Theory of Games II, volume 8 of Ann. of Math. Stud., pp. 51–73. Princeton University Press, 1953.
+
+Jorge Nocedal and Stephen Wright. Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer, 2006.
+
+Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in PyTorch. Autodiff Workshop, NIPS 2017, 2017.
+
+Deepak Pathak, Philipp Krahenb ¨ uhl, and Trevor Darrell. Constrained Convolutional Neural Net- ¨ works for Weakly Supervised Segmentation. In International Conference on Computer Vision (ICCV 2015), 2015.
+
+Bartolomeo Stellato, Goran Banjac, Paul Goulart, Alberto Bemporad, and Stephen Boyd. OSQP: An Operator Splitting Solver for Quadratic Programs. ArXiv e-prints, 2017.
+
+Ilya Sutskever, Oriol Vinyals, and Quoc V Le. Sequence to Sequence Learning with Neural Networks. In Proceedings of the 27th International Conference of Neural Information Processing Systems (NIPS 2014), 2014.
+
+Xingyi Zhou, Xiao Sun, Wei Zhang, Shuang Liang, and Yichen Wei. Deep Kinematic Pose Regression. Workshop on Geometry Meets Deep Learning, ECCV 2016, 2016.
\ No newline at end of file
diff --git a/parse/train/ByxXZpVtPB/ByxXZpVtPB_content_list.json b/parse/train/ByxXZpVtPB/ByxXZpVtPB_content_list.json
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@@ -0,0 +1,1079 @@
+[
+    {
+        "type": "text",
+        "text": "HOMOGENEOUS LINEAR INEQUALITY CONSTRAINTS FOR NEURAL NETWORK ACTIVATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            823,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            171,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose a method to impose homogeneous linear inequality constraints of the form $A x \\le 0$ on neural network activations. The proposed method allows a datadriven training approach to be combined with modeling prior knowledge about the task. One way to achieve this task is by means of a projection step at test time after unconstrained training. However, this is an expensive operation. By directly incorporating the constraints into the architecture, we can significantly speed-up inference at test time; for instance, our experiments show a speed-up of up to two orders of magnitude over a projection method. Our algorithm computes a suitable parameterization of the feasible set at initialization and uses standard variants of stochastic gradient descent to find solutions to the constrained network. Thus, the modeling constraints are always satisfied during training. Crucially, our approach avoids to solve an optimization problem at each training step or to manually trade-off data and constraint fidelity with additional hyperparameters. We consider constrained generative modeling as an important application domain and experimentally demonstrate the proposed method by constraining a variational autoencoder. ",
+        "bbox": [
+            233,
+            267,
+            764,
+            488
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            516,
+            336,
+            531
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep learning models (LeCun et al., 2015) have demonstrated remarkable success in tasks that require exploitation of subtle correlations, such as computer vision (Krizhevsky et al., 2012) and sequence learning (Sutskever et al., 2014). Typically, humans have strong prior knowledge about a task, e.g., based on symmetry, geometry, or physics. Learning such a priori assumptions in a purely data-driven manner is inefficient and, in some situations, may not be feasible at all. While certain prior knowledge was successfully imposed – for example translational symmetry through convolutional architectures (LeCun et al., 1998) – incorporating more general modeling assumptions in the training of deep networks remains an open challenge. Recently, generative neural networks have advanced significantly (Goodfellow et al., 2014; Kingma & Welling, 2014). With such models, controlling the generative process beyond a data-driven, black-box approach is particularly important. ",
+        "bbox": [
+            174,
+            547,
+            825,
+            686
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we present a method to impose prior knowledge through homogeneous linear inequality constraints of the form $A x \\le 0$ on the activations of deep learning models. We directly impose these constraints through a suitable parameterization of the feasible set. This has several advantages: ",
+        "bbox": [
+            176,
+            694,
+            825,
+            736
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "• The constraints are hard-constraints in the sense that they are satisfied at any point during training and inference.   \n• Inference on the constrained network incurs no overhead compared to unconstrained inference.   \n• There is no manual trade-off between constraint satisfaction and data representation. ",
+        "bbox": [
+            215,
+            747,
+            825,
+            828
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In summary, the main contribution of our method is a reparameterization that incorporates homogeneous linear inequality hard-constraints on neural network activations and allows for efficient test time predictions, i.e., our method is faster up to two orders of magnitude. The model can be optimized by standard variants of stochastic gradient descent. As an application in generative modeling, we demonstrate that our method is able to produce authentic samples from a variational autoencoder while satisfying the imposed constraints. ",
+        "bbox": [
+            174,
+            840,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/6ff2de8059afd1546259716b3fa3bcc6f284f19dca82e98544dec5426da694f1.jpg",
+        "image_caption": [
+            "Figure 1: Samples drawn from a variational autoencoder trained on MNIST without constraints (left) and with a checkerboard constraint on the output domain (right). For a pixel intensity domain $[ - 1 , 1 ]$ , the checkerboard constraint forces the image tiles to have average positive or negative brightness. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            356,
+            101,
+            637,
+            203
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            310,
+            341,
+            327
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Various works have introduced methods to impose some type of hard constraint on neural network activations. This differs from a classical constrained optimization problem (Nocedal & Wright, 2006) in that the constraints are on the image of a parameterized function rather than on the neural network parameters. ",
+        "bbox": [
+            174,
+            349,
+            823,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Marquez-Neila et al. (2017) formulated generic differentiable equality constraints as soft constraints ´ and employed a Lagrangian approach to train their model. While this is a principled approach to constrained optimization, it does not scale well to practical deep neural network models with their vast number of parameters. To make their method computationally tractable, a subset of the constraints is selected at each training step. In addition, these constraints are locally linearized; thus, there is no guarantee that this subset will be satisfied after a parameter update. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "For the specific problem of weakly supervised segmentation, Pathak et al. (2015) proposed an optimization scheme that alternates between optimizing the deep learning model and fitting a constrained distribution to these intermediate models. However, this method involves solving a (convex) optimization problem at each training step. Furthermore, the overall convergence path depends on how the alternating optimization steps are combined, which introduces an additional hyperparameter that must be tuned. Briq et al. (2018) approached the weakly supervised segmentation problem with a layer that implements the orthogonal projection onto a simplex, thereby directly constraining the activations to a probability distribution. This optimization problem can be solved efficiently, but does not generalize to other types of inequality constraints. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "OptNet, an approach to solve a generic quadratic program as a differentiable network layer, was proposed by Amos & Kolter (2017). OptNet backpropagates through the first-order optimality conditions of the quadratic program, and linear inequality constraints can be enforced as a special case. The formulation is flexible; however, it scales cubically with the number of variables and constraints. Thus, it becomes prohibitively expensive to train large-scale deep learning models. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Finally, several works have proposed handcrafted solutions for specific applications, such as skeleton prediction (Zhou et al., 2016) and prediction of rigid body motion (Byravan & Fox, 2017). In contrast, to avoid laborious architecture design, we argue for the value of generically modeling constraint classes. In practice, this makes constraint methods more accessible for a broader class of problems. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Contribution In this work, we tackle the problem of imposing homogeneous linear inequality constraints on neural network activations. Rather than solving an optimization problem during training, we split this task into a feasibility step at initialization and an optimality step during training. At initialization, we compute a suitable parameterization of the constraint set (a polyhedral cone) and use the neural network training algorithm to find a good solution within this feasible set. Conceptually, we are trading-off computational cost during initialization to obtain a model that has no overhead at test time. The proposed method is implemented as a neural network layer that is specified by a set of homogeneous linear inequalities and whose output parameterizes the feasible set. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We consider a generic $L$ layer neural network $F _ { \\theta }$ with model parameters $\\theta$ for inputs $x$ as follows: ",
+        "bbox": [
+            171,
+            132,
+            816,
+            147
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0ea9f946da75e03aa20d0e9e0bf4f7ce7c4abdbc95f203880c817c6911ffa619.jpg",
+        "text": "$$\nF _ { \\theta } ( x ) = f _ { \\theta _ { L } } ^ { ( L ) } ( \\sigma ( f _ { \\theta _ { L - 1 } } ^ { ( L - 1 ) } ( \\sigma ( \\dots f _ { \\theta _ { 1 } } ^ { ( 1 ) } ( x ) \\dots ) ) ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            151,
+            653,
+            174
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $f _ { \\theta _ { l } } ^ { ( l ) }$ are affine functions, e.g., a fully-connected or convolutional layer, and $\\sigma$ is an elementwise non-linearity1, e.g., a sigmoid or rectified linear unit (ReLU). In supervised learning, training targets $y$ are known and a loss $\\mathcal { L } _ { y } ( F _ { \\theta } ( x ) )$ is minimized as a function of the network parameters $\\theta$ . A typical loss for a classification task is the cross entropy between the network output and the empirical target distribution, while the mean-squared error is commonly used for a regression task. The proposed method can be applied to constrain any linear activations $z ^ { ( l ) } = f _ { \\theta _ { l } } ^ { ( l ) } ( a ^ { ( l - 1 ) } )$ or non-linear activations $a ^ { ( l ) } = \\sigma ( z ^ { ( l ) } )$ . In most cases, one would like to constrain the output $F _ { \\theta } ( x )$ . ",
+        "bbox": [
+            173,
+            179,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The feasible set for $m$ linear inequality constraints in $d$ dimensions is the convex polyhedron ",
+        "bbox": [
+            174,
+            295,
+            781,
+            310
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/948dee054635b23f991212ed43f04568c86d4c01a4fc9f9e26ec04531537dd24.jpg",
+        "text": "$$\n\\mathcal { C } : = \\left\\{ z \\bigg | A z \\leq b , A \\in \\mathbb { R } ^ { m \\times d } , b \\in \\mathbb { R } ^ { m } \\right\\} \\subseteq \\mathbb { R } ^ { d } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            339,
+            314,
+            656,
+            356
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A suitable description of the convex polyhedron $\\mathcal { C }$ is obtained by the decomposition theorem for polyhedra. ",
+        "bbox": [
+            173,
+            358,
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+            387
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1 (Decomposition of polyhedra, Minkowski-Weyl). $A$ set $\\mathcal { C } \\subset \\mathbb { R } ^ { d }$ is a convex polyhedron of the form (2) if and only $i f$ ",
+        "bbox": [
+            174,
+            388,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8ed8d31f28a2dfeb5fbe089d56a5410cabba0f133327c41c7397573ce679b99d.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\mathcal C } = \\mathrm { c o n v } \\big ( \\upsilon _ { 1 } , \\ldots , \\upsilon _ { n } \\big ) + \\mathrm { c o n e } \\big ( r _ { 1 } , \\ldots , r _ { s } \\big ) } \\ ~ } \\\\ { { \\displaystyle ~ = \\left\\{ \\sum _ { i = 1 } ^ { n } \\lambda _ { i } \\upsilon _ { i } + \\sum _ { j = 1 } ^ { s } \\mu _ { j } r _ { j } \\bigg | \\lambda _ { i } , \\mu _ { j } \\geq 0 , \\sum _ { i = 1 } ^ { n } \\lambda _ { i } = 1 \\right\\} } \\ } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            421,
+            673,
+            491
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "for finitely many vertices $\\{ v _ { 1 } , \\ldots , v _ { n } \\}$ and rays $\\{ r _ { 1 } , \\ldots , r _ { s } \\}$ ",
+        "bbox": [
+            171,
+            493,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, $\\mathcal { C } = \\left\\{ z | A z \\leq 0 , A \\in \\mathbb { R } ^ { m \\times d } \\right\\}$ if and only if ",
+        "bbox": [
+            173,
+            515,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/141c5be6d1de4af0a7e4ef915e90293fa419f7830888a5d6491eb626c5e933aa.jpg",
+        "text": "$$\n\\mathcal { C } = \\mathrm { c o n e } ( r _ { 1 } , . . . , r _ { s } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            426,
+            536,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "for finitely many rays $\\{ r _ { 1 } , \\ldots , r _ { s } \\}$ ",
+        "bbox": [
+            173,
+            556,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The theorem states that an intersection of half-spaces (half-space or $\\mathrm { H } \\cdot$ -representation) can be written as the Minkowski sum of a convex combination of the polyhedron’s vertices and a conical combination of some rays (vertex or $\\mathrm { V } .$ -representation). One can switch algorithmically between these two viewpoints via the double description method (Motzkin et al., 1953; Fukuda & Prodon, 1996), which we discuss in the following. Thus, the $\\mathrm { H }$ -representation, which is natural when modeling inequality constraints, can be transformed into the V-representation, which can be incorporated into gradient-based neural network training. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we focus on homogeneous constraints of the form (4), for which the feasible set is a polyhedral cone. Due to the special structure of this set, we can avoid to work with the convex combination parameters in (3), which is numerically advantageous (Section 3.5), and we can efficiently combine modeling constraints and domain constraints, such as a $[ - 1 , 1 ]$ -pixel domain for images (Section 3.3). Such a polyhedral cone is shown in Figure 2. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 DOUBLE DESCRIPTION METHOD ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            772,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The double description method converts between the half-space and vertex representation of a system of linear inequalities. It was originally proposed by Motzkin et al. (1953) and further refined by Fukuda & Prodon (1996).2 Here, we are only interested in the conversion from $\\mathrm { H } \\cdot$ -representation to V-representation for homogeneous constraints (4), ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/07953e3b7708217b360452246e0af32fd666d060d8d31f24becee732886e0ca0.jpg",
+        "text": "$$\n\\mathcal { H } \\to \\mathrm { c o n e } ( r _ { 1 } , \\dots , r _ { s } ) \\ .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            416,
+            858,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/e34f7d5efaaf63be07414ce913edd9a6616ef7e1dcd0ac505ab2653737bf016f.jpg",
+        "image_caption": [
+            "Figure 2: Diagram illustrating an iteration of the double description method. Adding a constraint to the $k$ -constraint set $A ^ { k }$ at iteration $k + 1$ introduces a hyperplane $H$ . The intersection points of $H$ with the boundary of the current polyhedron $R ^ { k }$ (marked by $\\circ ^ { \\prime }$ ) are added as rays $r _ { 6 }$ and $r _ { 7 }$ to the polyhedral cone. The ray $r _ { 2 }$ is cut-off by the hyperplane $H$ and is removed from $R ^ { k }$ . The result is the next iterate Rk+1. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            277,
+            103,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The core algorithm proceeds as follows. Let the rows of $A$ define a set of homogeneous inequalities and let $R = [ r _ { 1 } , \\ldots , r _ { s } ]$ be the matrix whose columns are the rays of the corresponding cone. Here, $( A , R )$ form a double description pair. The algorithm iteratively builds a double description pair $( A ^ { k + 1 } , R ^ { k + 1 } )$ from $( A ^ { k } , R ^ { k } )$ in the following manner. The rows in $A ^ { k }$ represent a $k$ -subset of the rows of $A$ and thus define a convex polyhedron associated with $R ^ { k }$ . Adding a single row to $A ^ { k }$ introduces an additional half-space constraint, which corresponds to a hyperplane. If the vector $r _ { i } - r _ { j }$ for two columns $r _ { i } , r _ { j }$ of $\\bar { \\boldsymbol { R } } ^ { k }$ intersects with this hyperplane then this intersection point is added to $R ^ { k }$ . Existing rays that are cut-off by the additional hyperplane are removed from $R ^ { k }$ . The result is the double description pair $( A ^ { k + 1 } , \\dot { R } ^ { k + 1 } )$ . This procedure is shown in Figure 2. ",
+        "bbox": [
+            174,
+            416,
+            825,
+            544
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Adding a hyperplane might drastically increase the number of rays in intermediate representations, which, in turn, contribute combinatorically in the subsequent iteration. In fact, there exist worst case polyhedra for which the algorithm has exponential run time as a function of the number of inequalities and the input dimension, as well as the number of rays (Dyer, 1983; Bremner, 1999). Overall, one can expect the algorithm to be efficient only for problems with a reasonably small number $m$ of inequalities and dimension $d$ . ",
+        "bbox": [
+            174,
+            550,
+            825,
+            633
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 INTEGRATION IN NEURAL NETWORK ARCHITECTURES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            660,
+            588,
+            672
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We parameterize the homogeneous form (4) via a neural network layer. This layer takes as input some (latent) representation of the data, which is mapped to activations satisfying the desired hard constraints. The algorithm is provided with the $_ \\mathrm { H }$ -representation of linear inequality constraints, i.e., a matrix $A \\in \\breve { \\mathbb { R } } ^ { m \\times d }$ for $m$ constraints in $d$ dimensions to specify the feasible set (4). At initialization, we convert this to the $\\mathrm { V } .$ -representation via the double description method (Section 3.1). This corresponds to computing the set of rays $\\{ r _ { 1 } , \\ldots , r _ { s } \\}$ to represent the polyhedral cone. During training, the neural network training algorithm is used to optimize within in the feasible set. There are two critical aspects in this procedure. First, as outlined in Section 3.1, the run-time complexity of the double description method may be prohibitive. Conceptually, the proposed approach allows for significant compute time at initialization to obtain an algorithm that is very efficient at training and test time. Second, we must ensure that the mapping from the latent representation to the parameters integrates well with the training algorithm. We assume that the model is trained with gradient-based backpropagation, as is common for current deep learning applications. The constraint layer comprises a batch normalization layer and an affine mapping (fully-connected layer with biases) followed by the element-wise absolute value function that ensures the non-negativity required by the conical combination parameters. In theory, any function $f : \\mathbb { R } \\to \\mathbb { R } _ { \\geq 0 }$ would fulfill this requirement; however, care must be taken to not interfere with backpropagated gradients. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 COMBINING MODELING AND DOMAIN CONSTRAINTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            104,
+            575,
+            117
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Domain constraints are often formulated as unit box constraints, $B : = \\{ x \\in \\mathbb { R } ^ { d } | - 1 \\leq x _ { i } \\leq 1 \\}$ , such as a pixel domain for images. Box constraints are particularly unfit to be converted using the double description method because the number of vertices is exponential in the dimension. Therefore, we distinguish modeling constraints and domain constraints and only convert the former into $\\mathrm { V } .$ - representation. Based on this representation, we obtain a point in the modeling constraint set, $x \\in { \\mathcal { C } }$ . However, this point may not be in the unit box $\\boldsymbol { B }$ . To arrive at a point in the intersection $\\mathcal { C } \\cap \\mathcal { B }$ , we normalize $x$ by its infinity norm if $x \\notin B$ , $\\hat { x } = x / \\operatorname* { m a x } \\{ \\| x \\| _ { \\infty } , 1 \\bar  \\}$ . Indeed, $\\hat { x } \\in \\mathcal { C } \\cap B$ since scaling by a positive constant remains in the cone, i.e., if $x \\in { \\mathcal { C } }$ , then $\\alpha x \\in \\mathcal { C } \\forall \\alpha \\geq 0$ . ",
+        "bbox": [
+            173,
+            128,
+            825,
+            241
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4 APPLICATIONS OF HOMOGENEOUS LINEAR INEQUALITY CONSTRAINTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            260,
+            704,
+            272
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A natural application of constraints of the form $A x \\le 0$ is a parameterization of a set of binary classifiers. If each row $a _ { i }$ of $A$ is such a binary classifier, then the method presented in this paper parameterizes the set $\\{ x | a _ { i } ^ { T } x \\le 0 \\ \\forall i \\}$ . Consequently, it can be guaranteed that neural network activations satisfy a set of binary criteria. Another domain is to express certain direct relations between neural network activations. Notably, one can guarantee mathematical properties such as monotonicity via $x _ { i + 1 } \\geq x _ { i }$ and convexity via $x _ { i + 1 } - 2 x _ { i } + x _ { i - 1 } \\geq 0$ . ",
+        "bbox": [
+            174,
+            284,
+            825,
+            368
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.5 EXTENSION TO GENERAL LINEAR INEQUALITY CONSTRAINTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            387,
+            640,
+            400
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The proposed method takes advantage of the special structure of a polyhedral cone to efficiently combine modeling and domain constraints (Section 3.3). General linear inequality constraints of the form $A x \\leq b$ without restrictions on $A$ and $b$ possibly require the conic and convex component of (3) for their $\\mathrm { v } .$ -representation. The main approach of this paper may be used in this case, i.e., our layer additionally needs to predict convex combination parameters. However, we observed slow convergence, which we ascribe to the simplex parameterization for the convex combination parameters. We used a softmax function $f ( x ) _ { i } = { \\mathrm { { \\bar { e x p } } } ( x _ { i } ) } / { \\sum _ { j = 1 } ^ { m } \\exp ( x _ { j } ) }$ to enforce the constraints $\\begin{array} { r } { \\lambda _ { i } \\ge 0 , \\sum _ { i = 1 } ^ { m } \\lambda _ { i } = 1 } \\end{array}$ of the convex combination parameters in (3). This function has vanishing gradients when one $x _ { i }$ is significantly greater than the other vector entries. Furthermore, this most general setting does not allow for efficient incorporation of domain constraints, as this would require an efficient parameterization of the intersection of a general convex polyhedron and the unit box. ",
+        "bbox": [
+            173,
+            411,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 NUMERICAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            588,
+            390,
+            604
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We compare the proposed constraint parameterization algorithm with an algorithm that trains without constraints, but requires a projection step at test time. We call this latter algorithm test time projection. We analyze these algorithms in two different settings. In an initial experiment, we learn the orthogonal projection onto a constraint set to demonstrate properties of these algorithms. Here, the result can be compared to the optimal solution of the convex optimization problem. In a second experiment, consistent with our motivation to constrain the output of generative models, we apply these algorithms to a variational autoencoder. Finally, we evaluate the running time of inference for these problems and show that the proposed algorithm is significantly more efficient compared to the test time projection method. ",
+        "bbox": [
+            173,
+            621,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We used the MNIST dataset (LeCun et al.) for both experiments (59000 training, 1000 validation, and 10000 test samples). We chose PyTorch (Paszke et al., 2017) for our implementation3 and all experiments were performed on a single Nvidia Titan X GPU. All networks were optimized with the Adam optimizer and we evaluated learning rates in the range $[ 1 0 ^ { - 5 } , 1 0 ^ { - 3 } ]$ . The initial learning rate was annealed by a factor of $1 / 2$ if progress on the validation loss stagnated for more than 5 epochs. We used OSQP (Stellato et al., 2017) as an efficient solver to compute orthogonal projections. ",
+        "bbox": [
+            174,
+            752,
+            823,
+            837
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Both experiments were performed with a checkerboard constraint with 16 tiles, where neighboring tiles are constrained to be on average either below or above pixel domain midpoint. For a $[ - 1 , 1 ]$ -pixel domain, the tiles’ average intensity is positive or negative, respectively. The initial computational cost of converting these constraints into $\\mathrm { v } .$ -representation via the double description method is negligible (less than 1s). We observed that it is numerically advantageous to activate unit box scaling after the constraint parameterization model was initially optimized only with modeling constraints for a specified number of epochs. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "One might consider OptNet (Amos & Kolter, 2017) and an analogous version of the method introduced by Pathak et al. (2015) as baselines. However, these approaches incur a significant drawback for the setup presented in this paper as they are are computationally expensive at training time. An OptNet layer solves a generic quadratic program as a differentiable network layer, which scales cubically with the number of variables and constraints. The method by Pathak et al. (2015) for the regression problems in this paper alternates between optimization steps in the network parameters via a variant of stochastic gradient descent and projecting the network output onto the constraints, which is computationally expensive. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 ORTHOGONAL PROJECTION ONTO A CONSTRAINT SET ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            311,
+            586,
+            325
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We learn an orthogonal projection to demonstrate general properties of both algorithms. For given linear inequalities specified in H-representation, we solve the following problem: ",
+        "bbox": [
+            174,
+            348,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/74fc678c3f170e82d36e5079493783b48948763b7547bae099077e8fe93f45c2.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { z \\in \\mathbb { R } ^ { d } } \\left\\| z - y \\right\\| _ { 2 } \\quad \\mathrm { s . t . } A z \\leq 0 \\qquad ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "where $y$ is an MNIST image. Here, the problem is convex; therefore, the global optimum can be readily computed and compared to the performance of the learning algorithms. In this setting, we can expect that training an unconstrained network with subsequent projection onto the constraint set at test time yields good results, which can be seen as follows. Let $\\begin{array} { r } { \\mathcal { P } c ( y ) : = \\arg \\operatorname* { m i n } _ { z \\in \\mathcal { C } } \\| z - y \\| _ { 2 } } \\end{array}$ be the orthogonal projection onto the constraint set $\\mathcal { C }$ and denote the mean-squared error as $\\mathcal { L } _ { y } ( x ) : =$ $\\| x - y \\| _ { 2 }$ . Both mappings are Lipschitz continuous with Lipschitz constant $L = 1$ . Consequently, for an output $\\hat { y }$ of an unconstrained model, ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/920f3f8c46ba6ecacc05ead4128f3ba6ca78c605e3362ed6634c6044f0a99b5d.jpg",
+        "text": "$$\n\\begin{array} { r } { \\displaystyle \\Big | \\mathcal { L } _ { y } ( \\mathcal { P } _ { \\mathcal { C } } ( \\hat { y } ) ) - \\mathcal { L } _ { y } ( \\mathcal { P } _ { \\mathcal { C } } ( y ) ) \\Big | \\le \\left\\| \\mathcal { P } _ { \\mathcal { C } } ( \\hat { y } ) - \\mathcal { P } _ { \\mathcal { C } } ( y ) \\right\\| _ { 2 } \\le \\left\\| \\hat { y } - y \\right\\| _ { 2 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            598,
+            707,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where, by definition, the term $\\mathcal { L } _ { y } ( \\mathcal { P } c ( y ) )$ is the optimal value of problem (6). The training algorithm fits $\\hat { y }$ to $y$ ; therefore, projecting the unconstrained output $\\hat { y }$ onto the constraint set will yield an objective value that is close to the optimal value of the constrained optimization problem. ",
+        "bbox": [
+            173,
+            665,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To have a comparable number of parameters for both methods, we use a single fully-connected layer in both cases. For the unconstrained model, we employ an $F C ( 7 8 4 , 7 8 4 )$ layer, and for the constrained model we employ an $F C ( 7 8 4 , n _ { r } )$ layer with $n _ { r } = 1 5 5 2$ many rays to represent the constraint set in V-representation. Additionally, the constraint layer first applies a batch normalization operation (Ioffe & Szegedy, 2015). Both models were optimized with an initial learning rate of $1 0 ^ { - 4 }$ , which was annealed by a factor of 0.1 if progress on the validation loss stagnated for more than 5 epochs. The batch size was chosen to be 256. The unit box constraints were activated after 25 epochs. Additionally, the data for training the model with all constraints being active is shown. This mode eventually results in worse generalization. Figure 3 shows that the mean-squared validation objective for both algorithms converges close to the average optimum. The constraint parameterization method has a larger variance and optimality gap, which hints at the numerical difficulty of training the constrained network. To be precise, the best average validation error during training is within $9 \\%$ of the optimum for the constraint parameterization method and within $1 \\%$ of the optimum for the test time projection method. Figure 4 shows a test set sample and the respective output of the learned models. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/ccb26edf90e7c914ec8c93ff4de1521e67d3e69261ccf2961a943d7f3e77649f.jpg",
+        "image_caption": [
+            "Figure 3: Mean-squared validation loss averaged over all pixels for 10 runs; shaded area denotes standard deviation. The objective function (6) is computed on a held-out validation set for the proposed constraint parameterization method and unconstrained optimization with subsequent test time projection. The average optimum over the validation set is obtained as a solution to a convex optimization problem. For the box delay curve, the box constraints are activated after 25 epochs (after $\\sim \\mathrm { 3 0 s }$ ), which results in better generalization. The best average validation error during training is within $9 \\%$ of the optimum for the constraint parameterization method with box constraint delay and within $1 \\%$ of the optimum for the test time projection method. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            303,
+            103,
+            684,
+            237
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/49fc0106d7c90222b808e52dcf5a411c2b40977fc67fcc1408d6bbddb9492dbf.jpg",
+        "image_caption": [
+            "Figure 4: Learning to solve the orthogonal projection onto a constraint set as defined in (6). From left to right: MNIST sample from a test set, optimal projection by solving a quadratic program, constraint parameterization model inference, and test time projection model inference. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            207,
+            395,
+            787,
+            500
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 CONSTRAINED GENERATIVE MODELING WITH VARIATIONAL AUTOENCODERS",
+        "text_level": 1,
+        "bbox": [
+            173,
+            582,
+            750,
+            594
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Variational autoencoders (VAE) are a class of generative models that are jointly trained to encode observations into latent variables via an encoder or inference network and decode observations from latent variables using a decoder or generative network (Kingma & Welling, 2014). We base our implementation on (Baumgartner, 2018). The model has a fully-connected architecture: ¨ ",
+        "bbox": [
+            174,
+            606,
+            825,
+            662
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Here, $\\mathrm { R e L U } ( x ) = \\operatorname* { m a x } ( 0 , x )$ and the sigmoid non-linearity takes the form $\\sigma ( x ) = 1 / ( 1 { + } \\mathrm { e x p } ( - x ) )$ . In contrast to a standard VAE, we constrain the samples generated by the model to obey a checkerboard constraint. The model was optimized with an initial learning rate of $1 0 ^ { - 4 }$ , which was annealed by a factor of 0.1 if progress on the validation loss stagnated for more than 5 epochs. The batch size was chosen to be 64. The model was trained for 200 epochs while the unit box constraints were activated after 100 epochs. To generate images, we sample the latent space prior $z \\sim \\mathcal { N } ( 0 , I )$ and evaluate the decoding neural network (Figure 5). The model is able to sample authentic digits while obeying the checkerboard constraint. ",
+        "bbox": [
+            174,
+            712,
+            825,
+            824
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 FAST INFERENCE WITH CONSTRAINED NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            179,
+            842,
+            625,
+            856
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The main advantage of the proposed method over a simple projection method is a vast speed-up at test time. Since the constraint is incorporated into the neural network architecture, a forward pass has almost no overhead compared to an unconstrained network. On the other hand, for a network that was trained without constraints, a final projection step is necessary; this requires solving a convex optimization problem, which is relatively costly. Table 1 shows inference times for both models for the above numerical experiments. The constraint parameterization approach is up to two orders of magnitude faster at test time compared to the test time projection algorithm. ",
+        "bbox": [
+            174,
+            867,
+            823,
+            924
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/02f0cfe4745a3f1d0401396a1e80304dc0a37a9588d7f9d2d50cf658cde58b72.jpg",
+        "image_caption": [
+            "Figure 5: Samples from a constrained variational autoencoder trained with the test time projection method and our constraint parameterization method. The images represent authentic digits while satisfying the imposed checkerboard constraint. Inference is significantly faster using our method. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            99,
+            803,
+            308
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/bdbfdf545c0c3896e0aaa1f7c2727179f15affae2568d69cb68ede176ba81380.jpg",
+        "table_caption": [
+            "Table 1: Inference time for test time projection and constraint parameterization methods. Mean and standard deviation of running times are computed over 100 runs of 59000 samples with a batch size of 256. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>METHOD</td><td>PROJECTION</td><td>VAE</td></tr><tr><td>Test time projection</td><td>82±1 s</td><td>40±1s</td></tr><tr><td>Constraint parameterization (ours)</td><td>0.46 ± 0.02 s</td><td>0.75 ± 0.04 s</td></tr></table>",
+        "bbox": [
+            250,
+            434,
+            746,
+            479
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            503,
+            826,
+            546
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            566,
+            318,
+            583
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To combine a data-driven task with modeling constraints, we have developed a method to impose homogeneous linear inequality constraints on neural network activations. At initialization, a suitable parameterization is computed and subsequently a standard variant of stochastic gradient descent is used to train the reparameterized network. In this way, we can efficiently guarantee that network activations – in the final or any intermediate layer – satisfy the constraints at any point during training. The main advantage of our method over simply projecting onto the feasible set after unconstrained training is a significant speed-up at test time of up to two orders of magnitude. An important application of the proposed method is generative modeling with prior assumptions. Therefore, we demonstrated experimentally that the proposed method can be used successfully to constrain the output of a variational autoencoder. Our method is implemented as a layer, which is simple to combine with existing and novel neural network architectures in modern deep learning frameworks and is therefore readily available in practice. ",
+        "bbox": [
+            173,
+            598,
+            825,
+            765
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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\ No newline at end of file
diff --git a/parse/train/ByxXZpVtPB/ByxXZpVtPB_middle.json b/parse/train/ByxXZpVtPB/ByxXZpVtPB_middle.json
new file mode 100644
index 0000000000000000000000000000000000000000..3c1e63a86eb9eb538be672ab8c0dac1e614529fc
--- /dev/null
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+                                    "content": "must be tuned. Briq et al. (2018) approached the weakly supervised segmentation problem with a",
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+                                    "content": "Various works have introduced methods to impose some type of hard constraint on neural network",
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+                                    "content": "2006) in that the constraints are on the image of a parameterized function rather than on the neural",
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+                                    "content": "For the specific problem of weakly supervised segmentation, Pathak et al. (2015) proposed an opti-",
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+                                    "content": "distribution to these intermediate models. However, this method involves solving a (convex) opti-",
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+                                    "score": 1.0,
+                                    "content": ". This procedure is shown in Figure 2.",
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+                                    "content": "Overall, one can expect the algorithm to be efficient only for problems with a reasonably small",
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+                                    "score": 1.0,
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+                                    "content": "for significant compute time at initialization to obtain an algorithm that is very efficient at train-",
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+                                    "content": "parameters integrates well with the training algorithm. We assume that the model is trained with",
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+                                    "content": ". This procedure is shown in Figure 2.",
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+                                    "content": "Overall, one can expect the algorithm to be efficient only for problems with a reasonably small",
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+                                    "content": "A \\in \\breve { \\mathbb { R } } ^ { m \\times d }",
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+                                    "score": 1.0,
+                                    "content": "tialization, we convert this to the",
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+                                    "content": "\\mathrm { V } .",
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+                                    "content": "for significant compute time at initialization to obtain an algorithm that is very efficient at train-",
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+                                    "content": "parameters integrates well with the training algorithm. We assume that the model is trained with",
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+                                    "score": 1.0,
+                                    "content": "with biases) followed by the element-wise absolute value function that ensures the non-negativity",
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+                                    "content": "would fulfill",
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+                                    "score": 0.93,
+                                    "content": "f ( x ) _ { i } = { \\mathrm { { \\bar { e x p } } } ( x _ { i } ) } / { \\sum _ { j = 1 } ^ { m } \\exp ( x _ { j } ) }",
+                                    "type": "inline_equation"
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+                                    "content": "to enforce the constraints",
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+                                    "score": 1.0,
+                                    "content": "of the convex combination parameters in (3). This function has vanishing",
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "is significantly greater than the other vector entries. Furthermore, this most",
+                                    "type": "text"
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+                        {
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+                                    "score": 1.0,
+                                    "content": "general setting does not allow for efficient incorporation of domain constraints, as this would require",
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+                                    "content": "an efficient parameterization of the intersection of a general convex polyhedron and the unit box.",
+                                    "type": "text"
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+                            ],
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+                    ],
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+                                    "content": "4 NUMERICAL RESULTS",
+                                    "type": "text"
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+                    ],
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+                                    "score": 1.0,
+                                    "content": "the orthogonal projection onto a constraint set to demonstrate properties of these algorithms. Here,",
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "the result can be compared to the optimal solution of the convex optimization problem. In a second",
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+                            ],
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "these algorithms to a variational autoencoder. Finally, we evaluate the running time of inference for",
+                                    "type": "text"
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "these problems and show that the proposed algorithm is significantly more efficient compared to the",
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+                            ],
+                            "index": 36
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "test time projection method.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 37
+                        }
+                    ],
+                    "index": 33
+                },
+                {
+                    "type": "text",
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+                                    "score": 1.0,
+                                    "content": "We used the MNIST dataset (LeCun et al.) for both experiments (59000 training, 1000 validation,",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "and 10000 test samples). We chose PyTorch (Paszke et al., 2017) for our implementation3 and all",
+                                    "type": "text"
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+                                    "content": "experiments were performed on a single Nvidia Titan X GPU. All networks were optimized with the",
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+                                    "score": 0.91,
+                                    "content": "[ 1 0 ^ { - 5 } , 1 0 ^ { - 3 } ]",
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+                                    "score": 1.0,
+                                    "content": ". The initial learning rate",
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+                                    "score": 1.0,
+                                    "content": "was annealed by a factor of",
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+                                    "score": 1.0,
+                                    "content": "if progress on the validation loss stagnated for more than 5 epochs.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 42
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+                        {
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+                                    "score": 1.0,
+                                    "content": "We used OSQP (Stellato et al., 2017) as an efficient solver to compute orthogonal projections.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 43
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+                    ],
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+                                    "score": 1.0,
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+                                    "score": 1.0,
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+                                    "content": "is significantly greater than the other vector entries. Furthermore, this most",
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+                                    "score": 1.0,
+                                    "content": "general setting does not allow for efficient incorporation of domain constraints, as this would require",
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+                                    "score": 1.0,
+                                    "content": "an efficient parameterization of the intersection of a general convex polyhedron and the unit box.",
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+                    ],
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+                                    "content": "4 NUMERICAL RESULTS",
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+                                    "content": "the orthogonal projection onto a constraint set to demonstrate properties of these algorithms. Here,",
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+                                    "score": 1.0,
+                                    "content": "and 10000 test samples). We chose PyTorch (Paszke et al., 2017) for our implementation3 and all",
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+                                    "content": "for the setup presented in this paper as they are are computationally expensive at training time. An",
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+                                    "score": 1.0,
+                                    "content": "bically with the number of variables and constraints. The method by Pathak et al. (2015) for the",
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+                                    "score": 1.0,
+                                    "content": "regression problems in this paper alternates between optimization steps in the network parameters",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": ", which was annealed by a factor of 0.1 if progress on the validation loss stagnated for more",
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+                                    "score": 1.0,
+                                    "content": "than 5 epochs. The batch size was chosen to be 256. The unit box constraints were activated after 25",
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+                                    "score": 1.0,
+                                    "content": "epochs. Additionally, the data for training the model with all constraints being active is shown. This",
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+                                    "score": 1.0,
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+                                    "content": ", which was annealed by a factor of 0.1 if progress on the validation loss stagnated for more",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "for the test time projection method. Figure 4 shows a test set sample and the respective output of the",
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+                                    "content": "\\mathrm { R e L U } ( x ) = \\operatorname* { m a x } ( 0 , x )",
+                                    "type": "inline_equation"
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+                                    "content": "optimization problem, which is relatively costly. Table 1 shows inference times for both models for",
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diff --git a/parse/train/H8VDvtm1ij8/H8VDvtm1ij8.md b/parse/train/H8VDvtm1ij8/H8VDvtm1ij8.md
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+# NORMALIZING FLOWS FOR CALIBRATION AND RECALIBRATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+In machine learning, due to model misspecification and overfitting, estimates of the aleatoric uncertainty are often inaccurate. One approach to fix this is isotonic regression, in which a monotonic function is fit on a validation set to map the model’s CDF to an optimally calibrated CDF. However, this makes it infeasible to compute additional statistics of interest on the model distribution (such as the mean). In this paper, through a reframing of recalibration as MLE, we replace isotonic regression with normalizing flows. This allows us to retain the ability to compute the statistical properties of the model (such as closed-form likelihoods, mean, correlation, etc.) and provides an opportunity for additional capacity at the cost of possible overfitting. Most importantly, the fundamental properties of normalizing flows allow us to generalize recalibration to conditional and multivariate distributions. To aid in detecting miscalibration and measuring our success at fixing it, we use a simple extension of the calibration Q-Q plot.
+
+# 1 INTRODUCTION
+
+Recent advances in deep learning have led to models with significantly higher overall accuracy on both classification and regression tasks compared to what was achievable in the past. However, an important component in conjunction with accuracy is a model’s ability to accurately assess the uncertainty in its prediction. Most taxonomies classify uncertainty into three sources: approximation, aleatoric, and epistemic uncertainty (Der Kiureghian & Ditlevsen, 2009). Approximation uncertainty quantifies the error from fitting a simple model to complex data. Aleatoric uncertainty quantifies the uncertainty of the conditional distribution of the target variable given features. This uncertainty arises from hidden variables or measurement errors and cannot be reduced through collecting more data under the same experimental conditions. Epistemic uncertainty quantifies the uncertainty arising from fitting a model utilizing finite data, i.e. it is inversely proportional to the density of the training examples and can be reduced by collecting data in the low density regions.
+
+These different sources of uncertainty have different techniques for handling them. Using high capacity models such as neural networks removes a large part of the approximation uncertainty. By fitting a full distribution on the target conditional on features, we can model the aleatoric uncertainty from observations. Inaccurate estimates of aleatoric uncertainty can be explained by underfitting (insufficient complexity in the conditional distributions) or overfitting (models with sufficient capacity can memorize the data, leading to the distributions collapsing to deltas). Though epistemic uncertainty is important for the model to answer what it does not know, the focus of this paper is on improving estimates of the aleatoric uncertainty.
+
+Our approach in this paper is to handle both model fit and calibration using normalizing flows. Normalizing flows can be used in conjunction with amortized inference to improve the flexibility of the output distribution, and further, through a reframing of recalibration as maximum likelihood estimation (MLE), normalizing flows can be used to handle any miscalibration found on a validation set. Further, we use a simple extension of the calibration plot from Kuleshov et al. (2018) to help with the the analysis of the calibration of a model across different regions of the data.
+
+# 2 RELATED WORK
+
+One method for handling aleatoric uncertainty is amortized inference with Gaussians (Lakshminarayanan et al., 2017; Nix & Weigend, 1994; Kendall & Gal, 2017) where a model, such as a neural network, maps from features to the parameters of a Gaussian. This approach models aleatoric uncertainty directly but suffers from approximation uncertainty as a Gaussian cannot model complex targets.
+
+Another approach is Bayesian methods such as Bayesian Ridge Regression (Tipping, 2001) and MC Dropout (Gal & Ghahramani, 2016). Similar to amortized inference with Gaussians, the output distribution limits the capacity of the model. Full Bayesian techniques with neural networks are often too computationally expensive in practice, and approximate methods often fail to capture the full complexity of the uncertainty (Lakshminarayanan et al., 2017).
+
+Another family of methods uses quantile regression with non-linear techniques such as decision trees or neural networks. Some of these methods require a predefined set of quantiles (Takeuchi et al., 2006; Wen et al., 2017; Rodrigues & Pereira, 2018; Taylor, 2000). Simultaneous Quantile Regression (SQR, Tagasovska & Lopez-Paz (2019)) trains one model on all quantiles and is able to learn complex shaped distributions. However the training procedure requires the model to learn to be monotonic instead of being constrained to be so and is not trivial to extend to multidimensional outputs. Pearce et al. (2018) learns a finite set of quantiles by using quality metrics for predictive intervals.
+
+Normalizing flows have been used in the contexts of variational inference and generative modeling. The approaches to normalizing flows can be categorized into autoregressive methods (Kingma et al., 2016; Papamakarios et al., 2017; Huang et al., 2018; Cao et al., 2019), coupling layers (Dinh et al., 2014; 2016; Kingma & Dhariwal, 2018; Ho et al., 2019), residual networks (Rezende & Mohamed, 2015; van den Berg et al., 2018; Gopal, 2020) and continuous flows (Grathwohl et al., 2018).
+
+# 3 BACKGROUND
+
+# 3.1 RECALIBRATION
+
+An important goal in modeling is to have well-calibrated distributions as this allows users of the model to understand the confidence the model places on its prediction. In other words, with a well-calibrated model, we can better ascertain the uncertainty in the model’s predictions and respond differently to those predictions depending on the uncertainty surrounding them.
+
+Guo et al. (2017) showed that unlike techniques used decades ago such as Bayesian Ridge Regression, modern neural network-based classifiers are very poorly calibrated. A simple variant of Platt scaling and other histogram based techniques applied to a validation set were shown to help alleviate the calibration problem where perfect calibration is defined as
+
+$$
+\mathbb { P } ( \hat { Y } = Y | \hat { P } = p ) = p , \quad \forall p \in [ 0 , 1 ]
+$$
+
+where $\hat { Y }$ is a class prediction and $\hat { P }$ is its predicted probability of correctness.
+
+Kuleshov et al. (2018) extended the analysis in Guo et al. (2017) to neural network-based regressors; isotonic regression, a method for learning monotonic univariate functions, was applied to map from $\mathcal { F } _ { x _ { j } } ( y _ { j } ) = \hat { p } _ { j }$ to $| \{ y _ { n } | \mathcal { F } _ { x _ { n } } ( y _ { n } ) < \hat { p } _ { j } , n \stackrel { - } { = } 1 , . . . , N \} | / N$ (the fraction of the data where the model CDF is less than $\hat { p }$ ) to improve calibration where perfect calibration is defined as
+
+$$
+\mathbb { P } ( Y < \mathcal { F } _ { X } ^ { - 1 } ( p ) ) = p , \quad \forall p \in [ 0 , 1 ]
+$$
+
+where ${ \mathcal { F } } _ { X }$ is the predicted CDF function.
+
+Kuleshov et al. (2018) further introduced calibration error as a metric to quantitatively measure how well the quantiles are aligned:
+
+$$
+\begin{array} { r } { \hat { p } _ { j } = \vert \{ y _ { n } \vert \mathcal { F } _ { x _ { n } } ( y _ { n } ) < p _ { j } , n = 1 , \ldots , N \} \vert / N } \\ { \operatorname { c a l } ( y _ { 1 } , \ldots , y _ { N } ) = \displaystyle \sum _ { j = 1 } ^ { M } ( p _ { j } - \hat { p } _ { j } ) ^ { 2 } } \end{array}
+$$
+
+where $M$ is the number of quantiles that are evaluated. In this paper, we set this to 100 evenly spaced quantiles.
+
+![](images/52aa6c065ffeced6d996d72cde20955f69b0c8da9a20d8031cebe127f210d0b4.jpg)  
+Figure 1: A comparison between using a hypernetwork approach versus conditional approach to creating Conditional QuAR Flows to model $p ( X | c )$ where $W _ { 1 }$ and $W _ { 2 }$ parametrize the layers in the residual connection.
+
+# 3.2 NORMALIZING FLOWS
+
+A crucial component in improving uncertainty estimates in this paper is to fit normalizing flows, a generative model for density estimation using invertible functions.
+
+Suppose that we wish to formulate a joint distribution on an $n$ -dimensional real vector $x$ . A flowbased approach treats $x$ as the result of a transformation $g$ applied to an underlying vector $z$ sampled from a base distribution $p _ { z } ( z )$ . The generative process for flows is defined as:
+
+$$
+\begin{array} { l } { z \sim p _ { z } ( z ) } \\ { x = g ( z ) } \end{array}
+$$
+
+where $p _ { z }$ is often a Normal distribution and $g$ is an invertible function. Notationally, we will use $f = g ^ { - 1 }$ . Using change of variables, the log likelihood of $x$ is
+
+$$
+\log p _ { x } ( x ) = \log p _ { z } \left( f ( x ) \right) + \log \left| { \operatorname* { d e t } \left( \frac { \partial f ( x ) } { \partial x } \right) } \right|
+$$
+
+To train flows (i.e. maximize the log likelihood of data points), we need to be able to compute the logarithm of the absolute value of the determinant of the Jacobian of $f$ , also called the log-determinant. To construct large normalizing flows, we can compose smaller ones as this is still invertible and the log-determinant of this composition is the sum of the individual log-determinants.
+
+An important observation is that every univariate distribution can be viewed as a flow in which the base distribution $p _ { z } ( z )$ is a Uniform distribution over $[ 0 , 1 ]$ and $g$ is the inverse CDF of the univariate distribution. This can be generalized to multivariate distributions using the chain rule.
+
+# 3.3 CONDITIONAL QUAR FLOWS
+
+In this paper, the specific normalizing flow used is Quasi-Autoregressive Residual Flows (QuAR Flows, Gopal (2020)). QuAR Flows were shown to handle modeling complex distributions while having nice optimization properties and computationally efficient log-likelihoods. Other commonly used normalizing flows have drawbacks. Coupling layers do not work for one-dimensional inputs. Autoregressive flows are often conditional Gaussians which means in the one-dimensional case, the flow is simply a Gaussian. Residual Flows (Chen et al., 2019) are expressive but, though equivalent to QuAR Flows in the one-dimensional case, are computationally expensive for high-dimensional distributions.
+
+If we would like to condition a QuAR Flow given some features $c$ , we could use a hypernetwork that maps $c$ to the parameters of the QuAR Flow (Figure 1a). However, this can be computationally expensive if the QuAR Flow is parameterized with thousands of parameters. Instead, conditional information can be incorporated by concatenating the conditions $c$ before applying the first layer in the residual connection within a QuAR Flow (Figure 1b).
+
+# 4 REGRESSION USING NORMALIZING FLOWS
+
+The goal of regression is to model the conditional probability distribution $p ( y | x )$ . Frequently, this task is reduced to obtaining point estimates of the mean or median by minimizing the mean squared error or mean absolute error respectively. In order to access uncertainties, we must model the full conditional distribution; in addition to this, we can still retrieve point estimates of the mean, median, or any other distributional statistic. In this section, we discuss how normalizing flows can be used to this end as well as how recalibration can be reframed as maximum likelihood estimation (MLE).
+
+# 4.1 CALIBRATION WITH NORMALIZING FLOWS
+
+One approach to modeling aleatoric uncertainty is to use amortized inference with a Normal distribution as its output distribution, conditioned on features (Lakshminarayanan et al., 2017). However, if a Gaussian is not appropriate to fit the target, then the error in miscalibration can be attributed to approximation error, i.e. insufficient capacity to learn the distribution. A simple example of this is where we have no features to condition on (or all the features we have contain no relevant information) with a one-dimensional target. In Figure 2, we can see that since the target is bimodal, the Gaussian fit does not well describe the data.
+
+To improve this model, we replace the Gaussian with a normalizing flow conditioned on our data, similar to Figure 1b. In this way, the distribution fit to the targets has sufficient capacity to learn complex distributions, including multimodal distributions (Figure 2).
+
+# 4.2 RECALIBRATION WITH NORMALIZING FLOWS
+
+To handle overfitting of the uncertainty, we recalibrate the results of the original model where we assume the original model outputs a full distribution that has closed-form likelihoods. Since the only requirement is a full distribution with a closed-form likelihood, the original model that we recalibrate can be as simple as a Bayesian Linear Regression model or as complicated as a conditional normalizing flow. We recalibrate the original model by fitting a normalizing flow on top of it. In this way, the training process is the following two-step procedure:
+
+![](images/ad181b91bb85b484eab2f815ac5f66a3057be04180b0b5783e57d91cc0f6099a.jpg)  
+Figure 2: One-dimensional example of Gaussian versus Flow
+
+$$
+\begin{array} { r l } & { 1 . \ f _ { \theta } = \arg \operatorname* { m a x } _ { f _ { \theta } } \sum _ { i = 1 } ^ { N _ { t r } } \log p _ { f _ { \theta } } ( y _ { i } ^ { t r } | x _ { i } ^ { t r } ) } \\ & { 2 . \ g _ { \phi } = \arg \operatorname* { m a x } _ { g _ { \phi } } \sum _ { i = 1 } ^ { N _ { v a l } } \log p _ { g _ { \phi } } ( F _ { \theta } ( y _ { i } ^ { v a l } | x _ { i } ^ { v a l } ) ) } \end{array}
+$$
+
+where the tra $f _ { \boldsymbol { \theta } } ( \boldsymbol { y } | \boldsymbol { x } )$ is thet and nction given is the valid $x$ , i $F _ { \theta } ( y | x )$ is the CDF of hough the for $f _ { \theta } ( y | x ) , \{ x ^ { t r } , y ^ { t r } \} _ { i = 1 } ^ { N _ { t r } }$ ison $\{ x ^ { v a l } , y ^ { v a l } \} _ { i = 1 } ^ { N _ { v a l } }$ above shows we can use any likelihood model for $g _ { \phi }$ , since isotonic regression is a method to fit a free form function to the CDFs, we choose to use normalizing flows, specifically QuAR Flows, as these are more flexible than using autoregressive flows such as MAFs (Papamakarios et al., 2017).
+
+After the two-step training, the new generative process is:
+
+$$
+\begin{array} { l } { z \sim \mathrm { U n i f o r m } ( 0 , 1 ) } \\ { u = G _ { \phi } ^ { - 1 } ( z ) } \\ { y = F _ { \theta } ^ { - 1 } ( u | x ) } \end{array}
+$$
+
+where $G _ { \phi }$ is the CDF of $g _ { \phi }$ . The density function of the above generative process is simply the product of the individual density functions: $f _ { \theta }$ and $g _ { \phi }$ . Given this view of the generative process, one interpretation of the two-step process is to first fit $\theta$ on the training set where $\phi$ is initialized so that $G _ { \phi } ^ { - 1 }$ is the identity function, and then to fit $\phi$ on the validation set while keeping $\theta$ frozen. Though $\theta$ and $\phi$ could be trained end-to-end, similar to the approach in Section 4.1, we show in our experiments (Section 6.2) that there are benefits to this two-step approach.
+
+![](images/9212fa4438adc3e36c2e75cfd149f1d9ffde426fb5aa80f4de671bd843cb558c.jpg)  
+Figure 3: Comparison plots of isotonic regression and flow recalibration on multiple outputs. The two axes are different targets being modeled, and the scatter plot shows that the targets are correlated. The top and right sides show the marginal histogram as well as the fit from the recalibrated distribution. The contour plot shows the two-dimensional contours of the recalibrated distribution which can be seen to not align well for isotonic regression.
+
+Given this log-likelihood viewpoint of recalibration, we can trivially incorporate conditioning of the normalizing flow on additional information. For example, say $f _ { \theta }$ outputs $\mu$ and $\sigma$ that parameterize a Normal distribution, we could then condition $g _ { \phi }$ on these two outputs. If we were to use many features to condition on, we would return to the first situation where we could overfit to the finite number of observations we have. We can think of the first model $f _ { \theta }$ as compressing/embedding the data into two dimensions, and the second model utilizing the embedding to create a meaningful and well-calibrated distribution on the target. Compared to isotonic regression, we can easily incorporate conditional information while retaining monotonicity as normalizing flows are monotonic by construction.
+
+Recalibration is interesting in that it is useful for both overfitting as well as underfitting. From Guo et al. (2017), we see that multinomial classification models have sufficient capacity to overfit uncertainties and benefit from recalibration. On the other hand, the simple example in Figure 2 suffers from miscalibrated uncertainties due to approximation uncertainty (underfitting), and recalibration reduces the approximation error.
+
+One solution to handling overfitting is to early-stop on a validation set. Whether early stopping occurs or not, our solution incorporates the information of the validation set to correct potential overfitting. However, if we try to correct for multiple sources of overfitting, then we could introduce a new overfitting problem. In essence, there is a tradeoff between the validation size (which in turn controls training size) and how many sources of overfitting one can try to correct after training.
+
+# 4.3 MULTIVARIATE RECALIBRATION
+
+Whereas Platt scaling (Guo et al., 2017) and isotonic regression (Kuleshov et al., 2018) have been shown to improve the calibration of distributions when we have one-dimensional outputs, scaling these solutions to high-dimensional targets is non-trivial.
+
+Say we are modeling two-dimensional real-valued targets and use isotonic regression to calibrate the targets individually. For simplicity, say we use amortized inference with independent output distributions. If the targets are not conditionally independent, the distribution will not be wellcalibrated, even after recalibration with isotonic regression (Figure 3a). However, using normalizing flows, we can compute the CDF of the observed targets per output distribution and recalibrate this two-dimensional distribution (Figure 3b).
+
+The general procedure of recalibrating $N$ -dimensional distributions is to fit a normalizing flow on the $N$ -dimensional vector $x _ { i } = F ( y _ { i } | y _ { < i } )$ .
+
+![](images/bd31ec7371b5fbab3c41c4af090e7e01d02a0effd5315ff0fe11e911a5356c31.jpg)  
+Figure 4: Calibration Q-Q plots for recalibration using isotonic regression and recalibration using flows for the simulated multivariate case in Figure 3a. The legend contains the calibration error for each curve.
+
+# 5 MULTIVARIATE RECALIBRATION EXAMPLE
+
+In multiclass classification, we often focus on individual metrics such as accuracy, precision, recall, F1, etc. However, exploring the accuracy of the separate classes using a confusion matrix can show if the model is biased even if the performance on average is good. Similar to this view for classification, we extend the calibration plot introduced in (Kuleshov et al., 2018) to show calibration performance on different subsets of the data, binned by either a feature or a prediction, e.g. Figure 4. The only restriction is the binning cannot be a function of the target we are evaluating.
+
+Say we are modeling two-dimensional real-valued targets $( Y _ { 1 } , Y _ { 2 } )$ shown in Figure 3a. If we fit independent Gaussians to the target, the resulting distribution will not align with the contours of the data.
+
+We first recalibrate the distribution by fittting isotonic regression models on $F ( Y _ { 1 } )$ and $F ( Y _ { 2 } | Y _ { 1 } )$ The results for $F ( Y _ { 2 } | Y _ { 1 } )$ are shown in Figure 4a where we can see that, though across the full dataset the calibration error is zero, when we bucket by $F ( Y _ { 1 } )$ , the calibration error achieved is both significant and a strong function of $Y _ { 1 }$ as it is unable to account for correlations in $Y _ { 1 }$ and $Y _ { 2 }$ . In other words, the model is not calibrated in any given region of the data but only in aggregate.
+
+However, if we recalibrate using normalizing flows by fitting a flow on the pair $( F ( Y _ { 1 } ) , F ( Y _ { 2 } | Y _ { 1 } ) )$ , we can see in Figure 4b that even if we bucket by $F ( Y _ { 1 } )$ , the calibration error stays close to zero. Unlike isotonic regression, the flow can learn the correlation between the CDFs by being trained jointly on all of them. In addition to this, had we used a more complex calibration model that could model the correlations directly, we would not be able to compute the correlation after recalibrating with isotonic regression. By recalibrating with flows, we obtain a result that is still a distribution with a closed-form likelihood; this allows us to still be able to use the model to compute the correlation and any other distributional statistics.
+
+In Figure 5, we show visually the difference between recalibration with flows and with isotonic regression. The key differences being that flows can recalibrate the CDFs jointly and be conditioned on additional information $c$ , while isotonic regression is fit on each CDF independently and is unable to incorporate any additional information. The reason is that flows are monotonic by construction and can therefore handle multiple variables and conditions.
+
+# 6 EXPERIMENTS
+
+In this section, we analyze the performance of flows on multiple UCI datasets. Importantly, we do not aim for state-of-the-art performance, nor do we expect normalizing flows to outperform isotonic
+
+![](images/875700dffcddbe4a75f5e5d09bdaf048c24fbe3c4bd6dcd6bef0495db4def2fa.jpg)  
+Figure 5: A diagram for multivariate recalibration using the same notation as that used in Section 4.2. The first step for recalibration is always the calibration step, shown in the blue box, where any likelihood model $f _ { \theta }$ can be used. On the left, isotonic regression is used to recalibrate the CDFs from $f _ { \theta }$ ; on the right, normalizing flows $g _ { \phi }$ are used to recalibrate the CDFs from $f _ { \theta }$ conditioned on some $c$ .
+
+regression in aggregate. In terms of calibration, we show that the increase in expressivity can be beneficial; in terms of recalibration, we show specific failure points of isotonic regression that binning uncovers and how normalizing flows fixes these issues.
+
+# 6.1 CALIBRATION
+
+To test the effect of increased capacity in distributions, we tested how well normalizing flows improve performance in distributional estimates compared to a few other popular approaches for estimating uncertainties. We do not compare directly against other methods for conditional distributions such as autoregressive flows for the reason specified in Section 3.3: in the one dimensional case, those are often the same as ConditionalGaussian. Instead of replacing the output distribution with a complex distribution created by hand, such as an explicit mixture, we rely on using more expressive flows which can model mixture behavior as well as more complicated behavior such as skew and kurtosis.
+
+Table 1: Results before and after recalibration on the UCI dataset “naval-propulsion-plant”. Details on the model methodology as well as performance on other UCI datasets are in Appendix B. The mean squared error (MSE) and its uncertainty is computed using 20 seeds. “Calib” is the calibration error (Equation 1) using 100 quantiles across 20 seeds instead of per seed.   
+
+<table><tr><td rowspan="3"></td><td colspan="2">Before Recalib.</td><td colspan="2">Isotonic Recalib.</td><td colspan="2">Flow Recalib.</td></tr><tr><td>MSE</td><td>Calib</td><td>MSE</td><td>Calib</td><td>MSE</td><td>Calib</td></tr><tr><td>ConditionalFlow</td><td>0.009 ±0.004</td><td>0.45</td><td>N.A.</td><td>&lt;0.01</td><td>0.016 ± 0.012</td><td>&lt;0.01</td></tr><tr><td>ConditionalGaussian</td><td>0.012 ±0.006</td><td>2.35</td><td>N.A.</td><td>&lt;0.01</td><td>0.012 ±0.006</td><td>&lt;0.01</td></tr><tr><td>BayesianRidgeRegression</td><td>0.055 ±0.004</td><td>8.03</td><td>N.A.</td><td>0.01</td><td>0.48 ± 0.02</td><td>&lt;0.01</td></tr><tr><td>MC Dropout</td><td>0.005 ±0.002</td><td>0.51</td><td>N.A.</td><td>N.A.</td><td>N.A.</td><td>N.A.</td></tr><tr><td>SQR</td><td>0.0052 ± 0.0012</td><td>1.44</td><td>N.A.</td><td>N.A.</td><td>N.A.</td><td>N.A.</td></tr></table>
+
+All results were evaluated on a held out test set that was not used for calibration or recalibration. Due to limited space, the full extent of the results and details of the models are left for Appendix A and Appendix B. To show the usefulness of the plot we introduced in Section 5, we focus on the results on the UCI dataset “naval-propulsion-plant” (Dheeru & Karra Taniskidou, 2017) in Table 1 and Section 6.2.
+
+The models with consistent performance across UCI datasets in terms of calibration are Bayesian Ridge Regression, Conditional Gaussian, and Conditional Flow, with Conditional Flow often performing well in cases where Conditional Gaussian is unable to. Whereas these three models are well calibrated, the mean performance is much weaker for Bayesian Ridge Regression due to the limitations of linear functions.
+
+# 6.2 RECALIBRATION
+
+Of the models we tested, we recalibrated the Bayesian Ridge Regression, Conditional Gaussian, and Conditional Flow models. MC Dropout and SQR could also be recalibrated, though we chose not to as they lack closed form CDFs, meaning the technique would be approximate. In Table 1, we see the calibration errors after recalibrating with normalizing flows are comparable to those from isotonic regression. However, after isotonic regression is applied, only the calibration error can be computed. In contrast, we can still compute ordinary distribution statistics such as the mean after applying normalizing flows. Due to this property, we can even detect cases in which an improvement in calibration error coincides with a degradation of mean performance, suggesting if the goal were to only get point estimates of the mean, recalibration could be detrimental. For example, in Table 1 this was the case for Bayesian Ridge Regression.
+
+In Figure 6a, whereas the calibration error metric shows the flow recalibrated model is well-calibrated averaged across the full dataset, we can see that see that, when bucketing by the predicted mean, the calibration error is significant.
+
+To improve this, we recalibrate with a normalizing flow conditioned on the mean predicted by the original model. In this way, we see in Figure 6b that though the aggregate performance does not change, the performance is improved across different buckets. In essence we see that if we were analyzing the distribution across the dataset, we would be accurate; if we started to look closer to individual distributions, the performance is worse suggesting the quantiles of individual distributions cannot be trusted.
+
+![](images/5ba0a74e3b6aa56744216a122f89938ce61fa7461edf3b64016e47551252acc8.jpg)  
+Figure 6: Calibration Q-Q plots for recalibration with an unconditional flow and recalibration with a conditional flow on the dataset “naval-propulsion-plant”. The legend contains the calibration error for each curve.
+
+![](images/96799e60b702cd7228ab4ea95bae04d75bb639f63db7286ef906822c86a386ab.jpg)  
+Figure 7: Calibration Q-Q plots for recalibration with isotonic regression and recalibration with normalizing flows on the second time step on the dataset “electricity”. The legend contains the calibration error for each curve.
+
+# 6.3 TIME SERIES
+
+For our time series example, we trained a model on the UCI Dataset “electricity”. We predict two time steps and show that even though we used an autoregressive model for calibration (the model could learn correlated distributions), there is still some correlation not captured that recalibration can help with.
+
+Figure 7a shows that the calibration error of $F ( Y _ { 2 } | Y _ { 1 } )$ when recalibrated independently with isotonic regression is significant. Figure 7b shows that a flow trained jointly on the CDFs significantly reduces the errors. Although the calibration error of the marginal distribution for the second step is comparable between the two approaches, the conditional performance favors normalizing flows. A possible explanantion of the still imperfect calibration of the flow model, though better than isotonic regression, is distribution shift.
+
+# 7 CONCLUSION AND FUTURE WORK
+
+In this paper, we addressed the calibration issue in modern neural networks through using higher capacity distributions via normalizing flows. We showed that normalizing flows can not only help during calibration, but also in recalibrating miscalibrated distributions, even improving the fit in the underfitting case. Though isotonic regression also uses a two-step producedure, we reframe the problem as MLE through which we can trivially generalize recalibration to the multivariate case. Though isotonic regression could be extended such that we could compute the log likelihood implied by its change of variables, normalizing flows are a more natural way to handle this with more research backing this technique, and are easily extendable to conditional and joint distributions. Interestingly, though the Conditional Flow has all it needs to be well calibrated, recalibration helps the overall calibration which was similarly found for classification in Guo et al. (2017); it remains for future work to better understand why and when this is useful.
+
+Further, we extend the plot introduced in Kuleshov et al. (2018) by showing the calibration error on different subsets of the data, for example bucketed by mean prediction. Though this plot is able to show the calibration error across buckets, for future work, we will explore ways to visually show any trends in the miscalibration as well as whether the misalignments of the quantiles are statistically significant.
+
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+
+![](images/c75db8aacd9d4a38c987062b87207c17d500f45fc7262577a8766d51b64af6c7.jpg)  
+Figure 8: In this figure, we show our conditional flow model. A neural network takes $X$ and outputs the scale and shift of an affine flow and a condition vector for QuAR Flows. The condition used in the QuAR Flow is the same for both $\mathrm { Q u A R }$ Flows. After the first conditional affine flow, there are two QuAR Flows with unconditional affine flows placed in between these two and at the end. The Conditional Flow model is equivalent to a Conditional Gaussian if we removed the QuAR Flows.
+
+# A MODEL TRAINING DETAILS
+
+# A.1 CALIBRATION
+
+We compare the performance of our model and multiple other popular approaches on 8 UCI datasets (Dheeru & Karra Taniskidou, 2017): boston-housing, concrete, energy, kin8nm, naval-propulsionplant, power-plant, wine-quality-red and yacht. The models we compared Conditional Flows to were: Conditional Gaussian (amortized inference with Gaussians, Lakshminarayanan et al. (2017)), Bayesian Ridge Regression (Tipping, 2001), MC Dropout (Gal & Ghahramani, 2016), Simultaneous Quantile Regression (SQR) (Tagasovska & Lopez-Paz, 2019), Conditional Quantile (SQR where we only learn prespecified quantiles), Quality Driven (Pearce et al., 2018), and a Gradient Boosted Decision Tree with LightGBM (Ke et al., 2017).
+
+Figure 8 shows a representation of the Conditional Flow network we used. The QuAR Flows are conditioned on a 64-dimensional representation from the condition network. Each Conditional QuAR Flow has two hidden layers in the residual connection with a hidden size 64 with ELU (Clevert et al., 2015) as our activation. The condition network has a single hidden layer with size 64 and ReLU as our activation. The QuAR Flows are conditioned on the 64-dimensional representation from the condition network.
+
+For the other neural network approaches, we used networks with two hidden layers with size 128 and used ReLU as our nonlinearity. For hyperparameter tuning on all our neural networks, we searched the following grid:
+
+• Learning rate: 1e-2, 1e-3, 1e-4   
+• Weight decay: 0, 1e-3, 1e-2, 1e-1   
+• Dropout Rates: 0.1 0.25, 0.5 0.75
+
+The hyperparameters we used were taken from Tagasovska & Lopez-Paz (2019). We trained each of our neural networks for 2000 epochs with batch size 128.
+
+For our LightGBM tree model (Ke et al., 2017), we used mean squared error as our loss to learn the mean of the output distribution. The grid of hyperparameters we searched was:
+
+• Number of Trees: 100, 1000   
+• Maximum Depths: 3, 5, 7, 9   
+• Learning Rates: 1e-1, 1e-2, 1e-3, 1e-4   
+• Minimum Child Samples: 1, 20   
+• Dropout Rates: 0.1 0.25, 0.5 0.75
+
+To make the tree model distributional, assuming homoscedasticity, we computed the standard deviation of the predicted normal distribution by computing the standard deviation of the residuals (observations minus predictions).
+
+For SQR and MC Dropout, we computed the estimated mean and CDF from 1000 samples. For Conditional Flows, we computed the estimated mean by using the Gauss-Legendre quadrature with 100 points to approximate the integral.
+
+# A.2 RECALIBRATION
+
+For recalibration, we trained the recalibration model on the validation set of each of the 20 seeds. For the flow architecture, we stacked 8 QuAR Flows with affine flows in between each one as well as at the beginning and end. Since the input to the flow is a $\mathrm { C D F } \in [ 0 , 1 ]$ , we also transform the data with a Logit Function as the first layer of the flow and account for the change of variables in the loss. In addition to this, we used a dropout rate of 0.25, two hidden layers with size 64 in each residual connection, and used Polyak averaging (Polyak & Juditsky, 1992) with a decay of 0.999 for evaluation.
+
+# A.3 TIME SERIES
+
+For our time series experiments, we tested the performance of flow recalibration on the UCI dataset “electricity”. We filtered the data down to observations after 2014-01-01. We split the data by date, where the first $72 \%$ of dates are in the train set, the next $18 \%$ are in validation set, and the last $10 \%$ are in the test set.
+
+The data is given in 15 minute increments, and so similarly to Deep AR (Salinas et al., 2017), we averaged the values per hour to have hourly data. Since we were trying to analyze the effect of correlated targets, we predicted two time steps conditioned on 48 hours of data using an autoregressive model.
+
+We included three numeric covariates: the time step within the sampled time series, the hour of the day, and the day of the week. Each of the covariates were normalized to have zero mean and one standard deviation. In addition to this, since there are 370 different types of time series, we also added an embedding layer from 370 to an embedding dimension of 20, same as that used in Deep AR (Salinas et al., 2017). We concatenated this vector with the observations and covariates, leading to a 27-dimensional vector.
+
+Since all the observations are positive, we model the target with a conditional Gamma distribution (i.e. amortized inference with Gamma distributiet al., 2017), we normalized each time series $\{ z _ { i , t } \} _ { t = 0 } ^ { T - 1 }$ of by $\begin{array} { r } { v _ { i } = 1 + 1 / T \sum _ { t = 0 } ^ { T - 1 } z _ { i , t } . } \end{array}$ eep AR (Salinas. We account for the scaling in the output distribution, i.e. Gamma $( k , \theta * v _ { i } )$ .
+
+The model we used was a 1 layer LSTM, and then two linear layers with a ReLU and Dropout (Srivastava et al., 2014) in between were applied to the hidden representation. We set the dropout rate to 0.5 and used a hidden size of 48 everywhere. For evaluation, we used Polyak averaging (Polyak & Juditsky, 1992) with a decay of 0.999.
+
+For recalibration with flows, we trained on the CDFs for both time steps with a flow with the same architecture as that used in Appendix A.2.
+
+# B EXPERIMENTAL RESULTS
+
+# B.1 CALIBRATION
+
+In Table 2, we compare Conditional Flows against multiple other approaches for estimating uncertainty, and we measure performance using Prediction Interval Coverage Probability (PICP), Mean Squared Error (MSE), and Calibration Error.
+
+For each model, we trained the model on 20 different seeds where each seed randomly splits the data in $72 \%$ train, $18 \%$ validation, and $10 \%$ test. We picked the best hyperparameter by choosing the hyperparameters that gave the best average validation loss across the 20 seeds where the loss used is the loss defined by the training procedure for the model, e.g. for Conditional Gaussian, we used log likelihood; for LightGBM, we used MSE; for QualityDriven, we used the Quality Driven Loss, etc.
+
+Table 2: Results on percentage of data captured from 0.025 to 0.975 quantiles (PICP), MSE, and calibration error with 100 buckets. The results are from minimizing the average validation loss across 20 seeds. The optimal value of PICP is 0.95, and the optimal value for MSE and Calib is 0.00.   
+
+<table><tr><td></td><td colspan="3">bostonHousing</td><td colspan="3">concrete</td></tr><tr><td></td><td>PICP</td><td>MSE</td><td>Calib</td><td>PICP</td><td>MSE</td><td>Calib</td></tr><tr><td>ConditionalFlow</td><td>0.93 ± 0.04</td><td>0.13 ± 0.06</td><td>0.06</td><td>0.92 ± 0.03</td><td>0.09 ±0.03</td><td>0.01</td></tr><tr><td>ConditionalGaussian</td><td>0.93 ± 0.03</td><td>0.14±0.07</td><td>0.13</td><td>0.91 ± 0.02</td><td>0.09 ±0.03</td><td>0.06</td></tr><tr><td>BayesianRidgeRegression</td><td>0.95 ± 0.03</td><td>0.28±0.08</td><td>0.34</td><td>0.95 ± 0.03</td><td>0.40±0.07</td><td>0.01</td></tr><tr><td>MC Dropout</td><td>0.50 ± 0.08</td><td>0.11 ± 0.05</td><td>2.99</td><td>0.69 ± 0.05</td><td>0.07±0.03</td><td>0.89</td></tr><tr><td>SQR</td><td>0.73 ± 0.08</td><td>0.11 ± 0.05</td><td>0.48</td><td>0.74± 0.04</td><td>0.07± 0.03</td><td>0.47</td></tr><tr><td>ConditionalQuantile</td><td>0.92 ± 0.05</td><td>N.A.</td><td>N.A.</td><td>0.93 ± 0.03</td><td>N.A.</td><td>N.A.</td></tr><tr><td>QualityDriven</td><td>0.83 ± 0.06</td><td>N.A.</td><td>N.A.</td><td>0.83± 0.04</td><td>N.A.</td><td>N.A.</td></tr><tr><td>LightGBM</td><td>0.04±0.03</td><td>0.12 ± 0.06</td><td>7.74</td><td>0.63±0.07</td><td>0.07±0.02</td><td>1.25</td></tr><tr><td></td><td colspan="3">energy</td><td colspan="3">kin8nm</td></tr><tr><td></td><td>PICP</td><td>MSE</td><td>Calib</td><td>PICP</td><td>MSE</td><td>Calib</td></tr><tr><td>ConditionalFlow</td><td>0.96 ± 0.02</td><td>0.0022 ± 0.0006</td><td>0.12</td><td>0.952 ± 0.006</td><td>0.081 ± 0.013</td><td>0.01</td></tr><tr><td>ConditionalGaussian</td><td>0.97 ± 0.03</td><td>0.0030± 0.0009</td><td>0.26</td><td>0.947 ± 0.007</td><td>0.071 ± 0.005</td><td>&lt;0.01</td></tr><tr><td>BayesianRidgeRegression</td><td>0.89 ± 0.04</td><td>0.09 ±0.02</td><td>0.24</td><td>0.951 ± 0.007</td><td>0.59 ± 0.02</td><td>0.09</td></tr><tr><td>MC Dropout</td><td>0.97± 0.02</td><td>0.0026 ± 0.0010</td><td>0.17</td><td>0.69 ± 0.02</td><td>0.069 ± 0.005</td><td>1.14</td></tr><tr><td>SQR</td><td>0.97± 0.02</td><td>0.0028 ± 0.0012</td><td>0.37</td><td>0.892 ± 0.010</td><td>0.072 ± 0.005</td><td>0.01</td></tr><tr><td>ConditionalQuantile</td><td>0.98 ±0.02</td><td>N.A.</td><td>N.A.</td><td>0.943 ± 0.011</td><td>N.A.</td><td>N.A.</td></tr><tr><td>QualityDriven</td><td>0.96 ± 0.04</td><td>N.A.</td><td>N.A.</td><td>0.88±0.02</td><td>N.A.</td><td>N.A.</td></tr><tr><td>LightGBM</td><td>0.24±0.05</td><td>0.0013 ± 0.0006</td><td>5.18</td><td>0.205 ± 0.010</td><td>0.190 ± 0.011</td><td>5.59</td></tr><tr><td></td><td colspan="3">naval-propulsion-plant</td><td colspan="3">power-plant</td></tr><tr><td></td><td>PICP</td><td>MSE</td><td>Calib</td><td>PICP</td><td>MSE</td><td>Calib</td></tr><tr><td>ConditionalFlow</td><td>0.98 ± 0.03</td><td>0.009 ±0.004</td><td>0.45</td><td>0.952 ± 0.007</td><td>0.048 ±0.007</td><td>0.01</td></tr><tr><td>ConditionalGaussian</td><td>0.9995 ± 0.0007</td><td>0.012 ± 0.006</td><td>2.35</td><td>0.947 ± 0.010</td><td>0.048 ±0.006</td><td>&lt;0.01</td></tr><tr><td>BayesianRidgeRegression</td><td>1.0 ±0.0</td><td>0.055 ± 0.004</td><td>8.03</td><td>0.964 ± 0.007</td><td>0.072 ± 0.007</td><td>0.01</td></tr><tr><td>MC Dropout</td><td>0.98 ±0.02</td><td>0.005±0.002</td><td>0.51</td><td>0.59 ±0.02</td><td>0.046±0.007</td><td>1.88</td></tr><tr><td>SQR</td><td>0.998 ± 0.002</td><td>0.0052 ± 0.0012</td><td>1.44</td><td>0.93 ± 0.02</td><td>0.048 ±0.007</td><td>0.02</td></tr><tr><td>ConditionalQuantile</td><td>0.97±0.02</td><td>N.A.</td><td>N.A.</td><td>0.937 ± 0.014</td><td>N.A.</td><td>N.A.</td></tr><tr><td>QualityDriven</td><td>0.949 ± 0.010</td><td>N.A.</td><td>N.A.</td><td>0.89 ±0.02</td><td>N.A.</td><td>N.A.</td></tr><tr><td>LightGBM</td><td>0.714 ± 0.011</td><td>0.0056 ± 0.0009</td><td>0.79</td><td>0.17 ± 0.02</td><td>0.035 ± 0.007</td><td>5.95</td></tr><tr><td></td><td colspan="3">wine-quality-red</td><td colspan="3">yacht</td></tr><tr><td></td><td>PICP</td><td>MSE</td><td>Calib</td><td>PICP</td><td>MSE</td><td>Calib</td></tr><tr><td>ConditionalFlow</td><td>0.90 ±0.02</td><td>0.6±0.2</td><td>0.04</td><td>0.95 ± 0.04</td><td>0.004±0.003</td><td>0.27</td></tr><tr><td>ConditionalGaussian</td><td>0.95 ± 0.02</td><td>0.59 ± 0.09</td><td>0.22</td><td>0.99 ±0.02</td><td>0.004±0.002</td><td>0.62</td></tr><tr><td>BayesianRidgeRegression</td><td>0.95 ± 0.02</td><td>0.63 ± 0.11</td><td>0.05</td><td>0.94± 0.03</td><td>0.35 ± 0.10</td><td>0.25</td></tr><tr><td>MC Dropout</td><td>0.35 ± 0.04</td><td>0.57 ±0.09</td><td>4.01</td><td>0.97 ± 0.04</td><td>0.004±0.002</td><td>0.21</td></tr><tr><td>SQR</td><td>0.81±0.09</td><td>0.59 ±0.10</td><td>0.15</td><td>0.96 ±0.04</td><td>0.004±0.004</td><td>0.49</td></tr><tr><td>ConditionalQuantile</td><td>0.95 ± 0.02</td><td>N.A.</td><td>N.A.</td><td>0.99 ± 0.02</td><td>N.A.</td><td>N.A.</td></tr><tr><td>QualityDriven</td><td>0.84 ±0.03</td><td>N.A.</td><td>N.A.</td><td>0.97 ± 0.04</td><td>N.A.</td><td>N.A.</td></tr><tr><td>LightGBM</td><td>0.40 ± 0.05</td><td>0.55± 0.10</td><td>2.93</td><td>0.45 ± 0.08</td><td>0.003±0.003</td><td>3.04</td></tr></table>
+
+Table 3: Calibration error (with 100 buckets) before and after recalibration.   
+
+<table><tr><td></td><td colspan="3">BayesianRidgeRegression</td><td colspan="3">ConditionalGaussian</td><td colspan="3">ConditionalFlow</td></tr><tr><td></td><td>Before</td><td>Flow</td><td>Iso</td><td>Before</td><td>Flow</td><td>Iso</td><td>Before</td><td>Flow</td><td>Iso</td></tr><tr><td>bostonHousing</td><td>0.34</td><td>0.02</td><td>0.02</td><td>0.13</td><td>0.04</td><td>0.02</td><td>0.06</td><td>0.02</td><td>0.01</td></tr><tr><td>concrete</td><td>0.01</td><td>0.01</td><td>0.01</td><td>0.06</td><td>0.02</td><td>0.02</td><td>0.01</td><td>0.02</td><td>0.03</td></tr><tr><td>energy</td><td>0.24</td><td>0.02</td><td>0.01</td><td>0.26</td><td>&lt;0.01</td><td>0.01</td><td>0.12</td><td>0.01</td><td>0.01</td></tr><tr><td>kin8nm</td><td>0.09</td><td>0.01</td><td>&lt;0.01</td><td>&lt;0.01</td><td>&lt;0.01</td><td>&lt;0.01</td><td>0.01</td><td>&lt;0.01</td><td>&lt;0.01</td></tr><tr><td>naval-propulsion-plant</td><td>8.03</td><td>0.01</td><td>&lt;0.01</td><td>2.35</td><td>&lt;0.01</td><td>&lt;0.01</td><td>0.45</td><td>&lt;0.01</td><td>&lt;0.01</td></tr><tr><td>power-plant</td><td>0.01</td><td>&lt;0.01</td><td>&lt;0.01</td><td>&lt;0.01</td><td>&lt;0.01</td><td>&lt;0.01</td><td>0.01</td><td>&lt;0.01</td><td>&lt;0.01</td></tr><tr><td>wine-quality-red</td><td>0.05</td><td>0.02</td><td>&lt;0.01</td><td>0.22</td><td>0.04</td><td>0.02</td><td>0.04</td><td>0.01</td><td>0.01</td></tr><tr><td>yacht</td><td>0.25</td><td>0.05</td><td>0.07</td><td>0.62</td><td>0.03</td><td>0.03</td><td>0.27</td><td>0.02</td><td>0.03</td></tr></table>
+
+Table 4: MSE performance before and after recalibration. We do not show the MSE for isotonic regression as the mean cannot be computed after recalibrating with isotonic regression.   
+
+<table><tr><td rowspan="3"></td><td colspan="2">BayesianRidgeRegression</td><td colspan="2">ConditionalGaussian</td></tr><tr><td>Before</td><td>Flow</td><td>Before</td><td>Flow</td></tr><tr><td>bostonHousing</td><td>0.28 ±0.08</td><td>0.28 ±0.08</td><td>0.14 ± 0.07</td><td>0.14 ± 0.07</td></tr><tr><td>concrete</td><td>0.40 ± 0.07</td><td>0.40 ± 0.07</td><td>0.09 ±0.03</td><td>0.09 ± 0.03</td></tr><tr><td>energy</td><td>0.09 ±0.02</td><td>0.09 ±0.02</td><td>0.0030 ±0.0009</td><td>0.0032 ± 0.0011</td></tr><tr><td>kin8nm</td><td>0.59 ± 0.02</td><td>0.59 ± 0.02</td><td>0.071 ± 0.005</td><td>0.072 ± 0.005</td></tr><tr><td>naval-propulsion-plant</td><td>0.055 ± 0.004</td><td>0.48 ±0.02</td><td>0.012 ± 0.006</td><td>0.012 ± 0.007</td></tr><tr><td>power-plant</td><td>0.072 ± 0.007</td><td>0.072 ± 0.007</td><td>0.048 ± 0.006</td><td>0.049 ± 0.006</td></tr><tr><td>wine-quality-red</td><td>0.63 ± 0.11</td><td>0.63 ± 0.11</td><td>0.59 ± 0.09</td><td>0.59 ± 0.10</td></tr><tr><td>yacht</td><td>0.35 ± 0.10</td><td>0.36 ± 0.10</td><td>0.004 ±0.002</td><td>0.004 ± 0.003</td></tr><tr><td colspan="2"></td><td colspan="2">ConditionalFlow</td><td></td></tr><tr><td rowspan="10"></td><td></td><td>Before</td><td>Flow</td><td></td></tr><tr><td>bostonHousing</td><td>0.13 ± 0.06</td><td></td><td></td></tr><tr><td>concrete</td><td>0.09 ± 0.03</td><td>0.13 ± 0.06 0.09 ± 0.03</td><td></td></tr><tr><td></td><td>0.0022 ± 0.0006</td><td>0.0027 ± 0.0007</td><td></td></tr><tr><td>energy kin8nm</td><td>0.081 ± 0.013</td><td>0.081 ± 0.013</td><td></td></tr><tr><td>naval-propulsion-plant</td><td>0.009 ± 0.004</td><td>0.016 ± 0.012</td><td></td></tr><tr><td>power-plant</td><td>0.048 ± 0.007</td><td>0.048 ± 0.007</td><td></td></tr><tr><td>wine-quality-red</td><td>0.6± 0.2</td><td>0.64 ± 0.15</td><td></td></tr><tr><td>yacht</td><td></td><td></td><td></td></tr><tr><td></td><td>0.004 ± 0.003</td><td>0.008 ± 0.012</td><td></td></tr></table>
+
+Reported in Table 2 are the mean and standard deviation of PICP and MSE across the 20 seeds. Due to the limited data, for calibration error with 100 bins, we computed it over the CDFs from all 20 seeds. We included PICP (the percentage of true observations between the quantiles 0.025 and 0.975) to compare against alternative approaches where we prespecify which quantiles we need.
+
+# B.2 RECALIBRATION
+
+From the above section, we recalibrate the best Bayesian Ridge Regression, Conditional Gaussian, and Conditional Flow. In Table 3, we compare the calibration error before recalibration, after recalibration with isotonic regression, and after recalibration with normalizing flows.
+
+As we can see, across all 8 UCI datasets, the calibration errors after recalibrating with normalizing flows are comparable to those from isotonic regression. More interestingly though, improvement in calibration error can be detrimental to mean performance.
\ No newline at end of file
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new file mode 100644
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+++ b/parse/train/H8VDvtm1ij8/H8VDvtm1ij8_content_list.json
@@ -0,0 +1,1799 @@
+[
+    {
+        "type": "text",
+        "text": "NORMALIZING FLOWS FOR CALIBRATION AND RECALIBRATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
+            826,
+            145
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            171,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            236,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In machine learning, due to model misspecification and overfitting, estimates of the aleatoric uncertainty are often inaccurate. One approach to fix this is isotonic regression, in which a monotonic function is fit on a validation set to map the model’s CDF to an optimally calibrated CDF. However, this makes it infeasible to compute additional statistics of interest on the model distribution (such as the mean). In this paper, through a reframing of recalibration as MLE, we replace isotonic regression with normalizing flows. This allows us to retain the ability to compute the statistical properties of the model (such as closed-form likelihoods, mean, correlation, etc.) and provides an opportunity for additional capacity at the cost of possible overfitting. Most importantly, the fundamental properties of normalizing flows allow us to generalize recalibration to conditional and multivariate distributions. To aid in detecting miscalibration and measuring our success at fixing it, we use a simple extension of the calibration Q-Q plot. ",
+        "bbox": [
+            233,
+            280,
+            766,
+            459
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            518,
+            334,
+            535
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent advances in deep learning have led to models with significantly higher overall accuracy on both classification and regression tasks compared to what was achievable in the past. However, an important component in conjunction with accuracy is a model’s ability to accurately assess the uncertainty in its prediction. Most taxonomies classify uncertainty into three sources: approximation, aleatoric, and epistemic uncertainty (Der Kiureghian & Ditlevsen, 2009). Approximation uncertainty quantifies the error from fitting a simple model to complex data. Aleatoric uncertainty quantifies the uncertainty of the conditional distribution of the target variable given features. This uncertainty arises from hidden variables or measurement errors and cannot be reduced through collecting more data under the same experimental conditions. Epistemic uncertainty quantifies the uncertainty arising from fitting a model utilizing finite data, i.e. it is inversely proportional to the density of the training examples and can be reduced by collecting data in the low density regions. ",
+        "bbox": [
+            174,
+            561,
+            825,
+            714
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "These different sources of uncertainty have different techniques for handling them. Using high capacity models such as neural networks removes a large part of the approximation uncertainty. By fitting a full distribution on the target conditional on features, we can model the aleatoric uncertainty from observations. Inaccurate estimates of aleatoric uncertainty can be explained by underfitting (insufficient complexity in the conditional distributions) or overfitting (models with sufficient capacity can memorize the data, leading to the distributions collapsing to deltas). Though epistemic uncertainty is important for the model to answer what it does not know, the focus of this paper is on improving estimates of the aleatoric uncertainty. ",
+        "bbox": [
+            174,
+            722,
+            825,
+            833
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Our approach in this paper is to handle both model fit and calibration using normalizing flows. Normalizing flows can be used in conjunction with amortized inference to improve the flexibility of the output distribution, and further, through a reframing of recalibration as maximum likelihood estimation (MLE), normalizing flows can be used to handle any miscalibration found on a validation set. Further, we use a simple extension of the calibration plot from Kuleshov et al. (2018) to help with the the analysis of the calibration of a model across different regions of the data. ",
+        "bbox": [
+            174,
+            840,
+            825,
+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            344,
+            118
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "One method for handling aleatoric uncertainty is amortized inference with Gaussians (Lakshminarayanan et al., 2017; Nix & Weigend, 1994; Kendall & Gal, 2017) where a model, such as a neural network, maps from features to the parameters of a Gaussian. This approach models aleatoric uncertainty directly but suffers from approximation uncertainty as a Gaussian cannot model complex targets. ",
+        "bbox": [
+            174,
+            132,
+            825,
+            203
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Another approach is Bayesian methods such as Bayesian Ridge Regression (Tipping, 2001) and MC Dropout (Gal & Ghahramani, 2016). Similar to amortized inference with Gaussians, the output distribution limits the capacity of the model. Full Bayesian techniques with neural networks are often too computationally expensive in practice, and approximate methods often fail to capture the full complexity of the uncertainty (Lakshminarayanan et al., 2017). ",
+        "bbox": [
+            174,
+            209,
+            825,
+            279
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Another family of methods uses quantile regression with non-linear techniques such as decision trees or neural networks. Some of these methods require a predefined set of quantiles (Takeuchi et al., 2006; Wen et al., 2017; Rodrigues & Pereira, 2018; Taylor, 2000). Simultaneous Quantile Regression (SQR, Tagasovska & Lopez-Paz (2019)) trains one model on all quantiles and is able to learn complex shaped distributions. However the training procedure requires the model to learn to be monotonic instead of being constrained to be so and is not trivial to extend to multidimensional outputs. Pearce et al. (2018) learns a finite set of quantiles by using quality metrics for predictive intervals. ",
+        "bbox": [
+            173,
+            285,
+            825,
+            383
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Normalizing flows have been used in the contexts of variational inference and generative modeling. The approaches to normalizing flows can be categorized into autoregressive methods (Kingma et al., 2016; Papamakarios et al., 2017; Huang et al., 2018; Cao et al., 2019), coupling layers (Dinh et al., 2014; 2016; Kingma & Dhariwal, 2018; Ho et al., 2019), residual networks (Rezende & Mohamed, 2015; van den Berg et al., 2018; Gopal, 2020) and continuous flows (Grathwohl et al., 2018). ",
+        "bbox": [
+            174,
+            390,
+            826,
+            460
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            479,
+            326,
+            496
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3.1 RECALIBRATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            510,
+            331,
+            525
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "An important goal in modeling is to have well-calibrated distributions as this allows users of the model to understand the confidence the model places on its prediction. In other words, with a well-calibrated model, we can better ascertain the uncertainty in the model’s predictions and respond differently to those predictions depending on the uncertainty surrounding them. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Guo et al. (2017) showed that unlike techniques used decades ago such as Bayesian Ridge Regression, modern neural network-based classifiers are very poorly calibrated. A simple variant of Platt scaling and other histogram based techniques applied to a validation set were shown to help alleviate the calibration problem where perfect calibration is defined as ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "equation",
+        "img_path": "images/e43808e66d2bfbf69972fead871980cc742ab0f6c00f45c4152c55d07f3044f3.jpg",
+        "text": "$$\n\\mathbb { P } ( \\hat { Y } = Y | \\hat { P } = p ) = p , \\quad \\forall p \\in [ 0 , 1 ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { Y }$ is a class prediction and $\\hat { P }$ is its predicted probability of correctness. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Kuleshov et al. (2018) extended the analysis in Guo et al. (2017) to neural network-based regressors; isotonic regression, a method for learning monotonic univariate functions, was applied to map from $\\mathcal { F } _ { x _ { j } } ( y _ { j } ) = \\hat { p } _ { j }$ to $| \\{ y _ { n } | \\mathcal { F } _ { x _ { n } } ( y _ { n } ) < \\hat { p } _ { j } , n \\stackrel { - } { = } 1 , . . . , N \\} | / N$ (the fraction of the data where the model CDF is less than $\\hat { p }$ ) to improve calibration where perfect calibration is defined as ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f8cfef23ac0b22f3f9c42993e6b5b55e99c0a224ebf71a0cd384b8b878d64f09.jpg",
+        "text": "$$\n\\mathbb { P } ( Y < \\mathcal { F } _ { X } ^ { - 1 } ( p ) ) = p , \\quad \\forall p \\in [ 0 , 1 ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            380,
+            758,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where ${ \\mathcal { F } } _ { X }$ is the predicted CDF function. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Kuleshov et al. (2018) further introduced calibration error as a metric to quantitatively measure how well the quantiles are aligned: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dcb4f26b646c6177de95703c877c629775389f8a8b0e1d7feaa1897790b407a4.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { p } _ { j } = \\vert \\{ y _ { n } \\vert \\mathcal { F } _ { x _ { n } } ( y _ { n } ) < p _ { j } , n = 1 , \\ldots , N \\} \\vert / N } \\\\ { \\operatorname { c a l } ( y _ { 1 } , \\ldots , y _ { N } ) = \\displaystyle \\sum _ { j = 1 } ^ { M } ( p _ { j } - \\hat { p } _ { j } ) ^ { 2 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $M$ is the number of quantiles that are evaluated. In this paper, we set this to 100 evenly spaced quantiles. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/52aa6c065ffeced6d996d72cde20955f69b0c8da9a20d8031cebe127f210d0b4.jpg",
+        "image_caption": [
+            "Figure 1: A comparison between using a hypernetwork approach versus conditional approach to creating Conditional QuAR Flows to model $p ( X | c )$ where $W _ { 1 }$ and $W _ { 2 }$ parametrize the layers in the residual connection. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 NORMALIZING FLOWS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "A crucial component in improving uncertainty estimates in this paper is to fit normalizing flows, a generative model for density estimation using invertible functions. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Suppose that we wish to formulate a joint distribution on an $n$ -dimensional real vector $x$ . A flowbased approach treats $x$ as the result of a transformation $g$ applied to an underlying vector $z$ sampled from a base distribution $p _ { z } ( z )$ . The generative process for flows is defined as: ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/ee13df7ff53df7ed3c8c5c5dfae1d84c3938600f51fcccaafc04540b8869278f.jpg",
+        "text": "$$\n\\begin{array} { l } { z \\sim p _ { z } ( z ) } \\\\ { x = g ( z ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $p _ { z }$ is often a Normal distribution and $g$ is an invertible function. Notationally, we will use $f = g ^ { - 1 }$ . Using change of variables, the log likelihood of $x$ is ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/09787bde3ddfbf23fec9c4cdeb1c41f2934a30cd91950c65b558857aa9ce255b.jpg",
+        "text": "$$\n\\log p _ { x } ( x ) = \\log p _ { z } \\left( f ( x ) \\right) + \\log \\left| { \\operatorname* { d e t } \\left( \\frac { \\partial f ( x ) } { \\partial x } \\right) } \\right|\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To train flows (i.e. maximize the log likelihood of data points), we need to be able to compute the logarithm of the absolute value of the determinant of the Jacobian of $f$ , also called the log-determinant. To construct large normalizing flows, we can compose smaller ones as this is still invertible and the log-determinant of this composition is the sum of the individual log-determinants. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "An important observation is that every univariate distribution can be viewed as a flow in which the base distribution $p _ { z } ( z )$ is a Uniform distribution over $[ 0 , 1 ]$ and $g$ is the inverse CDF of the univariate distribution. This can be generalized to multivariate distributions using the chain rule. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.3 CONDITIONAL QUAR FLOWS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this paper, the specific normalizing flow used is Quasi-Autoregressive Residual Flows (QuAR Flows, Gopal (2020)). QuAR Flows were shown to handle modeling complex distributions while having nice optimization properties and computationally efficient log-likelihoods. Other commonly used normalizing flows have drawbacks. Coupling layers do not work for one-dimensional inputs. Autoregressive flows are often conditional Gaussians which means in the one-dimensional case, the flow is simply a Gaussian. Residual Flows (Chen et al., 2019) are expressive but, though equivalent to QuAR Flows in the one-dimensional case, are computationally expensive for high-dimensional distributions. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "If we would like to condition a QuAR Flow given some features $c$ , we could use a hypernetwork that maps $c$ to the parameters of the QuAR Flow (Figure 1a). However, this can be computationally expensive if the QuAR Flow is parameterized with thousands of parameters. Instead, conditional information can be incorporated by concatenating the conditions $c$ before applying the first layer in the residual connection within a QuAR Flow (Figure 1b). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 REGRESSION USING NORMALIZING FLOWS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The goal of regression is to model the conditional probability distribution $p ( y | x )$ . Frequently, this task is reduced to obtaining point estimates of the mean or median by minimizing the mean squared error or mean absolute error respectively. In order to access uncertainties, we must model the full conditional distribution; in addition to this, we can still retrieve point estimates of the mean, median, or any other distributional statistic. In this section, we discuss how normalizing flows can be used to this end as well as how recalibration can be reframed as maximum likelihood estimation (MLE). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 CALIBRATION WITH NORMALIZING FLOWS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "One approach to modeling aleatoric uncertainty is to use amortized inference with a Normal distribution as its output distribution, conditioned on features (Lakshminarayanan et al., 2017). However, if a Gaussian is not appropriate to fit the target, then the error in miscalibration can be attributed to approximation error, i.e. insufficient capacity to learn the distribution. A simple example of this is where we have no features to condition on (or all the features we have contain no relevant information) with a one-dimensional target. In Figure 2, we can see that since the target is bimodal, the Gaussian fit does not well describe the data. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To improve this model, we replace the Gaussian with a normalizing flow conditioned on our data, similar to Figure 1b. In this way, the distribution fit to the targets has sufficient capacity to learn complex distributions, including multimodal distributions (Figure 2). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2 RECALIBRATION WITH NORMALIZING FLOWS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            467,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To handle overfitting of the uncertainty, we recalibrate the results of the original model where we assume the original model outputs a full distribution that has closed-form likelihoods. Since the only requirement is a full distribution with a closed-form likelihood, the original model that we recalibrate can be as simple as a Bayesian Linear Regression model or as complicated as a conditional normalizing flow. We recalibrate the original model by fitting a normalizing flow on top of it. In this way, the training process is the following two-step procedure: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/ad181b91bb85b484eab2f815ac5f66a3057be04180b0b5783e57d91cc0f6099a.jpg",
+        "image_caption": [
+            "Figure 2: One-dimensional example of Gaussian versus Flow "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c1646d62cd1f3dda7b41ed6bad0e7e343abd4ada3ef43285bf596837f42266e8.jpg",
+        "text": "$$\n\\begin{array} { r l } & { 1 . \\ f _ { \\theta } = \\arg \\operatorname* { m a x } _ { f _ { \\theta } } \\sum _ { i = 1 } ^ { N _ { t r } } \\log p _ { f _ { \\theta } } ( y _ { i } ^ { t r } | x _ { i } ^ { t r } ) } \\\\ & { 2 . \\ g _ { \\phi } = \\arg \\operatorname* { m a x } _ { g _ { \\phi } } \\sum _ { i = 1 } ^ { N _ { v a l } } \\log p _ { g _ { \\phi } } ( F _ { \\theta } ( y _ { i } ^ { v a l } | x _ { i } ^ { v a l } ) ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where the tra $f _ { \\boldsymbol { \\theta } } ( \\boldsymbol { y } | \\boldsymbol { x } )$ is thet and nction given is the valid $x$ , i $F _ { \\theta } ( y | x )$ is the CDF of hough the for $f _ { \\theta } ( y | x ) , \\{ x ^ { t r } , y ^ { t r } \\} _ { i = 1 } ^ { N _ { t r } }$ ison $\\{ x ^ { v a l } , y ^ { v a l } \\} _ { i = 1 } ^ { N _ { v a l } }$ above shows we can use any likelihood model for $g _ { \\phi }$ , since isotonic regression is a method to fit a free form function to the CDFs, we choose to use normalizing flows, specifically QuAR Flows, as these are more flexible than using autoregressive flows such as MAFs (Papamakarios et al., 2017). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "After the two-step training, the new generative process is: ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/80e6d83ca80da3b9239424445b97d3f04f662d18bb6ae77cac0633b9b50a4f85.jpg",
+        "text": "$$\n\\begin{array} { l } { z \\sim \\mathrm { U n i f o r m } ( 0 , 1 ) } \\\\ { u = G _ { \\phi } ^ { - 1 } ( z ) } \\\\ { y = F _ { \\theta } ^ { - 1 } ( u | x ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $G _ { \\phi }$ is the CDF of $g _ { \\phi }$ . The density function of the above generative process is simply the product of the individual density functions: $f _ { \\theta }$ and $g _ { \\phi }$ . Given this view of the generative process, one interpretation of the two-step process is to first fit $\\theta$ on the training set where $\\phi$ is initialized so that $G _ { \\phi } ^ { - 1 }$ is the identity function, and then to fit $\\phi$ on the validation set while keeping $\\theta$ frozen. Though $\\theta$ and $\\phi$ could be trained end-to-end, similar to the approach in Section 4.1, we show in our experiments (Section 6.2) that there are benefits to this two-step approach. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/9212fa4438adc3e36c2e75cfd149f1d9ffde426fb5aa80f4de671bd843cb558c.jpg",
+        "image_caption": [
+            "Figure 3: Comparison plots of isotonic regression and flow recalibration on multiple outputs. The two axes are different targets being modeled, and the scatter plot shows that the targets are correlated. The top and right sides show the marginal histogram as well as the fit from the recalibrated distribution. The contour plot shows the two-dimensional contours of the recalibrated distribution which can be seen to not align well for isotonic regression. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Given this log-likelihood viewpoint of recalibration, we can trivially incorporate conditioning of the normalizing flow on additional information. For example, say $f _ { \\theta }$ outputs $\\mu$ and $\\sigma$ that parameterize a Normal distribution, we could then condition $g _ { \\phi }$ on these two outputs. If we were to use many features to condition on, we would return to the first situation where we could overfit to the finite number of observations we have. We can think of the first model $f _ { \\theta }$ as compressing/embedding the data into two dimensions, and the second model utilizing the embedding to create a meaningful and well-calibrated distribution on the target. Compared to isotonic regression, we can easily incorporate conditional information while retaining monotonicity as normalizing flows are monotonic by construction. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Recalibration is interesting in that it is useful for both overfitting as well as underfitting. From Guo et al. (2017), we see that multinomial classification models have sufficient capacity to overfit uncertainties and benefit from recalibration. On the other hand, the simple example in Figure 2 suffers from miscalibrated uncertainties due to approximation uncertainty (underfitting), and recalibration reduces the approximation error. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "One solution to handling overfitting is to early-stop on a validation set. Whether early stopping occurs or not, our solution incorporates the information of the validation set to correct potential overfitting. However, if we try to correct for multiple sources of overfitting, then we could introduce a new overfitting problem. In essence, there is a tradeoff between the validation size (which in turn controls training size) and how many sources of overfitting one can try to correct after training. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.3 MULTIVARIATE RECALIBRATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Whereas Platt scaling (Guo et al., 2017) and isotonic regression (Kuleshov et al., 2018) have been shown to improve the calibration of distributions when we have one-dimensional outputs, scaling these solutions to high-dimensional targets is non-trivial. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Say we are modeling two-dimensional real-valued targets and use isotonic regression to calibrate the targets individually. For simplicity, say we use amortized inference with independent output distributions. If the targets are not conditionally independent, the distribution will not be wellcalibrated, even after recalibration with isotonic regression (Figure 3a). However, using normalizing flows, we can compute the CDF of the observed targets per output distribution and recalibrate this two-dimensional distribution (Figure 3b). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The general procedure of recalibrating $N$ -dimensional distributions is to fit a normalizing flow on the $N$ -dimensional vector $x _ { i } = F ( y _ { i } | y _ { < i } )$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/bd31ec7371b5fbab3c41c4af090e7e01d02a0effd5315ff0fe11e911a5356c31.jpg",
+        "image_caption": [
+            "Figure 4: Calibration Q-Q plots for recalibration using isotonic regression and recalibration using flows for the simulated multivariate case in Figure 3a. The legend contains the calibration error for each curve. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 MULTIVARIATE RECALIBRATION EXAMPLE",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In multiclass classification, we often focus on individual metrics such as accuracy, precision, recall, F1, etc. However, exploring the accuracy of the separate classes using a confusion matrix can show if the model is biased even if the performance on average is good. Similar to this view for classification, we extend the calibration plot introduced in (Kuleshov et al., 2018) to show calibration performance on different subsets of the data, binned by either a feature or a prediction, e.g. Figure 4. The only restriction is the binning cannot be a function of the target we are evaluating. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Say we are modeling two-dimensional real-valued targets $( Y _ { 1 } , Y _ { 2 } )$ shown in Figure 3a. If we fit independent Gaussians to the target, the resulting distribution will not align with the contours of the data. ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "We first recalibrate the distribution by fittting isotonic regression models on $F ( Y _ { 1 } )$ and $F ( Y _ { 2 } | Y _ { 1 } )$ The results for $F ( Y _ { 2 } | Y _ { 1 } )$ are shown in Figure 4a where we can see that, though across the full dataset the calibration error is zero, when we bucket by $F ( Y _ { 1 } )$ , the calibration error achieved is both significant and a strong function of $Y _ { 1 }$ as it is unable to account for correlations in $Y _ { 1 }$ and $Y _ { 2 }$ . In other words, the model is not calibrated in any given region of the data but only in aggregate. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "However, if we recalibrate using normalizing flows by fitting a flow on the pair $( F ( Y _ { 1 } ) , F ( Y _ { 2 } | Y _ { 1 } ) )$ , we can see in Figure 4b that even if we bucket by $F ( Y _ { 1 } )$ , the calibration error stays close to zero. Unlike isotonic regression, the flow can learn the correlation between the CDFs by being trained jointly on all of them. In addition to this, had we used a more complex calibration model that could model the correlations directly, we would not be able to compute the correlation after recalibrating with isotonic regression. By recalibrating with flows, we obtain a result that is still a distribution with a closed-form likelihood; this allows us to still be able to use the model to compute the correlation and any other distributional statistics. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In Figure 5, we show visually the difference between recalibration with flows and with isotonic regression. The key differences being that flows can recalibrate the CDFs jointly and be conditioned on additional information $c$ , while isotonic regression is fit on each CDF independently and is unable to incorporate any additional information. The reason is that flows are monotonic by construction and can therefore handle multiple variables and conditions. ",
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+    {
+        "type": "text",
+        "text": "6 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "In this section, we analyze the performance of flows on multiple UCI datasets. Importantly, we do not aim for state-of-the-art performance, nor do we expect normalizing flows to outperform isotonic ",
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+        "img_path": "images/875700dffcddbe4a75f5e5d09bdaf048c24fbe3c4bd6dcd6bef0495db4def2fa.jpg",
+        "image_caption": [
+            "Figure 5: A diagram for multivariate recalibration using the same notation as that used in Section 4.2. The first step for recalibration is always the calibration step, shown in the blue box, where any likelihood model $f _ { \\theta }$ can be used. On the left, isotonic regression is used to recalibrate the CDFs from $f _ { \\theta }$ ; on the right, normalizing flows $g _ { \\phi }$ are used to recalibrate the CDFs from $f _ { \\theta }$ conditioned on some $c$ . "
+        ],
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+    {
+        "type": "text",
+        "text": "regression in aggregate. In terms of calibration, we show that the increase in expressivity can be beneficial; in terms of recalibration, we show specific failure points of isotonic regression that binning uncovers and how normalizing flows fixes these issues. ",
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+    {
+        "type": "text",
+        "text": "6.1 CALIBRATION ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "To test the effect of increased capacity in distributions, we tested how well normalizing flows improve performance in distributional estimates compared to a few other popular approaches for estimating uncertainties. We do not compare directly against other methods for conditional distributions such as autoregressive flows for the reason specified in Section 3.3: in the one dimensional case, those are often the same as ConditionalGaussian. Instead of replacing the output distribution with a complex distribution created by hand, such as an explicit mixture, we rely on using more expressive flows which can model mixture behavior as well as more complicated behavior such as skew and kurtosis. ",
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+    {
+        "type": "table",
+        "img_path": "images/7abeef572c6f552073f92457b1bb61bd0b47b3ab73a6d809ea69090876e017c1.jpg",
+        "table_caption": [
+            "Table 1: Results before and after recalibration on the UCI dataset “naval-propulsion-plant”. Details on the model methodology as well as performance on other UCI datasets are in Appendix B. The mean squared error (MSE) and its uncertainty is computed using 20 seeds. “Calib” is the calibration error (Equation 1) using 100 quantiles across 20 seeds instead of per seed. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\"></td><td colspan=\"2\">Before Recalib.</td><td colspan=\"2\">Isotonic Recalib.</td><td colspan=\"2\">Flow Recalib.</td></tr><tr><td>MSE</td><td>Calib</td><td>MSE</td><td>Calib</td><td>MSE</td><td>Calib</td></tr><tr><td>ConditionalFlow</td><td>0.009 ±0.004</td><td>0.45</td><td>N.A.</td><td>&lt;0.01</td><td>0.016 ± 0.012</td><td>&lt;0.01</td></tr><tr><td>ConditionalGaussian</td><td>0.012 ±0.006</td><td>2.35</td><td>N.A.</td><td>&lt;0.01</td><td>0.012 ±0.006</td><td>&lt;0.01</td></tr><tr><td>BayesianRidgeRegression</td><td>0.055 ±0.004</td><td>8.03</td><td>N.A.</td><td>0.01</td><td>0.48 ± 0.02</td><td>&lt;0.01</td></tr><tr><td>MC Dropout</td><td>0.005 ±0.002</td><td>0.51</td><td>N.A.</td><td>N.A.</td><td>N.A.</td><td>N.A.</td></tr><tr><td>SQR</td><td>0.0052 ± 0.0012</td><td>1.44</td><td>N.A.</td><td>N.A.</td><td>N.A.</td><td>N.A.</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "All results were evaluated on a held out test set that was not used for calibration or recalibration. Due to limited space, the full extent of the results and details of the models are left for Appendix A and Appendix B. To show the usefulness of the plot we introduced in Section 5, we focus on the results on the UCI dataset “naval-propulsion-plant” (Dheeru & Karra Taniskidou, 2017) in Table 1 and Section 6.2. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The models with consistent performance across UCI datasets in terms of calibration are Bayesian Ridge Regression, Conditional Gaussian, and Conditional Flow, with Conditional Flow often performing well in cases where Conditional Gaussian is unable to. Whereas these three models are well calibrated, the mean performance is much weaker for Bayesian Ridge Regression due to the limitations of linear functions. ",
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+    },
+    {
+        "type": "text",
+        "text": "6.2 RECALIBRATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Of the models we tested, we recalibrated the Bayesian Ridge Regression, Conditional Gaussian, and Conditional Flow models. MC Dropout and SQR could also be recalibrated, though we chose not to as they lack closed form CDFs, meaning the technique would be approximate. In Table 1, we see the calibration errors after recalibrating with normalizing flows are comparable to those from isotonic regression. However, after isotonic regression is applied, only the calibration error can be computed. In contrast, we can still compute ordinary distribution statistics such as the mean after applying normalizing flows. Due to this property, we can even detect cases in which an improvement in calibration error coincides with a degradation of mean performance, suggesting if the goal were to only get point estimates of the mean, recalibration could be detrimental. For example, in Table 1 this was the case for Bayesian Ridge Regression. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In Figure 6a, whereas the calibration error metric shows the flow recalibrated model is well-calibrated averaged across the full dataset, we can see that see that, when bucketing by the predicted mean, the calibration error is significant. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To improve this, we recalibrate with a normalizing flow conditioned on the mean predicted by the original model. In this way, we see in Figure 6b that though the aggregate performance does not change, the performance is improved across different buckets. In essence we see that if we were analyzing the distribution across the dataset, we would be accurate; if we started to look closer to individual distributions, the performance is worse suggesting the quantiles of individual distributions cannot be trusted. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/5ba0a74e3b6aa56744216a122f89938ce61fa7461edf3b64016e47551252acc8.jpg",
+        "image_caption": [
+            "Figure 6: Calibration Q-Q plots for recalibration with an unconditional flow and recalibration with a conditional flow on the dataset “naval-propulsion-plant”. The legend contains the calibration error for each curve. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/96799e60b702cd7228ab4ea95bae04d75bb639f63db7286ef906822c86a386ab.jpg",
+        "image_caption": [
+            "Figure 7: Calibration Q-Q plots for recalibration with isotonic regression and recalibration with normalizing flows on the second time step on the dataset “electricity”. The legend contains the calibration error for each curve. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6.3 TIME SERIES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "For our time series example, we trained a model on the UCI Dataset “electricity”. We predict two time steps and show that even though we used an autoregressive model for calibration (the model could learn correlated distributions), there is still some correlation not captured that recalibration can help with. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Figure 7a shows that the calibration error of $F ( Y _ { 2 } | Y _ { 1 } )$ when recalibrated independently with isotonic regression is significant. Figure 7b shows that a flow trained jointly on the CDFs significantly reduces the errors. Although the calibration error of the marginal distribution for the second step is comparable between the two approaches, the conditional performance favors normalizing flows. A possible explanantion of the still imperfect calibration of the flow model, though better than isotonic regression, is distribution shift. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSION AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we addressed the calibration issue in modern neural networks through using higher capacity distributions via normalizing flows. We showed that normalizing flows can not only help during calibration, but also in recalibrating miscalibrated distributions, even improving the fit in the underfitting case. Though isotonic regression also uses a two-step producedure, we reframe the problem as MLE through which we can trivially generalize recalibration to the multivariate case. Though isotonic regression could be extended such that we could compute the log likelihood implied by its change of variables, normalizing flows are a more natural way to handle this with more research backing this technique, and are easily extendable to conditional and joint distributions. Interestingly, though the Conditional Flow has all it needs to be well calibrated, recalibration helps the overall calibration which was similarly found for classification in Guo et al. (2017); it remains for future work to better understand why and when this is useful. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "Further, we extend the plot introduced in Kuleshov et al. (2018) by showing the calibration error on different subsets of the data, for example bucketed by mean prediction. Though this plot is able to show the calibration error across buckets, for future work, we will explore ways to visually show any trends in the miscalibration as well as whether the misalignments of the quantiles are statistically significant. ",
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+    {
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+        "text": "Rianne van den Berg, Leonard Hasenclever, Jakub M. Tomczak, and Max Welling. Sylvester Normalizing Flows for Variational Inference. arXiv e-prints, art. arXiv:1803.05649, March 2018. ",
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+    {
+        "type": "text",
+        "text": "Ruofeng Wen, Kari Torkkola, Balakrishnan Narayanaswamy, and Dhruv Madeka. A Multi-Horizon Quantile Recurrent Forecaster. arXiv e-prints, art. arXiv:1711.11053, November 2017. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/c75db8aacd9d4a38c987062b87207c17d500f45fc7262577a8766d51b64af6c7.jpg",
+        "image_caption": [
+            "Figure 8: In this figure, we show our conditional flow model. A neural network takes $X$ and outputs the scale and shift of an affine flow and a condition vector for QuAR Flows. The condition used in the QuAR Flow is the same for both $\\mathrm { Q u A R }$ Flows. After the first conditional affine flow, there are two QuAR Flows with unconditional affine flows placed in between these two and at the end. The Conditional Flow model is equivalent to a Conditional Gaussian if we removed the QuAR Flows. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            271,
+            98,
+            725,
+            190
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A MODEL TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            297,
+            442,
+            314
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1 CALIBRATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            330,
+            316,
+            345
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We compare the performance of our model and multiple other popular approaches on 8 UCI datasets (Dheeru & Karra Taniskidou, 2017): boston-housing, concrete, energy, kin8nm, naval-propulsionplant, power-plant, wine-quality-red and yacht. The models we compared Conditional Flows to were: Conditional Gaussian (amortized inference with Gaussians, Lakshminarayanan et al. (2017)), Bayesian Ridge Regression (Tipping, 2001), MC Dropout (Gal & Ghahramani, 2016), Simultaneous Quantile Regression (SQR) (Tagasovska & Lopez-Paz, 2019), Conditional Quantile (SQR where we only learn prespecified quantiles), Quality Driven (Pearce et al., 2018), and a Gradient Boosted Decision Tree with LightGBM (Ke et al., 2017). ",
+        "bbox": [
+            173,
+            357,
+            825,
+            469
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Figure 8 shows a representation of the Conditional Flow network we used. The QuAR Flows are conditioned on a 64-dimensional representation from the condition network. Each Conditional QuAR Flow has two hidden layers in the residual connection with a hidden size 64 with ELU (Clevert et al., 2015) as our activation. The condition network has a single hidden layer with size 64 and ReLU as our activation. The QuAR Flows are conditioned on the 64-dimensional representation from the condition network. ",
+        "bbox": [
+            173,
+            476,
+            825,
+            560
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For the other neural network approaches, we used networks with two hidden layers with size 128 and used ReLU as our nonlinearity. For hyperparameter tuning on all our neural networks, we searched the following grid: ",
+        "bbox": [
+            176,
+            566,
+            825,
+            609
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• Learning rate: 1e-2, 1e-3, 1e-4   \n• Weight decay: 0, 1e-3, 1e-2, 1e-1   \n• Dropout Rates: 0.1 0.25, 0.5 0.75 ",
+        "bbox": [
+            215,
+            622,
+            454,
+            679
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The hyperparameters we used were taken from Tagasovska & Lopez-Paz (2019). We trained each of our neural networks for 2000 epochs with batch size 128. ",
+        "bbox": [
+            171,
+            693,
+            825,
+            722
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For our LightGBM tree model (Ke et al., 2017), we used mean squared error as our loss to learn the mean of the output distribution. The grid of hyperparameters we searched was: ",
+        "bbox": [
+            169,
+            728,
+            823,
+            757
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• Number of Trees: 100, 1000   \n• Maximum Depths: 3, 5, 7, 9   \n• Learning Rates: 1e-1, 1e-2, 1e-3, 1e-4   \n• Minimum Child Samples: 1, 20   \n• Dropout Rates: 0.1 0.25, 0.5 0.75 ",
+        "bbox": [
+            215,
+            770,
+            485,
+            867
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To make the tree model distributional, assuming homoscedasticity, we computed the standard deviation of the predicted normal distribution by computing the standard deviation of the residuals (observations minus predictions). ",
+        "bbox": [
+            176,
+            882,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For SQR and MC Dropout, we computed the estimated mean and CDF from 1000 samples. For Conditional Flows, we computed the estimated mean by using the Gauss-Legendre quadrature with 100 points to approximate the integral. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            146
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.2 RECALIBRATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            167,
+            334,
+            183
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For recalibration, we trained the recalibration model on the validation set of each of the 20 seeds. For the flow architecture, we stacked 8 QuAR Flows with affine flows in between each one as well as at the beginning and end. Since the input to the flow is a $\\mathrm { C D F } \\in [ 0 , 1 ]$ , we also transform the data with a Logit Function as the first layer of the flow and account for the change of variables in the loss. In addition to this, we used a dropout rate of 0.25, two hidden layers with size 64 in each residual connection, and used Polyak averaging (Polyak & Juditsky, 1992) with a decay of 0.999 for evaluation. ",
+        "bbox": [
+            174,
+            195,
+            826,
+            294
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.3 TIME SERIES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            315,
+            310,
+            330
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For our time series experiments, we tested the performance of flow recalibration on the UCI dataset “electricity”. We filtered the data down to observations after 2014-01-01. We split the data by date, where the first $72 \\%$ of dates are in the train set, the next $18 \\%$ are in validation set, and the last $10 \\%$ are in the test set. ",
+        "bbox": [
+            174,
+            343,
+            825,
+            398
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The data is given in 15 minute increments, and so similarly to Deep AR (Salinas et al., 2017), we averaged the values per hour to have hourly data. Since we were trying to analyze the effect of correlated targets, we predicted two time steps conditioned on 48 hours of data using an autoregressive model. ",
+        "bbox": [
+            174,
+            406,
+            825,
+            462
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We included three numeric covariates: the time step within the sampled time series, the hour of the day, and the day of the week. Each of the covariates were normalized to have zero mean and one standard deviation. In addition to this, since there are 370 different types of time series, we also added an embedding layer from 370 to an embedding dimension of 20, same as that used in Deep AR (Salinas et al., 2017). We concatenated this vector with the observations and covariates, leading to a 27-dimensional vector. ",
+        "bbox": [
+            174,
+            469,
+            825,
+            553
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Since all the observations are positive, we model the target with a conditional Gamma distribution (i.e. amortized inference with Gamma distributiet al., 2017), we normalized each time series $\\{ z _ { i , t } \\} _ { t = 0 } ^ { T - 1 }$ of by $\\begin{array} { r } { v _ { i } = 1 + 1 / T \\sum _ { t = 0 } ^ { T - 1 } z _ { i , t } . } \\end{array}$ eep AR (Salinas. We account for the scaling in the output distribution, i.e. Gamma $( k , \\theta * v _ { i } )$ . ",
+        "bbox": [
+            174,
+            559,
+            825,
+            619
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The model we used was a 1 layer LSTM, and then two linear layers with a ReLU and Dropout (Srivastava et al., 2014) in between were applied to the hidden representation. We set the dropout rate to 0.5 and used a hidden size of 48 everywhere. For evaluation, we used Polyak averaging (Polyak & Juditsky, 1992) with a decay of 0.999. ",
+        "bbox": [
+            174,
+            625,
+            825,
+            681
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For recalibration with flows, we trained on the CDFs for both time steps with a flow with the same architecture as that used in Appendix A.2. ",
+        "bbox": [
+            174,
+            688,
+            823,
+            717
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            742,
+            421,
+            758
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B.1 CALIBRATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            776,
+            316,
+            791
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In Table 2, we compare Conditional Flows against multiple other approaches for estimating uncertainty, and we measure performance using Prediction Interval Coverage Probability (PICP), Mean Squared Error (MSE), and Calibration Error. ",
+        "bbox": [
+            176,
+            805,
+            823,
+            847
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For each model, we trained the model on 20 different seeds where each seed randomly splits the data in $72 \\%$ train, $18 \\%$ validation, and $10 \\%$ test. We picked the best hyperparameter by choosing the hyperparameters that gave the best average validation loss across the 20 seeds where the loss used is the loss defined by the training procedure for the model, e.g. for Conditional Gaussian, we used log likelihood; for LightGBM, we used MSE; for QualityDriven, we used the Quality Driven Loss, etc. ",
+        "bbox": [
+            176,
+            853,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/ee857c7ba0f01bf9efb745de65bc42756ed607cf38e79d06640bf392e208a62a.jpg",
+        "table_caption": [
+            "Table 2: Results on percentage of data captured from 0.025 to 0.975 quantiles (PICP), MSE, and calibration error with 100 buckets. The results are from minimizing the average validation loss across 20 seeds. The optimal value of PICP is 0.95, and the optimal value for MSE and Calib is 0.00. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">bostonHousing</td><td colspan=\"3\">concrete</td></tr><tr><td></td><td>PICP</td><td>MSE</td><td>Calib</td><td>PICP</td><td>MSE</td><td>Calib</td></tr><tr><td>ConditionalFlow</td><td>0.93 ± 0.04</td><td>0.13 ± 0.06</td><td>0.06</td><td>0.92 ± 0.03</td><td>0.09 ±0.03</td><td>0.01</td></tr><tr><td>ConditionalGaussian</td><td>0.93 ± 0.03</td><td>0.14±0.07</td><td>0.13</td><td>0.91 ± 0.02</td><td>0.09 ±0.03</td><td>0.06</td></tr><tr><td>BayesianRidgeRegression</td><td>0.95 ± 0.03</td><td>0.28±0.08</td><td>0.34</td><td>0.95 ± 0.03</td><td>0.40±0.07</td><td>0.01</td></tr><tr><td>MC Dropout</td><td>0.50 ± 0.08</td><td>0.11 ± 0.05</td><td>2.99</td><td>0.69 ± 0.05</td><td>0.07±0.03</td><td>0.89</td></tr><tr><td>SQR</td><td>0.73 ± 0.08</td><td>0.11 ± 0.05</td><td>0.48</td><td>0.74± 0.04</td><td>0.07± 0.03</td><td>0.47</td></tr><tr><td>ConditionalQuantile</td><td>0.92 ± 0.05</td><td>N.A.</td><td>N.A.</td><td>0.93 ± 0.03</td><td>N.A.</td><td>N.A.</td></tr><tr><td>QualityDriven</td><td>0.83 ± 0.06</td><td>N.A.</td><td>N.A.</td><td>0.83± 0.04</td><td>N.A.</td><td>N.A.</td></tr><tr><td>LightGBM</td><td>0.04±0.03</td><td>0.12 ± 0.06</td><td>7.74</td><td>0.63±0.07</td><td>0.07±0.02</td><td>1.25</td></tr><tr><td></td><td colspan=\"3\">energy</td><td colspan=\"3\">kin8nm</td></tr><tr><td></td><td>PICP</td><td>MSE</td><td>Calib</td><td>PICP</td><td>MSE</td><td>Calib</td></tr><tr><td>ConditionalFlow</td><td>0.96 ± 0.02</td><td>0.0022 ± 0.0006</td><td>0.12</td><td>0.952 ± 0.006</td><td>0.081 ± 0.013</td><td>0.01</td></tr><tr><td>ConditionalGaussian</td><td>0.97 ± 0.03</td><td>0.0030± 0.0009</td><td>0.26</td><td>0.947 ± 0.007</td><td>0.071 ± 0.005</td><td>&lt;0.01</td></tr><tr><td>BayesianRidgeRegression</td><td>0.89 ± 0.04</td><td>0.09 ±0.02</td><td>0.24</td><td>0.951 ± 0.007</td><td>0.59 ± 0.02</td><td>0.09</td></tr><tr><td>MC Dropout</td><td>0.97± 0.02</td><td>0.0026 ± 0.0010</td><td>0.17</td><td>0.69 ± 0.02</td><td>0.069 ± 0.005</td><td>1.14</td></tr><tr><td>SQR</td><td>0.97± 0.02</td><td>0.0028 ± 0.0012</td><td>0.37</td><td>0.892 ± 0.010</td><td>0.072 ± 0.005</td><td>0.01</td></tr><tr><td>ConditionalQuantile</td><td>0.98 ±0.02</td><td>N.A.</td><td>N.A.</td><td>0.943 ± 0.011</td><td>N.A.</td><td>N.A.</td></tr><tr><td>QualityDriven</td><td>0.96 ± 0.04</td><td>N.A.</td><td>N.A.</td><td>0.88±0.02</td><td>N.A.</td><td>N.A.</td></tr><tr><td>LightGBM</td><td>0.24±0.05</td><td>0.0013 ± 0.0006</td><td>5.18</td><td>0.205 ± 0.010</td><td>0.190 ± 0.011</td><td>5.59</td></tr><tr><td></td><td colspan=\"3\">naval-propulsion-plant</td><td colspan=\"3\">power-plant</td></tr><tr><td></td><td>PICP</td><td>MSE</td><td>Calib</td><td>PICP</td><td>MSE</td><td>Calib</td></tr><tr><td>ConditionalFlow</td><td>0.98 ± 0.03</td><td>0.009 ±0.004</td><td>0.45</td><td>0.952 ± 0.007</td><td>0.048 ±0.007</td><td>0.01</td></tr><tr><td>ConditionalGaussian</td><td>0.9995 ± 0.0007</td><td>0.012 ± 0.006</td><td>2.35</td><td>0.947 ± 0.010</td><td>0.048 ±0.006</td><td>&lt;0.01</td></tr><tr><td>BayesianRidgeRegression</td><td>1.0 ±0.0</td><td>0.055 ± 0.004</td><td>8.03</td><td>0.964 ± 0.007</td><td>0.072 ± 0.007</td><td>0.01</td></tr><tr><td>MC Dropout</td><td>0.98 ±0.02</td><td>0.005±0.002</td><td>0.51</td><td>0.59 ±0.02</td><td>0.046±0.007</td><td>1.88</td></tr><tr><td>SQR</td><td>0.998 ± 0.002</td><td>0.0052 ± 0.0012</td><td>1.44</td><td>0.93 ± 0.02</td><td>0.048 ±0.007</td><td>0.02</td></tr><tr><td>ConditionalQuantile</td><td>0.97±0.02</td><td>N.A.</td><td>N.A.</td><td>0.937 ± 0.014</td><td>N.A.</td><td>N.A.</td></tr><tr><td>QualityDriven</td><td>0.949 ± 0.010</td><td>N.A.</td><td>N.A.</td><td>0.89 ±0.02</td><td>N.A.</td><td>N.A.</td></tr><tr><td>LightGBM</td><td>0.714 ± 0.011</td><td>0.0056 ± 0.0009</td><td>0.79</td><td>0.17 ± 0.02</td><td>0.035 ± 0.007</td><td>5.95</td></tr><tr><td></td><td colspan=\"3\">wine-quality-red</td><td colspan=\"3\">yacht</td></tr><tr><td></td><td>PICP</td><td>MSE</td><td>Calib</td><td>PICP</td><td>MSE</td><td>Calib</td></tr><tr><td>ConditionalFlow</td><td>0.90 ±0.02</td><td>0.6±0.2</td><td>0.04</td><td>0.95 ± 0.04</td><td>0.004±0.003</td><td>0.27</td></tr><tr><td>ConditionalGaussian</td><td>0.95 ± 0.02</td><td>0.59 ± 0.09</td><td>0.22</td><td>0.99 ±0.02</td><td>0.004±0.002</td><td>0.62</td></tr><tr><td>BayesianRidgeRegression</td><td>0.95 ± 0.02</td><td>0.63 ± 0.11</td><td>0.05</td><td>0.94± 0.03</td><td>0.35 ± 0.10</td><td>0.25</td></tr><tr><td>MC Dropout</td><td>0.35 ± 0.04</td><td>0.57 ±0.09</td><td>4.01</td><td>0.97 ± 0.04</td><td>0.004±0.002</td><td>0.21</td></tr><tr><td>SQR</td><td>0.81±0.09</td><td>0.59 ±0.10</td><td>0.15</td><td>0.96 ±0.04</td><td>0.004±0.004</td><td>0.49</td></tr><tr><td>ConditionalQuantile</td><td>0.95 ± 0.02</td><td>N.A.</td><td>N.A.</td><td>0.99 ± 0.02</td><td>N.A.</td><td>N.A.</td></tr><tr><td>QualityDriven</td><td>0.84 ±0.03</td><td>N.A.</td><td>N.A.</td><td>0.97 ± 0.04</td><td>N.A.</td><td>N.A.</td></tr><tr><td>LightGBM</td><td>0.40 ± 0.05</td><td>0.55± 0.10</td><td>2.93</td><td>0.45 ± 0.08</td><td>0.003±0.003</td><td>3.04</td></tr></table>",
+        "bbox": [
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+            233,
+            875,
+            837
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/d978e71823dd45f48ecdfeb91085639f8de04ae9198cc4086ae3aa1241745175.jpg",
+        "table_caption": [
+            "Table 3: Calibration error (with 100 buckets) before and after recalibration. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">BayesianRidgeRegression</td><td colspan=\"3\">ConditionalGaussian</td><td colspan=\"3\">ConditionalFlow</td></tr><tr><td></td><td>Before</td><td>Flow</td><td>Iso</td><td>Before</td><td>Flow</td><td>Iso</td><td>Before</td><td>Flow</td><td>Iso</td></tr><tr><td>bostonHousing</td><td>0.34</td><td>0.02</td><td>0.02</td><td>0.13</td><td>0.04</td><td>0.02</td><td>0.06</td><td>0.02</td><td>0.01</td></tr><tr><td>concrete</td><td>0.01</td><td>0.01</td><td>0.01</td><td>0.06</td><td>0.02</td><td>0.02</td><td>0.01</td><td>0.02</td><td>0.03</td></tr><tr><td>energy</td><td>0.24</td><td>0.02</td><td>0.01</td><td>0.26</td><td>&lt;0.01</td><td>0.01</td><td>0.12</td><td>0.01</td><td>0.01</td></tr><tr><td>kin8nm</td><td>0.09</td><td>0.01</td><td>&lt;0.01</td><td>&lt;0.01</td><td>&lt;0.01</td><td>&lt;0.01</td><td>0.01</td><td>&lt;0.01</td><td>&lt;0.01</td></tr><tr><td>naval-propulsion-plant</td><td>8.03</td><td>0.01</td><td>&lt;0.01</td><td>2.35</td><td>&lt;0.01</td><td>&lt;0.01</td><td>0.45</td><td>&lt;0.01</td><td>&lt;0.01</td></tr><tr><td>power-plant</td><td>0.01</td><td>&lt;0.01</td><td>&lt;0.01</td><td>&lt;0.01</td><td>&lt;0.01</td><td>&lt;0.01</td><td>0.01</td><td>&lt;0.01</td><td>&lt;0.01</td></tr><tr><td>wine-quality-red</td><td>0.05</td><td>0.02</td><td>&lt;0.01</td><td>0.22</td><td>0.04</td><td>0.02</td><td>0.04</td><td>0.01</td><td>0.01</td></tr><tr><td>yacht</td><td>0.25</td><td>0.05</td><td>0.07</td><td>0.62</td><td>0.03</td><td>0.03</td><td>0.27</td><td>0.02</td><td>0.03</td></tr></table>",
+        "bbox": [
+            142,
+            126,
+            851,
+            275
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/0fc0d10d28bfe001a56686f5c0b27f829ffc23a2fccbd3c80e31c43e5c515719.jpg",
+        "table_caption": [
+            "Table 4: MSE performance before and after recalibration. We do not show the MSE for isotonic regression as the mean cannot be computed after recalibrating with isotonic regression. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\"></td><td colspan=\"2\">BayesianRidgeRegression</td><td colspan=\"2\">ConditionalGaussian</td></tr><tr><td>Before</td><td>Flow</td><td>Before</td><td>Flow</td></tr><tr><td>bostonHousing</td><td>0.28 ±0.08</td><td>0.28 ±0.08</td><td>0.14 ± 0.07</td><td>0.14 ± 0.07</td></tr><tr><td>concrete</td><td>0.40 ± 0.07</td><td>0.40 ± 0.07</td><td>0.09 ±0.03</td><td>0.09 ± 0.03</td></tr><tr><td>energy</td><td>0.09 ±0.02</td><td>0.09 ±0.02</td><td>0.0030 ±0.0009</td><td>0.0032 ± 0.0011</td></tr><tr><td>kin8nm</td><td>0.59 ± 0.02</td><td>0.59 ± 0.02</td><td>0.071 ± 0.005</td><td>0.072 ± 0.005</td></tr><tr><td>naval-propulsion-plant</td><td>0.055 ± 0.004</td><td>0.48 ±0.02</td><td>0.012 ± 0.006</td><td>0.012 ± 0.007</td></tr><tr><td>power-plant</td><td>0.072 ± 0.007</td><td>0.072 ± 0.007</td><td>0.048 ± 0.006</td><td>0.049 ± 0.006</td></tr><tr><td>wine-quality-red</td><td>0.63 ± 0.11</td><td>0.63 ± 0.11</td><td>0.59 ± 0.09</td><td>0.59 ± 0.10</td></tr><tr><td>yacht</td><td>0.35 ± 0.10</td><td>0.36 ± 0.10</td><td>0.004 ±0.002</td><td>0.004 ± 0.003</td></tr><tr><td colspan=\"2\"></td><td colspan=\"2\">ConditionalFlow</td><td></td></tr><tr><td rowspan=\"10\"></td><td></td><td>Before</td><td>Flow</td><td></td></tr><tr><td>bostonHousing</td><td>0.13 ± 0.06</td><td></td><td></td></tr><tr><td>concrete</td><td>0.09 ± 0.03</td><td>0.13 ± 0.06 0.09 ± 0.03</td><td></td></tr><tr><td></td><td>0.0022 ± 0.0006</td><td>0.0027 ± 0.0007</td><td></td></tr><tr><td>energy kin8nm</td><td>0.081 ± 0.013</td><td>0.081 ± 0.013</td><td></td></tr><tr><td>naval-propulsion-plant</td><td>0.009 ± 0.004</td><td>0.016 ± 0.012</td><td></td></tr><tr><td>power-plant</td><td>0.048 ± 0.007</td><td>0.048 ± 0.007</td><td></td></tr><tr><td>wine-quality-red</td><td>0.6± 0.2</td><td>0.64 ± 0.15</td><td></td></tr><tr><td>yacht</td><td></td><td></td><td></td></tr><tr><td></td><td>0.004 ± 0.003</td><td>0.008 ± 0.012</td><td></td></tr></table>",
+        "bbox": [
+            173,
+            328,
+            830,
+            650
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Reported in Table 2 are the mean and standard deviation of PICP and MSE across the 20 seeds. Due to the limited data, for calibration error with 100 bins, we computed it over the CDFs from all 20 seeds. We included PICP (the percentage of true observations between the quantiles 0.025 and 0.975) to compare against alternative approaches where we prespecify which quantiles we need. ",
+        "bbox": [
+            173,
+            674,
+            826,
+            731
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "B.2 RECALIBRATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            747,
+            334,
+            762
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "From the above section, we recalibrate the best Bayesian Ridge Regression, Conditional Gaussian, and Conditional Flow. In Table 3, we compare the calibration error before recalibration, after recalibration with isotonic regression, and after recalibration with normalizing flows. ",
+        "bbox": [
+            174,
+            773,
+            826,
+            815
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "As we can see, across all 8 UCI datasets, the calibration errors after recalibrating with normalizing flows are comparable to those from isotonic regression. More interestingly though, improvement in calibration error can be detrimental to mean performance. ",
+        "bbox": [
+            174,
+            821,
+            825,
+            864
+        ],
+        "page_idx": 15
+    }
+]
\ No newline at end of file
diff --git a/parse/train/H8VDvtm1ij8/H8VDvtm1ij8_middle.json b/parse/train/H8VDvtm1ij8/H8VDvtm1ij8_middle.json
new file mode 100644
index 0000000000000000000000000000000000000000..003562bc626ab57b32445e8d627ae4f51bb8f4df
--- /dev/null
+++ b/parse/train/H8VDvtm1ij8/H8VDvtm1ij8_middle.json
@@ -0,0 +1,35878 @@
+{
+    "pdf_info": [
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+                                    "score": 1.0,
+                                    "content": "In machine learning, due to model misspecification and overfitting, estimates",
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+                                    "content": "of the aleatoric uncertainty are often inaccurate. One approach to fix this is",
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+                                    "content": "isotonic regression, in which a monotonic function is fit on a validation set to",
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+                                    "content": "infeasible to compute additional statistics of interest on the model distribution",
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+                                    "score": 1.0,
+                                    "content": "(such as the mean). In this paper, through a reframing of recalibration as MLE,",
+                                    "type": "text"
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+                        {
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+                                    "content": "we replace isotonic regression with normalizing flows. This allows us to retain",
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+                                    "content": "the ability to compute the statistical properties of the model (such as closed-form",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "likelihoods, mean, correlation, etc.) and provides an opportunity for additional",
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+                                    "score": 1.0,
+                                    "content": "capacity at the cost of possible overfitting. Most importantly, the fundamental",
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+                            "index": 14
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+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "properties of normalizing flows allow us to generalize recalibration to conditional",
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+                                    "score": 1.0,
+                                    "content": "and multivariate distributions. To aid in detecting miscalibration and measuring",
+                                    "type": "text"
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+                            "index": 16
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+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "our success at fixing it, we use a simple extension of the calibration Q-Q plot.",
+                                    "type": "text"
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+                            ],
+                            "index": 17
+                        }
+                    ],
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+                },
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+                                    "score": 1.0,
+                                    "content": "uncertainty in its prediction. Most taxonomies classify uncertainty into three sources: approximation,",
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+                                    "content": "narayanan et al., 2017; Nix & Weigend, 1994; Kendall & Gal, 2017) where a model, such as a",
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+                                    "content": "used normalizing flows have drawbacks. Coupling layers do not work for one-dimensional inputs.",
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+                                    "content": "to the parameters of the QuAR Flow (Figure 1a). However, this can be computationally",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "error or mean absolute error respectively. In order to access uncertainties, we must model the full",
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+                                    "score": 1.0,
+                                    "content": "tion as its output distribution, conditioned on features (Lakshminarayanan et al., 2017). However,",
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+                                    "content": "approximation error, i.e. insufficient capacity to learn the distribution. A simple example of this is",
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+                                    "score": 1.0,
+                                    "content": "where we have no features to condition on (or all the features we have contain no relevant information)",
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+                                    "content": "with a one-dimensional target. In Figure 2, we can see that since the target is bimodal, the Gaussian",
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+                                    "score": 1.0,
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+                                    "content": "we extend the calibration plot introduced in (Kuleshov et al., 2018) to show calibration performance",
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+                                    "score": 1.0,
+                                    "content": "on different subsets of the data, binned by either a feature or a prediction, e.g. Figure 4. The only",
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+                                    "content": "restriction is the binning cannot be a function of the target we are evaluating.",
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+                                    "score": 1.0,
+                                    "content": "are shown in Figure 4a where we can see that, though across the full",
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+                                    "content": ", the calibration error achieved is both",
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+                                    "content": ". In",
+                                    "type": "text"
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+                            ],
+                            "index": 19
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+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "other words, the model is not calibrated in any given region of the data but only in aggregate.",
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+                    ],
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+                    "type": "text",
+                    "bbox": [
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+                        {
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+                                    "score": 1.0,
+                                    "content": "we can see in Figure 4b that even if we bucket by",
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+                                    "score": 1.0,
+                                    "content": ", the calibration error stays close to zero.",
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+                            ],
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+                        },
+                        {
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+                                    "content": "Unlike isotonic regression, the flow can learn the correlation between the CDFs by being trained",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "jointly on all of them. In addition to this, had we used a more complex calibration model that could",
+                                    "type": "text"
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "model the correlations directly, we would not be able to compute the correlation after recalibrating",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "with isotonic regression. By recalibrating with flows, we obtain a result that is still a distribution with",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "a closed-form likelihood; this allows us to still be able to use the model to compute the correlation",
+                                    "type": "text"
+                                }
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+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "and any other distributional statistics.",
+                                    "type": "text"
+                                }
+                            ],
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+                        }
+                    ],
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+                {
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+                                    "content": "In Figure 5, we show visually the difference between recalibration with flows and with isotonic",
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "on additional information",
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+                                    "score": 1.0,
+                                    "content": ", while isotonic regression is fit on each CDF independently and is unable",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 31
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "to incorporate any additional information. The reason is that flows are monotonic by construction",
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+                                }
+                            ],
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+                                    "content": "6 EXPERIMENTS",
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+                                            "html": "<table><tr><td rowspan=\"3\"></td><td colspan=\"2\">Before Recalib.</td><td colspan=\"2\">Isotonic Recalib.</td><td colspan=\"2\">Flow Recalib.</td></tr><tr><td>MSE</td><td>Calib</td><td>MSE</td><td>Calib</td><td>MSE</td><td>Calib</td></tr><tr><td>ConditionalFlow</td><td>0.009 ±0.004</td><td>0.45</td><td>N.A.</td><td>&lt;0.01</td><td>0.016 ± 0.012</td><td>&lt;0.01</td></tr><tr><td>ConditionalGaussian</td><td>0.012 ±0.006</td><td>2.35</td><td>N.A.</td><td>&lt;0.01</td><td>0.012 ±0.006</td><td>&lt;0.01</td></tr><tr><td>BayesianRidgeRegression</td><td>0.055 ±0.004</td><td>8.03</td><td>N.A.</td><td>0.01</td><td>0.48 ± 0.02</td><td>&lt;0.01</td></tr><tr><td>MC Dropout</td><td>0.005 ±0.002</td><td>0.51</td><td>N.A.</td><td>N.A.</td><td>N.A.</td><td>N.A.</td></tr><tr><td>SQR</td><td>0.0052 ± 0.0012</td><td>1.44</td><td>N.A.</td><td>N.A.</td><td>N.A.</td><td>N.A.</td></tr></table>",
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+                                    "content": "performance in distributional estimates compared to a few other popular approaches for estimating",
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+                                            "content": "Table 1: Results before and after recalibration on the UCI dataset “naval-propulsion-plant”. Details",
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+                                            "content": "on the model methodology as well as performance on other UCI datasets are in Appendix B. The",
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+                                            "content": "mean squared error (MSE) and its uncertainty is computed using 20 seeds. “Calib” is the calibration",
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+                                            "html": "<table><tr><td rowspan=\"3\"></td><td colspan=\"2\">Before Recalib.</td><td colspan=\"2\">Isotonic Recalib.</td><td colspan=\"2\">Flow Recalib.</td></tr><tr><td>MSE</td><td>Calib</td><td>MSE</td><td>Calib</td><td>MSE</td><td>Calib</td></tr><tr><td>ConditionalFlow</td><td>0.009 ±0.004</td><td>0.45</td><td>N.A.</td><td>&lt;0.01</td><td>0.016 ± 0.012</td><td>&lt;0.01</td></tr><tr><td>ConditionalGaussian</td><td>0.012 ±0.006</td><td>2.35</td><td>N.A.</td><td>&lt;0.01</td><td>0.012 ±0.006</td><td>&lt;0.01</td></tr><tr><td>BayesianRidgeRegression</td><td>0.055 ±0.004</td><td>8.03</td><td>N.A.</td><td>0.01</td><td>0.48 ± 0.02</td><td>&lt;0.01</td></tr><tr><td>MC Dropout</td><td>0.005 ±0.002</td><td>0.51</td><td>N.A.</td><td>N.A.</td><td>N.A.</td><td>N.A.</td></tr><tr><td>SQR</td><td>0.0052 ± 0.0012</td><td>1.44</td><td>N.A.</td><td>N.A.</td><td>N.A.</td><td>N.A.</td></tr></table>",
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+                                    "content": "and Appendix B. To show the usefulness of the plot we introduced in Section 5, we focus on the",
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+                                    "content": "computed. In contrast, we can still compute ordinary distribution statistics such as the mean after",
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+                                    "content": "In Figure 6a, whereas the calibration error metric shows the flow recalibrated model is well-calibrated",
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+                                            "content": "normalizing flows on the second time step on the dataset “electricity”. The legend contains the",
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+                                    "score": 1.0,
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+                                    "content": "Figure 7a shows that the calibration error of",
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+                                    "score": 0.93,
+                                    "content": "F ( Y _ { 2 } | Y _ { 1 } )",
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+                                    "score": 1.0,
+                                    "content": "regression is significant. Figure 7b shows that a flow trained jointly on the CDFs significantly",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "reduces the errors. Although the calibration error of the marginal distribution for the second step is",
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+                            "index": 13
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "comparable between the two approaches, the conditional performance favors normalizing flows. A",
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+                            "index": 14
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+                        {
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+                                    "score": 1.0,
+                                    "content": "possible explanantion of the still imperfect calibration of the flow model, though better than isotonic",
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+                                    "score": 1.0,
+                                    "content": "regression, is distribution shift.",
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+                                    "content": "7 CONCLUSION AND FUTURE WORK",
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+                                    "content": "In this paper, we addressed the calibration issue in modern neural networks through using higher",
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+                                    "score": 1.0,
+                                    "content": "capacity distributions via normalizing flows. We showed that normalizing flows can not only help",
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+                                    "score": 1.0,
+                                    "content": "during calibration, but also in recalibrating miscalibrated distributions, even improving the fit in",
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+                                    "score": 1.0,
+                                    "content": "the underfitting case. Though isotonic regression also uses a two-step producedure, we reframe the",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "problem as MLE through which we can trivially generalize recalibration to the multivariate case.",
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+                                    "score": 1.0,
+                                    "content": "Though isotonic regression could be extended such that we could compute the log likelihood implied",
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "by its change of variables, normalizing flows are a more natural way to handle this with more research",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "backing this technique, and are easily extendable to conditional and joint distributions. Interestingly,",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "though the Conditional Flow has all it needs to be well calibrated, recalibration helps the overall",
+                                    "type": "text"
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+                            ],
+                            "index": 26
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "calibration which was similarly found for classification in Guo et al. (2017); it remains for future",
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+                                    "score": 1.0,
+                                    "content": "work to better understand why and when this is useful.",
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+                                            "content": "normalizing flows on the second time step on the dataset “electricity”. The legend contains the",
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+                                    "content": "For our time series example, we trained a model on the UCI Dataset “electricity”. We predict two",
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+                                    "content": "Figure 7a shows that the calibration error of",
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+                                    "content": "F ( Y _ { 2 } | Y _ { 1 } )",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "regression is significant. Figure 7b shows that a flow trained jointly on the CDFs significantly",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "reduces the errors. Although the calibration error of the marginal distribution for the second step is",
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "comparable between the two approaches, the conditional performance favors normalizing flows. A",
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+                                    "content": "possible explanantion of the still imperfect calibration of the flow model, though better than isotonic",
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+                                    "score": 1.0,
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+                                    "content": "7 CONCLUSION AND FUTURE WORK",
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+                                    "content": "In this paper, we addressed the calibration issue in modern neural networks through using higher",
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+                                    "content": "capacity distributions via normalizing flows. We showed that normalizing flows can not only help",
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+                                    "score": 1.0,
+                                    "content": "during calibration, but also in recalibrating miscalibrated distributions, even improving the fit in",
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+                                    "score": 1.0,
+                                    "content": "the underfitting case. Though isotonic regression also uses a two-step producedure, we reframe the",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "problem as MLE through which we can trivially generalize recalibration to the multivariate case.",
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+                                    "score": 1.0,
+                                    "content": "Though isotonic regression could be extended such that we could compute the log likelihood implied",
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "by its change of variables, normalizing flows are a more natural way to handle this with more research",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "backing this technique, and are easily extendable to conditional and joint distributions. Interestingly,",
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+                                    "score": 1.0,
+                                    "content": "though the Conditional Flow has all it needs to be well calibrated, recalibration helps the overall",
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+                                    "score": 1.0,
+                                    "content": "calibration which was similarly found for classification in Guo et al. (2017); it remains for future",
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+                                    "score": 1.0,
+                                    "content": "work to better understand why and when this is useful.",
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+                                            "content": "two QuAR Flows with unconditional affine flows placed in between these two and at the end. The",
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+                                            "content": "Conditional Flow model is equivalent to a Conditional Gaussian if we removed the QuAR Flows.",
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+                                    "content": "A MODEL TRAINING DETAILS",
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+                                    "score": 1.0,
+                                    "content": "(Dheeru & Karra Taniskidou, 2017): boston-housing, concrete, energy, kin8nm, naval-propulsion-",
+                                    "type": "text"
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+                            ],
+                            "index": 11
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+                                    "score": 1.0,
+                                    "content": "plant, power-plant, wine-quality-red and yacht. The models we compared Conditional Flows to",
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+                            "index": 12
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+                                    "score": 1.0,
+                                    "content": "were: Conditional Gaussian (amortized inference with Gaussians, Lakshminarayanan et al. (2017)),",
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+                            "index": 13
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+                                    "score": 1.0,
+                                    "content": "Bayesian Ridge Regression (Tipping, 2001), MC Dropout (Gal & Ghahramani, 2016), Simultaneous",
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+                                }
+                            ],
+                            "index": 14
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+                                    "score": 1.0,
+                                    "content": "Quantile Regression (SQR) (Tagasovska & Lopez-Paz, 2019), Conditional Quantile (SQR where",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 15
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+                                    "score": 1.0,
+                                    "content": "we only learn prespecified quantiles), Quality Driven (Pearce et al., 2018), and a Gradient Boosted",
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+                            "index": 16
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+                                    "score": 1.0,
+                                    "content": "Decision Tree with LightGBM (Ke et al., 2017).",
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+                                }
+                            ],
+                            "index": 17
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+                    ],
+                    "index": 13.5
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+                                    "score": 1.0,
+                                    "content": "Figure 8 shows a representation of the Conditional Flow network we used. The QuAR Flows are",
+                                    "type": "text"
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+                                    "content": "Flow has two hidden layers in the residual connection with a hidden size 64 with ELU (Clevert et al.,",
+                                    "type": "text"
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+                            ],
+                            "index": 20
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+                                    "score": 1.0,
+                                    "content": "2015) as our activation. The condition network has a single hidden layer with size 64 and ReLU",
+                                    "type": "text"
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "as our activation. The QuAR Flows are conditioned on the 64-dimensional representation from the",
+                                    "type": "text"
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "condition network.",
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+                            ],
+                            "index": 23
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+                    ],
+                    "index": 20.5
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+                                    "score": 1.0,
+                                    "content": "For the other neural network approaches, we used networks with two hidden layers with size 128 and",
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+                        },
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+                                    "score": 1.0,
+                                    "content": "used ReLU as our nonlinearity. For hyperparameter tuning on all our neural networks, we searched",
+                                    "type": "text"
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+                            ],
+                            "index": 25
+                        },
+                        {
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+                                    "content": "the following grid:",
+                                    "type": "text"
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+                            ],
+                            "index": 26
+                        }
+                    ],
+                    "index": 25
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+                                    "content": "• Learning rate: 1e-2, 1e-3, 1e-4",
+                                    "type": "text"
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+                            ],
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+                                    "content": "• Weight decay: 0, 1e-3, 1e-2, 1e-1",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 28
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+                        {
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+                                    "content": "• Dropout Rates: 0.1 0.25, 0.5 0.75",
+                                    "type": "text"
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+                            "index": 29
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+                                    "score": 1.0,
+                                    "content": "The hyperparameters we used were taken from Tagasovska & Lopez-Paz (2019). We trained each of",
+                                    "type": "text"
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+                                    "content": "our neural networks for 2000 epochs with batch size 128.",
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+                    "index": 30.5
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+                                    "score": 1.0,
+                                    "content": "For our LightGBM tree model (Ke et al., 2017), we used mean squared error as our loss to learn the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 32
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+                                    "score": 1.0,
+                                    "content": "mean of the output distribution. The grid of hyperparameters we searched was:",
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+                            ],
+                            "index": 33
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+                    ],
+                    "index": 32.5
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+                                    "score": 1.0,
+                                    "content": "• Number of Trees: 100, 1000",
+                                    "type": "text"
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+                            "index": 34
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+                                    "content": "• Maximum Depths: 3, 5, 7, 9",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 35
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+                                    "content": "• Learning Rates: 1e-1, 1e-2, 1e-3, 1e-4",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 36
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+                                    "score": 1.0,
+                                    "content": "• Minimum Child Samples: 1, 20",
+                                    "type": "text"
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+                            "index": 37
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+                                    "content": "• Dropout Rates: 0.1 0.25, 0.5 0.75",
+                                    "type": "text"
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+                                    "content": "viation of the predicted normal distribution by computing the standard deviation of the residuals",
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+                                    "score": 1.0,
+                                    "content": "(observations minus predictions).",
+                                    "type": "text"
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+                        }
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+                                    "content": "(Dheeru & Karra Taniskidou, 2017): boston-housing, concrete, energy, kin8nm, naval-propulsion-",
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+                                    "score": 1.0,
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+                            "index": 12
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+                                    "score": 1.0,
+                                    "content": "were: Conditional Gaussian (amortized inference with Gaussians, Lakshminarayanan et al. (2017)),",
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+                            "index": 13
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+                                    "content": "Bayesian Ridge Regression (Tipping, 2001), MC Dropout (Gal & Ghahramani, 2016), Simultaneous",
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+                            "index": 14
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+                                    "score": 1.0,
+                                    "content": "Quantile Regression (SQR) (Tagasovska & Lopez-Paz, 2019), Conditional Quantile (SQR where",
+                                    "type": "text"
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+                            ],
+                            "index": 15
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+                                    "score": 1.0,
+                                    "content": "we only learn prespecified quantiles), Quality Driven (Pearce et al., 2018), and a Gradient Boosted",
+                                    "type": "text"
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+                            "index": 16
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+                                    "score": 1.0,
+                                    "content": "Decision Tree with LightGBM (Ke et al., 2017).",
+                                    "type": "text"
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+                            ],
+                            "index": 17
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+                    ],
+                    "index": 13.5,
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+                                    "score": 1.0,
+                                    "content": "Figure 8 shows a representation of the Conditional Flow network we used. The QuAR Flows are",
+                                    "type": "text"
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+                            ],
+                            "index": 18
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+                                    "score": 1.0,
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+                                    "type": "text"
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+                            "index": 20
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+                                    "score": 1.0,
+                                    "content": "2015) as our activation. The condition network has a single hidden layer with size 64 and ReLU",
+                                    "type": "text"
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+                            "index": 21
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+                                    "score": 1.0,
+                                    "content": "as our activation. The QuAR Flows are conditioned on the 64-dimensional representation from the",
+                                    "type": "text"
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+                            ],
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "used ReLU as our nonlinearity. For hyperparameter tuning on all our neural networks, we searched",
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+                            ],
+                            "index": 25
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+                                    "content": "the following grid:",
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+                                    "content": "• Learning rate: 1e-2, 1e-3, 1e-4",
+                                    "type": "text"
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+                                    "content": "• Weight decay: 0, 1e-3, 1e-2, 1e-1",
+                                    "type": "text"
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+                                    "content": "• Dropout Rates: 0.1 0.25, 0.5 0.75",
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+                                    "score": 1.0,
+                                    "content": "The hyperparameters we used were taken from Tagasovska & Lopez-Paz (2019). We trained each of",
+                                    "type": "text"
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+                                    "content": "our neural networks for 2000 epochs with batch size 128.",
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+                                    "score": 1.0,
+                                    "content": "For our LightGBM tree model (Ke et al., 2017), we used mean squared error as our loss to learn the",
+                                    "type": "text"
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+                            ],
+                            "index": 32
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+                                    "score": 1.0,
+                                    "content": "mean of the output distribution. The grid of hyperparameters we searched was:",
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+                            "index": 33
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+                                    "score": 1.0,
+                                    "content": "• Number of Trees: 100, 1000",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "• Maximum Depths: 3, 5, 7, 9",
+                                    "type": "text"
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+                                    "type": "text"
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+                    ],
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+                            ],
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+                                    "content": "“electricity”. We filtered the data down to observations after 2014-01-01. We split the data by date,",
+                                    "type": "text"
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+                            ],
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+                                    "content": "of dates are in the train set, the next",
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+                                    "score": 0.87,
+                                    "content": "18 \\%",
+                                    "type": "inline_equation"
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+                                    "score": 0.84,
+                                    "content": "10 \\%",
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+                                    "score": 1.0,
+                                    "content": "are in the test set.",
+                                    "type": "text"
+                                }
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+                        }
+                    ],
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+                                    "score": 1.0,
+                                    "content": "The data is given in 15 minute increments, and so similarly to Deep AR (Salinas et al., 2017), we",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 16
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "averaged the values per hour to have hourly data. Since we were trying to analyze the effect of",
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "correlated targets, we predicted two time steps conditioned on 48 hours of data using an autoregressive",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 18
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "model.",
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+                                }
+                            ],
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+                        }
+                    ],
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+                },
+                {
+                    "type": "text",
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+                    ],
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+                                    "score": 1.0,
+                                    "content": "We included three numeric covariates: the time step within the sampled time series, the hour of the",
+                                    "type": "text"
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+                            ],
+                            "index": 20
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "day, and the day of the week. Each of the covariates were normalized to have zero mean and one",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 21
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "standard deviation. In addition to this, since there are 370 different types of time series, we also",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 22
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "added an embedding layer from 370 to an embedding dimension of 20, same as that used in Deep AR",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 23
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "(Salinas et al., 2017). We concatenated this vector with the observations and covariates, leading to a",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 24
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "27-dimensional vector.",
+                                    "type": "text"
+                                }
+                            ],
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+                        }
+                    ],
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+                },
+                {
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+                                    "score": 1.0,
+                                    "content": "Since all the observations are positive, we model the target with a conditional Gamma distribution (i.e.",
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+                            ],
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+                                    "content": "amortized inference with Gamma distributiet al., 2017), we normalized each time series",
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+                                    "bbox": [
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+                                    "score": 0.92,
+                                    "content": "\\{ z _ { i , t } \\} _ { t = 0 } ^ { T - 1 }",
+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    "score": 1.0,
+                                    "content": "of by",
+                                    "type": "text"
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+                                    "score": 0.92,
+                                    "content": "\\begin{array} { r } { v _ { i } = 1 + 1 / T \\sum _ { t = 0 } ^ { T - 1 } z _ { i , t } . } \\end{array}",
+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    "score": 1.0,
+                                    "content": "eep AR (Salinas. We account for",
+                                    "type": "text"
+                                }
+                            ],
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+                                    "score": 1.0,
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+                                    "score": 1.0,
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+                                    "content": "\\mathrm { C D F } \\in [ 0 , 1 ]",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": ", we also transform the",
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+                                    "score": 1.0,
+                                    "content": "data with a Logit Function as the first layer of the flow and account for the change of variables in",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "the loss. In addition to this, we used a dropout rate of 0.25, two hidden layers with size 64 in each",
+                                    "type": "text"
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+                            "index": 8
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+                                    "score": 1.0,
+                                    "content": "residual connection, and used Polyak averaging (Polyak & Juditsky, 1992) with a decay of 0.999 for",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 9
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+                                    "content": "evaluation.",
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+                                    "content": "A.3 TIME SERIES",
+                                    "type": "text"
+                                }
+                            ],
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+                        }
+                    ],
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+                                    "score": 1.0,
+                                    "content": "For our time series experiments, we tested the performance of flow recalibration on the UCI dataset",
+                                    "type": "text"
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "“electricity”. We filtered the data down to observations after 2014-01-01. We split the data by date,",
+                                    "type": "text"
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+                        },
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+                                    "score": 1.0,
+                                    "content": "where the first",
+                                    "type": "text"
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+                                    "score": 0.9,
+                                    "content": "72 \\%",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "of dates are in the train set, the next",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.87,
+                                    "content": "18 \\%",
+                                    "type": "inline_equation"
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "are in validation set, and the last",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 0.84,
+                                    "content": "10 \\%",
+                                    "type": "inline_equation"
+                                }
+                            ],
+                            "index": 14
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "are in the test set.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 15
+                        }
+                    ],
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+                    "type": "text",
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "The data is given in 15 minute increments, and so similarly to Deep AR (Salinas et al., 2017), we",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 16
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "averaged the values per hour to have hourly data. Since we were trying to analyze the effect of",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "correlated targets, we predicted two time steps conditioned on 48 hours of data using an autoregressive",
+                                    "type": "text"
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+                            ],
+                            "index": 18
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+                        {
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+                                    "score": 1.0,
+                                    "content": "model.",
+                                    "type": "text"
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+                    ],
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+                                    "score": 1.0,
+                                    "content": "We included three numeric covariates: the time step within the sampled time series, the hour of the",
+                                    "type": "text"
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+                            ],
+                            "index": 20
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+                                    "score": 1.0,
+                                    "content": "day, and the day of the week. Each of the covariates were normalized to have zero mean and one",
+                                    "type": "text"
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+                            ],
+                            "index": 21
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+                        {
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+                                    "score": 1.0,
+                                    "content": "standard deviation. In addition to this, since there are 370 different types of time series, we also",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 22
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "added an embedding layer from 370 to an embedding dimension of 20, same as that used in Deep AR",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 23
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "(Salinas et al., 2017). We concatenated this vector with the observations and covariates, leading to a",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 24
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "27-dimensional vector.",
+                                    "type": "text"
+                                }
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+                            "index": 25
+                        }
+                    ],
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+                                    "score": 1.0,
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+                                    "type": "text"
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+                                    "content": "amortized inference with Gamma distributiet al., 2017), we normalized each time series",
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+                                    "content": "\\{ z _ { i , t } \\} _ { t = 0 } ^ { T - 1 }",
+                                    "type": "inline_equation"
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+                                    "score": 0.92,
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+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    "score": 1.0,
+                                    "content": "eep AR (Salinas. We account for",
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+                                }
+                            ],
+                            "index": 27
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+                        {
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+                                    "score": 1.0,
+                                    "content": "the scaling in the output distribution, i.e. Gamma",
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+                                },
+                                {
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+                                    "score": 0.66,
+                                    "content": "( k , \\theta * v _ { i } )",
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+                                {
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+                                    "score": 1.0,
+                                    "content": ".",
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+                                }
+                            ],
+                            "index": 28
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+                    ],
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+                },
+                {
+                    "type": "text",
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+                    "lines": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "The model we used was a 1 layer LSTM, and then two linear layers with a ReLU and Dropout",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 29
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "(Srivastava et al., 2014) in between were applied to the hidden representation. We set the dropout rate",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 30
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "to 0.5 and used a hidden size of 48 everywhere. For evaluation, we used Polyak averaging (Polyak &",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 31
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Juditsky, 1992) with a decay of 0.999.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 32
+                        }
+                    ],
+                    "index": 30.5,
+                    "bbox_fs": [
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+                },
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+                    ],
+                    "lines": [
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+                                    "score": 1.0,
+                                    "content": "For recalibration with flows, we trained on the CDFs for both time steps with a flow with the same",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 33
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "architecture as that used in Appendix A.2.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 34
+                        }
+                    ],
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+                },
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+                    "type": "title",
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+                    ],
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+                                    "score": 1.0,
+                                    "content": "B EXPERIMENTAL RESULTS",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 35
+                        }
+                    ],
+                    "index": 35
+                },
+                {
+                    "type": "title",
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+                                    "score": 1.0,
+                                    "content": "B.1 CALIBRATION",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 36
+                        }
+                    ],
+                    "index": 36
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "In Table 2, we compare Conditional Flows against multiple other approaches for estimating uncer-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 37
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "tainty, and we measure performance using Prediction Interval Coverage Probability (PICP), Mean",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 38
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "Squared Error (MSE), and Calibration Error.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 39
+                        }
+                    ],
+                    "index": 38,
+                    "bbox_fs": [
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+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                    ],
+                    "lines": [
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+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "For each model, we trained the model on 20 different seeds where each seed randomly splits the data",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 40
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                        700
+                                    ],
+                                    "score": 1.0,
+                                    "content": "in",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                        688,
+                                        137,
+                                        698
+                                    ],
+                                    "score": 0.88,
+                                    "content": "72 \\%",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        137,
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+                                        700
+                                    ],
+                                    "score": 1.0,
+                                    "content": "train,",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                        183,
+                                        698
+                                    ],
+                                    "score": 0.86,
+                                    "content": "18 \\%",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                        248,
+                                        700
+                                    ],
+                                    "score": 1.0,
+                                    "content": "validation, and",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        248,
+                                        688,
+                                        268,
+                                        698
+                                    ],
+                                    "score": 0.87,
+                                    "content": "10 \\%",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        268,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "test. We picked the best hyperparameter by choosing the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 41
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "hyperparameters that gave the best average validation loss across the 20 seeds where the loss used is",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 42
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        708,
+                                        505,
+                                        723
+                                    ],
+                                    "score": 1.0,
+                                    "content": "the loss defined by the training procedure for the model, e.g. for Conditional Gaussian, we used log",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 43
+                        },
+                        {
+                            "bbox": [
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0.02</td><td>0.0026 ± 0.0010</td><td>0.17</td><td>0.69 ± 0.02</td><td>0.069 ± 0.005</td><td>1.14</td></tr><tr><td>SQR</td><td>0.97± 0.02</td><td>0.0028 ± 0.0012</td><td>0.37</td><td>0.892 ± 0.010</td><td>0.072 ± 0.005</td><td>0.01</td></tr><tr><td>ConditionalQuantile</td><td>0.98 ±0.02</td><td>N.A.</td><td>N.A.</td><td>0.943 ± 0.011</td><td>N.A.</td><td>N.A.</td></tr><tr><td>QualityDriven</td><td>0.96 ± 0.04</td><td>N.A.</td><td>N.A.</td><td>0.88±0.02</td><td>N.A.</td><td>N.A.</td></tr><tr><td>LightGBM</td><td>0.24±0.05</td><td>0.0013 ± 0.0006</td><td>5.18</td><td>0.205 ± 0.010</td><td>0.190 ± 0.011</td><td>5.59</td></tr><tr><td></td><td colspan=\"3\">naval-propulsion-plant</td><td colspan=\"3\">power-plant</td></tr><tr><td></td><td>PICP</td><td>MSE</td><td>Calib</td><td>PICP</td><td>MSE</td><td>Calib</td></tr><tr><td>ConditionalFlow</td><td>0.98 ± 0.03</td><td>0.009 ±0.004</td><td>0.45</td><td>0.952 ± 0.007</td><td>0.048 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0.007</td><td>5.95</td></tr><tr><td></td><td colspan=\"3\">wine-quality-red</td><td colspan=\"3\">yacht</td></tr><tr><td></td><td>PICP</td><td>MSE</td><td>Calib</td><td>PICP</td><td>MSE</td><td>Calib</td></tr><tr><td>ConditionalFlow</td><td>0.90 ±0.02</td><td>0.6±0.2</td><td>0.04</td><td>0.95 ± 0.04</td><td>0.004±0.003</td><td>0.27</td></tr><tr><td>ConditionalGaussian</td><td>0.95 ± 0.02</td><td>0.59 ± 0.09</td><td>0.22</td><td>0.99 ±0.02</td><td>0.004±0.002</td><td>0.62</td></tr><tr><td>BayesianRidgeRegression</td><td>0.95 ± 0.02</td><td>0.63 ± 0.11</td><td>0.05</td><td>0.94± 0.03</td><td>0.35 ± 0.10</td><td>0.25</td></tr><tr><td>MC Dropout</td><td>0.35 ± 0.04</td><td>0.57 ±0.09</td><td>4.01</td><td>0.97 ± 0.04</td><td>0.004±0.002</td><td>0.21</td></tr><tr><td>SQR</td><td>0.81±0.09</td><td>0.59 ±0.10</td><td>0.15</td><td>0.96 ±0.04</td><td>0.004±0.004</td><td>0.49</td></tr><tr><td>ConditionalQuantile</td><td>0.95 ± 0.02</td><td>N.A.</td><td>N.A.</td><td>0.99 ± 0.02</td><td>N.A.</td><td>N.A.</td></tr><tr><td>QualityDriven</td><td>0.84 ±0.03</td><td>N.A.</td><td>N.A.</td><td>0.97 ± 0.04</td><td>N.A.</td><td>N.A.</td></tr><tr><td>LightGBM</td><td>0.40 ± 0.05</td><td>0.55± 0.10</td><td>2.93</td><td>0.45 ± 0.08</td><td>0.003±0.003</td><td>3.04</td></tr></table>",
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+                                            "html": "<table><tr><td rowspan=\"3\"></td><td colspan=\"2\">BayesianRidgeRegression</td><td colspan=\"2\">ConditionalGaussian</td></tr><tr><td>Before</td><td>Flow</td><td>Before</td><td>Flow</td></tr><tr><td>bostonHousing</td><td>0.28 ±0.08</td><td>0.28 ±0.08</td><td>0.14 ± 0.07</td><td>0.14 ± 0.07</td></tr><tr><td>concrete</td><td>0.40 ± 0.07</td><td>0.40 ± 0.07</td><td>0.09 ±0.03</td><td>0.09 ± 0.03</td></tr><tr><td>energy</td><td>0.09 ±0.02</td><td>0.09 ±0.02</td><td>0.0030 ±0.0009</td><td>0.0032 ± 0.0011</td></tr><tr><td>kin8nm</td><td>0.59 ± 0.02</td><td>0.59 ± 0.02</td><td>0.071 ± 0.005</td><td>0.072 ± 0.005</td></tr><tr><td>naval-propulsion-plant</td><td>0.055 ± 0.004</td><td>0.48 ±0.02</td><td>0.012 ± 0.006</td><td>0.012 ± 0.007</td></tr><tr><td>power-plant</td><td>0.072 ± 0.007</td><td>0.072 ± 0.007</td><td>0.048 ± 0.006</td><td>0.049 ± 0.006</td></tr><tr><td>wine-quality-red</td><td>0.63 ± 0.11</td><td>0.63 ± 0.11</td><td>0.59 ± 0.09</td><td>0.59 ± 0.10</td></tr><tr><td>yacht</td><td>0.35 ± 0.10</td><td>0.36 ± 0.10</td><td>0.004 ±0.002</td><td>0.004 ± 0.003</td></tr><tr><td colspan=\"2\"></td><td colspan=\"2\">ConditionalFlow</td><td></td></tr><tr><td rowspan=\"10\"></td><td></td><td>Before</td><td>Flow</td><td></td></tr><tr><td>bostonHousing</td><td>0.13 ± 0.06</td><td></td><td></td></tr><tr><td>concrete</td><td>0.09 ± 0.03</td><td>0.13 ± 0.06 0.09 ± 0.03</td><td></td></tr><tr><td></td><td>0.0022 ± 0.0006</td><td>0.0027 ± 0.0007</td><td></td></tr><tr><td>energy kin8nm</td><td>0.081 ± 0.013</td><td>0.081 ± 0.013</td><td></td></tr><tr><td>naval-propulsion-plant</td><td>0.009 ± 0.004</td><td>0.016 ± 0.012</td><td></td></tr><tr><td>power-plant</td><td>0.048 ± 0.007</td><td>0.048 ± 0.007</td><td></td></tr><tr><td>wine-quality-red</td><td>0.6± 0.2</td><td>0.64 ± 0.15</td><td></td></tr><tr><td>yacht</td><td></td><td></td><td></td></tr><tr><td></td><td>0.004 ± 0.003</td><td>0.008 ± 0.012</td><td></td></tr></table>",
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+                                    "score": 1.0,
+                                    "content": "Reported in Table 2 are the mean and standard deviation of PICP and MSE across the 20 seeds. Due",
+                                    "type": "text"
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+                                    "content": "to compare against alternative approaches where we prespecify which quantiles we need.",
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "From the above section, we recalibrate the best Bayesian Ridge Regression, Conditional Gaussian,",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "and Conditional Flow. In Table 3, we compare the calibration error before recalibration, after",
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+                                    "content": "recalibration with isotonic regression, and after recalibration with normalizing flows.",
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+                                    "content": "As we can see, across all 8 UCI datasets, the calibration errors after recalibrating with normalizing",
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "to compare against alternative approaches where we prespecify which quantiles we need.",
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+                                    "content": "B.2 RECALIBRATION",
+                                    "type": "text"
+                                }
+                            ],
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+                    "index": 13
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "From the above section, we recalibrate the best Bayesian Ridge Regression, Conditional Gaussian,",
+                                    "type": "text"
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+                            ],
+                            "index": 14
+                        },
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+                                    "score": 1.0,
+                                    "content": "and Conditional Flow. In Table 3, we compare the calibration error before recalibration, after",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "calibration error can be detrimental to mean performance.",
+                                    "type": "text"
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+}
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diff --git a/parse/train/H8VDvtm1ij8/H8VDvtm1ij8_model.json b/parse/train/H8VDvtm1ij8/H8VDvtm1ij8_model.json
new file mode 100644
index 0000000000000000000000000000000000000000..eb82a3ad82aceb4befc4fe5d17aa5719fd1c8be5
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diff --git a/parse/train/S1ANxQW0b/S1ANxQW0b.md b/parse/train/S1ANxQW0b/S1ANxQW0b.md
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+# MAXIMUM A POSTERIORI POLICY OPTIMISATION
+
+Abbas Abdolmaleki, Jost Tobias Springenberg, Yuval Tassa, Remi Munos,   
+Nicolas Heess, Martin Riedmiller   
+DeepMind, London, UK   
+{aabdolmaleki,springenberg,tassa,munos,heess,riedmiller}@google.com
+
+# ABSTRACT
+
+We introduce a new algorithm for reinforcement learning called Maximum aposteriori Policy Optimisation (MPO) based on coordinate ascent on a relativeentropy objective. We show that several existing methods can directly be related to our derivation. We develop two off-policy algorithms and demonstrate that they are competitive with the state-of-the-art in deep reinforcement learning. In particular, for continuous control, our method outperforms existing methods with respect to sample efficiency, premature convergence and robustness to hyperparameter settings.
+
+# 1 INTRODUCTION
+
+Model free reinforcement learning algorithms can acquire sophisticated behaviours by interacting with the environment while receiving simple rewards. Recent experiments (Mnih et al., 2015; Jaderberg et al., 2016; Heess et al., 2017) successfully combined these algorithms with powerful deep neural-network approximators while benefiting from the increase of compute capacity.
+
+Unfortunately, the generality and flexibility of these algorithms comes at a price: They can require a large number of samples and – especially in continuous action spaces – suffer from high gradient variance. Taken together these issues can lead to unstable learning and/or slow convergence. Nonetheless, recent years have seen significant progress, with improvements to different aspects of learning algorithms including stability, data-efficiency and speed, enabling notable results on a variety of domains, including locomotion (Heess et al., 2017; Peng et al., 2016), multi-agent behaviour (Bansal et al., 2017) and classical control (Duan et al., 2016).
+
+Two types of algorithms currently dominate scalable learning for continuous control problems: First, Trust-Region Policy Optimisation (TRPO; Schulman et al. 2015) and the derivative family of Proximal Policy Optimisation algorithms (PPO; Schulman et al. 2017b). These policy-gradient algorithms are on-policy by design, reducing gradient variance through large batches and limiting the allowed change in parameters. They are robust, applicable to high-dimensional problems, and require moderate parameter tuning, making them a popular first choice (Ho & Ermon, 2016). However, as on-policy algorithms, they suffer from poor sample efficiency.
+
+In contrast, off-policy value-gradient algorithms such as the Deep Deterministic Policy Gradient (DDPG, Silver et al. 2014; Lillicrap et al. 2016), Stochastic Value Gradient (SVG, Heess et al. 2015), and the related Normalized Advantage Function formulation (NAF, Gu et al. 2016b) rely on experience replay and learned (action-)value functions. These algorithms exhibit much better data efficiency, approaching the regime where experiments with real robots are possible (Gu et al., 2016a; Andrychowicz et al., 2017). While also popular, these algorithms can be difficult to tune, especially for high-dimensional domains like general robot manipulation tasks.
+
+In this paper we propose a novel off-policy algorithm that benefits from the best properties of both classes. It exhibits the scalability, robustness and hyperparameter insensitivity of on-policy algorithms, while offering the data-efficiency of off-policy, value-based methods.
+
+To derive our algorithm, we take advantage of the duality between control and estimation by using Expectation Maximisation (EM), a powerful tool from the probabilistic estimation toolbox, in order to solve control problems. This duality can be understood as replacing the question “what are the actions which maximise future rewards?” with the question “assuming future success in maximising rewards, what are the actions most likely to have been taken?”. By using this estimation objective we have more control over the policy change in both E and M steps, yielding robust learning. We show below that several algorithms, including TRPO, can be directly related to this perspective. We leverage the fast convergence properties of EM-style coordinate ascent by alternating a nonparametric data-based E-step which re-weights state-action samples, with a supervised, parametric M-step using deep neural networks.
+
+We evaluate our algorithm on a broad spectrum of continuous control problems including a 56 DoF humanoid body. All experiments used the same optimisation hyperparameters 1. Our algorithm shows remarkable data efficiency often solving the tasks we consider an order of magnitude faster than the state-of-the-art. A video of some resulting behaviours can be found here dropbox.com/s/pgcmjst7t0zwm4y/MPO.mp4.
+
+# 2 BACKGROUND AND NOTATION
+
+# 2.1 RELATED WORK
+
+Casting Reinforcement Learning (RL) as an inference problem has a long history dating back at least two decades (Dayan & Hinton, 1997). The framework presented here is inspired by a variational inference perspective on RL that has previously been utilised in multiple studies; c.f. Dayan & Hinton (1997); Neumann (2011); Deisenroth et al. (2013); Rawlik et al. (2012); Levine & Koltun (2013); Florensa et al. (2017).
+
+Particular attention has been paid to obtaining maximum entropy policies as the solution to an inference problem. The penalisation of determinism can be seen encouraging both robustness and simplicity. Among these are methods that perform trajectory optimisation using either linearised dynamics (Todorov, 2008; Toussaint, 2009; Levine & Koltun, 2013) or general dynamics as in path integral control (Kappen, 2005; Theodorou et al., 2010). In contrast to these algorithms, here we do not assume the availability of a transition model and avoid on-policy optimisation. A number of other authors have considered the same perspective but in a model-free RL setting (Neumann, 2011; Peters et al., 2010a; Florensa et al., 2017; Daniel et al., 2016) or inverse RL problems (Ziebart et al., 2008). These algorithms are more directly related to our work and can be cast in the same (EM-like) alternating optimisation scheme on which we base our algorithm. However, they typically lack the maximisation (M)-step – with the prominent exception of REPS, AC-REPS, $\mathrm { P \bar { I } ^ { 2 } }$ -GPS and MDGPS (Peters et al., 2010a; Wirth et al., 2016; Chebotar et al., 2016; Montgomery & Levine, 2016) to which our algorithm is closely related as outlined below. An interesting recent addition to these approaches is an EM-perspective on the PoWER algorithm (Roux, 2016) which uses the same iterative policy improvement employed here, but commits to parametric inference distributions and avoids an exponential reward transformation, resulting in a harder to optimise lower bound.
+
+As an alternative to these policy gradient inspired algorithms, the class of recent algorithms for soft Q-learning (e.g. Rawlik et al. (2012); Haarnoja et al. (2017); Fox et al. (2016) parameterise and estimate a so called “soft” Q-function directly, implicitly inducing a maximum entropy policy. A perspective that can also be extended to hierarchical policies (Florensa et al., 2017), and has recently been used to establish connections between Q-learning and policy gradient methods (O’Donoghue et al., 2016; Schulman et al., 2017a). In contrast, we here rely on a parametric policy, our bound and derivation is however closely related to the definition of the soft (entropy regularised) Q-function.
+
+A line of work, that is directly related to the “RL as inference” perspective, has focused on using information theoretic regularisers such as the entropy of the policy or the Kullback-Leibler divergence (KL) between policies to stabilise standard RL objectives. In fact, most state-of-the-art policy gradient algorithms fall into this category. For example see the entropy regularization terms used in Mnih et al. (2016) or the KL constraints employed by work on trust-region based methods (Schulman et al., 2015; 2017b; Gu et al., 2017; Wang et al., 2017). The latter methods introduce a trust region constraint, defined by the KL divergence between the new policy and the old policy, so that the expected KL divergence over state space is bounded. From the perspective of this paper these trust-region based methods can be seen as optimising a parametric E-step, as in our algorithm, but are “missing” an explicit M-step.
+
+Finally, the connection between RL and inference has been invoked to motivate work on exploration. The most prominent examples for this are formed by work on Boltzmann exploration such as Kaelbling et al. (1996); Perkins & Precup (2002); Sutton (1990); O’Donoghue et al. (2017), which can be connected back to soft Q-learning (and thus to our approach) as shown in Haarnoja et al. (2017).
+
+# 2.2 MARKOV DECISION PROCESSES
+
+We consider the problem of finding an optimal policy $\pi$ for a discounted reinforcement learning (RL) problem; formally characterized by a Markov decision process (MDP). The MDP consists of: continuous states $s$ , actions $a$ , transition probabilities $p ( s _ { t + 1 } | s _ { t } , a _ { t } )$ – specifying the probability of transitioning from state $s _ { t }$ to $s _ { t + 1 }$ under action $a _ { t } \textrm { -- }$ , a reward function $r ( s , a ) \in \mathbb { R }$ as well as the discounting factor $\gamma \in [ 0 , 1 )$ . The policy $\pi ( a | s , \theta )$ (with parameters $\pmb { \theta }$ ) is assumed to specify a probability distribution over action choices given any state and – together with the transition probabilities – gives rise to the stationary distribution $\mu _ { \pi } ( s )$ .
+
+Using these basic quantities we can now define the notion of a Markov sequence or trajec$\overline { { \tau } } _ { \pi } ~ = ~ \{ ( s _ { 0 } , a _ { 0 } ) \} \ldots ( s _ { T } , a _ { T } ) \}$ by following the policy ; and the expected return $\pi$ $\begin{array} { r } { \tau _ { \pi } ~ \sim ~ p _ { \pi } ( \tau ) } \end{array}$ with. We $\begin{array} { r } { p _ { \pi } ( \tau ) = p ( s _ { 0 } ) \prod _ { t > 0 } p ( s _ { t + 1 } \vert s _ { t } , a _ { t } ) \pi ( a _ { t } \vert s _ { t } ) ; } \end{array}$ $\begin{array} { r } { \mathbb { E } _ { \tau _ { \pi } } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r ( s _ { t } , s _ { t } ) ] } \end{array}$ will use the shorthand $r _ { t } = r ( s _ { t } , a _ { t } )$ .
+
+# 3 MAXIMUM A POSTERIORI POLICY OPTIMISATION
+
+Our approach is motivated by the well established connection between RL and probabilistic inference. This connection casts the reinforcement learning problem as that of inference in a particular probabilistic model. Conventional formulations of RL aim to find a trajectory that maximizes expected reward. In contrast, inference formulations start from a prior distribution over trajectories, condition a desired outcome such as achieving a goal state, and then estimate the posterior distribution over trajectories consistent with this outcome.
+
+A finite-horizon undiscounted reward formulation can be cast as inference problem by constructing a suitable probabilistic model via a likelihood function $p ( O = 1 | \tau ) \propto \mathrm { e x p } ( \Sigma _ { t } r _ { t } / \alpha )$ , where $\alpha$ is a temperature parameter. Intuitively, $O$ can be interpreted as the event of obtaining maximum reward by choosing an action; or the event of succeeding at the RL task (Toussaint, 2009; Neumann, 2011). With this definition we can define the following lower bound on the likelihood of optimality for the policy $\pi$ :
+
+$$
+\begin{array} { l } { \displaystyle \log p _ { \pi } ( O = 1 ) = \log \int p _ { \pi } ( \tau ) p ( O = 1 | \tau ) d \tau \geq \int q ( \tau ) \big [ \log p ( O = 1 | \tau ) + \log \frac { p _ { \pi } ( \tau ) } { q ( \tau ) } \big ] d \tau } \\ { \displaystyle \qquad = \mathbb { E } _ { q } \Big [ \sum _ { t } r _ { t } / \alpha \Big ] - \mathrm { K L } \Big ( q ( \tau ) | | p _ { \pi } ( \tau ) \Big ) = \mathcal { I } ( q , \pi ) , } \end{array}
+$$
+
+where $p _ { \pi }$ is the trajectory distribution induced by policy $\pi ( a | s )$ as described in section 2.2 and $q ( \tau )$ is an auxiliary distribution over trajectories that will discussed in more detail below. The lower bound $\mathcal { I }$ is the evidence lower bound (ELBO) which plays an important role in the probabilistic modeling literature. It is worth already noting here that optimizing (2) with respect to $q$ can be seen as a regularized RL problem.
+
+An important motivation for transforming a RL problem into an inference problem is that this allows us draw from the rich toolbox of inference methods: For instance, $\mathcal { I }$ can be optimized with the familiy of expectation maximization (EM) algorithms which alternate between improving $\mathcal { I }$ with respect to $q$ and $\pi$ . In this paper we follow classical (Dayan & Hinton, 1997) and more recent works (e.g. Peters et al. 2010b; Levine & Koltun 2013; Daniel et al. 2016; Wirth et al. 2016) and cast policy search as a particular instance of this family. Our algorithm then combines properties of existing approaches in this family with properties of recent off-policy algorithms for neural networks.
+
+The algorithm alternates between two phases which we refer to as E and M step in reference to an EM-algorithm. The E-step improves $\mathcal { I }$ with respect to $q$ . Existing EM policy search approaches perform this step typically by reweighting trajectories with sample returns (Kober & Peters, 2009) or via local trajectory optimization (Levine & Koltun, 2013). We show how off-policy deep RL techniques and value-function approximation can be used to make this step both scalable as well as data efficient. The M-step then updates the parametric policy in a supervised learning step using the reweighted state-action samples from the E-step as targets.
+
+These choices lead to the following desirable properties: (a) low-variance estimates of the expected return via function approximation; (b) low-sample complexity of value function estimate via robust off-policy learning; (c) minimal parametric assumption about the form of the trajectory distribution in the E-step; (d) policy updates via supervised learning in the M step; (e) robust updates via hard trust-region constraints in both the E and the M step.
+
+# 3.1 POLICY IMPROVEMENT
+
+The derivation of our algorithm then starts from the infinite-horizon analogue of the KL-regularized expected reward objective from Equation (2). In particular, we consider variational distributions $q ( \tau )$ that factor in the same way as $p _ { \pi }$ , i.e. $\begin{array} { r } { q ( \tau ) = \bar { p } ( s _ { 0 } ) \prod _ { t > 0 } p ( s _ { t + 1 } | s _ { t } , a _ { t } ) q ( a _ { t } | s _ { t } ) } \end{array}$ which yields:
+
+$$
+\mathcal { I } ( \boldsymbol { q } , \theta ) = \mathbb { E } _ { \boldsymbol { q } } \Big [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } \big [ r _ { t } - \alpha \mathbf { K L } \big ( \boldsymbol { q } ( \boldsymbol { a } | s _ { t } ) \| \pi ( \boldsymbol { a } | s _ { t } , \theta ) \big ) \big ] \Big ] + \log p ( \theta ) .
+$$
+
+Note that due to the assumption about the structure of $q ( \tau )$ the $\mathrm { K L }$ over trajectories decomposes into a KL over the individual state-conditional action distributions. This objective has also been considered e.g. by Haarnoja et al. (2017); Schulman et al. (2017a). The additional $\log p ( \pmb \theta )$ term is a prior over policy parameters and can be motivated by a maximum a-posteriori estimation problem (see appendix for more details).
+
+We also define the regularized Q-value function associated with (3) as
+
+$$
+Q _ { \theta } ^ { q } ( s , a ) = r _ { 0 } + \mathbb { E } _ { q ( \tau ) , s _ { 0 } = s , a _ { 0 } = a } \left[ \sum _ { t \geq 1 } ^ { \infty } \gamma ^ { t } \big [ r _ { t } - \alpha \mathrm { K L } ( q _ { t } \| \pi _ { t } ) \big ] \right] ,
+$$
+
+with $\mathrm { K L } \big ( q _ { t } | | \pi _ { t } \big ) = \mathrm { K L } \big ( q ( a | s _ { t } ) \big ) \| \pi ( a | s _ { t } , \pmb { \theta } ) \big )$ . Note that $\mathrm { K L } \big ( q _ { 0 } | | \pi _ { 0 } \big )$ and $p ( \theta )$ are not part of the Q-function as they are not a function of the action.
+
+We observe that optimizing $\mathcal { I }$ with respect to $q$ is equivalent to solving an expected reward RL problem with augmented reward $\begin{array} { r } { \tilde { r } _ { t } = r _ { t } - \alpha \log \frac { q ( a _ { t } | s _ { t } ) } { \pi ( a _ { t } | s _ { t } , \pmb { \theta } ) } } \end{array}$ . In this view $\pi$ represents a default policy towards which $q$ is regularized – i.e. the current best policy. The MPO algorithm treats $\pi$ as the primary object of interest. In this case $q$ serves as an auxiliary distribution that allows optimizing $\mathcal { I }$ via alternate coordinate ascent in $q$ and $\pi _ { \pmb { \theta } }$ , analogous to the expectation-maximization algorithm in the probabilistic modelling literature. In our case, the E-step optimizes $\mathcal { I }$ with respect to $q$ while the M-step optimizes $\mathcal { I }$ with respect to $\pi$ . Different optimizations in the $\mathrm { E }$ -step and M-step lead to different algorithms. In particular, we note that for the case where $p ( \pmb \theta )$ is an uninformative prior a variant of our algorithm has a monotonic improvement guarantee as show in the Appendix A.
+
+# 3.2 E-STEP
+
+In the E-step of iteration $i$ we perform a partial maximization of $\mathcal { I } ( \boldsymbol { q } , \boldsymbol { \theta } )$ with respect to $q$ given $\theta = \theta _ { i }$ . We start by setting $q = \pi _ { \pmb { \theta } _ { i } }$ and estimate the unregularized action-value function:
+
+$$
+Q _ { \theta _ { i } } ^ { q } ( s , a ) = Q _ { \theta _ { i } } ( s , a ) = \mathbb { E } _ { \tau _ { \pi _ { i } } , s _ { 0 } = s , a _ { 0 } = a } \left[ \sum _ { t } ^ { \infty } \gamma ^ { t } r _ { t } \right] ,
+$$
+
+since $\mathrm { K L } ( q | | \pi _ { i } ) = 0$ . In practice we estimate $Q _ { \pmb { \theta } _ { i } }$ from off-policy data (we refer to Section 4 for details about the policy evaluation step). This greatly increases the data efficiency of our algorithm. Given $Q _ { \pmb { \theta } _ { i } }$ we improve the lower bound $\mathcal { I }$ w.r.t. $q$ by first expanding $Q _ { \theta _ { i } } ( s , a )$ via the regularized Bellman operator $\begin{array} { r } { T ^ { \pi , q } = \mathbb { E } _ { q ( a \mid s ) } \Big [ r ( s , a ) - \alpha \mathbf { K L } ( q \| \pi _ { i } ) + \gamma \mathbb { E } _ { p ( s ^ { \prime } \mid s , a ) } [ V _ { \pmb { \theta } _ { i } } ( s ^ { \prime } ) ] \Big ] . } \end{array}$ , and optimize the “one-step” KL regularised objective
+
+$$
+\begin{array} { l } { \displaystyle \mathop { \operatorname* { m a x } } _ { \boldsymbol { q } } \bar { \mathcal { T } } _ { s } ( \boldsymbol { q } , \theta _ { i } ) = \mathop { \operatorname* { m a x } } _ { \boldsymbol { q } } T ^ { \pi , { q } } Q _ { \theta _ { i } } ( s , \boldsymbol { a } ) } \\ { \displaystyle \mathop { \quad = } \mathop { \operatorname* { m a x } } _ { \boldsymbol { q } } \mathbb { E } _ { \boldsymbol { \mu } ( s ) } \Big [ \mathbb { E } _ { \boldsymbol { q } ( \cdot \vert s ) } [ Q _ { \theta _ { i } } ( s , \boldsymbol { a } ) ] - \alpha \mathbf { K } \mathbf { L } ( \boldsymbol { q } \| \pi _ { i } ) \Big ] , } \end{array}
+$$
+
+since $V _ { \pmb { \theta } _ { i } } ( s ) = \mathbb { E } _ { q ( a | s } [ Q _ { \pmb { \theta } _ { i } } ( s , a ) ]$ and thus $Q _ { \pmb { \theta } _ { i } } ( s , a ) = r ( s , a ) + \gamma V _ { \pmb { \theta } _ { i } } ( s ) .$ .
+
+Maximizing Equation (6), thus obtaining $q _ { i } = \arg \operatorname* { m a x } \bar { \mathcal { I } } ( q , \theta _ { i } )$ , does not fully optimize $\mathcal { I }$ since we treat $Q _ { \theta _ { i } }$ as constant with respect to $q$ . An intuitive interpretation $q _ { i }$ is that it chooses the softoptimal action for one step and then resorts to executing policy $\pi$ . In the language of the EM algorithm this optimization implements a partial E-step. In practice we also choose $\mu _ { q }$ to be the stationary distribution as given through samples from the replay buffer.
+
+# CONSTRAINED E-STEP
+
+The reward and the KL terms are on an arbitray relative scale. This can make it difficult to choose $\alpha$ . We therefore replace the soft KL regularization with a hard constraint with parameter $\epsilon$ , i.e,
+
+$$
+\begin{array} { r l } & { \underset { q } { \operatorname* { m a x } } \mathbb { E } _ { \boldsymbol { \mu } ( s ) } \left[ \mathbb { E } _ { q ( a \mid s ) } \left[ Q _ { \boldsymbol { \theta } _ { i } } ( s , a ) \right] \right] } \\ & { s . t . \mathbb { E } _ { \boldsymbol { \mu } ( s ) } \left[ { \mathbf { K L } } ( q ( a \mid s ) , \pi ( a \mid s , \boldsymbol { \theta } _ { i } ) ) \right] < \epsilon . } \end{array}
+$$
+
+If we choose to explicitly parameterize $q ( a | s ) -$ option 1 below – the resulting optimisation is similar to that performed by the recent TRPO algorithm for continuous control (Schulman et al., 2015); only in an off-policy setting. Analogously, the unconstrained objective (6) is similar to the objective used by PPO (Schulman et al., 2017b). We note, however, that the KL is reversed when compared to the KL used by TRPO and PPO.
+
+To implement (7) we need to choose a form for the variational policy $q ( a | s )$ . Two options arise:
+
+1. We can use a parametric variational distribution $q ( a | s , \pmb \theta ^ { q } )$ , with parameters $\pmb { \theta } ^ { q }$ , and optimise Equation (7) via the likelihood ratio or action-value gradients. This leads to an algorithm similar to TRPO/PPO and an explicit M-step becomes unnecessary (see. Alg. 3). 2. We can choose a non-parametric representation of $q ( a | s )$ given by one probability factor per sample. To achieve generalization in state space we then fit a parametric policy in the M-step.
+
+Fitting a parametric policy in the M-step is a supervised learning problem, allowing us to employ various regularization techniques at that point. It also makes it easier to enforce the hard KL constraint.
+
+# NON PARAMETRIC VARIATIONAL DISTRIBUTION
+
+In the non-parametric case we can obtain the optimal sample based $q$ distribution – the solution to Equation (7) – in closed form (see the appendix for a full derivation), as,
+
+$$
+q _ { i } ( a | s ) \propto \pi ( a | s , \pmb { \theta } _ { i } ) \exp \Big ( \frac { Q _ { \theta _ { i } } ( s , a ) } { \eta ^ { * } } \Big ) ,
+$$
+
+where we can obtain $\eta ^ { * }$ by minimising the following convex dual function,
+
+$$
+g ( \eta ) = \eta \epsilon + \eta \int \mu ( s ) \log \int \pi ( a | s , \pmb { \theta } _ { i } ) \exp \Big ( \frac { Q _ { \theta _ { i } } ( s , a ) } { \eta } \Big ) d a d s ,
+$$
+
+after the optimisation of which we can evaluate $q _ { i } ( a | s )$ on given samples.
+
+This optimization problem is similar to the one solved by relative entropy policy search (REPS) (Peters et al., 2010a) with the difference that we optimise only for the conditional variational distribution $q ( a | s )$ instead of a joint distribution $\scriptstyle q ( a , s )$ – effectively fixing $\mu _ { q } ( s )$ to the stationary distribution given by previously collected experience – and we use the $\mathrm { Q }$ function of the old policy to evaluate the integral over $a$ . While this might seem unimportant it is crucial as it allows us to estimate the integral over actions with multiple samples without additional environment interaction. This greatly reduces the variance of the estimate and allows for fully off-policy learning at the cost of performing only a partial optimization of $\mathcal { I }$ as described above.
+
+# 3.3 M-STEP
+
+Given $q _ { i }$ from the E-step we can optimize the lower bound $\mathcal { I }$ with respect to $\pmb \theta$ to obtain an updated policy $\pmb { \theta } _ { i + 1 } = \arg \operatorname* { m a x } _ { \pmb { \theta } } \mathcal { I } ( q _ { i } , \pmb { \theta } )$ . Dropping terms independent of $\pmb { \theta }$ this entails solving for the solution of
+
+$$
+\operatorname* { m a x } _ { \theta } \mathcal { I } ( q _ { i } , \theta ) = \operatorname* { m a x } _ { \theta } \mathbb { E } _ { \mu _ { q } ( s ) } \Big [ \mathbb { E } _ { q ( a | s ) } \Big [ \log \pi ( a | s , \theta ) \Big ] \Big ] + \log p ( \theta ) ,
+$$
+
+which corresponds to a weighted maximum a-posteriroi estimation (MAP) problem where samples are weighted by the variational distribution from the E-step. Since this is essentially a supervised learning step we can choose any policy representation in combination with any prior for regularisation. In this paper we set $p ( \pmb \theta )$ to a Gaussian prior around the current policy, i.e, $\begin{array} { r } { p ( \pmb { \theta } ) \approx \mathcal { N } \Big ( \mu = \pmb { \theta } _ { i } , \Sigma = \frac { F _ { \pmb { \theta } _ { i } } } { \lambda } \Big ) } \end{array}$ , where $\pmb \theta _ { i }$ are the parameters of the current policy distribution, $F _ { \pmb { \theta } _ { i } }$ is the empirical Fisher information matrix and $\lambda$ is a positive scalar. As shown in the appendix this suggests the following generalized M-step:
+
+$$
+\operatorname* { m a x } _ { \pi } \mathbb { E } _ { \mu _ { q } ( s ) } \Big [ \mathbb { E } _ { q ( a | s ) } \Big [ \log \pi ( a | s , \pmb { \theta } ) \Big ] - \lambda \mathrm { K L } \Big ( \pi ( a | s , \pmb { \theta } _ { i } ) , \pi ( a | s , \pmb { \theta } ) \Big ) \Big ]
+$$
+
+which can be re-written as the hard constrained version:
+
+$$
+\begin{array} { r } { \underset { \pi } { \operatorname* { m a x } } \mathbb { E } _ { \mu _ { q } ( s ) } \left[ \mathbb { E } _ { q ( a | s ) } \left[ \log \pi ( a | s , \pmb { \theta } ) \right] \right] } \\ { s . t . \mathbb { E } _ { \mu _ { q } ( s ) } \left[ { \mathbf { K L } } ( \pi ( a | s , \pmb { \theta } _ { i } ) , \pi ( a | s , \pmb { \theta } ) ) \right] < \epsilon . } \end{array}
+$$
+
+This additional constraint minimises the risk of overfitting the samples, i.e. it helps us to obtain a policy that generalises beyond the state-action samples used for the optimisation. In practice we have found the KL constraint in the M step to greatly increase stability of the algorithm. We also note that in the E-step we are using the reverse, mode-seeking, KL while in the M-step we are using the forward, moment-matching, KL which reduces the tendency of the entropy of the parametric policy to collapse. This is in contrast to other RL algorithms that use M-projection without KL constraint to fit a parametric policy (Peters et al., 2010a; Wirth et al., 2016; Chebotar et al., 2016; Montgomery & Levine, 2016). Using KL constraint in M-step has also been shown effective for stochastic search algorithms (Abdolmaleki et al., 2017).
+
+# 4 POLICY EVALUATION
+
+Our method is directly applicable in an off-policy setting. For this, we have to rely on a stable policy evaluation operator to obtain a parametric representation of the Q-function $Q _ { \theta } ( s , a )$ . We make use of the policy evaluation operator from the Retrace algorithm Munos et al. (2016), which we found to yield stable policy evaluation in practice2. Concretely, we fit the Q-function $Q _ { \theta _ { i } } ( s , a , \phi )$ as represented by a neural network, with parameters $\phi$ , by minimising the squared loss:
+
+$$
+\begin{array} { l } { \displaystyle \mathop { \operatorname* { m i n } } _ { \phi } L ( \phi ) = \mathop { \operatorname* { m i n } } _ { \phi } \mathbb { E } _ { \mu _ { b } ( s ) , b ( a | s ) } \left[ \left( Q _ { \theta _ { i } } ( s _ { t } , a _ { t } , \phi ) - Q _ { t } ^ { \mathrm { r e t } } \right) ^ { 2 } \right] , \mathrm { w i t h } } \\ { \displaystyle Q _ { t } ^ { \mathrm { r e t } } = Q _ { \phi ^ { \prime } } ( s _ { t } , a _ { t } ) + \sum _ { j = t } ^ { \infty } \gamma ^ { j - t } \Big ( \prod _ { k = t + 1 } ^ { j } c _ { k } \Big ) \left[ r ( s _ { j } , a _ { j } ) + \mathbb { E } _ { \pi ( a | s _ { j + 1 } ) } [ Q _ { \phi ^ { \prime } } ( s _ { j + 1 } , a ) ] - Q _ { \phi ^ { \prime } } ( s _ { j } , a _ { j } ) \right] , } \\ { \displaystyle c _ { k } = \operatorname* { m i n } \Big ( 1 , \frac { \pi ( a _ { k } | s _ { k } ) } { b ( a _ { k } | s _ { k } ) } \Big ) , } \end{array}
+$$
+
+where $Q _ { \phi ^ { \prime } } ( s , a )$ denotes the output of a target Q-network, with parameters $\phi ^ { \prime }$ , that we copy from the current parameters $\phi$ after each M-step. We truncate the infinite sum after $N$ steps by bootstrapping with $Q _ { \phi ^ { \prime } }$ (rather than considering a $\lambda$ return). Additionally, $b ( a | s )$ denotes the probabilities of an arbitrary behaviour policy. In our case we use an experience replay buffer and hence $b$ is given by the action probabilities stored in the buffer; which correspond to the action probabilities at the time of action selection.
+
+# 5 EXPERIMENTS
+
+For our experiments we evaluate our MPO algorithm across a wide range of tasks. Specifically, we start by looking at the continuous control tasks of the DeepMind Control Suite (Tassa et al. (2018), see Figure 1), and then consider the challenging parkour environments recently published in Heess et al. (2017). In both cases we use a Gaussian distribution for the policy whose mean and covariance are parameterized by a neural network (see appendix for details). In addition, we present initial experiments for discrete control using ATARI environments using a categorical policy distribution (whose logits are again parameterized by a neural network) in the appendix.
+
+# 5.1 EVALUATION ON CONTROL SUITE
+
+![](images/754c06df74f11ccb9772163bdb2d3c935cb2cba2b5b65def9b07cd08ab62b078.jpg)  
+Figure 1: Control Suite domains used for benchmarking. Top: Acrobot, Ball-in-cup, Cart-pole, Cheetah, Finger, Fish, Hopper. Bottom: Humanoid, Manipulator, Pendulum, Point-mass, Reacher, Swimmers (6 and 15 links), Walker.
+
+The suite of continuous control tasks that we are evaluating against contains 18 tasks, comprising a wide range of domains including well known tasks from the literature. For example, the classical cart-pole and acrobot dynamical systems, 2D and Humanoid walking as well as simple low-dimensional planar reaching and manipulation tasks. This suite of tasks was built in python on top of mujoco and will also be open sourced to the public by the time of publication.
+
+While we include plots depicting the performance of our algorithm on all tasks below; comparing it against the state-of-the-art algorithms in terms of data-efficiency. We want to start by directing the attention of the reader to a more detailed evaluation on three of the harder tasks from the suite.
+
+5.1.1 DETAILED ANALYSIS ON WALKER-2D, ACROBOT, HOPPER
+
+We start by looking at the results for the classical Acrobot task (two degrees of freedom, one continuous action dimension) as well as the 2D walker (which has 12 degrees of freedom and thus a 12 dimensional action space and a 21 dimensional state space) and the hopper standing task. The reward in the Acrobot task is the distance of the robots end-effector to an upright position of the underactuated system. For the walker task it is given by the forward velocity, whereas in the hopper the requirement is to stand still.
+
+Figure 2 shows the results for this task obtained by applying our algorithm MPO as well as several ablations – in which different parts were removed from the MPO optimization – and two baselines: our implementation of Proximal Policy Optimization (PPO) (Schulman et al., 2017b) and DDPG. The hyperparameters for MPO were kept fixed for all experiments in the paper (see the appendix for hyperparameter settings).
+
+As a first observation, we can see that MPO gives stable learning on all tasks and, thanks to its fully off-policy implementation, is significantly more sample efficient than the on-policy PPO baseline. Furthermore, we can observe that changing from the non-parametric variational distribution to a parametric distribution3 (which, as described above, can be related to PPO) results in only a minor asymptotic performance loss but slowed down optimisation and thus hampered sample efficiency; which can be attributed to the fact that the parametric $q$ distribution required a stricter KL constraint. Removing the automatically tuned KL constraint and replacing it with a manually set entropy regulariser then yields an off-policy actor-critic method with Retrace. This policy gradient method still uses the idea of estimating the integral over actions – and thus, for a gradient based optimiser, its likelihood ratio derivative – via multiple action samples (as judged by a Q-Retrace critic). This idea has previously been coined as using the expected policy gradient (EPG) (Ciosek & Whiteson, 2017) and we hence denote the corresponding algorithm with EPG $^ +$ Retrace, which no-longer follows the intuitions of the MPO perspective. $\mathrm { E P G } + .$ Retrace performed well when the correct entropy regularisation scale is used. This, however, required task specific tuning (c.f. Figure 4 where this hyperparameter was set to the one that performed best in average across tasks). Finally using only a single sample to estimate the integral (and hence the likelihood ratio gradient) results in an actor-critic variant with Retrace that is the least performant off-policy algorithm in our comparison.
+
+![](images/c77b362823bc448bf6268c862a2c6ae8c0cbe92c11eed9cb0fed52cbb3223fbf.jpg)  
+Figure 2: Ablation study of the MPO algorithm and comparison to common baselines from the literature on three domains from the control suite. We plot the median performance over 10 experiments with different random seeds.
+
+# 5.1.2 COMPLETE RESULTS ON THE CONTROL SUITE
+
+The results for MPO (non-parameteric) – and a comparison to an implementation of state-of-the-art algorithms from the literature in our framework – on all the environments from the control suite that we tested on are shown in Figure 4. All tasks have rewards that are scaled to be between 0 and 1000. We note that in order to ensure a fair comparison all algorithms ran with exactly the same network configuration, used a single learner (no distributed computation), used the same optimizer and were tuned w.r.t. their hyperparameters for best performance across all tasks. We refer to the appendix for a complete description of the hyperparameters. Our comparison is made in terms of data-efficiency.
+
+From the plot a few trends are readily apparent: i) We can clearly observe the advantage in terms of data-efficiency that methods relying on a Q-critic obtain over the PPO baseline. This difference is so extreme that in several instances the PPO baseline converges an order of magnitude slower than the off-policy algorithms and we thus indicate the asymptotic performance of each algorithm of PPO and DDPG (which also improved significantly later during training in some instances) with a colored star in the plot; ii) the difference between the MPO results and the (expected) policy gradient (EPG) with entropy regularisation confirm our suspicion from Section 5.1.1: finding a good setting for the entropy regulariser that transfers across environments without additional constraints on the policy distribution is very difficult, leading to instabilities in the learning curves. In contrast to this the MPO results appear to be stable across all environments; iii) Finally, in terms of data-efficiency the methods utilising Retrace obtain a clear advantage over DDPG. The single learner vanilla DDPG implementation learns the lower dimensional environments quickly but suffers in terms of learning speed in environments with sparse rewards (finger, acrobot) and higher dimensional action spaces. Overall, MPO is able to solve all environments using surprisingly moderate amounts of data. On average less than 1000 trajectories (or $1 0 ^ { 6 }$ samples) are needed to reach the best performance.
+
+# 5.2 HIGH-DIMENSIONAL CONTINUOUS CONTROL
+
+Next we turn to evaluating our algorithm on two higher-dimensional continuous control problems; humanoid and walker. To make computation time bearable in these more complicated domains we utilize a parallel variant of our algorithm: in this implementation K learners are all independently collecting data from an instance of the environment. Updates are performed at the end of each collected trajectory using distributed synchronous gradient descent on a shared set of policy and Q-function parameters (we refer to the appendix for an algorithm description). The results of this experiment are depicted in Figure 3.
+
+For the Humanoid running domain we can observe a similar trend to the experiments from the previous section: MPO quickly finds a stable running policy, outperforming all other algorithms in terms of sample efficiency also in this high-dimensional control problem.
+
+The case for the Walker-2D parkour domain (where we compare against a PPO baseline) is even more striking: where standard PPO requires approximately 1M trajectories to find a good policy MPO finds a solution that is asymptotically no worse than the PPO solution in in about 70k trajectories (or 60M samples), resulting in an order of magnitude improvement. In addition to the walker experiment we have also evaluated MPO on the Parkour domain using a humanoid body (with 22 degrees of freedom) which was learned successfully (not shown in the plot, please see the supplementary video).
+
+![](images/3f0db1cdbd6cb97bb8fdb4b7ec7943a5ae55699354d37055db91d206400596e2.jpg)  
+Figure 3: MPO on high-dimensional control problems (Parkour Walker2D and Humanoid walking from control suite).
+
+![](images/b39b3f4bf39eeac87f2d9d9433c59fe69a73415f4747eb95d2ef18eb773ea4b6.jpg)
+
+# 5.3 DISCRETE CONTROL
+
+As a proof of concept – showcasing the robustness of our algorithm and its hyperparameters – we performed an experiment on a subset of the games contained contained in the "Arcade Learning Environment" (ALE) where we used the same hyperparameter settings for the KL constraints as for the continuous control experiments. The results of this experiment can be found in the Appendix.
+
+# 6 CONCLUSION
+
+We have presented a new off-policy reinforcement learning algorithm called Maximum a-posteriori Policy Optimisation (MPO). The algorithm is motivated by the connection between RL and inference and it consists of an alternating optimisation scheme that has a direct relation to several existing algorithms from the literature. Overall, we arrive at a novel, off-policy algorithm that is highly data efficient, robust to hyperparameter choices and applicable to complex control problems. We demonstrated the effectiveness of MPO on a large set of continuous control problems.
+
+![](images/1fb493111520008f89fe9fd5525df44f3315390e08daaf3a2e48f3e75e24a2a1.jpg)  
+Figure 4: Complete comparison of results for the control suite. We plot the median performance over 10 random seeds together with 5 and $95 \%$ quantiles (shaded area). Note that for DDPG we only plot the median to avoid clutter in the plots. For DDPG and PPO final performance is marked by a star).
+
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+
+# A PROOF OF MONOTONIC IMPROVEMENT FOR THE KL-REGULARIZED POLICY OPTIMIZATION PROCEDURE
+
+In this section we prove a monotonic improvement guarantee for KL-regularized policy optimization via alternating updates on $\pi$ and $q$ under the assumption that the prior on $\theta$ is uninformative.
+
+# A.1 REGULARIZED REINFORCEMENT LEARNING
+
+Let $\pi$ be an arbitrary policy. For any other policy $q$ such that, for all $x , a , \{ \pi ( a | x ) > 0 \} \implies$ $\{ q ( a | x ) > 0 \}$ , define the $\pi$ -regularized reward for policy $q$ :
+
+$$
+r _ { \alpha } ^ { \pi , q } ( x , a ) = r ( x , a ) - \alpha \log \frac { q ( a | x ) } { \pi ( a | x ) } ,
+$$
+
+where $\alpha > 0$ .
+
+Bellman operators: Define the $\pi$ -regularized Bellman operator for policy $q$
+
+$$
+\begin{array} { r } { T _ { \alpha } ^ { \pi , q } V ( x ) = \mathbb { E } _ { a \sim q ( \cdot | x ) } \Big [ r _ { \alpha } ^ { \pi , q } ( x , a ) + \gamma \mathbb { E } _ { y \sim p ( \cdot | x , a ) } V ( y ) \Big ] , } \end{array}
+$$
+
+and the non-regularized Bellman operator for policy $q$
+
+$$
+T ^ { q } V ( x ) = \mathbb { E } _ { a \sim q ( \cdot | x ) } \Big [ r ( x , a ) + \gamma \mathbb { E } _ { y \sim p ( \cdot | x , a ) } V ( y ) \Big ] .
+$$
+
+Value function: Define the $\pi$ -regularized value function for policy $q$ as
+
+$$
+V _ { \alpha } ^ { \pi , q } ( \boldsymbol { x } ) = \mathbb { E } _ { q } \Big [ \sum _ { t \geq 0 } \gamma ^ { t } r _ { \alpha } ^ { \pi , q } ( \boldsymbol { x } _ { t } , a _ { t } ) | \boldsymbol { x } _ { 0 } = \boldsymbol { x } , q \Big ] .
+$$
+
+and the non-regularized value function
+
+$$
+V ^ { q } ( x ) = \mathbb { E } _ { q } \Big [ \sum _ { t \geq 0 } \gamma ^ { t } r ( x _ { t } , a _ { t } ) | x _ { 0 } = x , q \Big ] .
+$$
+
+Proposition 1. For any $q , \pi , V$ , we have $V _ { \alpha } ^ { \pi , q } \leq V ^ { q }$ and $T _ { \alpha } ^ { \pi , q } V \le T ^ { q } V$ . Indeed
+
+$$
+\mathbb { E } _ { q } \Big [ \log \frac { q \big ( a _ { t } | x _ { t } \big ) } { \pi \big ( a _ { t } | x _ { t } \big ) } \Big ] = K L \big ( q ( \cdot | x _ { t } ) \| \pi ( \cdot | x _ { t } ) \big ) \geq 0 .
+$$
+
+Optimal value function and policy Define the optimal regularized value function: $V _ { \alpha } ^ { \pi , * } ( x ) =$ $\operatorname* { m a x } _ { q } V _ { \alpha } ^ { \pi , q } ( x )$ , and the optimal (non-regularized) value function: $V ^ { * } ( x ) = \operatorname* { m a x } _ { q } V ^ { q } ( x )$ .
+
+The optimal policy of the $\pi$ -regularized problem $q _ { \alpha } ^ { \pi , * } ( \cdot | x ) = \arg \operatorname* { m a x } _ { q } V _ { \alpha } ^ { \pi , q } ( x )$ and the optimal policy of the non-regularized problem $q ^ { * } ( \cdot | x ) = \arg \operatorname* { m a x } _ { q } V ^ { q }$ .
+
+Proposition 2. We have that $V _ { \alpha } ^ { \pi , q }$ is the unique fixed point of $T _ { \alpha } ^ { \pi , q }$ , and $V ^ { q }$ is the unique fixed point of $T ^ { q }$ . Thus we have the following Bellman equations: For all $x \in X$ ,
+
+$$
+\begin{array} { r c l } { { V _ { \alpha } ^ { \pi , q } ( x ) } } & { { = } } & { { \displaystyle \sum _ { a } q ( a | x ) \Big [ r _ { \alpha } ^ { \pi , q } ( x , a ) + \gamma \mathbb { E } _ { y \sim p ( \cdot | x , a ) } \big [ V _ { \alpha } ^ { \pi , q } ( y ) \big ] \Big ] } } \\ { { } } & { { } } & { { } } \\ { { V ^ { q } ( x ) } } & { { = } } & { { \displaystyle \sum _ { a } q ( a | x ) \Big [ r ( x , a ) + \gamma \mathbb { E } _ { y \sim p ( \cdot | x , a ) } \big [ V ^ { q } ( y ) \big ] \Big ] } } \\ { { } } & { { } } & { { } } \\ { { V _ { \alpha } ^ { \pi , * } ( x ) } } & { { = } } & { { \displaystyle r _ { \alpha } ^ { \pi , q _ { \alpha } ^ { \pi , * } } ( x , a ) + \gamma \mathbb { E } _ { y \sim p ( \cdot | x , a ) } \big [ V _ { \alpha } ^ { \pi , * } ( y ) \big ] f o r a l l a \in A , } } \\ { { } } & { { } } & { { } } \\ { { V ^ { * } ( x ) } } & { { = } } & { { \displaystyle \operatorname* { m a x } _ { a \in A } \Big [ r ( x , a ) + \gamma \mathbb { E } _ { y \sim p ( \cdot | x , a ) } \big [ V ^ { * } ( y ) \big ] \Big ] . } } \end{array}
+$$
+
+Notice that (16) holds for all actions $a \in A$ , and not in expectation w.r.t. $a \sim q ( \cdot | x )$ only.
+
+# A.2 REGULARIZED JOINT POLICY GRADIENT
+
+We now consider a parametrized policy $\pi _ { \theta }$ and consider maximizing the regularized joint policy optimization problem for a given initial state $x _ { 0 }$ (this could be a distribution over initial states). Thus we want to find a parameter $\theta$ that (locally) maximizes
+
+$$
+\mathcal { I } ( \theta , q ) = V ^ { \pi _ { \theta } , q } ( \boldsymbol { x } _ { 0 } ) = \mathbb { E } _ { q } \Big [ \sum _ { t \geq 0 } \gamma ^ { t } \big ( r ( \boldsymbol { x } _ { t } , a _ { t } ) - \alpha \mathrm { K L } ( q ( \cdot | \boldsymbol { x } _ { t } ) | | \pi _ { \theta } ( \cdot | \boldsymbol { x } _ { t } ) ) \big ) \big | \boldsymbol { x } _ { 0 } , q \Big ] .
+$$
+
+We start with an initial parameter $\theta _ { 0 }$ and define a sequence of policies $\pi _ { i } = \pi _ { \theta _ { i } }$ parametrized by $\theta _ { i }$ , in the following way:
+
+• Given $\theta _ { i }$ , define
+
+$$
+q _ { i } = \arg \operatorname* { m a x } _ { q } T _ { \alpha } ^ { \pi _ { \theta _ { i } } , q } V ^ { \pi _ { \theta _ { i } } } ,
+$$
+
+• Define $\theta _ { i + 1 }$ as
+
+$$
+\theta _ { i + 1 } = \theta _ { i } - \beta \nabla _ { \theta } \mathbb { E } _ { \pi _ { i } } \left[ \sum _ { t \geq 0 } \gamma ^ { t } \mathrm { K L } \big ( q _ { k } ( \cdot | x _ { t } ) | | \pi _ { \theta } ( \cdot | x _ { t } ) \big ) _ { | \theta = \theta _ { i } } \big | x _ { 0 } , \pi _ { i } \right] .
+$$
+
+Proposition 3. We have the following properties:
+
+• The policy $q _ { i }$ satisfies:
+
+$$
+q _ { i } ( a | x ) = \frac { \pi _ { i } ( a | x ) e ^ { \frac { 1 } { \alpha } Q ^ { \pi _ { i } } ( x , a ) } } { \mathbb { E } _ { b \sim \pi _ { i } ( \cdot | x ) } \big [ e ^ { \frac { 1 } { \alpha } Q ^ { \pi _ { i } } ( x , b ) } \big ] } ,
+$$
+
+where $Q ^ { \pi } ( x , a ) = r ( x , a ) + \gamma \mathbb { E } _ { y \sim p ( \cdot | x , a ) } V ^ { \pi } ( y ) .$
+
+• We have
+
+$$
+V _ { \alpha } ^ { \pi _ { i } , q _ { i } } \geq V ^ { \pi _ { i } } .
+$$
+
+• For η sufficiently small, we have
+
+$$
+\mathcal { I } ( \theta _ { i + 1 } , q _ { i + 1 } ) \geq \mathcal { I } ( \theta _ { i } , q _ { i } ) + c g _ { i } ,
+$$
+
+where c is a numerical constant, and $g _ { i }$ is the norm of the gradient (minimized by the algorithm):
+
+$$
+g _ { i } = \Big \| \nabla _ { \theta } \mathbb { E } _ { \pi _ { i } } \Big [ \sum _ { t \geq 0 } \gamma ^ { t } K L \big ( q _ { i } ( \cdot | x _ { t } ) \| \pi _ { \theta } ( \cdot | x _ { t } ) \big ) _ { | \theta = \theta _ { i } } \big | x _ { 0 } , \pi _ { i } \Big ] \Big \| .
+$$
+
+Thus we build a sequence of policies $( \pi _ { \theta _ { i } } , q _ { i } )$ whose values $\mathcal { I } ( \theta _ { i } , q _ { i } )$ are non-decreasing thus converge to a local maximum. In addition, the improvement is lower-bounded by $a$ constant times the norm of the gradient, thus the algorithm keeps improving the performance until the gradient vanishes (when we reach the limit of the capacity of our representation).
+
+Proof. We have
+
+$$
+\begin{array} { r l r } { q _ { i } ( \cdot | x ) } & { = } & { \arg \underset { q } { \operatorname* { m a x } } \mathbb { E } _ { a \sim q ( \cdot | x ) } \Big [ \underset { Q ^ { \pi _ { i } } ( x , a ) } { \underbrace { r \big ( x , a ) + \gamma \mathbb { E } _ { y \sim p ( \cdot | x , a ) } V ^ { \pi _ { i } } ( y ) } } - \alpha \log \frac { q \big ( a | x \big ) } { \pi _ { i } \big ( a | x \big ) } \Big ] , } \end{array}
+$$
+
+from which we deduce (19). Now, from the definition of $q _ { i }$ , we have
+
+$$
+T _ { \alpha } ^ { \pi _ { i } , q _ { i } } V ^ { \pi _ { i } } \geq T _ { \alpha } ^ { \pi _ { i } , \pi _ { i } } V ^ { \pi _ { i } } = T ^ { \pi _ { i } } V ^ { \pi _ { i } } = V ^ { \pi _ { i } } .
+$$
+
+Now, since and its fixe $T _ { \alpha } ^ { \pi _ { i } , q _ { i } }$ is is otone operator (i.e. if , we have $V _ { 1 } \geq V _ { 2 }$ elementwise, then $T _ { \alpha } ^ { \pi _ { i } , q _ { i } } V _ { 1 } \geq T _ { \alpha } ^ { \pi _ { i } , q _ { i } } V _ { 2 } )$ $V _ { \alpha } ^ { \pi _ { i } , q _ { i } }$
+
+$$
+V _ { \alpha } ^ { \pi _ { i } , q _ { i } } = \operatorname* { l i m } _ { t \to \infty } ( T _ { \alpha } ^ { \pi _ { i } , q _ { i } } ) ^ { t } V ^ { \pi _ { i } } \geq V ^ { \pi _ { i } } ,
+$$
+
+which proves (20).
+
+Now, in order to prove (21) we derive the following steps.
+
+Step 1: From the definition of $q _ { i + 1 }$ we have, for any $x$
+
+$$
+\begin{array} { r } { \tilde { \Sigma } _ { a \sim q _ { 4 } + 1 } \left[ Q ^ { \pi _ { i + 1 } } ( x , a ) \right] - \alpha \mathbf { K L } \left( q _ { i + 1 } ( \cdot | x ) \| \pi _ { i + 1 } ( \cdot | x ) \right) \geq \mathbb { E } _ { a \sim q _ { 4 } } \left[ Q ^ { \pi _ { i + 1 } } ( x , a ) \right] - \alpha \mathbf { K L } \left( q _ { i } ( \cdot | x ) \| \pi _ { i + 1 } ( \cdot | x ) \right) } \end{array}
+$$
+
+Writing the functional that we minimize
+
+$$
+f ( \pi , q , \theta ) = \mathbb { E } _ { \pi } \Big [ \sum _ { t \geq 0 } \gamma ^ { t } \mathrm { K L } \big ( q ( \cdot | x _ { t } ) \| \pi _ { \theta } ( \cdot | x _ { t } ) \big ) \big | x _ { 0 } , \pi \Big ] ,
+$$
+
+the update rule is $\theta _ { i + 1 } = \theta _ { i } - \beta \nabla _ { \theta } f ( \pi _ { i } , q _ { i } , \theta _ { i } )$ . Thus we have that for sufficiently small $\beta$ ,
+
+$$
+f ( \pi _ { i } , q _ { i } , \theta _ { i + 1 } ) \leq f ( \pi _ { i } , q _ { i } , \theta _ { i } ) - \beta g _ { i } ,
+$$
+
+where $\begin{array} { r } { g _ { i } = \frac { 1 } { 2 } \| \nabla _ { \theta } f ( \pi _ { i } , q _ { i } , \theta _ { i } ) \| } \end{array}$ .
+
+Step 2: Now define $\mathcal { F }$ :
+
+$$
+\begin{array} { r c l } { \mathcal { F } ( \pi , q , \theta , \pi ^ { \prime } ) } & { = } & { \displaystyle \mathbb { E } _ { \pi } \left[ \sum _ { t \geq 0 } \gamma ^ { t } \big ( \mathbb { E } _ { a \sim q } \big [ Q ^ { \pi ^ { \prime } } ( x _ { t } , a ) \big ] - \alpha \mathrm { K L } \big ( q ( \cdot | x _ { t } ) \| \pi _ { \theta } ( \cdot | x _ { t } ) \big ) \big ) \big | x _ { 0 } , \pi \right] } \\ & { = } & { \displaystyle \delta _ { x _ { 0 } } \big ( I - \gamma P ^ { \pi } \big ) ^ { - 1 } T _ { \alpha } ^ { \pi _ { \theta } , q } V ^ { \pi ^ { \prime } } } \\ & { = } & { \delta _ { x _ { 0 } } \big ( I - \gamma P ^ { \pi } \big ) ^ { - 1 } T ^ { q } V ^ { \pi ^ { \prime } } - f ( \pi , q , \theta ) , } \end{array}
+$$
+
+where $\delta _ { x _ { 0 } }$ is a Dirac (in the row vector $x _ { 0 }$ ), and $P ^ { \pi }$ is the transition matrix for policy $\pi$
+
+From (22) and (23) we deduce that
+
+$$
+\begin{array} { r c l } { \mathcal { F } ( \pi _ { i } , q _ { i + 1 } , \theta _ { i + 1 } , \pi _ { i + 1 } ) } & { \geq } & { \mathcal { F } ( \pi _ { i } , q _ { i } , \theta _ { i + 1 } , \pi _ { i + 1 } ) } \\ & { \geq } & { \mathcal { F } ( \pi _ { i } , q _ { i } , \theta _ { i } , \pi _ { i + 1 } ) + \beta g _ { i } . } \end{array}
+$$
+
+We deduce
+
+$$
+\begin{array} { r l } & { \mathcal { F } ( \pi _ { i } , q _ { i + 1 } , \theta _ { i + 1 } , \pi _ { i } ) } \\ { \geq } & { \mathcal { F } ( \pi _ { i } , q _ { i } , \theta _ { i } , \pi _ { i } ) + \beta g _ { i } } \\ & { + \mathcal { F } ( \pi _ { i } , q _ { i + 1 } , \theta _ { i + 1 } , \pi _ { i } ) - \mathcal { F } ( \pi _ { i } , q _ { i + 1 } , \theta _ { i + 1 } , \pi _ { i + 1 } ) + \mathcal { F } ( \pi _ { i } , q _ { i } , \theta _ { i } , \pi _ { i + 1 } ) - \mathcal { F } ( \pi _ { i } , q _ { i } , \theta _ { i } , \pi _ { i } ) } \\ { = } & { \mathcal { F } ( \pi _ { i } , q _ { i } , \theta _ { i } , \pi _ { i } ) + \beta g _ { i } + } \\ & { \mathbb { E } _ { \pi _ { i } } \Big [ \displaystyle \sum _ { t \geq 0 } \gamma ^ { t } \big ( \mathbb { E } _ { a \sim q _ { i + 1 } } \big [ Q ^ { \pi _ { i } } ( x _ { t } , a ) - Q ^ { \pi _ { i + 1 } } ( x _ { t } , a ) \big ] - \mathbb { E } _ { a \sim q _ { i } } \big [ Q ^ { \pi _ { i } } ( x _ { t } , a ) - Q ^ { \pi _ { i + 1 } } ( x _ { t } , a ) \big ] \big ) \Big ] } \end{array}
+$$
+
+This rewrites:
+
+$$
+\delta _ { x _ { 0 } } ( I - \gamma ^ { \pi _ { i } } ) ^ { - 1 } \big ( T _ { \alpha } ^ { q _ { i + 1 } , \pi _ { i + 1 } } V ^ { \pi _ { i } } - T _ { \alpha } ^ { q _ { i } , \pi _ { i } } V ^ { \pi _ { i } } \big ) \geq \eta g _ { i } + O ( \beta ^ { 2 } ) .
+$$
+
+Step 3: Now a bit of algebra. For two stochastic matrices $P$ and $P ^ { \prime }$ , we have
+
+$$
+\begin{array} { r l } & { ( I - \gamma P ) ^ { - 1 } } \\ { = } & { ( I - \gamma P ^ { \prime } ) ^ { - 1 } + \gamma ( I - \gamma P ) ^ { - 1 } ( P - P ^ { \prime } ) ( I - \gamma P ^ { \prime } ) ^ { - 1 } } \\ { = } & { ( I - \gamma P ^ { \prime } ) ^ { - 1 } + \gamma \big [ ( I - \gamma P ^ { \prime } ) ^ { - 1 } + \gamma ( I - \gamma P ) ^ { - 1 } ( P - P ^ { \prime } ) ( I - \gamma P ^ { \prime } ) ^ { - 1 } \big ] ( P - P ^ { \prime } ) ( I - \gamma P ^ { \prime } ) ^ { - 1 } } \\ { = } & { ( I - \gamma P ^ { \prime } ) ^ { - 1 } + \gamma ( I - \gamma P ^ { \prime } ) ^ { - 1 } ( P - P ^ { \prime } ) ( I - \gamma P ^ { \prime } ) ^ { - 1 } } \\ & { + \gamma ^ { 2 } ( I - \gamma P ) ^ { - 1 } ( P - P ^ { \prime } ) ( I - \gamma P ^ { \prime } ) ^ { - 1 } ( P - P ^ { \prime } ) ( I - \gamma P ^ { \prime } ) ^ { - 1 } . } \end{array}
+$$
+
+Applying this equality to the transition matrices $P ^ { \pi _ { k } }$ and $P ^ { \pi _ { k + 1 } }$ and since $\lVert P ^ { \pi _ { k + 1 } } - P ^ { \pi _ { k } } \rVert = O ( \eta )$ , we have:
+
+$$
+\begin{array} { r l } & { V _ { \alpha } ^ { q _ { i + 1 } , \pi _ { i + 1 } } } \\ { = } & { ( I - \gamma P ^ { \pi _ { i } } ) ^ { - 1 } r _ { \alpha } ^ { q _ { i + 1 } , \pi _ { i + 1 } } } \\ { = } & { ( I - \gamma P ^ { \pi _ { i } } ) ^ { - 1 } r _ { \alpha } ^ { q _ { i + 1 } , \pi _ { i + 1 } } + \gamma ( I - \gamma P ^ { \pi _ { i } } ) ^ { - 1 } ( P ^ { \pi _ { i + 1 } } - P ^ { \pi _ { i } } ) ( I - \gamma P ^ { \pi _ { i } } ) ^ { - 1 } r _ { \alpha } ^ { q _ { i + 1 } , \pi _ { i + 1 } } + O ( \beta ^ { 2 } ) } \\ { = } & { ( I - \gamma P ^ { \pi _ { i } } ) ^ { - 1 } r _ { \alpha } ^ { q _ { i } , \pi _ { i } } + ( I - \gamma P ^ { \pi _ { i } } ) ^ { - 1 } ( r _ { \alpha } ^ { q _ { i + 1 } , \pi _ { i + 1 } } - r _ { \alpha } ^ { q _ { i } , \pi _ { i } } + \gamma P ^ { \pi _ { i + 1 } } - \gamma P ^ { \pi _ { i } } ) ( I - \gamma P ^ { \pi _ { i } } ) ^ { - 1 } r _ { \alpha } ^ { q _ { i } , \pi _ { i } } } \\ { = } & { V _ { \alpha } ^ { q _ { i } , \pi _ { i } } + ( I - \gamma P ^ { \pi _ { i } } ) ^ { - 1 } ( T _ { \alpha } ^ { q _ { i + 1 } , \pi _ { i } + 1 } V ^ { \pi _ { i } } - T _ { \alpha } ^ { q _ { i } , \pi _ { i } } V ^ { \pi _ { k } } ) + O ( \beta ^ { 2 } ) . } \end{array}
+$$
+
+Table 1: Results on a subset of the ALE environments in comparison to baselines taken from (Bellemare et al., 2017)   
+
+<table><tr><td>Game/Agent</td><td>Human</td><td>DQN</td><td>Prior. Dueling</td><td>C51</td><td>MPO</td></tr><tr><td></td><td></td><td>19.5</td><td>20.9</td><td></td><td></td></tr><tr><td>Pong Breakout</td><td>14.6 30.5</td><td>385.5</td><td>366.0</td><td>20.9 748</td><td>20.9 360.5</td></tr><tr><td></td><td>13,455.0</td><td>13,117.3</td><td>18,760.3</td><td>23,784</td><td></td></tr><tr><td>Q*bert Tennis</td><td>-8.3</td><td>12.2</td><td>0.0</td><td>23.1</td><td>10,317.0</td></tr><tr><td></td><td></td><td></td><td>98.9</td><td></td><td>22.2</td></tr><tr><td>Boxing</td><td>12.1</td><td>88.0</td><td></td><td>97.8</td><td>82.0</td></tr></table>
+
+Finally, using (24), we deduce that
+
+$$
+\begin{array} { l l l } { \mathcal { I } ( \theta _ { i + 1 } , q _ { i + 1 } ) } & { = } & { V _ { \alpha } ^ { q _ { i } + 1 , \pi _ { i } + 1 } \big ( x _ { 0 } \big ) } \\ & { = } & { V _ { \alpha } ^ { q _ { i } , \pi _ { i } } \big ( x _ { 0 } \big ) + \delta _ { x _ { 0 } } \big ( I - \gamma P ^ { \pi _ { i } } \big ) ^ { - 1 } \big ( T _ { \alpha } ^ { q _ { i } + 1 , \pi _ { i + 1 } } V ^ { \pi _ { i } } - T _ { \alpha } ^ { q _ { i } , \pi _ { i } } V ^ { \pi _ { i } } \big ) + O ( \beta ^ { 2 } ) } \\ & { \geq } & { \mathcal { I } ( \theta _ { i } , q _ { i } ) + \eta g _ { i } + O ( \beta ^ { 2 } ) } \\ & { \geq } & { \mathcal { I } ( \theta _ { i } , q _ { i } ) + \frac { 1 } { 2 } \eta g _ { i } , } \end{array}
+$$
+
+for small enough $\eta$ .
+
+# B ADDITIONAL EXPERIMENT: DISCRETE CONTROL
+
+As a proof of concept – showcasing the robustness of our algorithm and its hyperparameters – we performed an experiment on a subset of the games contained contained in the "Arcade Learning Environment" (ALE). For this experiment we used the same hyperparameter settings for the KL constraints as for the continuous control experiments as well as the same learning rate and merely altered the network architecture to the standard network structure used by DQN Mnih et al. (2015) – and created a seperate network with the same architecture, but predicting the parameters of the policy distribution. A comparison between our algorithm and well established baselines from the literature, in terms of the mean performance, is listed in Table 1. While we do not obtain state-ofthe-art performance in this experiment, the fact that MPO is competitive, out-of-the-box in these domains suggests that combining the ideas presented in this paper with recent advances for RL with discrete actions (Bellemare et al., 2017) could be a fruitful avenue for future work.
+
+# C EXPERIMENT DETAILS
+
+In this section we give the details on the hyper-parameters used for each experiment. All the continuous control experiments use a feed-forward network except for Parkour-2d were we used the same network architecture as in Heess et al. (2017). Other hyper parameters for MPO with non parametric variational distribution were set as follows,
+
+Table 2: Parameters for non-parametric variational distribution   
+
+<table><tr><td>Hyperparameter</td><td>control suite</td><td>humanoid</td></tr><tr><td>Policy net</td><td>100-100</td><td>200-200</td></tr><tr><td>Q function net</td><td>200-200</td><td>300-300</td></tr><tr><td>E</td><td>0.1</td><td>1</td></tr><tr><td>Eμ</td><td>0.1</td><td>&quot;</td></tr><tr><td>£</td><td>0.0001</td><td>&quot;</td></tr><tr><td>Discount factor()</td><td>0.99</td><td>1</td></tr><tr><td>Adam learning rate</td><td>0.0005</td><td>1</td></tr></table>
+
+Hyperparameters for MPO with parametric variational distribution were as follows,
+
+Table 3: Parameters for parametric variational distribution   
+
+<table><tr><td>Hyperparameter</td><td>control suite tasks</td><td>humanoid</td></tr><tr><td>Policy net</td><td>100-100</td><td>200-200</td></tr><tr><td>Q function net</td><td>200-200</td><td>300-300</td></tr><tr><td>Eμ</td><td>0.1</td><td>1</td></tr><tr><td>£</td><td>0.0001</td><td>1</td></tr><tr><td>Discount factor ()</td><td>0.99</td><td>1</td></tr><tr><td>Adam learning rate</td><td>0.0005</td><td>1</td></tr></table>
+
+# D DERIVATION OF UPDATE RULES FOR A GAUSSIAN POLICY
+
+For continuous control we assume that the policy is given by a Gaussian distribution with a full covariance matrix, i.e, $\pi ( \boldsymbol { a } | \boldsymbol { s } , \pmb { \theta } ) = \mathcal { N } \left( \mu , \pmb { \Sigma } \right)$ . Our neural network outputs the mean $\mu = \mu ( s )$ and Cholesky factor $A = A ( s )$ , such that $\Sigma = A A ^ { T }$ . The lower triagular factor $A$ has positive diagonal elements enforced by the softplus transform $A _ { i i }  \log ( 1 + \exp ( A _ { i i } ) )$ .
+
+# D.1 NON-PARAMETRIC VARIATIONAL DISTRIBUTION
+
+In this section we provide the derivations and implementation details for the non-parametric variational distribution case for both $\mathrm { E }$ -step and M-step.
+
+# D.2 E-STEP
+
+The E-step with a non-parametric variational solves the following program, where we have replaced expectations with integrals to simplify the following derivations:
+
+$$
+\begin{array} { l } { \displaystyle \operatorname* { m a x } _ { q } \int \mu _ { q } ( s ) \int q ( a | s ) Q _ { \theta _ { i } } ( s , a ) d a d s } \\ { \displaystyle s . t . \int \mu _ { q } ( s ) { \bf K L } ( q ( a | s ) , \pi ( a | s , \theta _ { i } ) ) d a < \epsilon , } \\ { \displaystyle \iint \mu _ { q } ( s ) q ( a | s ) d a d s = 1 . } \end{array}
+$$
+
+First we write the Lagrangian equation, i.e,
+
+$$
+\begin{array} { c } { { L ( q , \eta , \gamma ) = \displaystyle \int \mu _ { q } ( s ) \int q ( a | s ) Q _ { \theta _ { i } } ( s , a ) d a d s + } } \\ { { \displaystyle \eta \left( \epsilon - \int \mu _ { q } ( s ) \int q ( a | s ) \log \frac { q ( a | s ) } { \pi ( a | s , \theta _ { i } ) } \right) + \gamma \left( 1 - \displaystyle \iint \mu _ { q } ( s ) q ( a | s ) d a d s \right) . } } \end{array}
+$$
+
+Next we maximise the Lagrangian $L$ w.r.t the primal variable $q$ . The derivative w.r.t $q$ reads,
+
+$$
+\partial q L ( q , \eta , \gamma ) = Q _ { \theta _ { i } } ( a , s ) - \eta \log q ( a | s ) + \eta \log \pi ( a | s , \pmb \theta _ { i } ) - ( \eta - \gamma ) .
+$$
+
+Setting it to zero and rearranging terms we get
+
+$$
+q ( a | s ) = \pi ( a | s , \pmb { \theta } _ { i } ) \exp \left( \frac { Q _ { \pmb { \theta } _ { i } } ( a , s ) } { \eta } \right) \exp \left( - \frac { \eta - \gamma } { \eta } \right) .
+$$
+
+However the last exponential term is a normalisation constant for $q$ . Therefore we can write,
+
+$$
+\exp ( - \frac { \eta - \gamma } { \eta } ) = \int \pi ( a | s , \pmb { \theta } _ { i } ) \exp ( \frac { Q _ { \pmb { \theta } _ { i } } ( a , s ) } { \eta } ) d a ,
+$$
+
+$$
+\gamma = \eta - \eta \log \left( \int \pi ( a | s , \pmb { \theta } _ { i } ) \exp ( \frac { Q _ { \pmb { \theta } _ { i } } ( a , s ) } { \eta } ) d a \right) .
+$$
+
+Note that we could write $\gamma$ based on $\pi$ and $\eta$ . At this point we can derive the dual function,
+
+$$
+g ( \eta ) = \eta \epsilon + \eta \int \mu _ { q } ( s ) \log \left( \int \pi ( a | s , \pmb { \theta } _ { i } ) \exp ( \frac { Q ( a , s ) } { \eta } ) d a \right) .
+$$
+
+# D.3 M-STEP
+
+To obtain the KL constraint in the $\mathbf { M }$ step we set $p ( \theta )$ to a Gaussian prior around the current policy, i.e,
+
+$$
+p ( \pmb \theta ) \approx \mathcal { N } \Big ( \mu = \pmb \theta _ { i } , \Sigma = \frac { F _ { \pmb \theta _ { i } } } { \lambda } \Big ) ,
+$$
+
+where $\theta _ { i }$ are the parameters of the current policy distribution, $F _ { \pmb { \theta } _ { i } }$ is the empirical Fisher information matrix and $\lambda$ .
+
+With this, and dropping constant terms our optimization program becomes
+
+$$
+\operatorname* { m a x } _ { \pi } \int \mu _ { q } ( s ) \int q ( a | s ) \log \pi ( a | s , \pmb { \theta } ) d a d s - \lambda ( \pmb { \theta } - \pmb { \theta } _ { i } ) ^ { T } F _ { \pmb { \theta _ { i } } } ^ { - 1 } ( \pmb { \theta } - \pmb { \theta } _ { i } ) .
+$$
+
+We can observe that $( \pmb \theta - \pmb \theta _ { i } ) ^ { T } F _ { \pmb \theta _ { i } } ^ { - 1 } ( \pmb \theta - \pmb \theta _ { i } )$ is the second order Taylor approximation of $\begin{array} { r } { \int \mu _ { q } ( s ) \mathrm { K L } ( \pi ( a | s , \pmb { \theta } _ { i } ) , \pi ( a | s , \pmb { \theta } ) ) d s } \end{array}$ which leads us to the generalized M-step objective:
+
+$$
+\operatorname* { m a x } _ { \pi } \int \mu _ { q } ( s ) \int q ( a | s ) \log \pi ( a | s , \theta ) d a d s - \lambda \int \mu _ { q } ( s ) \mathbf { K L } ( \pi ( a | s , \theta _ { i } ) , \pi ( a | s , \theta ) ) d s
+$$
+
+which corresponds to Equation (11) from the main text, where expectations are replaced by integrals.
+
+After obtaining the non parametric variational distribution in the $\mathbf { M }$ step with a Gaussian policy we empirically observed that better results could be achieved by decoupling the KL constraint into two terms such that we can constrain the contribution of the mean and covariance separately i.e.
+
+$$
+\int \mu _ { q } ( s ) \mathrm { K L } ( \pi _ { i } ( a | s , \pmb \theta ) , \pi ( a | s , \pmb \theta ) ) = C _ { \mu } + C _ { \Sigma } ,
+$$
+
+where
+
+$$
+\begin{array} { c } { { C _ { \mu } = \displaystyle \int \mu _ { q } ( s ) \frac 1 2 \big ( \mathrm { t r } \big ( \Sigma ^ { - 1 } \Sigma _ { i } \big ) - n + \mathrm { l n } ( \frac { \Sigma } { \Sigma _ { i } } ) \big ) d s , } } \\ { { { } } } \\ { { C _ { \Sigma } = \displaystyle \int \mu _ { q } ( s ) \frac 1 2 \big ( \mu - \mu _ { i } \big ) ^ { T } \Sigma ^ { - 1 } ( \mu - \mu _ { i } ) d s . } } \end{array}
+$$
+
+This decoupling allows us to set different $\epsilon$ values for each component, i.e., $\epsilon _ { \mu } , \epsilon _ { \Sigma }$ for the mean, the covariance matrix respectively. Different $\epsilon$ lead to different learning rates. The effectivness of this decoupling has also been shown in Abdolmaleki et al. (2017). We always set a much smaller epsilon for covariance than the mean. The intuition is that while we would like the distribution moves fast in the action space, we also want to keep the exploration to avoid premature convergence.
+
+In order to solve the constrained optimisation in the M-step, we first write the generalised Lagrangian equation, i.e,
+
+$$
+L ( \pmb \theta , \eta _ { \mu } , \eta _ { \Sigma } ) = \int \mu _ { q } ( s ) \int q ( a | s ) \log \pi ( a | s , \pmb \theta ) d a d s + \eta _ { \mu } ( \epsilon _ { \mu } - C _ { \mu } ) + \eta _ { \Sigma } ( \epsilon _ { \Sigma } - C _ { \Sigma } )
+$$
+
+Where $\eta _ { \mu }$ and $\eta _ { \Sigma }$ are Lagrangian multipliers. Following prior work on constraint optimisation, we formulate the following primal problem,
+
+$$
+\operatorname* { m a x } _ { \pmb { \theta } } \operatorname* { m i n } _ { \eta _ { \mu } > 0 , \eta _ { \Sigma } > 0 } L ( \pmb { \theta } , \eta _ { \mu } , \eta _ { \Sigma } ) .
+$$
+
+In order to solve for $\pmb \theta$ we iteratively solve the inner and outer optimisation programs independently: We fix the Lagrangian multipliers to their current value and optimise for $\pmb \theta$ (outer maximisation) and then fix the parameters $\pmb \theta$ to their current value and optimise for the Lagrangian multipliers (inner minimisation). We continue this procedure until policy parameters $\pmb { \theta }$ and Lagrangian multipliers converge. Please note that the same approach can be employed to bound the KL explicitly instead of decoupling the contribution of mean and covariance matrix to the KL.
+
+# D.4 PARAMETRIC VARIATIONAL DISTRIBUTION
+
+In this case we assume our variational distribution also uses a Gaussian distribution over the action space and use the same structure as our policy $\pi$ .
+
+Similar to the non-parametric case for a Gaussian distribution in the M-step we also use a decoupled KL but this time in the E-step for a Gaussian variational distribution. Using the same reasoning as in the previous section we can obtain the following generalized Lagrangian equation:
+
+$$
+L ( \theta ^ { q } , \eta _ { \mu } , \eta _ { \Sigma } ) = \int \mu _ { q } ( s ) \int q ( a | s ; \theta ^ { q } ) A _ { i } ( a , s ) d a d s + \eta _ { \mu } ( \epsilon _ { \mu } - C _ { \mu } ) + \eta _ { \Sigma } ( \epsilon _ { \Sigma } - C _ { \Sigma } ) .
+$$
+
+Where $\eta _ { \mu }$ and $\eta _ { \Sigma }$ are Lagrangian multipliers. And where we use the advantage function $A ( a , s )$ instead of the $\mathrm { Q }$ function $Q ( a , s )$ , as it empirically gave better performance. Please note that the KL in the $\mathrm { E }$ -step is different than the one used in the M-step. Following prior works on constraint optimisation, we can formulate the following primal problem,
+
+$$
+\operatorname* { m a x } _ { \pmb { \theta } ^ { q } } \operatorname* { m i n } _ { \eta _ { \mu } > 0 , \eta _ { \Sigma } > 0 } L ( \pmb { \theta } ^ { q } , \eta _ { \mu } , \eta _ { \Sigma } )
+$$
+
+In order to solve for $\pmb { \theta } ^ { q }$ we iteratively solve the inner and outer optimisation programs independently. In order to that we fix the Lagrangian multipliers to their current value and optimise for $\pmb { \theta } ^ { q }$ (outer maximisation), in this case we use the likelihood ratio gradient to compute the gradient w.r.t $\pmb { \theta } ^ { q }$ . Subsequently we fix the parameters $\pmb { \theta } ^ { q }$ to their current value and optimise for Lagrangian multipliers (inner minimisation). We iteratively continue this procedure until the policy parameters $\pmb { \theta } ^ { q }$ and the Lagrangian multipliers converges. Please note that the same approach can be used to bound the KL explicitly instead of decoupling the contribution of mean and covariance matrix to the KL. As our policy has the same structure as the parametric variational distribution, the M step in this case reduce to set the policy parameters $\pmb { \theta }$ to the parameters $\pmb { \theta } ^ { q }$ we obtained in E-step, i.e,
+
+$$
+\pmb \theta _ { i + 1 } = \pmb \theta ^ { q }
+$$
+
+# E IMPLEMENTATION DETAILS
+
+While we ran most of our experiments using a single learner, we implemented a scalable variant of the presented method in which multiple workers collect data independently in an instance of the considered environment, compute gradients and send them to a chief (or parameter server) that performs parameter update by averaging gradients. That is we use distributed synchronous gradient descent. These procedures are described in Algorithms 1 and 2 for the non-parametric case and 3 for the parametric case.
+
+# Algorithm 1 MPO (chief)
+
+1: Input $G$ number of gradients to average   
+2: while True do   
+3: initialize $\Nu = 0$   
+4: initialize gradient store $s _ { \phi } = \{ \} , s _ { \eta } = \{ \} , s _ { \eta _ { \mu } } = \{ \} , s _ { \eta _ { \Sigma } } = \{ \} \ : s _ { \theta } = \{ \}$   
+5: while $N < G$ do   
+6: receive next gradient from worker $w$   
+7: $\begin{array} { l } { { s _ { \phi } = s _ { \phi } + [ \bar { \delta } \phi ^ { w } ] } } \\ { { s _ { \phi } = s _ { \theta } + [ \delta \theta ^ { w } ] } } \\ { { s _ { \eta } = s _ { \eta } + [ \delta \eta ^ { w } ] } } \\ { { s _ { \eta _ { \mu } } = s _ { \eta _ { \mu } } + [ \delta \eta _ { \mu } ^ { w } ] } } \\ { { s _ { \eta _ { \theta } } = s _ { \eta _ { \theta } } + [ \delta \eta _ { \theta } ^ { w } ] } } \end{array}$   
+8:   
+9:   
+10:   
+11:   
+12: update parameters with average gradient from   
+13: $s _ { \phi }$ , sη, sηµ, sηΣ sθ   
+14: send new parameters to workers
+
+# Algorithm 2 MPO (worker) - Non parametric variational distribution
+
+1: Input = , Σ, µ, Lmax   
+2: $i = 0$ , $L _ { \mathrm { c u r r } } = 0$   
+3: Initialise $Q _ { \omega _ { i } } ( a , s ) , \pi ( a | s , \pmb { \theta } _ { i } ) , \eta , \eta _ { \mu } , \eta _ { \Sigma }$   
+4: for each worker do   
+5: while $L _ { \mathrm { c u r r } } > L _ { \mathrm { m a x } }$ do   
+6: update replay buffer $\boldsymbol { B }$ with $\mathrm { L }$ trajectories from the environment   
+7: $\bar { k = 0 }$   
+8: // Find better policy by gradient descent   
+9: while $k < 1 0 0 0$ do   
+10: sample a mini-batch $\boldsymbol { B }$ of $N$ $( s , a , r )$ pairs from replay   
+11: sample $M$ additional actions for each state from $\boldsymbol { B }$ , $\bar { \pi } ( \bar { a } | s , \pmb \theta _ { i } )$ for estimating integrals   
+12: compute gradients, estimating integrals using samples   
+13: // Q-function gradient:   
+14: $\delta _ { \phi } = \partial _ { \phi } L _ { \phi } ^ { \prime } ( \phi )$   
+15: // E-Step gradient:   
+16: $\delta \eta = \bar { \partial _ { \eta } g ( \eta ) }$   
+17: Let: q(a|s) ∝ π(a|s, θi) exp( Qθt (a,s,φ0)η )   
+18: // M-Step gradient:   
+19: $[ \delta _ { \eta _ { \mu } } , \delta _ { \eta _ { \Sigma } } ] \stackrel { - } { = } \alpha \partial _ { \eta _ { \mu } , \eta _ { \Sigma } } { \cal L } ( \theta _ { k } , \eta _ { \mu } , \eta _ { \Sigma } )$   
+20: $\delta _ { \theta } = \partial _ { \theta } L ( \theta , \eta _ { \mu _ { k + 1 } } , \eta _ { \Sigma k + 1 } )$   
+21: send gradients to chief worker   
+22: wait for gradient update by chief   
+23: fetch new parameters $\phi , \theta , \eta , \eta _ { \mu } , \eta _ { \Sigma }$   
+24: $\begin{array} { r } { k = k + 1 ^ { \circ } } \\ { i = i + 1 , L _ { \mathrm { c u r r } } = L _ { \mathrm { c u r r } } + L } \\ { \pmb { \theta } _ { i } = \pmb { \theta } , \phi ^ { \prime } = \phi } \end{array}$   
+25:   
+26:
+
+# Algorithm 3 MPO (worker) - parametric variational distribution
+
+1: Input = Σ, µ, Lmax   
+2: $i = 0$ , Lcurr = 0   
+3: Initialise $Q _ { \omega _ { i } } ( a , s ) , \pi ( a | s , \pmb { \theta } _ { i } ) , \eta , \eta _ { \mu } , \eta _ { \Sigma }$   
+4: for each worker do   
+5: while $L _ { \mathrm { c u r r } } < L _ { \mathrm { m a x } }$ do   
+6: update replay buffer $\boldsymbol { B }$ with $\mathrm { L }$ trajectories from the environment   
+7: $k = 0$   
+8: // Find better policy by gradient descent   
+9: while $k < 1 0 0 0$ do   
+10: sample a mini-batch $\boldsymbol { B }$ of $\mathrm { ~ V ~ } ( s , a , r )$ pairs from replay   
+11: sample $M$ additional actions for each state from $\boldsymbol { B }$ , $\dot { \pi } ( a | s , \theta _ { k } )$ for estimating inte  
+grals   
+12: compute gradients, estimating integrals using samples   
+13: // Q-function gradient:   
+14: $\delta _ { \phi } = \partial _ { \phi } L _ { \phi } ^ { \prime } ( \phi )$   
+15: // E-Step gradient:   
+16: $\begin{array} { r l } & { \bigl [ \delta _ { \eta _ { \mu } } , \delta _ { \eta _ { \Sigma } } \bigr ] ^ { \circ } = \alpha \partial _ { \eta _ { \mu } , \eta _ { \Sigma } } L ( \pmb \theta _ { k } , \eta _ { \mu } , \eta _ { \Sigma } ) } \\ & { \delta _ { \theta } = \partial _ { \theta } L ( \pmb \theta , \eta _ { \mu _ { k + 1 } } , \eta _ { \Sigma k + 1 , \eta } , } \\ & { \dots , \dots } \end{array}$   
+17:   
+18: // M-Step gradient: In practice there is no M-step in this case as policy and variatinal   
+distribution $q$ use a same structure.   
+19: send gradients to chief worker   
+20: wait for gradient update by chief   
+21: fetch new parameters $\phi , \theta , \eta , \eta _ { \mu } , \eta _ { \Sigma }$   
+22: $\begin{array} { r } { k = k + 1 ^ { \mathbf { \complement } } } \\ { i = i + 1 , L _ { \mathrm { c u r r } } = L _ { \mathrm { c u r r } } + L } \\ { \pmb { \theta } _ { i } = \pmb { \theta } , \phi ^ { \prime } = \phi } \end{array}$   
+23:   
+24:
\ No newline at end of file
diff --git a/parse/train/S1ANxQW0b/S1ANxQW0b_content_list.json b/parse/train/S1ANxQW0b/S1ANxQW0b_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..03b3945de3beb76355de4aae2fb91e3ba426fb56
--- /dev/null
+++ b/parse/train/S1ANxQW0b/S1ANxQW0b_content_list.json
@@ -0,0 +1,3294 @@
+[
+    {
+        "type": "text",
+        "text": "MAXIMUM A POSTERIORI POLICY OPTIMISATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
+            769,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abbas Abdolmaleki, Jost Tobias Springenberg, Yuval Tassa, Remi Munos,   \nNicolas Heess, Martin Riedmiller   \nDeepMind, London, UK   \n{aabdolmaleki,springenberg,tassa,munos,heess,riedmiller}@google.com ",
+        "bbox": [
+            184,
+            143,
+            841,
+            200
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            237,
+            544,
+            252
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We introduce a new algorithm for reinforcement learning called Maximum aposteriori Policy Optimisation (MPO) based on coordinate ascent on a relativeentropy objective. We show that several existing methods can directly be related to our derivation. We develop two off-policy algorithms and demonstrate that they are competitive with the state-of-the-art in deep reinforcement learning. In particular, for continuous control, our method outperforms existing methods with respect to sample efficiency, premature convergence and robustness to hyperparameter settings. ",
+        "bbox": [
+            233,
+            270,
+            764,
+            381
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            410,
+            336,
+            426
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Model free reinforcement learning algorithms can acquire sophisticated behaviours by interacting with the environment while receiving simple rewards. Recent experiments (Mnih et al., 2015; Jaderberg et al., 2016; Heess et al., 2017) successfully combined these algorithms with powerful deep neural-network approximators while benefiting from the increase of compute capacity. ",
+        "bbox": [
+            174,
+            443,
+            825,
+            498
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Unfortunately, the generality and flexibility of these algorithms comes at a price: They can require a large number of samples and – especially in continuous action spaces – suffer from high gradient variance. Taken together these issues can lead to unstable learning and/or slow convergence. Nonetheless, recent years have seen significant progress, with improvements to different aspects of learning algorithms including stability, data-efficiency and speed, enabling notable results on a variety of domains, including locomotion (Heess et al., 2017; Peng et al., 2016), multi-agent behaviour (Bansal et al., 2017) and classical control (Duan et al., 2016). ",
+        "bbox": [
+            174,
+            506,
+            825,
+            603
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Two types of algorithms currently dominate scalable learning for continuous control problems: First, Trust-Region Policy Optimisation (TRPO; Schulman et al. 2015) and the derivative family of Proximal Policy Optimisation algorithms (PPO; Schulman et al. 2017b). These policy-gradient algorithms are on-policy by design, reducing gradient variance through large batches and limiting the allowed change in parameters. They are robust, applicable to high-dimensional problems, and require moderate parameter tuning, making them a popular first choice (Ho & Ermon, 2016). However, as on-policy algorithms, they suffer from poor sample efficiency. ",
+        "bbox": [
+            174,
+            609,
+            825,
+            707
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In contrast, off-policy value-gradient algorithms such as the Deep Deterministic Policy Gradient (DDPG, Silver et al. 2014; Lillicrap et al. 2016), Stochastic Value Gradient (SVG, Heess et al. 2015), and the related Normalized Advantage Function formulation (NAF, Gu et al. 2016b) rely on experience replay and learned (action-)value functions. These algorithms exhibit much better data efficiency, approaching the regime where experiments with real robots are possible (Gu et al., 2016a; Andrychowicz et al., 2017). While also popular, these algorithms can be difficult to tune, especially for high-dimensional domains like general robot manipulation tasks. ",
+        "bbox": [
+            174,
+            714,
+            823,
+            811
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper we propose a novel off-policy algorithm that benefits from the best properties of both classes. It exhibits the scalability, robustness and hyperparameter insensitivity of on-policy algorithms, while offering the data-efficiency of off-policy, value-based methods. ",
+        "bbox": [
+            174,
+            819,
+            823,
+            861
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To derive our algorithm, we take advantage of the duality between control and estimation by using Expectation Maximisation (EM), a powerful tool from the probabilistic estimation toolbox, in order to solve control problems. This duality can be understood as replacing the question “what are the actions which maximise future rewards?” with the question “assuming future success in maximising rewards, what are the actions most likely to have been taken?”. By using this estimation objective we have more control over the policy change in both E and M steps, yielding robust learning. We show below that several algorithms, including TRPO, can be directly related to this perspective. We leverage the fast convergence properties of EM-style coordinate ascent by alternating a nonparametric data-based E-step which re-weights state-action samples, with a supervised, parametric M-step using deep neural networks. ",
+        "bbox": [
+            176,
+            867,
+            823,
+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
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+        "text": "We evaluate our algorithm on a broad spectrum of continuous control problems including a 56 DoF humanoid body. All experiments used the same optimisation hyperparameters 1. Our algorithm shows remarkable data efficiency often solving the tasks we consider an order of magnitude faster than the state-of-the-art. A video of some resulting behaviours can be found here dropbox.com/s/pgcmjst7t0zwm4y/MPO.mp4. ",
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+        "text": "2 BACKGROUND AND NOTATION",
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+        "text": "2.1 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Casting Reinforcement Learning (RL) as an inference problem has a long history dating back at least two decades (Dayan & Hinton, 1997). The framework presented here is inspired by a variational inference perspective on RL that has previously been utilised in multiple studies; c.f. Dayan & Hinton (1997); Neumann (2011); Deisenroth et al. (2013); Rawlik et al. (2012); Levine & Koltun (2013); Florensa et al. (2017). ",
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+        "type": "text",
+        "text": "Particular attention has been paid to obtaining maximum entropy policies as the solution to an inference problem. The penalisation of determinism can be seen encouraging both robustness and simplicity. Among these are methods that perform trajectory optimisation using either linearised dynamics (Todorov, 2008; Toussaint, 2009; Levine & Koltun, 2013) or general dynamics as in path integral control (Kappen, 2005; Theodorou et al., 2010). In contrast to these algorithms, here we do not assume the availability of a transition model and avoid on-policy optimisation. A number of other authors have considered the same perspective but in a model-free RL setting (Neumann, 2011; Peters et al., 2010a; Florensa et al., 2017; Daniel et al., 2016) or inverse RL problems (Ziebart et al., 2008). These algorithms are more directly related to our work and can be cast in the same (EM-like) alternating optimisation scheme on which we base our algorithm. However, they typically lack the maximisation (M)-step – with the prominent exception of REPS, AC-REPS, $\\mathrm { P \\bar { I } ^ { 2 } }$ -GPS and MDGPS (Peters et al., 2010a; Wirth et al., 2016; Chebotar et al., 2016; Montgomery & Levine, 2016) to which our algorithm is closely related as outlined below. An interesting recent addition to these approaches is an EM-perspective on the PoWER algorithm (Roux, 2016) which uses the same iterative policy improvement employed here, but commits to parametric inference distributions and avoids an exponential reward transformation, resulting in a harder to optimise lower bound. ",
+        "bbox": [
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+        "type": "text",
+        "text": "As an alternative to these policy gradient inspired algorithms, the class of recent algorithms for soft Q-learning (e.g. Rawlik et al. (2012); Haarnoja et al. (2017); Fox et al. (2016) parameterise and estimate a so called “soft” Q-function directly, implicitly inducing a maximum entropy policy. A perspective that can also be extended to hierarchical policies (Florensa et al., 2017), and has recently been used to establish connections between Q-learning and policy gradient methods (O’Donoghue et al., 2016; Schulman et al., 2017a). In contrast, we here rely on a parametric policy, our bound and derivation is however closely related to the definition of the soft (entropy regularised) Q-function. ",
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+    {
+        "type": "text",
+        "text": "A line of work, that is directly related to the “RL as inference” perspective, has focused on using information theoretic regularisers such as the entropy of the policy or the Kullback-Leibler divergence (KL) between policies to stabilise standard RL objectives. In fact, most state-of-the-art policy gradient algorithms fall into this category. For example see the entropy regularization terms used in Mnih et al. (2016) or the KL constraints employed by work on trust-region based methods (Schulman et al., 2015; 2017b; Gu et al., 2017; Wang et al., 2017). The latter methods introduce a trust region constraint, defined by the KL divergence between the new policy and the old policy, so that the expected KL divergence over state space is bounded. From the perspective of this paper these trust-region based methods can be seen as optimising a parametric E-step, as in our algorithm, but are “missing” an explicit M-step. ",
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+    {
+        "type": "text",
+        "text": "Finally, the connection between RL and inference has been invoked to motivate work on exploration. The most prominent examples for this are formed by work on Boltzmann exploration such as Kaelbling et al. (1996); Perkins & Precup (2002); Sutton (1990); O’Donoghue et al. (2017), which can be connected back to soft Q-learning (and thus to our approach) as shown in Haarnoja et al. (2017). ",
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+        "type": "text",
+        "text": "2.2 MARKOV DECISION PROCESSES ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We consider the problem of finding an optimal policy $\\pi$ for a discounted reinforcement learning (RL) problem; formally characterized by a Markov decision process (MDP). The MDP consists of: continuous states $s$ , actions $a$ , transition probabilities $p ( s _ { t + 1 } | s _ { t } , a _ { t } )$ – specifying the probability of transitioning from state $s _ { t }$ to $s _ { t + 1 }$ under action $a _ { t } \\textrm { -- }$ , a reward function $r ( s , a ) \\in \\mathbb { R }$ as well as the discounting factor $\\gamma \\in [ 0 , 1 )$ . The policy $\\pi ( a | s , \\theta )$ (with parameters $\\pmb { \\theta }$ ) is assumed to specify a probability distribution over action choices given any state and – together with the transition probabilities – gives rise to the stationary distribution $\\mu _ { \\pi } ( s )$ . ",
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+        "type": "text",
+        "text": "Using these basic quantities we can now define the notion of a Markov sequence or trajec$\\overline { { \\tau } } _ { \\pi } ~ = ~ \\{ ( s _ { 0 } , a _ { 0 } ) \\} \\ldots ( s _ { T } , a _ { T } ) \\}$ by following the policy ; and the expected return $\\pi$ $\\begin{array} { r } { \\tau _ { \\pi } ~ \\sim ~ p _ { \\pi } ( \\tau ) } \\end{array}$ with. We $\\begin{array} { r } { p _ { \\pi } ( \\tau ) = p ( s _ { 0 } ) \\prod _ { t > 0 } p ( s _ { t + 1 } \\vert s _ { t } , a _ { t } ) \\pi ( a _ { t } \\vert s _ { t } ) ; } \\end{array}$ $\\begin{array} { r } { \\mathbb { E } _ { \\tau _ { \\pi } } [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r ( s _ { t } , s _ { t } ) ] } \\end{array}$ will use the shorthand $r _ { t } = r ( s _ { t } , a _ { t } )$ . ",
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+        "type": "text",
+        "text": "3 MAXIMUM A POSTERIORI POLICY OPTIMISATION ",
+        "text_level": 1,
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+        "text": "Our approach is motivated by the well established connection between RL and probabilistic inference. This connection casts the reinforcement learning problem as that of inference in a particular probabilistic model. Conventional formulations of RL aim to find a trajectory that maximizes expected reward. In contrast, inference formulations start from a prior distribution over trajectories, condition a desired outcome such as achieving a goal state, and then estimate the posterior distribution over trajectories consistent with this outcome. ",
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+        "text": "A finite-horizon undiscounted reward formulation can be cast as inference problem by constructing a suitable probabilistic model via a likelihood function $p ( O = 1 | \\tau ) \\propto \\mathrm { e x p } ( \\Sigma _ { t } r _ { t } / \\alpha )$ , where $\\alpha$ is a temperature parameter. Intuitively, $O$ can be interpreted as the event of obtaining maximum reward by choosing an action; or the event of succeeding at the RL task (Toussaint, 2009; Neumann, 2011). With this definition we can define the following lower bound on the likelihood of optimality for the policy $\\pi$ : ",
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+        "img_path": "images/c585372fbc484f565501eb5fe988b323a1410d0d19fe5e98590a652145b9366d.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\log p _ { \\pi } ( O = 1 ) = \\log \\int p _ { \\pi } ( \\tau ) p ( O = 1 | \\tau ) d \\tau \\geq \\int q ( \\tau ) \\big [ \\log p ( O = 1 | \\tau ) + \\log \\frac { p _ { \\pi } ( \\tau ) } { q ( \\tau ) } \\big ] d \\tau } \\\\ { \\displaystyle \\qquad = \\mathbb { E } _ { q } \\Big [ \\sum _ { t } r _ { t } / \\alpha \\Big ] - \\mathrm { K L } \\Big ( q ( \\tau ) | | p _ { \\pi } ( \\tau ) \\Big ) = \\mathcal { I } ( q , \\pi ) , } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $p _ { \\pi }$ is the trajectory distribution induced by policy $\\pi ( a | s )$ as described in section 2.2 and $q ( \\tau )$ is an auxiliary distribution over trajectories that will discussed in more detail below. The lower bound $\\mathcal { I }$ is the evidence lower bound (ELBO) which plays an important role in the probabilistic modeling literature. It is worth already noting here that optimizing (2) with respect to $q$ can be seen as a regularized RL problem. ",
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+        "type": "text",
+        "text": "An important motivation for transforming a RL problem into an inference problem is that this allows us draw from the rich toolbox of inference methods: For instance, $\\mathcal { I }$ can be optimized with the familiy of expectation maximization (EM) algorithms which alternate between improving $\\mathcal { I }$ with respect to $q$ and $\\pi$ . In this paper we follow classical (Dayan & Hinton, 1997) and more recent works (e.g. Peters et al. 2010b; Levine & Koltun 2013; Daniel et al. 2016; Wirth et al. 2016) and cast policy search as a particular instance of this family. Our algorithm then combines properties of existing approaches in this family with properties of recent off-policy algorithms for neural networks. ",
+        "bbox": [
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+        "type": "text",
+        "text": "The algorithm alternates between two phases which we refer to as E and M step in reference to an EM-algorithm. The E-step improves $\\mathcal { I }$ with respect to $q$ . Existing EM policy search approaches perform this step typically by reweighting trajectories with sample returns (Kober & Peters, 2009) or via local trajectory optimization (Levine & Koltun, 2013). We show how off-policy deep RL techniques and value-function approximation can be used to make this step both scalable as well as data efficient. The M-step then updates the parametric policy in a supervised learning step using the reweighted state-action samples from the E-step as targets. ",
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+        "text": "These choices lead to the following desirable properties: (a) low-variance estimates of the expected return via function approximation; (b) low-sample complexity of value function estimate via robust off-policy learning; (c) minimal parametric assumption about the form of the trajectory distribution in the E-step; (d) policy updates via supervised learning in the M step; (e) robust updates via hard trust-region constraints in both the E and the M step. ",
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+        "text": "3.1 POLICY IMPROVEMENT ",
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+        "text": "The derivation of our algorithm then starts from the infinite-horizon analogue of the KL-regularized expected reward objective from Equation (2). In particular, we consider variational distributions $q ( \\tau )$ that factor in the same way as $p _ { \\pi }$ , i.e. $\\begin{array} { r } { q ( \\tau ) = \\bar { p } ( s _ { 0 } ) \\prod _ { t > 0 } p ( s _ { t + 1 } | s _ { t } , a _ { t } ) q ( a _ { t } | s _ { t } ) } \\end{array}$ which yields: ",
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+        "text": "$$\n\\mathcal { I } ( \\boldsymbol { q } , \\theta ) = \\mathbb { E } _ { \\boldsymbol { q } } \\Big [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\big [ r _ { t } - \\alpha \\mathbf { K L } \\big ( \\boldsymbol { q } ( \\boldsymbol { a } | s _ { t } ) \\| \\pi ( \\boldsymbol { a } | s _ { t } , \\theta ) \\big ) \\big ] \\Big ] + \\log p ( \\theta ) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Note that due to the assumption about the structure of $q ( \\tau )$ the $\\mathrm { K L }$ over trajectories decomposes into a KL over the individual state-conditional action distributions. This objective has also been considered e.g. by Haarnoja et al. (2017); Schulman et al. (2017a). The additional $\\log p ( \\pmb \\theta )$ term is a prior over policy parameters and can be motivated by a maximum a-posteriori estimation problem (see appendix for more details). ",
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+        "type": "text",
+        "text": "We also define the regularized Q-value function associated with (3) as ",
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+        "img_path": "images/94b398d8581f4b49760d89cd721f5bc7a45444c60483374fc1cfd24d10e845ba.jpg",
+        "text": "$$\nQ _ { \\theta } ^ { q } ( s , a ) = r _ { 0 } + \\mathbb { E } _ { q ( \\tau ) , s _ { 0 } = s , a _ { 0 } = a } \\left[ \\sum _ { t \\geq 1 } ^ { \\infty } \\gamma ^ { t } \\big [ r _ { t } - \\alpha \\mathrm { K L } ( q _ { t } \\| \\pi _ { t } ) \\big ] \\right] ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "with $\\mathrm { K L } \\big ( q _ { t } | | \\pi _ { t } \\big ) = \\mathrm { K L } \\big ( q ( a | s _ { t } ) \\big ) \\| \\pi ( a | s _ { t } , \\pmb { \\theta } ) \\big )$ . Note that $\\mathrm { K L } \\big ( q _ { 0 } | | \\pi _ { 0 } \\big )$ and $p ( \\theta )$ are not part of the Q-function as they are not a function of the action. ",
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+        "type": "text",
+        "text": "We observe that optimizing $\\mathcal { I }$ with respect to $q$ is equivalent to solving an expected reward RL problem with augmented reward $\\begin{array} { r } { \\tilde { r } _ { t } = r _ { t } - \\alpha \\log \\frac { q ( a _ { t } | s _ { t } ) } { \\pi ( a _ { t } | s _ { t } , \\pmb { \\theta } ) } } \\end{array}$ . In this view $\\pi$ represents a default policy towards which $q$ is regularized – i.e. the current best policy. The MPO algorithm treats $\\pi$ as the primary object of interest. In this case $q$ serves as an auxiliary distribution that allows optimizing $\\mathcal { I }$ via alternate coordinate ascent in $q$ and $\\pi _ { \\pmb { \\theta } }$ , analogous to the expectation-maximization algorithm in the probabilistic modelling literature. In our case, the E-step optimizes $\\mathcal { I }$ with respect to $q$ while the M-step optimizes $\\mathcal { I }$ with respect to $\\pi$ . Different optimizations in the $\\mathrm { E }$ -step and M-step lead to different algorithms. In particular, we note that for the case where $p ( \\pmb \\theta )$ is an uninformative prior a variant of our algorithm has a monotonic improvement guarantee as show in the Appendix A. ",
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+        "type": "text",
+        "text": "3.2 E-STEP ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In the E-step of iteration $i$ we perform a partial maximization of $\\mathcal { I } ( \\boldsymbol { q } , \\boldsymbol { \\theta } )$ with respect to $q$ given $\\theta = \\theta _ { i }$ . We start by setting $q = \\pi _ { \\pmb { \\theta } _ { i } }$ and estimate the unregularized action-value function: ",
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+        "text": "$$\nQ _ { \\theta _ { i } } ^ { q } ( s , a ) = Q _ { \\theta _ { i } } ( s , a ) = \\mathbb { E } _ { \\tau _ { \\pi _ { i } } , s _ { 0 } = s , a _ { 0 } = a } \\left[ \\sum _ { t } ^ { \\infty } \\gamma ^ { t } r _ { t } \\right] ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "since $\\mathrm { K L } ( q | | \\pi _ { i } ) = 0$ . In practice we estimate $Q _ { \\pmb { \\theta } _ { i } }$ from off-policy data (we refer to Section 4 for details about the policy evaluation step). This greatly increases the data efficiency of our algorithm. Given $Q _ { \\pmb { \\theta } _ { i } }$ we improve the lower bound $\\mathcal { I }$ w.r.t. $q$ by first expanding $Q _ { \\theta _ { i } } ( s , a )$ via the regularized Bellman operator $\\begin{array} { r } { T ^ { \\pi , q } = \\mathbb { E } _ { q ( a \\mid s ) } \\Big [ r ( s , a ) - \\alpha \\mathbf { K L } ( q \\| \\pi _ { i } ) + \\gamma \\mathbb { E } _ { p ( s ^ { \\prime } \\mid s , a ) } [ V _ { \\pmb { \\theta } _ { i } } ( s ^ { \\prime } ) ] \\Big ] . } \\end{array}$ , and optimize the “one-step” KL regularised objective ",
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+        "img_path": "images/99844cbc259f5753c811b6ea1e6c148c0f2e81f9d3e10080bfbaebbdccf26843.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathop { \\operatorname* { m a x } } _ { \\boldsymbol { q } } \\bar { \\mathcal { T } } _ { s } ( \\boldsymbol { q } , \\theta _ { i } ) = \\mathop { \\operatorname* { m a x } } _ { \\boldsymbol { q } } T ^ { \\pi , { q } } Q _ { \\theta _ { i } } ( s , \\boldsymbol { a } ) } \\\\ { \\displaystyle \\mathop { \\quad = } \\mathop { \\operatorname* { m a x } } _ { \\boldsymbol { q } } \\mathbb { E } _ { \\boldsymbol { \\mu } ( s ) } \\Big [ \\mathbb { E } _ { \\boldsymbol { q } ( \\cdot \\vert s ) } [ Q _ { \\theta _ { i } } ( s , \\boldsymbol { a } ) ] - \\alpha \\mathbf { K } \\mathbf { L } ( \\boldsymbol { q } \\| \\pi _ { i } ) \\Big ] , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "since $V _ { \\pmb { \\theta } _ { i } } ( s ) = \\mathbb { E } _ { q ( a | s } [ Q _ { \\pmb { \\theta } _ { i } } ( s , a ) ]$ and thus $Q _ { \\pmb { \\theta } _ { i } } ( s , a ) = r ( s , a ) + \\gamma V _ { \\pmb { \\theta } _ { i } } ( s ) .$ . ",
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+    {
+        "type": "text",
+        "text": "Maximizing Equation (6), thus obtaining $q _ { i } = \\arg \\operatorname* { m a x } \\bar { \\mathcal { I } } ( q , \\theta _ { i } )$ , does not fully optimize $\\mathcal { I }$ since we treat $Q _ { \\theta _ { i } }$ as constant with respect to $q$ . An intuitive interpretation $q _ { i }$ is that it chooses the softoptimal action for one step and then resorts to executing policy $\\pi$ . In the language of the EM algorithm this optimization implements a partial E-step. In practice we also choose $\\mu _ { q }$ to be the stationary distribution as given through samples from the replay buffer. ",
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+        "type": "text",
+        "text": "CONSTRAINED E-STEP ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The reward and the KL terms are on an arbitray relative scale. This can make it difficult to choose $\\alpha$ . We therefore replace the soft KL regularization with a hard constraint with parameter $\\epsilon$ , i.e, ",
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+        "text": "$$\n\\begin{array} { r l } & { \\underset { q } { \\operatorname* { m a x } } \\mathbb { E } _ { \\boldsymbol { \\mu } ( s ) } \\left[ \\mathbb { E } _ { q ( a \\mid s ) } \\left[ Q _ { \\boldsymbol { \\theta } _ { i } } ( s , a ) \\right] \\right] } \\\\ & { s . t . \\mathbb { E } _ { \\boldsymbol { \\mu } ( s ) } \\left[ { \\mathbf { K L } } ( q ( a \\mid s ) , \\pi ( a \\mid s , \\boldsymbol { \\theta } _ { i } ) ) \\right] < \\epsilon . } \\end{array}\n$$",
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+        "text": "If we choose to explicitly parameterize $q ( a | s ) -$ option 1 below – the resulting optimisation is similar to that performed by the recent TRPO algorithm for continuous control (Schulman et al., 2015); only in an off-policy setting. Analogously, the unconstrained objective (6) is similar to the objective used by PPO (Schulman et al., 2017b). We note, however, that the KL is reversed when compared to the KL used by TRPO and PPO. ",
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+        "type": "text",
+        "text": "To implement (7) we need to choose a form for the variational policy $q ( a | s )$ . Two options arise: ",
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+        "type": "text",
+        "text": "1. We can use a parametric variational distribution $q ( a | s , \\pmb \\theta ^ { q } )$ , with parameters $\\pmb { \\theta } ^ { q }$ , and optimise Equation (7) via the likelihood ratio or action-value gradients. This leads to an algorithm similar to TRPO/PPO and an explicit M-step becomes unnecessary (see. Alg. 3). 2. We can choose a non-parametric representation of $q ( a | s )$ given by one probability factor per sample. To achieve generalization in state space we then fit a parametric policy in the M-step. ",
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+    {
+        "type": "text",
+        "text": "Fitting a parametric policy in the M-step is a supervised learning problem, allowing us to employ various regularization techniques at that point. It also makes it easier to enforce the hard KL constraint. ",
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+    {
+        "type": "text",
+        "text": "NON PARAMETRIC VARIATIONAL DISTRIBUTION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the non-parametric case we can obtain the optimal sample based $q$ distribution – the solution to Equation (7) – in closed form (see the appendix for a full derivation), as, ",
+        "bbox": [
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+        "img_path": "images/e2cc243c4fce56c4fd8e0417866e6c3dfe80f34b88867a5673bc94a2ae845caa.jpg",
+        "text": "$$\nq _ { i } ( a | s ) \\propto \\pi ( a | s , \\pmb { \\theta } _ { i } ) \\exp \\Big ( \\frac { Q _ { \\theta _ { i } } ( s , a ) } { \\eta ^ { * } } \\Big ) ,\n$$",
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+        "type": "text",
+        "text": "where we can obtain $\\eta ^ { * }$ by minimising the following convex dual function, ",
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+        "img_path": "images/c92f4e82fd7f9b818244b56be050da21e0f55ba3880b30c6c94bfd61a49caf39.jpg",
+        "text": "$$\ng ( \\eta ) = \\eta \\epsilon + \\eta \\int \\mu ( s ) \\log \\int \\pi ( a | s , \\pmb { \\theta } _ { i } ) \\exp \\Big ( \\frac { Q _ { \\theta _ { i } } ( s , a ) } { \\eta } \\Big ) d a d s ,\n$$",
+        "text_format": "latex",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "after the optimisation of which we can evaluate $q _ { i } ( a | s )$ on given samples. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This optimization problem is similar to the one solved by relative entropy policy search (REPS) (Peters et al., 2010a) with the difference that we optimise only for the conditional variational distribution $q ( a | s )$ instead of a joint distribution $\\scriptstyle q ( a , s )$ – effectively fixing $\\mu _ { q } ( s )$ to the stationary distribution given by previously collected experience – and we use the $\\mathrm { Q }$ function of the old policy to evaluate the integral over $a$ . While this might seem unimportant it is crucial as it allows us to estimate the integral over actions with multiple samples without additional environment interaction. This greatly reduces the variance of the estimate and allows for fully off-policy learning at the cost of performing only a partial optimization of $\\mathcal { I }$ as described above. ",
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+    {
+        "type": "text",
+        "text": "3.3 M-STEP ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Given $q _ { i }$ from the E-step we can optimize the lower bound $\\mathcal { I }$ with respect to $\\pmb \\theta$ to obtain an updated policy $\\pmb { \\theta } _ { i + 1 } = \\arg \\operatorname* { m a x } _ { \\pmb { \\theta } } \\mathcal { I } ( q _ { i } , \\pmb { \\theta } )$ . Dropping terms independent of $\\pmb { \\theta }$ this entails solving for the solution of ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/126b9a3c732b4985eaa8d9188c13a3d13893c384351538cd65543e255a10c66f.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\theta } \\mathcal { I } ( q _ { i } , \\theta ) = \\operatorname* { m a x } _ { \\theta } \\mathbb { E } _ { \\mu _ { q } ( s ) } \\Big [ \\mathbb { E } _ { q ( a | s ) } \\Big [ \\log \\pi ( a | s , \\theta ) \\Big ] \\Big ] + \\log p ( \\theta ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "which corresponds to a weighted maximum a-posteriroi estimation (MAP) problem where samples are weighted by the variational distribution from the E-step. Since this is essentially a supervised learning step we can choose any policy representation in combination with any prior for regularisation. In this paper we set $p ( \\pmb \\theta )$ to a Gaussian prior around the current policy, i.e, $\\begin{array} { r } { p ( \\pmb { \\theta } ) \\approx \\mathcal { N } \\Big ( \\mu = \\pmb { \\theta } _ { i } , \\Sigma = \\frac { F _ { \\pmb { \\theta } _ { i } } } { \\lambda } \\Big ) } \\end{array}$ , where $\\pmb \\theta _ { i }$ are the parameters of the current policy distribution, $F _ { \\pmb { \\theta } _ { i } }$ is the empirical Fisher information matrix and $\\lambda$ is a positive scalar. As shown in the appendix this suggests the following generalized M-step: ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/e3414a54c816aac2322f74663302b60046b419e9c4fa26ac9745e0714899d392.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\pi } \\mathbb { E } _ { \\mu _ { q } ( s ) } \\Big [ \\mathbb { E } _ { q ( a | s ) } \\Big [ \\log \\pi ( a | s , \\pmb { \\theta } ) \\Big ] - \\lambda \\mathrm { K L } \\Big ( \\pi ( a | s , \\pmb { \\theta } _ { i } ) , \\pi ( a | s , \\pmb { \\theta } ) \\Big ) \\Big ]\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "which can be re-written as the hard constrained version: ",
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+        "img_path": "images/326bacdbf4aabc3a5b055834d5937f2e8ed54ad75664f7b06e32c83160d55f29.jpg",
+        "text": "$$\n\\begin{array} { r } { \\underset { \\pi } { \\operatorname* { m a x } } \\mathbb { E } _ { \\mu _ { q } ( s ) } \\left[ \\mathbb { E } _ { q ( a | s ) } \\left[ \\log \\pi ( a | s , \\pmb { \\theta } ) \\right] \\right] } \\\\ { s . t . \\mathbb { E } _ { \\mu _ { q } ( s ) } \\left[ { \\mathbf { K L } } ( \\pi ( a | s , \\pmb { \\theta } _ { i } ) , \\pi ( a | s , \\pmb { \\theta } ) ) \\right] < \\epsilon . } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "This additional constraint minimises the risk of overfitting the samples, i.e. it helps us to obtain a policy that generalises beyond the state-action samples used for the optimisation. In practice we have found the KL constraint in the M step to greatly increase stability of the algorithm. We also note that in the E-step we are using the reverse, mode-seeking, KL while in the M-step we are using the forward, moment-matching, KL which reduces the tendency of the entropy of the parametric policy to collapse. This is in contrast to other RL algorithms that use M-projection without KL constraint to fit a parametric policy (Peters et al., 2010a; Wirth et al., 2016; Chebotar et al., 2016; Montgomery & Levine, 2016). Using KL constraint in M-step has also been shown effective for stochastic search algorithms (Abdolmaleki et al., 2017). ",
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+    {
+        "type": "text",
+        "text": "4 POLICY EVALUATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our method is directly applicable in an off-policy setting. For this, we have to rely on a stable policy evaluation operator to obtain a parametric representation of the Q-function $Q _ { \\theta } ( s , a )$ . We make use of the policy evaluation operator from the Retrace algorithm Munos et al. (2016), which we found to yield stable policy evaluation in practice2. Concretely, we fit the Q-function $Q _ { \\theta _ { i } } ( s , a , \\phi )$ as represented by a neural network, with parameters $\\phi$ , by minimising the squared loss: ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/ddd41d4e1eeeddfb6b26cf9d237375ea93c7ad203c8bd727946d1235131b9f6b.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathop { \\operatorname* { m i n } } _ { \\phi } L ( \\phi ) = \\mathop { \\operatorname* { m i n } } _ { \\phi } \\mathbb { E } _ { \\mu _ { b } ( s ) , b ( a | s ) } \\left[ \\left( Q _ { \\theta _ { i } } ( s _ { t } , a _ { t } , \\phi ) - Q _ { t } ^ { \\mathrm { r e t } } \\right) ^ { 2 } \\right] , \\mathrm { w i t h } } \\\\ { \\displaystyle Q _ { t } ^ { \\mathrm { r e t } } = Q _ { \\phi ^ { \\prime } } ( s _ { t } , a _ { t } ) + \\sum _ { j = t } ^ { \\infty } \\gamma ^ { j - t } \\Big ( \\prod _ { k = t + 1 } ^ { j } c _ { k } \\Big ) \\left[ r ( s _ { j } , a _ { j } ) + \\mathbb { E } _ { \\pi ( a | s _ { j + 1 } ) } [ Q _ { \\phi ^ { \\prime } } ( s _ { j + 1 } , a ) ] - Q _ { \\phi ^ { \\prime } } ( s _ { j } , a _ { j } ) \\right] , } \\\\ { \\displaystyle c _ { k } = \\operatorname* { m i n } \\Big ( 1 , \\frac { \\pi ( a _ { k } | s _ { k } ) } { b ( a _ { k } | s _ { k } ) } \\Big ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $Q _ { \\phi ^ { \\prime } } ( s , a )$ denotes the output of a target Q-network, with parameters $\\phi ^ { \\prime }$ , that we copy from the current parameters $\\phi$ after each M-step. We truncate the infinite sum after $N$ steps by bootstrapping with $Q _ { \\phi ^ { \\prime } }$ (rather than considering a $\\lambda$ return). Additionally, $b ( a | s )$ denotes the probabilities of an arbitrary behaviour policy. In our case we use an experience replay buffer and hence $b$ is given by the action probabilities stored in the buffer; which correspond to the action probabilities at the time of action selection. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "For our experiments we evaluate our MPO algorithm across a wide range of tasks. Specifically, we start by looking at the continuous control tasks of the DeepMind Control Suite (Tassa et al. (2018), see Figure 1), and then consider the challenging parkour environments recently published in Heess et al. (2017). In both cases we use a Gaussian distribution for the policy whose mean and covariance are parameterized by a neural network (see appendix for details). In addition, we present initial experiments for discrete control using ATARI environments using a categorical policy distribution (whose logits are again parameterized by a neural network) in the appendix. ",
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+    {
+        "type": "text",
+        "text": "5.1 EVALUATION ON CONTROL SUITE ",
+        "text_level": 1,
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+    {
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+        "img_path": "images/754c06df74f11ccb9772163bdb2d3c935cb2cba2b5b65def9b07cd08ab62b078.jpg",
+        "image_caption": [
+            "Figure 1: Control Suite domains used for benchmarking. Top: Acrobot, Ball-in-cup, Cart-pole, Cheetah, Finger, Fish, Hopper. Bottom: Humanoid, Manipulator, Pendulum, Point-mass, Reacher, Swimmers (6 and 15 links), Walker. "
+        ],
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+    {
+        "type": "text",
+        "text": "The suite of continuous control tasks that we are evaluating against contains 18 tasks, comprising a wide range of domains including well known tasks from the literature. For example, the classical cart-pole and acrobot dynamical systems, 2D and Humanoid walking as well as simple low-dimensional planar reaching and manipulation tasks. This suite of tasks was built in python on top of mujoco and will also be open sourced to the public by the time of publication. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While we include plots depicting the performance of our algorithm on all tasks below; comparing it against the state-of-the-art algorithms in terms of data-efficiency. We want to start by directing the attention of the reader to a more detailed evaluation on three of the harder tasks from the suite. ",
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+    {
+        "type": "text",
+        "text": "5.1.1 DETAILED ANALYSIS ON WALKER-2D, ACROBOT, HOPPER ",
+        "bbox": [
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+        "type": "text",
+        "text": "We start by looking at the results for the classical Acrobot task (two degrees of freedom, one continuous action dimension) as well as the 2D walker (which has 12 degrees of freedom and thus a 12 dimensional action space and a 21 dimensional state space) and the hopper standing task. The reward in the Acrobot task is the distance of the robots end-effector to an upright position of the underactuated system. For the walker task it is given by the forward velocity, whereas in the hopper the requirement is to stand still. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 2 shows the results for this task obtained by applying our algorithm MPO as well as several ablations – in which different parts were removed from the MPO optimization – and two baselines: our implementation of Proximal Policy Optimization (PPO) (Schulman et al., 2017b) and DDPG. The hyperparameters for MPO were kept fixed for all experiments in the paper (see the appendix for hyperparameter settings). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As a first observation, we can see that MPO gives stable learning on all tasks and, thanks to its fully off-policy implementation, is significantly more sample efficient than the on-policy PPO baseline. Furthermore, we can observe that changing from the non-parametric variational distribution to a parametric distribution3 (which, as described above, can be related to PPO) results in only a minor asymptotic performance loss but slowed down optimisation and thus hampered sample efficiency; which can be attributed to the fact that the parametric $q$ distribution required a stricter KL constraint. Removing the automatically tuned KL constraint and replacing it with a manually set entropy regulariser then yields an off-policy actor-critic method with Retrace. This policy gradient method still uses the idea of estimating the integral over actions – and thus, for a gradient based optimiser, its likelihood ratio derivative – via multiple action samples (as judged by a Q-Retrace critic). This idea has previously been coined as using the expected policy gradient (EPG) (Ciosek & Whiteson, 2017) and we hence denote the corresponding algorithm with EPG $^ +$ Retrace, which no-longer follows the intuitions of the MPO perspective. $\\mathrm { E P G } + .$ Retrace performed well when the correct entropy regularisation scale is used. This, however, required task specific tuning (c.f. Figure 4 where this hyperparameter was set to the one that performed best in average across tasks). Finally using only a single sample to estimate the integral (and hence the likelihood ratio gradient) results in an actor-critic variant with Retrace that is the least performant off-policy algorithm in our comparison. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/c77b362823bc448bf6268c862a2c6ae8c0cbe92c11eed9cb0fed52cbb3223fbf.jpg",
+        "image_caption": [
+            "Figure 2: Ablation study of the MPO algorithm and comparison to common baselines from the literature on three domains from the control suite. We plot the median performance over 10 experiments with different random seeds. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5.1.2 COMPLETE RESULTS ON THE CONTROL SUITE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The results for MPO (non-parameteric) – and a comparison to an implementation of state-of-the-art algorithms from the literature in our framework – on all the environments from the control suite that we tested on are shown in Figure 4. All tasks have rewards that are scaled to be between 0 and 1000. We note that in order to ensure a fair comparison all algorithms ran with exactly the same network configuration, used a single learner (no distributed computation), used the same optimizer and were tuned w.r.t. their hyperparameters for best performance across all tasks. We refer to the appendix for a complete description of the hyperparameters. Our comparison is made in terms of data-efficiency. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "From the plot a few trends are readily apparent: i) We can clearly observe the advantage in terms of data-efficiency that methods relying on a Q-critic obtain over the PPO baseline. This difference is so extreme that in several instances the PPO baseline converges an order of magnitude slower than the off-policy algorithms and we thus indicate the asymptotic performance of each algorithm of PPO and DDPG (which also improved significantly later during training in some instances) with a colored star in the plot; ii) the difference between the MPO results and the (expected) policy gradient (EPG) with entropy regularisation confirm our suspicion from Section 5.1.1: finding a good setting for the entropy regulariser that transfers across environments without additional constraints on the policy distribution is very difficult, leading to instabilities in the learning curves. In contrast to this the MPO results appear to be stable across all environments; iii) Finally, in terms of data-efficiency the methods utilising Retrace obtain a clear advantage over DDPG. The single learner vanilla DDPG implementation learns the lower dimensional environments quickly but suffers in terms of learning speed in environments with sparse rewards (finger, acrobot) and higher dimensional action spaces. Overall, MPO is able to solve all environments using surprisingly moderate amounts of data. On average less than 1000 trajectories (or $1 0 ^ { 6 }$ samples) are needed to reach the best performance. ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "5.2 HIGH-DIMENSIONAL CONTINUOUS CONTROL ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Next we turn to evaluating our algorithm on two higher-dimensional continuous control problems; humanoid and walker. To make computation time bearable in these more complicated domains we utilize a parallel variant of our algorithm: in this implementation K learners are all independently collecting data from an instance of the environment. Updates are performed at the end of each collected trajectory using distributed synchronous gradient descent on a shared set of policy and Q-function parameters (we refer to the appendix for an algorithm description). The results of this experiment are depicted in Figure 3. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "For the Humanoid running domain we can observe a similar trend to the experiments from the previous section: MPO quickly finds a stable running policy, outperforming all other algorithms in terms of sample efficiency also in this high-dimensional control problem. ",
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+    {
+        "type": "text",
+        "text": "The case for the Walker-2D parkour domain (where we compare against a PPO baseline) is even more striking: where standard PPO requires approximately 1M trajectories to find a good policy MPO finds a solution that is asymptotically no worse than the PPO solution in in about 70k trajectories (or 60M samples), resulting in an order of magnitude improvement. In addition to the walker experiment we have also evaluated MPO on the Parkour domain using a humanoid body (with 22 degrees of freedom) which was learned successfully (not shown in the plot, please see the supplementary video). ",
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+        "img_path": "images/3f0db1cdbd6cb97bb8fdb4b7ec7943a5ae55699354d37055db91d206400596e2.jpg",
+        "image_caption": [
+            "Figure 3: MPO on high-dimensional control problems (Parkour Walker2D and Humanoid walking from control suite). "
+        ],
+        "image_footnote": [],
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+        "img_path": "images/b39b3f4bf39eeac87f2d9d9433c59fe69a73415f4747eb95d2ef18eb773ea4b6.jpg",
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+    {
+        "type": "text",
+        "text": "5.3 DISCRETE CONTROL ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "As a proof of concept – showcasing the robustness of our algorithm and its hyperparameters – we performed an experiment on a subset of the games contained contained in the \"Arcade Learning Environment\" (ALE) where we used the same hyperparameter settings for the KL constraints as for the continuous control experiments. The results of this experiment can be found in the Appendix. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We have presented a new off-policy reinforcement learning algorithm called Maximum a-posteriori Policy Optimisation (MPO). The algorithm is motivated by the connection between RL and inference and it consists of an alternating optimisation scheme that has a direct relation to several existing algorithms from the literature. Overall, we arrive at a novel, off-policy algorithm that is highly data efficient, robust to hyperparameter choices and applicable to complex control problems. We demonstrated the effectiveness of MPO on a large set of continuous control problems. ",
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+        "image_caption": [
+            "Figure 4: Complete comparison of results for the control suite. We plot the median performance over 10 random seeds together with 5 and $95 \\%$ quantiles (shaded area). Note that for DDPG we only plot the median to avoid clutter in the plots. For DDPG and PPO final performance is marked by a star). "
+        ],
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+    {
+        "type": "text",
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+        "text": "In this section we prove a monotonic improvement guarantee for KL-regularized policy optimization via alternating updates on $\\pi$ and $q$ under the assumption that the prior on $\\theta$ is uninformative. ",
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+        "type": "text",
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+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Let $\\pi$ be an arbitrary policy. For any other policy $q$ such that, for all $x , a , \\{ \\pi ( a | x ) > 0 \\} \\implies$ $\\{ q ( a | x ) > 0 \\}$ , define the $\\pi$ -regularized reward for policy $q$ : ",
+        "bbox": [
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+        "page_idx": 13
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+    {
+        "type": "equation",
+        "img_path": "images/19b71895a29fda80f1b8591fef2b5e192fca3e05c9ba486025caa6f7d721c698.jpg",
+        "text": "$$\nr _ { \\alpha } ^ { \\pi , q } ( x , a ) = r ( x , a ) - \\alpha \\log \\frac { q ( a | x ) } { \\pi ( a | x ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha > 0$ . ",
+        "bbox": [
+            174,
+            304,
+            264,
+            319
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Bellman operators: Define the $\\pi$ -regularized Bellman operator for policy $q$ ",
+        "bbox": [
+            173,
+            335,
+            686,
+            352
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/508ad5a3c5fa9928bfe550dfd50ee50545843b2d286cf673ab0ef7fbced2c085.jpg",
+        "text": "$$\n\\begin{array} { r } { T _ { \\alpha } ^ { \\pi , q } V ( x ) = \\mathbb { E } _ { a \\sim q ( \\cdot | x ) } \\Big [ r _ { \\alpha } ^ { \\pi , q } ( x , a ) + \\gamma \\mathbb { E } _ { y \\sim p ( \\cdot | x , a ) } V ( y ) \\Big ] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            312,
+            359,
+            683,
+            386
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "and the non-regularized Bellman operator for policy $q$ ",
+        "bbox": [
+            174,
+            393,
+            529,
+            409
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/681f737615b202146cd9513dce7825ec03b9da0f1e4390b5dfe43f695e4a0eed.jpg",
+        "text": "$$\nT ^ { q } V ( x ) = \\mathbb { E } _ { a \\sim q ( \\cdot | x ) } \\Big [ r ( x , a ) + \\gamma \\mathbb { E } _ { y \\sim p ( \\cdot | x , a ) } V ( y ) \\Big ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            326,
+            416,
+            669,
+            444
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Value function: Define the $\\pi$ -regularized value function for policy $q$ as ",
+        "bbox": [
+            173,
+            458,
+            653,
+            473
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4f5602e99a52b3bbbfced2c9d8c1c6cec1445d1690ec985743dde120900363df.jpg",
+        "text": "$$\nV _ { \\alpha } ^ { \\pi , q } ( \\boldsymbol { x } ) = \\mathbb { E } _ { q } \\Big [ \\sum _ { t \\geq 0 } \\gamma ^ { t } r _ { \\alpha } ^ { \\pi , q } ( \\boldsymbol { x } _ { t } , a _ { t } ) | \\boldsymbol { x } _ { 0 } = \\boldsymbol { x } , q \\Big ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            481,
+            650,
+            517
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "and the non-regularized value function ",
+        "bbox": [
+            173,
+            525,
+            429,
+            540
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2f7fd48b4d6c6c7288755235956d88617509e88daeb0635db7ca46060c49b50b.jpg",
+        "text": "$$\nV ^ { q } ( x ) = \\mathbb { E } _ { q } \\Big [ \\sum _ { t \\geq 0 } \\gamma ^ { t } r ( x _ { t } , a _ { t } ) | x _ { 0 } = x , q \\Big ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            361,
+            547,
+            635,
+            585
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. For any $q , \\pi , V$ , we have $V _ { \\alpha } ^ { \\pi , q } \\leq V ^ { q }$ and $T _ { \\alpha } ^ { \\pi , q } V \\le T ^ { q } V$ . Indeed ",
+        "bbox": [
+            173,
+            592,
+            705,
+            608
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a64cd95e6d312561d3379d282bc2d9cd023b8307018318681d5a181e94947f9e.jpg",
+        "text": "$$\n\\mathbb { E } _ { q } \\Big [ \\log \\frac { q \\big ( a _ { t } | x _ { t } \\big ) } { \\pi \\big ( a _ { t } | x _ { t } \\big ) } \\Big ] = K L \\big ( q ( \\cdot | x _ { t } ) \\| \\pi ( \\cdot | x _ { t } ) \\big ) \\geq 0 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
+            616,
+            658,
+            650
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Optimal value function and policy Define the optimal regularized value function: $V _ { \\alpha } ^ { \\pi , * } ( x ) =$ $\\operatorname* { m a x } _ { q } V _ { \\alpha } ^ { \\pi , q } ( x )$ , and the optimal (non-regularized) value function: $V ^ { * } ( x ) = \\operatorname* { m a x } _ { q } V ^ { q } ( x )$ . ",
+        "bbox": [
+            171,
+            665,
+            823,
+            695
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The optimal policy of the $\\pi$ -regularized problem $q _ { \\alpha } ^ { \\pi , * } ( \\cdot | x ) = \\arg \\operatorname* { m a x } _ { q } V _ { \\alpha } ^ { \\pi , q } ( x )$ and the optimal policy of the non-regularized problem $q ^ { * } ( \\cdot | x ) = \\arg \\operatorname* { m a x } _ { q } V ^ { q }$ . ",
+        "bbox": [
+            171,
+            700,
+            823,
+            731
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2. We have that $V _ { \\alpha } ^ { \\pi , q }$ is the unique fixed point of $T _ { \\alpha } ^ { \\pi , q }$ , and $V ^ { q }$ is the unique fixed point of $T ^ { q }$ . Thus we have the following Bellman equations: For all $x \\in X$ , ",
+        "bbox": [
+            171,
+            734,
+            820,
+            763
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/66a75351c24b47e29e8ad0475f6feb00abfe1985e8f74a3c359f94cd4b27f79c.jpg",
+        "text": "$$\n\\begin{array} { r c l } { { V _ { \\alpha } ^ { \\pi , q } ( x ) } } & { { = } } & { { \\displaystyle \\sum _ { a } q ( a | x ) \\Big [ r _ { \\alpha } ^ { \\pi , q } ( x , a ) + \\gamma \\mathbb { E } _ { y \\sim p ( \\cdot | x , a ) } \\big [ V _ { \\alpha } ^ { \\pi , q } ( y ) \\big ] \\Big ] } } \\\\ { { } } & { { } } & { { } } \\\\ { { V ^ { q } ( x ) } } & { { = } } & { { \\displaystyle \\sum _ { a } q ( a | x ) \\Big [ r ( x , a ) + \\gamma \\mathbb { E } _ { y \\sim p ( \\cdot | x , a ) } \\big [ V ^ { q } ( y ) \\big ] \\Big ] } } \\\\ { { } } & { { } } & { { } } \\\\ { { V _ { \\alpha } ^ { \\pi , * } ( x ) } } & { { = } } & { { \\displaystyle r _ { \\alpha } ^ { \\pi , q _ { \\alpha } ^ { \\pi , * } } ( x , a ) + \\gamma \\mathbb { E } _ { y \\sim p ( \\cdot | x , a ) } \\big [ V _ { \\alpha } ^ { \\pi , * } ( y ) \\big ] f o r a l l a \\in A , } } \\\\ { { } } & { { } } & { { } } \\\\ { { V ^ { * } ( x ) } } & { { = } } & { { \\displaystyle \\operatorname* { m a x } _ { a \\in A } \\Big [ r ( x , a ) + \\gamma \\mathbb { E } _ { y \\sim p ( \\cdot | x , a ) } \\big [ V ^ { * } ( y ) \\big ] \\Big ] . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            770,
+            720,
+            895
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Notice that (16) holds for all actions $a \\in A$ , and not in expectation w.r.t. $a \\sim q ( \\cdot | x )$ only. ",
+        "bbox": [
+            168,
+            909,
+            754,
+            925
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.2 REGULARIZED JOINT POLICY GRADIENT ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            498,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We now consider a parametrized policy $\\pi _ { \\theta }$ and consider maximizing the regularized joint policy optimization problem for a given initial state $x _ { 0 }$ (this could be a distribution over initial states). Thus we want to find a parameter $\\theta$ that (locally) maximizes ",
+        "bbox": [
+            173,
+            128,
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+            171
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5fb3af47931f1e293c60f069a4937af38754fd0e6032f29c13b2c5a0ed8f9a2c.jpg",
+        "text": "$$\n\\mathcal { I } ( \\theta , q ) = V ^ { \\pi _ { \\theta } , q } ( \\boldsymbol { x } _ { 0 } ) = \\mathbb { E } _ { q } \\Big [ \\sum _ { t \\geq 0 } \\gamma ^ { t } \\big ( r ( \\boldsymbol { x } _ { t } , a _ { t } ) - \\alpha \\mathrm { K L } ( q ( \\cdot | \\boldsymbol { x } _ { t } ) | | \\pi _ { \\theta } ( \\cdot | \\boldsymbol { x } _ { t } ) ) \\big ) \\big | \\boldsymbol { x } _ { 0 } , q \\Big ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            235,
+            175,
+            761,
+            213
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We start with an initial parameter $\\theta _ { 0 }$ and define a sequence of policies $\\pi _ { i } = \\pi _ { \\theta _ { i } }$ parametrized by $\\theta _ { i }$ , in the following way: ",
+        "bbox": [
+            171,
+            223,
+            825,
+            252
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "• Given $\\theta _ { i }$ , define ",
+        "bbox": [
+            215,
+            262,
+            339,
+            279
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a231be82b298c49b858d95221233146488ec6ad66e7a6e4f2f7085b7f45643eb.jpg",
+        "text": "$$\nq _ { i } = \\arg \\operatorname* { m a x } _ { q } T _ { \\alpha } ^ { \\pi _ { \\theta _ { i } } , q } V ^ { \\pi _ { \\theta _ { i } } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            439,
+            273,
+            614,
+            299
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "• Define $\\theta _ { i + 1 }$ as ",
+        "bbox": [
+            215,
+            303,
+            331,
+            319
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1bee90d5ff4b76d736775fa10d00f95f791d305affcfca00910bdf2205cf6e2f.jpg",
+        "text": "$$\n\\theta _ { i + 1 } = \\theta _ { i } - \\beta \\nabla _ { \\theta } \\mathbb { E } _ { \\pi _ { i } } \\left[ \\sum _ { t \\geq 0 } \\gamma ^ { t } \\mathrm { K L } \\big ( q _ { k } ( \\cdot | x _ { t } ) | | \\pi _ { \\theta } ( \\cdot | x _ { t } ) \\big ) _ { | \\theta = \\theta _ { i } } \\big | x _ { 0 } , \\pi _ { i } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            321,
+            745,
+            359
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3. We have the following properties: ",
+        "bbox": [
+            173,
+            359,
+            501,
+            376
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "• The policy $q _ { i }$ satisfies: ",
+        "bbox": [
+            215,
+            386,
+            380,
+            401
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c85329b4c21ece37a61527077e5df064603bd4e849bef91be85db126d0c9a8ec.jpg",
+        "text": "$$\nq _ { i } ( a | x ) = \\frac { \\pi _ { i } ( a | x ) e ^ { \\frac { 1 } { \\alpha } Q ^ { \\pi _ { i } } ( x , a ) } } { \\mathbb { E } _ { b \\sim \\pi _ { i } ( \\cdot | x ) } \\big [ e ^ { \\frac { 1 } { \\alpha } Q ^ { \\pi _ { i } } ( x , b ) } \\big ] } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            413,
+            405,
+            642,
+            444
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "where $Q ^ { \\pi } ( x , a ) = r ( x , a ) + \\gamma \\mathbb { E } _ { y \\sim p ( \\cdot | x , a ) } V ^ { \\pi } ( y ) .$ ",
+        "bbox": [
+            232,
+            448,
+            552,
+            465
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "• We have ",
+        "bbox": [
+            215,
+            472,
+            290,
+            487
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f79df8ab2ba40bba18fbcac661d65d3983e2c2668c1a540bbe267aeb7f8f8b95.jpg",
+        "text": "$$\nV _ { \\alpha } ^ { \\pi _ { i } , q _ { i } } \\geq V ^ { \\pi _ { i } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            478,
+            486,
+            576,
+            502
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "• For η sufficiently small, we have ",
+        "bbox": [
+            217,
+            511,
+            446,
+            526
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ac411152811a0f9ec2767761960d1140849565463498b556c838ffc1a9c268fb.jpg",
+        "text": "$$\n\\mathcal { I } ( \\theta _ { i + 1 } , q _ { i + 1 } ) \\geq \\mathcal { I } ( \\theta _ { i } , q _ { i } ) + c g _ { i } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            416,
+            530,
+            638,
+            547
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "where c is a numerical constant, and $g _ { i }$ is the norm of the gradient (minimized by the algorithm): ",
+        "bbox": [
+            232,
+            551,
+            823,
+            580
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3beb61cab24715a6c37c66b8506906fb14e67845ec9f6243b0d72435b8db5c51.jpg",
+        "text": "$$\ng _ { i } = \\Big \\| \\nabla _ { \\theta } \\mathbb { E } _ { \\pi _ { i } } \\Big [ \\sum _ { t \\geq 0 } \\gamma ^ { t } K L \\big ( q _ { i } ( \\cdot | x _ { t } ) \\| \\pi _ { \\theta } ( \\cdot | x _ { t } ) \\big ) _ { | \\theta = \\theta _ { i } } \\big | x _ { 0 } , \\pi _ { i } \\Big ] \\Big \\| .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            583,
+            722,
+            621
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Thus we build a sequence of policies $( \\pi _ { \\theta _ { i } } , q _ { i } )$ whose values $\\mathcal { I } ( \\theta _ { i } , q _ { i } )$ are non-decreasing thus converge to a local maximum. In addition, the improvement is lower-bounded by $a$ constant times the norm of the gradient, thus the algorithm keeps improving the performance until the gradient vanishes (when we reach the limit of the capacity of our representation). ",
+        "bbox": [
+            232,
+            632,
+            825,
+            703
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Proof. We have ",
+        "bbox": [
+            173,
+            718,
+            281,
+            733
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b7f089d1a8523c5975d650c2cbc3c73cec952033c50a4f0c6b61d739dac54719.jpg",
+        "text": "$$\n\\begin{array} { r l r } { q _ { i } ( \\cdot | x ) } & { = } & { \\arg \\underset { q } { \\operatorname* { m a x } } \\mathbb { E } _ { a \\sim q ( \\cdot | x ) } \\Big [ \\underset { Q ^ { \\pi _ { i } } ( x , a ) } { \\underbrace { r \\big ( x , a ) + \\gamma \\mathbb { E } _ { y \\sim p ( \\cdot | x , a ) } V ^ { \\pi _ { i } } ( y ) } } - \\alpha \\log \\frac { q \\big ( a | x \\big ) } { \\pi _ { i } \\big ( a | x \\big ) } \\Big ] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            232,
+            738,
+            764,
+            784
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "from which we deduce (19). Now, from the definition of $q _ { i }$ , we have ",
+        "bbox": [
+            173,
+            787,
+            622,
+            803
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/04f39d59eb33e246cbee27b1aada1d93ea563fdcd209acdee360483348da0a8c.jpg",
+        "text": "$$\nT _ { \\alpha } ^ { \\pi _ { i } , q _ { i } } V ^ { \\pi _ { i } } \\geq T _ { \\alpha } ^ { \\pi _ { i } , \\pi _ { i } } V ^ { \\pi _ { i } } = T ^ { \\pi _ { i } } V ^ { \\pi _ { i } } = V ^ { \\pi _ { i } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            808,
+            642,
+            824
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Now, since and its fixe $T _ { \\alpha } ^ { \\pi _ { i } , q _ { i } }$ is is otone operator (i.e. if , we have $V _ { 1 } \\geq V _ { 2 }$ elementwise, then $T _ { \\alpha } ^ { \\pi _ { i } , q _ { i } } V _ { 1 } \\geq T _ { \\alpha } ^ { \\pi _ { i } , q _ { i } } V _ { 2 } )$ $V _ { \\alpha } ^ { \\pi _ { i } , q _ { i } }$ ",
+        "bbox": [
+            174,
+            828,
+            823,
+            858
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b5406ec4758472f538cfbf40ab29a64c9cbd02afff188b0f7c935c8b0aff3228.jpg",
+        "text": "$$\nV _ { \\alpha } ^ { \\pi _ { i } , q _ { i } } = \\operatorname* { l i m } _ { t \\to \\infty } ( T _ { \\alpha } ^ { \\pi _ { i } , q _ { i } } ) ^ { t } V ^ { \\pi _ { i } } \\geq V ^ { \\pi _ { i } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            380,
+            861,
+            616,
+            883
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "which proves (20). ",
+        "bbox": [
+            174,
+            887,
+            297,
+            902
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Now, in order to prove (21) we derive the following steps. ",
+        "bbox": [
+            173,
+            909,
+            553,
+            925
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Step 1: From the definition of $q _ { i + 1 }$ we have, for any $x$ ",
+        "bbox": [
+            173,
+            103,
+            547,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/552747ed5a6e20240a82619613d59708d93a9469b61b5717ae3776d2b71e4cb3.jpg",
+        "text": "$$\n\\begin{array} { r } { \\tilde { \\Sigma } _ { a \\sim q _ { 4 } + 1 } \\left[ Q ^ { \\pi _ { i + 1 } } ( x , a ) \\right] - \\alpha \\mathbf { K L } \\left( q _ { i + 1 } ( \\cdot | x ) \\| \\pi _ { i + 1 } ( \\cdot | x ) \\right) \\geq \\mathbb { E } _ { a \\sim q _ { 4 } } \\left[ Q ^ { \\pi _ { i + 1 } } ( x , a ) \\right] - \\alpha \\mathbf { K L } \\left( q _ { i } ( \\cdot | x ) \\| \\pi _ { i + 1 } ( \\cdot | x ) \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            119,
+            839,
+            140
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Writing the functional that we minimize ",
+        "bbox": [
+            174,
+            156,
+            439,
+            171
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1f4b282a29f644d2f9bc8341e93165d2579b6e4b081ef15b9b49301cfa6be4ca.jpg",
+        "text": "$$\nf ( \\pi , q , \\theta ) = \\mathbb { E } _ { \\pi } \\Big [ \\sum _ { t \\geq 0 } \\gamma ^ { t } \\mathrm { K L } \\big ( q ( \\cdot | x _ { t } ) \\| \\pi _ { \\theta } ( \\cdot | x _ { t } ) \\big ) \\big | x _ { 0 } , \\pi \\Big ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            174,
+            656,
+            210
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "the update rule is $\\theta _ { i + 1 } = \\theta _ { i } - \\beta \\nabla _ { \\theta } f ( \\pi _ { i } , q _ { i } , \\theta _ { i } )$ . Thus we have that for sufficiently small $\\beta$ , ",
+        "bbox": [
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+            212,
+            769,
+            228
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b66ed1027d29baee2cee300451ac53827045f254feacb981ae17b3e6a814444a.jpg",
+        "text": "$$\nf ( \\pi _ { i } , q _ { i } , \\theta _ { i + 1 } ) \\leq f ( \\pi _ { i } , q _ { i } , \\theta _ { i } ) - \\beta g _ { i } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            231,
+            619,
+            247
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { g _ { i } = \\frac { 1 } { 2 } \\| \\nabla _ { \\theta } f ( \\pi _ { i } , q _ { i } , \\theta _ { i } ) \\| } \\end{array}$ . ",
+        "bbox": [
+            174,
+            250,
+            387,
+            268
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Step 2: Now define $\\mathcal { F }$ : ",
+        "bbox": [
+            173,
+            280,
+            338,
+            295
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/993b4298f6e970d0f2e4670bc59e30344f4ff25766c36ccb063c73e26cbd300e.jpg",
+        "text": "$$\n\\begin{array} { r c l } { \\mathcal { F } ( \\pi , q , \\theta , \\pi ^ { \\prime } ) } & { = } & { \\displaystyle \\mathbb { E } _ { \\pi } \\left[ \\sum _ { t \\geq 0 } \\gamma ^ { t } \\big ( \\mathbb { E } _ { a \\sim q } \\big [ Q ^ { \\pi ^ { \\prime } } ( x _ { t } , a ) \\big ] - \\alpha \\mathrm { K L } \\big ( q ( \\cdot | x _ { t } ) \\| \\pi _ { \\theta } ( \\cdot | x _ { t } ) \\big ) \\big ) \\big | x _ { 0 } , \\pi \\right] } \\\\ & { = } & { \\displaystyle \\delta _ { x _ { 0 } } \\big ( I - \\gamma P ^ { \\pi } \\big ) ^ { - 1 } T _ { \\alpha } ^ { \\pi _ { \\theta } , q } V ^ { \\pi ^ { \\prime } } } \\\\ & { = } & { \\delta _ { x _ { 0 } } \\big ( I - \\gamma P ^ { \\pi } \\big ) ^ { - 1 } T ^ { q } V ^ { \\pi ^ { \\prime } } - f ( \\pi , q , \\theta ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            215,
+            296,
+            781,
+            376
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "where $\\delta _ { x _ { 0 } }$ is a Dirac (in the row vector $x _ { 0 }$ ), and $P ^ { \\pi }$ is the transition matrix for policy $\\pi$ ",
+        "bbox": [
+            176,
+            377,
+            746,
+            392
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "From (22) and (23) we deduce that ",
+        "bbox": [
+            173,
+            398,
+            405,
+            412
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8281c3095dc4a6dd13ecf9d780cf86802e6f3b12a3c26359dd90592202f86a3e.jpg",
+        "text": "$$\n\\begin{array} { r c l } { \\mathcal { F } ( \\pi _ { i } , q _ { i + 1 } , \\theta _ { i + 1 } , \\pi _ { i + 1 } ) } & { \\geq } & { \\mathcal { F } ( \\pi _ { i } , q _ { i } , \\theta _ { i + 1 } , \\pi _ { i + 1 } ) } \\\\ & { \\geq } & { \\mathcal { F } ( \\pi _ { i } , q _ { i } , \\theta _ { i } , \\pi _ { i + 1 } ) + \\beta g _ { i } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            415,
+            684,
+            452
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We deduce ",
+        "bbox": [
+            173,
+            460,
+            250,
+            473
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b6f5c5e6ee822b58dd2984629f7c3e6e4c8226af264d9b586eb8d198ae26c22c.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { F } ( \\pi _ { i } , q _ { i + 1 } , \\theta _ { i + 1 } , \\pi _ { i } ) } \\\\ { \\geq } & { \\mathcal { F } ( \\pi _ { i } , q _ { i } , \\theta _ { i } , \\pi _ { i } ) + \\beta g _ { i } } \\\\ & { + \\mathcal { F } ( \\pi _ { i } , q _ { i + 1 } , \\theta _ { i + 1 } , \\pi _ { i } ) - \\mathcal { F } ( \\pi _ { i } , q _ { i + 1 } , \\theta _ { i + 1 } , \\pi _ { i + 1 } ) + \\mathcal { F } ( \\pi _ { i } , q _ { i } , \\theta _ { i } , \\pi _ { i + 1 } ) - \\mathcal { F } ( \\pi _ { i } , q _ { i } , \\theta _ { i } , \\pi _ { i } ) } \\\\ { = } & { \\mathcal { F } ( \\pi _ { i } , q _ { i } , \\theta _ { i } , \\pi _ { i } ) + \\beta g _ { i } + } \\\\ & { \\mathbb { E } _ { \\pi _ { i } } \\Big [ \\displaystyle \\sum _ { t \\geq 0 } \\gamma ^ { t } \\big ( \\mathbb { E } _ { a \\sim q _ { i + 1 } } \\big [ Q ^ { \\pi _ { i } } ( x _ { t } , a ) - Q ^ { \\pi _ { i + 1 } } ( x _ { t } , a ) \\big ] - \\mathbb { E } _ { a \\sim q _ { i } } \\big [ Q ^ { \\pi _ { i } } ( x _ { t } , a ) - Q ^ { \\pi _ { i + 1 } } ( x _ { t } , a ) \\big ] \\big ) \\Big ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            191,
+            476,
+            815,
+            585
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "This rewrites: ",
+        "bbox": [
+            174,
+            616,
+            266,
+            631
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8f25845cb16cee1896d898bedf65d0cec51090548fd89b16c452956e7e7d89fd.jpg",
+        "text": "$$\n\\delta _ { x _ { 0 } } ( I - \\gamma ^ { \\pi _ { i } } ) ^ { - 1 } \\big ( T _ { \\alpha } ^ { q _ { i + 1 } , \\pi _ { i + 1 } } V ^ { \\pi _ { i } } - T _ { \\alpha } ^ { q _ { i } , \\pi _ { i } } V ^ { \\pi _ { i } } \\big ) \\geq \\eta g _ { i } + O ( \\beta ^ { 2 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            290,
+            632,
+            707,
+            651
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Step 3: Now a bit of algebra. For two stochastic matrices $P$ and $P ^ { \\prime }$ , we have ",
+        "bbox": [
+            173,
+            661,
+            691,
+            676
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6833b9af9c4885f7be6be6eaabc73f30680d58070bd1fbf2d24a0be37741cfa4.jpg",
+        "text": "$$\n\\begin{array} { r l } & { ( I - \\gamma P ) ^ { - 1 } } \\\\ { = } & { ( I - \\gamma P ^ { \\prime } ) ^ { - 1 } + \\gamma ( I - \\gamma P ) ^ { - 1 } ( P - P ^ { \\prime } ) ( I - \\gamma P ^ { \\prime } ) ^ { - 1 } } \\\\ { = } & { ( I - \\gamma P ^ { \\prime } ) ^ { - 1 } + \\gamma \\big [ ( I - \\gamma P ^ { \\prime } ) ^ { - 1 } + \\gamma ( I - \\gamma P ) ^ { - 1 } ( P - P ^ { \\prime } ) ( I - \\gamma P ^ { \\prime } ) ^ { - 1 } \\big ] ( P - P ^ { \\prime } ) ( I - \\gamma P ^ { \\prime } ) ^ { - 1 } } \\\\ { = } & { ( I - \\gamma P ^ { \\prime } ) ^ { - 1 } + \\gamma ( I - \\gamma P ^ { \\prime } ) ^ { - 1 } ( P - P ^ { \\prime } ) ( I - \\gamma P ^ { \\prime } ) ^ { - 1 } } \\\\ & { + \\gamma ^ { 2 } ( I - \\gamma P ) ^ { - 1 } ( P - P ^ { \\prime } ) ( I - \\gamma P ^ { \\prime } ) ^ { - 1 } ( P - P ^ { \\prime } ) ( I - \\gamma P ^ { \\prime } ) ^ { - 1 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            184,
+            683,
+            838,
+            776
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Applying this equality to the transition matrices $P ^ { \\pi _ { k } }$ and $P ^ { \\pi _ { k + 1 } }$ and since $\\lVert P ^ { \\pi _ { k + 1 } } - P ^ { \\pi _ { k } } \\rVert = O ( \\eta )$ , we have: ",
+        "bbox": [
+            173,
+            799,
+            825,
+            829
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8bdebb9723dc34c32a537e413c81a04ddbece818013055ae73b233f81c4e75cd.jpg",
+        "text": "$$\n\\begin{array} { r l } & { V _ { \\alpha } ^ { q _ { i + 1 } , \\pi _ { i + 1 } } } \\\\ { = } & { ( I - \\gamma P ^ { \\pi _ { i } } ) ^ { - 1 } r _ { \\alpha } ^ { q _ { i + 1 } , \\pi _ { i + 1 } } } \\\\ { = } & { ( I - \\gamma P ^ { \\pi _ { i } } ) ^ { - 1 } r _ { \\alpha } ^ { q _ { i + 1 } , \\pi _ { i + 1 } } + \\gamma ( I - \\gamma P ^ { \\pi _ { i } } ) ^ { - 1 } ( P ^ { \\pi _ { i + 1 } } - P ^ { \\pi _ { i } } ) ( I - \\gamma P ^ { \\pi _ { i } } ) ^ { - 1 } r _ { \\alpha } ^ { q _ { i + 1 } , \\pi _ { i + 1 } } + O ( \\beta ^ { 2 } ) } \\\\ { = } & { ( I - \\gamma P ^ { \\pi _ { i } } ) ^ { - 1 } r _ { \\alpha } ^ { q _ { i } , \\pi _ { i } } + ( I - \\gamma P ^ { \\pi _ { i } } ) ^ { - 1 } ( r _ { \\alpha } ^ { q _ { i + 1 } , \\pi _ { i + 1 } } - r _ { \\alpha } ^ { q _ { i } , \\pi _ { i } } + \\gamma P ^ { \\pi _ { i + 1 } } - \\gamma P ^ { \\pi _ { i } } ) ( I - \\gamma P ^ { \\pi _ { i } } ) ^ { - 1 } r _ { \\alpha } ^ { q _ { i } , \\pi _ { i } } } \\\\ { = } & { V _ { \\alpha } ^ { q _ { i } , \\pi _ { i } } + ( I - \\gamma P ^ { \\pi _ { i } } ) ^ { - 1 } ( T _ { \\alpha } ^ { q _ { i + 1 } , \\pi _ { i } + 1 } V ^ { \\pi _ { i } } - T _ { \\alpha } ^ { q _ { i } , \\pi _ { i } } V ^ { \\pi _ { k } } ) + O ( \\beta ^ { 2 } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            186,
+            830,
+            839,
+            929
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/3d049027fa95b82742e76aaca76dec4b746d1e3c2894cc45a64f5325b8029996.jpg",
+        "table_caption": [
+            "Table 1: Results on a subset of the ALE environments in comparison to baselines taken from (Bellemare et al., 2017) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Game/Agent</td><td>Human</td><td>DQN</td><td>Prior. Dueling</td><td>C51</td><td>MPO</td></tr><tr><td></td><td></td><td>19.5</td><td>20.9</td><td></td><td></td></tr><tr><td>Pong Breakout</td><td>14.6 30.5</td><td>385.5</td><td>366.0</td><td>20.9 748</td><td>20.9 360.5</td></tr><tr><td></td><td>13,455.0</td><td>13,117.3</td><td>18,760.3</td><td>23,784</td><td></td></tr><tr><td>Q*bert Tennis</td><td>-8.3</td><td>12.2</td><td>0.0</td><td>23.1</td><td>10,317.0</td></tr><tr><td></td><td></td><td></td><td>98.9</td><td></td><td>22.2</td></tr><tr><td>Boxing</td><td>12.1</td><td>88.0</td><td></td><td>97.8</td><td>82.0</td></tr></table>",
+        "bbox": [
+            235,
+            143,
+            761,
+            246
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Finally, using (24), we deduce that ",
+        "bbox": [
+            174,
+            279,
+            403,
+            292
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b21e45fd65a916467315c45c1c84e9892ac2f711ac43978fbb51d0544f98608e.jpg",
+        "text": "$$\n\\begin{array} { l l l } { \\mathcal { I } ( \\theta _ { i + 1 } , q _ { i + 1 } ) } & { = } & { V _ { \\alpha } ^ { q _ { i } + 1 , \\pi _ { i } + 1 } \\big ( x _ { 0 } \\big ) } \\\\ & { = } & { V _ { \\alpha } ^ { q _ { i } , \\pi _ { i } } \\big ( x _ { 0 } \\big ) + \\delta _ { x _ { 0 } } \\big ( I - \\gamma P ^ { \\pi _ { i } } \\big ) ^ { - 1 } \\big ( T _ { \\alpha } ^ { q _ { i } + 1 , \\pi _ { i + 1 } } V ^ { \\pi _ { i } } - T _ { \\alpha } ^ { q _ { i } , \\pi _ { i } } V ^ { \\pi _ { i } } \\big ) + O ( \\beta ^ { 2 } ) } \\\\ & { \\geq } & { \\mathcal { I } ( \\theta _ { i } , q _ { i } ) + \\eta g _ { i } + O ( \\beta ^ { 2 } ) } \\\\ & { \\geq } & { \\mathcal { I } ( \\theta _ { i } , q _ { i } ) + \\frac { 1 } { 2 } \\eta g _ { i } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            194,
+            300,
+            803,
+            388
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "for small enough $\\eta$ . ",
+        "bbox": [
+            174,
+            395,
+            303,
+            410
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "B ADDITIONAL EXPERIMENT: DISCRETE CONTROL ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            430,
+            616,
+            446
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "As a proof of concept – showcasing the robustness of our algorithm and its hyperparameters – we performed an experiment on a subset of the games contained contained in the \"Arcade Learning Environment\" (ALE). For this experiment we used the same hyperparameter settings for the KL constraints as for the continuous control experiments as well as the same learning rate and merely altered the network architecture to the standard network structure used by DQN Mnih et al. (2015) – and created a seperate network with the same architecture, but predicting the parameters of the policy distribution. A comparison between our algorithm and well established baselines from the literature, in terms of the mean performance, is listed in Table 1. While we do not obtain state-ofthe-art performance in this experiment, the fact that MPO is competitive, out-of-the-box in these domains suggests that combining the ideas presented in this paper with recent advances for RL with discrete actions (Bellemare et al., 2017) could be a fruitful avenue for future work. ",
+        "bbox": [
+            173,
+            463,
+            825,
+            616
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "C EXPERIMENT DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            638,
+            397,
+            655
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "In this section we give the details on the hyper-parameters used for each experiment. All the continuous control experiments use a feed-forward network except for Parkour-2d were we used the same network architecture as in Heess et al. (2017). Other hyper parameters for MPO with non parametric variational distribution were set as follows, ",
+        "bbox": [
+            176,
+            670,
+            823,
+            727
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/9aea7df29472b2a734d9b8fb63ea3ef395c0a4ded780c11b28640be851b0d3df.jpg",
+        "table_caption": [
+            "Table 2: Parameters for non-parametric variational distribution "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter</td><td>control suite</td><td>humanoid</td></tr><tr><td>Policy net</td><td>100-100</td><td>200-200</td></tr><tr><td>Q function net</td><td>200-200</td><td>300-300</td></tr><tr><td>E</td><td>0.1</td><td>1</td></tr><tr><td>Eμ</td><td>0.1</td><td>&quot;</td></tr><tr><td>£</td><td>0.0001</td><td>&quot;</td></tr><tr><td>Discount factor()</td><td>0.99</td><td>1</td></tr><tr><td>Adam learning rate</td><td>0.0005</td><td>1</td></tr></table>",
+        "bbox": [
+            330,
+            739,
+            666,
+            856
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Hyperparameters for MPO with parametric variational distribution were as follows, ",
+        "bbox": [
+            173,
+            909,
+            720,
+            924
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/265f9d410660612f734f42d9c5e5275ae2403351d3285b07876e818a9a07114b.jpg",
+        "table_caption": [
+            "Table 3: Parameters for parametric variational distribution "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter</td><td>control suite tasks</td><td>humanoid</td></tr><tr><td>Policy net</td><td>100-100</td><td>200-200</td></tr><tr><td>Q function net</td><td>200-200</td><td>300-300</td></tr><tr><td>Eμ</td><td>0.1</td><td>1</td></tr><tr><td>£</td><td>0.0001</td><td>1</td></tr><tr><td>Discount factor ()</td><td>0.99</td><td>1</td></tr><tr><td>Adam learning rate</td><td>0.0005</td><td>1</td></tr></table>",
+        "bbox": [
+            312,
+            99,
+            684,
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "D DERIVATION OF UPDATE RULES FOR A GAUSSIAN POLICY ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            256,
+            689,
+            273
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "For continuous control we assume that the policy is given by a Gaussian distribution with a full covariance matrix, i.e, $\\pi ( \\boldsymbol { a } | \\boldsymbol { s } , \\pmb { \\theta } ) = \\mathcal { N } \\left( \\mu , \\pmb { \\Sigma } \\right)$ . Our neural network outputs the mean $\\mu = \\mu ( s )$ and Cholesky factor $A = A ( s )$ , such that $\\Sigma = A A ^ { T }$ . The lower triagular factor $A$ has positive diagonal elements enforced by the softplus transform $A _ { i i }  \\log ( 1 + \\exp ( A _ { i i } ) )$ . ",
+        "bbox": [
+            173,
+            287,
+            825,
+            344
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "D.1 NON-PARAMETRIC VARIATIONAL DISTRIBUTION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            362,
+            553,
+            377
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "In this section we provide the derivations and implementation details for the non-parametric variational distribution case for both $\\mathrm { E }$ -step and M-step. ",
+        "bbox": [
+            173,
+            388,
+            823,
+            417
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "D.2 E-STEP ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            434,
+            272,
+            449
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "The E-step with a non-parametric variational solves the following program, where we have replaced expectations with integrals to simplify the following derivations: ",
+        "bbox": [
+            173,
+            460,
+            823,
+            489
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/79abb19cfc5e0d689e7bde00318662a5dfa7e557ca8e3279e6a5da350ba71ac7.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\operatorname* { m a x } _ { q } \\int \\mu _ { q } ( s ) \\int q ( a | s ) Q _ { \\theta _ { i } } ( s , a ) d a d s } \\\\ { \\displaystyle s . t . \\int \\mu _ { q } ( s ) { \\bf K L } ( q ( a | s ) , \\pi ( a | s , \\theta _ { i } ) ) d a < \\epsilon , } \\\\ { \\displaystyle \\iint \\mu _ { q } ( s ) q ( a | s ) d a d s = 1 . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            506,
+            640,
+            607
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "First we write the Lagrangian equation, i.e, ",
+        "bbox": [
+            174,
+            614,
+            457,
+            631
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/486183ca7f5b2729ebda3c5343d5c3c8e310496b9f5e9601dcd9761c13bf9f97.jpg",
+        "text": "$$\n\\begin{array} { c } { { L ( q , \\eta , \\gamma ) = \\displaystyle \\int \\mu _ { q } ( s ) \\int q ( a | s ) Q _ { \\theta _ { i } } ( s , a ) d a d s + } } \\\\ { { \\displaystyle \\eta \\left( \\epsilon - \\int \\mu _ { q } ( s ) \\int q ( a | s ) \\log \\frac { q ( a | s ) } { \\pi ( a | s , \\theta _ { i } ) } \\right) + \\gamma \\left( 1 - \\displaystyle \\iint \\mu _ { q } ( s ) q ( a | s ) d a d s \\right) . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            724
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Next we maximise the Lagrangian $L$ w.r.t the primal variable $q$ . The derivative w.r.t $q$ reads, ",
+        "bbox": [
+            168,
+            729,
+            776,
+            746
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0723148fe81a97fbe26688adf5c0c326fc468c1bc06907d2fbe2d5df23530a92.jpg",
+        "text": "$$\n\\partial q L ( q , \\eta , \\gamma ) = Q _ { \\theta _ { i } } ( a , s ) - \\eta \\log q ( a | s ) + \\eta \\log \\pi ( a | s , \\pmb \\theta _ { i } ) - ( \\eta - \\gamma ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            763,
+            728,
+            782
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Setting it to zero and rearranging terms we get ",
+        "bbox": [
+            173,
+            794,
+            480,
+            809
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/419ba691557a8dd5a84dae08d83546c35a08d6bc39c863a20e1930df77232286.jpg",
+        "text": "$$\nq ( a | s ) = \\pi ( a | s , \\pmb { \\theta } _ { i } ) \\exp \\left( \\frac { Q _ { \\pmb { \\theta } _ { i } } ( a , s ) } { \\eta } \\right) \\exp \\left( - \\frac { \\eta - \\gamma } { \\eta } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            825,
+            683,
+            861
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "However the last exponential term is a normalisation constant for $q$ . Therefore we can write, ",
+        "bbox": [
+            169,
+            863,
+            777,
+            878
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/acd9b0e1d27167627e3254634727a03a260fe88cf8dafe9fe7c1f053dcdc760b.jpg",
+        "text": "$$\n\\exp ( - \\frac { \\eta - \\gamma } { \\eta } ) = \\int \\pi ( a | s , \\pmb { \\theta } _ { i } ) \\exp ( \\frac { Q _ { \\pmb { \\theta } _ { i } } ( a , s ) } { \\eta } ) d a ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            895,
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+            929
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "equation",
+        "img_path": "images/45584bcea1f3f117f3096dc8366d9774d4c64dbda4de636bfefdd2b1fc9f7dbb.jpg",
+        "text": "$$\n\\gamma = \\eta - \\eta \\log \\left( \\int \\pi ( a | s , \\pmb { \\theta } _ { i } ) \\exp ( \\frac { Q _ { \\pmb { \\theta } _ { i } } ( a , s ) } { \\eta } ) d a \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            99,
+            668,
+            135
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Note that we could write $\\gamma$ based on $\\pi$ and $\\eta$ . At this point we can derive the dual function, ",
+        "bbox": [
+            171,
+            143,
+            769,
+            160
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a448e228678b7bb1f2eb43e1ce3a4008cd43608e136500ca96d3727a51d9c816.jpg",
+        "text": "$$\ng ( \\eta ) = \\eta \\epsilon + \\eta \\int \\mu _ { q } ( s ) \\log \\left( \\int \\pi ( a | s , \\pmb { \\theta } _ { i } ) \\exp ( \\frac { Q ( a , s ) } { \\eta } ) d a \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            292,
+            165,
+            704,
+            200
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "D.3 M-STEP ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            214,
+            274,
+            229
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "To obtain the KL constraint in the $\\mathbf { M }$ step we set $p ( \\theta )$ to a Gaussian prior around the current policy, i.e, ",
+        "bbox": [
+            176,
+            241,
+            823,
+            268
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "img_path": "images/953f1d4dd059437ad180fba7715e206d9437adfdafcb37c9d10cbf016ed52105.jpg",
+        "text": "$$\np ( \\pmb \\theta ) \\approx \\mathcal { N } \\Big ( \\mu = \\pmb \\theta _ { i } , \\Sigma = \\frac { F _ { \\pmb \\theta _ { i } } } { \\lambda } \\Big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            267,
+            602,
+            297
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "where $\\theta _ { i }$ are the parameters of the current policy distribution, $F _ { \\pmb { \\theta } _ { i } }$ is the empirical Fisher information matrix and $\\lambda$ . ",
+        "bbox": [
+            174,
+            299,
+            825,
+            328
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "With this, and dropping constant terms our optimization program becomes ",
+        "bbox": [
+            176,
+            334,
+            663,
+            351
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1298caec4b94c89c434739404123167a1e8e4e6bd9d01d67eb4ba2e682b4adce.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\pi } \\int \\mu _ { q } ( s ) \\int q ( a | s ) \\log \\pi ( a | s , \\pmb { \\theta } ) d a d s - \\lambda ( \\pmb { \\theta } - \\pmb { \\theta } _ { i } ) ^ { T } F _ { \\pmb { \\theta _ { i } } } ^ { - 1 } ( \\pmb { \\theta } - \\pmb { \\theta } _ { i } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            356,
+            732,
+            388
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "We can observe that $( \\pmb \\theta - \\pmb \\theta _ { i } ) ^ { T } F _ { \\pmb \\theta _ { i } } ^ { - 1 } ( \\pmb \\theta - \\pmb \\theta _ { i } )$ is the second order Taylor approximation of $\\begin{array} { r } { \\int \\mu _ { q } ( s ) \\mathrm { K L } ( \\pi ( a | s , \\pmb { \\theta } _ { i } ) , \\pi ( a | s , \\pmb { \\theta } ) ) d s } \\end{array}$ which leads us to the generalized M-step objective: ",
+        "bbox": [
+            173,
+            402,
+            826,
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e5275e1e0cf0f2c9bfddf9241707ccdb4a2151b0b352705a569cd196af07678a.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\pi } \\int \\mu _ { q } ( s ) \\int q ( a | s ) \\log \\pi ( a | s , \\theta ) d a d s - \\lambda \\int \\mu _ { q } ( s ) \\mathbf { K L } ( \\pi ( a | s , \\theta _ { i } ) , \\pi ( a | s , \\theta ) ) d s\n$$",
+        "text_format": "latex",
+        "bbox": [
+            209,
+            443,
+            759,
+            477
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "which corresponds to Equation (11) from the main text, where expectations are replaced by integrals. ",
+        "bbox": [
+            171,
+            482,
+            823,
+            497
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "After obtaining the non parametric variational distribution in the $\\mathbf { M }$ step with a Gaussian policy we empirically observed that better results could be achieved by decoupling the KL constraint into two terms such that we can constrain the contribution of the mean and covariance separately i.e. ",
+        "bbox": [
+            174,
+            502,
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+            546
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "img_path": "images/23b9d5fa791c4c2bd8b5aa2f89398646a3528bffe20aa5b8a0a0fd3f44d63ede.jpg",
+        "text": "$$\n\\int \\mu _ { q } ( s ) \\mathrm { K L } ( \\pi _ { i } ( a | s , \\pmb \\theta ) , \\pi ( a | s , \\pmb \\theta ) ) = C _ { \\mu } + C _ { \\Sigma } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            571,
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "where ",
+        "bbox": [
+            173,
+            611,
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5e9cecdb584af75d226dc2b9ebded0f7485cc323b36c04fbea13f2add88eaddc.jpg",
+        "text": "$$\n\\begin{array} { c } { { C _ { \\mu } = \\displaystyle \\int \\mu _ { q } ( s ) \\frac 1 2 \\big ( \\mathrm { t r } \\big ( \\Sigma ^ { - 1 } \\Sigma _ { i } \\big ) - n + \\mathrm { l n } ( \\frac { \\Sigma } { \\Sigma _ { i } } ) \\big ) d s , } } \\\\ { { { } } } \\\\ { { C _ { \\Sigma } = \\displaystyle \\int \\mu _ { q } ( s ) \\frac 1 2 \\big ( \\mu - \\mu _ { i } \\big ) ^ { T } \\Sigma ^ { - 1 } ( \\mu - \\mu _ { i } ) d s . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            619,
+            655,
+            689
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "This decoupling allows us to set different $\\epsilon$ values for each component, i.e., $\\epsilon _ { \\mu } , \\epsilon _ { \\Sigma }$ for the mean, the covariance matrix respectively. Different $\\epsilon$ lead to different learning rates. The effectivness of this decoupling has also been shown in Abdolmaleki et al. (2017). We always set a much smaller epsilon for covariance than the mean. The intuition is that while we would like the distribution moves fast in the action space, we also want to keep the exploration to avoid premature convergence. ",
+        "bbox": [
+            173,
+            696,
+            826,
+            768
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "In order to solve the constrained optimisation in the M-step, we first write the generalised Lagrangian equation, i.e, ",
+        "bbox": [
+            173,
+            773,
+            825,
+            804
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "img_path": "images/841f2c7aa058eb67d06c4f85f4daddf3b071bc49ee06632b4c8f73aaf33a6cfe.jpg",
+        "text": "$$\nL ( \\pmb \\theta , \\eta _ { \\mu } , \\eta _ { \\Sigma } ) = \\int \\mu _ { q } ( s ) \\int q ( a | s ) \\log \\pi ( a | s , \\pmb \\theta ) d a d s + \\eta _ { \\mu } ( \\epsilon _ { \\mu } - C _ { \\mu } ) + \\eta _ { \\Sigma } ( \\epsilon _ { \\Sigma } - C _ { \\Sigma } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            217,
+            819,
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Where $\\eta _ { \\mu }$ and $\\eta _ { \\Sigma }$ are Lagrangian multipliers. Following prior work on constraint optimisation, we formulate the following primal problem, ",
+        "bbox": [
+            173,
+            854,
+            825,
+            883
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b93b94f99817576cd36409fa558ec24e83f02944d691e76051e954cc61c676b6.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\pmb { \\theta } } \\operatorname* { m i n } _ { \\eta _ { \\mu } > 0 , \\eta _ { \\Sigma } > 0 } L ( \\pmb { \\theta } , \\eta _ { \\mu } , \\eta _ { \\Sigma } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            904,
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+            928
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "In order to solve for $\\pmb \\theta$ we iteratively solve the inner and outer optimisation programs independently: We fix the Lagrangian multipliers to their current value and optimise for $\\pmb \\theta$ (outer maximisation) and then fix the parameters $\\pmb \\theta$ to their current value and optimise for the Lagrangian multipliers (inner minimisation). We continue this procedure until policy parameters $\\pmb { \\theta }$ and Lagrangian multipliers converge. Please note that the same approach can be employed to bound the KL explicitly instead of decoupling the contribution of mean and covariance matrix to the KL. ",
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "D.4 PARAMETRIC VARIATIONAL DISTRIBUTION ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            204,
+            513,
+            218
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "In this case we assume our variational distribution also uses a Gaussian distribution over the action space and use the same structure as our policy $\\pi$ . ",
+        "bbox": [
+            171,
+            229,
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+            258
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Similar to the non-parametric case for a Gaussian distribution in the M-step we also use a decoupled KL but this time in the E-step for a Gaussian variational distribution. Using the same reasoning as in the previous section we can obtain the following generalized Lagrangian equation: ",
+        "bbox": [
+            173,
+            265,
+            826,
+            309
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "img_path": "images/de4b2919525f4d6b623529678194f028b8600a1252310bf3abf4bdee14ee6788.jpg",
+        "text": "$$\nL ( \\theta ^ { q } , \\eta _ { \\mu } , \\eta _ { \\Sigma } ) = \\int \\mu _ { q } ( s ) \\int q ( a | s ; \\theta ^ { q } ) A _ { i } ( a , s ) d a d s + \\eta _ { \\mu } ( \\epsilon _ { \\mu } - C _ { \\mu } ) + \\eta _ { \\Sigma } ( \\epsilon _ { \\Sigma } - C _ { \\Sigma } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            215,
+            324,
+            781,
+            357
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Where $\\eta _ { \\mu }$ and $\\eta _ { \\Sigma }$ are Lagrangian multipliers. And where we use the advantage function $A ( a , s )$ instead of the $\\mathrm { Q }$ function $Q ( a , s )$ , as it empirically gave better performance. Please note that the KL in the $\\mathrm { E }$ -step is different than the one used in the M-step. Following prior works on constraint optimisation, we can formulate the following primal problem, ",
+        "bbox": [
+            174,
+            359,
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+            415
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4b234296dd6ac24b07d0dee1307a6b8b8791da36b2d98c8193a85ed6bc4271a2.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\pmb { \\theta } ^ { q } } \\operatorname* { m i n } _ { \\eta _ { \\mu } > 0 , \\eta _ { \\Sigma } > 0 } L ( \\pmb { \\theta } ^ { q } , \\eta _ { \\mu } , \\eta _ { \\Sigma } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            398,
+            434,
+            598,
+            460
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "In order to solve for $\\pmb { \\theta } ^ { q }$ we iteratively solve the inner and outer optimisation programs independently. In order to that we fix the Lagrangian multipliers to their current value and optimise for $\\pmb { \\theta } ^ { q }$ (outer maximisation), in this case we use the likelihood ratio gradient to compute the gradient w.r.t $\\pmb { \\theta } ^ { q }$ . Subsequently we fix the parameters $\\pmb { \\theta } ^ { q }$ to their current value and optimise for Lagrangian multipliers (inner minimisation). We iteratively continue this procedure until the policy parameters $\\pmb { \\theta } ^ { q }$ and the Lagrangian multipliers converges. Please note that the same approach can be used to bound the KL explicitly instead of decoupling the contribution of mean and covariance matrix to the KL. As our policy has the same structure as the parametric variational distribution, the M step in this case reduce to set the policy parameters $\\pmb { \\theta }$ to the parameters $\\pmb { \\theta } ^ { q }$ we obtained in E-step, i.e, ",
+        "bbox": [
+            173,
+            469,
+            825,
+            597
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3335988042fcacda2ddb21904a29a953d9a17fbbbd5d1c6bf3c8258d315d9930.jpg",
+        "text": "$$\n\\pmb \\theta _ { i + 1 } = \\pmb \\theta ^ { q }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            460,
+            603,
+            535,
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "E IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            640,
+            434,
+            655
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "While we ran most of our experiments using a single learner, we implemented a scalable variant of the presented method in which multiple workers collect data independently in an instance of the considered environment, compute gradients and send them to a chief (or parameter server) that performs parameter update by averaging gradients. That is we use distributed synchronous gradient descent. These procedures are described in Algorithms 1 and 2 for the non-parametric case and 3 for the parametric case. ",
+        "bbox": [
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+            670,
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 MPO (chief) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            151,
+            348,
+            166
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "1: Input $G$ number of gradients to average   \n2: while True do   \n3: initialize $\\Nu = 0$   \n4: initialize gradient store $s _ { \\phi } = \\{ \\} , s _ { \\eta } = \\{ \\} , s _ { \\eta _ { \\mu } } = \\{ \\} , s _ { \\eta _ { \\Sigma } } = \\{ \\} \\ : s _ { \\theta } = \\{ \\}$   \n5: while $N < G$ do   \n6: receive next gradient from worker $w$   \n7: $\\begin{array} { l } { { s _ { \\phi } = s _ { \\phi } + [ \\bar { \\delta } \\phi ^ { w } ] } } \\\\ { { s _ { \\phi } = s _ { \\theta } + [ \\delta \\theta ^ { w } ] } } \\\\ { { s _ { \\eta } = s _ { \\eta } + [ \\delta \\eta ^ { w } ] } } \\\\ { { s _ { \\eta _ { \\mu } } = s _ { \\eta _ { \\mu } } + [ \\delta \\eta _ { \\mu } ^ { w } ] } } \\\\ { { s _ { \\eta _ { \\theta } } = s _ { \\eta _ { \\theta } } + [ \\delta \\eta _ { \\theta } ^ { w } ] } } \\end{array}$   \n8:   \n9:   \n10:   \n11:   \n12: update parameters with average gradient from   \n13: $s _ { \\phi }$ , sη, sηµ, sηΣ sθ   \n14: send new parameters to workers ",
+        "bbox": [
+            178,
+            169,
+            697,
+            373
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2 MPO (worker) - Non parametric variational distribution ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            482,
+            629,
+            496
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "1: Input = \u000f, \u000fΣ, \u000fµ, Lmax   \n2: $i = 0$ , $L _ { \\mathrm { c u r r } } = 0$   \n3: Initialise $Q _ { \\omega _ { i } } ( a , s ) , \\pi ( a | s , \\pmb { \\theta } _ { i } ) , \\eta , \\eta _ { \\mu } , \\eta _ { \\Sigma }$   \n4: for each worker do   \n5: while $L _ { \\mathrm { c u r r } } > L _ { \\mathrm { m a x } }$ do   \n6: update replay buffer $\\boldsymbol { B }$ with $\\mathrm { L }$ trajectories from the environment   \n7: $\\bar { k = 0 }$   \n8: // Find better policy by gradient descent   \n9: while $k < 1 0 0 0$ do   \n10: sample a mini-batch $\\boldsymbol { B }$ of $N$ $( s , a , r )$ pairs from replay   \n11: sample $M$ additional actions for each state from $\\boldsymbol { B }$ , $\\bar { \\pi } ( \\bar { a } | s , \\pmb \\theta _ { i } )$ for estimating integrals   \n12: compute gradients, estimating integrals using samples   \n13: // Q-function gradient:   \n14: $\\delta _ { \\phi } = \\partial _ { \\phi } L _ { \\phi } ^ { \\prime } ( \\phi )$   \n15: // E-Step gradient:   \n16: $\\delta \\eta = \\bar { \\partial _ { \\eta } g ( \\eta ) }$   \n17: Let: q(a|s) ∝ π(a|s, θi) exp( Qθt (a,s,φ0)η )   \n18: // M-Step gradient:   \n19: $[ \\delta _ { \\eta _ { \\mu } } , \\delta _ { \\eta _ { \\Sigma } } ] \\stackrel { - } { = } \\alpha \\partial _ { \\eta _ { \\mu } , \\eta _ { \\Sigma } } { \\cal L } ( \\theta _ { k } , \\eta _ { \\mu } , \\eta _ { \\Sigma } )$   \n20: $\\delta _ { \\theta } = \\partial _ { \\theta } L ( \\theta , \\eta _ { \\mu _ { k + 1 } } , \\eta _ { \\Sigma k + 1 } )$   \n21: send gradients to chief worker   \n22: wait for gradient update by chief   \n23: fetch new parameters $\\phi , \\theta , \\eta , \\eta _ { \\mu } , \\eta _ { \\Sigma }$   \n24: $\\begin{array} { r } { k = k + 1 ^ { \\circ } } \\\\ { i = i + 1 , L _ { \\mathrm { c u r r } } = L _ { \\mathrm { c u r r } } + L } \\\\ { \\pmb { \\theta } _ { i } = \\pmb { \\theta } , \\phi ^ { \\prime } = \\phi } \\end{array}$   \n25:   \n26: ",
+        "bbox": [
+            173,
+            501,
+            826,
+            872
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 3 MPO (worker) - parametric variational distribution ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            320,
+            598,
+            335
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "text",
+        "text": "1: Input = \u000fΣ, \u000fµ, Lmax   \n2: $i = 0$ , Lcurr = 0   \n3: Initialise $Q _ { \\omega _ { i } } ( a , s ) , \\pi ( a | s , \\pmb { \\theta } _ { i } ) , \\eta , \\eta _ { \\mu } , \\eta _ { \\Sigma }$   \n4: for each worker do   \n5: while $L _ { \\mathrm { c u r r } } < L _ { \\mathrm { m a x } }$ do   \n6: update replay buffer $\\boldsymbol { B }$ with $\\mathrm { L }$ trajectories from the environment   \n7: $k = 0$   \n8: // Find better policy by gradient descent   \n9: while $k < 1 0 0 0$ do   \n10: sample a mini-batch $\\boldsymbol { B }$ of $\\mathrm { ~ V ~ } ( s , a , r )$ pairs from replay   \n11: sample $M$ additional actions for each state from $\\boldsymbol { B }$ , $\\dot { \\pi } ( a | s , \\theta _ { k } )$ for estimating inte  \ngrals   \n12: compute gradients, estimating integrals using samples   \n13: // Q-function gradient:   \n14: $\\delta _ { \\phi } = \\partial _ { \\phi } L _ { \\phi } ^ { \\prime } ( \\phi )$   \n15: // E-Step gradient:   \n16: $\\begin{array} { r l } & { \\bigl [ \\delta _ { \\eta _ { \\mu } } , \\delta _ { \\eta _ { \\Sigma } } \\bigr ] ^ { \\circ } = \\alpha \\partial _ { \\eta _ { \\mu } , \\eta _ { \\Sigma } } L ( \\pmb \\theta _ { k } , \\eta _ { \\mu } , \\eta _ { \\Sigma } ) } \\\\ & { \\delta _ { \\theta } = \\partial _ { \\theta } L ( \\pmb \\theta , \\eta _ { \\mu _ { k + 1 } } , \\eta _ { \\Sigma k + 1 , \\eta } , } \\\\ & { \\dots , \\dots } \\end{array}$   \n17:   \n18: // M-Step gradient: In practice there is no M-step in this case as policy and variatinal   \ndistribution $q$ use a same structure.   \n19: send gradients to chief worker   \n20: wait for gradient update by chief   \n21: fetch new parameters $\\phi , \\theta , \\eta , \\eta _ { \\mu } , \\eta _ { \\Sigma }$   \n22: $\\begin{array} { r } { k = k + 1 ^ { \\mathbf { \\complement } } } \\\\ { i = i + 1 , L _ { \\mathrm { c u r r } } = L _ { \\mathrm { c u r r } } + L } \\\\ { \\pmb { \\theta } _ { i } = \\pmb { \\theta } , \\phi ^ { \\prime } = \\phi } \\end{array}$   \n23:   \n24: ",
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+        "page_idx": 21
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+]
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diff --git a/parse/train/S1ANxQW0b/S1ANxQW0b_middle.json b/parse/train/S1ANxQW0b/S1ANxQW0b_middle.json
new file mode 100644
index 0000000000000000000000000000000000000000..d5e07d21df5ab9b0d577a191e823d5a6893c555e
--- /dev/null
+++ b/parse/train/S1ANxQW0b/S1ANxQW0b_middle.json
@@ -0,0 +1,59235 @@
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+                                    "content": "MAXIMUM A POSTERIORI POLICY OPTIMISATION",
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+                                    "content": "Abbas Abdolmaleki, Jost Tobias Springenberg, Yuval Tassa, Remi Munos,",
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+                                    "content": "Nicolas Heess, Martin Riedmiller",
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+                                    "score": 1.0,
+                                    "content": "to our derivation. We develop two off-policy algorithms and demonstrate that",
+                                    "type": "text"
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+                        {
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+                                    "score": 1.0,
+                                    "content": "they are competitive with the state-of-the-art in deep reinforcement learning. In",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "ety of domains, including locomotion (Heess et al., 2017; Peng et al., 2016), multi-agent behaviour",
+                                    "type": "text"
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+                            "index": 24
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+                    ],
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+                                    "content": "et al., 2008). These algorithms are more directly related to our work and can be cast in the same",
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+                                    "score": 1.0,
+                                    "content": "et al., 2016; Schulman et al., 2017a). In contrast, we here rely on a parametric policy, our bound and",
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+                                    "content": ". Existing EM policy search approaches",
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+                                    "score": 1.0,
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+                                    "content": "\\mathrm { K L } \\big ( q _ { t } | | \\pi _ { t } \\big ) = \\mathrm { K L } \\big ( q ( a | s _ { t } ) \\big ) \\| \\pi ( a | s _ { t } , \\pmb { \\theta } ) \\big )",
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+                                    "content": "\\begin{array} { r } { \\tilde { r } _ { t } = r _ { t } - \\alpha \\log \\frac { q ( a _ { t } | s _ { t } ) } { \\pi ( a _ { t } | s _ { t } , \\pmb { \\theta } ) } } \\end{array}",
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+                                    "score": 1.0,
+                                    "content": ". In this view",
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+                                    "content": "represents a default policy",
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+                                    "score": 1.0,
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+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "and estimate the unregularized action-value function:",
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+                                    "content": ". While this might seem unimportant it is crucial as it allows us to",
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+                                    "content": "Montgomery & Levine, 2016). Using KL constraint in M-step has also been shown effective for",
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+                                    "score": 1.0,
+                                    "content": "constraint. Removing the automatically tuned KL constraint and replacing it with a manually set",
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+                            ],
+                            "index": 6
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "entropy regulariser then yields an off-policy actor-critic method with Retrace. This policy gradient",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "method still uses the idea of estimating the integral over actions – and thus, for a gradient based",
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "optimiser, its likelihood ratio derivative – via multiple action samples (as judged by a Q-Retrace",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "critic). This idea has previously been coined as using the expected policy gradient (EPG) (Ciosek",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "& Whiteson, 2017) and we hence denote the corresponding algorithm with EPG",
+                                    "type": "text"
+                                },
+                                {
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+                                {
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+                                    "score": 1.0,
+                                    "content": "Retrace, which",
+                                    "type": "text"
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+                            ],
+                            "index": 11
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+                        {
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+                                    "score": 1.0,
+                                    "content": "no-longer follows the intuitions of the MPO perspective.",
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+                                    "score": 0.26,
+                                    "content": "\\mathrm { E P G } + .",
+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    "score": 1.0,
+                                    "content": "Retrace performed well when the",
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+                                    "score": 1.0,
+                                    "content": "correct entropy regularisation scale is used. This, however, required task specific tuning (c.f. Fig-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 13
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "ure 4 where this hyperparameter was set to the one that performed best in average across tasks).",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "Finally using only a single sample to estimate the integral (and hence the likelihood ratio gradient)",
+                                    "type": "text"
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+                            ],
+                            "index": 15
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "results in an actor-critic variant with Retrace that is the least performant off-policy algorithm in our",
+                                    "type": "text"
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+                                    "content": "algorithms from the literature in our framework – on all the environments from the control suite that",
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+                                    "score": 1.0,
+                                    "content": "we tested on are shown in Figure 4. All tasks have rewards that are scaled to be between 0 and 1000.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 27
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "We note that in order to ensure a fair comparison all algorithms ran with exactly the same network",
+                                    "type": "text"
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+                            ],
+                            "index": 28
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "configuration, used a single learner (no distributed computation), used the same optimizer and were",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 29
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "tuned w.r.t. their hyperparameters for best performance across all tasks. We refer to the appendix for",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 30
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "a complete description of the hyperparameters. Our comparison is made in terms of data-efficiency.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 31
+                        }
+                    ],
+                    "index": 28
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                    ],
+                    "lines": [
+                        {
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "From the plot a few trends are readily apparent: i) We can clearly observe the advantage in terms",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 32
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "of data-efficiency that methods relying on a Q-critic obtain over the PPO baseline. This difference",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 33
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "is so extreme that in several instances the PPO baseline converges an order of magnitude slower",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 34
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "than the off-policy algorithms and we thus indicate the asymptotic performance of each algorithm",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 35
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "of PPO and DDPG (which also improved significantly later during training in some instances) with a",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 36
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    "score": 1.0,
+                                    "content": "colored star in the plot; ii) the difference between the MPO results and the (expected) policy gradient",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 37
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "(EPG) with entropy regularisation confirm our suspicion from Section 5.1.1: finding a good setting",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 38
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "for the entropy regulariser that transfers across environments without additional constraints on the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 39
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "policy distribution is very difficult, leading to instabilities in the learning curves. In contrast to this",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 40
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "the MPO results appear to be stable across all environments; iii) Finally, in terms of data-efficiency",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 41
+                        },
+                        {
+                            "bbox": [
+                                105,
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                    "score": 1.0,
+                                    "content": "the methods utilising Retrace obtain a clear advantage over DDPG. The single learner vanilla DDPG",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 42
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "implementation learns the lower dimensional environments quickly but suffers in terms of learning",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 43
+                        }
+                    ],
+                    "index": 37.5
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+                                    "score": 1.0,
+                                    "content": "3We note that we use a value function baseline",
+                                    "type": "text"
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+                                    "bbox": [
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+                                    "content": "in this setup. See appendix for details.",
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+                            ]
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+                    ]
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+            ],
+            "para_blocks": [
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+                    ],
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+                                    "score": 1.0,
+                                    "content": "As a first observation, we can see that MPO gives stable learning on all tasks and, thanks to its",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 0
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "fully off-policy implementation, is significantly more sample efficient than the on-policy PPO base-",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "line. Furthermore, we can observe that changing from the non-parametric variational distribution",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 2
+                        },
+                        {
+                            "bbox": [
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+                            ],
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+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "to a parametric distribution3 (which, as described above, can be related to PPO) results in only a",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 3
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "minor asymptotic performance loss but slowed down optimisation and thus hampered sample effi-",
+                                    "type": "text"
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+                            ],
+                            "index": 4
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "ciency; which can be attributed to the fact that the parametric",
+                                    "type": "text"
+                                },
+                                {
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+                                    "score": 0.79,
+                                    "content": "q",
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "distribution required a stricter KL",
+                                    "type": "text"
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+                            ],
+                            "index": 5
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "constraint. Removing the automatically tuned KL constraint and replacing it with a manually set",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 6
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        172
+                                    ],
+                                    "score": 1.0,
+                                    "content": "entropy regulariser then yields an off-policy actor-critic method with Retrace. This policy gradient",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 7
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "method still uses the idea of estimating the integral over actions – and thus, for a gradient based",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 8
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        505,
+                                        193
+                                    ],
+                                    "score": 1.0,
+                                    "content": "optimiser, its likelihood ratio derivative – via multiple action samples (as judged by a Q-Retrace",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        205
+                                    ],
+                                    "score": 1.0,
+                                    "content": "critic). This idea has previously been coined as using the expected policy gradient (EPG) (Ciosek",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        203,
+                                        433,
+                                        215
+                                    ],
+                                    "score": 1.0,
+                                    "content": "& Whiteson, 2017) and we hence denote the corresponding algorithm with EPG",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        433,
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+                                    ],
+                                    "score": 0.61,
+                                    "content": "^ +",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        442,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "Retrace, which",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 11
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
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+                                    "bbox": [
+                                        105,
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+                                        338,
+                                        226
+                                    ],
+                                    "score": 1.0,
+                                    "content": "no-longer follows the intuitions of the MPO perspective.",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 0.26,
+                                    "content": "\\mathrm { E P G } + .",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "Retrace performed well when the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 12
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
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+                                        505,
+                                        239
+                                    ],
+                                    "score": 1.0,
+                                    "content": "correct entropy regularisation scale is used. This, however, required task specific tuning (c.f. Fig-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 13
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "ure 4 where this hyperparameter was set to the one that performed best in average across tasks).",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 14
+                        },
+                        {
+                            "bbox": [
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+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        260
+                                    ],
+                                    "score": 1.0,
+                                    "content": "Finally using only a single sample to estimate the integral (and hence the likelihood ratio gradient)",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 15
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "results in an actor-critic variant with Retrace that is the least performant off-policy algorithm in our",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 16
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "comparison.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 17
+                        }
+                    ],
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+                                    "content": "we tested on are shown in Figure 4. All tasks have rewards that are scaled to be between 0 and 1000.",
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+                                    "content": "configuration, used a single learner (no distributed computation), used the same optimizer and were",
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+                                    "score": 1.0,
+                                    "content": "tuned w.r.t. their hyperparameters for best performance across all tasks. We refer to the appendix for",
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+                                    "content": "a complete description of the hyperparameters. Our comparison is made in terms of data-efficiency.",
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+                                    "score": 1.0,
+                                    "content": "than the off-policy algorithms and we thus indicate the asymptotic performance of each algorithm",
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+                                    "content": "of PPO and DDPG (which also improved significantly later during training in some instances) with a",
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+                                    "content": "colored star in the plot; ii) the difference between the MPO results and the (expected) policy gradient",
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+                                    "content": "(EPG) with entropy regularisation confirm our suspicion from Section 5.1.1: finding a good setting",
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+                                    "score": 1.0,
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+                            "index": 40
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+                                    "content": "literature, in terms of the mean performance, is listed in Table 1. While we do not obtain state-of-",
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+                                    "score": 1.0,
+                                    "content": "literature, in terms of the mean performance, is listed in Table 1. While we do not obtain state-of-",
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+                                    "content": "uous control experiments use a feed-forward network except for Parkour-2d were we used the same",
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+                                    "content": "network architecture as in Heess et al. (2017). Other hyper parameters for MPO with non parametric",
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+                                    "score": 1.0,
+                                    "content": "to their current value and optimise for the Lagrangian multipliers (inner",
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+                                    "content": "minimisation). We continue this procedure until policy parameters",
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+                                    "score": 1.0,
+                                    "content": "and Lagrangian multipliers",
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+                                    "score": 1.0,
+                                    "content": "converge. Please note that the same approach can be employed to bound the KL explicitly instead",
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+                            ],
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+                                    "content": "In this case we assume our variational distribution also uses a Gaussian distribution over the action",
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+                                    "score": 1.0,
+                                    "content": "Similar to the non-parametric case for a Gaussian distribution in the M-step we also use a decoupled",
+                                    "type": "text"
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+                                    "content": "KL but this time in the E-step for a Gaussian variational distribution. Using the same reasoning as",
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+                                    "score": 1.0,
+                                    "content": "in the previous section we can obtain the following generalized Lagrangian equation:",
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+                                }
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+                        }
+                    ],
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+                                    "content": "L ( \\theta ^ { q } , \\eta _ { \\mu } , \\eta _ { \\Sigma } ) = \\int \\mu _ { q } ( s ) \\int q ( a | s ; \\theta ^ { q } ) A _ { i } ( a , s ) d a d s + \\eta _ { \\mu } ( \\epsilon _ { \\mu } - C _ { \\mu } ) + \\eta _ { \\Sigma } ( \\epsilon _ { \\Sigma } - C _ { \\Sigma } ) .",
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+                                    "score": 1.0,
+                                    "content": ", as it empirically gave better performance. Please note that the",
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+                                    "content": "-step is different than the one used in the M-step. Following prior works on constraint",
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+                        },
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+                                    "score": 1.0,
+                                    "content": "optimisation, we can formulate the following primal problem,",
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+                                    "content": "\\operatorname* { m a x } _ { \\pmb { \\theta } ^ { q } } \\operatorname* { m i n } _ { \\eta _ { \\mu } > 0 , \\eta _ { \\Sigma } > 0 } L ( \\pmb { \\theta } ^ { q } , \\eta _ { \\mu } , \\eta _ { \\Sigma } )",
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+                                    "content": "Subsequently we fix the parameters",
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+                                    "score": 1.0,
+                                    "content": "to their current value and optimise for Lagrangian multipliers",
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+                        },
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+                                    "score": 1.0,
+                                    "content": "(inner minimisation). We iteratively continue this procedure until the policy parameters",
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+                                    "score": 1.0,
+                                    "content": "and the",
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "Lagrangian multipliers converges. Please note that the same approach can be used to bound the KL",
+                                    "type": "text"
+                                }
+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "explicitly instead of decoupling the contribution of mean and covariance matrix to the KL. As our",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 24
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "policy has the same structure as the parametric variational distribution, the M step in this case reduce",
+                                    "type": "text"
+                                }
+                            ],
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+                        },
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+                                    "content": "we obtained in E-step, i.e,",
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+                                    "content": "E IMPLEMENTATION DETAILS",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "the considered environment, compute gradients and send them to a chief (or parameter server) that",
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+                                    "score": 1.0,
+                                    "content": "performs parameter update by averaging gradients. That is we use distributed synchronous gradient",
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+                                    "score": 1.0,
+                                    "content": "descent. These procedures are described in Algorithms 1 and 2 for the non-parametric case and 3",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "for the parametric case.",
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+                                    "score": 1.0,
+                                    "content": "minimisation). We continue this procedure until policy parameters",
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+                                    "score": 1.0,
+                                    "content": "and Lagrangian multipliers",
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+                                    "content": "converge. Please note that the same approach can be employed to bound the KL explicitly instead",
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+                                    "score": 1.0,
+                                    "content": "in the previous section we can obtain the following generalized Lagrangian equation:",
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+                                    "score": 1.0,
+                                    "content": ", as it empirically gave better performance. Please note that the",
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+                                    "content": "(inner minimisation). We iteratively continue this procedure until the policy parameters",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "we obtained in E-step, i.e,",
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+                                    "content": "descent. These procedures are described in Algorithms 1 and 2 for the non-parametric case and 3",
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+                                    "content": "update parameters with average gradient from",
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+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": ", sη, sηµ, sηΣ sθ",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "send new parameters to workers",
+                                    "type": "text"
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+                            ],
+                            "index": 14
+                        }
+                    ],
+                    "index": 7.5
+                },
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+                                    "content": "Algorithm 2 MPO (worker) - Non parametric variational distribution",
+                                    "type": "text"
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+                                    "type": "text"
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+                                    "content": "L _ { \\mathrm { c u r r } } = 0",
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+                                    "score": 0.4,
+                                    "content": "Q _ { \\omega _ { i } } ( a , s ) , \\pi ( a | s , \\pmb { \\theta } _ { i } ) , \\eta , \\eta _ { \\mu } , \\eta _ { \\Sigma }",
+                                    "type": "inline_equation"
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+                                    "content": "4: for each worker do",
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+                                    "score": 0.87,
+                                    "content": "L _ { \\mathrm { c u r r } } > L _ { \\mathrm { m a x } }",
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+                                    "content": "6:",
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+                                    "score": 1.0,
+                                    "content": "update replay buffer",
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+                                    "score": 0.44,
+                                    "content": "\\mathrm { L }",
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+                                    "type": "text"
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+                                    "score": 0.42,
+                                    "content": "( s , a , r )",
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+                                    "content": "additional actions for each state from",
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+                        {
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "// E-Step gradient:",
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+                                    "score": 0.76,
+                                    "content": "\\delta \\eta = \\bar { \\partial _ { \\eta } g ( \\eta ) }",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
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+                                {
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+                                    "score": 1.0,
+                                    "content": "Let: q(a|s) ∝ π(a|s, θi) exp( Qθt (a,s,φ0)η )",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 32
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+                        {
+                            "bbox": [
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+                                        599
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+                                    "score": 1.0,
+                                    "content": "18:",
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+                                {
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+                                    "content": "// M-Step gradient:",
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+                                    "score": 1.0,
+                                    "content": "19:",
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+                                {
+                                    "bbox": [
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+                                    "score": 0.37,
+                                    "content": "[ \\delta _ { \\eta _ { \\mu } } , \\delta _ { \\eta _ { \\Sigma } } ] \\stackrel { - } { = } \\alpha \\partial _ { \\eta _ { \\mu } , \\eta _ { \\Sigma } } { \\cal L } ( \\theta _ { k } , \\eta _ { \\mu } , \\eta _ { \\Sigma } )",
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+                                    "score": 1.0,
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+                                    "bbox": [
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+                                    "score": 0.34,
+                                    "content": "\\delta _ { \\theta } = \\partial _ { \\theta } L ( \\theta , \\eta _ { \\mu _ { k + 1 } } , \\eta _ { \\Sigma k + 1 } )",
+                                    "type": "inline_equation"
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+                            "index": 35
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+                                    "score": 1.0,
+                                    "content": "send gradients to chief worker",
+                                    "type": "text"
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+                                    "score": 1.0,
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+                                    "score": 1.0,
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+                                    "score": 1.0,
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+                                    "bbox": [
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+                                    "score": 0.87,
+                                    "content": "\\begin{array} { r } { k = k + 1 ^ { \\circ } } \\\\ { i = i + 1 , L _ { \\mathrm { c u r r } } = L _ { \\mathrm { c u r r } } + L } \\\\ { \\pmb { \\theta } _ { i } = \\pmb { \\theta } , \\phi ^ { \\prime } = \\phi } \\end{array}",
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+                                    "score": 1.0,
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+                                    "content": "s _ { \\phi } = \\{ \\} , s _ { \\eta } = \\{ \\} , s _ { \\eta _ { \\mu } } = \\{ \\} , s _ { \\eta _ { \\Sigma } } = \\{ \\} \\ : s _ { \\theta } = \\{ \\}",
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+                                    "content": "6:",
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+                                    "score": 1.0,
+                                    "content": "receive next gradient from worker",
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+                                    "content": "7:",
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+                                    "content": "\\begin{array} { l } { { s _ { \\phi } = s _ { \\phi } + [ \\bar { \\delta } \\phi ^ { w } ] } } \\\\ { { s _ { \\phi } = s _ { \\theta } + [ \\delta \\theta ^ { w } ] } } \\\\ { { s _ { \\eta } = s _ { \\eta } + [ \\delta \\eta ^ { w } ] } } \\\\ { { s _ { \\eta _ { \\mu } } = s _ { \\eta _ { \\mu } } + [ \\delta \\eta _ { \\mu } ^ { w } ] } } \\\\ { { s _ { \\eta _ { \\theta } } = s _ { \\eta _ { \\theta } } + [ \\delta \\eta _ { \\theta } ^ { w } ] } } \\end{array}",
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+                                    "score": 1.0,
+                                    "content": ", sη, sηµ, sηΣ sθ",
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+                                    "score": 1.0,
+                                    "content": "send new parameters to workers",
+                                    "type": "text"
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+                            "is_list_start_line": true
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+                    ],
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+                                    "content": "Algorithm 2 MPO (worker) - Non parametric variational distribution",
+                                    "type": "text"
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+                        }
+                    ],
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+                                    "type": "text"
+                                }
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+                                    "content": "L _ { \\mathrm { c u r r } } = 0",
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+                                    "content": "3: Initialise",
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+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "4: for each worker do",
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+                                    "score": 0.87,
+                                    "content": "L _ { \\mathrm { c u r r } } > L _ { \\mathrm { m a x } }",
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+                                    "score": 1.0,
+                                    "content": "update replay buffer",
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+                                    "score": 0.81,
+                                    "content": "\\boldsymbol { B }",
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+                                    "score": 0.44,
+                                    "content": "\\mathrm { L }",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
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+                            "is_list_start_line": true
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+                        {
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+                                    "score": 1.0,
+                                    "content": "// Find better policy by gradient descent",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "additional actions for each state from",
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+                                    "score": 0.7,
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+                                    "type": "inline_equation"
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+                                    "score": 0.92,
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+                                    "score": 0.88,
+                                    "content": "\\delta _ { \\phi } = \\partial _ { \\phi } L _ { \\phi } ^ { \\prime } ( \\phi )",
+                                    "type": "inline_equation"
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+                            ],
+                            "index": 29,
+                            "is_list_start_line": true
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+                                    "score": 1.0,
+                                    "content": "15:",
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+                                    "score": 1.0,
+                                    "content": "// E-Step gradient:",
+                                    "type": "text"
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+                            "index": 30,
+                            "is_list_start_line": true
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "16:",
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+                                    "score": 0.76,
+                                    "content": "\\delta \\eta = \\bar { \\partial _ { \\eta } g ( \\eta ) }",
+                                    "type": "inline_equation"
+                                }
+                            ],
+                            "index": 31,
+                            "is_list_start_line": true
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+                                    "score": 1.0,
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "Let: q(a|s) ∝ π(a|s, θi) exp( Qθt (a,s,φ0)η )",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 32,
+                            "is_list_start_line": true
+                        },
+                        {
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "// M-Step gradient:",
+                                    "type": "text"
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+                            "is_list_start_line": true
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+                        {
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+                                    "score": 0.37,
+                                    "content": "[ \\delta _ { \\eta _ { \\mu } } , \\delta _ { \\eta _ { \\Sigma } } ] \\stackrel { - } { = } \\alpha \\partial _ { \\eta _ { \\mu } , \\eta _ { \\Sigma } } { \\cal L } ( \\theta _ { k } , \\eta _ { \\mu } , \\eta _ { \\Sigma } )",
+                                    "type": "inline_equation"
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+                                    "score": 0.34,
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+                                    "score": 1.0,
+                                    "content": "send gradients to chief worker",
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+                            ],
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+                            "is_list_start_line": true
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+                        {
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+                                    "score": 1.0,
+                                    "content": "wait for gradient update by chief",
+                                    "type": "text"
+                                }
+                            ],
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+                            "is_list_start_line": true
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "fetch new parameters",
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+                                    "content": "\\phi , \\theta , \\eta , \\eta _ { \\mu } , \\eta _ { \\Sigma }",
+                                    "type": "inline_equation"
+                                }
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+                            "index": 38,
+                            "is_list_start_line": true
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+                                    "bbox": [
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+                                    "score": 0.87,
+                                    "content": "\\begin{array} { r } { k = k + 1 ^ { \\circ } } \\\\ { i = i + 1 , L _ { \\mathrm { c u r r } } = L _ { \\mathrm { c u r r } } + L } \\\\ { \\pmb { \\theta } _ { i } = \\pmb { \\theta } , \\phi ^ { \\prime } = \\phi } \\end{array}",
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+                                    "content": "26:",
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+                                    "content": "Algorithm 3 MPO (worker) - parametric variational distribution",
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+                                    "score": 1.0,
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+                                    "score": 0.4,
+                                    "content": "Q _ { \\omega _ { i } } ( a , s ) , \\pi ( a | s , \\pmb { \\theta } _ { i } ) , \\eta , \\eta _ { \\mu } , \\eta _ { \\Sigma }",
+                                    "type": "inline_equation"
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+                            "index": 3
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+                                    "score": 1.0,
+                                    "content": "4: for each worker do",
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+                                    "content": "5:",
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+                                    "score": 0.87,
+                                    "content": "L _ { \\mathrm { c u r r } } < L _ { \\mathrm { m a x } }",
+                                    "type": "inline_equation"
+                                },
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+                                    "content": "6:",
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+                                        334
+                                    ],
+                                    "score": 1.0,
+                                    "content": "update replay buffer",
+                                    "type": "text"
+                                },
+                                {
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+                                        244,
+                                        332
+                                    ],
+                                    "score": 0.8,
+                                    "content": "\\boldsymbol { B }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        245,
+                                        321,
+                                        265,
+                                        334
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+                                    "content": "with",
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+                                        332
+                                    ],
+                                    "score": 0.43,
+                                    "content": "\\mathrm { L }",
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+                                        334
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+                                    "score": 1.0,
+                                    "content": "trajectories from the environment",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 6
+                        },
+                        {
+                            "bbox": [
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+                                        122,
+                                        345
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+                                    "content": "7:",
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+                                    "type": "text"
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+                                    "content": "sample a mini-batch",
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+                                    "content": "\\boldsymbol { B }",
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+                                    "score": 0.55,
+                                    "content": "\\mathrm { ~ V ~ } ( s , a , r )",
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+                                    "content": "pairs from replay",
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+                            "index": 10
+                        },
+                        {
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+                                    "score": 0.77,
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+                                    "score": 1.0,
+                                    "content": "additional actions for each state from",
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+                                    "score": 0.7,
+                                    "content": "\\boldsymbol { B }",
+                                    "type": "inline_equation"
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": ",",
+                                    "type": "text"
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+                                {
+                                    "bbox": [
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+                                        377,
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+                                        389
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+                                    "score": 0.92,
+                                    "content": "\\dot { \\pi } ( a | s , \\theta _ { k } )",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        423,
+                                        376,
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+                                    "score": 1.0,
+                                    "content": "for estimating inte-",
+                                    "type": "text"
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+                            "index": 11
+                        },
+                        {
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+                                120,
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+                                401
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+                                    "score": 1.0,
+                                    "content": "grals",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "compute gradients, estimating integrals using samples",
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "// Q-function gradient:",
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+                                    "score": 0.84,
+                                    "content": "\\delta _ { \\phi } = \\partial _ { \\phi } L _ { \\phi } ^ { \\prime } ( \\phi )",
+                                    "type": "inline_equation"
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+                            "index": 15
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+                                    "score": 1.0,
+                                    "content": "// E-Step gradient:",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 16
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "16:",
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+                                    "score": 0.39,
+                                    "content": "\\begin{array} { r l } & { \\bigl [ \\delta _ { \\eta _ { \\mu } } , \\delta _ { \\eta _ { \\Sigma } } \\bigr ] ^ { \\circ } = \\alpha \\partial _ { \\eta _ { \\mu } , \\eta _ { \\Sigma } } L ( \\pmb \\theta _ { k } , \\eta _ { \\mu } , \\eta _ { \\Sigma } ) } \\\\ & { \\delta _ { \\theta } = \\partial _ { \\theta } L ( \\pmb \\theta , \\eta _ { \\mu _ { k + 1 } } , \\eta _ { \\Sigma k + 1 , \\eta } , } \\\\ & { \\dots , \\dots } \\end{array}",
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+                                    "score": 1.0,
+                                    "content": "17:",
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+                            ],
+                            "index": 17
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+                        {
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+                                    "content": "18:",
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+                                    "score": 1.0,
+                                    "content": "// M-Step gradient: In practice there is no M-step in this case as policy and variatinal",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "distribution",
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+                                    "score": 0.75,
+                                    "content": "q",
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "use a same structure.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 20
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "19:",
+                                    "type": "text"
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+                                {
+                                    "bbox": [
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+                                        498
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+                                    "score": 1.0,
+                                    "content": "send gradients to chief worker",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 21
+                        },
+                        {
+                            "bbox": [
+                                106,
+                                497,
+                                302,
+                                510
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "20:",
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+                                },
+                                {
+                                    "bbox": [
+                                        167,
+                                        497,
+                                        302,
+                                        510
+                                    ],
+                                    "score": 1.0,
+                                    "content": "wait for gradient update by chief",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 22
+                        },
+                        {
+                            "bbox": [
+                                105,
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "21:",
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+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "fetch new parameters",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                        509,
+                                        312,
+                                        521
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+                                    "score": 0.91,
+                                    "content": "\\phi , \\theta , \\eta , \\eta _ { \\mu } , \\eta _ { \\Sigma }",
+                                    "type": "inline_equation"
+                                }
+                            ],
+                            "index": 23
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                        531
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+                                    "score": 1.0,
+                                    "content": "22:",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 0.85,
+                                    "content": "\\begin{array} { r } { k = k + 1 ^ { \\mathbf { \\complement } } } \\\\ { i = i + 1 , L _ { \\mathrm { c u r r } } = L _ { \\mathrm { c u r r } } + L } \\\\ { \\pmb { \\theta } _ { i } = \\pmb { \\theta } , \\phi ^ { \\prime } = \\phi } \\end{array}",
+                                    "type": "inline_equation",
+                                    "image_path": "3c9f26eb59da1807a518de932e9c7e4425bc4623f215bc90bce1357701b1aca8.jpg"
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+                                    "score": 1.0,
+                                    "content": "23:",
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+                                }
+                            ],
+                            "index": 24
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+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "24:",
+                                    "type": "text"
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+                            "index": 26
+                        }
+                    ],
+                    "index": 13.5
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+            "page_idx": 21,
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+                                    "content": "Published as a conference paper at ICLR 2018",
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+                                    "score": 1.0,
+                                    "content": "Algorithm 3 MPO (worker) - parametric variational distribution",
+                                    "type": "text"
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+                            ],
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+                    "index": 0
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+                                    "content": "1: Input = \u000fΣ, \u000fµ, Lmax",
+                                    "type": "text"
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+                            ],
+                            "index": 1,
+                            "is_list_start_line": true
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+                                    "score": 1.0,
+                                    "content": "2:",
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+                                    "score": 1.0,
+                                    "content": ", Lcurr = 0",
+                                    "type": "text"
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+                            "is_list_start_line": true
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "3: Initialise",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                        286,
+                                        302
+                                    ],
+                                    "score": 0.4,
+                                    "content": "Q _ { \\omega _ { i } } ( a , s ) , \\pi ( a | s , \\pmb { \\theta } _ { i } ) , \\eta , \\eta _ { \\mu } , \\eta _ { \\Sigma }",
+                                    "type": "inline_equation"
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+                            "index": 3,
+                            "is_list_start_line": true
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "4: for each worker do",
+                                    "type": "text"
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+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                        123,
+                                        324
+                                    ],
+                                    "score": 1.0,
+                                    "content": "5:",
+                                    "type": "text"
+                                },
+                                {
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "while",
+                                    "type": "text"
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+                                    "bbox": [
+                                        164,
+                                        312,
+                                        216,
+                                        323
+                                    ],
+                                    "score": 0.87,
+                                    "content": "L _ { \\mathrm { c u r r } } < L _ { \\mathrm { m a x } }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        217,
+                                        311,
+                                        231,
+                                        324
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+                                    "score": 1.0,
+                                    "content": "do",
+                                    "type": "text"
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+                            "index": 5,
+                            "is_list_start_line": true
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+                        {
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "6:",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                        321,
+                                        236,
+                                        334
+                                    ],
+                                    "score": 1.0,
+                                    "content": "update replay buffer",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                        323,
+                                        244,
+                                        332
+                                    ],
+                                    "score": 0.8,
+                                    "content": "\\boldsymbol { B }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        245,
+                                        321,
+                                        265,
+                                        334
+                                    ],
+                                    "score": 1.0,
+                                    "content": "with",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                        273,
+                                        332
+                                    ],
+                                    "score": 0.43,
+                                    "content": "\\mathrm { L }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        273,
+                                        321,
+                                        411,
+                                        334
+                                    ],
+                                    "score": 1.0,
+                                    "content": "trajectories from the environment",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 6,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
+                                110,
+                                333,
+                                179,
+                                345
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        110,
+                                        333,
+                                        122,
+                                        345
+                                    ],
+                                    "score": 1.0,
+                                    "content": "7:",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 0.54,
+                                    "content": "k = 0",
+                                    "type": "inline_equation",
+                                    "image_path": "d2d609a59c7e6a80c3fcfb4b8d070099e24d4ecf350a3b8e307875d5f5d07c27.jpg"
+                                }
+                            ],
+                            "index": 7,
+                            "is_list_start_line": true
+                        },
+                        {
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+                            "spans": [
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+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "8:",
+                                    "type": "text"
+                                },
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "// Find better policy by gradient descent",
+                                    "type": "text"
+                                }
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+                            "index": 8,
+                            "is_list_start_line": true
+                        },
+                        {
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+                                    "score": 0.85,
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+                                    "score": 1.0,
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+                                    "type": "text"
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+                                        376
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+                                    "score": 0.82,
+                                    "content": "\\boldsymbol { B }",
+                                    "type": "inline_equation"
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+                                {
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+                                    "content": "of",
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+                                    "score": 0.55,
+                                    "content": "\\mathrm { ~ V ~ } ( s , a , r )",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "pairs from replay",
+                                    "type": "text"
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+                            "is_list_start_line": true
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+                                    "score": 0.77,
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+                                        367,
+                                        389
+                                    ],
+                                    "score": 1.0,
+                                    "content": "additional actions for each state from",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        368,
+                                        378,
+                                        376,
+                                        387
+                                    ],
+                                    "score": 0.7,
+                                    "content": "\\boldsymbol { B }",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        376,
+                                        376,
+                                        380,
+                                        389
+                                    ],
+                                    "score": 1.0,
+                                    "content": ",",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        380,
+                                        377,
+                                        423,
+                                        389
+                                    ],
+                                    "score": 0.92,
+                                    "content": "\\dot { \\pi } ( a | s , \\theta _ { k } )",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        423,
+                                        376,
+                                        505,
+                                        389
+                                    ],
+                                    "score": 1.0,
+                                    "content": "for estimating inte-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 11,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
+                                120,
+                                387,
+                                148,
+                                401
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        120,
+                                        387,
+                                        148,
+                                        401
+                                    ],
+                                    "score": 1.0,
+                                    "content": "grals",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 12,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
+                                105,
+                                398,
+                                385,
+                                412
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        398,
+                                        123,
+                                        412
+                                    ],
+                                    "score": 1.0,
+                                    "content": "12:",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        167,
+                                        398,
+                                        385,
+                                        411
+                                    ],
+                                    "score": 1.0,
+                                    "content": "compute gradients, estimating integrals using samples",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 13,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
+                                106,
+                                409,
+                                261,
+                                423
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        410,
+                                        122,
+                                        422
+                                    ],
+                                    "score": 1.0,
+                                    "content": "13:",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        167,
+                                        409,
+                                        261,
+                                        423
+                                    ],
+                                    "score": 1.0,
+                                    "content": "// Q-function gradient:",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 14,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
+                                106,
+                                421,
+                                229,
+                                434
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        421,
+                                        122,
+                                        433
+                                    ],
+                                    "score": 1.0,
+                                    "content": "14:",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        168,
+                                        421,
+                                        229,
+                                        434
+                                    ],
+                                    "score": 0.84,
+                                    "content": "\\delta _ { \\phi } = \\partial _ { \\phi } L _ { \\phi } ^ { \\prime } ( \\phi )",
+                                    "type": "inline_equation"
+                                }
+                            ],
+                            "index": 15,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
+                                106,
+                                430,
+                                245,
+                                446
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        432,
+                                        122,
+                                        444
+                                    ],
+                                    "score": 1.0,
+                                    "content": "15:",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        166,
+                                        430,
+                                        245,
+                                        446
+                                    ],
+                                    "score": 1.0,
+                                    "content": "// E-Step gradient:",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 16,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
+                                106,
+                                442,
+                                310,
+                                468
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        443,
+                                        122,
+                                        455
+                                    ],
+                                    "score": 1.0,
+                                    "content": "16:",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        168,
+                                        442,
+                                        310,
+                                        468
+                                    ],
+                                    "score": 0.39,
+                                    "content": "\\begin{array} { r l } & { \\bigl [ \\delta _ { \\eta _ { \\mu } } , \\delta _ { \\eta _ { \\Sigma } } \\bigr ] ^ { \\circ } = \\alpha \\partial _ { \\eta _ { \\mu } , \\eta _ { \\Sigma } } L ( \\pmb \\theta _ { k } , \\eta _ { \\mu } , \\eta _ { \\Sigma } ) } \\\\ & { \\delta _ { \\theta } = \\partial _ { \\theta } L ( \\pmb \\theta , \\eta _ { \\mu _ { k + 1 } } , \\eta _ { \\Sigma k + 1 , \\eta } , } \\\\ & { \\dots , \\dots } \\end{array}",
+                                    "type": "inline_equation",
+                                    "image_path": "1180ddf38bef6422e4afff17e1457f9c31d841b654b4372c5a7524524e2344b6.jpg"
+                                }
+                            ],
+                            "index": 18,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
+                                106,
+                                453,
+                                122,
+                                466
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        453,
+                                        122,
+                                        466
+                                    ],
+                                    "score": 1.0,
+                                    "content": "17:",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 17,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
+                                106,
+                                464,
+                                505,
+                                477
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        465,
+                                        122,
+                                        477
+                                    ],
+                                    "score": 1.0,
+                                    "content": "18:",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        167,
+                                        464,
+                                        505,
+                                        477
+                                    ],
+                                    "score": 1.0,
+                                    "content": "// M-Step gradient: In practice there is no M-step in this case as policy and variatinal",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 19,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
+                                123,
+                                475,
+                                264,
+                                488
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        123,
+                                        475,
+                                        171,
+                                        488
+                                    ],
+                                    "score": 1.0,
+                                    "content": "distribution",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        172,
+                                        478,
+                                        178,
+                                        487
+                                    ],
+                                    "score": 0.75,
+                                    "content": "q",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
+                                        178,
+                                        475,
+                                        264,
+                                        488
+                                    ],
+                                    "score": 1.0,
+                                    "content": "use a same structure.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 20,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
+                                106,
+                                486,
+                                291,
+                                499
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        486,
+                                        123,
+                                        499
+                                    ],
+                                    "score": 1.0,
+                                    "content": "19:",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        167,
+                                        487,
+                                        291,
+                                        498
+                                    ],
+                                    "score": 1.0,
+                                    "content": "send gradients to chief worker",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 21,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
+                                106,
+                                497,
+                                302,
+                                510
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        497,
+                                        122,
+                                        509
+                                    ],
+                                    "score": 1.0,
+                                    "content": "20:",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        167,
+                                        497,
+                                        302,
+                                        510
+                                    ],
+                                    "score": 1.0,
+                                    "content": "wait for gradient update by chief",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 22,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
+                                105,
+                                506,
+                                312,
+                                523
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        508,
+                                        122,
+                                        520
+                                    ],
+                                    "score": 1.0,
+                                    "content": "21:",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        167,
+                                        506,
+                                        255,
+                                        523
+                                    ],
+                                    "score": 1.0,
+                                    "content": "fetch new parameters",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        256,
+                                        509,
+                                        312,
+                                        521
+                                    ],
+                                    "score": 0.91,
+                                    "content": "\\phi , \\theta , \\eta , \\eta _ { \\mu } , \\eta _ { \\Sigma }",
+                                    "type": "inline_equation"
+                                }
+                            ],
+                            "index": 23,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
+                                106,
+                                518,
+                                267,
+                                556
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        106,
+                                        519,
+                                        122,
+                                        531
+                                    ],
+                                    "score": 1.0,
+                                    "content": "22:",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
+                                        151,
+                                        518,
+                                        267,
+                                        556
+                                    ],
+                                    "score": 0.85,
+                                    "content": "\\begin{array} { r } { k = k + 1 ^ { \\mathbf { \\complement } } } \\\\ { i = i + 1 , L _ { \\mathrm { c u r r } } = L _ { \\mathrm { c u r r } } + L } \\\\ { \\pmb { \\theta } _ { i } = \\pmb { \\theta } , \\phi ^ { \\prime } = \\phi } \\end{array}",
+                                    "type": "inline_equation",
+                                    "image_path": "3c9f26eb59da1807a518de932e9c7e4425bc4623f215bc90bce1357701b1aca8.jpg"
+                                }
+                            ],
+                            "index": 25,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
+                                105,
+                                531,
+                                122,
+                                543
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        531,
+                                        122,
+                                        543
+                                    ],
+                                    "score": 1.0,
+                                    "content": "23:",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 24,
+                            "is_list_start_line": true
+                        },
+                        {
+                            "bbox": [
+                                105,
+                                542,
+                                122,
+                                554
+                            ],
+                            "spans": [
+                                {
+                                    "bbox": [
+                                        105,
+                                        542,
+                                        122,
+                                        554
+                                    ],
+                                    "score": 1.0,
+                                    "content": "24:",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 26,
+                            "is_list_start_line": true
+                        }
+                    ],
+                    "index": 13.5,
+                    "bbox_fs": [
+                        105,
+                        266,
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+                        556
+                    ]
+                }
+            ]
+        }
+    ],
+    "_backend": "pipeline",
+    "_version_name": "2.2.2"
+}
\ No newline at end of file
diff --git a/parse/train/S1ANxQW0b/S1ANxQW0b_model.json b/parse/train/S1ANxQW0b/S1ANxQW0b_model.json
new file mode 100644
index 0000000000000000000000000000000000000000..69e48c1ac3e078596b165e1fc64c45b8bc4fd026
--- /dev/null
+++ b/parse/train/S1ANxQW0b/S1ANxQW0b_model.json
@@ -0,0 +1,30136 @@
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diff --git a/parse/train/SJg4Y3VFPS/SJg4Y3VFPS.md b/parse/train/SJg4Y3VFPS/SJg4Y3VFPS.md
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+# GROUP-CONNECTED MULTILAYER PERCEPTRON NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Despite the success of deep learning in domains such as image, voice, and graphs, there has been little progress in deep representation learning for domains without a known structure between features. For instance, a tabular dataset of different demographic and clinical factors where the feature interactions are not given as a prior. In this paper, we propose Group-Connected Multilayer Perceptron (GMLP) networks to enable deep representation learning in these domains. GMLP is based on the idea of learning expressive feature combinations (groups) and exploiting them to reduce the network complexity by defining local group-wise operations. During the training phase, GMLP learns a sparse feature grouping matrix using temperature annealing softmax with an added entropy loss term to encourage the sparsity. Furthermore, an architecture is suggested which resembles binary trees, where group-wise operations are followed by pooling operations to combine information; reducing the number of groups as the network grows in depth. To evaluate the proposed method, we conducted experiments on five different real-world datasets covering various application areas. Additionally, we provide visualizations on MNIST and synthesized data. According to the results, GMLP is able to successfully learn and exploit expressive feature combinations and achieve state-of-the-art classification performance on different datasets.
+
+# 1 INTRODUCTION
+
+Deep neural networks have been quite successful across various machine learning tasks. However, this advancement has been mostly limited to certain domains. For example in image and voice data, one can leverage domain properties such as location invariance, scale invariance, coherence, etc. via using convolutional layers (Goodfellow et al., 2016). Alternatively, for graph data, graph convolutional networks were suggested to leverage adjacency patterns present in datasets structured as a graph (Kipf & Welling, 2016; Xu et al., 2019).
+
+However, there has been little progress in learning deep representations for datasets that do not follow a particular known structure in the feature domain. Take for instance the case of a simple tabular dataset for disease diagnosis. Such a dataset may consist of features from different categories such as demographics (e.g., age, gender, income, etc.), examinations (e.g., blood pressure, lab results, etc.), and other clinical conditions. In this scenario, the lack of any known structure between features to be used as a prior would lead to the use of a fully-connected multilayer perceptron network (MLP). Nonetheless, it has been known in the literature that MLP architectures, due to their huge complexity, do not usually admit efficient training and generalization for networks of more than a few layers.
+
+In this paper, we propose Group-Connected Multiplayer Perceptron (GMLP) networks. The main idea behind GMLP is to learn and leverage expressive feature subsets, henceforth referred to as feature groups. A feature group is defined as a subset of features that provides a meaningful representation or high-level concept that would help the downstream task. For instance, in the disease diagnosis example, the combination of a certain blood factor and age might be the indicator of a higher level clinical condition which would help the final classification task. Furthermore, GMLP leverages feature groups limiting network connections to local group-wise connections and builds a feature hierarchy via merging groups as the network grows in depth. GMLP can be seen as an architecture that learns expressive feature combinations and leverages them via group-wise operations.
+
+The main contributions of this paper are as follows: $( i )$ proposing a method for end-to-end learning of expressive feature combinations, $( i i )$ suggesting a network architecture to utilize feature groups and local connections to build deep representations, (iii) conducting extensive experiments demonstrating the effectiveness of GMLP as well as visualizations and ablation studies for better understanding of the suggested architecture.
+
+We evaluated the proposed method on five different real-world datasets in various application domains and demonstrated the effectiveness of GMLP compared to state-of-the-art methods in the literature. Furthermore, we conducted ablation studies and comparisons to study different architectural and training factors as well as visualizations on MNIST and synthesized data. To help to reproduce the results and encouraging future studies on group-connected architectures, we made the source code related to this paper available online 1.
+
+# 2 RELATED WORK
+
+Fully-connected MLPs are the most widely-used neural models for datasets in which no prior assumption is made on the relationship between features. However, due to the huge complexity of fully-connected layers, MLPs are prone to overfitting resulting in shallow architectures limited to a few layers in depth (Goodfellow et al., 2016). Various techniques have been suggested to improve training these models which include regularization techniques such as L-1/L-2 regularization, dropout, etc. and normalization techniques such as layer normalization, weigh normalization, batch normalization, etc.(Srivastava et al., 2014; Ba et al., 2016; Salimans & Kingma, 2016; Ioffe & Szegedy, 2015). For instance, self-normalizing neural networks (SNNs) have been recently suggested as state of the art normalization methods that prevent vanishing or exploding gradients which help training feed-forward networks with higher depths (Klambauer et al., 2017).
+
+From the architectural perspective, there has been great attention toward networks consisting of sparse connections between layers rather than having dense fully-connected layers (Dey et al., 2018). Sparse connected neural networks are usually trained based on either a sparse prior structure over the network architecture (Richter & Wattenhofer, 2018) or based on pruning a fully-connected network to a sparse network (Yun et al., 2019; Tartaglione et al., 2018; Mocanu et al., 2018). However, it should be noted that the main objective of most sparse neural network literature has been focused on improving the memory and compute requirements while maintaining competitive accuracies compared to MLPs.
+
+As a parallel line of research, the idea of using expressive feature combinations or groups has been suggested as a prior over the feature domain. Perhaps, the most successful and widespread use of this idea is in creating random forest models in which different trees are trained based on different feature subsets in order to deal with high-dimensional and high-variance data (Breiman, 2001). More recently, feature grouping is suggested by Aydore et al. (2019) as a statistical regularization technique to learn from datasets of large feature size and a small number of training samples. They do the forward network computation by projecting input features using samples taken from a bank of feature grouping matrices, reducing the input layer complexity and regularizing the model. In another recent study, Ke et al. (2018) used expressive feature combinations to learn from tabular datasets using a recursive encoder with a shared embedding network. They suggest a recursive architecture in which more important feature groups have a more direct impact on the final prediction.
+
+While promising results have been reported using these methods, feature grouping has been mostly considered as a preprocessing step. For instance, Aydore et al. (2019) uses the recursive nearest agglomeration (ReNA) (Hoyos-Idrobo et al., 2018) clustering to determine feature groups prior to the analysis. Alternatively, Ke et al. (2018) defined feature groups based on a pre-trained gradient boosting decision tree (GBDT) (Friedman, 2001). Feature grouping as a preprocessing step not only increases the complexity and raises practical considerations, but also limits the optimality of the selected features in subsequent analysis. In this study, we propose an end-to-end solution to learn expressive feature groups. Moreover, we introduce a network architecture to exploit interrelations within the feature groups to reduce the network complexity and to train deeper representations.
+
+![](images/c1d29b2cea3d35501f1bef3d00700d77364f884f1a3ac447f57fd5a8e7118523.jpg)  
+Figure 1: The GMLP network architecture.
+
+# 3 PROPOSED METHOD
+
+# 3.1 ARCHITECTURE OVERVIEW
+
+In this paper, we propose GMLP which intuitively can be broken down to three stages: $( i )$ selecting expressive feature groups, $( i i )$ learning dynamics within each group individually, and (iii) merging information between groups as the network grows in depth (see Figure 1). In this architecture, expressive groups are jointly selected during the training phase. Furthermore, GMLP is leveraging feature groups and using local group-wise weight layers to significantly reduce the number of parameters. While the suggested idea can be materialized as different architectures, in the current study, we suggest organization of the network as architectures resembling a binary tree spanning from leaves (i.e., features) to a certain abstraction depth closer to the root2. As the network grows deeper, after each local group-wise weight layer, half of the groups are merged using pooling operations, effectively reducing the width of the network while increasing the receptive field. At the last layer, all features within all groups are concatenated into a dense feature vector fed to the output layer.
+
+# 3.2 NOTATION
+
+We consider the generic problem of supervised classification based on a dataset of feature and target pairs, $\mathcal { D }$ : $\left( \pmb { x } _ { 1 : N } , \pmb { y } _ { 1 : N } \right)$ , where $\pmb { x } _ { i } \in \Re ^ { d }$ , $y _ { i } \in \{ 1 \ldots C \}$ , and $N$ is the number of dataset samples. Furthermore, we define group size, $m$ , as the number of neurons or elements within each group, and group count, $k$ , as the number of selected groups which are essentially subsets of input features. Also, $L$ is used to refer to the total depth of a network. We use $\pmb { z } _ { i } ^ { l } \in \Re ^ { m }$ to refer to activation values of group $i$ in layer $l$ . In this paper, we define all vectors as column vectors.
+
+# 3.3 NETWORK LAYERS
+
+In this section, we present the formal definition of different GMLP network layers. The very first layer of the network, Group-Select, is responsible for organizing features into $k$ groups of size $m$ each. A routing matrix, $\Psi$ , is used for connecting each neuron within each group to exactly one feature in the feature set:
+
+$$
+z _ { 1 : k } ^ { 0 } = \Psi x ,
+$$
+
+where $\Psi \in \{ 0 , 1 \} ^ { k m \times d }$ is a sparse matrix determining features that are present in each group. As we are interested in jointly learning $\Psi$ during the training phase, we use the following continuous relaxation:
+
+$$
+\Psi _ { i , j } \approx \frac { \exp ( \psi _ { i , j } / \tau ) } { \sum _ { j ^ { \prime } = 1 } ^ { j ^ { \prime } = d } \exp ( \psi _ { i , j ^ { \prime } } / \tau ) } .
+$$
+
+In this equation, $\psi$ is a real-valued matrix reparameterizing the routing matrix through a softmax operation with temperature, $\tau$ . The lower the temperature, the more (2) converges to the desired discrete and sparse binary routing matrix. Note that, in the continuous relaxation, the matrix $\psi$ can be optimized via the backpropagation of gradients from classification loss terms. In the next section, we provide further detail on temperature annealing schedules as well as other techniques to enhance the $\Psi$ approximation.
+
+Based on selected groups, we suggest local fully-connected weight layers for each group: Group-FC. The goal of Group-FC is to extract higher-level representations using the selected expressive feature subsets. This operation is usually followed by non-linearity functions (e.g., ReLU), normalization operations (e.g, Batch Norm), and dropout. Formally, Group-FC can be defined as:
+
+$$
+\begin{array} { r } { \pmb { z } _ { i } ^ { l + 1 } = f ( W _ { i } ^ { l } \pmb { z } _ { i } ^ { l } + \pmb { b } _ { i } ^ { l } ) , } \end{array}
+$$
+
+where $W _ { i } ^ { l } \in \Re ^ { m \times m }$ and $\pmb { b } _ { i } ^ { l } \in \Re ^ { m }$ are the weight matrix and bias vector, applied on group $i$ at layer $l$ Here, $f$ represents other subsequent operations such as non-linearity, normalization, and dropout.
+
+Lastly, Group-Pool is defined as an operation which merges representations of two groups into a single group, reducing network width by half while increasing the effective receptive field:
+
+$$
+\begin{array} { r } { z _ { i } ^ { l + 1 } = p o o l ( z _ { i } ^ { l } , z _ { i + k / 2 ^ { l + 1 } } ^ { l } ) , } \end{array}
+$$
+
+where $z _ { i } ^ { l }$ and $z _ { i + k / 2 } ^ { l }$ are the ith group from the first and second halves, respectively; and pool is a pooling function from $\Re ^ { 2 m }$ to $\Re ^ { m }$ . In this study, we explore different variants of pooling functions such as max pooling, average pooling, or using linear weight layers as transformations from $\Re ^ { 2 m }$ to $\Re ^ { m }$ . Please note that while we use a similar terminology as pooling in convolutional networks, the pooling operation explained here is not applied location-wise, but instead it is applied feature-wise, between different groups pairs.
+
+The values of $m$ and $k$ are closely related to the number and order of feature interactions for a certain task. Using proper $m$ and $k$ values enables us to reduce the parameter space while maintaining the model complexity required to solve the task. However, finding the ideal $m$ and $k$ directly from a given dataset is a very challenging problem. In this work, we treat $m$ and $k$ as hyperparameters to be found by a hyperparameter search.
+
+# 3.4 TRAINING
+
+We define the objective function to be used for end-to-end training of weights as well as the routing matrix as:
+
+$$
+L = - \frac { 1 } { N } \sum _ { i } \sum _ { c } y _ { i , c } \log ( F _ { \theta } ( \pmb { x } _ { i } ) ) + \lambda H ( \psi ) + \alpha \sum _ { \omega \in \theta } | | \omega | | _ { 2 } ^ { 2 } .
+$$
+
+In this objective function, the first term is the standard cross-entropy classification loss where $F _ { \theta }$ denotes the GMLP network as a function with parameters $\theta$ , and $N$ is the number of training samples used. The second term is an entropy loss over the distribution of the routing matrix that is weighted by the hyperparameter $\lambda$ :
+
+$$
+H ( \psi ) = - \frac { 1 } { d } \sum _ { j = 1 } ^ { j = d } \sum _ { i = 1 } ^ { i = k m } \frac { \exp ( \psi _ { i , j } ) } { \sum _ { j ^ { \prime } = 1 } ^ { j ^ { \prime } = d } \exp ( \psi _ { i , j ^ { \prime } } ) } \mathrm { l o g } ( \frac { \exp ( \psi _ { i , j } ) } { \sum _ { j ^ { \prime } = 1 } ^ { j ^ { \prime } = d } \exp ( \psi _ { i , j ^ { \prime } } ) } ) .
+$$
+
+$H ( \psi )$ is minimizing the entropy corresponding to the distribution of $\psi$ regardless of the temperature used for $\Psi$ approximation. Accordingly, $\lambda$ can be viewed as a hyperparameter and as an additional method for encouraging sparse $\Psi$ matrices. The last term in (5) is an L-2 regularization term with the hyperparameter $\alpha$ to control the magnitude of parameters in layer weights and in $\psi$ . Note that without the L-2 regularization term, $\psi$ elements may keep increasing during the optimization loop, since $\psi$ only appears in normalized form in the objective function of (5).
+
+We use Adam (Kingma & Ba, 2014) optimization algorithm starting from the default 0.001 learning rate and reducing the learning rate by a factor of 5 as the validation accuracy stops improving. Regarding the temperature annealing, during the training, the temperature is exponentially decayed from 1.0 to 0.01. In order to initialize the Group-FC weights, we used Xavier initialization (Glorot & Bengio, 2010) with m for both fan-in and fan-out values. Similarly, the $\psi$ matrix is initialized by setting the fan-in equal to $d$ and fan-out to $k m$ .
+
+Further detail on architectures and hyperparameters used for each specific experiment as well as details on the software implementation are provided as appendices to this paper.
+
+# 3.5 ANALYSIS
+
+The computational complexity of GMLP at the prediction time can be written as (for simplicity, ignoring bias and pooling terms):
+
+$$
+k m + k m ^ { 2 } + \frac { k m ^ { 2 } } { 2 } + \frac { k m ^ { 2 } } { 4 } + . . . + \frac { k m ^ { 2 } } { 2 ^ { L - 1 } } + C \frac { k m } { 2 ^ { L - 1 } } .
+$$
+
+In this series, the first term, $k m$ , is the work required to organize features to groups. The subsequent terms, except the last term, are representing the computational cost of local fully-connected operations at each layer. The last term is the complexity of the output layer transformation from the concatenated features to the number of classes. Therefore, the computational complexity of GMLP at the prediction time can be written as with a similar network $\begin{array} { r } { \mathcal { O } ( k m ^ { 2 } + \frac { C k m } { 2 L - 1 } ) } \end{array}$ . In comparison, the computational complexity of an MLP
+
+$$
+k m d + k ^ { 2 } m ^ { 2 } + \frac { k ^ { 2 } m ^ { 2 } } { 2 } + \frac { k ^ { 2 } m ^ { 2 } } { 4 } + . . . + \frac { k ^ { 2 } m ^ { 2 } } { 2 ^ { L - 1 } } + C \frac { k m } { 2 ^ { L - 1 } } ,
+$$
+
+where the first term is the work required for the first network layer from $d$ to $k m$ neurons, the second term is corresponding to a hidden layer of size $k m$ , and so forth. The last term is the complexity $\begin{array} { r } { \mathcal { O } ( k m d + { k ^ { 2 } } { m ^ { 2 } } + \frac { C k m } { 2 { L - 1 } } ) } \end{array}$ r to the case of GMLP. The overall work required from this equa) complexity. This is substantially higher than GMLP, for typical $k , d .$ is of, and $C$ values.
+
+Additionally, the density of the Group-FC layer connections can be calculated as: $\begin{array} { r } { \frac { k m ^ { 2 } } { k ^ { 2 } m ^ { 2 } } = \frac { 1 } { k } } \end{array}$ , which is very small for reasonably large number of $k$ values used in our experiments. Also, assuming pooling operations in every other layer, the receptive field size or the maximum number of features impacting a neuron at layer $l$ can be written as $2 ^ { { \hat { l } } - 1 } m$ . For instance, a neuron in the first layer of the network is only connected to $m$ features, and a neuron in the second layer is connected to two groups or $2 m$ features and so forth.
+
+# 4 EXPERIMENTS
+
+# 4.1 EXPERIMENTAL SETUP
+
+The proposed method is evaluated on five different real-world datasets, covering various domains and applications: permutation invariant CIFAR-10 (Krizhevsky et al., 2009), human activity recognition (HAPT) (Anguita et al., 2013), diabetes classification (Kachuee et al., 2019), UCI Landsat (Dua & Graff, 2017), and MIT-BIH arrhythmia classification (Moody & Mark, 2001) datasets. Additionally, we use MNIST (LeCun et al., 2010) and a synthesized dataset to provide further insight into the operation of GMLP (see Section 4.4). Table 1 presents a summary of datasets used in this study. Regarding the CIFAR-10 dataset, we permute the image pixels to discard pixel coordinates in our experiments. Note that the permutation is not changing across samples, it is merely a fixed random ordering used to remove pixel coordinates for each experiment. For all datasets, basic statistical normalization with $\mu = 0$ and $\sigma = 1$ is used to normalize features as a preprocessing step. The only exception is CIFAR-10 for which we used the standard channel-wise normalization and standard data augmentation (i.e., random crops and random horizontal flips). The standard test and train data splits were used as dictated by dataset publishers. In cases that the separated sets are not provided, test and train subsets are created by randomly splitting samples to $2 0 \%$ for test and the rest for training/validation.
+
+Table 1: Summary of datasets used in our experiments.   
+
+<table><tr><td>Dataset</td><td># Train Samples</td><td># Test Samples</td><td>#Features</td><td># Classes</td><td>Domain</td></tr><tr><td>CIFAR-10a (Krizhevsky et al.,2009)</td><td>50,000</td><td>10.000</td><td>3072</td><td>10</td><td>Image Classification</td></tr><tr><td>HAPT (Anguita et al., 2013)</td><td>6,002</td><td>2,451</td><td>561</td><td>5</td><td>Activity Recognition</td></tr><tr><td>Diabetes (Kachuee et al., 2019)</td><td>73,649</td><td>18,413</td><td>45</td><td>3</td><td>Disease Diagnosis</td></tr><tr><td>Landsat (Dua &amp; Graff,2017)</td><td>4,435</td><td>2,000</td><td>36</td><td>6</td><td>Satellite Imaging</td></tr><tr><td>MIT-BIH(Moody &amp; Mark,2001)</td><td>87,554</td><td>21,892</td><td>187</td><td>5</td><td>Time Series</td></tr><tr><td>MNIST (LeCun et al., 2010)</td><td>60,000</td><td>10,000</td><td>784</td><td>10</td><td>Digit Classification</td></tr><tr><td>Synthesized (Section 4.4)</td><td>5,120</td><td>1,280</td><td>6</td><td>2</td><td>See Figure 8</td></tr></table>
+
+aPermuted version, i.e. pixel coordinates are ignored.
+
+We compare the performance of the proposed method with recent related work including SelfNormalizing Neural Networks (SNN) (Klambauer et al., 2017), Sparse Evolutionary Training (SET) (Mocanu et al., $2 0 1 8 ) ^ { 3 }$ , Feature Grouping as a Stochastic Regularizer (in this paper, denoted as FGR) (Aydore et al., 2019)4 as well as the basic dropout regularized and batch normalized MLPs. In order to ensure a fair comparison, we adapted source codes provided by other work to be compatible with our data loader and preprocessing modules.
+
+Furthermore, for each method, we conducted an extensive hyperparameter search using Microsoft Neural Network Intelligence (NNI) toolkit5 and the Tree-structured Parzen Estimator (TPE) tuner (Bergstra et al., 2011) covering different architectural and learning hyperparameters for each case. More detail on hyperparameter search spaces and specific architectures used in this paper is provided in Appendix A and Appendix B. We run each case using the best hyperparameter configuration eight times and report mean and standard deviation values.
+
+# 4.2 RESULTS
+
+Table 2 presents a comparison between the proposed method (GMLP) and 4 other baselines: MLP, SNN (Klambauer et al., 2017), SET (Mocanu et al., 2018), and FGR (Aydore et al., 2019). As it can be seen from this comparison, GMLP outperforms other work, achieving state-of-the-art classification accuracies. Concerning the CIFAR-10 results, to the best of our knowledge, GMLP achieves a new state-of-the-art performance on permutation invariant CIFAR-10 augmented using the standard data augmentation. We believe that leveraging expressive feature groups enables GMLP to consistently perform better across different datasets.
+
+To compare model complexity and performance we conduct an experiment by changing the number of model parameters and reporting the resulting test accuracies. Here, we reduce the number of parameters by reducing the width of each network; i.e. reducing the number of groups and hidden neurons for GMLP and MLP, respectively. Figure 2 shows accuracy versus the number of parameters for the GMLP and MLP baseline on CIFAR-10 and MIT-BIH datasets. Based on this figure, GMLP is able to achieve higher accuracies using significantly less number of parameters. It is consistent with the complexity analysis provided in Section 3.5. Note that in this comparison, we consider the number of parameters involved at the prediction time.
+
+# 4.3 ABLATION STUDY
+
+Figure 3 presents an ablation study comparing the performance of GMLP on CIFAR-10 dataset for networks trained: $( i )$ using both the temperature annealing and the entropy loss objective, (ii) using only temperature annealing without the entropy loss objective, $( i i i )$ using no temperature annealing but using the entropy loss objective, $( i v )$ not using any of the temperature annealing or the entropy loss objective. From this figure, it can be seen that excluding both techniques leads to a significantly lower performance. However, using any of the two techniques leads to relatively similar high accuracies. It is consistent with the intuition that the functionality of these techniques is to encourage learning sparse routing matrices, either using softmax temperatures or entropy regularization to achieve this. In this paper, in order to ensure sparse and low complexity routing matrices, we use both techniques simultaneously as in case $( i )$ .
+
+Table 2: Comparison of top-1 test accuracies for GMLP and other work.   
+
+<table><tr><td></td><td colspan="5">Accuracy (%)</td></tr><tr><td>Dataset</td><td>GMLP</td><td>MLP</td><td>SNNa</td><td>SETb</td><td>FGRc</td></tr><tr><td>CIFAR-10d (Krizhevsky et al., 2009)</td><td>73.76 (±0.14)</td><td>68.15 (±0.56)</td><td>66.88 (±0.30)</td><td>72.71 (±0.29)</td><td>45.90 (±0.40)</td></tr><tr><td>HAPT (Anguita et al., 2013)</td><td>96.34 (±0.19)</td><td>95.73 (±0.38)</td><td>95.47 (±0.09)</td><td>71.35 (±0.74)</td><td>91.57 (±0.31)</td></tr><tr><td>Diabetes (Kachuee et al.,2019)</td><td>90.92 (±0.07)</td><td>90.68 (±0.06)</td><td>90.31 (±0.10)</td><td>85.07 (±0.61)</td><td>89.24 (±0.08)</td></tr><tr><td>Landsat (Dua &amp; Graff,2017)</td><td>91.54 (±0.16)</td><td>91.21 (±0.44)</td><td>91.37 (±0.21)</td><td>91.03 (±0.56)</td><td>90.70 (±0.17)</td></tr><tr><td>MIT-BIH(Moody &amp; Mark,2001)</td><td>98.74 (±0.04)</td><td>98.65 (±0.01)</td><td>98.56 (±0.02)</td><td>98.10 (±0.01)</td><td>98.13 (±0.03)</td></tr></table>
+
+aSelf-Normalizing Neural Networks (Klambauer et al., 2017) bSparse Evolutionary Training (Mocanu et al., 2018) cFeature Grouping as a Stochastic Regularizer (Aydore et al., 2019) dPermuted version, i.e. pixel coordinates are ignored.
+
+![](images/d26c4b320c2c398b02009e440c592db21896536976c571200b75f1d20484e65b.jpg)  
+Figure 2: Accuracy versus number of parameters for this work (GMLP) and the MLP baseline: (a) CIFAR-10 dataset, (b) MIT-BIH dataset. The $\mathbf { X }$ -axis is in a logarithmic scale
+
+Figure 4 shows a comparison between GMLP models trained on CIFAR-10 using different pooling types: $( i )$ linear transformation, $( i i )$ max pooling, and $( i i i )$ average pooling. As it can be seen from this comparison, while there are slight differences in the convergence speed of using different pooling types, all of them achieve relatively similar accuracies. In our experiments, we decided to use max pooling and average pooling as they provide reasonable results without the need to introduce additional parameters required for the linear pooling method.
+
+Figure 5 shows learning curves for training CIFAR-10 GMLP models using different group sizes. As it can be seen from this figure, using very small group sizes would cause a reduction in the final accuracy. At the other extreme, the improvement achieved using larger values is negligible for $m$ values more than 16. Finally, Figure 6 shows a comparison between learning curves for using a different number of groups. Using very small $k$ values result in a significant reduction in performance. However, the rate of performance gains for using more groups is very small for $k$ of more than 1536. Note that the number of model parameters and compute scales linearly with $k$ and quadratically with $m$ (see Section 3.5).
+
+![](images/ec451b5dcc3d15a4b055447f394e6b8be9a5fa4bfe08e4a086a4fdc284d82bb0.jpg)  
+Figure 3: Ablation study demonstrating the impact of temperature annealing and entropy loss terms.
+
+![](images/31eb22c5a65b11f8db055dab3da98be8c4fff744e3c6ce3eb7019fda6b37313a.jpg)  
+Figure 5: Ablation study on the impact of using different group sizes $( m )$ . For this experiment, we used $k { = } 1 5 3 6$ .
+
+![](images/f5d07291db61bdf6e202b4af44e455999a04c9bd2e363b9f778a6200f99f31b6.jpg)  
+Figure 4: Ablation study demonstrating the impact of different pooling functions.   
+Figure 6: Ablation study on the impact of using different number of groups $( k )$ . For this experiment, we used $m { = } 1 6$ .
+
+# 4.4 EXPERIMENTS ON MNIST AND SYNTHESIZED DATA
+
+MNIST dataset (LeCun et al., 2010) is used to visually inspect the performance of the Group-Select layer. Figure 7 shows a heat-map of how frequently each pixel is selected across all feature groups for: $( a )$ original MNIST samples, (b) MNIST samples where the lower-half is replaced by Gaussian noise. From Figure 7a, it can be seen that most groups are selecting pixels within the center of the frame, effectively discarding margin pixels. This is consistent with other work which show the importance of different locations for MNIST images (Kachuee et al., 2018). Apart from this, in Figure 7b, a version of the MNIST dataset is used in which half of the frame does not provide any useful information for the downstream classification task. From this figure, GMLP is not selecting any features to be used from the lower region.
+
+In order to show the effectiveness of GMLP, we synthesized a dataset which has intrinsic and known expressive feature groups. Specifically, we used a simple Bayesian network as depicted in Figure 8. This network consists of six binary features, A to F, interacting with each other as specified by the graph edges, which determine the distribution of the target node, J. The graph and conditionals are designed such that each of the nodes in the second level take the XOR value of their parents with a $9 9 \%$ probability. The target node, J, is essentially one with a high probability if at least two of the second level nodes are one. We synthesized dataset by sampling 6,400 samples from the network (1,280 samples for test and the rest of training/evaluation). On this dataset, we trained a very simple GMLP consisting of four groups of size two, one group-wise fully-connected layer, and an output layer. Figure 9 shows the features selected for each group after the training phase (i.e., the $\Psi$ matrix).
+
+![](images/be34d4ab1e56d0743796f3b972d79052ac656ff4def94a3af94f04280b784cb3.jpg)  
+Figure 7: MNIST visualization of pixels selected by the Group-Select layer: (a) using complete images as input, (b) using images that the lower half is replaced by Gaussian noise. In this figure, warmer colors represent pixels being being present in more groups.
+
+From this figure, the Group-Select layer successfully learns to detect the feature pairs that are interacting, enabling the Group-FC layers to decode the non-linear XOR relations.
+
+![](images/3028e5791c3271e74e91ac8dd61f67f7a45ce591feb1177f4985246c80008720.jpg)
+
+$\begin{array} { l l } { P ( G = A \oplus B ) = 0 . 9 9 } & { , P ( G \neq A \oplus B ) = 0 . 0 1 } \\ { P ( H = C \oplus D ) = 0 . 9 9 } & { , P ( H \neq C \oplus D ) = 0 . 0 1 } \\ { P ( I = E \oplus F ) = 0 . 9 9 } & { , P ( I \neq E \oplus F ) = 0 . 0 1 } \end{array}$ $\begin{array} { l } { P [ J = ( G \land H ) \lor ( G \land I ) \lor ( H \land I ) ] = 0 . 9 9 } \\ { P [ J \neq ( G \land H ) \lor ( G \land I ) \lor ( H \land I ) ] = 0 . 0 1 } \end{array}$
+
+![](images/98b30235aff7dc90e1118b185b0ecf3d8c34a1bc9f40e599bc9dde9753a54ae8.jpg)  
+Figure 8: The Bayesian network and conditionals used to generate the synthesized dataset of binary features A-F and target J.   
+Figure 9: Visualization of the selected features within each group. Every two consecutive rows show features selected for a certain group.
+
+# 5 DISCUSSION
+
+Intuitively, training a GMLP model with certain groups can be viewed as a prior assumption over the number and order of interactions between the features. It is a reasonable prior assumption as in many natural datasets, a conceptual hierarchy exists where only a limited number of features interact with each other. Furthermore, it is consistent with the discoveries made in understanding the human decision-making process; finding that we are only able to consider at most nine factors at the same time during a decision making process (Cowan, 2001; Baddeley, 1994).
+
+Additionally, GMLP can be considered as a more general neural counterpart of random forests. Both models use different subsets of features (i.e., groups) and learn interactions within each group. One major difference between the two methods is the fact that GMLP combines information between different groups using pooling operations, while random forest uses the selected features to train an ensemble of independent trees on each group. From another perspective, the idea of studying feature groups is closely related to causal models such as Bayesian networks and factor graphs (Darwiche, 2009; Neapolitan et al., 2004; Clifford, 1990). These methods are often impractical for large-scale problems, because without a prior over the causal graph, they require an architecture search of the NP-complete complexity or more.
+
+# 6 CONCLUSION
+
+In this paper, we proposed GMLP as a solution for deep learning in domains where the feature interactions are not known as prior and do not admit the use of convolutional or other techniques leveraging domain priors. GMLP jointly learns expressive feature combinations and employs groupwise operations to reduce the network complexity. We conducted extensive experiments demonstrating the effectiveness of the proposed idea and compared the achieved performances with state-of-the-art methods in the literature.
+
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+
+# A HYPERPARAMETER SEARCH SPACE
+
+Tables 3-5 present the hyperparameter search space considered for experiments on GMLP, MLP, SNN, and FGR, respectively. For the GMLP search space, the number of groups is adjusted based on the number of features and samples in each specific task. Also, the number of layers is adjusted to be compatible with the number of groups being used. Regarding the FGR experiments, due to scalability issues of the published source provided by the original authors, we were only able to train networks with at most two hidden layers. For SET, as their architecture is evolutionary i.e., prunes certain weights and adds new ones, we only explored using a different number of hidden neurons in the range of 500 to 4000.
+
+Regarding the number of epochs, we used 2000 epochs for CIFAR-10, 1000 epochs for HAPT, 100 epochs for Diabetes, 300 epochs for Landsat, and 300 epochs for MIT-BIH experiments. The only exception is the SNN experiments where we had to reduce the learning rate to increase the stability of the training resulting in more epochs required to converge.
+
+Table 3: Hyperparameter search space used for GMLP experiments.   
+
+<table><tr><td>Hyperparameter</td><td>Considered Values</td></tr><tr><td>Number of hidden layers</td><td>{1,2,..,8]a</td></tr><tr><td>Number of groups</td><td>16 - 4096b</td></tr><tr><td>Size of groups</td><td>4-32</td></tr><tr><td>Lambda</td><td>[10-²,104]</td></tr><tr><td>Alpha</td><td>[10-12,10-1]</td></tr><tr><td>Dropout rate</td><td>[0,1]</td></tr></table>
+
+aThe range is adjusted based on the number of groups. bThe range is adjusted based on the number of features and sample size.
+
+Table 4: Hyperparameter search space used for MLP and SNN experiments.   
+
+<table><tr><td>Hyperparameter</td><td>Considered Values</td></tr><tr><td>Number of hidden layers</td><td>{1,2,..,6}</td></tr><tr><td>Size of hidden layers</td><td>[0.05 × nfeatures] - [20 X nfeatures]</td></tr><tr><td>Alpha</td><td>[10-12,10-1]</td></tr><tr><td>Dropout rate</td><td>[0,1]</td></tr></table>
+
+Table 5: Hyperparameter search space used for FGR experiments.   
+
+<table><tr><td>Hyperparameter</td><td>Considered Values</td></tr><tr><td>Number of hidden layers</td><td>{1,2}</td></tr><tr><td>Size of hidden layers</td><td>nfeatures - [50 X nfeatures]</td></tr><tr><td>Number of groups</td><td>2- nfeatures</td></tr><tr><td>Alpha</td><td>[10-12,10-1]</td></tr></table>
+
+# B ARCHITECTURES
+
+Table 6,7,8,9,10 show the selected architectures for GMLP, MLP, SNN, SET, and FGR, respectively. We used the following notation to indicate different layer types and parameters: $\mathtt { G S e l - k - m }$ represents a Group-Select layer selecting k groups of m features each. GFC indicates Group-FC layers, and $\operatorname { F C - x }$ represents fully-connected layer with $_ \textrm { x }$ hidden neurons. $\mathtt { G P O O 1 - x }$ is a Group-Pool layer of type $_ \textrm { x }$ (max, mean, linear, etc.). Concat is concatenation of groups used prior to the output layer in GMLP architectures. ${ \mathrm { S C } } ^ { - } \mathrm { x }$ refers to SET sparse evolutionary layer of size $_ \textrm { x }$ .
+
+Table 6: GMLP architectures used in our experiments.   
+
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>Architecture</td></tr><tr><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>GSel-1536-16，GFC，ReLU，BNorm，GPool-maX，GFC，ReLU，BNorm,GPool-maX，GFC，ReLU，BNorm，GPool-max，GFC，ReLU，BNorm,Concat，FC-10，Softmax</td></tr><tr><td rowspan=1 colspan=1>HAPT</td><td rowspan=1 colspan=1>GSel-288-12，GFC，ReLU，BNorm，GPool-mean，GFC，ReLU，BNorm,Concat，FC-5，Softmax</td></tr><tr><td rowspan=1 colspan=1>Diabetes</td><td rowspan=1 colspan=1>GSel-16-16，GFC，ReLU，BNorm，GPool-mean，GFC，ReLU，BNorm, GPool-mean，GFC，ReLU，BNorm，GPool-mean，GFC，ReLU，BNorm,GPool-mean，Concat，FC-3，Softmax</td></tr><tr><td rowspan=1 colspan=1>Landsat</td><td rowspan=1 colspan=1>GSel-88-16，GFC，ReLU，BNorm，GPool-mean，GFC，ReLU，BNorm,GPool-mean，GFC，ReLU，BNorm，GPool-mean，Concat，FC-6，Softmax</td></tr><tr><td rowspan=1 colspan=1>MIT-BIH</td><td rowspan=1 colspan=1>GSel-240-24，GFC，ReLU，BNorm，GPool-mean，GFC，ReLU，BNorm,GPool-mean，GFC，ReLU，BNorm，GPool-mean，Concat，FC-5，Softmax</td></tr><tr><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>GSel-64-16，GFC，ReLU，BNorm，GPool-mean，GFC，ReLU，BNorm,GPool-mean，GFC，ReLU，BNorm，GPool-mean，GFC，ReLU，BNorm,Concat，FC-10，Softmax</td></tr><tr><td rowspan=1 colspan=1>Synthesized</td><td rowspan=1 colspan=1>GSel-4-2，GFC，ReLU，BNorm，Concat，FC-2，Softmax</td></tr></table>
+
+Table 7: MLP architectures used in our experiments.   
+
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>Architecture</td></tr><tr><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>FC-3072，ReLU，BNorm，FC-2764，ReLU，BNorm,FC-2488，ReLU，BNorm，FC-10，Softmax</td></tr><tr><td rowspan=1 colspan=1>HAPT</td><td rowspan=1 colspan=1>FC-106，ReLU，BNorm，FC-21，ReLU，BNorm，FC-5，SoftmaX</td></tr><tr><td rowspan=1 colspan=1>Diabetes</td><td rowspan=1 colspan=1>FC-45，I ReLU，BNorm，FC-45，ReLU，BNorm，FC-45，ReLU，BNorm,FC-45，ReLU，BNorm，FC-3，Softmax</td></tr><tr><td rowspan=1 colspan=1>Landsat</td><td rowspan=1 colspan=1>FC-68， ReLU，BNorm，FC-68，ReLU，BNorm，FC-68，ReLU，BNorm,FC-68，ReLU，BNorm，FC-6，Softmax</td></tr><tr><td rowspan=1 colspan=1>MIT-BIH</td><td rowspan=1 colspan=1>FC-1737，ReLU，BNorm，FC-1737，ReLU，BNorm,FC-1737，ReLU，BNorm，FC-5，Softmax</td></tr></table>
+
+Table 8: SNN architectures used in our experiments.   
+
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>Architecture</td></tr><tr><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>FC-3901，SeLU，BNorm，FC-3901，SeLU，BNorm,FC-3901，SeLU，BNorm，FC-10，Softmax</td></tr><tr><td rowspan=1 colspan=1>HAPT</td><td rowspan=1 colspan=1>FC-510，ReLU，BNorm，FC-510，SeLU，BNorm,FC-510，SeLU，FC-5，Softmax</td></tr><tr><td rowspan=1 colspan=1>Diabetes</td><td rowspan=1 colspan=1>FC-105，SeLU，BNorm，FC-105，SeLU，BNorm,FC-105，SeLU，BNorm，FC-3，Softmax</td></tr><tr><td rowspan=1 colspan=1>Landsat</td><td rowspan=1 colspan=1>FC-816，SeLU，BNorm，FC-816，SeLU，BNorm，FC-6，Softmax</td></tr><tr><td rowspan=1 colspan=1>MIT-BIH</td><td rowspan=1 colspan=1>FC-1140，SeLU，BNorm，FC-1140，SeLU，BNorm,FC-1140，SeLU，BNorm，FC-5，Softmax</td></tr></table>
+
+Table 9: SET architectures used in our experiments.   
+
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1> Architecture</td></tr><tr><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1> SC-4000，SReLU，SC-1000，SReLU，SC-4000，SReLU，FC-10，Softmax</td></tr><tr><td rowspan=1 colspan=1>HAPT</td><td rowspan=1 colspan=1>SC-500，SReLU，SC-500，SReLU，SC-500，SReLU，FC-5，Softmax</td></tr><tr><td rowspan=1 colspan=1>Diabetes</td><td rowspan=1 colspan=1>SC-1000，SReLU，SC-1000，SReLU，SC-1000，SReLU，FC-3，Softmax</td></tr><tr><td rowspan=1 colspan=1>Landsat</td><td rowspan=1 colspan=1>SC-1000，SReLU，SC-1000，SReLU，SC-1000，SReLU，FC-6，Softmax</td></tr><tr><td rowspan=1 colspan=1>MIT-BIH</td><td rowspan=1 colspan=1> SC-1000，SReLU，SC-1000，SReLU，SC-1000，SReLU，FC-5，SoftmaX</td></tr></table>
+
+Table 10: FGR architectures used in our experiments.   
+
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>Architecture</td></tr><tr><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>Group-256，FC-3072，ReLU，FC-10，Softmax</td></tr><tr><td rowspan=1 colspan=1>HAPT</td><td rowspan=1 colspan=1>Group-104，FC-12173，ReLU，FC-5，Softmax</td></tr><tr><td rowspan=1 colspan=1>Diabetes</td><td rowspan=1 colspan=1>Group-44，FC-337，ReLU，FC-3，Softmax</td></tr><tr><td rowspan=1 colspan=1>Landsat</td><td rowspan=1 colspan=1>Group-32，FC-1577，ReLU，FC-6，Softmax</td></tr><tr><td rowspan=1 colspan=1>MIT-BIH</td><td rowspan=1 colspan=1>Group-160，FC-3444，ReLU，FC-5，Softmax</td></tr></table>
+
+# C SOFTWARE IMPLEMENTATION
+
+Table 11 presents the list of software dependencies and versions used in our implementation. To produce results related to this paper, we used a workstation with 4 NVIDIA GeForce RTX-2080Ti GPUs, a 12 core Intel Core i9-7920X processor, and 128 GB memory. Each experiment took between about 30 minutes to 72 hours, based on the task and method being tested.
+
+Table 11: Software dependencies.   
+
+<table><tr><td>Dependency</td><td>Version</td></tr><tr><td>python</td><td>3.7.1</td></tr><tr><td>pytorch</td><td>1.1.0</td></tr><tr><td>torchvision</td><td>0.2.1</td></tr><tr><td>cuda100</td><td>1.0</td></tr><tr><td>ipython</td><td>6.5.0</td></tr><tr><td>jupyter</td><td>1.0.0</td></tr><tr><td>numpy</td><td>1.15.4</td></tr><tr><td>nni</td><td>0.9.1.1</td></tr><tr><td>pandas</td><td>0.23.4</td></tr><tr><td>scikit-learn</td><td>0.19.2</td></tr><tr><td>scipy</td><td>1.1.0</td></tr><tr><td>pomegranate</td><td>0.11.1</td></tr><tr><td>tqdm</td><td>4.32.1</td></tr><tr><td>matplotlib</td><td>3.0.1</td></tr></table>
+
+# D VISUAL ANALYSIS
+
+In Figure 10, we present a visualization of the selected feature for 25 randomly selected groups in our final CIFAR-10 architecture. Red, green, and blue colors indicate which channel is selected for each location. Compared to visualizations that are frequently used for convolutional networks, as GMLP has the flexibility to select pixels at different locations and different color channels, it is not easy to find explicit patterns in this visualization. However, one noticeable pattern is that features selected from a certain color channel usually appear in clusters resembling irregularly shaped patches.
+
+![](images/f18122c74dd123d377ebf80f5e3ca8a9a8d30d8b9b1e3229f4a82137393c758e.jpg)  
+Figure 10: Visualization of pixels selected by each group for the CIFAR-10 GMLP architecture. Red, green, and blue colors indicate which channel is selected for each location. Due to space limitations, 25 random groups out of 1536 total groups visualized here.
+
+Figure 11 shows the frequency in which each CIFAR-10 location is selected by the GMLP network. From this visualization, GMLP is mostly ignoring the border areas which can be a result of the data augmentation process used to train the network i.e., randomly cropping the center area and padding the margins.
+
+![](images/c6de4f1b62f90a17c1b1e28a45ef256082d0acf42fbf91947838a92f007d06e2.jpg)  
+Figure 11: Visualization of pixels selected by the group-select layer for the CIFAR-10 GMLP model. Warmer colors represent features that are being selected more frequently.
\ No newline at end of file
diff --git a/parse/train/SJg4Y3VFPS/SJg4Y3VFPS_content_list.json b/parse/train/SJg4Y3VFPS/SJg4Y3VFPS_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..a3b84fc4ef4dce57a7de6c254c22ef373956f6f3
--- /dev/null
+++ b/parse/train/SJg4Y3VFPS/SJg4Y3VFPS_content_list.json
@@ -0,0 +1,1681 @@
+[
+    {
+        "type": "text",
+        "text": "GROUP-CONNECTED MULTILAYER PERCEPTRON NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            98,
+            905,
+            122
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            145,
+            400,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            209,
+            544,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Despite the success of deep learning in domains such as image, voice, and graphs, there has been little progress in deep representation learning for domains without a known structure between features. For instance, a tabular dataset of different demographic and clinical factors where the feature interactions are not given as a prior. In this paper, we propose Group-Connected Multilayer Perceptron (GMLP) networks to enable deep representation learning in these domains. GMLP is based on the idea of learning expressive feature combinations (groups) and exploiting them to reduce the network complexity by defining local group-wise operations. During the training phase, GMLP learns a sparse feature grouping matrix using temperature annealing softmax with an added entropy loss term to encourage the sparsity. Furthermore, an architecture is suggested which resembles binary trees, where group-wise operations are followed by pooling operations to combine information; reducing the number of groups as the network grows in depth. To evaluate the proposed method, we conducted experiments on five different real-world datasets covering various application areas. Additionally, we provide visualizations on MNIST and synthesized data. According to the results, GMLP is able to successfully learn and exploit expressive feature combinations and achieve state-of-the-art classification performance on different datasets. ",
+        "bbox": [
+            233,
+            242,
+            764,
+            492
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            522,
+            336,
+            537
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep neural networks have been quite successful across various machine learning tasks. However, this advancement has been mostly limited to certain domains. For example in image and voice data, one can leverage domain properties such as location invariance, scale invariance, coherence, etc. via using convolutional layers (Goodfellow et al., 2016). Alternatively, for graph data, graph convolutional networks were suggested to leverage adjacency patterns present in datasets structured as a graph (Kipf & Welling, 2016; Xu et al., 2019). ",
+        "bbox": [
+            174,
+            554,
+            825,
+            638
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, there has been little progress in learning deep representations for datasets that do not follow a particular known structure in the feature domain. Take for instance the case of a simple tabular dataset for disease diagnosis. Such a dataset may consist of features from different categories such as demographics (e.g., age, gender, income, etc.), examinations (e.g., blood pressure, lab results, etc.), and other clinical conditions. In this scenario, the lack of any known structure between features to be used as a prior would lead to the use of a fully-connected multilayer perceptron network (MLP). Nonetheless, it has been known in the literature that MLP architectures, due to their huge complexity, do not usually admit efficient training and generalization for networks of more than a few layers. ",
+        "bbox": [
+            174,
+            645,
+            825,
+            756
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose Group-Connected Multiplayer Perceptron (GMLP) networks. The main idea behind GMLP is to learn and leverage expressive feature subsets, henceforth referred to as feature groups. A feature group is defined as a subset of features that provides a meaningful representation or high-level concept that would help the downstream task. For instance, in the disease diagnosis example, the combination of a certain blood factor and age might be the indicator of a higher level clinical condition which would help the final classification task. Furthermore, GMLP leverages feature groups limiting network connections to local group-wise connections and builds a feature hierarchy via merging groups as the network grows in depth. GMLP can be seen as an architecture that learns expressive feature combinations and leverages them via group-wise operations. ",
+        "bbox": [
+            174,
+            762,
+            825,
+            888
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The main contributions of this paper are as follows: $( i )$ proposing a method for end-to-end learning of expressive feature combinations, $( i i )$ suggesting a network architecture to utilize feature groups and local connections to build deep representations, (iii) conducting extensive experiments demonstrating the effectiveness of GMLP as well as visualizations and ablation studies for better understanding of the suggested architecture. ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            103,
+            823,
+            145
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We evaluated the proposed method on five different real-world datasets in various application domains and demonstrated the effectiveness of GMLP compared to state-of-the-art methods in the literature. Furthermore, we conducted ablation studies and comparisons to study different architectural and training factors as well as visualizations on MNIST and synthesized data. To help to reproduce the results and encouraging future studies on group-connected architectures, we made the source code related to this paper available online 1. ",
+        "bbox": [
+            174,
+            152,
+            825,
+            236
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            275,
+            344,
+            291
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Fully-connected MLPs are the most widely-used neural models for datasets in which no prior assumption is made on the relationship between features. However, due to the huge complexity of fully-connected layers, MLPs are prone to overfitting resulting in shallow architectures limited to a few layers in depth (Goodfellow et al., 2016). Various techniques have been suggested to improve training these models which include regularization techniques such as L-1/L-2 regularization, dropout, etc. and normalization techniques such as layer normalization, weigh normalization, batch normalization, etc.(Srivastava et al., 2014; Ba et al., 2016; Salimans & Kingma, 2016; Ioffe & Szegedy, 2015). For instance, self-normalizing neural networks (SNNs) have been recently suggested as state of the art normalization methods that prevent vanishing or exploding gradients which help training feed-forward networks with higher depths (Klambauer et al., 2017). ",
+        "bbox": [
+            174,
+            318,
+            825,
+            458
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "From the architectural perspective, there has been great attention toward networks consisting of sparse connections between layers rather than having dense fully-connected layers (Dey et al., 2018). Sparse connected neural networks are usually trained based on either a sparse prior structure over the network architecture (Richter & Wattenhofer, 2018) or based on pruning a fully-connected network to a sparse network (Yun et al., 2019; Tartaglione et al., 2018; Mocanu et al., 2018). However, it should be noted that the main objective of most sparse neural network literature has been focused on improving the memory and compute requirements while maintaining competitive accuracies compared to MLPs. ",
+        "bbox": [
+            174,
+            464,
+            825,
+            561
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "As a parallel line of research, the idea of using expressive feature combinations or groups has been suggested as a prior over the feature domain. Perhaps, the most successful and widespread use of this idea is in creating random forest models in which different trees are trained based on different feature subsets in order to deal with high-dimensional and high-variance data (Breiman, 2001). More recently, feature grouping is suggested by Aydore et al. (2019) as a statistical regularization technique to learn from datasets of large feature size and a small number of training samples. They do the forward network computation by projecting input features using samples taken from a bank of feature grouping matrices, reducing the input layer complexity and regularizing the model. In another recent study, Ke et al. (2018) used expressive feature combinations to learn from tabular datasets using a recursive encoder with a shared embedding network. They suggest a recursive architecture in which more important feature groups have a more direct impact on the final prediction. ",
+        "bbox": [
+            174,
+            568,
+            825,
+            722
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "While promising results have been reported using these methods, feature grouping has been mostly considered as a preprocessing step. For instance, Aydore et al. (2019) uses the recursive nearest agglomeration (ReNA) (Hoyos-Idrobo et al., 2018) clustering to determine feature groups prior to the analysis. Alternatively, Ke et al. (2018) defined feature groups based on a pre-trained gradient boosting decision tree (GBDT) (Friedman, 2001). Feature grouping as a preprocessing step not only increases the complexity and raises practical considerations, but also limits the optimality of the selected features in subsequent analysis. In this study, we propose an end-to-end solution to learn expressive feature groups. Moreover, we introduce a network architecture to exploit interrelations within the feature groups to reduce the network complexity and to train deeper representations. ",
+        "bbox": [
+            174,
+            728,
+            825,
+            853
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/c1d29b2cea3d35501f1bef3d00700d77364f884f1a3ac447f57fd5a8e7118523.jpg",
+        "image_caption": [
+            "Figure 1: The GMLP network architecture. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            243,
+            99,
+            754,
+            373
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 PROPOSED METHOD ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            422,
+            377,
+            439
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 ARCHITECTURE OVERVIEW ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            455,
+            405,
+            469
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose GMLP which intuitively can be broken down to three stages: $( i )$ selecting expressive feature groups, $( i i )$ learning dynamics within each group individually, and (iii) merging information between groups as the network grows in depth (see Figure 1). In this architecture, expressive groups are jointly selected during the training phase. Furthermore, GMLP is leveraging feature groups and using local group-wise weight layers to significantly reduce the number of parameters. While the suggested idea can be materialized as different architectures, in the current study, we suggest organization of the network as architectures resembling a binary tree spanning from leaves (i.e., features) to a certain abstraction depth closer to the root2. As the network grows deeper, after each local group-wise weight layer, half of the groups are merged using pooling operations, effectively reducing the width of the network while increasing the receptive field. At the last layer, all features within all groups are concatenated into a dense feature vector fed to the output layer. ",
+        "bbox": [
+            173,
+            481,
+            825,
+            633
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 NOTATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            651,
+            287,
+            665
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We consider the generic problem of supervised classification based on a dataset of feature and target pairs, $\\mathcal { D }$ : $\\left( \\pmb { x } _ { 1 : N } , \\pmb { y } _ { 1 : N } \\right)$ , where $\\pmb { x } _ { i } \\in \\Re ^ { d }$ , $y _ { i } \\in \\{ 1 \\ldots C \\}$ , and $N$ is the number of dataset samples. Furthermore, we define group size, $m$ , as the number of neurons or elements within each group, and group count, $k$ , as the number of selected groups which are essentially subsets of input features. Also, $L$ is used to refer to the total depth of a network. We use $\\pmb { z } _ { i } ^ { l } \\in \\Re ^ { m }$ to refer to activation values of group $i$ in layer $l$ . In this paper, we define all vectors as column vectors. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.3 NETWORK LAYERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            777,
+            348,
+            792
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present the formal definition of different GMLP network layers. The very first layer of the network, Group-Select, is responsible for organizing features into $k$ groups of size $m$ each. A routing matrix, $\\Psi$ , is used for connecting each neuron within each group to exactly one feature in the feature set: ",
+        "bbox": [
+            174,
+            804,
+            823,
+            858
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d5d993d1abcea2f4089759f87e3c16e8a8bb121597d7718a42d7b270068a713d.jpg",
+        "text": "$$\nz _ { 1 : k } ^ { 0 } = \\Psi x ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            459,
+            857,
+            537,
+            876
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\Psi \\in \\{ 0 , 1 \\} ^ { k m \\times d }$ is a sparse matrix determining features that are present in each group. As we are interested in jointly learning $\\Psi$ during the training phase, we use the following continuous relaxation: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7ce04ac6714541927ef25a4e185a055abcdf7dd5f2de98d89a42fd4d31d13a71.jpg",
+        "text": "$$\n\\Psi _ { i , j } \\approx \\frac { \\exp ( \\psi _ { i , j } / \\tau ) } { \\sum _ { j ^ { \\prime } = 1 } ^ { j ^ { \\prime } = d } \\exp ( \\psi _ { i , j ^ { \\prime } } / \\tau ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "In this equation, $\\psi$ is a real-valued matrix reparameterizing the routing matrix through a softmax operation with temperature, $\\tau$ . The lower the temperature, the more (2) converges to the desired discrete and sparse binary routing matrix. Note that, in the continuous relaxation, the matrix $\\psi$ can be optimized via the backpropagation of gradients from classification loss terms. In the next section, we provide further detail on temperature annealing schedules as well as other techniques to enhance the $\\Psi$ approximation. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Based on selected groups, we suggest local fully-connected weight layers for each group: Group-FC. The goal of Group-FC is to extract higher-level representations using the selected expressive feature subsets. This operation is usually followed by non-linearity functions (e.g., ReLU), normalization operations (e.g, Batch Norm), and dropout. Formally, Group-FC can be defined as: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/528c5c51dea8db6e41a83e28bc85f0976ee266751478a07ba8097f4460fa27e5.jpg",
+        "text": "$$\n\\begin{array} { r } { \\pmb { z } _ { i } ^ { l + 1 } = f ( W _ { i } ^ { l } \\pmb { z } _ { i } ^ { l } + \\pmb { b } _ { i } ^ { l } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            421,
+            347,
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+            366
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $W _ { i } ^ { l } \\in \\Re ^ { m \\times m }$ and $\\pmb { b } _ { i } ^ { l } \\in \\Re ^ { m }$ are the weight matrix and bias vector, applied on group $i$ at layer $l$ Here, $f$ represents other subsequent operations such as non-linearity, normalization, and dropout. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Lastly, Group-Pool is defined as an operation which merges representations of two groups into a single group, reducing network width by half while increasing the effective receptive field: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5f3d3761878a518847f8b745a53dcd68eceb8935de4d9bc26c73c29baf7c9492.jpg",
+        "text": "$$\n\\begin{array} { r } { z _ { i } ^ { l + 1 } = p o o l ( z _ { i } ^ { l } , z _ { i + k / 2 ^ { l + 1 } } ^ { l } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            403,
+            445,
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+            467
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $z _ { i } ^ { l }$ and $z _ { i + k / 2 } ^ { l }$ are the ith group from the first and second halves, respectively; and pool is a pooling function from $\\Re ^ { 2 m }$ to $\\Re ^ { m }$ . In this study, we explore different variants of pooling functions such as max pooling, average pooling, or using linear weight layers as transformations from $\\Re ^ { 2 m }$ to $\\Re ^ { m }$ . Please note that while we use a similar terminology as pooling in convolutional networks, the pooling operation explained here is not applied location-wise, but instead it is applied feature-wise, between different groups pairs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The values of $m$ and $k$ are closely related to the number and order of feature interactions for a certain task. Using proper $m$ and $k$ values enables us to reduce the parameter space while maintaining the model complexity required to solve the task. However, finding the ideal $m$ and $k$ directly from a given dataset is a very challenging problem. In this work, we treat $m$ and $k$ as hyperparameters to be found by a hyperparameter search. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.4 TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We define the objective function to be used for end-to-end training of weights as well as the routing matrix as: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/45ae1efe1774739a1fb528c5640a32343bd17fde145c07952a96a552502cfed8.jpg",
+        "text": "$$\nL = - \\frac { 1 } { N } \\sum _ { i } \\sum _ { c } y _ { i , c } \\log ( F _ { \\theta } ( \\pmb { x } _ { i } ) ) + \\lambda H ( \\psi ) + \\alpha \\sum _ { \\omega \\in \\theta } | | \\omega | | _ { 2 } ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this objective function, the first term is the standard cross-entropy classification loss where $F _ { \\theta }$ denotes the GMLP network as a function with parameters $\\theta$ , and $N$ is the number of training samples used. The second term is an entropy loss over the distribution of the routing matrix that is weighted by the hyperparameter $\\lambda$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/80833478d96310b008fb4d7c5133b52fa8c2bd330ad915eb54322db0ba2e61ae.jpg",
+        "text": "$$\nH ( \\psi ) = - \\frac { 1 } { d } \\sum _ { j = 1 } ^ { j = d } \\sum _ { i = 1 } ^ { i = k m } \\frac { \\exp ( \\psi _ { i , j } ) } { \\sum _ { j ^ { \\prime } = 1 } ^ { j ^ { \\prime } = d } \\exp ( \\psi _ { i , j ^ { \\prime } } ) } \\mathrm { l o g } ( \\frac { \\exp ( \\psi _ { i , j } ) } { \\sum _ { j ^ { \\prime } = 1 } ^ { j ^ { \\prime } = d } \\exp ( \\psi _ { i , j ^ { \\prime } } ) } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$H ( \\psi )$ is minimizing the entropy corresponding to the distribution of $\\psi$ regardless of the temperature used for $\\Psi$ approximation. Accordingly, $\\lambda$ can be viewed as a hyperparameter and as an additional method for encouraging sparse $\\Psi$ matrices. The last term in (5) is an L-2 regularization term with the hyperparameter $\\alpha$ to control the magnitude of parameters in layer weights and in $\\psi$ . Note that without the L-2 regularization term, $\\psi$ elements may keep increasing during the optimization loop, since $\\psi$ only appears in normalized form in the objective function of (5). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We use Adam (Kingma & Ba, 2014) optimization algorithm starting from the default 0.001 learning rate and reducing the learning rate by a factor of 5 as the validation accuracy stops improving. Regarding the temperature annealing, during the training, the temperature is exponentially decayed from 1.0 to 0.01. In order to initialize the Group-FC weights, we used Xavier initialization (Glorot & Bengio, 2010) with m for both fan-in and fan-out values. Similarly, the $\\psi$ matrix is initialized by setting the fan-in equal to $d$ and fan-out to $k m$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Further detail on architectures and hyperparameters used for each specific experiment as well as details on the software implementation are provided as appendices to this paper. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.5 ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The computational complexity of GMLP at the prediction time can be written as (for simplicity, ignoring bias and pooling terms): ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/20d4348b2b09ac791f02e3029c19618100a3f737e512af23a7c3ed32ed1be77d.jpg",
+        "text": "$$\nk m + k m ^ { 2 } + \\frac { k m ^ { 2 } } { 2 } + \\frac { k m ^ { 2 } } { 4 } + . . . + \\frac { k m ^ { 2 } } { 2 ^ { L - 1 } } + C \\frac { k m } { 2 ^ { L - 1 } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this series, the first term, $k m$ , is the work required to organize features to groups. The subsequent terms, except the last term, are representing the computational cost of local fully-connected operations at each layer. The last term is the complexity of the output layer transformation from the concatenated features to the number of classes. Therefore, the computational complexity of GMLP at the prediction time can be written as with a similar network $\\begin{array} { r } { \\mathcal { O } ( k m ^ { 2 } + \\frac { C k m } { 2 L - 1 } ) } \\end{array}$ . In comparison, the computational complexity of an MLP ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4e717f6fd1b8c1e031deed818e8c38578330f5e5ed38fd0e41700bc75be38497.jpg",
+        "text": "$$\nk m d + k ^ { 2 } m ^ { 2 } + \\frac { k ^ { 2 } m ^ { 2 } } { 2 } + \\frac { k ^ { 2 } m ^ { 2 } } { 4 } + . . . + \\frac { k ^ { 2 } m ^ { 2 } } { 2 ^ { L - 1 } } + C \\frac { k m } { 2 ^ { L - 1 } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            464,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where the first term is the work required for the first network layer from $d$ to $k m$ neurons, the second term is corresponding to a hidden layer of size $k m$ , and so forth. The last term is the complexity $\\begin{array} { r } { \\mathcal { O } ( k m d + { k ^ { 2 } } { m ^ { 2 } } + \\frac { C k m } { 2 { L - 1 } } ) } \\end{array}$ r to the case of GMLP. The overall work required from this equa) complexity. This is substantially higher than GMLP, for typical $k , d .$ is of, and $C$ values. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Additionally, the density of the Group-FC layer connections can be calculated as: $\\begin{array} { r } { \\frac { k m ^ { 2 } } { k ^ { 2 } m ^ { 2 } } = \\frac { 1 } { k } } \\end{array}$ , which is very small for reasonably large number of $k$ values used in our experiments. Also, assuming pooling operations in every other layer, the receptive field size or the maximum number of features impacting a neuron at layer $l$ can be written as $2 ^ { { \\hat { l } } - 1 } m$ . For instance, a neuron in the first layer of the network is only connected to $m$ features, and a neuron in the second layer is connected to two groups or $2 m$ features and so forth. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The proposed method is evaluated on five different real-world datasets, covering various domains and applications: permutation invariant CIFAR-10 (Krizhevsky et al., 2009), human activity recognition (HAPT) (Anguita et al., 2013), diabetes classification (Kachuee et al., 2019), UCI Landsat (Dua & Graff, 2017), and MIT-BIH arrhythmia classification (Moody & Mark, 2001) datasets. Additionally, we use MNIST (LeCun et al., 2010) and a synthesized dataset to provide further insight into the operation of GMLP (see Section 4.4). Table 1 presents a summary of datasets used in this study. Regarding the CIFAR-10 dataset, we permute the image pixels to discard pixel coordinates in our experiments. Note that the permutation is not changing across samples, it is merely a fixed random ordering used to remove pixel coordinates for each experiment. For all datasets, basic statistical normalization with $\\mu = 0$ and $\\sigma = 1$ is used to normalize features as a preprocessing step. The only exception is CIFAR-10 for which we used the standard channel-wise normalization and standard data augmentation (i.e., random crops and random horizontal flips). The standard test and train data splits were used as dictated by dataset publishers. In cases that the separated sets are not provided, test and train subsets are created by randomly splitting samples to $2 0 \\%$ for test and the rest for training/validation. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/08c80a9795d1043d51615805605cd8dd69d06e3efea202634d736cea292c1a27.jpg",
+        "table_caption": [
+            "Table 1: Summary of datasets used in our experiments. "
+        ],
+        "table_footnote": [
+            "aPermuted version, i.e. pixel coordinates are ignored. "
+        ],
+        "table_body": "<table><tr><td>Dataset</td><td># Train Samples</td><td># Test Samples</td><td>#Features</td><td># Classes</td><td>Domain</td></tr><tr><td>CIFAR-10a (Krizhevsky et al.,2009)</td><td>50,000</td><td>10.000</td><td>3072</td><td>10</td><td>Image Classification</td></tr><tr><td>HAPT (Anguita et al., 2013)</td><td>6,002</td><td>2,451</td><td>561</td><td>5</td><td>Activity Recognition</td></tr><tr><td>Diabetes (Kachuee et al., 2019)</td><td>73,649</td><td>18,413</td><td>45</td><td>3</td><td>Disease Diagnosis</td></tr><tr><td>Landsat (Dua &amp; Graff,2017)</td><td>4,435</td><td>2,000</td><td>36</td><td>6</td><td>Satellite Imaging</td></tr><tr><td>MIT-BIH(Moody &amp; Mark,2001)</td><td>87,554</td><td>21,892</td><td>187</td><td>5</td><td>Time Series</td></tr><tr><td>MNIST (LeCun et al., 2010)</td><td>60,000</td><td>10,000</td><td>784</td><td>10</td><td>Digit Classification</td></tr><tr><td>Synthesized (Section 4.4)</td><td>5,120</td><td>1,280</td><td>6</td><td>2</td><td>See Figure 8</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "We compare the performance of the proposed method with recent related work including SelfNormalizing Neural Networks (SNN) (Klambauer et al., 2017), Sparse Evolutionary Training (SET) (Mocanu et al., $2 0 1 8 ) ^ { 3 }$ , Feature Grouping as a Stochastic Regularizer (in this paper, denoted as FGR) (Aydore et al., 2019)4 as well as the basic dropout regularized and batch normalized MLPs. In order to ensure a fair comparison, we adapted source codes provided by other work to be compatible with our data loader and preprocessing modules. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, for each method, we conducted an extensive hyperparameter search using Microsoft Neural Network Intelligence (NNI) toolkit5 and the Tree-structured Parzen Estimator (TPE) tuner (Bergstra et al., 2011) covering different architectural and learning hyperparameters for each case. More detail on hyperparameter search spaces and specific architectures used in this paper is provided in Appendix A and Appendix B. We run each case using the best hyperparameter configuration eight times and report mean and standard deviation values. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.2 RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 2 presents a comparison between the proposed method (GMLP) and 4 other baselines: MLP, SNN (Klambauer et al., 2017), SET (Mocanu et al., 2018), and FGR (Aydore et al., 2019). As it can be seen from this comparison, GMLP outperforms other work, achieving state-of-the-art classification accuracies. Concerning the CIFAR-10 results, to the best of our knowledge, GMLP achieves a new state-of-the-art performance on permutation invariant CIFAR-10 augmented using the standard data augmentation. We believe that leveraging expressive feature groups enables GMLP to consistently perform better across different datasets. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To compare model complexity and performance we conduct an experiment by changing the number of model parameters and reporting the resulting test accuracies. Here, we reduce the number of parameters by reducing the width of each network; i.e. reducing the number of groups and hidden neurons for GMLP and MLP, respectively. Figure 2 shows accuracy versus the number of parameters for the GMLP and MLP baseline on CIFAR-10 and MIT-BIH datasets. Based on this figure, GMLP is able to achieve higher accuracies using significantly less number of parameters. It is consistent with the complexity analysis provided in Section 3.5. Note that in this comparison, we consider the number of parameters involved at the prediction time. ",
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+    {
+        "type": "text",
+        "text": "4.3 ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Figure 3 presents an ablation study comparing the performance of GMLP on CIFAR-10 dataset for networks trained: $( i )$ using both the temperature annealing and the entropy loss objective, (ii) using only temperature annealing without the entropy loss objective, $( i i i )$ using no temperature annealing but using the entropy loss objective, $( i v )$ not using any of the temperature annealing or the entropy loss objective. From this figure, it can be seen that excluding both techniques leads to a significantly lower performance. However, using any of the two techniques leads to relatively similar high accuracies. It is consistent with the intuition that the functionality of these techniques is to encourage learning sparse routing matrices, either using softmax temperatures or entropy regularization to achieve this. In this paper, in order to ensure sparse and low complexity routing matrices, we use both techniques simultaneously as in case $( i )$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/4f729724ff918cdf5530e6d79c7d6dc399d912f4bbe63b0c42f7d3c7d3227ff5.jpg",
+        "table_caption": [
+            "Table 2: Comparison of top-1 test accuracies for GMLP and other work. "
+        ],
+        "table_footnote": [
+            "aSelf-Normalizing Neural Networks (Klambauer et al., 2017) bSparse Evolutionary Training (Mocanu et al., 2018) cFeature Grouping as a Stochastic Regularizer (Aydore et al., 2019) dPermuted version, i.e. pixel coordinates are ignored. "
+        ],
+        "table_body": "<table><tr><td></td><td colspan=\"5\">Accuracy (%)</td></tr><tr><td>Dataset</td><td>GMLP</td><td>MLP</td><td>SNNa</td><td>SETb</td><td>FGRc</td></tr><tr><td>CIFAR-10d (Krizhevsky et al., 2009)</td><td>73.76 (±0.14)</td><td>68.15 (±0.56)</td><td>66.88 (±0.30)</td><td>72.71 (±0.29)</td><td>45.90 (±0.40)</td></tr><tr><td>HAPT (Anguita et al., 2013)</td><td>96.34 (±0.19)</td><td>95.73 (±0.38)</td><td>95.47 (±0.09)</td><td>71.35 (±0.74)</td><td>91.57 (±0.31)</td></tr><tr><td>Diabetes (Kachuee et al.,2019)</td><td>90.92 (±0.07)</td><td>90.68 (±0.06)</td><td>90.31 (±0.10)</td><td>85.07 (±0.61)</td><td>89.24 (±0.08)</td></tr><tr><td>Landsat (Dua &amp; Graff,2017)</td><td>91.54 (±0.16)</td><td>91.21 (±0.44)</td><td>91.37 (±0.21)</td><td>91.03 (±0.56)</td><td>90.70 (±0.17)</td></tr><tr><td>MIT-BIH(Moody &amp; Mark,2001)</td><td>98.74 (±0.04)</td><td>98.65 (±0.01)</td><td>98.56 (±0.02)</td><td>98.10 (±0.01)</td><td>98.13 (±0.03)</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/d26c4b320c2c398b02009e440c592db21896536976c571200b75f1d20484e65b.jpg",
+        "image_caption": [
+            "Figure 2: Accuracy versus number of parameters for this work (GMLP) and the MLP baseline: (a) CIFAR-10 dataset, (b) MIT-BIH dataset. The $\\mathbf { X }$ -axis is in a logarithmic scale "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Figure 4 shows a comparison between GMLP models trained on CIFAR-10 using different pooling types: $( i )$ linear transformation, $( i i )$ max pooling, and $( i i i )$ average pooling. As it can be seen from this comparison, while there are slight differences in the convergence speed of using different pooling types, all of them achieve relatively similar accuracies. In our experiments, we decided to use max pooling and average pooling as they provide reasonable results without the need to introduce additional parameters required for the linear pooling method. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Figure 5 shows learning curves for training CIFAR-10 GMLP models using different group sizes. As it can be seen from this figure, using very small group sizes would cause a reduction in the final accuracy. At the other extreme, the improvement achieved using larger values is negligible for $m$ values more than 16. Finally, Figure 6 shows a comparison between learning curves for using a different number of groups. Using very small $k$ values result in a significant reduction in performance. However, the rate of performance gains for using more groups is very small for $k$ of more than 1536. Note that the number of model parameters and compute scales linearly with $k$ and quadratically with $m$ (see Section 3.5). ",
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+    {
+        "type": "image",
+        "img_path": "images/ec451b5dcc3d15a4b055447f394e6b8be9a5fa4bfe08e4a086a4fdc284d82bb0.jpg",
+        "image_caption": [
+            "Figure 3: Ablation study demonstrating the impact of temperature annealing and entropy loss terms. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
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+        "img_path": "images/31eb22c5a65b11f8db055dab3da98be8c4fff744e3c6ce3eb7019fda6b37313a.jpg",
+        "image_caption": [
+            "Figure 5: Ablation study on the impact of using different group sizes $( m )$ . For this experiment, we used $k { = } 1 5 3 6$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/f5d07291db61bdf6e202b4af44e455999a04c9bd2e363b9f778a6200f99f31b6.jpg",
+        "image_caption": [
+            "Figure 4: Ablation study demonstrating the impact of different pooling functions. ",
+            "Figure 6: Ablation study on the impact of using different number of groups $( k )$ . For this experiment, we used $m { = } 1 6$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.4 EXPERIMENTS ON MNIST AND SYNTHESIZED DATA ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "MNIST dataset (LeCun et al., 2010) is used to visually inspect the performance of the Group-Select layer. Figure 7 shows a heat-map of how frequently each pixel is selected across all feature groups for: $( a )$ original MNIST samples, (b) MNIST samples where the lower-half is replaced by Gaussian noise. From Figure 7a, it can be seen that most groups are selecting pixels within the center of the frame, effectively discarding margin pixels. This is consistent with other work which show the importance of different locations for MNIST images (Kachuee et al., 2018). Apart from this, in Figure 7b, a version of the MNIST dataset is used in which half of the frame does not provide any useful information for the downstream classification task. From this figure, GMLP is not selecting any features to be used from the lower region. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In order to show the effectiveness of GMLP, we synthesized a dataset which has intrinsic and known expressive feature groups. Specifically, we used a simple Bayesian network as depicted in Figure 8. This network consists of six binary features, A to F, interacting with each other as specified by the graph edges, which determine the distribution of the target node, J. The graph and conditionals are designed such that each of the nodes in the second level take the XOR value of their parents with a $9 9 \\%$ probability. The target node, J, is essentially one with a high probability if at least two of the second level nodes are one. We synthesized dataset by sampling 6,400 samples from the network (1,280 samples for test and the rest of training/evaluation). On this dataset, we trained a very simple GMLP consisting of four groups of size two, one group-wise fully-connected layer, and an output layer. Figure 9 shows the features selected for each group after the training phase (i.e., the $\\Psi$ matrix). ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/be34d4ab1e56d0743796f3b972d79052ac656ff4def94a3af94f04280b784cb3.jpg",
+        "image_caption": [
+            "Figure 7: MNIST visualization of pixels selected by the Group-Select layer: (a) using complete images as input, (b) using images that the lower half is replaced by Gaussian noise. In this figure, warmer colors represent pixels being being present in more groups. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "From this figure, the Group-Select layer successfully learns to detect the feature pairs that are interacting, enabling the Group-FC layers to decode the non-linear XOR relations. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "image",
+        "img_path": "images/3028e5791c3271e74e91ac8dd61f67f7a45ce591feb1177f4985246c80008720.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+    },
+    {
+        "type": "text",
+        "text": "$\\begin{array} { l l } { P ( G = A \\oplus B ) = 0 . 9 9 } & { , P ( G \\neq A \\oplus B ) = 0 . 0 1 } \\\\ { P ( H = C \\oplus D ) = 0 . 9 9 } & { , P ( H \\neq C \\oplus D ) = 0 . 0 1 } \\\\ { P ( I = E \\oplus F ) = 0 . 9 9 } & { , P ( I \\neq E \\oplus F ) = 0 . 0 1 } \\end{array}$ $\\begin{array} { l } { P [ J = ( G \\land H ) \\lor ( G \\land I ) \\lor ( H \\land I ) ] = 0 . 9 9 } \\\\ { P [ J \\neq ( G \\land H ) \\lor ( G \\land I ) \\lor ( H \\land I ) ] = 0 . 0 1 } \\end{array}$ ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/98b30235aff7dc90e1118b185b0ecf3d8c34a1bc9f40e599bc9dde9753a54ae8.jpg",
+        "image_caption": [
+            "Figure 8: The Bayesian network and conditionals used to generate the synthesized dataset of binary features A-F and target J. ",
+            "Figure 9: Visualization of the selected features within each group. Every two consecutive rows show features selected for a certain group. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Intuitively, training a GMLP model with certain groups can be viewed as a prior assumption over the number and order of interactions between the features. It is a reasonable prior assumption as in many natural datasets, a conceptual hierarchy exists where only a limited number of features interact with each other. Furthermore, it is consistent with the discoveries made in understanding the human decision-making process; finding that we are only able to consider at most nine factors at the same time during a decision making process (Cowan, 2001; Baddeley, 1994). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Additionally, GMLP can be considered as a more general neural counterpart of random forests. Both models use different subsets of features (i.e., groups) and learn interactions within each group. One major difference between the two methods is the fact that GMLP combines information between different groups using pooling operations, while random forest uses the selected features to train an ensemble of independent trees on each group. From another perspective, the idea of studying feature groups is closely related to causal models such as Bayesian networks and factor graphs (Darwiche, 2009; Neapolitan et al., 2004; Clifford, 1990). These methods are often impractical for large-scale problems, because without a prior over the causal graph, they require an architecture search of the NP-complete complexity or more. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we proposed GMLP as a solution for deep learning in domains where the feature interactions are not known as prior and do not admit the use of convolutional or other techniques leveraging domain priors. GMLP jointly learns expressive feature combinations and employs groupwise operations to reduce the network complexity. We conducted extensive experiments demonstrating the effectiveness of the proposed idea and compared the achieved performances with state-of-the-art methods in the literature. ",
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+    {
+        "type": "text",
+        "text": "A HYPERPARAMETER SEARCH SPACE ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 11
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+    {
+        "type": "text",
+        "text": "Tables 3-5 present the hyperparameter search space considered for experiments on GMLP, MLP, SNN, and FGR, respectively. For the GMLP search space, the number of groups is adjusted based on the number of features and samples in each specific task. Also, the number of layers is adjusted to be compatible with the number of groups being used. Regarding the FGR experiments, due to scalability issues of the published source provided by the original authors, we were only able to train networks with at most two hidden layers. For SET, as their architecture is evolutionary i.e., prunes certain weights and adds new ones, we only explored using a different number of hidden neurons in the range of 500 to 4000. ",
+        "bbox": [
+            174,
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Regarding the number of epochs, we used 2000 epochs for CIFAR-10, 1000 epochs for HAPT, 100 epochs for Diabetes, 300 epochs for Landsat, and 300 epochs for MIT-BIH experiments. The only exception is the SNN experiments where we had to reduce the learning rate to increase the stability of the training resulting in more epochs required to converge. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/df7b75211edc15ca96c8f9e50dcd25e7b6449637f6b805138e78cfa2c33b33f9.jpg",
+        "table_caption": [
+            "Table 3: Hyperparameter search space used for GMLP experiments. "
+        ],
+        "table_footnote": [
+            "aThe range is adjusted based on the number of groups. bThe range is adjusted based on the number of features and sample size. "
+        ],
+        "table_body": "<table><tr><td>Hyperparameter</td><td>Considered Values</td></tr><tr><td>Number of hidden layers</td><td>{1,2,..,8]a</td></tr><tr><td>Number of groups</td><td>16 - 4096b</td></tr><tr><td>Size of groups</td><td>4-32</td></tr><tr><td>Lambda</td><td>[10-²,104]</td></tr><tr><td>Alpha</td><td>[10-12,10-1]</td></tr><tr><td>Dropout rate</td><td>[0,1]</td></tr></table>",
+        "bbox": [
+            330,
+            347,
+            665,
+            493
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/7c4ef75dbc564ee535449d29e5bf4b668d37663b820d718142ddf28c94ca0257.jpg",
+        "table_caption": [
+            "Table 4: Hyperparameter search space used for MLP and SNN experiments. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter</td><td>Considered Values</td></tr><tr><td>Number of hidden layers</td><td>{1,2,..,6}</td></tr><tr><td>Size of hidden layers</td><td>[0.05 × nfeatures] - [20 X nfeatures]</td></tr><tr><td>Alpha</td><td>[10-12,10-1]</td></tr><tr><td>Dropout rate</td><td>[0,1]</td></tr></table>",
+        "bbox": [
+            271,
+            578,
+            725,
+            686
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/cb6f93ec8332dd22ee97a46187b5bb083e473c0129e0db9513a280d929fec982.jpg",
+        "table_caption": [
+            "Table 5: Hyperparameter search space used for FGR experiments. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter</td><td>Considered Values</td></tr><tr><td>Number of hidden layers</td><td>{1,2}</td></tr><tr><td>Size of hidden layers</td><td>nfeatures - [50 X nfeatures]</td></tr><tr><td>Number of groups</td><td>2- nfeatures</td></tr><tr><td>Alpha</td><td>[10-12,10-1]</td></tr></table>",
+        "bbox": [
+            299,
+            733,
+            697,
+            842
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B ARCHITECTURES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            352,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 6,7,8,9,10 show the selected architectures for GMLP, MLP, SNN, SET, and FGR, respectively. We used the following notation to indicate different layer types and parameters: $\\mathtt { G S e l - k - m }$ represents a Group-Select layer selecting k groups of m features each. GFC indicates Group-FC layers, and $\\operatorname { F C - x }$ represents fully-connected layer with $_ \\textrm { x }$ hidden neurons. $\\mathtt { G P O O 1 - x }$ is a Group-Pool layer of type $_ \\textrm { x }$ (max, mean, linear, etc.). Concat is concatenation of groups used prior to the output layer in GMLP architectures. ${ \\mathrm { S C } } ^ { - } \\mathrm { x }$ refers to SET sparse evolutionary layer of size $_ \\textrm { x }$ . ",
+        "bbox": [
+            173,
+            133,
+            826,
+            218
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/b72a23e662f3fdd92762a6fe13068edad5e5365c8213fa89b96993c8bb464f26.jpg",
+        "table_caption": [
+            "Table 6: GMLP architectures used in our experiments. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>Architecture</td></tr><tr><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>GSel-1536-16，GFC，ReLU，BNorm，GPool-maX，GFC，ReLU，BNorm,GPool-maX，GFC，ReLU，BNorm，GPool-max，GFC，ReLU，BNorm,Concat，FC-10，Softmax</td></tr><tr><td rowspan=1 colspan=1>HAPT</td><td rowspan=1 colspan=1>GSel-288-12，GFC，ReLU，BNorm，GPool-mean，GFC，ReLU，BNorm,Concat，FC-5，Softmax</td></tr><tr><td rowspan=1 colspan=1>Diabetes</td><td rowspan=1 colspan=1>GSel-16-16，GFC，ReLU，BNorm，GPool-mean，GFC，ReLU，BNorm, GPool-mean，GFC，ReLU，BNorm，GPool-mean，GFC，ReLU，BNorm,GPool-mean，Concat，FC-3，Softmax</td></tr><tr><td rowspan=1 colspan=1>Landsat</td><td rowspan=1 colspan=1>GSel-88-16，GFC，ReLU，BNorm，GPool-mean，GFC，ReLU，BNorm,GPool-mean，GFC，ReLU，BNorm，GPool-mean，Concat，FC-6，Softmax</td></tr><tr><td rowspan=1 colspan=1>MIT-BIH</td><td rowspan=1 colspan=1>GSel-240-24，GFC，ReLU，BNorm，GPool-mean，GFC，ReLU，BNorm,GPool-mean，GFC，ReLU，BNorm，GPool-mean，Concat，FC-5，Softmax</td></tr><tr><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>GSel-64-16，GFC，ReLU，BNorm，GPool-mean，GFC，ReLU，BNorm,GPool-mean，GFC，ReLU，BNorm，GPool-mean，GFC，ReLU，BNorm,Concat，FC-10，Softmax</td></tr><tr><td rowspan=1 colspan=1>Synthesized</td><td rowspan=1 colspan=1>GSel-4-2，GFC，ReLU，BNorm，Concat，FC-2，Softmax</td></tr></table>",
+        "bbox": [
+            174,
+            257,
+            823,
+            556
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/caeaf4ca5c1ffedd5f0a0a23fc391383e20a0aad09e92a6724280d43754fd070.jpg",
+        "table_caption": [
+            "Table 7: MLP architectures used in our experiments. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>Architecture</td></tr><tr><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>FC-3072，ReLU，BNorm，FC-2764，ReLU，BNorm,FC-2488，ReLU，BNorm，FC-10，Softmax</td></tr><tr><td rowspan=1 colspan=1>HAPT</td><td rowspan=1 colspan=1>FC-106，ReLU，BNorm，FC-21，ReLU，BNorm，FC-5，SoftmaX</td></tr><tr><td rowspan=1 colspan=1>Diabetes</td><td rowspan=1 colspan=1>FC-45，I ReLU，BNorm，FC-45，ReLU，BNorm，FC-45，ReLU，BNorm,FC-45，ReLU，BNorm，FC-3，Softmax</td></tr><tr><td rowspan=1 colspan=1>Landsat</td><td rowspan=1 colspan=1>FC-68， ReLU，BNorm，FC-68，ReLU，BNorm，FC-68，ReLU，BNorm,FC-68，ReLU，BNorm，FC-6，Softmax</td></tr><tr><td rowspan=1 colspan=1>MIT-BIH</td><td rowspan=1 colspan=1>FC-1737，ReLU，BNorm，FC-1737，ReLU，BNorm,FC-1737，ReLU，BNorm，FC-5，Softmax</td></tr></table>",
+        "bbox": [
+            176,
+            606,
+            821,
+            797
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/75f80422c9a796bded563caa5246b3a2877c91162e47a23f23c745013c2a76aa.jpg",
+        "table_caption": [
+            "Table 8: SNN architectures used in our experiments. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>Architecture</td></tr><tr><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>FC-3901，SeLU，BNorm，FC-3901，SeLU，BNorm,FC-3901，SeLU，BNorm，FC-10，Softmax</td></tr><tr><td rowspan=1 colspan=1>HAPT</td><td rowspan=1 colspan=1>FC-510，ReLU，BNorm，FC-510，SeLU，BNorm,FC-510，SeLU，FC-5，Softmax</td></tr><tr><td rowspan=1 colspan=1>Diabetes</td><td rowspan=1 colspan=1>FC-105，SeLU，BNorm，FC-105，SeLU，BNorm,FC-105，SeLU，BNorm，FC-3，Softmax</td></tr><tr><td rowspan=1 colspan=1>Landsat</td><td rowspan=1 colspan=1>FC-816，SeLU，BNorm，FC-816，SeLU，BNorm，FC-6，Softmax</td></tr><tr><td rowspan=1 colspan=1>MIT-BIH</td><td rowspan=1 colspan=1>FC-1140，SeLU，BNorm，FC-1140，SeLU，BNorm,FC-1140，SeLU，BNorm，FC-5，Softmax</td></tr></table>",
+        "bbox": [
+            176,
+            174,
+            821,
+            377
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/0a868bee8c28257311ab48facc8e4c3379f4b29c5ca62284254ede93a0f84ebd.jpg",
+        "table_caption": [
+            "Table 9: SET architectures used in our experiments. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1> Architecture</td></tr><tr><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1> SC-4000，SReLU，SC-1000，SReLU，SC-4000，SReLU，FC-10，Softmax</td></tr><tr><td rowspan=1 colspan=1>HAPT</td><td rowspan=1 colspan=1>SC-500，SReLU，SC-500，SReLU，SC-500，SReLU，FC-5，Softmax</td></tr><tr><td rowspan=1 colspan=1>Diabetes</td><td rowspan=1 colspan=1>SC-1000，SReLU，SC-1000，SReLU，SC-1000，SReLU，FC-3，Softmax</td></tr><tr><td rowspan=1 colspan=1>Landsat</td><td rowspan=1 colspan=1>SC-1000，SReLU，SC-1000，SReLU，SC-1000，SReLU，FC-6，Softmax</td></tr><tr><td rowspan=1 colspan=1>MIT-BIH</td><td rowspan=1 colspan=1> SC-1000，SReLU，SC-1000，SReLU，SC-1000，SReLU，FC-5，SoftmaX</td></tr></table>",
+        "bbox": [
+            176,
+            503,
+            821,
+            619
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/98386d91bd180b20d5da54f606bd1a619855c8fa97e3bede1fc758b1973e1e15.jpg",
+        "table_caption": [
+            "Table 10: FGR architectures used in our experiments. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>Architecture</td></tr><tr><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>Group-256，FC-3072，ReLU，FC-10，Softmax</td></tr><tr><td rowspan=1 colspan=1>HAPT</td><td rowspan=1 colspan=1>Group-104，FC-12173，ReLU，FC-5，Softmax</td></tr><tr><td rowspan=1 colspan=1>Diabetes</td><td rowspan=1 colspan=1>Group-44，FC-337，ReLU，FC-3，Softmax</td></tr><tr><td rowspan=1 colspan=1>Landsat</td><td rowspan=1 colspan=1>Group-32，FC-1577，ReLU，FC-6，Softmax</td></tr><tr><td rowspan=1 colspan=1>MIT-BIH</td><td rowspan=1 colspan=1>Group-160，FC-3444，ReLU，FC-5，Softmax</td></tr></table>",
+        "bbox": [
+            240,
+            744,
+            756,
+            875
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C SOFTWARE IMPLEMENTATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            457,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Table 11 presents the list of software dependencies and versions used in our implementation. To produce results related to this paper, we used a workstation with 4 NVIDIA GeForce RTX-2080Ti GPUs, a 12 core Intel Core i9-7920X processor, and 128 GB memory. Each experiment took between about 30 minutes to 72 hours, based on the task and method being tested. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            189
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/0cf8a598c183371666eb5e48a2082b17b23228c411f430c66938c73e6a9bcaea.jpg",
+        "table_caption": [
+            "Table 11: Software dependencies. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dependency</td><td>Version</td></tr><tr><td>python</td><td>3.7.1</td></tr><tr><td>pytorch</td><td>1.1.0</td></tr><tr><td>torchvision</td><td>0.2.1</td></tr><tr><td>cuda100</td><td>1.0</td></tr><tr><td>ipython</td><td>6.5.0</td></tr><tr><td>jupyter</td><td>1.0.0</td></tr><tr><td>numpy</td><td>1.15.4</td></tr><tr><td>nni</td><td>0.9.1.1</td></tr><tr><td>pandas</td><td>0.23.4</td></tr><tr><td>scikit-learn</td><td>0.19.2</td></tr><tr><td>scipy</td><td>1.1.0</td></tr><tr><td>pomegranate</td><td>0.11.1</td></tr><tr><td>tqdm</td><td>4.32.1</td></tr><tr><td>matplotlib</td><td>3.0.1</td></tr></table>",
+        "bbox": [
+            387,
+            228,
+            617,
+            508
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D VISUAL ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            366,
+            117
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In Figure 10, we present a visualization of the selected feature for 25 randomly selected groups in our final CIFAR-10 architecture. Red, green, and blue colors indicate which channel is selected for each location. Compared to visualizations that are frequently used for convolutional networks, as GMLP has the flexibility to select pixels at different locations and different color channels, it is not easy to find explicit patterns in this visualization. However, one noticeable pattern is that features selected from a certain color channel usually appear in clusters resembling irregularly shaped patches. ",
+        "bbox": [
+            174,
+            133,
+            826,
+            218
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/f18122c74dd123d377ebf80f5e3ca8a9a8d30d8b9b1e3229f4a82137393c758e.jpg",
+        "image_caption": [
+            "Figure 10: Visualization of pixels selected by each group for the CIFAR-10 GMLP architecture. Red, green, and blue colors indicate which channel is selected for each location. Due to space limitations, 25 random groups out of 1536 total groups visualized here. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            202,
+            239,
+            812,
+            714
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Figure 11 shows the frequency in which each CIFAR-10 location is selected by the GMLP network. From this visualization, GMLP is mostly ignoring the border areas which can be a result of the data augmentation process used to train the network i.e., randomly cropping the center area and padding the margins. ",
+        "bbox": [
+            173,
+            103,
+            826,
+            160
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/c6de4f1b62f90a17c1b1e28a45ef256082d0acf42fbf91947838a92f007d06e2.jpg",
+        "image_caption": [
+            "Figure 11: Visualization of pixels selected by the group-select layer for the CIFAR-10 GMLP model. Warmer colors represent features that are being selected more frequently. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            377,
+            181,
+            619,
+            367
+        ],
+        "page_idx": 16
+    }
+]
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new file mode 100644
index 0000000000000000000000000000000000000000..ced6f468a4c90d1f27b2de1ed64828f1c08f31d2
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+# JOINTLY LEARNING TO CONSTRUCT AND CONTROL AGENTS USING DEEP REINFORCEMENT LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+The physical design of a robot and the policy that controls its motion are inherently coupled. However, existing approaches largely ignore this coupling, instead choosing to alternate between separate design and control phases, which requires expert intuition throughout and risks convergence to suboptimal designs. In this work, we propose a method that jointly optimizes over the physical design of a robot and the corresponding control policy in a model-free fashion, without any need for expert supervision. Given an arbitrary robot morphology, our method maintains a distribution over the design parameters and uses reinforcement learning to train a neural network controller. Throughout training, we refine the robot distribution to maximize the expected reward. This results in an assignment to the robot parameters and neural network policy that are jointly optimal. We evaluate our approach in the context of legged locomotion, and demonstrate that it discovers novel robot designs and walking gaits for several different morphologies, achieving performance comparable to or better than that of hand-crafted designs.
+
+# 1 INTRODUCTION
+
+An agent’s ability to navigate through and interact with its environment depends not just on its skill at planning and controlling its motion, but also on its physical design. Different physical designs are inherently better suited to different tasks and environments. By making appropriate choices during fabrication, mechanical elements can be designed to improve robustness to non-idealities such as errors in perception, delays in actuation, etc., and indeed, make control problem an easier one to solve. At the same time, robots that take different forms may find completely different control strategies to be optimal to complete the same task. Therefore, the physical and computational design of an agent are inherently coupled, and must ideally be jointly optimized if the robot is to successfully complete a task in a particular environment.
+
+Consider the development of a legged robot for locomotion. Variations in physical design will require changes to the joint torques in order to preserve a particular locomotion behavior (e.g., a heavier torso requires greater torque at the ankle), and will likely result in completely different walking gaits, even when the morphology is preserved. In fact, some changes to design may render locomotion impossible for the target operating environment (e.g., a robot with long feet may be unable to locomote up an incline). Meanwhile, careful choice of bipedal design enables passive walking (McGeer, 1990; Goswami et al., 1998; Collins et al., 2001). It is therefore beneficial to not simply consider the robot’s design or gait to be fixed, but to optimize both jointly for the target environment and task. Similar co-design can be beneficial in other settings—for example for the control policy and physical characteristics of digits in robotic grippers for grasping.
+
+While a robot’s physical design and the corresponding control policy are inherently coupled, most existing methods ignore this coupling, instead choosing to alternate between separate design and control phases. Existing approaches that jointly reason over design and control (Digumarti et al., 2014; Ha et al., 2017; Spielberg et al., 2017) assume knowledge of an accurate model of the robot dynamics and require expert supervision (e.g., to provide a suitable initial design and guide the optimization process). However, these restrictive assumptions limits their applicability to a handful of specific settings, and often yield solutions heavily influenced by expert intuition.
+
+In this work, we seek a general approach—one that can optimize a robot’s physical characteristics jointly with controllers of a desired complexity (Fig. 1), that can be applied to general tasks in some given environment, and that can explore the joint search space of physical design and computational control in a purely data-driven way, without a model of the robot dynamics and independent of the biases of expert intuition. We develop this approach in the context of determining the physical parameters of an articulated agent—the lengths and thicknesses of each limbs in a given morphology— through joint training with a neural network for control, with the objective of achieving locomotion. Our method maintains a distribution over these physical parameters, and simultaneously trains the parameters of this distribution with those of a neural network controller, using deep reinforcement learning. In this way, we pursue a design distribution and control policy that are jointly optimal for the given task and environment. Experimental results show that starting from random initializations, our approach is able to find novel designs and walking gaits that match or exceed the performance of manually designed agents. To the best of our knowledge, our method is the first to successfully carry out such a joint optimization of design and control in a completely model-free manner.
+
+![](images/601586609236628de9cee35fd71c942e14a74590dd246e90589f10ecbea5ea80.jpg)  
+Figure 1: Our algorithm learns a robot’s physical design jointly with the control policy. Here we show the learned designs evolving over time for the Hopper (top left), the Walker2d (top right) and the Ant (bottom), each with the default Roboschool design for comparison. Scale is fixed for each robot. Note that these designs correspond to modes of the distribution over robot designs that our algorithm maintains during training.
+
+# 2 RELATED WORK
+
+Attention has been paid recently to the problem of jointly optimizing the parameters that specify a robot’s design and motion (e.g., gaits) or control. Early work in this area takes an evolutionary approach to optimizing a robot’s design and controller, typically parameterized as a neural network, for virtual agents (Sims, 1994; Paul & Bongard, 2001; Agrawal et al., 2013) and physical robots (Lipson & Pollack, 2000; Bongard, 2011). Particularly relevant to our approach is the work of Ha et al. (2017), who relate design and motion parameters via a set of implicit functions that express robot dynamics, desired trajectories, and actuation limits. These functions encode a manifold that is then linearized to model the relationship between design and motion via the implicit function theorem. The method then solves for the desired parameters in a local fashion via constraint-based optimization. Similarly, Spielberg et al. (2017) describe an approach that jointly reasons over physical design and motion parameters for robots with articulated degrees of freedom (e.g., legged robots). They formulate the problem in the framework of trajectory optimization by incorporating parameters involved in the design of articulated robots, including dimensions, masses, mass distribution, and moments of inertia, together with the contact force and torque variables typically associated with trajectory optimization. They use weighted actuation cost as the objective, subject to a regularization penalty on parameters. Their method is guaranteed to find a feasible design so long as the problem is initialized within a small neighborhood of a feasible solution. Unlike our approach, their method requires that the user provide an estimate of the robot configuration at each time step, as well as an accurate analytic model of constraint dynamics (e.g., foot contact), which is computationally expensive to enforce. Geijtenbeek et al. (2013) propose a derivative-free strategy that optimizes over muscle routing and muscle-based control for simulated bipeds to realize a design that incorporates biomechanical constraints. Meanwhile, Digumarti et al. (2014) describe an evolutionary method that jointly reasons over design and motion parameters for legged robots, while also tuning the parameters of a robust controller that tracks these motions. Their method is limited to biologically inspired quadrupedal foothold patterns, and does not account for contact dynamics. In contrast, our approach applies to arbitrary morphologies and does not require that we model contact dynamics.
+
+While the focus on simultaneous optimization over robot design and motion parameters is relatively new, there is a long history of research focused on the related problem of co-design of physical structure and control (Reyer & Papalambros, 2002; Ravichandran et al., 2006). Park & Asada (1994) jointly optimize the link geometry of a high-speed robot arm along with the parameters of a PD joint controller. These methods often rely upon access to an accurate model of the robot dynamics in order to design the controller. In order to reduce the dependence upon a detailed analytical model, Pil & Asada (1996) use on-robot experiments to refine their model as part of the iterative design process. More recently, Villarreal-Cervantes et al. (2013) allow for some degree of uncertainty in the model by including the sensitivity of the design and control parameters to model uncertainty along with the task-specific optimization objectives. However, existing methods still rely upon access to an analytical model of the robot dynamics and are typically limited to simple (e.g., linear) control designs. Our method assumes only that the controller can be modeled via a convolutional neural network, and can thereby learn complex, highly nonlinear control policies with no a priori knowledge of the robot dynamics.
+
+Far more attention has been paid to the individual problem of task-driven optimization of robot motion and control. Given an arbitrary robot design chosen by novice users, Megaro et al. (2015) describe an interactive design framework that solves for a realizable walking gait that results in stable locomotion. Similarly, Mordatch et al. (2012) synthesize emergent behaviors (motion) for arbitrary morphologies and tasks from high-level specifications, by jointly optimizing over contact and motion. Mordatch et al. (2015) build upon this work by training recurrent neural networks to serve as feedback controllers capable of producing complex, stable behaviors for a variety of dynamical systems. Their method interleaves supervised learning with trajectory optimization, and incorporates noise to improve generalizability. Meanwhile, there is a large body of literature that formulates motion synthesis as a trajectory optimization problem. This approach has proven effective at respecting contact constraints (e.g., between the foot and ground), which make controlling dynamic motion particularly challenging (Dai & Tedrake, 2016; Posa et al., 2016; Griffin & Grizzle, 2016). These approaches have been shown to generate sophisticated behaviors for complex robot designs (e.g., humanoids) (Tassa et al., 2012), and for robots of arbitrary morphologies using only a high-level specification of the robot’s shape, gait, and task (Wampler et al., 2013). Related, a number of methods interleave trajectory optimization and supervised learning with neural network regression (Levine & Abbeel, 2014; Levine & Koltun, 2014; Mordatch & Todorov, 2014; Mordatch et al., 2015). Unlike our method, the use of trajectory optimization makes these approaches reliant upon knowledge of the model.
+
+A great deal of attention of-late has focused on the problem of learning complex control policies directly from low-level sensory input, without any knowledge of the system dynamics. Methods that have proven particularly effective combine neural networks that learn representations of the highdimensional raw sensor input with deep reinforcement learning (Riedmiller, 2005; Mnih et al., 2013; 2015; Schulman et al., 2015a). While much of the work in this area focuses on low-dimensional, discrete action spaces, several methods have been recently proposed that learn continuous control policies through deep reinforcement learning. These techniques have been applied to control simple, simulated robots (Wawrzynski, 2009; Wawrzy ´ nski & Tanwani, 2013; Watter et al., 2015; Lillicrap ´ et al., 2015), perform robot manipulation (Levine et al., 2016a;b; Gu et al., 2017), and control legged robots (Schulman et al., 2015b; Peng et al., 2016).
+
+Black-box optimization (Schmidhuber et al., 2007; Risi & Togelius, 2017) is an alternative to using reinforcement learning to training the control policy. These approaches have the advantage that they do not involve backpropagating gradients, are insensitive to reward sparsity, and can handle long time horizons. While black-box optimization strategies have traditionally been thought of as illsuited to difficult reinforcement learning problems, Salimans et al. (2017) recently showed that they perform similarly to state-of-the-art RL methods on difficult continuous control problems, including those that involve locomotion. Closely related is the policy-gradient method of Sehnke et al. (2010), who define the policy as a distribution over the parameters of a controller, which they then sample over. This results in gradient estimates that are far less noisy than is typical of policy gradient algorithms. These approaches are similar to the way in which we learn the robot design, which we formulate as a Gaussian mixture model over design parameters. Indeed, the two referenced methods yield the same gradient estimate for Gaussian parameter distributions (Salimans et al., 2017).
+
+Meanwhile, much work has focused on the problem of determining robot designs that meet the requirements of a particular task. Given a user demonstration of the desired behaviors, Coros et al. (2013) learn optimum kinematic linkages that are capable of reproducing these motions. Mehta et al. (2014) synthesize electromechanical robot designs in a compositional fashion based upon a complete user-specified structural specification of the robot. Mehta et al. (2016) build upon this work, allowing the user to specify functional objectives via structured English, which is parsed to a formal specification using linear temporal logic. Censi (2017) describes a theory for co-design that includes the ability to select discrete robot parts according to functional constraints, but do not reason over geometry or motion.
+
+Related to our approach is recent work that jointly optimizes sensor design and inference algorithms for perception systems. Chakrabarti (2016) considers the problem of jointly learning a camera sensor’s multiplexing pattern along with reconstruction methods for imaging tasks. They model inference as a neural network together and use stochastic gradient descent to backpropagate the loss to a neural layer representation of the multiplexing pattern. Related, Schaff et al. (2017) jointly learn design and inference for beacon-based localization. They encode beacon allocation (spatially and across transmission channels) as a differential neural layer that interfaces with a neural network for inference. Joint optimization then follows from standard techniques for training neural networks.
+
+# 3 APPROACH
+
+In this section, we begin by describing the standard reinforcement learning framework for training agent policies, and then describe how we extend this to also learn the physical design of the agent.
+
+# 3.1 REINFORCEMENT LEARNING BACKGROUND
+
+In the standard reinforcement learning setting, an agent interacts with its environment, usually a Markov Decision Process, over a number of discrete timesteps. At each time step $t$ , the agent receives a state $s _ { t } \in S$ and takes action $a _ { t } \in { \mathcal { A } }$ according to a policy $\pi : { \mathcal { S } }  A$ . Then, the agent receives a scalar reward $r _ { t }$ and the next state $s _ { t + 1 }$ from the environment. This process continues until a terminal state is reached. The goal of reinforcement learning is to then find a policy $\pi ^ { * }$ that maximizes the expected return $\mathbb { E } [ \mathcal { R } _ { t } ]$ , where $\begin{array} { r } { \mathcal { R } _ { t } = \sum _ { i = 0 } ^ { \infty } \gamma ^ { i } r _ { t + i } } \end{array}$ and $\gamma \in [ 0 , 1 )$ is a discount factor.
+
+Policy gradient methods are a class of algorithms often used to optimize reinforcement learning problems, due to their ability to optimize cumulative reward and the ease with which the can be used with neural networks and other nonlinear function approximators. Consequently, they are commonly used for reinforcement learning problems that involve complex, continuous action spaces. Policy gradient methods directly parameterize a stochastic policy $\bar { \pi _ { \boldsymbol { \theta } } } ( a _ { t } | \boldsymbol { s } _ { t } )$ and perform stochastic gradient ascent on the expected return. “Vanilla” policy gradient methods compute an estimate of the gradient $\nabla _ { \boldsymbol { \theta } } \mathbb { E } [ \mathcal { R } _ { t } ]$ using a sample-based mean computed over $\nabla _ { \theta } \log \pi _ { \theta } ( \bar { a _ { t } } \vert s _ { t } ) \mathcal { R } _ { t }$ (Williams, 1992), which yields an unbiased gradient estimate (Sutton et al., 2000). While the variance in the resulting estimate decreases with the number of samples, sampling is computationally expensive. Another way to reduce variance while maintaining an unbiased estimate is to estimate the gradient by comparing the reward to a “baseline” reward $b ( s _ { t } )$ .
+
+These methods are effective but can be unstable, especially when used for deep reinforcement learning. Small changes to the policy parameters may cause large changes in the distribution of visited states. Several methods have been proposed to mitigate these effects. Among them, Schulman et al. (2015a) introduce Trust Region Policy Optimization (TRPO), which imposes a constraint on the KLdivergence between policies before and after an update. Recently, Schulman et al. (2017) proposed proximal policy optimization (PPO), a first-order class of methods similar to TRPO. PPO alternates between sampling data through interaction with the environment and optimizing the objective
+
+$$
+\hat { \mathbb { E } } _ { t } \left[ \operatorname* { m i n } ( r _ { t } ( \theta ) \hat { A } _ { t } , \operatorname { c l i p } ( r _ { t } ( \theta ) , 1 - \epsilon , 1 + \epsilon ) ) \right] ,
+$$
+
+where $\begin{array} { r } { r _ { t } ( \theta ) = \frac { \pi _ { \theta } ( a _ { t } | s _ { t } ) } { \pi _ { \theta _ { \mathrm { o l d } } } ( a _ { t } | s _ { t } ) } } \end{array}$ and $\hat { \mathbb { E } } _ { t }$ represents an empirical average over a finite sample set. This objective seeks to maximize the expected return while encouraging a small step in policy space. The clipping within the objective removes any incentive for moving $r _ { t } ( \theta )$ outside the interval $[ 1 - \epsilon , 1 + \epsilon ]$ . The net effect of this objective is a simple, robust policy gradient algorithm that attains or matches state-of-the-art results on a wide array of tasks (Schulman et al., 2017).
+
+<table><tr><td>Algorithm1Joint Optimization</td><td></td></tr><tr><td>Initialize πθ(a|s,r),PΦ(r)</td><td></td></tr><tr><td>while True do</td><td></td></tr><tr><td>for i ∈[1..np] do</td><td></td></tr><tr><td> Sample ri,.., rn from pΦ</td><td></td></tr><tr><td>Collect rollouts with each design</td><td></td></tr><tr><td>Update 0 to maximize PPO&#x27;s surrogate objective. end for</td><td></td></tr><tr><td>for i∈[1..nr] do</td><td></td></tr><tr><td>Sample ri,.*, rm from pΦ</td><td></td></tr><tr><td>Compute returns R1，., Rm·</td><td></td></tr><tr><td></td><td>Update  to maximize PPO&#x27;s surrogate objective.</td></tr><tr><td>end for</td><td></td></tr><tr><td>end while</td><td></td></tr></table>
+
+# 3.2 JOINT OPTIMIZATION OF DESIGN AND CONTROL
+
+We extend the standard reinforcement learning formulation by considering the space of possible robot designs $\Omega$ . Specifically, we assume that for every design $\mathcal { E } \in \Omega$ , we can define common state and action spaces $s$ and $\mathcal { A }$ , reward function $r ( s , a )$ , and initial state distribution $p _ { 0 } ( s )$ , that are meaningful to all designs. The designs differ only in the transition dynamics they induce $p \varepsilon ( s ^ { \prime } | s , a )$ and share a common action space $A$ —to achieve this, we assume a common morphology for all possible designs.
+
+Our goal then is to find the optimal design $\mathcal { E } ^ { * }$ and policy $\pi _ { \mathcal { E } ^ { * } } ^ { * }$ pair that maximizes the expected value of a given reward function. However, this is a non-linear, non-convex optimization problem over the spaces of all possible designs and all possible policies. Solving it exactly would require enumerating all possible designs (possibly through discretization of the design space), learning policies for each, and comparing the resulting expected rewards. This is computationally infeasible for all but the simplest of cases. Instead, we develop a gradient-based approach to solving this optimization problem with the following key components: (1) we maintain a multi-modal distribution over the space of physical designs and update this distribution using policy gradient methods in parameter space, similar to Sehnke et al. (2010) and Salimans et al. (2017); and (2) we train a single controller to act on all sampled designs during training, providing this controller with the design parameters of the specific sample it is controlling. We find that the multi-modal stochastic parameterization of the design space allows our method to explore the space more thoroughly (where a uni-modal Gaussian distribution would frequently get trapped in local minima). Moreover, a common controller makes optimization tractable, allowing efficient evaluation of unseen designs. We additionally benefit from learning common strategies across diverse designs, while still adapting different policies to different parts of the design space given the sample parameters as input. Together, these components enable efficient joint exploration of the design and policy spaces.
+
+Formally, let $p ( r ; \phi )$ denote the distribution over designs, and $\pi ( a _ { t } | s _ { t } , r ; \theta )$ a stochastic control policy parameterized by $\phi$ and $\theta$ respectively. In our experiments, we use a neural network to model the control policy $\pi$ , and a Gaussian mixture model as the parametric form of the distribution $p ( r ; \phi )$ . Our goal is then to solve the following optimization problem:
+
+$$
+\phi ^ { * } , \theta ^ { * } = \underset { \phi , \theta } { \arg \operatorname* { m a x } } \mathbb { E } _ { r , t } \left[ \mathcal { R } _ { t } \right] ,
+$$
+
+where $\mathbb { E } _ { \boldsymbol { r } , t } [ \cdot ]$ is the expectation over robots and trajectories those robots induce.
+
+We use stochastic gradient-based updates to optimize Eqn. 2. Our method (Algorithm 3.2) alternates between updating the parameters of the policy and design distributions, $\theta$ and $\phi$ , respectively. We empirically find this to yield convergence to better solutions in a reasonable number of iterations compared to performing simultaneous updates. We optimize the policy parameters $\theta$ using Proximal Policy Optimization. However, instead of collecting data from a single design, we sample a design $r$ after a fixed number $T$ of timesteps according to the distribution $p ( r ; \phi )$ . After $n$ iterations of PPO, we freeze the policy parameters and proceed to optimize the parameters of the design distribution. Without knowledge of the model, we optimize the design parameters $\phi$ via policy gradient over parameter space, similar to Sehnke et al. (2010). This is equivalent to black-box optimization methods that have proven effective for complex learning problems (Salimans et al., 2017). We sample $m$ different designs and compute their returns for a single episode acting under policy $\pi ( \cdot ; \theta )$ . We then use this data to shift the design distribution $p ( \cdot ; \phi )$ in a direction maximizing expected return. These iterations are repeated until convergence.
+
+![](images/0babea94ff71820dd39bdf9c33c7081a0ac16e1f9df8bc584653a1c4ea8568a5.jpg)  
+Figure 2: From left to right: the default Hopper, Walker2d, and Ant robots. The Hopper and Walker2d are constrained to walk in a line, while the Ant can walk anywhere in the plane.
+
+# 4 RESULTS
+
+We validate our approach with three commonly used robots provided in OpenAI’s Roboschool (Schulman et al., 2017). Environments in Roboschool are built on top of Bullet Physics, a popular open-source physics engine. In a series of locomotion experiments, we show that our approach not only discovers novel robot designs and gaits, but also outperforms two out of three of the hand-designed robots trained on the same task and yields performance comparable to the third.
+
+# 4.1 EXPERIMENT DETAILS
+
+We evaluate our approach on three legged locomotion tasks within Roboschool: RoboschoolHopper, RoboschoolWalker2d, and RoboschoolAnt (note that we subsequently drop the Roboschool prefix for brevity). Figure 2 depicts the default Roboschool design for each robot. These environments describe a locomotion task that has the robots moving along the positive $\mathbf { X }$ -axis (to the right in the figures) towards a distant goal. The reward function defined in these environments is a weighted sum of rewards for forward progress and staying upright, and penalties for applying torques and for joints that reach their rotational limits. The episode ends when the robot falls over or reaches a maximum number of timesteps.
+
+We learn robot designs using the morphologies specified by the standard Roboschool designs. For each morphology, we parameterize the robot in terms of the length and radius (e.g., mass) of each link (or just the radius in the case of the sphere for the ant body). We impose symmetry, and share parameters across each leg for the Walker2d and Ant designs. This parameterization permits a wide variety of robots of different shapes and sizes, some of which are better suited to the locomotion objective than others, and some designs that do not permit a control policy that results in locomotion.
+
+We model the control policy $\pi ( a _ { t } | s _ { t } , \mathcal { E } ; \theta _ { p } )$ as a feed forward neural network consisting of three fully-connected layers with 64 units and tanh activation functions. A final layer maps the output to the robot action space. With the exception of the last robot-specific layer, the architecture is the same for all experiments. Meanwhile, we represent the distribution over the parameterized robot design $p ( \cdot ; \theta _ { r } )$ using a Gaussian mixture model with four mixture components (each with a diagonal covariance matrix over the different parameters). We initialize the means of each component randomly within a wide range, and initialize the variances in order to span the range of each parameter. We find that our approach maintains high variance distributions during early iterations—thereby continuing exploration of the design space—before committing to a chosen design. The appendix provides further details regardng the evolution of these distributions.
+
+![](images/87d05764a68b4483dd235162ccd22ce42ae5244a22dfc99728015163faafa544.jpg)  
+Figure 3: Training curves that show the evolution of reward across training iterations for the Hopper, Walker2d, and Ant environments over eight different random seeds. The black dashed line corresponds to the highest achievable performance of the corresponding baseline robot. In other figures, we display the best-performing checkpoint of each run. Therefore, we make each training curve transparent after it stops improving.
+
+We train our method with eight threads of Proximal Policy Optimization for a total of 300M environment timesteps across all threads. We use the publicly available implementation of PPO provided in OpenAI Baselines (Hesse et al., 2017). We alternate between 50 policy PPO iterations and 2 design iterations, finding this ratio to yield convergence to good solutions. We set this and the other hyper-parameters on the Hopper and used the same settings for all experiments.
+
+# 4.2 EXPERIMENTS
+
+We evaluate the performance of our approach to joint optimization over robot design and control on the Hopper, Walker2d, and Ant robots, and compare against a policy trained for the standard Roboschool design. We evaluate the consistency and robustness of our approach relative to the initial robot parameter distribution using several different random seeds. For each robot morphology, we find that our method learns robot designs and the corresponding control policy that are either comparable (Walker2d) or exceed the performance (Hopper and Ant) achievable using the default designs (Fig. 3). We also see that our method achieves these levels of performance by discovering unique robot designs (Fig. 1) together with novel walking gaits (Fig. 4) that can not be achieved with the default designs. Note that our method was able to learn these designs and control policies from a random initialization, without access to a dynamics model or any expert supervision.
+
+Our method learns a joint design and controller for the Hopper that outperforms the baseline by as much as $5 0 \%$ . Our learned robot exploits the small size of the foot to achieve faster, more precise walking gaits. Meanwhile, the longer torso of the learned robot improves stability, allowing it to maintain balance while locomoting at a faster pace. We found this behavior to be consistent across several different Hopper experiments, with the method converging to designs with a small foot and long, thin torso. In the appendix, we explore the stability of this design with respect to variations in the environment using the coefficient of friction as an example, and find the improvement to be consistent. For the ant, our optimization yields a physical design that is significantly different from the default design (Fig. 2). Consequently, the learned Ant drastically outperforms the baseline, improving reward by up to $1 1 6 \%$ . Our method learns a design with a small, lightweight body and extremely long legs. The long legs enable the ant to apply large torque at contact, allowing it to move at a fast pace.
+
+Our framework learned different design-control pairs for the Walker2d that perform similarly to the default design. Across several different experiments, we see two distinct, successful designs and walking gaits. Interestingly, neither agent walks primarily on its feet. The first design has small, thick legs and long feet. The controller is then able to exploit the thick middle leg link, which protrudes slightly past the foot, to push off the ground. The long foot then provides additional balance. The second design is similar in geometry to the baseline Walker2d, but moves in a very different way. By lowering the knee joint and lengthening the foot, the Walker2d is able to efficiently balance on its knees and toes. This low stance allows the Walker2d to fully extend its leg backwards, creating a long, powerful stride, similar to that of a sprinter using starting blocks.
+
+![](images/97b6ba644983f0da7af6c1a0a705d198e90586ea3304b1179bfe57ed0dfdfbc5.jpg)  
+Figure 4: Here, we compare locomotion of default and learned robots, visualizing both their physical design and corresponding learned gaits. We pick the best results for Hopper and Ant, and two of the best results for Walker2d (due to diversity in gaits). Note that for each robot type, we show a blend of a fixed number of frames in the same time interval, allowing direct comparison between the speed with which different designs are able to locomote.
+
+# 5 CONCLUSION
+
+We proposed what is, to the best of our knowledge, the first model-free algorithm that jointly optimizes over the physical design of a robot and the corresponding control policy, without any need for expert supervision. Given an arbitrary morphology, our robot maintains a distribution over the robot design parameters and learns these parameters together with a neural network controller using policy gradient-based reinforcement learning. This results in an assignment to the policy over robot parameters and the control policy that are jointly optimal. We evaluated our approach on a series of different legged robot morphologies, demonstrating that it results in novel robot designs and walking gaits, achieving performance that either matches or exceeds that of manually defined designs.
+
+Our findings suggest several avenues for future work. The most direct is extending the current approach to find optimized designs for uneven terrain, the presence of obstacles, changes in slope, variations in friction, etc. We are also interested in extending our framework to relax the assumption that the morphology is pre-defined. Finally, we are investigating applications to different types of agents and design spaces beyond legged robots (e.g., end-effectors), and exploring appropriate stochastic parameterization for such designs.
+
+# REFERENCES
+
+Shailen Agrawal, Shuo Shen, and Michiel van de Panne. Diverse motion variations for physicsbased character animation. In Proc. ACM SIGGRAPH/Eurographics Symposium on Computer Animation, pp. 37–44, 2013.
+
+Josh Bongard. Morphological change in machines accelerates the evolution of robust behavior. Proc. National Academy of Sciences, 108(4):1234–1239, 2011.
+
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+
+# 6 APPENDIX
+
+The following provides further evaluation of our framework, including the behavior of the design distribution during training and the robustness of the learned designs to environment variations.
+
+# 6.1 A - EVOLUTION OF GMM THROUGHOUT TRAINING
+
+To provide insight into the training process of our algorithm, we evaluate the evolution of the Gaussian mixture model distribution throughout training of the best performing Hopper experiment. We initialize each component of the mixture model with random means and a diagonal covariance matrix chosen to cover the parameter space. The initial mixture weights are uniform. As shown in Figure 5 (left), roughly one third of the way through training, our algorithm converges to the most successful component. Additionally, we find that modes generally do not collapse (Figure 5 (right)). We find this behavior to be consistent across random seeds and different robots.
+
+![](images/31445f3810e5a39d1253d521aef51b22276ca400f4dae3d24e97f4bee29e0cef.jpg)  
+Figure 5: Plots of (left) the entropy of the mixture weights and (right) the marginal variance of each parameter throughout training. The curves are normalized such that all initial variances are one.
+
+# 6.2 B - ROBUSTNESS OF LEARNED DESIGNS
+
+It is often desirable for a robot to be able to operate in a variety of environments. In this section, we consider the robustness of a learned design to changes in friction. We conducted experiments on the Hopper in which we first learned the design and controller for one friction setting (0.8). We finetuned the controller in environments with different friction settings, while leaving the design fixed. We find the learned design to be reasonably robust with variability comparable to controllers finetuned for the default hand-crafted design (Fig. 6). Additionally, the learned design outperforms the hand-crafted one across the full range of friction values (although, for very low friction values, both designs essentially were unable to learn a successful gait).
+
+Note that our framework can incorporate this goal of generalization by simply sampling from a diverse set of environments during training. But at the same time, it may be useful in some applications to seek out solutions that are specifically adapted to a relatively narrower set of environments, gaining better performance within this set at the cost of more general performance.
+
+![](images/6950886e33eae5aebfefebe1362c093a2ef2d79ce2a27d5c428bd80267a0f480.jpg)  
+Figure 6: A visualization of accumulated reward for different friction values. The controllers and learned design were trained in an environment with a friction value of 0.8. We finetuned the controllers (but not the designs) for both the learned and hand-crafted designs for 10M timesteps for each friction value. Rewards are reported as an average over 100 episodes.
\ No newline at end of file
diff --git a/parse/train/SyfiiMZA-/SyfiiMZA-_content_list.json b/parse/train/SyfiiMZA-/SyfiiMZA-_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..3196cd36833635182b607269abf6169914189648
--- /dev/null
+++ b/parse/train/SyfiiMZA-/SyfiiMZA-_content_list.json
@@ -0,0 +1,1181 @@
+[
+    {
+        "type": "text",
+        "text": "JOINTLY LEARNING TO CONSTRUCT AND CONTROL AGENTS USING DEEP REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            821,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            171,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The physical design of a robot and the policy that controls its motion are inherently coupled. However, existing approaches largely ignore this coupling, instead choosing to alternate between separate design and control phases, which requires expert intuition throughout and risks convergence to suboptimal designs. In this work, we propose a method that jointly optimizes over the physical design of a robot and the corresponding control policy in a model-free fashion, without any need for expert supervision. Given an arbitrary robot morphology, our method maintains a distribution over the design parameters and uses reinforcement learning to train a neural network controller. Throughout training, we refine the robot distribution to maximize the expected reward. This results in an assignment to the robot parameters and neural network policy that are jointly optimal. We evaluate our approach in the context of legged locomotion, and demonstrate that it discovers novel robot designs and walking gaits for several different morphologies, achieving performance comparable to or better than that of hand-crafted designs. ",
+        "bbox": [
+            233,
+            265,
+            764,
+            458
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            482,
+            336,
+            498
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "An agent’s ability to navigate through and interact with its environment depends not just on its skill at planning and controlling its motion, but also on its physical design. Different physical designs are inherently better suited to different tasks and environments. By making appropriate choices during fabrication, mechanical elements can be designed to improve robustness to non-idealities such as errors in perception, delays in actuation, etc., and indeed, make control problem an easier one to solve. At the same time, robots that take different forms may find completely different control strategies to be optimal to complete the same task. Therefore, the physical and computational design of an agent are inherently coupled, and must ideally be jointly optimized if the robot is to successfully complete a task in a particular environment. ",
+        "bbox": [
+            174,
+            512,
+            825,
+            638
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Consider the development of a legged robot for locomotion. Variations in physical design will require changes to the joint torques in order to preserve a particular locomotion behavior (e.g., a heavier torso requires greater torque at the ankle), and will likely result in completely different walking gaits, even when the morphology is preserved. In fact, some changes to design may render locomotion impossible for the target operating environment (e.g., a robot with long feet may be unable to locomote up an incline). Meanwhile, careful choice of bipedal design enables passive walking (McGeer, 1990; Goswami et al., 1998; Collins et al., 2001). It is therefore beneficial to not simply consider the robot’s design or gait to be fixed, but to optimize both jointly for the target environment and task. Similar co-design can be beneficial in other settings—for example for the control policy and physical characteristics of digits in robotic grippers for grasping. ",
+        "bbox": [
+            174,
+            645,
+            825,
+            784
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "While a robot’s physical design and the corresponding control policy are inherently coupled, most existing methods ignore this coupling, instead choosing to alternate between separate design and control phases. Existing approaches that jointly reason over design and control (Digumarti et al., 2014; Ha et al., 2017; Spielberg et al., 2017) assume knowledge of an accurate model of the robot dynamics and require expert supervision (e.g., to provide a suitable initial design and guide the optimization process). However, these restrictive assumptions limits their applicability to a handful of specific settings, and often yield solutions heavily influenced by expert intuition. ",
+        "bbox": [
+            174,
+            791,
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+            888
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we seek a general approach—one that can optimize a robot’s physical characteristics jointly with controllers of a desired complexity (Fig. 1), that can be applied to general tasks in some given environment, and that can explore the joint search space of physical design and computational control in a purely data-driven way, without a model of the robot dynamics and independent of the biases of expert intuition. We develop this approach in the context of determining the physical parameters of an articulated agent—the lengths and thicknesses of each limbs in a given morphology— through joint training with a neural network for control, with the objective of achieving locomotion. Our method maintains a distribution over these physical parameters, and simultaneously trains the parameters of this distribution with those of a neural network controller, using deep reinforcement learning. In this way, we pursue a design distribution and control policy that are jointly optimal for the given task and environment. Experimental results show that starting from random initializations, our approach is able to find novel designs and walking gaits that match or exceed the performance of manually designed agents. To the best of our knowledge, our method is the first to successfully carry out such a joint optimization of design and control in a completely model-free manner. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/601586609236628de9cee35fd71c942e14a74590dd246e90589f10ecbea5ea80.jpg",
+        "image_caption": [
+            "Figure 1: Our algorithm learns a robot’s physical design jointly with the control policy. Here we show the learned designs evolving over time for the Hopper (top left), the Walker2d (top right) and the Ant (bottom), each with the default Roboschool design for comparison. Scale is fixed for each robot. Note that these designs correspond to modes of the distribution over robot designs that our algorithm maintains during training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            555,
+            344,
+            571
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Attention has been paid recently to the problem of jointly optimizing the parameters that specify a robot’s design and motion (e.g., gaits) or control. Early work in this area takes an evolutionary approach to optimizing a robot’s design and controller, typically parameterized as a neural network, for virtual agents (Sims, 1994; Paul & Bongard, 2001; Agrawal et al., 2013) and physical robots (Lipson & Pollack, 2000; Bongard, 2011). Particularly relevant to our approach is the work of Ha et al. (2017), who relate design and motion parameters via a set of implicit functions that express robot dynamics, desired trajectories, and actuation limits. These functions encode a manifold that is then linearized to model the relationship between design and motion via the implicit function theorem. The method then solves for the desired parameters in a local fashion via constraint-based optimization. Similarly, Spielberg et al. (2017) describe an approach that jointly reasons over physical design and motion parameters for robots with articulated degrees of freedom (e.g., legged robots). They formulate the problem in the framework of trajectory optimization by incorporating parameters involved in the design of articulated robots, including dimensions, masses, mass distribution, and moments of inertia, together with the contact force and torque variables typically associated with trajectory optimization. They use weighted actuation cost as the objective, subject to a regularization penalty on parameters. Their method is guaranteed to find a feasible design so long as the problem is initialized within a small neighborhood of a feasible solution. Unlike our approach, their method requires that the user provide an estimate of the robot configuration at each time step, as well as an accurate analytic model of constraint dynamics (e.g., foot contact), which is computationally expensive to enforce. Geijtenbeek et al. (2013) propose a derivative-free strategy that optimizes over muscle routing and muscle-based control for simulated bipeds to realize a design that incorporates biomechanical constraints. Meanwhile, Digumarti et al. (2014) describe an evolutionary method that jointly reasons over design and motion parameters for legged robots, while also tuning the parameters of a robust controller that tracks these motions. Their method is limited to biologically inspired quadrupedal foothold patterns, and does not account for contact dynamics. In contrast, our approach applies to arbitrary morphologies and does not require that we model contact dynamics. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "While the focus on simultaneous optimization over robot design and motion parameters is relatively new, there is a long history of research focused on the related problem of co-design of physical structure and control (Reyer & Papalambros, 2002; Ravichandran et al., 2006). Park & Asada (1994) jointly optimize the link geometry of a high-speed robot arm along with the parameters of a PD joint controller. These methods often rely upon access to an accurate model of the robot dynamics in order to design the controller. In order to reduce the dependence upon a detailed analytical model, Pil & Asada (1996) use on-robot experiments to refine their model as part of the iterative design process. More recently, Villarreal-Cervantes et al. (2013) allow for some degree of uncertainty in the model by including the sensitivity of the design and control parameters to model uncertainty along with the task-specific optimization objectives. However, existing methods still rely upon access to an analytical model of the robot dynamics and are typically limited to simple (e.g., linear) control designs. Our method assumes only that the controller can be modeled via a convolutional neural network, and can thereby learn complex, highly nonlinear control policies with no a priori knowledge of the robot dynamics. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Far more attention has been paid to the individual problem of task-driven optimization of robot motion and control. Given an arbitrary robot design chosen by novice users, Megaro et al. (2015) describe an interactive design framework that solves for a realizable walking gait that results in stable locomotion. Similarly, Mordatch et al. (2012) synthesize emergent behaviors (motion) for arbitrary morphologies and tasks from high-level specifications, by jointly optimizing over contact and motion. Mordatch et al. (2015) build upon this work by training recurrent neural networks to serve as feedback controllers capable of producing complex, stable behaviors for a variety of dynamical systems. Their method interleaves supervised learning with trajectory optimization, and incorporates noise to improve generalizability. Meanwhile, there is a large body of literature that formulates motion synthesis as a trajectory optimization problem. This approach has proven effective at respecting contact constraints (e.g., between the foot and ground), which make controlling dynamic motion particularly challenging (Dai & Tedrake, 2016; Posa et al., 2016; Griffin & Grizzle, 2016). These approaches have been shown to generate sophisticated behaviors for complex robot designs (e.g., humanoids) (Tassa et al., 2012), and for robots of arbitrary morphologies using only a high-level specification of the robot’s shape, gait, and task (Wampler et al., 2013). Related, a number of methods interleave trajectory optimization and supervised learning with neural network regression (Levine & Abbeel, 2014; Levine & Koltun, 2014; Mordatch & Todorov, 2014; Mordatch et al., 2015). Unlike our method, the use of trajectory optimization makes these approaches reliant upon knowledge of the model. ",
+        "bbox": [
+            174,
+            339,
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+            603
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A great deal of attention of-late has focused on the problem of learning complex control policies directly from low-level sensory input, without any knowledge of the system dynamics. Methods that have proven particularly effective combine neural networks that learn representations of the highdimensional raw sensor input with deep reinforcement learning (Riedmiller, 2005; Mnih et al., 2013; 2015; Schulman et al., 2015a). While much of the work in this area focuses on low-dimensional, discrete action spaces, several methods have been recently proposed that learn continuous control policies through deep reinforcement learning. These techniques have been applied to control simple, simulated robots (Wawrzynski, 2009; Wawrzy ´ nski & Tanwani, 2013; Watter et al., 2015; Lillicrap ´ et al., 2015), perform robot manipulation (Levine et al., 2016a;b; Gu et al., 2017), and control legged robots (Schulman et al., 2015b; Peng et al., 2016). ",
+        "bbox": [
+            174,
+            611,
+            825,
+            750
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Black-box optimization (Schmidhuber et al., 2007; Risi & Togelius, 2017) is an alternative to using reinforcement learning to training the control policy. These approaches have the advantage that they do not involve backpropagating gradients, are insensitive to reward sparsity, and can handle long time horizons. While black-box optimization strategies have traditionally been thought of as illsuited to difficult reinforcement learning problems, Salimans et al. (2017) recently showed that they perform similarly to state-of-the-art RL methods on difficult continuous control problems, including those that involve locomotion. Closely related is the policy-gradient method of Sehnke et al. (2010), who define the policy as a distribution over the parameters of a controller, which they then sample over. This results in gradient estimates that are far less noisy than is typical of policy gradient algorithms. These approaches are similar to the way in which we learn the robot design, which we formulate as a Gaussian mixture model over design parameters. Indeed, the two referenced methods yield the same gradient estimate for Gaussian parameter distributions (Salimans et al., 2017). ",
+        "bbox": [
+            174,
+            756,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Meanwhile, much work has focused on the problem of determining robot designs that meet the requirements of a particular task. Given a user demonstration of the desired behaviors, Coros et al. (2013) learn optimum kinematic linkages that are capable of reproducing these motions. Mehta et al. (2014) synthesize electromechanical robot designs in a compositional fashion based upon a complete user-specified structural specification of the robot. Mehta et al. (2016) build upon this work, allowing the user to specify functional objectives via structured English, which is parsed to a formal specification using linear temporal logic. Censi (2017) describes a theory for co-design that includes the ability to select discrete robot parts according to functional constraints, but do not reason over geometry or motion. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            229
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Related to our approach is recent work that jointly optimizes sensor design and inference algorithms for perception systems. Chakrabarti (2016) considers the problem of jointly learning a camera sensor’s multiplexing pattern along with reconstruction methods for imaging tasks. They model inference as a neural network together and use stochastic gradient descent to backpropagate the loss to a neural layer representation of the multiplexing pattern. Related, Schaff et al. (2017) jointly learn design and inference for beacon-based localization. They encode beacon allocation (spatially and across transmission channels) as a differential neural layer that interfaces with a neural network for inference. Joint optimization then follows from standard techniques for training neural networks. ",
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+    {
+        "type": "text",
+        "text": "3 APPROACH",
+        "text_level": 1,
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+        "text": "In this section, we begin by describing the standard reinforcement learning framework for training agent policies, and then describe how we extend this to also learn the physical design of the agent. ",
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+        "text": "3.1 REINFORCEMENT LEARNING BACKGROUND ",
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+        "text": "In the standard reinforcement learning setting, an agent interacts with its environment, usually a Markov Decision Process, over a number of discrete timesteps. At each time step $t$ , the agent receives a state $s _ { t } \\in S$ and takes action $a _ { t } \\in { \\mathcal { A } }$ according to a policy $\\pi : { \\mathcal { S } }  A$ . Then, the agent receives a scalar reward $r _ { t }$ and the next state $s _ { t + 1 }$ from the environment. This process continues until a terminal state is reached. The goal of reinforcement learning is to then find a policy $\\pi ^ { * }$ that maximizes the expected return $\\mathbb { E } [ \\mathcal { R } _ { t } ]$ , where $\\begin{array} { r } { \\mathcal { R } _ { t } = \\sum _ { i = 0 } ^ { \\infty } \\gamma ^ { i } r _ { t + i } } \\end{array}$ and $\\gamma \\in [ 0 , 1 )$ is a discount factor. ",
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+        "text": "Policy gradient methods are a class of algorithms often used to optimize reinforcement learning problems, due to their ability to optimize cumulative reward and the ease with which the can be used with neural networks and other nonlinear function approximators. Consequently, they are commonly used for reinforcement learning problems that involve complex, continuous action spaces. Policy gradient methods directly parameterize a stochastic policy $\\bar { \\pi _ { \\boldsymbol { \\theta } } } ( a _ { t } | \\boldsymbol { s } _ { t } )$ and perform stochastic gradient ascent on the expected return. “Vanilla” policy gradient methods compute an estimate of the gradient $\\nabla _ { \\boldsymbol { \\theta } } \\mathbb { E } [ \\mathcal { R } _ { t } ]$ using a sample-based mean computed over $\\nabla _ { \\theta } \\log \\pi _ { \\theta } ( \\bar { a _ { t } } \\vert s _ { t } ) \\mathcal { R } _ { t }$ (Williams, 1992), which yields an unbiased gradient estimate (Sutton et al., 2000). While the variance in the resulting estimate decreases with the number of samples, sampling is computationally expensive. Another way to reduce variance while maintaining an unbiased estimate is to estimate the gradient by comparing the reward to a “baseline” reward $b ( s _ { t } )$ . ",
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+        "text": "These methods are effective but can be unstable, especially when used for deep reinforcement learning. Small changes to the policy parameters may cause large changes in the distribution of visited states. Several methods have been proposed to mitigate these effects. Among them, Schulman et al. (2015a) introduce Trust Region Policy Optimization (TRPO), which imposes a constraint on the KLdivergence between policies before and after an update. Recently, Schulman et al. (2017) proposed proximal policy optimization (PPO), a first-order class of methods similar to TRPO. PPO alternates between sampling data through interaction with the environment and optimizing the objective ",
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+        "type": "equation",
+        "img_path": "images/b0404ccd7847b3087ede7fdf66114a8fcc5dbdb719766f039eb092b8ecde360c.jpg",
+        "text": "$$\n\\hat { \\mathbb { E } } _ { t } \\left[ \\operatorname* { m i n } ( r _ { t } ( \\theta ) \\hat { A } _ { t } , \\operatorname { c l i p } ( r _ { t } ( \\theta ) , 1 - \\epsilon , 1 + \\epsilon ) ) \\right] ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $\\begin{array} { r } { r _ { t } ( \\theta ) = \\frac { \\pi _ { \\theta } ( a _ { t } | s _ { t } ) } { \\pi _ { \\theta _ { \\mathrm { o l d } } } ( a _ { t } | s _ { t } ) } } \\end{array}$ and $\\hat { \\mathbb { E } } _ { t }$ represents an empirical average over a finite sample set. This objective seeks to maximize the expected return while encouraging a small step in policy space. The clipping within the objective removes any incentive for moving $r _ { t } ( \\theta )$ outside the interval $[ 1 - \\epsilon , 1 + \\epsilon ]$ . The net effect of this objective is a simple, robust policy gradient algorithm that attains or matches state-of-the-art results on a wide array of tasks (Schulman et al., 2017). ",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm1Joint Optimization</td><td></td></tr><tr><td>Initialize πθ(a|s,r),PΦ(r)</td><td></td></tr><tr><td>while True do</td><td></td></tr><tr><td>for i ∈[1..np] do</td><td></td></tr><tr><td> Sample ri,.., rn from pΦ</td><td></td></tr><tr><td>Collect rollouts with each design</td><td></td></tr><tr><td>Update 0 to maximize PPO&#x27;s surrogate objective. end for</td><td></td></tr><tr><td>for i∈[1..nr] do</td><td></td></tr><tr><td>Sample ri,.*, rm from pΦ</td><td></td></tr><tr><td>Compute returns R1，., Rm·</td><td></td></tr><tr><td></td><td>Update  to maximize PPO&#x27;s surrogate objective.</td></tr><tr><td>end for</td><td></td></tr><tr><td>end while</td><td></td></tr></table>",
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+    {
+        "type": "text",
+        "text": "3.2 JOINT OPTIMIZATION OF DESIGN AND CONTROL ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We extend the standard reinforcement learning formulation by considering the space of possible robot designs $\\Omega$ . Specifically, we assume that for every design $\\mathcal { E } \\in \\Omega$ , we can define common state and action spaces $s$ and $\\mathcal { A }$ , reward function $r ( s , a )$ , and initial state distribution $p _ { 0 } ( s )$ , that are meaningful to all designs. The designs differ only in the transition dynamics they induce $p \\varepsilon ( s ^ { \\prime } | s , a )$ and share a common action space $A$ —to achieve this, we assume a common morphology for all possible designs. ",
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+        "text": "Our goal then is to find the optimal design $\\mathcal { E } ^ { * }$ and policy $\\pi _ { \\mathcal { E } ^ { * } } ^ { * }$ pair that maximizes the expected value of a given reward function. However, this is a non-linear, non-convex optimization problem over the spaces of all possible designs and all possible policies. Solving it exactly would require enumerating all possible designs (possibly through discretization of the design space), learning policies for each, and comparing the resulting expected rewards. This is computationally infeasible for all but the simplest of cases. Instead, we develop a gradient-based approach to solving this optimization problem with the following key components: (1) we maintain a multi-modal distribution over the space of physical designs and update this distribution using policy gradient methods in parameter space, similar to Sehnke et al. (2010) and Salimans et al. (2017); and (2) we train a single controller to act on all sampled designs during training, providing this controller with the design parameters of the specific sample it is controlling. We find that the multi-modal stochastic parameterization of the design space allows our method to explore the space more thoroughly (where a uni-modal Gaussian distribution would frequently get trapped in local minima). Moreover, a common controller makes optimization tractable, allowing efficient evaluation of unseen designs. We additionally benefit from learning common strategies across diverse designs, while still adapting different policies to different parts of the design space given the sample parameters as input. Together, these components enable efficient joint exploration of the design and policy spaces. ",
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+    {
+        "type": "text",
+        "text": "Formally, let $p ( r ; \\phi )$ denote the distribution over designs, and $\\pi ( a _ { t } | s _ { t } , r ; \\theta )$ a stochastic control policy parameterized by $\\phi$ and $\\theta$ respectively. In our experiments, we use a neural network to model the control policy $\\pi$ , and a Gaussian mixture model as the parametric form of the distribution $p ( r ; \\phi )$ . Our goal is then to solve the following optimization problem: ",
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+    {
+        "type": "equation",
+        "img_path": "images/3cba5431354f3ab665ec88bbb63f904897e4ef18653b9e0b0f2a6ec18f4d91dd.jpg",
+        "text": "$$\n\\phi ^ { * } , \\theta ^ { * } = \\underset { \\phi , \\theta } { \\arg \\operatorname* { m a x } } \\mathbb { E } _ { r , t } \\left[ \\mathcal { R } _ { t } \\right] ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathbb { E } _ { \\boldsymbol { r } , t } [ \\cdot ]$ is the expectation over robots and trajectories those robots induce. ",
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+    {
+        "type": "text",
+        "text": "We use stochastic gradient-based updates to optimize Eqn. 2. Our method (Algorithm 3.2) alternates between updating the parameters of the policy and design distributions, $\\theta$ and $\\phi$ , respectively. We empirically find this to yield convergence to better solutions in a reasonable number of iterations compared to performing simultaneous updates. We optimize the policy parameters $\\theta$ using Proximal Policy Optimization. However, instead of collecting data from a single design, we sample a design $r$ after a fixed number $T$ of timesteps according to the distribution $p ( r ; \\phi )$ . After $n$ iterations of PPO, we freeze the policy parameters and proceed to optimize the parameters of the design distribution. Without knowledge of the model, we optimize the design parameters $\\phi$ via policy gradient over parameter space, similar to Sehnke et al. (2010). This is equivalent to black-box optimization methods that have proven effective for complex learning problems (Salimans et al., 2017). We sample $m$ different designs and compute their returns for a single episode acting under policy $\\pi ( \\cdot ; \\theta )$ . We then use this data to shift the design distribution $p ( \\cdot ; \\phi )$ in a direction maximizing expected return. These iterations are repeated until convergence. ",
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+    {
+        "type": "image",
+        "img_path": "images/0babea94ff71820dd39bdf9c33c7081a0ac16e1f9df8bc584653a1c4ea8568a5.jpg",
+        "image_caption": [
+            "Figure 2: From left to right: the default Hopper, Walker2d, and Ant robots. The Hopper and Walker2d are constrained to walk in a line, while the Ant can walk anywhere in the plane. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "4 RESULTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We validate our approach with three commonly used robots provided in OpenAI’s Roboschool (Schulman et al., 2017). Environments in Roboschool are built on top of Bullet Physics, a popular open-source physics engine. In a series of locomotion experiments, we show that our approach not only discovers novel robot designs and gaits, but also outperforms two out of three of the hand-designed robots trained on the same task and yields performance comparable to the third. ",
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+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENT DETAILS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We evaluate our approach on three legged locomotion tasks within Roboschool: RoboschoolHopper, RoboschoolWalker2d, and RoboschoolAnt (note that we subsequently drop the Roboschool prefix for brevity). Figure 2 depicts the default Roboschool design for each robot. These environments describe a locomotion task that has the robots moving along the positive $\\mathbf { X }$ -axis (to the right in the figures) towards a distant goal. The reward function defined in these environments is a weighted sum of rewards for forward progress and staying upright, and penalties for applying torques and for joints that reach their rotational limits. The episode ends when the robot falls over or reaches a maximum number of timesteps. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We learn robot designs using the morphologies specified by the standard Roboschool designs. For each morphology, we parameterize the robot in terms of the length and radius (e.g., mass) of each link (or just the radius in the case of the sphere for the ant body). We impose symmetry, and share parameters across each leg for the Walker2d and Ant designs. This parameterization permits a wide variety of robots of different shapes and sizes, some of which are better suited to the locomotion objective than others, and some designs that do not permit a control policy that results in locomotion. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We model the control policy $\\pi ( a _ { t } | s _ { t } , \\mathcal { E } ; \\theta _ { p } )$ as a feed forward neural network consisting of three fully-connected layers with 64 units and tanh activation functions. A final layer maps the output to the robot action space. With the exception of the last robot-specific layer, the architecture is the same for all experiments. Meanwhile, we represent the distribution over the parameterized robot design $p ( \\cdot ; \\theta _ { r } )$ using a Gaussian mixture model with four mixture components (each with a diagonal covariance matrix over the different parameters). We initialize the means of each component randomly within a wide range, and initialize the variances in order to span the range of each parameter. We find that our approach maintains high variance distributions during early iterations—thereby continuing exploration of the design space—before committing to a chosen design. The appendix provides further details regardng the evolution of these distributions. ",
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+    {
+        "type": "image",
+        "img_path": "images/87d05764a68b4483dd235162ccd22ce42ae5244a22dfc99728015163faafa544.jpg",
+        "image_caption": [
+            "Figure 3: Training curves that show the evolution of reward across training iterations for the Hopper, Walker2d, and Ant environments over eight different random seeds. The black dashed line corresponds to the highest achievable performance of the corresponding baseline robot. In other figures, we display the best-performing checkpoint of each run. Therefore, we make each training curve transparent after it stops improving. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "We train our method with eight threads of Proximal Policy Optimization for a total of 300M environment timesteps across all threads. We use the publicly available implementation of PPO provided in OpenAI Baselines (Hesse et al., 2017). We alternate between 50 policy PPO iterations and 2 design iterations, finding this ratio to yield convergence to good solutions. We set this and the other hyper-parameters on the Hopper and used the same settings for all experiments. ",
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+    {
+        "type": "text",
+        "text": "4.2 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the performance of our approach to joint optimization over robot design and control on the Hopper, Walker2d, and Ant robots, and compare against a policy trained for the standard Roboschool design. We evaluate the consistency and robustness of our approach relative to the initial robot parameter distribution using several different random seeds. For each robot morphology, we find that our method learns robot designs and the corresponding control policy that are either comparable (Walker2d) or exceed the performance (Hopper and Ant) achievable using the default designs (Fig. 3). We also see that our method achieves these levels of performance by discovering unique robot designs (Fig. 1) together with novel walking gaits (Fig. 4) that can not be achieved with the default designs. Note that our method was able to learn these designs and control policies from a random initialization, without access to a dynamics model or any expert supervision. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our method learns a joint design and controller for the Hopper that outperforms the baseline by as much as $5 0 \\%$ . Our learned robot exploits the small size of the foot to achieve faster, more precise walking gaits. Meanwhile, the longer torso of the learned robot improves stability, allowing it to maintain balance while locomoting at a faster pace. We found this behavior to be consistent across several different Hopper experiments, with the method converging to designs with a small foot and long, thin torso. In the appendix, we explore the stability of this design with respect to variations in the environment using the coefficient of friction as an example, and find the improvement to be consistent. For the ant, our optimization yields a physical design that is significantly different from the default design (Fig. 2). Consequently, the learned Ant drastically outperforms the baseline, improving reward by up to $1 1 6 \\%$ . Our method learns a design with a small, lightweight body and extremely long legs. The long legs enable the ant to apply large torque at contact, allowing it to move at a fast pace. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Our framework learned different design-control pairs for the Walker2d that perform similarly to the default design. Across several different experiments, we see two distinct, successful designs and walking gaits. Interestingly, neither agent walks primarily on its feet. The first design has small, thick legs and long feet. The controller is then able to exploit the thick middle leg link, which protrudes slightly past the foot, to push off the ground. The long foot then provides additional balance. The second design is similar in geometry to the baseline Walker2d, but moves in a very different way. By lowering the knee joint and lengthening the foot, the Walker2d is able to efficiently balance on its knees and toes. This low stance allows the Walker2d to fully extend its leg backwards, creating a long, powerful stride, similar to that of a sprinter using starting blocks. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/97b6ba644983f0da7af6c1a0a705d198e90586ea3304b1179bfe57ed0dfdfbc5.jpg",
+        "image_caption": [
+            "Figure 4: Here, we compare locomotion of default and learned robots, visualizing both their physical design and corresponding learned gaits. We pick the best results for Hopper and Ant, and two of the best results for Walker2d (due to diversity in gaits). Note that for each robot type, we show a blend of a fixed number of frames in the same time interval, allowing direct comparison between the speed with which different designs are able to locomote. "
+        ],
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We proposed what is, to the best of our knowledge, the first model-free algorithm that jointly optimizes over the physical design of a robot and the corresponding control policy, without any need for expert supervision. Given an arbitrary morphology, our robot maintains a distribution over the robot design parameters and learns these parameters together with a neural network controller using policy gradient-based reinforcement learning. This results in an assignment to the policy over robot parameters and the control policy that are jointly optimal. We evaluated our approach on a series of different legged robot morphologies, demonstrating that it results in novel robot designs and walking gaits, achieving performance that either matches or exceeds that of manually defined designs. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our findings suggest several avenues for future work. The most direct is extending the current approach to find optimized designs for uneven terrain, the presence of obstacles, changes in slope, variations in friction, etc. We are also interested in extending our framework to relax the assumption that the morphology is pre-defined. Finally, we are investigating applications to different types of agents and design spaces beyond legged robots (e.g., end-effectors), and exploring appropriate stochastic parameterization for such designs. ",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
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+        "text": "Tim Salimans, Jonathan Ho, Xi Chen, and Ilya Sutskever. Evolution strategies as a scalable alternative to reinforcement learning. arXiv preprint arXiv:1703.03864, 2017.   \nCharles Schaff, David Yunis, Ayan Chakrabarti, and Matthew R. Walter. Jointly optimizing placement and inference for beacon-based localization. In Proc. IEEE/RSJ Int’l Conf. on Intelligent Robots and Systems (IROS), September 2017.   \nJurgen Schmidhuber, Daan Wierstra, Matteo Gagliolo, and Faustino Gomez. Training recurrent ¨ networks by evolino. Neural Computation, 19(3):757–779, 2007.   \nJohn Schulman, Sergey Levine, Philipp Moritz, Michael I. Jordan, and Pieter Abbeel. Trust region policy optimization. arXiv preprint arXiv:1502.05477, 2015a.   \nJohn Schulman, Philipp Moritz, Sergey Levine, Michael Jordan, and Pieter Abbeel. Highdimensional continuous control using generalized advantage estimation. arXiv preprint arXiv:1506.02438, 2015b.   \nJohn Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347, 2017.   \nFrank Sehnke, Christian Osendorfer, Thomas Ruckstieß, Alex Graves, Jan Peters, and J ¨ urgen ¨ Schmidhuber. Parameter-exploring policy gradients. Neural Networks, 23(4):551–559, 2010.   \nKarl Sims. Evolving virtual creatures. In Proc. Int’l Conf. on Computer Graphics and Interactive Techniques (SIGGRAPH), pp. 15–22, 1994.   \nAndrew Spielberg, Brandon Araki, Cynthia Sung, Russ Tedrake, and Daniela Rus. Functional co-optimization of articulated robots. In Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), May 2017.   \nRichard S. Sutton, David A. McAllester, Satinder P. Singh, and Yishay Mansour. Policy gradient methods for reinforcement learning with function approximation. In Advances in Neural Information Processing Systems (NIPS), 2000.   \nYuval Tassa, Tom Erez, and Emanuel Todorov. Synthesis and stabilization of complex behaviors through online trajectory optimization. In Proc. IEEE/RSJ Int’l Conf. on Intelligent Robots and Systems (IROS), 2012.   \nMiguel G. Villarreal-Cervantes, Carlos A. Cruz-Villar, Jaime Alvarez-Gallegos, and Edgar A. Portilla-Flores. Robust structure-control design approach for mechatronic systems. Trans. on Mechatronics, 18(5):1592–1601, 2013.   \nKevin Wampler, Jovan Popovic, and Zoran Popovi ´ c. Animal locomotion controllers from scratch. ´ In Computer Graphics Forum, volume 32, 2013.   \nManuel Watter, Jost Springenberg, Joschka Boedecker, and Martin Riedmiller. Embed to control: A locally linear latent dynamics model for control from raw images. In Advances in Neural Information Processing Systems (NIPS), 2015.   \nPaweł Wawrzynski. Real-time reinforcement learning by sequential actor-critics and experience ´ replay. Neural Networks, 22(10):1484–1497, 2009.   \nPaweł Wawrzynski and Ajay Kumar Tanwani. Autonomous reinforcement learning with experience ´ replay. Neural Networks, 41:156–167, 2013.   \nRonald J. Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning, 8(3–4), May 1992. ",
+        "bbox": [
+            169,
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+            838
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "6 APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            294,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The following provides further evaluation of our framework, including the behavior of the design distribution during training and the robustness of the learned designs to environment variations. ",
+        "bbox": [
+            176,
+            133,
+            823,
+            161
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "6.1 A - EVOLUTION OF GMM THROUGHOUT TRAINING",
+        "text_level": 1,
+        "bbox": [
+            171,
+            179,
+            566,
+            193
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "To provide insight into the training process of our algorithm, we evaluate the evolution of the Gaussian mixture model distribution throughout training of the best performing Hopper experiment. We initialize each component of the mixture model with random means and a diagonal covariance matrix chosen to cover the parameter space. The initial mixture weights are uniform. As shown in Figure 5 (left), roughly one third of the way through training, our algorithm converges to the most successful component. Additionally, we find that modes generally do not collapse (Figure 5 (right)). We find this behavior to be consistent across random seeds and different robots. ",
+        "bbox": [
+            173,
+            204,
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+            301
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/31445f3810e5a39d1253d521aef51b22276ca400f4dae3d24e97f4bee29e0cef.jpg",
+        "image_caption": [
+            "Figure 5: Plots of (left) the entropy of the mixture weights and (right) the marginal variance of each parameter throughout training. The curves are normalized such that all initial variances are one. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            315,
+            815,
+            506
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "6.2 B - ROBUSTNESS OF LEARNED DESIGNS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            585,
+            490,
+            599
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "It is often desirable for a robot to be able to operate in a variety of environments. In this section, we consider the robustness of a learned design to changes in friction. We conducted experiments on the Hopper in which we first learned the design and controller for one friction setting (0.8). We finetuned the controller in environments with different friction settings, while leaving the design fixed. We find the learned design to be reasonably robust with variability comparable to controllers finetuned for the default hand-crafted design (Fig. 6). Additionally, the learned design outperforms the hand-crafted one across the full range of friction values (although, for very low friction values, both designs essentially were unable to learn a successful gait). ",
+        "bbox": [
+            174,
+            611,
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+            723
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Note that our framework can incorporate this goal of generalization by simply sampling from a diverse set of environments during training. But at the same time, it may be useful in some applications to seek out solutions that are specifically adapted to a relatively narrower set of environments, gaining better performance within this set at the cost of more general performance. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/6950886e33eae5aebfefebe1362c093a2ef2d79ce2a27d5c428bd80267a0f480.jpg",
+        "image_caption": [
+            "Figure 6: A visualization of accumulated reward for different friction values. The controllers and learned design were trained in an environment with a friction value of 0.8. We finetuned the controllers (but not the designs) for both the learned and hand-crafted designs for 10M timesteps for each friction value. Rewards are reported as an average over 100 episodes. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            319,
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+            631
+        ],
+        "page_idx": 12
+    }
+]
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new file mode 100644
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+                                    "score": 1.0,
+                                    "content": "ing to train a neural network controller. Throughout training, we refine the robot",
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+                                    "content": "such as errors in perception, delays in actuation, etc., and indeed, make control problem an easier",
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+                                    "content": "one to solve. At the same time, robots that take different forms may find completely different",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "unable to locomote up an incline). Meanwhile, careful choice of bipedal design enables passive",
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+                                    "score": 1.0,
+                                    "content": "walking (McGeer, 1990; Goswami et al., 1998; Collins et al., 2001). It is therefore beneficial to",
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+                                    "content": "environment and task. Similar co-design can be beneficial in other settings—for example for the",
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+                                    "content": "control policy and physical characteristics of digits in robotic grippers for grasping.",
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+                                    "content": "optimization process). However, these restrictive assumptions limits their applicability to a handful",
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+                                    "content": "of specific settings, and often yield solutions heavily influenced by expert intuition.",
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+                                    "score": 1.0,
+                                    "content": "walking (McGeer, 1990; Goswami et al., 1998; Collins et al., 2001). It is therefore beneficial to",
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+                                    "content": "and motion parameters for robots with articulated degrees of freedom (e.g., legged robots). They",
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+                                    "score": 1.0,
+                                    "content": "et al., 2015). Unlike our method, the use of trajectory optimization makes these approaches reliant",
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+                                    "score": 1.0,
+                                    "content": "simulated robots (Wawrzynski, 2009; Wawrzy ´ nski & Tanwani, 2013; Watter et al., 2015; Lillicrap ´",
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+                                    "content": "those that involve locomotion. Closely related is the policy-gradient method of Sehnke et al. (2010),",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "namical systems. Their method interleaves supervised learning with trajectory optimization, and",
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+                                    "score": 1.0,
+                                    "content": "incorporates noise to improve generalizability. Meanwhile, there is a large body of literature that",
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+                                    "score": 1.0,
+                                    "content": "tive at respecting contact constraints (e.g., between the foot and ground), which make controlling",
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+                                    "score": 1.0,
+                                    "content": "2016). These approaches have been shown to generate sophisticated behaviors for complex robot",
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+                            "index": 28
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+                                    "content": "a high-level specification of the robot’s shape, gait, and task (Wampler et al., 2013). Related, a",
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+                            "index": 32
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+                                    "score": 1.0,
+                                    "content": "et al., 2015). Unlike our method, the use of trajectory optimization makes these approaches reliant",
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+                                    "content": "those that involve locomotion. Closely related is the policy-gradient method of Sehnke et al. (2010),",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
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+                                    "content": "for brevity). Figure 2 depicts the default Roboschool design for each robot. These environments",
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+                                    "score": 1.0,
+                                    "content": "We learn robot designs using the morphologies specified by the standard Roboschool designs. For",
+                                    "type": "text"
+                                }
+                            ],
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+                        },
+                        {
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+                        },
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+                                    "content": "link (or just the radius in the case of the sphere for the ant body). We impose symmetry, and share",
+                                    "type": "text"
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+                        },
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+                                    "type": "text"
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+                            ],
+                            "index": 28
+                        },
+                        {
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+                                    "content": "variety of robots of different shapes and sizes, some of which are better suited to the locomotion",
+                                    "type": "text"
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+                        {
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+                                    "score": 1.0,
+                                    "content": "objective than others, and some designs that do not permit a control policy that results in locomotion.",
+                                    "type": "text"
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+                    ],
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+                                    "score": 1.0,
+                                    "content": "onal covariance matrix over the different parameters). We initialize the means of each component",
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+                            "index": 36
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+                        {
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+                                    "score": 1.0,
+                                    "content": "randomly within a wide range, and initialize the variances in order to span the range of each parame-",
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+                                    "score": 1.0,
+                                    "content": "ter. We find that our approach maintains high variance distributions during early iterations—thereby",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "continuing exploration of the design space—before committing to a chosen design. The appendix",
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+                                            "type": "text"
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+                                    ],
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+                                },
+                                {
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+                                            "content": "Walker2d, and Ant environments over eight different random seeds. The black dashed line corre-",
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+                                    ],
+                                    "index": 4
+                                },
+                                {
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+                                            "content": "sponds to the highest achievable performance of the corresponding baseline robot. In other figures,",
+                                            "type": "text"
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+                                    ],
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+                                {
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+                                            "score": 1.0,
+                                            "content": "we display the best-performing checkpoint of each run. Therefore, we make each training curve",
+                                            "type": "text"
+                                        }
+                                    ],
+                                    "index": 6
+                                },
+                                {
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+                                            "score": 1.0,
+                                            "content": "transparent after it stops improving.",
+                                            "type": "text"
+                                        }
+                                    ],
+                                    "index": 7
+                                }
+                            ],
+                            "index": 5
+                        }
+                    ],
+                    "index": 3.0
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                    ],
+                    "lines": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "We train our method with eight threads of Proximal Policy Optimization for a total of 300M environ-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 8
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "ment timesteps across all threads. We use the publicly available implementation of PPO provided",
+                                    "type": "text"
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+                            ],
+                            "index": 9
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "in OpenAI Baselines (Hesse et al., 2017). We alternate between 50 policy PPO iterations and 2",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 10
+                        },
+                        {
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "design iterations, finding this ratio to yield convergence to good solutions. We set this and the other",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 11
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "hyper-parameters on the Hopper and used the same settings for all experiments.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 12
+                        }
+                    ],
+                    "index": 10
+                },
+                {
+                    "type": "title",
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+                                    "content": "4.2 EXPERIMENTS",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 13
+                        }
+                    ],
+                    "index": 13
+                },
+                {
+                    "type": "text",
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+                                    "score": 1.0,
+                                    "content": "We evaluate the performance of our approach to joint optimization over robot design and control",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 14
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "on the Hopper, Walker2d, and Ant robots, and compare against a policy trained for the standard",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 15
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "Roboschool design. We evaluate the consistency and robustness of our approach relative to the",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 16
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "initial robot parameter distribution using several different random seeds. For each robot morphology,",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 17
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "we find that our method learns robot designs and the corresponding control policy that are either",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 18
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "comparable (Walker2d) or exceed the performance (Hopper and Ant) achievable using the default",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "designs (Fig. 3). We also see that our method achieves these levels of performance by discovering",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 20
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "unique robot designs (Fig. 1) together with novel walking gaits (Fig. 4) that can not be achieved with",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 21
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "the default designs. Note that our method was able to learn these designs and control policies from",
+                                    "type": "text"
+                                }
+                            ],
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+                        },
+                        {
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+                                    "content": "from the default design (Fig. 2). Consequently, the learned Ant drastically outperforms the baseline,",
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+                                    "score": 1.0,
+                                    "content": "in OpenAI Baselines (Hesse et al., 2017). We alternate between 50 policy PPO iterations and 2",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "design iterations, finding this ratio to yield convergence to good solutions. We set this and the other",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "We evaluate the performance of our approach to joint optimization over robot design and control",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 14
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "on the Hopper, Walker2d, and Ant robots, and compare against a policy trained for the standard",
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+                                    "score": 1.0,
+                                    "content": "Roboschool design. We evaluate the consistency and robustness of our approach relative to the",
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+                                    "score": 1.0,
+                                    "content": "initial robot parameter distribution using several different random seeds. For each robot morphology,",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 17
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "score": 1.0,
+                                    "content": "we find that our method learns robot designs and the corresponding control policy that are either",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "comparable (Walker2d) or exceed the performance (Hopper and Ant) achievable using the default",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "designs (Fig. 3). We also see that our method achieves these levels of performance by discovering",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 20
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+                        {
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+                                    "score": 1.0,
+                                    "content": "unique robot designs (Fig. 1) together with novel walking gaits (Fig. 4) that can not be achieved with",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 21
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "the default designs. Note that our method was able to learn these designs and control policies from",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 22
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "a random initialization, without access to a dynamics model or any expert supervision.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 23
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+                    ],
+                    "index": 18.5,
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+                                    "score": 1.0,
+                                    "content": "Our method learns a joint design and controller for the Hopper that outperforms the baseline by as",
+                                    "type": "text"
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+                            ],
+                            "index": 24
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+                        {
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+                                    "score": 1.0,
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+                                    "content": "5 0 \\%",
+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": ". Our learned robot exploits the small size of the foot to achieve faster, more precise",
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+                                    "score": 1.0,
+                                    "content": "walking gaits. Meanwhile, the longer torso of the learned robot improves stability, allowing it to",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "maintain balance while locomoting at a faster pace. We found this behavior to be consistent across",
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "several different Hopper experiments, with the method converging to designs with a small foot and",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "long, thin torso. In the appendix, we explore the stability of this design with respect to variations",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 29
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "in the environment using the coefficient of friction as an example, and find the improvement to",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "be consistent. For the ant, our optimization yields a physical design that is significantly different",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 31
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "from the default design (Fig. 2). Consequently, the learned Ant drastically outperforms the baseline,",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "improving reward by up to",
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+                                    "score": 0.89,
+                                    "content": "1 1 6 \\%",
+                                    "type": "inline_equation"
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+                                {
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+                                    "score": 1.0,
+                                    "content": ". Our method learns a design with a small, lightweight body and",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 33
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "extremely long legs. The long legs enable the ant to apply large torque at contact, allowing it to",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 34
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "move at a fast pace.",
+                                    "type": "text"
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+                        }
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+                                    "score": 1.0,
+                                    "content": "Our framework learned different design-control pairs for the Walker2d that perform similarly to the",
+                                    "type": "text"
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+                            ],
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+                        },
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+                                    "score": 1.0,
+                                    "content": "default design. Across several different experiments, we see two distinct, successful designs and",
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+                            ],
+                            "index": 37
+                        },
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+                                    "score": 1.0,
+                                    "content": "walking gaits. Interestingly, neither agent walks primarily on its feet. The first design has small,",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 38
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "thick legs and long feet. The controller is then able to exploit the thick middle leg link, which",
+                                    "type": "text"
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+                            ],
+                            "index": 39
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "protrudes slightly past the foot, to push off the ground. The long foot then provides additional",
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+                            ],
+                            "index": 40
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "balance. The second design is similar in geometry to the baseline Walker2d, but moves in a very",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 41
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "different way. By lowering the knee joint and lengthening the foot, the Walker2d is able to efficiently",
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+                                    "score": 1.0,
+                                    "content": "balance on its knees and toes. This low stance allows the Walker2d to fully extend its leg backwards,",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 43
+                        },
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+                                    "score": 1.0,
+                                    "content": "creating a long, powerful stride, similar to that of a sprinter using starting blocks.",
+                                    "type": "text"
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+                                            "score": 1.0,
+                                            "content": "Figure 4: Here, we compare locomotion of default and learned robots, visualizing both their physical",
+                                            "type": "text"
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+                                    ],
+                                    "index": 3
+                                },
+                                {
+                                    "bbox": [
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+                                            "score": 1.0,
+                                            "content": "design and corresponding learned gaits. We pick the best results for Hopper and Ant, and two of the",
+                                            "type": "text"
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+                                    ],
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+                                },
+                                {
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+                                            "score": 1.0,
+                                            "content": "best results for Walker2d (due to diversity in gaits). Note that for each robot type, we show a blend",
+                                            "type": "text"
+                                        }
+                                    ],
+                                    "index": 5
+                                },
+                                {
+                                    "bbox": [
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+                                            "score": 1.0,
+                                            "content": "of a fixed number of frames in the same time interval, allowing direct comparison between the speed",
+                                            "type": "text"
+                                        }
+                                    ],
+                                    "index": 6
+                                },
+                                {
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+                                            "score": 1.0,
+                                            "content": "with which different designs are able to locomote.",
+                                            "type": "text"
+                                        }
+                                    ],
+                                    "index": 7
+                                }
+                            ],
+                            "index": 5
+                        }
+                    ],
+                    "index": 3.0
+                },
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+                                    "content": "Manuel Watter, Jost Springenberg, Joschka Boedecker, and Martin Riedmiller. Embed to control:",
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+                                    "content": "replay. Neural Networks, 22(10):1484–1497, 2009.",
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+                                    "content": "To provide insight into the training process of our algorithm, we evaluate the evolution of the Gaus-",
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+                                    "content": "sian mixture model distribution throughout training of the best performing Hopper experiment. We",
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+                                    "score": 1.0,
+                                    "content": "initialize each component of the mixture model with random means and a diagonal covariance ma-",
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+                                    "score": 1.0,
+                                    "content": "trix chosen to cover the parameter space. The initial mixture weights are uniform. As shown in",
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+                                    "score": 1.0,
+                                    "content": "Figure 5 (left), roughly one third of the way through training, our algorithm converges to the most",
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+                                    "content": "successful component. Additionally, we find that modes generally do not collapse (Figure 5 (right)).",
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+                                    "content": "We find this behavior to be consistent across random seeds and different robots.",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "on the Hopper in which we first learned the design and controller for one friction setting (0.8). We",
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diff --git a/parse/train/Z7xSQ3SXLQU/Z7xSQ3SXLQU.md b/parse/train/Z7xSQ3SXLQU/Z7xSQ3SXLQU.md
new file mode 100644
index 0000000000000000000000000000000000000000..baa90045e9f68f7e16795337d46cf65acabd2c15
--- /dev/null
+++ b/parse/train/Z7xSQ3SXLQU/Z7xSQ3SXLQU.md
@@ -0,0 +1,233 @@
+# Manipulating SGD with Data Ordering Attacks
+
+Zakhar Shumaylov University of Cambridge zs334@cam.ac.uk
+
+Ilia Shumailov University of Cambridge & University of Toronto & Vector Institute ilia.shumailov@cl.cam.ac.uk
+
+Dmitry Kazhdan University of Cambridge dk525@cam.ac.uk
+
+Yiren Zhao University of Cambridge yiren.zhao@cl.cam.ac.uk
+
+Nicolas Papernot University of Toronto & Vector Institute nicolas.papernot@utoronto.ca
+
+Murat A. Erdogdu University of Toronto & Vector Institute erdogdu@cs.toronto.edu
+
+Ross Anderson University of Cambridge & University of Edinburgh ross.anderson@cl.cam.ac.uk
+
+# Abstract
+
+Machine learning is vulnerable to a wide variety of attacks. It is now well understood that by changing the underlying data distribution, an adversary can poison the model trained with it or introduce backdoors. In this paper we present a novel class of training-time attacks that require no changes to the underlying dataset or model architecture, but instead only change the order in which data are supplied to the model. In particular, we find that the attacker can either prevent the model from learning, or poison it to learn behaviours specified by the attacker. Furthermore, we find that even a single adversarially-ordered epoch can be enough to slow down model learning, or even to reset all of the learning progress. Indeed, the attacks presented here are not specific to the model or dataset, but rather target the stochastic nature of modern learning procedures. We extensively evaluate our attacks on computer vision and natural language benchmarks to find that the adversary can disrupt model training and even introduce backdoors.
+
+# 1 Introduction
+
+The data-driven nature of modern machine learning (ML) training routines puts pressure on data supply pipelines, which become increasingly more complex. It is common to find separate disks or whole content distribution networks dedicated to servicing massive datasets. Training is often distributed across multiple workers. This emergent complexity gives a perfect opportunity for an attacker to disrupt ML training, while remaining covert. In the case of stochastic gradient descent (SGD), it assumes uniform random sampling of items from the training dataset, yet in practice this randomness is rarely tested or enforced. Here, we focus on adversarial data sampling.
+
+It is now well known that malicious actors can poison data and introduce backdoors, forcing ML models to behave differently in the presence of triggers [12]. While such attacks have been shown to pose a real threat, they have so far required the attacker to perturb the dataset used for training.
+
+We now show that by simply changing the order in which batches or data points are supplied to a model during training, an attacker can affect model behaviour. More precisely, we show that it is possible to perform integrity and availability attacks without adding or modifying any data points. For integrity, an attacker can reduce model accuracy or arbitrarily control its predictions in the presence of particular triggers. For availability, an attacker can increase the amount of time it takes for the model to train, or even reset the learning progress, forcing the model parameters into a meaningless state.
+
+<table><tr><td>Attack</td><td></td><td></td><td></td><td>Dataset knowledgeModel knowledgeModelspecificChangingdatasetAdding dataAddingperturbations</td><td></td><td></td></tr><tr><td>Batch Reorder</td><td></td><td></td><td></td><td></td><td>X</td><td></td></tr><tr><td>Batch Reshuffle</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>Batch Replace</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>Adversarial initialisation [11]</td><td></td><td>×</td><td>√</td><td>X</td><td>X</td><td></td></tr><tr><td>BadNets [12]</td><td>X</td><td></td><td>X</td><td></td><td>X</td><td>×</td></tr><tr><td>Dynamic triggers [28]</td><td>√</td><td>√</td><td>X</td><td></td><td>X</td><td>√</td></tr><tr><td>Poisoned frogs [32]</td><td></td><td>√</td><td>X</td><td>√</td><td>X</td><td>√</td></tr></table>
+
+Table 1: Taxonomy of training time attacks. In green, we highlight our attacks.
+
+We present three different types of attacks that exploit Batch Reordering, Reshuffling and Replacing – naming them BRRR attacks. We show that an attacker can significantly change model performance by (i) changing the order in which batches are supplied to models during training; (ii) changing the order in which individual data points are supplied to models during training; and (iii) replacing datapoints from batches with other points from the dataset to promote specific data biases. Furthermore, we introduce Batch-Order Poison (BOP) and Batch-Order Backdoor (BOB), the first techniques that enable poisoning and backdooring of neural networks using only clean data and clean labels; an attacker can control the parameter update of a model by appropriate choice of benign datapoints. Importantly, BRRR attacks require no underlying model access or knowledge of the dataset. Instead, they focus on the stochasticity of gradient descent, disrupting how well individual batches approximate the true distribution that a model is trying to learn.
+
+To summarise, we make the following contributions in this paper:
+
+• We present a novel class of attacks on ML models that target the data batching procedure used during training, affecting their integrity and availability. We present a theoretical analysis explaining how and why these attacks work, showing that they target fundamental assumptions of stochastic learning, and are therefore model and dataset agnostic1. • We evaluate these attacks on a set of common computer vision and language benchmarks, using a range of different hyper-parameter configurations, and find that an attacker can slow the progress of training, as well as reset it, with just a single epoch of intervention. • We show that data order can poison models and introduce backdoors, even in a blackbox setup. For a whitebox setup, we find that the adversary can introduce backdoors almost as well as if they used perturbed data. While a baseline CIFAR10 VGG16 model that uses perturbed data gets ${ \bar { 9 } } 9 \%$ trigger accuracy, the whitebox BOB attacker gets $9 1 \% \pm 1 3$ and the blackbox BOB attacker achieves $6 8 \% \pm 1 9$ .
+
+# 2 Related Work
+
+Attacks on integrity: Szegedy et al. [34] and Biggio et al. [6] concurrently discovered the existence of adversarial examples. These samples, containing human imperceptible perturbations, cause models to output incorrect results during inference. The original whitebox attacks require the adversary to access the models and use gradient information to perform conditioned optimisation to maximise the model loss [34, 6, 10, 22]. The attack later generalised to blackbox setups, where the adversary trains a surrogate model and hopes the generated adversarial samples transfer to the target model [24].
+
+The data-poisoning attack aims at using data manipulation to cause DNNs to fail on specific test-time instances [17]. Chen et al. demonstrated that manipulation of the labels of around 50 training samples is enough to trigger failure [8]. Gu et al. showed that attackers can associate adversarial patterns with labelled images and cause DNNs to overfit to this pattern [12]. Shafahi et al. launched a poisoning attack using instances with clean labels [29]. A number of other works have since created more efficient triggers [28]. It was a common belief that poisoning attacks on DNNs have to contain a certain level of malicious manipulation of whether the data or label at train time. However, this paper shows how poisoning is possible with clean data and clean labels, with the only manipulation being of the batching process at training time.
+
+Attacks on availability: Shumailov et al. first attacked the availability of computer vision and natural language processing models at inference time with sponge examples [30]. They pessimized over energy utilisation and inference latency to target hardware and internal model optimisations. By contrast, this paper targets availability at training time. We show that the attacker can reset or slow down training progress by reordering or reshuffling natural batches. Finally, we note that unlike Shumailov et al., our attacks do not target specific optimisations in hardware or individual models, but instead break the fundamental stochastic assumption of training.
+
+Note on terminology: As we study attacks on training, we use the terms integrity and availability in a slightly different sense from the rest of the literature. By availability, we mean the availability of a training procedure – the rate at which the model trains, as opposed to model performance on the task. By integrity attacks, we mean attacks that either stop the model from learning its intended task, or force it to learn an additional unrelated task.
+
+# 3 Methodology
+
+# 3.1 Threat model
+
+![](images/7350808cdc653df8e4d2f8c68bb3e0851adb28912ef8969280821c54c3a3fc85.jpg)  
+Figure 1: The attacker reorders the benign randomly supplied data based on the surrogate model outputs. Attacker co-trains the surrogate model with the data that is supplied to the source model.
+
+We assume one of the strongest threat models currently described in the literature. In particular, our blackbox attacker assumes no access to the model and no prior knowledge of the training data, whereas a whitebox attacker has access to the model under attack and can compute its loss directly. The attack specifically focuses on the batching part of the ML pipeline as is depicted in Figure 1. We discuss the related work in Section 2.
+
+This attack is realistic and can be instantiated in several ways. The attack code can be infiltrated into: the operating system handing file system requests; the disk handling individual data accesses; the software that determines the way random data sampling is performed; the distributed storage manager; or the machine learning pipeline itself handling prefetch operations. That is a substantial attack surface, and for large models these components may be controlled by different principals. The attack is also very stealthy. The attacker does not add any noise or perturbation to the data. There are no triggers or backdoors introduced into the dataset. All of the data points are natural. In two of four variants the attacker uses the whole dataset and does not oversample any given point, i.e. the sampling is without replacement. This makes it difficult to deploy simple countermeasures.
+
+# 3.2 Primer on stochastic learning and batching
+
+We assume that the defender is trying to train a deep neural network model with parameters $\theta$ operating over $X _ { i } \sim { \mathcal { X } } _ { \mathrm { t r a i n } }$ , solving a non-convex optimization problem with respect to parameters $\theta$ , corresponding to minimization of a given loss function $L ( \theta )$ . We will denote the training dataset $X \ = \ \{ X _ { i } \}$ . We assume a commonly-used loss function defined as the sample average of the loss per training data point $L _ { i } ( \theta ) \stackrel { \cdot } { = } \stackrel { \cdot } { L } ( X _ { i } , \theta )$ in $k$ -th batch over the training set, where $B$ is the batch size: $\begin{array} { r } { \hat { L } _ { k + 1 } ( \theta ) = \frac { 1 } { B } \sum _ { i = k B + 1 } ^ { k B + B } L _ { i } ( \theta ) } \end{array}$ . If we let $N \cdot B$ be the total number of items for training, then in a single epoch one aims to optimize: $\begin{array} { r } { \hat { L } ( \theta ) = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \hat { L } _ { i } ( \theta ) } \end{array}$ . Optimization with stochastic gradient descent (SGD) algorithm of $N \cdot B$ samples and a learning rate of $\eta$ leads to the following weight update rule over one epoch: $\theta _ { k + 1 } = \theta _ { k } + \eta \Delta \theta _ { k }$ ; $\Delta \theta _ { k } = - \nabla _ { \theta } \hat { L } _ { k } ( \theta _ { k } )$ . SGD is often implemented with momentum [25, 33], with $\mu$ and $v$ representing momentum and velocity respectively: $v _ { k + 1 } = \mu v _ { k } + \eta \Delta \theta _ { k }$ ; $\theta _ { k + 1 } = \theta _ { k } + v _ { k + 1 }$ .
+
+Given data, SGD’s stochasticity comes from the batch sampling procedure. Mini-batched gradients approximate the true gradients of $\hat { L }$ and the quality of this approximation can vary greatly. In fact, assuming an unbiased sampling procedure, i.e. when the $k$ ’th gradient step corresponds to $i _ { k }$ ’th batch with $\mathbb { P } ( i _ { k } = i ) = 1 / N$ , in expectation the batch gradient matches the true gradient:
+
+$$
+\mathbb { E } [ \nabla \hat { L } _ { i _ { k } } ( \theta ) ] = \sum _ { i = 1 } ^ { N } \mathbb { P } ( i _ { k } = i ) \nabla \hat { L } _ { i } ( \theta ) = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \nabla \hat { L } _ { i } ( \theta ) = \nabla \hat { L } ( \theta ) .
+$$
+
+Although this happens in expectation, a given batch taken in isolation can be very far from the mean. This variation has been exploited in the literature to aid training: there exists a field responsible for variance reduction techniques for stochastic optimisation [18], active learning [9, 35], (anti-) curriculum learning [5, 20, 13] and core-set construction [2]. Each area looks at identifying and scheduling data subsets that aid training and give a better true gradient approximation. In this paper, we turn things round and investigate how an attacker can exploit data order to break training. The explicit stochastic assumption opens a new attack surface for the attacker to influence the learning process. In particular, let us consider the effect of $N$ SGD steps over one epoch [31]:
+
+$$
+\begin{array} { l } { \displaystyle \theta _ { N + 1 } = \theta _ { 1 } - \eta \nabla \hat { L } _ { 1 } ( \theta _ { 1 } ) - \eta \nabla \hat { L } _ { 2 } ( \theta _ { 2 } ) - \cdot \cdot \cdot - \eta \nabla \hat { L } _ { N } ( \theta _ { N } ) } \\ { \displaystyle \qquad = \theta _ { 1 } - \eta \sum _ { j = 1 } ^ { N } \nabla \hat { L } _ { j } ( \theta _ { 1 } ) + \eta ^ { 2 } \sum _ { j = 1 } ^ { N } \sum _ { k < j } \nabla \nabla \hat { L } _ { j } ( \theta _ { 1 } ) \nabla \hat { L } _ { k } ( \theta _ { 1 } ) + O ( N ^ { 3 } \eta ^ { 3 } ) . } \end{array}
+$$
+
+As we can see, in this case the second order correction term is dependent on the order of the batches provided. The attacker we describe in this paper focuses on manipulating it i.e. finding a sequence of updates such that the first and second derivatives are misaligned with the true gradient step. In Appendix C we prove that under equally-distributed loss gradients, a change in data order can lead to an increase in the expected value of the term which is dependent on data order. We also derive a condition on the gradient distribution given a model for which it is guaranteed. Finally, we derive an attack objective to target the upper bound on the rate of SGD convergence explicitly in Appendix A.
+
+In this paper we assume the blackbox attacker has no access to the underlying model, and thus no way to monitor its errors or the progress of its training. Instead, we co-train a separate surrogate model, using the batches supplied to the target model. We find that in practice the losses produced by the surrogate model approximate the losses of the true model well enough to enable attacks on both integrity and availability. We empirically find that our blackbox reshuffle attacks perform as well as the whitebox one in Appendix D.
+
+Finally, although the attacker can maximise the term dependent on data order directly, in practice it is expensive to do so. Therefore, in the attack we use the loss magnitudes directly rather than their gradient. Intuitively, large prediction errors correspond to large loss gradient norms, while correct predictions produce near-zero gradients. Although this assumption may not hold for all loss landscapes, for the loss landscapes incurred by real machine learning models, local minima often cluster around the global minima as the loss satisfies certain dissipativity properties [26, 36], and consequently loss magnitudes correlate with the gradient of the loss. We emphasize that we use this approximation only for computational advantage.
+
+# 3.3 Taxonomy of batching attacks
+
+In this section we describe the taxonomy of batching attacks as shown in Figure 2. The overall attack algorithm is shown in Algorithm 2 in the Appendix, and a shortened attack flow is shown in Algorithm 1. We highlight that our attacks are the first to successfully poison the underlying
+
+![](images/9ecad19d92ad1568ab77942a28d92ec253ae771264a960b96261a0566ccd5b9a.jpg)  
+Figure 2: Taxonomy of BRRR attacks. Normal batching assumes randomly distributed data points and batches. Batch reordering assumes the batches appear to the model in a different order, but internal contents stay in the original random order. Batch reshuffling assumes that the individual datapoints within batches change order, but appear only once and do not repeat across batches. Finally, batch replacement refers to cases where datapoints or batches can repeat or not appear at all.
+
+# Algorithm 1: A high level description of the BRRR attack algorithm
+
+/\* –– Attack preparation: collecting data \*/   
+do get a new batch and add it to a list of unseen datapoints; train surrogate model on a batch and pass it on to the model;   
+while first epoch is not finished   
+/\* Attack: reorder based on surrogate loss \*/   
+while training do rank each data point from epoch one with a surrogate loss; reorder the data points according to the attack strategy; pass batches to model and train the surrogate at the same time.
+
+model without changing the underlying dataset. In this paper we use three attack policies – (1) batch reshuffling or changing the order of datapoints inside batches; (2) batch reordering or changing the order of batches; and (3) batch replacement or replacing both points and batches. We consider four reordering policies, motivated by research in the fields of curriculum learning [5] and core-set selection [2], which discovered that model training can be enhanced by scheduling how and what data is presented to the model. That can help the model to generalize and to avoid overfitting with memorization. This paper does the opposite – we promote memorization and overfitting, forcing the model to forget generalizable features. Anti-curriculum learning (e.g. [13]) is particularly close to our batch reordering setting.
+
+![](images/df105449880bbabdecd0854bc6cc57fd11ee600465a07e1058c5c441f8b34b55.jpg)  
+Figure 3: We use four different reorder and reshuffle policies based on the corresponding data point and batch losses. We color-code the loss values from bright to dark colors, to represent loss values from low to high. Low-high policy orders a sequence by the loss magnitude. High-low policy orders a sequence by the negative loss magnitude. Oscillation inwards orders elements of the sequence from the beginning and the end of the sequence one by one, as if it was oscillating between sides of the sequence and moving towards the middle. Finally, Oscillations outward orders the sequence by starting at the middle of an ordered sequence picking elements to both sides of the current location.
+
+Figure 3 shows attack policies. Low to high orders sequence items by their loss. High to low is an inverse of Low to high. Oscillations inwards picks elements from both sides in sequence. Oscillations outwards inverts the halves of the sequence and then picks elements from both sides.
+
+# 3.4 Batch-order poison and backdoor
+
+Machine-learning poisoning and backdooring techniques aim to manipulate the training of a given model to control its behavior during inference. In the classical setting, both involve either appending adversarial datapoints $\hat { X }$ to natural dataset $X$ or changing natural datapoints $X + \delta$ so as to change model behaviour. This makes the attack easier to detect, and to prevent. For example, an adversary may add a red pixel above every tank in the dataset to introduce the red pixel trigger and cause other objects under red pixels to be classified as tanks.
+
+We present batch-order poisoning (BOP) and batch-order backdooring (BOB) – the first poison and backdoor strategies that do not rely on adding adversarial datapoints or perturbations during training, but only on changing the order in which genuine data are presented. BOP and BOB are based on the idea that the stochastic gradient update rule used in DNN training is agnostic of the batch contents and is an aggregation. Indeed, consider a classical poison setting with an adversarial dataset $\hat { X }$ : $\theta _ { k + 1 } = \theta _ { k } \stackrel { \smile \sim } { + } \eta \bar { \Delta } \theta _ { k } ; \quad \Delta \theta _ { k } = - ( \nabla _ { \theta } \hat { L } ( X _ { k } , \theta _ { k } ) + \nabla _ { \theta } \hat { L } ( \hat { X } _ { k } , \theta _ { k } ) )$
+
+Order-agnostic aggregation with a sum makes it hard to reconstruct the individual datapoints $X _ { k }$ from just observing $\Delta \theta _ { k }$ . Indeed, the stochastic nature of optimisation allows one to find a set of datapoints $X _ { j } \neq X _ { i }$ such that $\nabla _ { \boldsymbol { \theta } } \hat { L } ( X _ { i } , { \boldsymbol { \theta } } _ { k } ) \approx \nabla _ { \boldsymbol { \theta } } \hat { L } ( X _ { j } , { \boldsymbol { \theta } } _ { k } )$ . Given a model and a dataset such that the gradient covariance matrix is non-singular, an attacker can approximate the gradient update from an adversarial dataset $\hat { X }$ using natural datapoints from the genuine dataset $X$ , enabling poisoning without having to change of underlying dataset in any way:
+
+$$
+\theta _ { k + 1 } = \theta _ { k } + \eta \hat { \Delta } \theta _ { k } , \mathrm { w h e r e \ } \left\{ \begin{array} { l l } { \hat { \Delta } \theta _ { k } = - \nabla _ { \theta } \hat { L } ( X _ { i } , \theta _ { k } ) } \\ { \nabla _ { \theta } \hat { L } ( X _ { i } , \theta _ { k } ) \approx \nabla _ { \theta } \hat { L } ( \hat { X } _ { k } , \theta _ { k } ) . } \end{array} \right.
+$$
+
+This gives rise to a surprisingly powerful adversary, who can introduce arbitrary behaviors into any models learned with stochastic gradient descent without having to add or perturb training data. This attack becomes more effective as training datasets become larger, further improving the attacker’s ability to shape the gradient update. We discuss its fidelity in Appendix B.
+
+![](images/429f59afc50256e77aa3ccf6dba59f0766f7a0748716b19ea920ca86f94cc5c2.jpg)  
+Figure 4: Examples of batches shown in (a) and (b) with similar gradient updates. Images shown in (a) are natural and are coming from the training set, whereas image in (b) is the poison datapoint. Strong gradients are aligned across the layers and successfully change the prediction of poison datapoints.
+
+We evaluated a number of different setups, and found that the attack works best when the attacker comprises the batch with $B - V$ natural data points and appends $V$ adversarially-chosen data points ${ \hat { X } } _ { i }$ to the batch. Aa larger $V$ is better for gradient approximation, but leads to more conflict with natural gradients; in the paper up to $30 \%$ of the batch is filled with natural datapoints to find a balance. Finding precise batch reconstruction is an intractable problem that scales with batch size and size of the dataset. However, we find that random sampling works well; even if mistakes are made, the network still learns the poison over the course of a few batches. Overall we try to minimize the following reconstruction error for a given poisoned batch $\hat { X _ { j } }$ :
+
+$$
+\operatorname* { m i n } _ { X _ { i } } \quad \left\| \nabla _ { \theta } \hat { L } ( \hat { X _ { j } } , \theta _ { k } ) - \nabla _ { \theta } \hat { L } ( X _ { i } , \theta _ { k } ) \right\| ^ { p } ; \quad \mathrm { s . t . } \quad X _ { i } \in X .
+$$
+
+Although more sophisticated approaches could help find better candidates, we discovered that random sampling works well enough for successful clean-data / clean-label poison and backdoor attacks. It also helps us strike a balance between speed of batch construction and impact on model performance. Figure 4 shows an example of natural data (a) that closely resembled the would-be update with batch in (b). Notably, such updates result in a prediction change towards the target class.
+
+# 4 Evaluation
+
+# 4.1 Experimental setup
+
+We evaluate our attacks using two computer vision and one natural language benchmarks: the CIFAR-10, CIFAR-100 [19] and AGNews [37] datasets. We use ResNet-18 and ResNet-50 as source models [14], and LeNet-5 [21] and MobileNet [16] as surrogate models, to train CIFAR-10 and CIFAR-100 respectively. For AGNews we used sparse mean EmbeddingBag followed by three fully connected layer from torchtext as source model and one fully connected layer for surrogate. Note that the surrogate model is significantly less performant than its corresponding source model in all cases, and cannot learn the dataset to the same degree of accuracy, limiting attacker capabilities. Thus our results represent a lower bound on attack performance.
+
+4.2 Integrity attacks with reshuffling and reordering of natural data   
+
+<table><tr><td></td><td colspan="2">CIFAR-10</td><td></td><td colspan="2">CIFAR-100</td><td></td><td></td><td>AGNews</td><td></td></tr><tr><td>Attack</td><td>Train acc</td><td>Test acc</td><td>△</td><td>Train acc</td><td>Test acc</td><td>△</td><td>Train acc</td><td>Test acc</td><td>△</td></tr><tr><td>Baseline</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>None</td><td>95.51</td><td>90.51</td><td>-0.0%</td><td>99.96</td><td>75.56</td><td>-0.0%</td><td>93.13</td><td>90.87</td><td>-0.0%</td></tr><tr><td>Batch reshuffle</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Oscillation outward</td><td>17.44</td><td>26.13</td><td>-64.38%</td><td>99.80</td><td>18.00</td><td>-57.56%</td><td>97.72</td><td>65.85</td><td>-25.02%</td></tr><tr><td>Oscillation inward</td><td>22.85</td><td>28.94</td><td>-61.57%</td><td>99.92</td><td>31.38</td><td>-44.18%</td><td>94.06</td><td>89.23</td><td>-1.64%</td></tr><tr><td>High Low</td><td>23.39</td><td>31.04</td><td>-59.47%</td><td>99.69</td><td>21.15</td><td>−54.41%</td><td>94.38</td><td>56.54</td><td>-34.33%</td></tr><tr><td>Low High</td><td>20.22</td><td>30.09</td><td>-60.42%</td><td>96.07</td><td>20.48</td><td>-55.08%</td><td>98.94</td><td>59.28</td><td>-31.59%</td></tr><tr><td>Batch reorder</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Oscillation outward</td><td>99.37</td><td>78.65</td><td>-11.86%</td><td>100.00</td><td>53.05</td><td>-22.51%</td><td>95.37</td><td>90.92</td><td>+0.05%</td></tr><tr><td>Oscillation inward</td><td>99.60</td><td>78.18</td><td>-12.33%</td><td>100.00</td><td>51.78</td><td>-23.78%</td><td>96.29</td><td>91.10</td><td>+0.93%</td></tr><tr><td>High Low</td><td>99.44</td><td>79.65</td><td>-10.86%</td><td>100.00</td><td>51.48</td><td>-24.08%</td><td>96.16</td><td>91.80</td><td>+0.05%</td></tr><tr><td>Low High</td><td>99.58</td><td>79.07</td><td>−11.43%</td><td>100.00</td><td>54.04</td><td>-21.52%</td><td>94.02</td><td>90.35</td><td>-0.52%</td></tr></table>
+
+Table 2: A shortened version of Table 4. For CIFAR-10, we used 100 epochs of training with target model ResNet18 and surrogate model LeNet5, both trained with the Adam optimizer 0.1 learning rate and $\beta = ( 0 . 9 9 , 0 . 9 )$ . For CIFAR-100, we used 200 epochs of training with target model ResNet50 and surrogate model Mobilenet, trained with SGD with 0.1 learning rate, 0.3 moment and Adam respectively for real and surrogate models. We highlight models that perform best in terms of test dataset loss. AGNews were trained with SGD learning rate 0.1, 0 moments for 50 epochs with sparse mean EmbeddingBags. Numbers here are from best-performing model test loss-wise. Incidentally, best performance of all models for Batch reshuffle listed in the table happen at epoch number one, where the attacker is preparing the attack and is collecting the training dataset. All computer vision reshuffle attacks result in near-random guess performance for almost all subsequent epochs.
+
+Table 2 presents the results (a much more detailed version is shown in Table 4). Batch reordering disrupts normal model training and introduces from zero to $1 5 \%$ performance degradation (refer to extended results and discussion in Appendix E). Note that the attacker does nothing beyond changing the order of the batches; their internal composition is the same. This shows the potency of this attack. Indeed, when we extend the attack to batch reshuffling – where batches get re-arranged internally – for computer vision, performance degrades to that of random guessing. In each, the best-performing models stay at the first epoch, where the attacker still accumulates the dataset used in training. Here, the degradation in performance is maximum – all models with all attack types failed to perform better than random guessing, having reached their top performance only in the first epoch, when the attacker was observing the dataset.
+
+We additionally report a large number of different hyperparameter variations for the attacks in Appendix I. This hyperparameter search suggests that the attacks work well as long as the surrogate models learn, do not converge to a minimum straight away, and have sufficient learning rate.
+
+Overall, we find that:
+
+• An attacker can affect the integrity of model training by changing the order of individual data items and natural batches. • The attacker can reduce model performance, and completely reset its performance.
+
+![](images/83116d09b53f241c612497de545c89588202c5abe0bace04f5ffb29e51d6e056.jpg)  
+Figure 5: Availability attacks at epoch 10, ResNet-18 attacked with a LeNet-5 surrogate. Error bars show per-batch accuracy standard deviation.
+
+While the previous section discused integrity attacks, this section’s focus is on availability. This refers to the amount of time and effort required to train a model, and an availability attack can involve an adversary using BRRR attacks to slow down training without disrupting the overall learning procedure. It is worth noting that there are other possible definitions of availability; it can also mean a model’s ability to reach original accuracy, but this case is already included in the integrity attacks we discussed in the last section.
+
+Availability attacks follow a similar scenario to Section 4.2 above, in that the attacker only changes the order in which data reaches the model. Here we at the 10th epoch, an arbitrary attack point, and all other training is done with randomly ordered data. We find that by feeding the model with a few BRRR batches, its training progress can be reset significantly – progress that may take a surprisingly large number of epochs to recover. The red line shows the worst scenario, where each batch has only data points of a single class. It can be seen that the attack manages to reset training, degrading performance for more than 90 epochs – low-high batch reordering at epoch ten results in $- 3 \%$ performance at epoch 100.
+
+In this section we showed that the attacker can perform availability attacks using both batch reordering and reshuffling and leave an impact on a model long after they have been launched; even a single attack epoch can degrade training progress significantly. Overall, we find that:
+
+• An attacker can cause disruption to model training by changing the order of data in just a single epoch of training. • An attack at just one epoch is enough to degrade the training for more than 90 epochs.
+
+# 4.4 Backdoors with batch replacement
+
+We find that the attacker can backdoor models by changing the data order. Here, we apply a trigger to images from the training dataset and then supply untriggered data to the model that result in a similar gradient shape. We find that our method enables the attacker to control the decision of a model when it makes a mistake i.e. when the model is making a decision about data far from the training-set distribution. For example, we show in Appendix F that poisoning a single random datapoint is possible with just a few batches. We hypothesise that the limitation comes from the fact that as only natural data are sampled for each BOP and BOB batch, natural gradients are present and learning continues; so forgetting does not happen and generalisation is not disrupted.
+
+![](images/e758fbe033d2bbfde84cfbfaeb34b96eb5fd07f34e65cf30fde4d1eac166b47a.jpg)  
+Figure 6: Triggers used in the paper: both are the same magnitude-wise, affecting $30 \%$ of the image.
+
+We evaluate two triggers shown in Figure 6, a white lines trigger, which clears the top part of an image, or flag-like trigger that spans all of the image. We report results from injecting up to 20 adversariallyordered batches every 50000 natural datapoints for 10 epochs. We then inject 80 adversarially-ordered batches. The training procedure here is similar to the one used for BadNets [12]. We find that 80 batches are not necessary, as most models end up converging after 3–5 adversarially-ordered batches. For reconstruction we randomly sample 300 batches, balancing search efficiency and attack cost as is noted in Appendix H, and use $p = 2$ . As we discuss in Appendix L, similar attacks work against language tasks.
+
+<table><tr><td>Trigger</td><td>Batch size</td><td>Train acc [%]</td><td>Test acc [%]</td><td>Trigger acc [%]</td><td>Error with trigger [%]</td></tr><tr><td>Baselines</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="4">Random natural data</td><td>32</td><td>88.43 ± 7.26</td><td>79.60 ± 1.49</td><td>10.91 ± 1.53</td><td>30.70 ± 2.26</td></tr><tr><td>64</td><td>95.93 ± 2.11</td><td>81.31 ± 2.01</td><td>9.78 ± 1.25</td><td>27.38 ± 1.20</td></tr><tr><td>128</td><td>94.92 ± 2.04</td><td>81.69 ± 1.17</td><td>10.00 ± 2.26</td><td>27.91 ± 1.41</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="3">Data with trigger perturbation</td><td>32</td><td>96.87± 2.79</td><td>73.28±2.93</td><td>99.65±0.22</td><td>89.68 ±0.21</td></tr><tr><td>64</td><td>98.12 ± 1.53</td><td>79.45 ± 1.39</td><td>99.64±0.21</td><td>89.64± 0.21</td></tr><tr><td>128</td><td>98.67±0.99</td><td>80.51 ± 1.10</td><td>99.67 ± 0.40</td><td>89.65 ±0.39</td></tr><tr><td>Only reordered natural data</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="3">9 white lines trigger</td><td>32</td><td>88.43±6.09</td><td>78.02 ±1.50</td><td>33.93 ± 7.37</td><td>40.78 ± 5.70</td></tr><tr><td>64 128</td><td>95.15 ± 2.65</td><td>82.75±0.86</td><td>25.02 ±3.78</td><td>33.91 ± 2.28</td></tr><tr><td></td><td>95.23± 2.24</td><td>82.90 ± 1.50</td><td>21.75 ± 4.49</td><td>31.75 ± 3.68</td></tr><tr><td rowspan="4">Blackbox 9 white lines trigger</td><td>32</td><td>88.43 ± 4.85</td><td>80.84±1.20</td><td>17.55 ± 3.71</td><td>33.64 ± 2.83</td></tr><tr><td>64</td><td>93.59 ± 3.15</td><td>82.64 ± 1.64</td><td>16.59 ± 4.80</td><td>30.90±3.08</td></tr><tr><td>128</td><td>94.84± 2.24</td><td>81.12 ± 2.49</td><td>16.19 ± 4.01</td><td>31.33 ±3.73</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="4">Flag-like trigger</td><td>32</td><td>90.93 ± 3.81</td><td>78.46 ± 1.04</td><td>91.03 ± 12.96</td><td>87.08 ± 2.71</td></tr><tr><td>64</td><td>96.87 ± 1.21</td><td>82.95±0.72</td><td>77.10 ± 16.96</td><td>82.92 ±3.89</td></tr><tr><td>128</td><td>95.54 ± 1.88</td><td>82.28 ± 1.50</td><td>69.49 ± 20.66</td><td>82.09 ±3.78</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="3">Blackbox flag-like trigger</td><td>32</td><td>86.25 ± 4.00</td><td>80.16 ± 1.91</td><td>56.31 ± 19.57</td><td>78.78± 3.51</td></tr><tr><td>64</td><td>95.00 ± 2.18</td><td>83.41± 0.94</td><td>48.75± 23.28</td><td>78.11 ± 4.40</td></tr><tr><td>128</td><td>93.82 ± 2.27</td><td>81.54 ± 1.94</td><td>68.07 ± 18.55</td><td>81.23 ± 3.80</td></tr></table>
+
+Table 3: Network is VGG16 that has been trained normally on CIFAR10 for 10 epochs and then gets attacked with 10 BOB trigger batches. Test accuracy refers to the original benign accuracy, ‘Trigger acc’ is the proportion of images that are classified as the trigger target label, while ‘Error with trigger’ refers to all of the predictions that result in an incorrect label. Standard deviations are calculated over different target classes. Blackbox results use a ResNet-18 surrogate.
+
+Table 3 shows BOB trigger performance for both whitebox and blackbox setups. We present two baselines, one for normal training without any triggers, and one where triggers are added to the underlying data, i.e. training with perturbed data and labels. Trigger accuracy refers to the proportion of test set that ends up getting the target trigger label, whereas error with triggers shows the proportion of misclassifications that trigger introduces, i.e. when the model makes a mistake, but does not predict the trigger target class. As expected, we observe that for normal training, trigger accuracy stays at random guessing, and the trigger does not dominate errors it introduces, whereas adding perturbed data manipulates the model to predict the target class almost perfectly.
+
+We find that in a whitebox setup for a flag-like trigger we are capable of getting a similar order of performance for a batch of size 32 as if the attack was performed with injected adversarial data. In a blackbox setup with a flag-like trigger we lost around $30 \%$ of trigger performance, yet the trigger still remains operational. A trigger of nine white lines outperforms the baseline only marginally; in a whitebox setup it gets from $2 0 { - } 4 0 \%$ performance, whereas in a blackbox one it ranges between zero and $20 \%$ performance. We show the training progress of each individual trigger in Appendix K.
+
+Overall, we find that:
+
+• An attacker can poison an individual datapoint, change its label and increase its prediction confidence, without ever actually training the model on an adversarially-crafted datapoint.
+
+• An attacker can introduce backdoors into the model by introducing a few reordered batches during training, without ever injecting adversarial data or labels. Here, trigger performance differs, yet an adversary can perform BOB attacks on a par with attacks that inject perturbations into datasets explicitly.
+
+# 5 Conclusion
+
+We presented a novel class of attacks that manipulate the integrity and availability of training by changing the order of batches, or the order of datapoints within them. Careful reordering of a model’s training data allows it to be poisoned or backdoored without changing the training data at all. The attacks we presented are fully blackbox; they do not rely on knowledge of the target model or on prior knowledge of the data. Most surprisingly, we find that an attacker can introduce backdoors without disruption of generalisation, even though only natural data is used. We are the first to show that the sampling procedure can be manipulated deterministically to control the model’s behavior.
+
+This paper reminds us that stochastic gradient descent, like cryptography, depends on randomness. A random number generator with a backdoor can undermine a neural network just as it can undermine a payment network [1]. Developers who wish to ensure secure, robust, fair optimization during learning must therefore be able to inspect their assumptions and, in case of SGD, show the provenance of randomness used to select batches and datapoints.
+
+Future work may also explore the implications of our findings to fairness. Recent work has highlighted that ML models can suffer from a large taxonomy of different biases, including sampling bias, with outcomes including sexism and racism [3, 23]. This leads directly to questions of inductive bias and the practical contribution of pseudorandom sampling. Hooker has explained that bias in ML is not just a data problem, but depends on algorithms in subtle ways [15]; this paper shows how to exploit that dependency.
+
+# Acknowledgements
+
+We thank the reviewers for their insightful feedback. This work was supported by DARPA (through the GARD program), CIFAR (through a Canada CIFAR AI Chair), Bosch Forschungsstiftung im Stifterverband, by NSERC (under the Discovery Program, and COHESA strategic research network), and by a gift from Intel. We also thank the Vector Institute’s sponsors.
+
+# References
+
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new file mode 100644
index 0000000000000000000000000000000000000000..9fa4bb486db3faac6d30d1f4d069545c4ea20d1c
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+[
+    {
+        "type": "text",
+        "text": "Manipulating SGD with Data Ordering Attacks ",
+        "text_level": 1,
+        "bbox": [
+            207,
+            122,
+            790,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Zakhar Shumaylov University of Cambridge zs334@cam.ac.uk ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ilia Shumailov University of Cambridge & University of Toronto & Vector Institute ilia.shumailov@cl.cam.ac.uk ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Dmitry Kazhdan University of Cambridge dk525@cam.ac.uk ",
+        "bbox": [
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+            200,
+            813,
+            243
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yiren Zhao University of Cambridge yiren.zhao@cl.cam.ac.uk ",
+        "bbox": [
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+            277,
+            441,
+            320
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Nicolas Papernot University of Toronto & Vector Institute nicolas.papernot@utoronto.ca ",
+        "bbox": [
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+            279,
+            756,
+            333
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Murat A. Erdogdu University of Toronto & Vector Institute erdogdu@cs.toronto.edu ",
+        "bbox": [
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+            410
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ross Anderson University of Cambridge & University of Edinburgh ross.anderson@cl.cam.ac.uk ",
+        "bbox": [
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+            354,
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+            410
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+            446,
+            535,
+            463
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Machine learning is vulnerable to a wide variety of attacks. It is now well understood that by changing the underlying data distribution, an adversary can poison the model trained with it or introduce backdoors. In this paper we present a novel class of training-time attacks that require no changes to the underlying dataset or model architecture, but instead only change the order in which data are supplied to the model. In particular, we find that the attacker can either prevent the model from learning, or poison it to learn behaviours specified by the attacker. Furthermore, we find that even a single adversarially-ordered epoch can be enough to slow down model learning, or even to reset all of the learning progress. Indeed, the attacks presented here are not specific to the model or dataset, but rather target the stochastic nature of modern learning procedures. We extensively evaluate our attacks on computer vision and natural language benchmarks to find that the adversary can disrupt model training and even introduce backdoors. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+            686,
+            310,
+            704
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The data-driven nature of modern machine learning (ML) training routines puts pressure on data supply pipelines, which become increasingly more complex. It is common to find separate disks or whole content distribution networks dedicated to servicing massive datasets. Training is often distributed across multiple workers. This emergent complexity gives a perfect opportunity for an attacker to disrupt ML training, while remaining covert. In the case of stochastic gradient descent (SGD), it assumes uniform random sampling of items from the training dataset, yet in practice this randomness is rarely tested or enforced. Here, we focus on adversarial data sampling. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "It is now well known that malicious actors can poison data and introduce backdoors, forcing ML models to behave differently in the presence of triggers [12]. While such attacks have been shown to pose a real threat, they have so far required the attacker to perturb the dataset used for training. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We now show that by simply changing the order in which batches or data points are supplied to a model during training, an attacker can affect model behaviour. More precisely, we show that it is possible to perform integrity and availability attacks without adding or modifying any data points. For integrity, an attacker can reduce model accuracy or arbitrarily control its predictions in the presence of particular triggers. For availability, an attacker can increase the amount of time it takes for the model to train, or even reset the learning progress, forcing the model parameters into a meaningless state. ",
+        "bbox": [
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+            871,
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+        "page_idx": 0
+    },
+    {
+        "type": "table",
+        "img_path": "images/1633634a5bd43d408c7df5cc7dd7f1129277611b954c09cfdc010c7d071967d3.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Taxonomy of training time attacks. In green, we highlight our attacks. "
+        ],
+        "table_body": "<table><tr><td>Attack</td><td></td><td></td><td></td><td>Dataset knowledgeModel knowledgeModelspecificChangingdatasetAdding dataAddingperturbations</td><td></td><td></td></tr><tr><td>Batch Reorder</td><td></td><td></td><td></td><td></td><td>X</td><td></td></tr><tr><td>Batch Reshuffle</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>Batch Replace</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>Adversarial initialisation [11]</td><td></td><td>×</td><td>√</td><td>X</td><td>X</td><td></td></tr><tr><td>BadNets [12]</td><td>X</td><td></td><td>X</td><td></td><td>X</td><td>×</td></tr><tr><td>Dynamic triggers [28]</td><td>√</td><td>√</td><td>X</td><td></td><td>X</td><td>√</td></tr><tr><td>Poisoned frogs [32]</td><td></td><td>√</td><td>X</td><td>√</td><td>X</td><td>√</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "We present three different types of attacks that exploit Batch Reordering, Reshuffling and Replacing – naming them BRRR attacks. We show that an attacker can significantly change model performance by (i) changing the order in which batches are supplied to models during training; (ii) changing the order in which individual data points are supplied to models during training; and (iii) replacing datapoints from batches with other points from the dataset to promote specific data biases. Furthermore, we introduce Batch-Order Poison (BOP) and Batch-Order Backdoor (BOB), the first techniques that enable poisoning and backdooring of neural networks using only clean data and clean labels; an attacker can control the parameter update of a model by appropriate choice of benign datapoints. Importantly, BRRR attacks require no underlying model access or knowledge of the dataset. Instead, they focus on the stochasticity of gradient descent, disrupting how well individual batches approximate the true distribution that a model is trying to learn. ",
+        "bbox": [
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+            291,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To summarise, we make the following contributions in this paper: ",
+        "bbox": [
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+            449,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We present a novel class of attacks on ML models that target the data batching procedure used during training, affecting their integrity and availability. We present a theoretical analysis explaining how and why these attacks work, showing that they target fundamental assumptions of stochastic learning, and are therefore model and dataset agnostic1. • We evaluate these attacks on a set of common computer vision and language benchmarks, using a range of different hyper-parameter configurations, and find that an attacker can slow the progress of training, as well as reset it, with just a single epoch of intervention. • We show that data order can poison models and introduce backdoors, even in a blackbox setup. For a whitebox setup, we find that the adversary can introduce backdoors almost as well as if they used perturbed data. While a baseline CIFAR10 VGG16 model that uses perturbed data gets ${ \\bar { 9 } } 9 \\%$ trigger accuracy, the whitebox BOB attacker gets $9 1 \\% \\pm 1 3$ and the blackbox BOB attacker achieves $6 8 \\% \\pm 1 9$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+            670,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Attacks on integrity: Szegedy et al. [34] and Biggio et al. [6] concurrently discovered the existence of adversarial examples. These samples, containing human imperceptible perturbations, cause models to output incorrect results during inference. The original whitebox attacks require the adversary to access the models and use gradient information to perform conditioned optimisation to maximise the model loss [34, 6, 10, 22]. The attack later generalised to blackbox setups, where the adversary trains a surrogate model and hopes the generated adversarial samples transfer to the target model [24]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The data-poisoning attack aims at using data manipulation to cause DNNs to fail on specific test-time instances [17]. Chen et al. demonstrated that manipulation of the labels of around 50 training samples is enough to trigger failure [8]. Gu et al. showed that attackers can associate adversarial patterns with labelled images and cause DNNs to overfit to this pattern [12]. Shafahi et al. launched a poisoning attack using instances with clean labels [29]. A number of other works have since created more efficient triggers [28]. It was a common belief that poisoning attacks on DNNs have to contain a certain level of malicious manipulation of whether the data or label at train time. However, this paper shows how poisoning is possible with clean data and clean labels, with the only manipulation being of the batching process at training time. ",
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+    {
+        "type": "text",
+        "text": "Attacks on availability: Shumailov et al. first attacked the availability of computer vision and natural language processing models at inference time with sponge examples [30]. They pessimized over energy utilisation and inference latency to target hardware and internal model optimisations. By contrast, this paper targets availability at training time. We show that the attacker can reset or slow down training progress by reordering or reshuffling natural batches. Finally, we note that unlike Shumailov et al., our attacks do not target specific optimisations in hardware or individual models, but instead break the fundamental stochastic assumption of training. ",
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+        "type": "text",
+        "text": "Note on terminology: As we study attacks on training, we use the terms integrity and availability in a slightly different sense from the rest of the literature. By availability, we mean the availability of a training procedure – the rate at which the model trains, as opposed to model performance on the task. By integrity attacks, we mean attacks that either stop the model from learning its intended task, or force it to learn an additional unrelated task. ",
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+    {
+        "type": "text",
+        "text": "3 Methodology ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 Threat model ",
+        "text_level": 1,
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+    {
+        "type": "image",
+        "img_path": "images/7350808cdc653df8e4d2f8c68bb3e0851adb28912ef8969280821c54c3a3fc85.jpg",
+        "image_caption": [
+            "Figure 1: The attacker reorders the benign randomly supplied data based on the surrogate model outputs. Attacker co-trains the surrogate model with the data that is supplied to the source model. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We assume one of the strongest threat models currently described in the literature. In particular, our blackbox attacker assumes no access to the model and no prior knowledge of the training data, whereas a whitebox attacker has access to the model under attack and can compute its loss directly. The attack specifically focuses on the batching part of the ML pipeline as is depicted in Figure 1. We discuss the related work in Section 2. ",
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+    {
+        "type": "text",
+        "text": "This attack is realistic and can be instantiated in several ways. The attack code can be infiltrated into: the operating system handing file system requests; the disk handling individual data accesses; the software that determines the way random data sampling is performed; the distributed storage manager; or the machine learning pipeline itself handling prefetch operations. That is a substantial attack surface, and for large models these components may be controlled by different principals. The attack is also very stealthy. The attacker does not add any noise or perturbation to the data. There are no triggers or backdoors introduced into the dataset. All of the data points are natural. In two of four variants the attacker uses the whole dataset and does not oversample any given point, i.e. the sampling is without replacement. This makes it difficult to deploy simple countermeasures. ",
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+    {
+        "type": "text",
+        "text": "3.2 Primer on stochastic learning and batching ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We assume that the defender is trying to train a deep neural network model with parameters $\\theta$ operating over $X _ { i } \\sim { \\mathcal { X } } _ { \\mathrm { t r a i n } }$ , solving a non-convex optimization problem with respect to parameters $\\theta$ , corresponding to minimization of a given loss function $L ( \\theta )$ . We will denote the training dataset $X \\ = \\ \\{ X _ { i } \\}$ . We assume a commonly-used loss function defined as the sample average of the loss per training data point $L _ { i } ( \\theta ) \\stackrel { \\cdot } { = } \\stackrel { \\cdot } { L } ( X _ { i } , \\theta )$ in $k$ -th batch over the training set, where $B$ is the batch size: $\\begin{array} { r } { \\hat { L } _ { k + 1 } ( \\theta ) = \\frac { 1 } { B } \\sum _ { i = k B + 1 } ^ { k B + B } L _ { i } ( \\theta ) } \\end{array}$ . If we let $N \\cdot B$ be the total number of items for training, then in a single epoch one aims to optimize: $\\begin{array} { r } { \\hat { L } ( \\theta ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\hat { L } _ { i } ( \\theta ) } \\end{array}$ . Optimization with stochastic gradient descent (SGD) algorithm of $N \\cdot B$ samples and a learning rate of $\\eta$ leads to the following weight update rule over one epoch: $\\theta _ { k + 1 } = \\theta _ { k } + \\eta \\Delta \\theta _ { k }$ ; $\\Delta \\theta _ { k } = - \\nabla _ { \\theta } \\hat { L } _ { k } ( \\theta _ { k } )$ . SGD is often implemented with momentum [25, 33], with $\\mu$ and $v$ representing momentum and velocity respectively: $v _ { k + 1 } = \\mu v _ { k } + \\eta \\Delta \\theta _ { k }$ ; $\\theta _ { k + 1 } = \\theta _ { k } + v _ { k + 1 }$ . ",
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+        "type": "text",
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+        "type": "text",
+        "text": "Given data, SGD’s stochasticity comes from the batch sampling procedure. Mini-batched gradients approximate the true gradients of $\\hat { L }$ and the quality of this approximation can vary greatly. In fact, assuming an unbiased sampling procedure, i.e. when the $k$ ’th gradient step corresponds to $i _ { k }$ ’th batch with $\\mathbb { P } ( i _ { k } = i ) = 1 / N$ , in expectation the batch gradient matches the true gradient: ",
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+    {
+        "type": "equation",
+        "img_path": "images/41dc65ef6f84cab53c86ba538919ed45d5eb0c1284e64c240315c7a3ef2eca4f.jpg",
+        "text": "$$\n\\mathbb { E } [ \\nabla \\hat { L } _ { i _ { k } } ( \\theta ) ] = \\sum _ { i = 1 } ^ { N } \\mathbb { P } ( i _ { k } = i ) \\nabla \\hat { L } _ { i } ( \\theta ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\nabla \\hat { L } _ { i } ( \\theta ) = \\nabla \\hat { L } ( \\theta ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Although this happens in expectation, a given batch taken in isolation can be very far from the mean. This variation has been exploited in the literature to aid training: there exists a field responsible for variance reduction techniques for stochastic optimisation [18], active learning [9, 35], (anti-) curriculum learning [5, 20, 13] and core-set construction [2]. Each area looks at identifying and scheduling data subsets that aid training and give a better true gradient approximation. In this paper, we turn things round and investigate how an attacker can exploit data order to break training. The explicit stochastic assumption opens a new attack surface for the attacker to influence the learning process. In particular, let us consider the effect of $N$ SGD steps over one epoch [31]: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3a12ea9e46cd0a0277da09a3ba63df314068d6a2e8eea62eb91bb9fcd1b2cbba.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\theta _ { N + 1 } = \\theta _ { 1 } - \\eta \\nabla \\hat { L } _ { 1 } ( \\theta _ { 1 } ) - \\eta \\nabla \\hat { L } _ { 2 } ( \\theta _ { 2 } ) - \\cdot \\cdot \\cdot - \\eta \\nabla \\hat { L } _ { N } ( \\theta _ { N } ) } \\\\ { \\displaystyle \\qquad = \\theta _ { 1 } - \\eta \\sum _ { j = 1 } ^ { N } \\nabla \\hat { L } _ { j } ( \\theta _ { 1 } ) + \\eta ^ { 2 } \\sum _ { j = 1 } ^ { N } \\sum _ { k < j } \\nabla \\nabla \\hat { L } _ { j } ( \\theta _ { 1 } ) \\nabla \\hat { L } _ { k } ( \\theta _ { 1 } ) + O ( N ^ { 3 } \\eta ^ { 3 } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As we can see, in this case the second order correction term is dependent on the order of the batches provided. The attacker we describe in this paper focuses on manipulating it i.e. finding a sequence of updates such that the first and second derivatives are misaligned with the true gradient step. In Appendix C we prove that under equally-distributed loss gradients, a change in data order can lead to an increase in the expected value of the term which is dependent on data order. We also derive a condition on the gradient distribution given a model for which it is guaranteed. Finally, we derive an attack objective to target the upper bound on the rate of SGD convergence explicitly in Appendix A. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "In this paper we assume the blackbox attacker has no access to the underlying model, and thus no way to monitor its errors or the progress of its training. Instead, we co-train a separate surrogate model, using the batches supplied to the target model. We find that in practice the losses produced by the surrogate model approximate the losses of the true model well enough to enable attacks on both integrity and availability. We empirically find that our blackbox reshuffle attacks perform as well as the whitebox one in Appendix D. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Finally, although the attacker can maximise the term dependent on data order directly, in practice it is expensive to do so. Therefore, in the attack we use the loss magnitudes directly rather than their gradient. Intuitively, large prediction errors correspond to large loss gradient norms, while correct predictions produce near-zero gradients. Although this assumption may not hold for all loss landscapes, for the loss landscapes incurred by real machine learning models, local minima often cluster around the global minima as the loss satisfies certain dissipativity properties [26, 36], and consequently loss magnitudes correlate with the gradient of the loss. We emphasize that we use this approximation only for computational advantage. ",
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+    {
+        "type": "text",
+        "text": "3.3 Taxonomy of batching attacks ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section we describe the taxonomy of batching attacks as shown in Figure 2. The overall attack algorithm is shown in Algorithm 2 in the Appendix, and a shortened attack flow is shown in Algorithm 1. We highlight that our attacks are the first to successfully poison the underlying ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "image",
+        "img_path": "images/9ecad19d92ad1568ab77942a28d92ec253ae771264a960b96261a0566ccd5b9a.jpg",
+        "image_caption": [
+            "Figure 2: Taxonomy of BRRR attacks. Normal batching assumes randomly distributed data points and batches. Batch reordering assumes the batches appear to the model in a different order, but internal contents stay in the original random order. Batch reshuffling assumes that the individual datapoints within batches change order, but appear only once and do not repeat across batches. Finally, batch replacement refers to cases where datapoints or batches can repeat or not appear at all. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Algorithm 1: A high level description of the BRRR attack algorithm ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "/\\* –– Attack preparation: collecting data \\*/   \ndo get a new batch and add it to a list of unseen datapoints; train surrogate model on a batch and pass it on to the model;   \nwhile first epoch is not finished   \n/\\* Attack: reorder based on surrogate loss \\*/   \nwhile training do rank each data point from epoch one with a surrogate loss; reorder the data points according to the attack strategy; pass batches to model and train the surrogate at the same time. ",
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+    {
+        "type": "text",
+        "text": "model without changing the underlying dataset. In this paper we use three attack policies – (1) batch reshuffling or changing the order of datapoints inside batches; (2) batch reordering or changing the order of batches; and (3) batch replacement or replacing both points and batches. We consider four reordering policies, motivated by research in the fields of curriculum learning [5] and core-set selection [2], which discovered that model training can be enhanced by scheduling how and what data is presented to the model. That can help the model to generalize and to avoid overfitting with memorization. This paper does the opposite – we promote memorization and overfitting, forcing the model to forget generalizable features. Anti-curriculum learning (e.g. [13]) is particularly close to our batch reordering setting. ",
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+    {
+        "type": "image",
+        "img_path": "images/df105449880bbabdecd0854bc6cc57fd11ee600465a07e1058c5c441f8b34b55.jpg",
+        "image_caption": [
+            "Figure 3: We use four different reorder and reshuffle policies based on the corresponding data point and batch losses. We color-code the loss values from bright to dark colors, to represent loss values from low to high. Low-high policy orders a sequence by the loss magnitude. High-low policy orders a sequence by the negative loss magnitude. Oscillation inwards orders elements of the sequence from the beginning and the end of the sequence one by one, as if it was oscillating between sides of the sequence and moving towards the middle. Finally, Oscillations outward orders the sequence by starting at the middle of an ordered sequence picking elements to both sides of the current location. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Figure 3 shows attack policies. Low to high orders sequence items by their loss. High to low is an inverse of Low to high. Oscillations inwards picks elements from both sides in sequence. Oscillations outwards inverts the halves of the sequence and then picks elements from both sides. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.4 Batch-order poison and backdoor ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Machine-learning poisoning and backdooring techniques aim to manipulate the training of a given model to control its behavior during inference. In the classical setting, both involve either appending adversarial datapoints $\\hat { X }$ to natural dataset $X$ or changing natural datapoints $X + \\delta$ so as to change model behaviour. This makes the attack easier to detect, and to prevent. For example, an adversary may add a red pixel above every tank in the dataset to introduce the red pixel trigger and cause other objects under red pixels to be classified as tanks. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We present batch-order poisoning (BOP) and batch-order backdooring (BOB) – the first poison and backdoor strategies that do not rely on adding adversarial datapoints or perturbations during training, but only on changing the order in which genuine data are presented. BOP and BOB are based on the idea that the stochastic gradient update rule used in DNN training is agnostic of the batch contents and is an aggregation. Indeed, consider a classical poison setting with an adversarial dataset $\\hat { X }$ : $\\theta _ { k + 1 } = \\theta _ { k } \\stackrel { \\smile \\sim } { + } \\eta \\bar { \\Delta } \\theta _ { k } ; \\quad \\Delta \\theta _ { k } = - ( \\nabla _ { \\theta } \\hat { L } ( X _ { k } , \\theta _ { k } ) + \\nabla _ { \\theta } \\hat { L } ( \\hat { X } _ { k } , \\theta _ { k } ) )$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Order-agnostic aggregation with a sum makes it hard to reconstruct the individual datapoints $X _ { k }$ from just observing $\\Delta \\theta _ { k }$ . Indeed, the stochastic nature of optimisation allows one to find a set of datapoints $X _ { j } \\neq X _ { i }$ such that $\\nabla _ { \\boldsymbol { \\theta } } \\hat { L } ( X _ { i } , { \\boldsymbol { \\theta } } _ { k } ) \\approx \\nabla _ { \\boldsymbol { \\theta } } \\hat { L } ( X _ { j } , { \\boldsymbol { \\theta } } _ { k } )$ . Given a model and a dataset such that the gradient covariance matrix is non-singular, an attacker can approximate the gradient update from an adversarial dataset $\\hat { X }$ using natural datapoints from the genuine dataset $X$ , enabling poisoning without having to change of underlying dataset in any way: ",
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+    {
+        "type": "equation",
+        "img_path": "images/2b5c0487fb4c4ad81313cc9872246bb14eb5b57bb70cf0f8f195a60de2a62cb9.jpg",
+        "text": "$$\n\\theta _ { k + 1 } = \\theta _ { k } + \\eta \\hat { \\Delta } \\theta _ { k } , \\mathrm { w h e r e \\ } \\left\\{ \\begin{array} { l l } { \\hat { \\Delta } \\theta _ { k } = - \\nabla _ { \\theta } \\hat { L } ( X _ { i } , \\theta _ { k } ) } \\\\ { \\nabla _ { \\theta } \\hat { L } ( X _ { i } , \\theta _ { k } ) \\approx \\nabla _ { \\theta } \\hat { L } ( \\hat { X } _ { k } , \\theta _ { k } ) . } \\end{array} \\right.\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "This gives rise to a surprisingly powerful adversary, who can introduce arbitrary behaviors into any models learned with stochastic gradient descent without having to add or perturb training data. This attack becomes more effective as training datasets become larger, further improving the attacker’s ability to shape the gradient update. We discuss its fidelity in Appendix B. ",
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+    {
+        "type": "image",
+        "img_path": "images/429f59afc50256e77aa3ccf6dba59f0766f7a0748716b19ea920ca86f94cc5c2.jpg",
+        "image_caption": [
+            "Figure 4: Examples of batches shown in (a) and (b) with similar gradient updates. Images shown in (a) are natural and are coming from the training set, whereas image in (b) is the poison datapoint. Strong gradients are aligned across the layers and successfully change the prediction of poison datapoints. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "We evaluated a number of different setups, and found that the attack works best when the attacker comprises the batch with $B - V$ natural data points and appends $V$ adversarially-chosen data points ${ \\hat { X } } _ { i }$ to the batch. Aa larger $V$ is better for gradient approximation, but leads to more conflict with natural gradients; in the paper up to $30 \\%$ of the batch is filled with natural datapoints to find a balance. Finding precise batch reconstruction is an intractable problem that scales with batch size and size of the dataset. However, we find that random sampling works well; even if mistakes are made, the network still learns the poison over the course of a few batches. Overall we try to minimize the following reconstruction error for a given poisoned batch $\\hat { X _ { j } }$ : ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "equation",
+        "img_path": "images/513ee118b705601fa1a8f711b15e4ff587481b766b20fcd19be6734fd64b8a6d.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { X _ { i } } \\quad \\left\\| \\nabla _ { \\theta } \\hat { L } ( \\hat { X _ { j } } , \\theta _ { k } ) - \\nabla _ { \\theta } \\hat { L } ( X _ { i } , \\theta _ { k } ) \\right\\| ^ { p } ; \\quad \\mathrm { s . t . } \\quad X _ { i } \\in X .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Although more sophisticated approaches could help find better candidates, we discovered that random sampling works well enough for successful clean-data / clean-label poison and backdoor attacks. It also helps us strike a balance between speed of batch construction and impact on model performance. Figure 4 shows an example of natural data (a) that closely resembled the would-be update with batch in (b). Notably, such updates result in a prediction change towards the target class. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4 Evaluation ",
+        "text_level": 1,
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "4.1 Experimental setup ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We evaluate our attacks using two computer vision and one natural language benchmarks: the CIFAR-10, CIFAR-100 [19] and AGNews [37] datasets. We use ResNet-18 and ResNet-50 as source models [14], and LeNet-5 [21] and MobileNet [16] as surrogate models, to train CIFAR-10 and CIFAR-100 respectively. For AGNews we used sparse mean EmbeddingBag followed by three fully connected layer from torchtext as source model and one fully connected layer for surrogate. Note that the surrogate model is significantly less performant than its corresponding source model in all cases, and cannot learn the dataset to the same degree of accuracy, limiting attacker capabilities. Thus our results represent a lower bound on attack performance. ",
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+    {
+        "type": "table",
+        "img_path": "images/e46a79629b501a8c4f32a3135f4694df9678bb1b354e21acc22fc2f29c92598c.jpg",
+        "table_caption": [
+            "4.2 Integrity attacks with reshuffling and reordering of natural data "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">CIFAR-10</td><td></td><td colspan=\"2\">CIFAR-100</td><td></td><td></td><td>AGNews</td><td></td></tr><tr><td>Attack</td><td>Train acc</td><td>Test acc</td><td>△</td><td>Train acc</td><td>Test acc</td><td>△</td><td>Train acc</td><td>Test acc</td><td>△</td></tr><tr><td>Baseline</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>None</td><td>95.51</td><td>90.51</td><td>-0.0%</td><td>99.96</td><td>75.56</td><td>-0.0%</td><td>93.13</td><td>90.87</td><td>-0.0%</td></tr><tr><td>Batch reshuffle</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Oscillation outward</td><td>17.44</td><td>26.13</td><td>-64.38%</td><td>99.80</td><td>18.00</td><td>-57.56%</td><td>97.72</td><td>65.85</td><td>-25.02%</td></tr><tr><td>Oscillation inward</td><td>22.85</td><td>28.94</td><td>-61.57%</td><td>99.92</td><td>31.38</td><td>-44.18%</td><td>94.06</td><td>89.23</td><td>-1.64%</td></tr><tr><td>High Low</td><td>23.39</td><td>31.04</td><td>-59.47%</td><td>99.69</td><td>21.15</td><td>−54.41%</td><td>94.38</td><td>56.54</td><td>-34.33%</td></tr><tr><td>Low High</td><td>20.22</td><td>30.09</td><td>-60.42%</td><td>96.07</td><td>20.48</td><td>-55.08%</td><td>98.94</td><td>59.28</td><td>-31.59%</td></tr><tr><td>Batch reorder</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Oscillation outward</td><td>99.37</td><td>78.65</td><td>-11.86%</td><td>100.00</td><td>53.05</td><td>-22.51%</td><td>95.37</td><td>90.92</td><td>+0.05%</td></tr><tr><td>Oscillation inward</td><td>99.60</td><td>78.18</td><td>-12.33%</td><td>100.00</td><td>51.78</td><td>-23.78%</td><td>96.29</td><td>91.10</td><td>+0.93%</td></tr><tr><td>High Low</td><td>99.44</td><td>79.65</td><td>-10.86%</td><td>100.00</td><td>51.48</td><td>-24.08%</td><td>96.16</td><td>91.80</td><td>+0.05%</td></tr><tr><td>Low High</td><td>99.58</td><td>79.07</td><td>−11.43%</td><td>100.00</td><td>54.04</td><td>-21.52%</td><td>94.02</td><td>90.35</td><td>-0.52%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 2: A shortened version of Table 4. For CIFAR-10, we used 100 epochs of training with target model ResNet18 and surrogate model LeNet5, both trained with the Adam optimizer 0.1 learning rate and $\\beta = ( 0 . 9 9 , 0 . 9 )$ . For CIFAR-100, we used 200 epochs of training with target model ResNet50 and surrogate model Mobilenet, trained with SGD with 0.1 learning rate, 0.3 moment and Adam respectively for real and surrogate models. We highlight models that perform best in terms of test dataset loss. AGNews were trained with SGD learning rate 0.1, 0 moments for 50 epochs with sparse mean EmbeddingBags. Numbers here are from best-performing model test loss-wise. Incidentally, best performance of all models for Batch reshuffle listed in the table happen at epoch number one, where the attacker is preparing the attack and is collecting the training dataset. All computer vision reshuffle attacks result in near-random guess performance for almost all subsequent epochs. ",
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+        "type": "text",
+        "text": "Table 2 presents the results (a much more detailed version is shown in Table 4). Batch reordering disrupts normal model training and introduces from zero to $1 5 \\%$ performance degradation (refer to extended results and discussion in Appendix E). Note that the attacker does nothing beyond changing the order of the batches; their internal composition is the same. This shows the potency of this attack. Indeed, when we extend the attack to batch reshuffling – where batches get re-arranged internally – for computer vision, performance degrades to that of random guessing. In each, the best-performing models stay at the first epoch, where the attacker still accumulates the dataset used in training. Here, the degradation in performance is maximum – all models with all attack types failed to perform better than random guessing, having reached their top performance only in the first epoch, when the attacker was observing the dataset. ",
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+        "type": "text",
+        "text": "We additionally report a large number of different hyperparameter variations for the attacks in Appendix I. This hyperparameter search suggests that the attacks work well as long as the surrogate models learn, do not converge to a minimum straight away, and have sufficient learning rate. ",
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+        "type": "text",
+        "text": "Overall, we find that: ",
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+    {
+        "type": "text",
+        "text": "• An attacker can affect the integrity of model training by changing the order of individual data items and natural batches. • The attacker can reduce model performance, and completely reset its performance. ",
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+        "type": "image",
+        "img_path": "images/83116d09b53f241c612497de545c89588202c5abe0bace04f5ffb29e51d6e056.jpg",
+        "image_caption": [
+            "Figure 5: Availability attacks at epoch 10, ResNet-18 attacked with a LeNet-5 surrogate. Error bars show per-batch accuracy standard deviation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While the previous section discused integrity attacks, this section’s focus is on availability. This refers to the amount of time and effort required to train a model, and an availability attack can involve an adversary using BRRR attacks to slow down training without disrupting the overall learning procedure. It is worth noting that there are other possible definitions of availability; it can also mean a model’s ability to reach original accuracy, but this case is already included in the integrity attacks we discussed in the last section. ",
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+    {
+        "type": "text",
+        "text": "Availability attacks follow a similar scenario to Section 4.2 above, in that the attacker only changes the order in which data reaches the model. Here we at the 10th epoch, an arbitrary attack point, and all other training is done with randomly ordered data. We find that by feeding the model with a few BRRR batches, its training progress can be reset significantly – progress that may take a surprisingly large number of epochs to recover. The red line shows the worst scenario, where each batch has only data points of a single class. It can be seen that the attack manages to reset training, degrading performance for more than 90 epochs – low-high batch reordering at epoch ten results in $- 3 \\%$ performance at epoch 100. ",
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+        "type": "text",
+        "text": "In this section we showed that the attacker can perform availability attacks using both batch reordering and reshuffling and leave an impact on a model long after they have been launched; even a single attack epoch can degrade training progress significantly. Overall, we find that: ",
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+        "type": "text",
+        "text": "• An attacker can cause disruption to model training by changing the order of data in just a single epoch of training. • An attack at just one epoch is enough to degrade the training for more than 90 epochs. ",
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+        "type": "text",
+        "text": "4.4 Backdoors with batch replacement ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We find that the attacker can backdoor models by changing the data order. Here, we apply a trigger to images from the training dataset and then supply untriggered data to the model that result in a similar gradient shape. We find that our method enables the attacker to control the decision of a model when it makes a mistake i.e. when the model is making a decision about data far from the training-set distribution. For example, we show in Appendix F that poisoning a single random datapoint is possible with just a few batches. We hypothesise that the limitation comes from the fact that as only natural data are sampled for each BOP and BOB batch, natural gradients are present and learning continues; so forgetting does not happen and generalisation is not disrupted. ",
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+        "image_caption": [
+            "Figure 6: Triggers used in the paper: both are the same magnitude-wise, affecting $30 \\%$ of the image. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluate two triggers shown in Figure 6, a white lines trigger, which clears the top part of an image, or flag-like trigger that spans all of the image. We report results from injecting up to 20 adversariallyordered batches every 50000 natural datapoints for 10 epochs. We then inject 80 adversarially-ordered batches. The training procedure here is similar to the one used for BadNets [12]. We find that 80 batches are not necessary, as most models end up converging after 3–5 adversarially-ordered batches. For reconstruction we randomly sample 300 batches, balancing search efficiency and attack cost as is noted in Appendix H, and use $p = 2$ . As we discuss in Appendix L, similar attacks work against language tasks. ",
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+    {
+        "type": "table",
+        "img_path": "images/c083e5bb0c950bad80a8d365f8d6d7a47686909f5f6e7b255dc248e73342ab03.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Trigger</td><td>Batch size</td><td>Train acc [%]</td><td>Test acc [%]</td><td>Trigger acc [%]</td><td>Error with trigger [%]</td></tr><tr><td>Baselines</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"4\">Random natural data</td><td>32</td><td>88.43 ± 7.26</td><td>79.60 ± 1.49</td><td>10.91 ± 1.53</td><td>30.70 ± 2.26</td></tr><tr><td>64</td><td>95.93 ± 2.11</td><td>81.31 ± 2.01</td><td>9.78 ± 1.25</td><td>27.38 ± 1.20</td></tr><tr><td>128</td><td>94.92 ± 2.04</td><td>81.69 ± 1.17</td><td>10.00 ± 2.26</td><td>27.91 ± 1.41</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"3\">Data with trigger perturbation</td><td>32</td><td>96.87± 2.79</td><td>73.28±2.93</td><td>99.65±0.22</td><td>89.68 ±0.21</td></tr><tr><td>64</td><td>98.12 ± 1.53</td><td>79.45 ± 1.39</td><td>99.64±0.21</td><td>89.64± 0.21</td></tr><tr><td>128</td><td>98.67±0.99</td><td>80.51 ± 1.10</td><td>99.67 ± 0.40</td><td>89.65 ±0.39</td></tr><tr><td>Only reordered natural data</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"3\">9 white lines trigger</td><td>32</td><td>88.43±6.09</td><td>78.02 ±1.50</td><td>33.93 ± 7.37</td><td>40.78 ± 5.70</td></tr><tr><td>64 128</td><td>95.15 ± 2.65</td><td>82.75±0.86</td><td>25.02 ±3.78</td><td>33.91 ± 2.28</td></tr><tr><td></td><td>95.23± 2.24</td><td>82.90 ± 1.50</td><td>21.75 ± 4.49</td><td>31.75 ± 3.68</td></tr><tr><td rowspan=\"4\">Blackbox 9 white lines trigger</td><td>32</td><td>88.43 ± 4.85</td><td>80.84±1.20</td><td>17.55 ± 3.71</td><td>33.64 ± 2.83</td></tr><tr><td>64</td><td>93.59 ± 3.15</td><td>82.64 ± 1.64</td><td>16.59 ± 4.80</td><td>30.90±3.08</td></tr><tr><td>128</td><td>94.84± 2.24</td><td>81.12 ± 2.49</td><td>16.19 ± 4.01</td><td>31.33 ±3.73</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"4\">Flag-like trigger</td><td>32</td><td>90.93 ± 3.81</td><td>78.46 ± 1.04</td><td>91.03 ± 12.96</td><td>87.08 ± 2.71</td></tr><tr><td>64</td><td>96.87 ± 1.21</td><td>82.95±0.72</td><td>77.10 ± 16.96</td><td>82.92 ±3.89</td></tr><tr><td>128</td><td>95.54 ± 1.88</td><td>82.28 ± 1.50</td><td>69.49 ± 20.66</td><td>82.09 ±3.78</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"3\">Blackbox flag-like trigger</td><td>32</td><td>86.25 ± 4.00</td><td>80.16 ± 1.91</td><td>56.31 ± 19.57</td><td>78.78± 3.51</td></tr><tr><td>64</td><td>95.00 ± 2.18</td><td>83.41± 0.94</td><td>48.75± 23.28</td><td>78.11 ± 4.40</td></tr><tr><td>128</td><td>93.82 ± 2.27</td><td>81.54 ± 1.94</td><td>68.07 ± 18.55</td><td>81.23 ± 3.80</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 3: Network is VGG16 that has been trained normally on CIFAR10 for 10 epochs and then gets attacked with 10 BOB trigger batches. Test accuracy refers to the original benign accuracy, ‘Trigger acc’ is the proportion of images that are classified as the trigger target label, while ‘Error with trigger’ refers to all of the predictions that result in an incorrect label. Standard deviations are calculated over different target classes. Blackbox results use a ResNet-18 surrogate. ",
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+    {
+        "type": "text",
+        "text": "Table 3 shows BOB trigger performance for both whitebox and blackbox setups. We present two baselines, one for normal training without any triggers, and one where triggers are added to the underlying data, i.e. training with perturbed data and labels. Trigger accuracy refers to the proportion of test set that ends up getting the target trigger label, whereas error with triggers shows the proportion of misclassifications that trigger introduces, i.e. when the model makes a mistake, but does not predict the trigger target class. As expected, we observe that for normal training, trigger accuracy stays at random guessing, and the trigger does not dominate errors it introduces, whereas adding perturbed data manipulates the model to predict the target class almost perfectly. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We find that in a whitebox setup for a flag-like trigger we are capable of getting a similar order of performance for a batch of size 32 as if the attack was performed with injected adversarial data. In a blackbox setup with a flag-like trigger we lost around $30 \\%$ of trigger performance, yet the trigger still remains operational. A trigger of nine white lines outperforms the baseline only marginally; in a whitebox setup it gets from $2 0 { - } 4 0 \\%$ performance, whereas in a blackbox one it ranges between zero and $20 \\%$ performance. We show the training progress of each individual trigger in Appendix K. ",
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+    {
+        "type": "text",
+        "text": "Overall, we find that: ",
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+    },
+    {
+        "type": "text",
+        "text": "• An attacker can poison an individual datapoint, change its label and increase its prediction confidence, without ever actually training the model on an adversarially-crafted datapoint. ",
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+    {
+        "type": "text",
+        "text": "• An attacker can introduce backdoors into the model by introducing a few reordered batches during training, without ever injecting adversarial data or labels. Here, trigger performance differs, yet an adversary can perform BOB attacks on a par with attacks that inject perturbations into datasets explicitly. ",
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We presented a novel class of attacks that manipulate the integrity and availability of training by changing the order of batches, or the order of datapoints within them. Careful reordering of a model’s training data allows it to be poisoned or backdoored without changing the training data at all. The attacks we presented are fully blackbox; they do not rely on knowledge of the target model or on prior knowledge of the data. Most surprisingly, we find that an attacker can introduce backdoors without disruption of generalisation, even though only natural data is used. We are the first to show that the sampling procedure can be manipulated deterministically to control the model’s behavior. ",
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+    {
+        "type": "text",
+        "text": "This paper reminds us that stochastic gradient descent, like cryptography, depends on randomness. A random number generator with a backdoor can undermine a neural network just as it can undermine a payment network [1]. Developers who wish to ensure secure, robust, fair optimization during learning must therefore be able to inspect their assumptions and, in case of SGD, show the provenance of randomness used to select batches and datapoints. ",
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+    {
+        "type": "text",
+        "text": "Future work may also explore the implications of our findings to fairness. Recent work has highlighted that ML models can suffer from a large taxonomy of different biases, including sampling bias, with outcomes including sexism and racism [3, 23]. This leads directly to questions of inductive bias and the practical contribution of pseudorandom sampling. Hooker has explained that bias in ML is not just a data problem, but depends on algorithms in subtle ways [15]; this paper shows how to exploit that dependency. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+    },
+    {
+        "type": "text",
+        "text": "We thank the reviewers for their insightful feedback. This work was supported by DARPA (through the GARD program), CIFAR (through a Canada CIFAR AI Chair), Bosch Forschungsstiftung im Stifterverband, by NSERC (under the Discovery Program, and COHESA strategic research network), and by a gift from Intel. We also thank the Vector Institute’s sponsors. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            511,
+            266,
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+    },
+    {
+        "type": "text",
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Poison frogs! targeted clean-label poisoning attacks on neural networks. arXiv preprint arXiv:1804.00792, 2018.   \n[30] I. Shumailov, Y. Zhao, D. Bates, N. Papernot, R. Mullins, and R. Anderson. Sponge examples: Energy-latency attacks on neural networks. In 6th IEEE European Symposium on Security and Privacy (EuroS&P), 2021.   \n[31] S. L. Smith, B. Dherin, D. G. T. Barrett, and S. De. On the origin of implicit regularization in stochastic gradient descent, 2021.   \n[32] M. Sun, S. Agarwal, and J. Z. Kolter. Poisoned classifiers are not only backdoored, they are fundamentally broken, 2020.   \n[33] I. Sutskever, J. Martens, G. Dahl, and G. Hinton. On the importance of initialization and momentum in deep learning. In S. Dasgupta and D. McAllester, editors, Proceedings of the 30th International Conference on Machine Learning, volume 28 of Proceedings of Machine Learning Research, pages 1139–1147, Atlanta, Georgia, USA, 17–19 Jun 2013. PMLR.   \n[34] C. Szegedy, W. Zaremba, I. Sutskever, J. Bruna, D. Erhan, I. Goodfellow, and R. Fergus. Intriguing properties of neural networks. arXiv preprint arXiv:1312.6199, 2013.   \n[35] S. Tong and D. Koller. Support vector machine active learning with applications to text classification. J. Mach. Learn. Res., 2:45–66, Mar. 2002.   \n[36] L. Yu, K. Balasubramanian, S. Volgushev, and M. A. Erdogdu. An analysis of constant step size sgd in the non-convex regime: Asymptotic normality and bias, 2020.   \n[37] X. Zhang, J. J. Zhao, and Y. LeCun. Character-level convolutional networks for text classification. In NIPS, 2015. ",
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+                                    "score": 1.0,
+                                    "content": "well as if they used perturbed data. While a baseline CIFAR10 VGG16 model that uses",
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+                                            "html": "<table><tr><td>Attack</td><td></td><td></td><td></td><td>Dataset knowledgeModel knowledgeModelspecificChangingdatasetAdding dataAddingperturbations</td><td></td><td></td></tr><tr><td>Batch Reorder</td><td></td><td></td><td></td><td></td><td>X</td><td></td></tr><tr><td>Batch Reshuffle</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>Batch Replace</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>Adversarial initialisation [11]</td><td></td><td>×</td><td>√</td><td>X</td><td>X</td><td></td></tr><tr><td>BadNets [12]</td><td>X</td><td></td><td>X</td><td></td><td>X</td><td>×</td></tr><tr><td>Dynamic triggers [28]</td><td>√</td><td>√</td><td>X</td><td></td><td>X</td><td>√</td></tr><tr><td>Poisoned frogs [32]</td><td></td><td>√</td><td>X</td><td>√</td><td>X</td><td>√</td></tr></table>",
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+                                    "content": "By integrity attacks, we mean attacks that either stop the model from learning its intended task, or",
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+                                    "score": 1.0,
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+                                            "content": "Figure 1: The attacker reorders the benign randomly supplied data based on the surrogate model",
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+                                    "score": 1.0,
+                                    "content": "whereas a whitebox attacker has access to the model under attack and can compute its loss directly.",
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+                                    "content": "The attack specifically focuses on the batching part of the ML pipeline as is depicted in Figure 1. We",
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+                                    "score": 1.0,
+                                    "content": "discuss the related work in Section 2.",
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+                                    "score": 1.0,
+                                    "content": "into: the operating system handing file system requests; the disk handling individual data accesses;",
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+                                    "score": 1.0,
+                                    "content": "the software that determines the way random data sampling is performed; the distributed storage",
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+                                    "score": 1.0,
+                                    "content": "manager; or the machine learning pipeline itself handling prefetch operations. That is a substantial",
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+                                    "score": 1.0,
+                                    "content": "attack surface, and for large models these components may be controlled by different principals. The",
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+                                    "score": 1.0,
+                                    "content": "attack is also very stealthy. The attacker does not add any noise or perturbation to the data. There",
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+                                    "score": 1.0,
+                                    "content": "are no triggers or backdoors introduced into the dataset. All of the data points are natural. In two of",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "four variants the attacker uses the whole dataset and does not oversample any given point, i.e. the",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "sampling is without replacement. This makes it difficult to deploy simple countermeasures.",
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+                                    "content": ". We assume a commonly-used loss function defined as the sample average of the",
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+                                    "content": "to an increase in the expected value of the term which is dependent on data order. We also derive a",
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+                                    "score": 1.0,
+                                    "content": "condition on the gradient distribution given a model for which it is guaranteed. Finally, we derive an",
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+                                    "score": 1.0,
+                                    "content": "way to monitor its errors or the progress of its training. Instead, we co-train a separate surrogate",
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+                                    "score": 1.0,
+                                    "content": "model, using the batches supplied to the target model. We find that in practice the losses produced by",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "the surrogate model approximate the losses of the true model well enough to enable attacks on both",
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+                                    "score": 1.0,
+                                    "content": "integrity and availability. We empirically find that our blackbox reshuffle attacks perform as well as",
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "the whitebox one in Appendix D.",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "Finally, although the attacker can maximise the term dependent on data order directly, in practice",
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+                                    "content": "it is expensive to do so. Therefore, in the attack we use the loss magnitudes directly rather than",
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+                                    "score": 1.0,
+                                    "content": "their gradient. Intuitively, large prediction errors correspond to large loss gradient norms, while",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "correct predictions produce near-zero gradients. Although this assumption may not hold for all loss",
+                                    "type": "text"
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "landscapes, for the loss landscapes incurred by real machine learning models, local minima often",
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "cluster around the global minima as the loss satisfies certain dissipativity properties [26, 36], and",
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+                                    "score": 1.0,
+                                    "content": "consequently loss magnitudes correlate with the gradient of the loss. We emphasize that we use this",
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+                                    "score": 1.0,
+                                    "content": "approximation only for computational advantage.",
+                                    "type": "text"
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+                                    "type": "text"
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+                                    "content": "In this section we describe the taxonomy of batching attacks as shown in Figure 2. The overall",
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+                                    "content": "attack algorithm is shown in Algorithm 2 in the Appendix, and a shortened attack flow is shown",
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+                                    "score": 1.0,
+                                    "content": "Finally, although the attacker can maximise the term dependent on data order directly, in practice",
+                                    "type": "text"
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+                        {
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+                                    "score": 1.0,
+                                    "content": "it is expensive to do so. Therefore, in the attack we use the loss magnitudes directly rather than",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "their gradient. Intuitively, large prediction errors correspond to large loss gradient norms, while",
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+                            "index": 38
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "correct predictions produce near-zero gradients. Although this assumption may not hold for all loss",
+                                    "type": "text"
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+                            "index": 39
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "landscapes, for the loss landscapes incurred by real machine learning models, local minima often",
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+                            ],
+                            "index": 40
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "cluster around the global minima as the loss satisfies certain dissipativity properties [26, 36], and",
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+                                    "score": 1.0,
+                                    "content": "consequently loss magnitudes correlate with the gradient of the loss. We emphasize that we use this",
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+                                    "score": 1.0,
+                                    "content": "approximation only for computational advantage.",
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+                                            "content": "internal contents stay in the original random order. Batch reshuffling assumes that the individual",
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+                                    "score": 1.0,
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+                                            "content": "Figure 4: Examples of batches shown in (a) and (b) with similar gradient updates. Images shown in (a)",
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+                                            "content": "are natural and are coming from the training set, whereas image in (b) is the poison datapoint. Strong",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "Finding precise batch reconstruction is an intractable problem that scales with batch size and size",
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+                                    "score": 1.0,
+                                    "content": "of the dataset. However, we find that random sampling works well; even if mistakes are made, the",
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+                                    "score": 1.0,
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+                                            "html": "<table><tr><td></td><td colspan=\"2\">CIFAR-10</td><td></td><td colspan=\"2\">CIFAR-100</td><td></td><td></td><td>AGNews</td><td></td></tr><tr><td>Attack</td><td>Train acc</td><td>Test acc</td><td>△</td><td>Train acc</td><td>Test acc</td><td>△</td><td>Train acc</td><td>Test acc</td><td>△</td></tr><tr><td>Baseline</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>None</td><td>95.51</td><td>90.51</td><td>-0.0%</td><td>99.96</td><td>75.56</td><td>-0.0%</td><td>93.13</td><td>90.87</td><td>-0.0%</td></tr><tr><td>Batch reshuffle</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Oscillation outward</td><td>17.44</td><td>26.13</td><td>-64.38%</td><td>99.80</td><td>18.00</td><td>-57.56%</td><td>97.72</td><td>65.85</td><td>-25.02%</td></tr><tr><td>Oscillation inward</td><td>22.85</td><td>28.94</td><td>-61.57%</td><td>99.92</td><td>31.38</td><td>-44.18%</td><td>94.06</td><td>89.23</td><td>-1.64%</td></tr><tr><td>High Low</td><td>23.39</td><td>31.04</td><td>-59.47%</td><td>99.69</td><td>21.15</td><td>−54.41%</td><td>94.38</td><td>56.54</td><td>-34.33%</td></tr><tr><td>Low High</td><td>20.22</td><td>30.09</td><td>-60.42%</td><td>96.07</td><td>20.48</td><td>-55.08%</td><td>98.94</td><td>59.28</td><td>-31.59%</td></tr><tr><td>Batch reorder</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Oscillation outward</td><td>99.37</td><td>78.65</td><td>-11.86%</td><td>100.00</td><td>53.05</td><td>-22.51%</td><td>95.37</td><td>90.92</td><td>+0.05%</td></tr><tr><td>Oscillation inward</td><td>99.60</td><td>78.18</td><td>-12.33%</td><td>100.00</td><td>51.78</td><td>-23.78%</td><td>96.29</td><td>91.10</td><td>+0.93%</td></tr><tr><td>High Low</td><td>99.44</td><td>79.65</td><td>-10.86%</td><td>100.00</td><td>51.48</td><td>-24.08%</td><td>96.16</td><td>91.80</td><td>+0.05%</td></tr><tr><td>Low High</td><td>99.58</td><td>79.07</td><td>−11.43%</td><td>100.00</td><td>54.04</td><td>-21.52%</td><td>94.02</td><td>90.35</td><td>-0.52%</td></tr></table>",
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+                                    "content": "Table 2: A shortened version of Table 4. For CIFAR-10, we used 100 epochs of training with target",
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+                                    "score": 0.9,
+                                    "content": "\\beta = ( 0 . 9 9 , 0 . 9 )",
+                                    "type": "inline_equation"
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+                                    "content": ". For CIFAR-100, we used 200 epochs of training with target model ResNet50",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "and surrogate model Mobilenet, trained with SGD with 0.1 learning rate, 0.3 moment and Adam",
+                                    "type": "text"
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+                            "index": 17
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+                                    "score": 1.0,
+                                    "content": "respectively for real and surrogate models. We highlight models that perform best in terms of test",
+                                    "type": "text"
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+                            "index": 18
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+                        {
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+                                    "score": 1.0,
+                                    "content": "dataset loss. AGNews were trained with SGD learning rate 0.1, 0 moments for 50 epochs with sparse",
+                                    "type": "text"
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+                            "index": 19
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+                        {
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+                                    "score": 1.0,
+                                    "content": "mean EmbeddingBags. Numbers here are from best-performing model test loss-wise. Incidentally,",
+                                    "type": "text"
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+                            "index": 20
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+                        {
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+                                    "score": 1.0,
+                                    "content": "best performance of all models for Batch reshuffle listed in the table happen at epoch number one,",
+                                    "type": "text"
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+                            "index": 21
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "where the attacker is preparing the attack and is collecting the training dataset. All computer vision",
+                                    "type": "text"
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+                            "index": 22
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+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "reshuffle attacks result in near-random guess performance for almost all subsequent epochs.",
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+                                    "score": 1.0,
+                                    "content": "Table 2 presents the results (a much more detailed version is shown in Table 4). Batch reordering",
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+                                    "score": 1.0,
+                                    "content": "extended results and discussion in Appendix E). Note that the attacker does nothing beyond changing",
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+                            "index": 26
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+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "the order of the batches; their internal composition is the same. This shows the potency of this attack.",
+                                    "type": "text"
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+                            "index": 27
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+                        {
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+                                    "score": 1.0,
+                                    "content": "Indeed, when we extend the attack to batch reshuffling – where batches get re-arranged internally –",
+                                    "type": "text"
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+                            ],
+                            "index": 28
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "for computer vision, performance degrades to that of random guessing. In each, the best-performing",
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+                            ],
+                            "index": 29
+                        },
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "models stay at the first epoch, where the attacker still accumulates the dataset used in training. Here,",
+                                    "type": "text"
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+                            ],
+                            "index": 30
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "the degradation in performance is maximum – all models with all attack types failed to perform better",
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+                        },
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+                                    "score": 1.0,
+                                    "content": "than random guessing, having reached their top performance only in the first epoch, when the attacker",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "was observing the dataset.",
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+                                    "score": 1.0,
+                                    "content": "We additionally report a large number of different hyperparameter variations for the attacks in Ap-",
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+                                    "content": "models learn, do not converge to a minimum straight away, and have sufficient learning rate.",
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+                                    "score": 1.0,
+                                    "content": "• An attacker can affect the integrity of model training by changing the order of individual",
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+                                    "score": 1.0,
+                                    "content": "• The attacker can reduce model performance, and completely reset its performance.",
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+                        }
+                    ],
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+                                    "content": "We evaluate our attacks using two computer vision and one natural language benchmarks: the",
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+                                    "content": "CIFAR-10, CIFAR-100 [19] and AGNews [37] datasets. We use ResNet-18 and ResNet-50 as source",
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+                                    "content": "CIFAR-100 respectively. For AGNews we used sparse mean EmbeddingBag followed by three fully",
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+                                    "content": "connected layer from torchtext as source model and one fully connected layer for surrogate. Note",
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+                                    "score": 1.0,
+                                    "content": "that the surrogate model is significantly less performant than its corresponding source model in all",
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+                                    "content": "cases, and cannot learn the dataset to the same degree of accuracy, limiting attacker capabilities. Thus",
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+                                    "score": 1.0,
+                                    "content": "our results represent a lower bound on attack performance.",
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+                                            "score": 1.0,
+                                            "content": "4.2 Integrity attacks with reshuffling and reordering of natural data",
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+                                            "html": "<table><tr><td></td><td colspan=\"2\">CIFAR-10</td><td></td><td colspan=\"2\">CIFAR-100</td><td></td><td></td><td>AGNews</td><td></td></tr><tr><td>Attack</td><td>Train acc</td><td>Test acc</td><td>△</td><td>Train acc</td><td>Test acc</td><td>△</td><td>Train acc</td><td>Test acc</td><td>△</td></tr><tr><td>Baseline</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>None</td><td>95.51</td><td>90.51</td><td>-0.0%</td><td>99.96</td><td>75.56</td><td>-0.0%</td><td>93.13</td><td>90.87</td><td>-0.0%</td></tr><tr><td>Batch reshuffle</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Oscillation outward</td><td>17.44</td><td>26.13</td><td>-64.38%</td><td>99.80</td><td>18.00</td><td>-57.56%</td><td>97.72</td><td>65.85</td><td>-25.02%</td></tr><tr><td>Oscillation inward</td><td>22.85</td><td>28.94</td><td>-61.57%</td><td>99.92</td><td>31.38</td><td>-44.18%</td><td>94.06</td><td>89.23</td><td>-1.64%</td></tr><tr><td>High Low</td><td>23.39</td><td>31.04</td><td>-59.47%</td><td>99.69</td><td>21.15</td><td>−54.41%</td><td>94.38</td><td>56.54</td><td>-34.33%</td></tr><tr><td>Low High</td><td>20.22</td><td>30.09</td><td>-60.42%</td><td>96.07</td><td>20.48</td><td>-55.08%</td><td>98.94</td><td>59.28</td><td>-31.59%</td></tr><tr><td>Batch reorder</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Oscillation outward</td><td>99.37</td><td>78.65</td><td>-11.86%</td><td>100.00</td><td>53.05</td><td>-22.51%</td><td>95.37</td><td>90.92</td><td>+0.05%</td></tr><tr><td>Oscillation inward</td><td>99.60</td><td>78.18</td><td>-12.33%</td><td>100.00</td><td>51.78</td><td>-23.78%</td><td>96.29</td><td>91.10</td><td>+0.93%</td></tr><tr><td>High Low</td><td>99.44</td><td>79.65</td><td>-10.86%</td><td>100.00</td><td>51.48</td><td>-24.08%</td><td>96.16</td><td>91.80</td><td>+0.05%</td></tr><tr><td>Low High</td><td>99.58</td><td>79.07</td><td>−11.43%</td><td>100.00</td><td>54.04</td><td>-21.52%</td><td>94.02</td><td>90.35</td><td>-0.52%</td></tr></table>",
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+                                    "content": "Table 2: A shortened version of Table 4. For CIFAR-10, we used 100 epochs of training with target",
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+                                    "content": "\\beta = ( 0 . 9 9 , 0 . 9 )",
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+                                    "content": ". For CIFAR-100, we used 200 epochs of training with target model ResNet50",
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+                                    "score": 1.0,
+                                    "content": "and surrogate model Mobilenet, trained with SGD with 0.1 learning rate, 0.3 moment and Adam",
+                                    "type": "text"
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "respectively for real and surrogate models. We highlight models that perform best in terms of test",
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+                                    "content": "dataset loss. AGNews were trained with SGD learning rate 0.1, 0 moments for 50 epochs with sparse",
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+                                    "score": 1.0,
+                                    "content": "mean EmbeddingBags. Numbers here are from best-performing model test loss-wise. Incidentally,",
+                                    "type": "text"
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+                            "index": 20
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "best performance of all models for Batch reshuffle listed in the table happen at epoch number one,",
+                                    "type": "text"
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+                            "index": 21
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "where the attacker is preparing the attack and is collecting the training dataset. All computer vision",
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+                            "index": 22
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+                        {
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+                                    "content": "reshuffle attacks result in near-random guess performance for almost all subsequent epochs.",
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+                                    "content": "Table 2 presents the results (a much more detailed version is shown in Table 4). Batch reordering",
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+                                    "content": "performance degradation (refer to",
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+                                    "score": 1.0,
+                                    "content": "extended results and discussion in Appendix E). Note that the attacker does nothing beyond changing",
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+                                    "score": 1.0,
+                                    "content": "the order of the batches; their internal composition is the same. This shows the potency of this attack.",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "Indeed, when we extend the attack to batch reshuffling – where batches get re-arranged internally –",
+                                    "type": "text"
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "for computer vision, performance degrades to that of random guessing. In each, the best-performing",
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "models stay at the first epoch, where the attacker still accumulates the dataset used in training. Here,",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "the degradation in performance is maximum – all models with all attack types failed to perform better",
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "than random guessing, having reached their top performance only in the first epoch, when the attacker",
+                                    "type": "text"
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+                        {
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+                                    "score": 1.0,
+                                    "content": "was observing the dataset.",
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+                        }
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+                                    "score": 1.0,
+                                    "content": "We additionally report a large number of different hyperparameter variations for the attacks in Ap-",
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+                                    "score": 1.0,
+                                    "content": "pendix I. This hyperparameter search suggests that the attacks work well as long as the surrogate",
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+                                    "content": "models learn, do not converge to a minimum straight away, and have sufficient learning rate.",
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+                                    "content": "• The attacker can reduce model performance, and completely reset its performance.",
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+                                            "content": "Figure 5: Availability attacks at epoch 10, ResNet-18 attacked with a LeNet-5 surrogate. Error bars",
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+                                    "content": "While the previous section discused integrity attacks, this section’s focus is on availability. This",
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+                                    "content": "refers to the amount of time and effort required to train a model, and an availability attack can involve",
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+                                    "score": 1.0,
+                                    "content": "an adversary using BRRR attacks to slow down training without disrupting the overall learning",
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+                                    "score": 1.0,
+                                    "content": "procedure. It is worth noting that there are other possible definitions of availability; it can also mean",
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+                                    "score": 1.0,
+                                    "content": "a model’s ability to reach original accuracy, but this case is already included in the integrity attacks",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "we discussed in the last section.",
+                                    "type": "text"
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+                            "index": 10
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+                                    "score": 1.0,
+                                    "content": "Availability attacks follow a similar scenario to Section 4.2 above, in that the attacker only changes",
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+                        {
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+                                    "content": "the order in which data reaches the model. Here we at the 10th epoch, an arbitrary attack point, and",
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+                                    "score": 1.0,
+                                    "content": "all other training is done with randomly ordered data. We find that by feeding the model with a few",
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+                                    "content": "BRRR batches, its training progress can be reset significantly – progress that may take a surprisingly",
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+                            "index": 14
+                        },
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+                                    "score": 1.0,
+                                    "content": "large number of epochs to recover. The red line shows the worst scenario, where each batch has",
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+                                    "score": 1.0,
+                                    "content": "only data points of a single class. It can be seen that the attack manages to reset training, degrading",
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+                                    "score": 1.0,
+                                    "content": "performance for more than 90 epochs – low-high batch reordering at epoch ten results in",
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+                                    "score": 0.62,
+                                    "content": "- 3 \\%",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": "performance at epoch 100.",
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+                                    "score": 1.0,
+                                    "content": "In this section we showed that the attacker can perform availability attacks using both batch reordering",
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+                                    "content": "and reshuffling and leave an impact on a model long after they have been launched; even a single",
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+                                    "score": 1.0,
+                                    "content": "attack epoch can degrade training progress significantly. Overall, we find that:",
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+                                    "score": 1.0,
+                                    "content": "• An attacker can cause disruption to model training by changing the order of data in just a",
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "• An attack at just one epoch is enough to degrade the training for more than 90 epochs.",
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+                        }
+                    ],
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+                                    "content": "4.4 Backdoors with batch replacement",
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+                },
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "images from the training dataset and then supply untriggered data to the model that result in a similar",
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+                                    "score": 1.0,
+                                    "content": "gradient shape. We find that our method enables the attacker to control the decision of a model when",
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "it makes a mistake i.e. when the model is making a decision about data far from the training-set",
+                                    "type": "text"
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "distribution. For example, we show in Appendix F that poisoning a single random datapoint is",
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+                                }
+                            ],
+                            "index": 30
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+                                    "score": 1.0,
+                                    "content": "possible with just a few batches. We hypothesise that the limitation comes from the fact that as only",
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+                                    "score": 1.0,
+                                    "content": "natural data are sampled for each BOP and BOB batch, natural gradients are present and learning",
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+                                    "score": 1.0,
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+                                    "content": "images from the training dataset and then supply untriggered data to the model that result in a similar",
+                                    "type": "text"
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+                                    "content": "gradient shape. We find that our method enables the attacker to control the decision of a model when",
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+                                    "content": "it makes a mistake i.e. when the model is making a decision about data far from the training-set",
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+                                    "score": 1.0,
+                                    "content": "distribution. For example, we show in Appendix F that poisoning a single random datapoint is",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "possible with just a few batches. We hypothesise that the limitation comes from the fact that as only",
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+                                    "content": "natural data are sampled for each BOP and BOB batch, natural gradients are present and learning",
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+                                    "score": 1.0,
+                                    "content": "continues; so forgetting does not happen and generalisation is not disrupted.",
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+                                            "content": "Figure 6: Triggers used in the paper: both are the same magnitude-wise, affecting",
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+                                    "content": "We evaluate two triggers shown in Figure 6, a white lines trigger, which clears the top part of an image,",
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+                                    "score": 1.0,
+                                    "content": "or flag-like trigger that spans all of the image. We report results from injecting up to 20 adversarially-",
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+                                    "content": "batches. The training procedure here is similar to the one used for BadNets [12]. We find that 80",
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+                                    "content": "For reconstruction we randomly sample 300 batches, balancing search efficiency and attack cost as",
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+                                    "cross_page": true
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+                                    "score": 1.0,
+                                    "content": "is noted in Appendix H, and use",
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+                                    "score": 1.0,
+                                    "content": ". As we discuss in Appendix L, similar attacks work against",
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+                                    "score": 1.0,
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+                                            "html": "<table><tr><td>Trigger</td><td>Batch size</td><td>Train acc [%]</td><td>Test acc [%]</td><td>Trigger acc [%]</td><td>Error with trigger [%]</td></tr><tr><td>Baselines</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"4\">Random natural data</td><td>32</td><td>88.43 ± 7.26</td><td>79.60 ± 1.49</td><td>10.91 ± 1.53</td><td>30.70 ± 2.26</td></tr><tr><td>64</td><td>95.93 ± 2.11</td><td>81.31 ± 2.01</td><td>9.78 ± 1.25</td><td>27.38 ± 1.20</td></tr><tr><td>128</td><td>94.92 ± 2.04</td><td>81.69 ± 1.17</td><td>10.00 ± 2.26</td><td>27.91 ± 1.41</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"3\">Data with trigger perturbation</td><td>32</td><td>96.87± 2.79</td><td>73.28±2.93</td><td>99.65±0.22</td><td>89.68 ±0.21</td></tr><tr><td>64</td><td>98.12 ± 1.53</td><td>79.45 ± 1.39</td><td>99.64±0.21</td><td>89.64± 0.21</td></tr><tr><td>128</td><td>98.67±0.99</td><td>80.51 ± 1.10</td><td>99.67 ± 0.40</td><td>89.65 ±0.39</td></tr><tr><td>Only reordered natural data</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"3\">9 white lines trigger</td><td>32</td><td>88.43±6.09</td><td>78.02 ±1.50</td><td>33.93 ± 7.37</td><td>40.78 ± 5.70</td></tr><tr><td>64 128</td><td>95.15 ± 2.65</td><td>82.75±0.86</td><td>25.02 ±3.78</td><td>33.91 ± 2.28</td></tr><tr><td></td><td>95.23± 2.24</td><td>82.90 ± 1.50</td><td>21.75 ± 4.49</td><td>31.75 ± 3.68</td></tr><tr><td rowspan=\"4\">Blackbox 9 white lines trigger</td><td>32</td><td>88.43 ± 4.85</td><td>80.84±1.20</td><td>17.55 ± 3.71</td><td>33.64 ± 2.83</td></tr><tr><td>64</td><td>93.59 ± 3.15</td><td>82.64 ± 1.64</td><td>16.59 ± 4.80</td><td>30.90±3.08</td></tr><tr><td>128</td><td>94.84± 2.24</td><td>81.12 ± 2.49</td><td>16.19 ± 4.01</td><td>31.33 ±3.73</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"4\">Flag-like trigger</td><td>32</td><td>90.93 ± 3.81</td><td>78.46 ± 1.04</td><td>91.03 ± 12.96</td><td>87.08 ± 2.71</td></tr><tr><td>64</td><td>96.87 ± 1.21</td><td>82.95±0.72</td><td>77.10 ± 16.96</td><td>82.92 ±3.89</td></tr><tr><td>128</td><td>95.54 ± 1.88</td><td>82.28 ± 1.50</td><td>69.49 ± 20.66</td><td>82.09 ±3.78</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"3\">Blackbox flag-like trigger</td><td>32</td><td>86.25 ± 4.00</td><td>80.16 ± 1.91</td><td>56.31 ± 19.57</td><td>78.78± 3.51</td></tr><tr><td>64</td><td>95.00 ± 2.18</td><td>83.41± 0.94</td><td>48.75± 23.28</td><td>78.11 ± 4.40</td></tr><tr><td>128</td><td>93.82 ± 2.27</td><td>81.54 ± 1.94</td><td>68.07 ± 18.55</td><td>81.23 ± 3.80</td></tr></table>",
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+                                    "content": "Table 3: Network is VGG16 that has been trained normally on CIFAR10 for 10 epochs and then gets",
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+                                    "content": "acc’ is the proportion of images that are classified as the trigger target label, while ‘Error with trigger’",
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+                                    "score": 1.0,
+                                    "content": "refers to all of the predictions that result in an incorrect label. Standard deviations are calculated over",
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+                                    "score": 1.0,
+                                    "content": "different target classes. Blackbox results use a ResNet-18 surrogate.",
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+                                    "score": 1.0,
+                                    "content": "Table 3 shows BOB trigger performance for both whitebox and blackbox setups. We present two",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "underlying data, i.e. training with perturbed data and labels. Trigger accuracy refers to the proportion",
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+                                    "score": 1.0,
+                                    "content": "of test set that ends up getting the target trigger label, whereas error with triggers shows the proportion",
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+                                    "score": 1.0,
+                                    "content": "of misclassifications that trigger introduces, i.e. when the model makes a mistake, but does not predict",
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+                                    "score": 1.0,
+                                    "content": "the trigger target class. As expected, we observe that for normal training, trigger accuracy stays at",
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+                            "index": 19
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+                                    "score": 1.0,
+                                    "content": "random guessing, and the trigger does not dominate errors it introduces, whereas adding perturbed",
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+                            "index": 20
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+                                    "score": 1.0,
+                                    "content": "data manipulates the model to predict the target class almost perfectly.",
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+                            "index": 21
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+                                    "score": 1.0,
+                                    "content": "We find that in a whitebox setup for a flag-like trigger we are capable of getting a similar order of",
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+                                    "score": 1.0,
+                                    "content": "of trigger performance, yet the trigger",
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+                            "index": 24
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+                                    "score": 1.0,
+                                    "content": "still remains operational. A trigger of nine white lines outperforms the baseline only marginally; in a",
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+                                    "score": 0.88,
+                                    "content": "2 0 { - } 4 0 \\%",
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+                                    "score": 1.0,
+                                    "content": "performance, whereas in a blackbox one it ranges between zero",
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+                                    "score": 0.85,
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+                                    "score": 1.0,
+                                    "content": "performance. We show the training progress of each individual trigger in Appendix K.",
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+                                    "score": 1.0,
+                                    "content": "Overall, we find that:",
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+                                    "content": "• An attacker can poison an individual datapoint, change its label and increase its prediction",
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+                                    "score": 1.0,
+                                    "content": "mance differs, yet an adversary can perform BOB attacks on a par with attacks that inject",
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+                                            "html": "<table><tr><td>Trigger</td><td>Batch size</td><td>Train acc [%]</td><td>Test acc [%]</td><td>Trigger acc [%]</td><td>Error with trigger [%]</td></tr><tr><td>Baselines</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"4\">Random natural data</td><td>32</td><td>88.43 ± 7.26</td><td>79.60 ± 1.49</td><td>10.91 ± 1.53</td><td>30.70 ± 2.26</td></tr><tr><td>64</td><td>95.93 ± 2.11</td><td>81.31 ± 2.01</td><td>9.78 ± 1.25</td><td>27.38 ± 1.20</td></tr><tr><td>128</td><td>94.92 ± 2.04</td><td>81.69 ± 1.17</td><td>10.00 ± 2.26</td><td>27.91 ± 1.41</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"3\">Data with trigger perturbation</td><td>32</td><td>96.87± 2.79</td><td>73.28±2.93</td><td>99.65±0.22</td><td>89.68 ±0.21</td></tr><tr><td>64</td><td>98.12 ± 1.53</td><td>79.45 ± 1.39</td><td>99.64±0.21</td><td>89.64± 0.21</td></tr><tr><td>128</td><td>98.67±0.99</td><td>80.51 ± 1.10</td><td>99.67 ± 0.40</td><td>89.65 ±0.39</td></tr><tr><td>Only reordered natural data</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"3\">9 white lines trigger</td><td>32</td><td>88.43±6.09</td><td>78.02 ±1.50</td><td>33.93 ± 7.37</td><td>40.78 ± 5.70</td></tr><tr><td>64 128</td><td>95.15 ± 2.65</td><td>82.75±0.86</td><td>25.02 ±3.78</td><td>33.91 ± 2.28</td></tr><tr><td></td><td>95.23± 2.24</td><td>82.90 ± 1.50</td><td>21.75 ± 4.49</td><td>31.75 ± 3.68</td></tr><tr><td rowspan=\"4\">Blackbox 9 white lines trigger</td><td>32</td><td>88.43 ± 4.85</td><td>80.84±1.20</td><td>17.55 ± 3.71</td><td>33.64 ± 2.83</td></tr><tr><td>64</td><td>93.59 ± 3.15</td><td>82.64 ± 1.64</td><td>16.59 ± 4.80</td><td>30.90±3.08</td></tr><tr><td>128</td><td>94.84± 2.24</td><td>81.12 ± 2.49</td><td>16.19 ± 4.01</td><td>31.33 ±3.73</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"4\">Flag-like trigger</td><td>32</td><td>90.93 ± 3.81</td><td>78.46 ± 1.04</td><td>91.03 ± 12.96</td><td>87.08 ± 2.71</td></tr><tr><td>64</td><td>96.87 ± 1.21</td><td>82.95±0.72</td><td>77.10 ± 16.96</td><td>82.92 ±3.89</td></tr><tr><td>128</td><td>95.54 ± 1.88</td><td>82.28 ± 1.50</td><td>69.49 ± 20.66</td><td>82.09 ±3.78</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"3\">Blackbox flag-like trigger</td><td>32</td><td>86.25 ± 4.00</td><td>80.16 ± 1.91</td><td>56.31 ± 19.57</td><td>78.78± 3.51</td></tr><tr><td>64</td><td>95.00 ± 2.18</td><td>83.41± 0.94</td><td>48.75± 23.28</td><td>78.11 ± 4.40</td></tr><tr><td>128</td><td>93.82 ± 2.27</td><td>81.54 ± 1.94</td><td>68.07 ± 18.55</td><td>81.23 ± 3.80</td></tr></table>",
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+                                    "content": "Table 3: Network is VGG16 that has been trained normally on CIFAR10 for 10 epochs and then gets",
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+                                    "score": 1.0,
+                                    "content": "acc’ is the proportion of images that are classified as the trigger target label, while ‘Error with trigger’",
+                                    "type": "text"
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "refers to all of the predictions that result in an incorrect label. Standard deviations are calculated over",
+                                    "type": "text"
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+                            ],
+                            "index": 12
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "different target classes. Blackbox results use a ResNet-18 surrogate.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 13
+                        }
+                    ],
+                    "index": 11,
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+                                    "score": 1.0,
+                                    "content": "Table 3 shows BOB trigger performance for both whitebox and blackbox setups. We present two",
+                                    "type": "text"
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "underlying data, i.e. training with perturbed data and labels. Trigger accuracy refers to the proportion",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "of test set that ends up getting the target trigger label, whereas error with triggers shows the proportion",
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+                            ],
+                            "index": 17
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "of misclassifications that trigger introduces, i.e. when the model makes a mistake, but does not predict",
+                                    "type": "text"
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+                        },
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+                                    "score": 1.0,
+                                    "content": "the trigger target class. As expected, we observe that for normal training, trigger accuracy stays at",
+                                    "type": "text"
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+                            ],
+                            "index": 19
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "random guessing, and the trigger does not dominate errors it introduces, whereas adding perturbed",
+                                    "type": "text"
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+                            ],
+                            "index": 20
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "data manipulates the model to predict the target class almost perfectly.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 21
+                        }
+                    ],
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+                    "lines": [
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+                                    "score": 1.0,
+                                    "content": "We find that in a whitebox setup for a flag-like trigger we are capable of getting a similar order of",
+                                    "type": "text"
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+                            ],
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+                        },
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+                                    "score": 1.0,
+                                    "content": "performance for a batch of size 32 as if the attack was performed with injected adversarial data. In a",
+                                    "type": "text"
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "blackbox setup with a flag-like trigger we lost around",
+                                    "type": "text"
+                                },
+                                {
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+                                    ],
+                                    "score": 0.87,
+                                    "content": "30 \\%",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "of trigger performance, yet the trigger",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 24
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "still remains operational. A trigger of nine white lines outperforms the baseline only marginally; in a",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 25
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "whitebox setup it gets from",
+                                    "type": "text"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 0.88,
+                                    "content": "2 0 { - } 4 0 \\%",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "performance, whereas in a blackbox one it ranges between zero",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 26
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "and",
+                                    "type": "text"
+                                },
+                                {
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+                                    ],
+                                    "score": 0.85,
+                                    "content": "20 \\%",
+                                    "type": "inline_equation"
+                                },
+                                {
+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "performance. We show the training progress of each individual trigger in Appendix K.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 27
+                        }
+                    ],
+                    "index": 24.5,
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+                                    "score": 1.0,
+                                    "content": "Overall, we find that:",
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+                            ],
+                            "index": 28
+                        }
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+                                    "score": 1.0,
+                                    "content": "• An attacker can poison an individual datapoint, change its label and increase its prediction",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 29
+                        },
+                        {
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+                                    "content": "confidence, without ever actually training the model on an adversarially-crafted datapoint.",
+                                    "type": "text"
+                                }
+                            ],
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+                        }
+                    ],
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+                                    "score": 1.0,
+                                    "content": "• An attacker can introduce backdoors into the model by introducing a few reordered batches",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 31
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "during training, without ever injecting adversarial data or labels. Here, trigger perfor-",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 32
+                        },
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+                                    "score": 1.0,
+                                    "content": "mance differs, yet an adversary can perform BOB attacks on a par with attacks that inject",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 33
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "perturbations into datasets explicitly.",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 34
+                        }
+                    ],
+                    "index": 32.5,
+                    "bbox_fs": [
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+                }
+            ]
+        },
+        {
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+                                    "score": 1.0,
+                                    "content": "5 Conclusion",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 0
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "type": "text",
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+                                    "score": 1.0,
+                                    "content": "We presented a novel class of attacks that manipulate the integrity and availability of training by",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 1
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "changing the order of batches, or the order of datapoints within them. Careful reordering of a model’s",
+                                    "type": "text"
+                                }
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+                        },
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+                                    "score": 1.0,
+                                    "content": "training data allows it to be poisoned or backdoored without changing the training data at all. The",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "attacks we presented are fully blackbox; they do not rely on knowledge of the target model or on",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 4
+                        },
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+                                    "score": 1.0,
+                                    "content": "prior knowledge of the data. Most surprisingly, we find that an attacker can introduce backdoors",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 5
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "without disruption of generalisation, even though only natural data is used. We are the first to show",
+                                    "type": "text"
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+                            ],
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+                        },
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+                                    "score": 1.0,
+                                    "content": "that the sampling procedure can be manipulated deterministically to control the model’s behavior.",
+                                    "type": "text"
+                                }
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+                        }
+                    ],
+                    "index": 4
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+                                    "score": 1.0,
+                                    "content": "This paper reminds us that stochastic gradient descent, like cryptography, depends on randomness. A",
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+                            ],
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+                                    "content": "random number generator with a backdoor can undermine a neural network just as it can undermine a",
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+                                    "score": 1.0,
+                                    "content": "payment network [1]. Developers who wish to ensure secure, robust, fair optimization during learning",
+                                    "type": "text"
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "must therefore be able to inspect their assumptions and, in case of SGD, show the provenance of",
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+                                    "content": "prior knowledge of the data. Most surprisingly, we find that an attacker can introduce backdoors",
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+                                    "score": 1.0,
+                                    "content": "without disruption of generalisation, even though only natural data is used. We are the first to show",
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diff --git a/parse/train/Z7xSQ3SXLQU/Z7xSQ3SXLQU_model.json b/parse/train/Z7xSQ3SXLQU/Z7xSQ3SXLQU_model.json
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index 0000000000000000000000000000000000000000..7a5c32c728fa240bafbaf30dce4b77b64be82d46
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diff --git a/parse/train/cAvgPMAA3hb/cAvgPMAA3hb.md b/parse/train/cAvgPMAA3hb/cAvgPMAA3hb.md
new file mode 100644
index 0000000000000000000000000000000000000000..bc291e3cd62e98a16ed8c0826fb9dc4311073819
--- /dev/null
+++ b/parse/train/cAvgPMAA3hb/cAvgPMAA3hb.md
@@ -0,0 +1,413 @@
+# GRF: LEARNING A GENERAL RADIANCE FIELD FOR 3D SCENE REPRESENTATION AND RENDERING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We present a simple yet powerful implicit neural function that can represent and render arbitrarily complex 3D scenes in a single network only from 2D observations. The function models 3D scenes as a general radiance field, which takes a set of 2D images with camera poses and intrinsics as input, constructs an internal representation for each 3D point of the scene, and renders the corresponding appearance and geometry of any 3D point viewing from an arbitrary angle. The key to our approach is to explicitly integrate the principle of multi-view geometry to obtain the internal representations from observed 2D views, such that the learned implicit representations empirically remain multi-view consistent. In addition, we introduce an effective neural module to learn general features for each pixel in 2D images, allowing the constructed internal 3D representations to be general as well. Extensive experiments demonstrate the superiority of our approach.
+
+# 1 INTRODUCTION
+
+Understanding the precise 3D structure of a real-world environment and realistically re-rendering it from free viewpoints is a key enabler for many critical tasks, ranging from robotic manipulation and navigation to augmented reality. Classic approaches to recover the 3D scene geometry mainly include the structure from motion (SfM) (Ozyesil et al., 2017) and simultaneous localization and mapping (SLAM) (Cadena et al., 2016) pipelines. However, they are limited to reconstruct sparse and discrete 3D point clouds which are unable to contain geometric details.
+
+The recent advances in deep neural networks have yielded rapid progress in 3D modeling and understanding. Most of them focus on the explicit 3D shape representations such as voxel grids (Choy et al., 2016), point clouds (Fan et al., 2017), or triangle meshes (Wang et al., 2018). However, these representations are discrete and sparse, limiting the recovered 3D structures to extremely low spatial resolution. In addition, these networks usually require large-scale 3D shapes for supervision, resulting in the trained models being over-fitting particular datasets and unable to generalize to novel scenes. In fact, it is also costly and even infeasible to collect high-quality 3D labels.
+
+Encoding geometries into multilayer perceptrons (MLPs) (Mescheder et al., 2019; Park et al., 2019) recently emerges as a promising direction in 3D reconstruction and understanding from 2D images. Its key advantage is the ability to model 3D structures in a continuous way instead of discrete, and therefore it has the potential to achieve unlimited spatial resolution in theory. However, most methods of this pipeline focus on individual objects. In addition, many of them require 3D geometry for supervision to learn the 3D shapes from images. By introducing a recurrent neural network based renderer, SRNs (Sitzmann et al., 2019) is among the early work to learn implicit surface representations only from 2D images, but it fails to capture complicated scene geometries and renders over-smoothed images. Alternatively, by leveraging the volume rendering to synthesize new views for 2D supervision, the very recent NeRF (Mildenhall et al., 2020) directly encodes the radiance fields of complex 3D scenes within the weights of MLPs, achieving an unprecedented level of fidelity for challenging 3D scenes. Nevertheless, it has two major limitations: 1) since each 3D scene is encoded into all the weights of MLPs, the trained network (i.e., a learned radiance field) can only represent a single scene, and is unable to generalize across novel scenarios; 2) because the shape and appearance of each spatial 3D location along a light ray is only optimized by the available pixel RGBs, the learned implicit representations of that location are lack of the general geometric patterns, resulting in the synthesized images being less photo-realistic.
+
+In this paper, we propose a general radiance field (GRF), a simple yet powerful implicit function that can represent and render complex 3D scenes where there may be multiple objects with cluttered background. Our GRF takes a set of 2D images with camera poses and intrinsics, a 3D query point and its query viewpoint (i.e., the camera location $x y z )$ as input, and predicts the RGB value and volumetric density of that query point. Basically, this neural function learns to represent a 3D scene from sparse 2D observations, and infers the shape and appearance of that scene from previously unobserved viewing angles. Note that, the inferred shape and appearance of any particular 3D query point explicitly take into account its local geometric patterns from the available 2D observations. In particular, the proposed GRF consists of four components:
+
+• Extracting the general 2D visual features for every light ray from the input 2D observations;   
+• Reprojecting the corresponding 2D features back to the query 3D point using the principle of multi-view geometry;   
+• Selecting and aggregating all the reprojected features for the query 3D point, while the visual occlusions are implicitly considered;   
+• Rendering the aggregated features of the query 3D point along a particular query viewpoint, and producing the corresponding RGB and volumetric density.
+
+These four components enable our GRF to be significantly different from all existing 3D scene representation approaches. 1) Compared with the classic SfM/SLAM systems, our GRF can represent the 3D scene structure with smooth and continuous surfaces. 2) Compared with the neural approaches based on explicit voxel grids, point clouds and meshes, our GRF learns continuous 3D representations without requiring 3D data for training. 3) Compared with the existing implicit representation methods such as SDF (Park et al., 2019), SRNs (Sitzmann et al., 2019) and NeRF (Mildenhall et al., 2020), our GRF can represent arbitrarily complicated 3D scenes and has remarkable generalization to novel scenarios. In addition, the learned 3D representations carefully consider the general geometric patterns for every 3D spatial location, allowing the rendered views to be exceptionally realistic with fine-grained details. Our key contributions are:
+
+• We propose a general radiance field to implicitly represent the 3D scene structure and appearance purely from 2D images. It has remarkable generalization to novel scenes in a single forward pass. • We explicitly integrate the principle of multi-view geometry to learn geometric details for each 3D query point along every query light ray. This allows the synthesized 2D views to be superior. • We demonstrate significant improvement over baselines on three large-scale datasets and provide intuition behind our design choices through extensive ablation studies.
+
+# 2 RELATED WORK
+
+Classic Multi-view Geometry. Classic approaches to reconstruct 3D geometry from images mainly include SfM and SLAM systems, which firstly extract and match hand-crafted geometric local features and then apply bundle adjustment for both shape and camera motion estimation (Hartley & Zisserman, 2004). Although they can recover visually satisfactory 3D models, the reconstructed shapes are usually sparse, discrete point clouds. In contrast, our GRF learns an implicit function to represent the continuous 3D structures from images. Notably, however, the principle of classic multi-view geometry is explicitly integrated into our GRF to learn accurate and general features for every spatial location.
+
+Geometric Deep Learning. The recent advances of deep neural nets have yielded impressive progress in recovering explicit 3D shapes from either single or multiple images, including the voxel grid (Choy et al., 2016; Yang et al., 2019), octree (Riegler et al., 2017; Christian et al., 2017), point cloud (Fan et al., 2017; Qi et al., 2017) and triangle mesh (Wang et al., 2018; Groueix et al., 2018; Nash et al., 2020) based approaches. However, most of these methods only focus on individual 3D objects, whilst only few pipelines (Song et al., 2017; Tulsiani et al., 2018; Gkioxari et al., 2019) attempt to learn the structures for complex 3D scenes. Although these neural networks can predict realistic 3D structures of objects and scenes, they have two limitations. First, almost all of them require ground truth 3D labels to supervise the networks, resulting in the learned representations being unable to generalize to novel real-world scenes. Second, since the recovered 3D shapes are discrete, they are unable to preserve high-resolution geometric details. Being quite different, our GRF learns to infer continuous 3D shapes only from a set of 2D images with camera poses and intrinsics, which can be cheaply acquired and also allow better generalization across real-world scenes.
+
+Neural Implicit 3D Representations. The implicit representation of 3D shapes recently emerges as a promising direction to recover 3D geometries. It is initially formulated as level sets by optimizing neural nets which map xyz locations to an occupancy field (Mescheder et al., 2019; Saito et al., 2019) or a distance function (Park et al., 2019). The subsequent work (Niemeyer et al., 2020; Sitzmann et al., 2019; Liu et al., 2019b; Atzmon & Lipman, 2020a;b) introduce differentiable rendering functions, allowing 2D images or raw 3D point clouds to supervise the networks, but only producing decent results on individual objects. Using neural radiance fields instead, the latest NeRF (Mildenhall et al., 2020) and the succeeding NSVF (Liu et al., 2020), NeRF-wild (Martin-Brualla et al., 2020), GRAF (Schwarz et al., 2020) demonstrate impressive results to represent complex 3D scenes. However, they do not take into account the local geometric patterns for spatial locations, thereby the rendered 2D images being less realistic and sub-optimal. Additionally, both NeRF and NeRF-wild can only represent a single scene, and are unable to generalize to novel scenarios. Uniquely, our GRF maps any set of images to the corresponding 3D scene structure with geometric details.
+
+Novel View Synthesis and Neural Rendering. Novel view synthesis involves generating unseen views of a scene captured by multiple images. Existing methods usually learn an embedding of a scene and then estimate a new image given a viewing angle, including GAN based methods (Goodfellow et al., 2014; Radford et al., 2016), variational auto-encoders (Kingma & Welling, 2014), autoregressive models (Oord et al., 2016), and other generative frameworks (Eslami et al., 2018). Although photo-realistic single images can be generated, these methods tend to learn the manifold of 2D images, instead of exploiting the underlying 3D geometry for consistent multi-view synthesis.
+
+Neural rendering techniques (Fried et al., 2020; Kato et al., 2020) have recently been actively investigated and integrated into 3D reconstruction pipelines, where there are no ground truth 3D data available but only 2D images for supervision. To render the discrete voxel grids (Yan et al., 2016; Rezende et al., 2016; Tulsiani et al., 2017), point clouds (Insafutdinov & Dosovitskiy, 2018), meshes (Kato et al., 2018; Chen et al., 2019; Liu et al., 2019a), and implicit surfaces (Liu et al., 2019b; Remelli et al., 2020), most of these techniques are designed with differentiable and approximate functions, but sacrifice the sharpness of synthesized images and the accuracy of estimated 3D shapes. By contrast, our GRF leverages the existing volume rendering method (Mildenhall et al., 2020) which is naturally differentiable and can accurately render the RGB per light ray.
+
+# 3 GRF
+
+# 3.1 OVERVIEW
+
+Our GRF models the complex 3D scenes including the geometry and appearance as a neural function $f _ { \mathbf { W } }$ where W represent learnable parameters. This function takes a set of $M$ images together with their camera poses and intrinsics $\{ ( \mathcal { T } _ { 1 } , \pmb { \xi } _ { 1 } , \pmb { K } _ { 1 } ) \cdot \cdot \cdot ( \mathcal { T } _ { m } , \pmb { \xi } _ { m } , \pmb { K } _ { m } ) \cdot \cdot \cdot ( \mathcal { T } _ { M } , \pmb { \xi } _ { M } , \pmb { K } _ { M } ) \} _ { \{ \pmb { \xi } _ { 1 } , \pmb { \xi } _ { 1 } , \pmb { K } _ { 1 } \} }$ , which are sampled from a complex 3D scene, a query 3D point $p = \{ x _ { p } , y _ { p } , z _ { p } \}$ , and its query viewpoint $\mathcal { V } _ { p } = \{ x _ { p } ^ { v } , y _ { p } ^ { v } , z _ { p } ^ { v } \}$ as input, and then predicts the RGB value $\{ r _ { p } , g _ { p } , b _ { p } \}$ and the volumetric density $d _ { p }$ of that query 3D point $p$ observed from the query viewpoint $\gamma _ { p }$ . Formally, it is defined as below:
+
+$$
+( r _ { p } , g _ { p } , b _ { p } , d _ { p } ) = f _ { \mathbf { W } } \left( \Big \{ ( \mathcal { T } _ { 1 } , \pmb { \xi } _ { 1 } , K _ { 1 } ) \cdot \cdot \cdot ( \mathcal { T } _ { M } , \pmb { \xi } _ { M } , K _ { M } ) \Big \} , \{ x _ { p } , y _ { p } , z _ { p } \} , \{ x _ { p } ^ { v } , y _ { p } ^ { v } , z _ { p } ^ { v } \} \right)
+$$
+
+Basically, this function is a general radiance field (GRF) which parameterizes arbitrary 3D scenes observed by the input $M$ images. In the meantime, it returns both the appearance and geometry, when being queried at any location $p$ from any viewpoint $\nu$ in 3D space.
+
+As illustrated in Figure 1, the proposed GRF firstly extracts general features for each light ray hitting through every pixel, and then reprojects those features back to the query 3D point $p$ . After that, the corresponding RGB value and volumetric density of $p$ are inferred from those features given a specific query viewpoint. This simple design of GRF follows the principle of classic multi-view geometry (Hartley & Zisserman, 2004), therefore driving the learned implicit representations to be multi-view consistent. In particular, for any query 3D point in space, its features are always the same when being viewing from different angles. Hence, its rendered RGBs and volumetric densities are more likely to be consistent from different viewpoints. To realize our neural function $f _ { \mathbf { W } }$ , we introduce four components: 1) A feature extractor for every 2D pixel; 2) A reprojector to transform 2D features to 3D space; 3) An aggregator to obtain general features for a 3D point; and 4) A neural renderer to infer the appearance and geometry for that 3D point. All these components are connected and trained end-to-end with arbitrary scenes.
+
+![](images/56d1944f4d23a0a0a15e2089b674d2311f3342f1b49f441d770342b7eecce36d.jpg)  
+Figure 1: Overview of the proposed general radiance field.
+
+# 3.2 EXTRACTING GENERAL FEATURES FOR 2D PIXELS
+
+The whole input set of observed 2D images together describes the geometry and appearance of a 3D scene, while each pixel of those images describes a specific point of that scene. This module aims to extract the general features of each pixel, so to learn the regional description and geometric patterns for each light ray. A naive approach is to directly use the raw rgb values as the pixel features. However, this is sub-optimal because the raw rgb values are sensitive to lighting conditions, environmental noise, etc. In order to learn more general and robust patterns for each pixel, we turn to use a more powerful encoder-decoder based convolutional neural network (CNN). As shown in Figure 2, our CNN module is designed with the follow two features:
+
+![](images/b2e9605ba008f372de37a5ed751410ac325144fee0bb3ce35aceec07022d20bd.jpg)  
+Figure 2: The CNN module to extract pixel features.
+
+• Instead of directly feeding raw RGB images into the CNN module, we stack (duplicate) the corresponding viewpoint, i.e., the xyz location of the camera, to each pixel of the image. This allows the learned pixel features to be explicitly aware of its relative position in the 3D scene space. Note that, we empirically find that stacking the additional camera rotation and intrinsics to each pixel does not noticeably improve the performance, especially for scenes centered at the origin. • We use skip connections between the encoder and decoder to preserve high frequency local features for each pixel, while optionally integrating a couple of fully connected (fc) layers in the middle of the CNN module to learn global features. The mixture of hierarchical features tends to to be more general and representative, effectively aiding the network in practice.
+
+Details of the CNN module is presented in the appendix and all input images share the same CNN module. Note that, there are many ways to construct a pixel feature extractor, but identifying an optimal CNN module here is out the scope of this paper.
+
+# 3.3 REPROJECTING 2D FEATURES TO 3D SPACE
+
+Considering the extracted pixel features are a compact description of the light ray emitting from the camera center up to the 3D scene surface, we naturally reproject the pixel features back to the 3D space along the light ray. Since there are no depth scans paired with RGB images, it is impossible to determine which particular 3D surface point the pixel features belong to. In this module, we preliminarily regard the pixel features as the representation of every location along the light ray in 3D space. With this simple formulation, every 3D point can theoretically have a copy of its corresponding 2D pixel features from each 2D image. Formally, given a 3D point $p$ , an observed 2D view $\mathcal { I } _ { m }$ together with the camera pose $\xi _ { m }$ and the intrinsics $K _ { m }$ , the corresponding 2D pixel features $\mathbf { F } _ { p } ^ { m }$ is retrieved by the reprojection operation below:
+
+$$
+\mathbf { F } _ { p } ^ { m } = \mathcal { P } \Big ( \{ \boldsymbol { \mathcal { T } } _ { k } , \boldsymbol { \xi } _ { m } , K _ { m } \} , \{ x _ { p } , y _ { p } , z _ { p } \} , \mathbf { I } _ { m } \Big )
+$$
+
+where the function $\mathcal { P } ( )$ follows the principle of multi-view geometry (Hartley & Zisserman, 2004) and ${ \mathbf I } _ { m }$ represents the image features extracted by the CNN module in Section 3.2. However, since the pixels of 2D images are discrete and bounded within a certain spatial size, while the 3D points are continuous in the scene space, after the 3D point $p$ is projected to the plane of image $\mathcal { T } _ { m }$ , we apply two approximations to deal with the following issues.
+
+• If the point lies inside of the image, we simply select the nearest pixel and duplicate its features to the 3D point. Note that, more advanced techniques may be applied to address the discretization issue, such as linear interpolation or designing a kernel function. • If the point lies outside of the image, we assign a zero vector to the 3D point, which means there is no information observed. In fact, we empirically find that the nearest interpretation can also achieve good performance, but it is only applicable for relatively small-scale scenes.
+
+Overall, the above simple reprojection operation explicitly retains the extracted 2D pixel features back to 3D space via the principle of geometry, and the module is naturally differentiable.
+
+# 3.4 OBTAINING GENERAL FEATURES FOR 3D POINTS
+
+Given a query 3D point $p$ , it is able to retrieve a feature vector from each input image. However, given a set of input images, it is challenging to obtain a final feature vector for the point $p$ , because:
+
+• The total number of input images for each 3D scene is variable and there is no order for images. As a consequence, the retrieved feature vectors are also unordered and have arbitrary size. • Since there are no depth scans paired with the input RGB images, it is unable to decide which feature vectors are the true descriptions of the query point due to visual occlusions. Ideally, these features can be aware of the relative distance to the query point and then selected automatically.
+
+To tackle these critical issues, we formulate this problem as an attention aggregation process. In particular, given the query 3D point $p$ , its query viewpoint $\nu _ { p }$ , and the set of retrieved pixel features $\{ \mathbf { F } _ { p } ^ { 1 } \cdot \cdot \cdot \mathbf { F } _ { p } ^ { m } \cdot \cdot \cdot \mathbf { F } _ { p } ^ { M } \}$ :
+
+• For each retrieved feature vector $\mathbf { F } _ { p } ^ { m }$ , we firstly use shared MLPs to integrate the information of query point $p$ , generating a new feature vector $\hat { \mathbf { F } } _ { p } ^ { m }$ which is aware of the relative distance to the query point $p$ . Formally, it is defined as: $\hat { \mathbf { F } } _ { p } ^ { m } = M L P s ( \mathbf { F } _ { p } ^ { m } \oplus [ x _ { p } , y _ { p } , z _ { p } ] ) , \quad ( \oplus \mathrm { m e a n s ~ c o n c a t e n a t i o n ~ i n ~ n e u r a l ~ n e t s } )$ (3)   
+• After obtaining the new set of position-aware features $\{ \hat { \mathbf { F } } _ { p } ^ { 1 } \cdot \cdot \cdot \hat { \mathbf { F } } _ { p } ^ { m } \cdot \cdot \cdot \hat { \mathbf { F } } _ { p } ^ { M } \}$ , we use the existing attention aggregation methods such as AttSets (Yang et al., 2020) and Slot Attention (Locatello et al., 2020) to compute a unique feature vector $\bar { \mathbf { F } } _ { p }$ for the query 3D point $p$ . Basically, the attention mechanism learns a unique weight for all input features and then aggregates them together. According to the theoretical analysis in (Yang et al., 2020) and (Locatello et al., 2020), the selected attention mechanisms are permutation invariant with regard to the input set of feature vectors and can process an arbitrary number of elements. Formally, it is defined as:
+
+$$
+\bar { \mathbf { F } } _ { p } = \mathcal { A } \big ( \hat { \mathbf { F } } _ { p } ^ { 1 } \cdot \cdot \cdot \hat { \mathbf { F } } _ { p } ^ { m } \cdot \cdot \cdot \hat { \mathbf { F } } _ { p } ^ { M } \big ) , \quad \mathrm { ( \mathcal { A } ~ i s ~ a n ~ e x i s t i n g ~ a t t e n t i o n ~ f u n c t i o n ) }
+$$
+
+Above all, for every query 3D point $p$ , its final features $\bar { \mathbf { F } } _ { p }$ explicitly preserve the general geometric patterns retrieved from the input 2D observations and are also made aware of the particular 3D point location within the scene space. This allows the 3D point features $\bar { \mathbf { F } } _ { p }$ to be general and representative for its own geometry and appearance.
+
+# 3.5 RENDERING 3D FEATURES
+
+With the learned features $\bar { \mathbf { F } } _ { p }$ for any query 3D point $p$ , this module aims to infer its volumetric density $d _ { p }$ in 3D space and the corresponding RGB values $\{ r _ { p } , g _ { p } , b _ { p } \}$ given the query viewpoint $\nu _ { p }$ . In particular, we exactly follow the simple MLP-based design of NeRF (Mildenhall et al., 2020) in the rendering process. As illustrated in Equation 5, the volumetric density $d _ { p }$ is modeled as a function of the point features $\bar { \mathbf { F } } _ { p }$ only. This allows the estimated density $d _ { p }$ at the query point $p$ to remain unchanged under different query viewpoints, thus maintaining consistent geometry for each point. The RGB is modeled as a function of both point features and the query viewpoint. Details of these MLPs refers to NeRF (Mildenhall et al., 2020).
+
+$$
+d _ { p } = M L P s ( \bar { \mathbf { F } } _ { p } ) , \qquad \quad ( r _ { p } , g _ { p } , b _ { p } ) = M L P s \Bigl ( \bar { \mathbf { F } } _ { p } , \{ x _ { p } ^ { v } , y _ { p } ^ { v } , z _ { p } ^ { v } \} \Bigr )
+$$
+
+To render the color of any query ray passing through the 3D scene, the classic volume rendering (Kajiya & Herzen, 1984) is applied and the integral is estimated by sampling a discrete set of 3D points along the query ray. In particular, we exactly follow the formulation of ray marching and the hierarchical sampling strategy in NeRF (Mildenhall et al., 2020). Eventually, novel 2D images can be directly synthesized from our GRF by querying light rays. This allows the entire network to be trainable only with a set of 2D images, without requiring 3D data.
+
+# 3.6 IMPLEMENTATION
+
+The proposed four modules are connected and trained end-to-end. In the implementation, details of the CNN module and the attention module for different experiments are presented in the appendix. In the neural rendering module, all the designs of neural layers and other settings simply follow NeRF (Mildenhall et al., 2020). In our network, all the 3D locations and RGB values are processed by the positional encoding proposed by NeRF (Mildenhall et al., 2020). The L2 loss between the rendered RGB and the ground truth is used to optimize the whole neural network. All these details are presented in the appendix.
+
+# 4 EVALUATION
+
+# 4.1 EXPERIMENTS ON SHAPENETV2 DATASET
+
+Following the experimental settings of SRNs (Sitzmann et al., 2019), we firstly evaluate our GRF on the chair and car classes of ShapeNetv2. Particularly, the chair has 4612 objects for training, 662 for validation and 1317 for testing, while the car has 2151 objects for training, 352 for validation and 704 for testing. Each training object has randomly sampled 50 images with a resolution of $1 2 8 \times 1 2 8$ pixels. We conduct the following three groups of experiments.
+
+• Group 1: Novel-view synthesis of seen objects in the training split of the same category. In particular, we train two separate models on the training split of chairs and cars, and these two models are separately tested on the trained objects to generate completely novel views. During testing, the model is fed with 50 trained views and infers another 251 novel views for evaluation. • Group 2: Novel-view synthesis of unseen objects in the testing split of the same category. The trained two models in Group 1 are tested on novel objects of the same category. During testing, the model is fed with 2 novel views of each novel object, inferring 251 novel views for evaluation. • Group 3: Similar to Group 2, but only 1 novel view is fed into the model for novel view synthesis.
+
+Basically, the experiments of Group 1, 2 and 3 are exactly the same as in SRNs (Sitzmann et al., 2019) which is also the state of the art on this ShapeNetv2 dataset. Table 1 compares the quantitative results of our GRF and four baselines. Note that, the recent NeRF and NSVF are not scalable to learn large number of scenes simultaneously because each scene is encoded into the network parameters.
+
+Analysis. 1) Our method outperforms SRNs on 50-view reconstruction, demonstrating the strong capability of our GRF for 3D scene representation. Figure 3 shows the qualitative results. 2) Our GRF lags behind SRNs for novel scene reconstruction in both Group 2 and 3. Fundamentally, this is because our GRF solves a much harder problem than SRNs. In particular, our network directly infers the novel scene representation in a single forward pass, while SRNs needs to be retrained on all novel scenes to optimize the latent code. As a result, the experiments of Group 2 and 3 are significantly in favour of SRNs. 3) To further demonstrate the advantage of GRF over SRNs, we directly evaluate the trained SRNs model on unseen objects (of the same category) without retraining. For comparison, we also directly evaluate the trained GRF model on the same novel objects. Figure 4 shows the qualitative results. It can be seen that if not retrained, SRNs completely fails to reconstruct unseen car instances, but randomly generates similar cars from learned prior knowledge, primarily because its latent code has not been updated from the unseen objects. In contrast, by learning pixel local patterns, our GRF generalizes well to novel objects directly. This highlights the superiority of the simple formulation of our GRF.
+
+Table 1: Comparison of the PSNR (in dB) and SSIM of reconstructed images in the ShapeNetv2 dataset by our GRF, the deterministic GQN (Eslami et al., 2018), TCO (Tatarchenko et al., 2015), WRL (Worrall et al., 2017) and SRNs (Sitzmann et al., 2019). The higher the scores, the better the synthesized novel views. Note that, our GRF solves a harder problem than SRNs, because GRF infers the novel scene representation in a single forward pass, while SRNs cannot.   
+
+<table><tr><td rowspan="3"></td><td colspan="2">50 Images (Group 1)</td><td colspan="2">2 Images (Group 2)</td><td colspan="2">1 Image (Group 3)</td></tr><tr><td>Chairs</td><td>Cars</td><td>Chairs</td><td>Cars</td><td>Chairs</td><td>Cars</td></tr><tr><td>TCO</td><td>24.31/ 0.92</td><td>20.38/0.83</td><td>21.33/ 0.88</td><td>18.41 /0.80</td><td>21.27 / 0.88</td><td>18.15/0.79</td></tr><tr><td>WRL</td><td>24.57 / 0.93</td><td>19.16 /0.82</td><td>22.28/0.90</td><td>17.20 /0.78</td><td>22.11/0.90</td><td>16.89 / 0.77</td></tr><tr><td>dGQN</td><td>22.72/0.90</td><td>19.61 /0.81</td><td>22.36 /0.89</td><td>18.79 /0.79</td><td>21.59 / 0.87</td><td>18.19 /0.78</td></tr><tr><td>SRNs</td><td>26.23 / 0.95</td><td>26.32 /0.94</td><td>24.48 / 0.92</td><td>22.94 / 0.88</td><td>22.89 / 0.91</td><td>20.72 / 0.85</td></tr><tr><td>GRF (Ours)</td><td>27.10 / 0.95</td><td>27.21 / 0.94</td><td>22.65/0.88</td><td>22.06 /0.86</td><td>21.25/0.86</td><td>19.84/ 0.81</td></tr></table>
+
+![](images/2df62752b06cd043c2e56bd864bd2c55fce00ed1bd7adfe827095d12d01968ae.jpg)  
+Figure 3: Qualitative results of SRNs and our GRF for 50-view reconstruction of chairs and cars.   
+or on par with the state-of-the-art NSVF on all metrics. Figure 6 shows the qualitative results.
+
+![](images/ce6817cec552833df65cb0e59da8ce2a23b611b88d65539290d9ff8fca51274d.jpg)  
+Figure 4: Qualitative results of SRNs (without retraining) and our GRF for 50-view reconstruction of unseen cars. SRNs fails to recover faithful shapes for unseen cars without retraining.
+
+# 4.2 EXPERIMENTS ON SYNTHETIC-NERF DATASET
+
+We further evaluate our approach on the more complex dataset Synthetic-NeRF (Mildenhall et al., 2020). It consists of pathtraced images of 8 synthetic scenes with much more complicated geometry and realistic non-Lambertian materials. Each scene has 100 views for training and 200 novel views for testing. Each image has a high resolution of $8 0 0 \times 8 0 0$ pixels.
+
+Single-scene Learning. To demonstrate the representational capability of our GRF, we train a single model for each synthetic scene, following the same experimental settings of NSVF (Liu et al., 2020). Table 2 compares the quantitative results. Our method is better
+
+Table 2: The average scores of PSNR, SSIM and LPIPS (Zhang et al., 2018) for our GRF, SRNs (Sitzmann et al., 2019), NV (Lombardi et al., 2019), LLFF (Mildenhall et al., 2019), NeRF (Mildenhall et al., 2020) and NSVF (Liu et al., 2020) on the Synthetic-NeRF dataset for single-scene learning.   
+
+<table><tr><td></td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td></tr><tr><td>SRNs</td><td>22.26</td><td>0.846</td><td>0.170</td></tr><tr><td>NV</td><td>26.05</td><td>0.893</td><td>0.160</td></tr><tr><td>LLFF</td><td>24.88</td><td>0.911</td><td>0.114</td></tr><tr><td>NeRF</td><td>31.01</td><td>0.947</td><td>0.081</td></tr><tr><td>NSVF</td><td>31.74</td><td>0.953</td><td>0.047</td></tr><tr><td>GRF(Ours)</td><td>32.06</td><td>0.960</td><td>0.048</td></tr></table>
+
+Novel-scene Generalization. To evaluate the generalization of GRF on complex scenes, we train a single model simultaneously from randomly selected 4 scenes of Synthetic-NeRF, i.e., Chair, Mic, Ship, and Hotdog, and then conduct the following two groups of experiments.
+
+Group 1: The trained model is directly tested on the remaining 4 novel scenes of Synthetic-NeRF dataset, i.e., Drums, Lego, Materials, and Ficus. Fundamentally, this experiment is to evaluate whether the learned limited features can truly generalize to new scenarios. This is extremely challenging because there are only 4 training scenes and the overall shapes of the 4 novel scenes are dramatically different from the trained ones.
+
+Group 2: The trained model is further finetuned on each of the four novel scenes with 100, 1k, and 10k iterations separately. In total, we obtain $4 \times 3 = 1 2$ new models. For comparison, we also train NeRF on each of the four novel scenes with 100, 1k, 10k iterations from scratch. This group of experiments evaluates how the initially-learned general features of our GRF improve novel scene learning.
+
+Analysis. Table 3 compares the quantitative results. We can see that: 1) In Group 1, our GRF can indeed generalize to novel scenes with complex and completely different overall geometries, demonstrating the effectiveness of learning point local features in GRF. As shown in Figure 5, the overall shape and appearance of Lego can be satisfactorily recovered, though it has never been seen before. 2) In Group 2, our GRF can quickly learn high-quality scene representations given a sparse number of training pix
+
+Table 3: The average scores of PSNR, SSIM and LPIPS for GRF and NeRF on four novel scenes of Synthetic-NeRF in Group 1&2.   
+
+<table><tr><td>PSNR↑ SSIM↑LPIPS↓</td></tr><tr><td>GRF (Group 1) 13.62 0.763 0.246</td></tr><tr><td>NeRF(Group 2, 100 iters) 15.15 0.752 0.359 NeRF(Group 2,1k iters) 19.81 0.809 0.228 NeRF(Group 2,10k iters) 23.35 0.875 0.137</td></tr><tr><td>GRF (Group 2, 100 iters) 19.69 0.835 0.169 GRF (Group 2, 1k iters) 22.00 0.876 0.128 GRF (Group 2,10k iters) 25.10 0.916 0.089</td></tr></table>
+
+els, thanks to the initially learned general features. Compared with the NeRF trained from scratch, the learned GRF significantly improves novel scene learning, achieving much better results given the same amount of training signals of new scenes. Figure 5 qualitatively shows the generalization of GRF and more results are presented in appendix.
+
+![](images/2b33473f8cad83e6fc6ade9aac3c829b58988eea96b3c91c1071bdc0382ad409.jpg)  
+Figure 5: Qualitative results of our GRF for novel scene generalization.
+
+4.3 EXPERIMENTS ON REAL-WORLD COMPLEX SCENES
+
+We evaluate the novel view synthesis on complex real-world scenes captured by cellphones. There are 8 scenes, 5 from LLFF (Mildenhall et al., 2019) and 3 from NeRF. Each scene has 20 to 62 images, 1/8 of which for testing. All images have high resolution of $1 0 0 8 \times 7 5 6$ pixels.
+
+Single-scene Learning. We train a single model for each real-world scene, following the same experimental settings of NeRF. Table 4 compares the quantitative results. Note that, the recent
+
+Table 4: Comparison of the average PSNR, SSIM and LPIPS scores of our GRF, SRNs, LLFF and NeRF in the challenging real-world dataset for single-scene learning.   
+
+<table><tr><td></td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td></tr><tr><td>SRNs</td><td>22.84</td><td>0.668</td><td>0.378</td></tr><tr><td>LLFF</td><td>24.13</td><td>0.798</td><td>0.212</td></tr><tr><td>NeRF</td><td>26.50</td><td>0.811</td><td>0.250</td></tr><tr><td>GRF(Ours)</td><td>26.64</td><td>0.837</td><td>0.178</td></tr></table>
+
+NSVF (Liu et al., 2020) is unable to process these forward-facing scenes because the predefined voxels cannot represent the unbounded 3D space.
+
+Analysis. Our method surpasses the state of the art by large margins, especially over the SSIM and LPIPS metrics. Compared with PSNR which only measures the average per-pixel accuracy, the metrics SSIM and LPIPS favor high quality of photorealism, highlighting the superiority of our GRF to generate truly realistic images. Figure 6 shows the qualitative results. As highlighted by the red circles, our GRF can generate fine-grained and natural geometries in pixel level, while NeRF produces many artifacts. This clearly demonstrates our GRF indeed learns general and precise pixel features from the observed 2D images for 3D scene representation and rendering.
+
+Compared with Synthetic-NeRF, this dataset is much harder because of the complicated geometries and the inaccuracy of estimated camera poses. Therefore, it is non-trivial to conduct real-world experiments for novel-scene generalization or large-scale scene reconstruction on ScanNet (Dai et al., 2017) or “Tanks and Temples” (Knapitsch et al., 2017). We leave these for future exploration.
+
+![](images/d2dd827b02e584e3f576cc7940a4fd39cb69b473fd797cdb28c8a8cf51466e19.jpg)  
+NeRF GRF (Ours) Ground Truth Figure 6: Qualitative results on Synthetic-NeRF (1st row) and real-world datasets (2nd row).
+
+# 4.4 ABLATION STUDY
+
+To evaluate the effectiveness of the key components of our GRF, we conduct 3 groups of ablation experiments on the car category in ShapeNetv2 dataset. In particular, we train on the entire training split then randomly select 500 objects from the training split for testing. During testing, the model infers 50 novel views for each of these 500 objects for evaluation. Basically, this setting is similar to the experiment of Group 1 in Section 4.1.
+
+Table 5: The average scores of PSNR and SSIM for ablated GRF in the subset of car category.   
+
+<table><tr><td></td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>(1)Remove Input Viewpoints</td><td>20.13</td><td>0.807</td></tr><tr><td>(2) Remove Pixel Local Features</td><td>20.23</td><td>0.818</td></tr><tr><td>(3) Replace Attention by Maxpool</td><td>24.88</td><td>0.914</td></tr><tr><td>(4) The Full Model</td><td>27.16</td><td>0.942</td></tr></table>
+
+• Group 1: The viewpoints of the input images are removed from the CNN module. The CNN module is not explicitly aware of the relative position of the pixel features. • Group 2: The decoder of the CNN module is removed and each input image is encoded as a global feature vector. For any 3D query point which is rightly projected into the image boundary, that global feature vector is retrieved and reprojected to the query point. Fundamentally, this modified CNN module can be regarded as a hyper-network that learns a conditional embedding from input images and then feed it into NeRF, but it sacrifices the precise pixel local features. • Group 3: The advanced attention module is replaced by max-pooling to aggregate the pixel features, aiming to investigate how the visual occlusion can be better addressed using soft attention. • Group 4: The full model is trained and tested with the same settings for comparison.
+
+Analysis. Table 5 compares the performance of all ablated models. We can see that: 1) The greatest impact is caused by the removal of image viewpoints from the CNN module and the lack of local pixel features to represent 3D points. It highlights that obtaining the position-aware and precise pixel features is crucial to represent 3D scenes from 2D images, while learning a simple hyper-network for NeRF cannot achieve comparable performance. 2) Using max-pooling to select the reprojected pixel features is sub-optimal to address the visual occlusions from different viewing angles.
+
+To further investigate the attention module, the maximal attention scores are retrieved and visualized (detailed in appendix). It shows that the attention module can truly drive the network to focus on the visible pixel local features among the multiple intersected light rays. We also conduct additional ablation experiments (detailed in appendix), demonstrating that our GRF can indeed generalize well to novel scenes with severe visual occlusions and is robust to a variable number of input images.
+
+# 5 CONCLUSION
+
+The proposed method models complex 3D scenes as a general radiance field. We demonstrated that it can learn general and robust 3D point features from a set of 2D observations. By using the principle of multi-view geometry to precisely map 2D pixel features back to 3D space and leveraging the attention mechanism to implicitly address the visual occlusions, our GRF can synthesize truly realistic 2D novel views. However, there are still limitations that lead to the future work. (1) More advanced CNN modules can be designed to learn better pixel features. (2) Depth scans can be integrated into the network to explicitly address visual occlusions.
+
+# REFERENCES
+
+Matan Atzmon and Yaron Lipman. SAL: Sign Agnostic Learning of Shapes from Raw Data. IEEE Conference on Computer Vision and Pattern Recognition, 2020a.
+
+Matan Atzmon and Yaron Lipman. SALD: Sign Agnostic Learning with Derivatives. arXiv, 2020b.
+
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+
+# A APPENDIX
+
+A.1 DETAILS OF NETWORK ARCHITECTURE
+
+Table 6: The CNN Module for Experiments on ShapeNetv2 Dataset.   
+
+<table><tr><td>Type Size/Channels Activation Stride</td></tr><tr><td>Input: embedding of RGB and viewpoint</td></tr><tr><td>1 - 64 ReLU 2</td></tr><tr><td>Li: Conv 7 × 7 L2: Conv 3 × 3 128 ReLU 2</td></tr><tr><td>L3: Conv 3 × 3 256 ReLU 2</td></tr><tr><td></td></tr><tr><td>L4: Conv 3 × 3 512 ReLU 2</td></tr><tr><td>L5: Conv 4 × 4 128 ReLU 4</td></tr><tr><td>L6: Flatten /Tile - 1</td></tr><tr><td>L7: Concat (L6,L4) - 1</td></tr><tr><td>L8:Dilated Conv 3×3 256 ReLU 4</td></tr><tr><td>L8: Concat (L8,L3) 1 1 =</td></tr><tr><td>L9:Dilated Conv 3 × 3 128 ReLU 8</td></tr><tr><td>L9: Concat (L9,L2) = 一 =</td></tr><tr><td>L10: Dilated Conv 3 × 3 64 ReLU 16</td></tr><tr><td>L10: Concat (L10, L1) - ■ = L11: Dilated Conv 3 × 3 128 ReLU 32</td></tr></table>
+
+Table 7: The CNN Module for Experiments on Synthetic-NeRF Dataset. The average pooling is added to aggressively downsample the feature maps.   
+
+<table><tr><td>Type Size/Channels Activation Stride</td></tr><tr><td>Input: embedding of RGB and viewpoint - -</td></tr><tr><td>L1: Conv 7 × 7 64 ReLU 2</td></tr><tr><td>L2: Conv 3 × 3 128 ReLU 2</td></tr><tr><td>L3: Conv 3 × 3 256 ReLU 2</td></tr><tr><td>L4: Conv 3 × 3 512 ReLU 2</td></tr><tr><td>L4: AveragePooling 5 × 5</td></tr><tr><td>= =</td></tr><tr><td>L5: Conv 5 × 5 128 ReLU 5</td></tr><tr><td>L6:Flatten /Tile - 1</td></tr><tr><td>L7: Concat (L6,L4) 1 1</td></tr><tr><td>L8: Deconv 3× 3 256 ReLU 2</td></tr><tr><td>L8: Concat (L8, L3) 1 1</td></tr><tr><td>L9: Deonv 3 × 3 128 ReLU 2</td></tr><tr><td>L9: Concat (L9,L2) 1 1</td></tr><tr><td>L10: Deconv 3×3 64 ReLU 2</td></tr><tr><td>L10: Concat (L10,L1) - 1 L11: Deconv 3 × 3 128 ReLU 2</td></tr></table>
+
+Table 8: The CNN Module for Experiments on the real-world Dataset. The average pooling is added to aggressively downsample the feature maps.   
+
+<table><tr><td>Type Size/Channels Activation Stride</td></tr><tr><td>Input: embedding of RGB and viewpoint 1 1</td></tr><tr><td>L1: Conv 7 × 7 64 ReLU</td></tr><tr><td>22 L2: Conv 3 × 3 128 ReLU</td></tr><tr><td>L3: Conv 3 × 3 256 ReLU 2</td></tr><tr><td>L4: Conv 3 × 3 512 ReLU 2</td></tr><tr><td></td></tr><tr><td>L4: AveragePooling 8 × 8 1 =</td></tr><tr><td>L5: Conv 4 × 4 128 ReLU 4</td></tr><tr><td>L6: Flatten /Tile 1 1</td></tr><tr><td>L7: Concat (L6,L4) 1 1</td></tr><tr><td>L8: Deconv 3 × 3 256 ReLU 2</td></tr><tr><td>L8: Concat (L8,L3) - 1</td></tr><tr><td>L9: Deconv 3 × 3 128 ReLU</td></tr><tr><td>L9: Concat (L9,L2) 1 =</td></tr><tr><td>L10: Deconv 3×3 64 ReLU 2</td></tr><tr><td>L10: Concat (L10, L1) - =</td></tr><tr><td>L11: Deconv 3 × 3 128 ReLU 2</td></tr></table>
+
+Table 9: The Attention Module, AttSets (Yang et al., 2020), for Experiments on the ShapeNetv2 Dataset. The simple AttSets is computationally efficient and we choose it to train the large-scale ShapeNetv2 dataset.   
+
+<table><tr><td>Type</td><td>Size/Channels Activation</td><td></td></tr><tr><td>Input: Concat(K × 128,embedding of viewpoint)</td><td>=</td><td>=</td></tr><tr><td>L1: fc</td><td>256</td><td>ReLU</td></tr><tr><td>L2: fc</td><td>256</td><td>ReLU</td></tr><tr><td>L3: fc</td><td>256</td><td>ReLU</td></tr><tr><td>L4: fc</td><td>512</td><td>ReLU</td></tr><tr><td>L5: fc</td><td>512</td><td>ReLU</td></tr><tr><td>L6: softmax(L5)</td><td>1</td><td>1</td></tr><tr><td>L7: sum(L6*L5,axis=-2)</td><td>1</td><td>1</td></tr><tr><td>L8: fc</td><td>512</td><td>ReLU</td></tr></table>
+
+We use Slot Attention as the pixel feature aggregation module for experiments on the SyntheticNeRF and real-world datasets. In particular, we use two slots, two iterations, and the hidden size is 128. The final output two slots are flattened and a 256 dimensional vector is obtained. For details of Slot Attention refer to the paper (Locatello et al., 2020).
+
+Details of the neural rendering layers and the volume rendering can be found in NeRF (Mildenhall et al., 2020). We set the positional embedding length $L = 5$ for all inputs to the CNN module, except the rotation, which we convert to quaternion and embed at $L = 4$ .
+
+During training, we feed the models between 2 and 6 views of each scene at each gradient step. We set the learning rate for the ShapeNetv2 models at 1e-4. We set the learning rate for leaves and orchids in the real-world dataset at 7e-5, and for the rest, we use 1e-4. For Synthetic-NeRF dataset, we use a learning rate of 1e-4. We use the Adam optimizer for all models, and train for $2 0 0 \mathbf { k } { - } 3 0 0 \mathbf { k }$ iterations. At each gradient step, we take 1000 rays for ShapeNetv2 with 32 coarse samples and 64 fine samples, and 800 rays for the real-world and Synthetic-NeRF datasets with 64 coarse samples and 192 fine samples. We train each model on a single Nvidia-V100 GPU with 32GB VRAM.
+
+During testing for the ShapeNetv2 reconstruction from 50 images, we feed the model the 5 closest views to the desired novel view by cosine similarity from the 50 training images. During testing for single-scene reconstruction, we feed the model the 4 closest views by cosine similarity to the desired novel view.
+
+# A.2 DETAILS OF EXPERIMENTAL RESULTS ON THE SYNTHETIC-NERF DATASET
+
+Table 10: Comparison of the PSNR (in dB), SSIM and LPIPS (Zhang et al., 2018) scores of our GRF, SRNs (Sitzmann et al., 2019), NV (Lombardi et al., 2019), NeRF (Mildenhall et al., 2020) and NSVF (Liu et al., 2020) in the Synthetic-NeRF dataset for single-scene learning.   
+
+<table><tr><td></td><td>Chair</td><td>Drums</td><td>Lego</td><td>Mic</td><td>Materials</td><td>Ship</td><td>Hotdog</td><td>Ficus</td><td>Mean</td></tr><tr><td colspan="10">PSNR↑</td></tr><tr><td>SRNs</td><td>26.96</td><td>17.18</td><td>20.85</td><td>26.85</td><td>18.09</td><td>20.60</td><td>26.81</td><td>20.73</td><td>22.26</td></tr><tr><td>NV</td><td>28.33</td><td>22.58</td><td>26.08</td><td>27.78</td><td>24.22</td><td>23.93</td><td>30.71</td><td>24.79</td><td>26.05</td></tr><tr><td>NeRF</td><td>33.00</td><td>25.01</td><td>32.54</td><td>32.91</td><td>29.62</td><td>28.65</td><td>36.18</td><td>30.13</td><td>31.01</td></tr><tr><td>NSVF</td><td>33.19</td><td>25.18</td><td>32.29</td><td>34.27</td><td>32.68</td><td>27.93</td><td>37.14</td><td>31.23</td><td>31.74</td></tr><tr><td>GRF(Ours)</td><td>34.51</td><td>25.83</td><td>32.92</td><td>33.94</td><td>30.91</td><td>30.12</td><td>37.47</td><td>30.75</td><td>32.06</td></tr><tr><td colspan="10">SSIM↑</td></tr><tr><td>SRNs</td><td>0.910</td><td>0.766</td><td>0.809</td><td>0.947</td><td>0.808</td><td>0.757</td><td>0.923</td><td>0.849</td><td>0.846</td></tr><tr><td>NV</td><td>0.916</td><td>0.873</td><td>0.880</td><td>0.946</td><td>0.888</td><td>0.784</td><td>0.944</td><td>0.910</td><td>0.893</td></tr><tr><td>NeRF</td><td>0.967</td><td>0.925</td><td>0.961</td><td>0.980</td><td>0.949</td><td>0.856</td><td>0.974</td><td>0.964</td><td>0.947</td></tr><tr><td>NSVF</td><td>0.968</td><td>0.931</td><td>0.960</td><td>0.987</td><td>0.973</td><td>0.854</td><td>0.980</td><td>0.973</td><td>0.953</td></tr><tr><td>GRF(Ours)</td><td>0.981</td><td>0.937</td><td>0.967</td><td>0.987</td><td>0.963</td><td>0.891</td><td>0.983</td><td>0.969</td><td>0.960</td></tr><tr><td colspan="10">LPIPS↓</td></tr><tr><td>SRNs</td><td>0.106</td><td>0.267</td><td>0.200</td><td>0.063</td><td>0.174</td><td>0.299</td><td>0.100</td><td>0.149</td><td>0.170</td></tr><tr><td>NV</td><td>0.109</td><td>0.214</td><td>0.175</td><td>0.107</td><td>0.130</td><td>0.276</td><td>0.109</td><td>0.162</td><td>0.160</td></tr><tr><td>NeRF</td><td>0.046</td><td>0.091</td><td>0.050</td><td>0.028</td><td>0.063</td><td>0.206</td><td>0.121</td><td>0.044</td><td>0.081</td></tr><tr><td>NSVF</td><td>0.043</td><td>0.069</td><td>0.029</td><td>0.010</td><td>0.021</td><td>0.162</td><td>0.025</td><td>0.017</td><td>0.047</td></tr><tr><td>GRF(Ours)</td><td>0.021</td><td>0.068</td><td>0.042</td><td>0.013</td><td>0.041</td><td>0.141</td><td>0.028</td><td>0.032</td><td>0.048</td></tr></table>
+
+Table 11: The PSNR, SSIM and LPIPS scores of our GRF simultaneously trained on 4 scenes of the Synthetic-NeRF dataset for multi-scene learning. The scores of SRNs, NeRF and NSVF trained on single scenes are included for comparison.   
+
+<table><tr><td></td><td>Chair</td><td>Mic</td><td>Ship</td><td>Hotdog</td></tr><tr><td colspan="5">PSNR↑</td></tr><tr><td>SRNs (Single-scene)</td><td>26.96</td><td>26.85</td><td>20.60</td><td>26.81</td></tr><tr><td>NeRF (Single-scene)</td><td>33.00</td><td>32.91</td><td>28.65</td><td>36.18</td></tr><tr><td>NSVF (Single-scene)</td><td>33.19</td><td>34.27</td><td>27.93</td><td>37.14</td></tr><tr><td>GRF (Multi-scene)</td><td>32.49</td><td>32.02</td><td>27.76</td><td>34.92</td></tr><tr><td colspan="5">SSIM↑</td></tr><tr><td>SRNs (Single-scene)</td><td>0.910</td><td>0.947</td><td>0.757</td><td>0.923</td></tr><tr><td>NeRF (Single-scene)</td><td>0.967</td><td>0.980</td><td>0.856</td><td>0.974</td></tr><tr><td>NSVF (Single-scene)</td><td>0.968</td><td>0.987</td><td>0.854</td><td>0.980</td></tr><tr><td>GRF (Multi-scene)</td><td>0.971</td><td>0.982</td><td>0.866</td><td>0.975</td></tr><tr><td colspan="5">LPIPS↓</td></tr><tr><td>SRNs (Single-scene)</td><td>0.106</td><td>0.063</td><td>0.299</td><td>0.100</td></tr><tr><td>NeRF (Single-scene)</td><td>0.046</td><td>0.028</td><td>0.206</td><td>0.121</td></tr><tr><td>NSVF (Single-scene)</td><td>0.043</td><td>0.010</td><td>0.162</td><td>0.025</td></tr><tr><td>GRF (Multi-scene)</td><td>0.032</td><td>0.019</td><td>0.167</td><td>0.040</td></tr></table>
+
+Table 12: The PSNR, SSIM and LPIPS scores of our GRF and NeRF on four novel scenes of Synthetic-NeRF in Group 1&2 experiments.   
+
+<table><tr><td colspan="4">Drums</td></tr><tr><td></td><td>Lego PSNR↑</td><td>Materials</td><td>Ficus</td></tr><tr><td colspan="2"></td></tr><tr><td>GRF (Group 1) 13.23 13.53</td><td>12.26 15.47 15.42 15.72</td></tr><tr><td>NeRF( F(Group2,100 iters) 14.54 14.92</td><td>20.04 20.40 20.81</td></tr><tr><td>NeRF( (Group2,1k iters) 18.01</td><td>19.81 23.36 23.47</td></tr><tr><td>NeRF (Group 2,10k iters) 21.57 24.99</td><td>23.35</td></tr><tr><td>GRF (Group 2,100 iters) 18.70 20.24</td><td>18.81 21.03 19.69</td></tr><tr><td>GRF (Group2,1k iters) 20.49 23.64</td><td>21.87 22.02 22.00</td></tr><tr><td>GRF (Group2,10k iters) 23.11 27.07</td><td>25.11 25.11 25.10</td></tr><tr><td colspan="2">SSIM↑</td></tr><tr><td>F (Group 1) 0.762</td><td>0.736 0.703 0.849</td></tr><tr><td>NeRF(Group 2,100 iters)</td><td>0.769 0.717 0.716</td></tr><tr><td>NeRF(Group 2,1k iters) 0.793 0.775</td><td>0.812 0.857 0.809 0.877 0.896 0.875</td></tr><tr><td>NeRF F(Group2,10k iters) 0.865</td><td>0.862</td></tr><tr><td>GRF(Group 2,100 iters) 0.822 0.813</td><td>0.829 0.878 0.835</td></tr><tr><td>GRF (Group2,1k iters) 0.856</td><td>0.877 0.878 0.894 0.876 0.916</td></tr><tr><td>GRF F(Group2,10k iters) 0.901</td><td>0.924 0.913 0.923</td></tr><tr><td colspan="2">LPIPS↓</td></tr><tr><td></td><td>0.256 0.273 0.301 0.150</td></tr><tr><td>GRF (Group 1) NeRF(Group 2,100 iters)</td><td>0.332 0.395 0.314</td></tr><tr><td>NeRF(Group 2,1k iters) 0.254 0.264</td><td>0.393 0.359 0.229 0.164 0.228 0.137</td></tr><tr><td>NeRF (Group2,10k iters) 0.157</td><td>0.154 0.128 0.110</td></tr><tr><td>GRF (Group 2,100 iters) 0.196</td><td>0.203 0.159 0.117 0.169 0.128</td></tr><tr><td>GRF F(Group2,1k iters) 0.154 GRF (Group 2,10k iters) 0.104</td><td>0.138 0.123 0.097 0.090 0.090 0.071</td></tr></table>
+
+A.3 DETAILS OF EXPERIMENTAL RESULTS ON THE REAL-WORLD DATASET
+
+Table 13: Comparison of the PSNR, SSIM and LPIPS scores of our GRF, SRNs (Sitzmann et al., 2019), LLFF (Mildenhall et al., 2019) and NeRF (Mildenhall et al., 2020) in the real-world dataset for single-scene learning.   
+
+<table><tr><td></td><td>Room</td><td>Fern</td><td>Leaves</td><td>Fortress</td><td>Orchids</td><td>Flower</td><td>T-Rex</td><td>Horns</td><td>Mean</td></tr><tr><td colspan="10">PSNR↑</td></tr><tr><td>SRNs</td><td>27.29</td><td>21.37</td><td>18.24</td><td>26.63</td><td>17.37</td><td>24.63</td><td>22.87</td><td>24.33</td><td>22.84</td></tr><tr><td>LLFF</td><td>28.42</td><td>22.85</td><td>19.52</td><td>29.40</td><td>18.52</td><td>25.46</td><td>24.15</td><td>24.70</td><td>24.13</td></tr><tr><td>NeRF</td><td>32.70</td><td>25.17</td><td>20.92</td><td>31.16</td><td>20.36</td><td>27.40</td><td>26.80</td><td>27.45</td><td>26.50</td></tr><tr><td>GRF(Ours)</td><td>31.74</td><td>25.72</td><td>21.16</td><td>31.28</td><td>20.88</td><td>27.83</td><td>27.01</td><td>27.50</td><td>26.64</td></tr><tr><td colspan="10">SSIM↑</td></tr><tr><td>SRNs</td><td>0.883</td><td>0.611</td><td>0.520</td><td>0.641</td><td>0.449</td><td>0.738</td><td>0.761</td><td>0.742</td><td>0.668</td></tr><tr><td>LLFF</td><td>0.932</td><td>0.753</td><td>0.697</td><td>0.872</td><td>0.588</td><td>0.844</td><td>0.857</td><td>0.840</td><td>0.798</td></tr><tr><td>NeRF</td><td>0.948</td><td>0.792</td><td>0.690</td><td>0.881</td><td>0.641</td><td>0.827</td><td>0.880</td><td>0.828</td><td>0.811</td></tr><tr><td>GRF(Ours)</td><td>0.951</td><td>0.827</td><td>0.727</td><td>0.898</td><td>0.667</td><td>0.852</td><td>0.901</td><td>0.873</td><td>0.837</td></tr><tr><td colspan="10">LPIPS↓</td></tr><tr><td>SRNs</td><td>0.240</td><td>0.459</td><td>0.440</td><td>0.453</td><td>0.467</td><td>0.288</td><td>0.298</td><td>0.376</td><td>0.378</td></tr><tr><td>LLFF</td><td>0.155</td><td>0.247</td><td>0.216</td><td>0.173</td><td>0.313</td><td>0.174</td><td>0.222</td><td>0.193</td><td>0.212</td></tr><tr><td>NeRF</td><td>0.178</td><td>0.280</td><td>0.316</td><td>0.171</td><td>0.321</td><td>0.219</td><td>0.249</td><td>0.268</td><td>0.250</td></tr><tr><td>GRF(Ours)</td><td>0.104</td><td>0.191</td><td>0.238</td><td>0.127</td><td>0.275</td><td>0.176</td><td>0.146</td><td>0.169</td><td>0.178</td></tr></table>
+
+# A.4 ANALYSIS OF ATTENTION MECHANISM(Generalizat
+
+The attention mechanism in our GRF aims to automatically select the correct pixel patch from multiple pixel patches where the light rays intersect at the same query 3D point in space.
+
+In order to investigate how the attention mechanism learns to select the useful information, we retrieve the maximal attention score from the observed multiple pixel patches for analysis. Intuitively, the higher the attention score is assigned to a particular pixel patch, the more important that patch for inferring the novel pixel RGB. In particular, we conduct the following experiment using our GRF model trained on ShapeNetv2 Cars. In this case, the AttSets attention module is used (details are in Table 9). Given a query light ray, multiple 3D points are sampled to query the network.
+
+We firstly try to find the 3D point which is near the surface according to the predicted volume density for points along the ray through a given pixel, if they exist. Otherwise, we ignore such pixels, making them white.   
+Then we compute the M feature vectors from the input M views for these surface points. Thirdly, the attention masks for those M feature vectors are computed. We identify the view whose sum of the attention mask along the feature axis is greatest as the main contributor for inferring the novel pixel RGB.   
+After querying light rays for each pixel, we obtain a rendered RGB image. At the same time, for each pixel of that image, we select the most important view from the M input views for the surfaceintersection point along the ray from the viewpoint through that pixel, according to the maximal attention score. Eventually, we obtain a Max Attention Map corresponding to the rendered RGB image.
+
+Figure 7 shows the qualitative results of the above experiment. In particular, we feed the three images (#1,#2,#3) of an unseen car into our GRF model which is well-trained on car category, and then render a new image (e.g., the 5th image in Figure 7). Note that, we carefully select the input 3 images and the rendered image with very large viewing baselines. In the mean time, we obtain and visualize the Max Attention Map corresponding to the rendered image.
+
+For each pixel of the rendered image, we retrieve the input image pixel that has the highest attention score. Specifically, the rendered pixels with purple color correspond to the input image #1, the green pixels correspond to the input image #2, while the blue pixels correspond to the input image #3.
+
+Analysis. It can be seen that, when inferring a new image, the attention module of our GRF focuses on the most informative pixel patch from the multiple input pixel patches. In addition, it is able to truly deal with the visual occlusion. For example, when inferring the windshield of the car, the attention module focuses on the input image #2 where the windshield is visible, while ignoring the image #1 and #3 where the windshield is self-occluded.
+
+![](images/20f93eecb76196ee8f164adc09b398e9f5dbb82c10a34c1453af75819e8f5736.jpg)  
+Figure 7: Visualization of Max Attention Map from Multiple Input Images of a Novel Object for Inferring a Novel View.
+
+A.5 GENERALIZATION TO VISUAL OCCLUSIONS AND VARIABLE INPUT IMAGES
+
+We carefully select the attention module, i.e., either AttSets or SlotAtt, to aggregate the features from an arbitrary number of input views. In order to evaluate how our GRF is able to generalize with a variable number of input views, especially when there is a very sparse number of views with severe visual occlusions, we conduct the following four groups of experiments.
+
+• 1-view Reconstruction. We feed the a single image of a novel car into our GRF model which is well-trained on car category (trained with 5 images per object), and then render 9 new images from vastly different viewing angles. This is the extreme case where the majority of the object is self-occluded.
+
+• 2-view Reconstruction. Similarly, we feed only two images of the novel car into the same model and render the same 9 novel views. In this case, more information is given to the network, but there are still many parts occluded.
+
+• 5-view / 10-view Reconstruction. The same GRF model is fed with 5 and 10 views of the novel object, rendering the same set of new images.
+
+Analysis. Figure 8 shows the qualitative results. It can be seen that: 1) In the extreme case, i.e., 1-view reconstruction, our GRF is still able to recover the general 3D shape of the unseen object, including the visually occluded parts, primarily because our CNN model learns the hierarchical features including the high-level shapes. 2) Given more input views, the originally occluded parts tend to be observed from some viewing angles, and then these parts can be reconstructed better and better. This shows that our GRF is indeed able to effectively identify the corresponding useful pixel features for more accurately recovering shape and appearance.
+
+![](images/452fb2908ba7c814b8da50727c0309cbd16427f8d4f7072fd5be8f665d791d99.jpg)  
+Figure 8: Qualitative results of our GRF when being fed with a variable number of views of a novel object. The red circle highlights that the tail of the car is able to be recovered given more visual cues from more input images.
+
+![](images/5e585a9c6f55e0f762c826bccc3ad70554c0e9a9023da31e86706efd23ce1569.jpg)  
+Figure 9: Qualitative results of SRNs and our GRF for 50-view reconstruction of chairs and cars.
+
+![](images/720b1b3f0a109fae40aade5f250bbe18fc1e9ff6bc9faf69eda08923c27ddadc.jpg)  
+Figure 10: Qualitative results of our GRF for novel view depth and RGB estimation on the SyntheticNeRF dataset.
+
+![](images/cadef3a36c5e380d2f99cc5b827f2e4ca75d2c8175fcc3a245639772f579210d.jpg)  
+Figure 11: Qualitative results of our GRF for novel view depth and RGB estimation on the SyntheticNeRF dataset.
+
+![](images/453a93119d60769e89c2b0a65f9b43d42efc589faba79975733f9e58b7782c58.jpg)  
+Figure 12: Qualitative results of our GRF for novel view depth and RGB estimation on the SyntheticNeRF dataset.
+
+![](images/beeb4046890bf5004b0f81769e12fa68b175541b587410c51e428be01d998227.jpg)  
+Figure 13: Qualitative results of our GRF for novel view depth and RGB estimation on the SyntheticNeRF dataset.
+
+![](images/f8157646fde3d77d5203c0f15aa9ddd82767ee191425ac2129cfeb332f153e56.jpg)  
+Figure 14: Qualitative results of our GRF for novel view depth and RGB estimation on the real-world dataset.
+
+![](images/88f1544d6d632c9c8a99ef63ec61bd9fd14e5db337eb9260544d7c69beee1ad9.jpg)  
+Figure 15: Qualitative results of our GRF for novel view depth and RGB estimation on the real-world dataset.
+
+![](images/caf80e9622eb47880990dde4703304c939c21246832958437a9bef0ab4ca34d0.jpg)  
+Figure 16: Qualitative results of our GRF for novel view depth and RGB estimation on the real-world dataset.
+
+![](images/e60781d9d56ccf57a254173588c4ff5552b76267158d5a3703af27473667497f.jpg)  
+Figure 17: Qualitative results of our GRF for novel view depth and RGB estimation on the real-world dataset.
\ No newline at end of file
diff --git a/parse/train/cAvgPMAA3hb/cAvgPMAA3hb_content_list.json b/parse/train/cAvgPMAA3hb/cAvgPMAA3hb_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..6f9593425573b9d584febaabebb074eba3654a0a
--- /dev/null
+++ b/parse/train/cAvgPMAA3hb/cAvgPMAA3hb_content_list.json
@@ -0,0 +1,2177 @@
+[
+    {
+        "type": "text",
+        "text": "GRF: LEARNING A GENERAL RADIANCE FIELD FOR 3D SCENE REPRESENTATION AND RENDERING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            821,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            184,
+            170,
+            400,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We present a simple yet powerful implicit neural function that can represent and render arbitrarily complex 3D scenes in a single network only from 2D observations. The function models 3D scenes as a general radiance field, which takes a set of 2D images with camera poses and intrinsics as input, constructs an internal representation for each 3D point of the scene, and renders the corresponding appearance and geometry of any 3D point viewing from an arbitrary angle. The key to our approach is to explicitly integrate the principle of multi-view geometry to obtain the internal representations from observed 2D views, such that the learned implicit representations empirically remain multi-view consistent. In addition, we introduce an effective neural module to learn general features for each pixel in 2D images, allowing the constructed internal 3D representations to be general as well. Extensive experiments demonstrate the superiority of our approach. ",
+        "bbox": [
+            233,
+            260,
+            764,
+            428
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            455,
+            336,
+            472
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Understanding the precise 3D structure of a real-world environment and realistically re-rendering it from free viewpoints is a key enabler for many critical tasks, ranging from robotic manipulation and navigation to augmented reality. Classic approaches to recover the 3D scene geometry mainly include the structure from motion (SfM) (Ozyesil et al., 2017) and simultaneous localization and mapping (SLAM) (Cadena et al., 2016) pipelines. However, they are limited to reconstruct sparse and discrete 3D point clouds which are unable to contain geometric details. ",
+        "bbox": [
+            174,
+            491,
+            825,
+            574
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The recent advances in deep neural networks have yielded rapid progress in 3D modeling and understanding. Most of them focus on the explicit 3D shape representations such as voxel grids (Choy et al., 2016), point clouds (Fan et al., 2017), or triangle meshes (Wang et al., 2018). However, these representations are discrete and sparse, limiting the recovered 3D structures to extremely low spatial resolution. In addition, these networks usually require large-scale 3D shapes for supervision, resulting in the trained models being over-fitting particular datasets and unable to generalize to novel scenes. In fact, it is also costly and even infeasible to collect high-quality 3D labels. ",
+        "bbox": [
+            174,
+            582,
+            825,
+            679
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Encoding geometries into multilayer perceptrons (MLPs) (Mescheder et al., 2019; Park et al., 2019) recently emerges as a promising direction in 3D reconstruction and understanding from 2D images. Its key advantage is the ability to model 3D structures in a continuous way instead of discrete, and therefore it has the potential to achieve unlimited spatial resolution in theory. However, most methods of this pipeline focus on individual objects. In addition, many of them require 3D geometry for supervision to learn the 3D shapes from images. By introducing a recurrent neural network based renderer, SRNs (Sitzmann et al., 2019) is among the early work to learn implicit surface representations only from 2D images, but it fails to capture complicated scene geometries and renders over-smoothed images. Alternatively, by leveraging the volume rendering to synthesize new views for 2D supervision, the very recent NeRF (Mildenhall et al., 2020) directly encodes the radiance fields of complex 3D scenes within the weights of MLPs, achieving an unprecedented level of fidelity for challenging 3D scenes. Nevertheless, it has two major limitations: 1) since each 3D scene is encoded into all the weights of MLPs, the trained network (i.e., a learned radiance field) can only represent a single scene, and is unable to generalize across novel scenarios; 2) because the shape and appearance of each spatial 3D location along a light ray is only optimized by the available pixel RGBs, the learned implicit representations of that location are lack of the general geometric patterns, resulting in the synthesized images being less photo-realistic. ",
+        "bbox": [
+            173,
+            685,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a general radiance field (GRF), a simple yet powerful implicit function that can represent and render complex 3D scenes where there may be multiple objects with cluttered background. Our GRF takes a set of 2D images with camera poses and intrinsics, a 3D query point and its query viewpoint (i.e., the camera location $x y z )$ as input, and predicts the RGB value and volumetric density of that query point. Basically, this neural function learns to represent a 3D scene from sparse 2D observations, and infers the shape and appearance of that scene from previously unobserved viewing angles. Note that, the inferred shape and appearance of any particular 3D query point explicitly take into account its local geometric patterns from the available 2D observations. In particular, the proposed GRF consists of four components: ",
+        "bbox": [
+            173,
+            103,
+            825,
+            233
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Extracting the general 2D visual features for every light ray from the input 2D observations;   \n• Reprojecting the corresponding 2D features back to the query 3D point using the principle of multi-view geometry;   \n• Selecting and aggregating all the reprojected features for the query 3D point, while the visual occlusions are implicitly considered;   \n• Rendering the aggregated features of the query 3D point along a particular query viewpoint, and producing the corresponding RGB and volumetric density. ",
+        "bbox": [
+            173,
+            238,
+            825,
+            348
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "These four components enable our GRF to be significantly different from all existing 3D scene representation approaches. 1) Compared with the classic SfM/SLAM systems, our GRF can represent the 3D scene structure with smooth and continuous surfaces. 2) Compared with the neural approaches based on explicit voxel grids, point clouds and meshes, our GRF learns continuous 3D representations without requiring 3D data for training. 3) Compared with the existing implicit representation methods such as SDF (Park et al., 2019), SRNs (Sitzmann et al., 2019) and NeRF (Mildenhall et al., 2020), our GRF can represent arbitrarily complicated 3D scenes and has remarkable generalization to novel scenarios. In addition, the learned 3D representations carefully consider the general geometric patterns for every 3D spatial location, allowing the rendered views to be exceptionally realistic with fine-grained details. Our key contributions are: ",
+        "bbox": [
+            173,
+            354,
+            825,
+            493
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We propose a general radiance field to implicitly represent the 3D scene structure and appearance purely from 2D images. It has remarkable generalization to novel scenes in a single forward pass. • We explicitly integrate the principle of multi-view geometry to learn geometric details for each 3D query point along every query light ray. This allows the synthesized 2D views to be superior. • We demonstrate significant improvement over baselines on three large-scale datasets and provide intuition behind our design choices through extensive ablation studies. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            617,
+            344,
+            633
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Classic Multi-view Geometry. Classic approaches to reconstruct 3D geometry from images mainly include SfM and SLAM systems, which firstly extract and match hand-crafted geometric local features and then apply bundle adjustment for both shape and camera motion estimation (Hartley & Zisserman, 2004). Although they can recover visually satisfactory 3D models, the reconstructed shapes are usually sparse, discrete point clouds. In contrast, our GRF learns an implicit function to represent the continuous 3D structures from images. Notably, however, the principle of classic multi-view geometry is explicitly integrated into our GRF to learn accurate and general features for every spatial location. ",
+        "bbox": [
+            173,
+            652,
+            825,
+            763
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Geometric Deep Learning. The recent advances of deep neural nets have yielded impressive progress in recovering explicit 3D shapes from either single or multiple images, including the voxel grid (Choy et al., 2016; Yang et al., 2019), octree (Riegler et al., 2017; Christian et al., 2017), point cloud (Fan et al., 2017; Qi et al., 2017) and triangle mesh (Wang et al., 2018; Groueix et al., 2018; Nash et al., 2020) based approaches. However, most of these methods only focus on individual 3D objects, whilst only few pipelines (Song et al., 2017; Tulsiani et al., 2018; Gkioxari et al., 2019) attempt to learn the structures for complex 3D scenes. Although these neural networks can predict realistic 3D structures of objects and scenes, they have two limitations. First, almost all of them require ground truth 3D labels to supervise the networks, resulting in the learned representations being unable to generalize to novel real-world scenes. Second, since the recovered 3D shapes are discrete, they are unable to preserve high-resolution geometric details. Being quite different, our GRF learns to infer continuous 3D shapes only from a set of 2D images with camera poses and intrinsics, which can be cheaply acquired and also allow better generalization across real-world scenes. ",
+        "bbox": [
+            173,
+            770,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Neural Implicit 3D Representations. The implicit representation of 3D shapes recently emerges as a promising direction to recover 3D geometries. It is initially formulated as level sets by optimizing neural nets which map xyz locations to an occupancy field (Mescheder et al., 2019; Saito et al., 2019) or a distance function (Park et al., 2019). The subsequent work (Niemeyer et al., 2020; Sitzmann et al., 2019; Liu et al., 2019b; Atzmon & Lipman, 2020a;b) introduce differentiable rendering functions, allowing 2D images or raw 3D point clouds to supervise the networks, but only producing decent results on individual objects. Using neural radiance fields instead, the latest NeRF (Mildenhall et al., 2020) and the succeeding NSVF (Liu et al., 2020), NeRF-wild (Martin-Brualla et al., 2020), GRAF (Schwarz et al., 2020) demonstrate impressive results to represent complex 3D scenes. However, they do not take into account the local geometric patterns for spatial locations, thereby the rendered 2D images being less realistic and sub-optimal. Additionally, both NeRF and NeRF-wild can only represent a single scene, and are unable to generalize to novel scenarios. Uniquely, our GRF maps any set of images to the corresponding 3D scene structure with geometric details. ",
+        "bbox": [
+            173,
+            138,
+            825,
+            335
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Novel View Synthesis and Neural Rendering. Novel view synthesis involves generating unseen views of a scene captured by multiple images. Existing methods usually learn an embedding of a scene and then estimate a new image given a viewing angle, including GAN based methods (Goodfellow et al., 2014; Radford et al., 2016), variational auto-encoders (Kingma & Welling, 2014), autoregressive models (Oord et al., 2016), and other generative frameworks (Eslami et al., 2018). Although photo-realistic single images can be generated, these methods tend to learn the manifold of 2D images, instead of exploiting the underlying 3D geometry for consistent multi-view synthesis. ",
+        "bbox": [
+            173,
+            343,
+            825,
+            441
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Neural rendering techniques (Fried et al., 2020; Kato et al., 2020) have recently been actively investigated and integrated into 3D reconstruction pipelines, where there are no ground truth 3D data available but only 2D images for supervision. To render the discrete voxel grids (Yan et al., 2016; Rezende et al., 2016; Tulsiani et al., 2017), point clouds (Insafutdinov & Dosovitskiy, 2018), meshes (Kato et al., 2018; Chen et al., 2019; Liu et al., 2019a), and implicit surfaces (Liu et al., 2019b; Remelli et al., 2020), most of these techniques are designed with differentiable and approximate functions, but sacrifice the sharpness of synthesized images and the accuracy of estimated 3D shapes. By contrast, our GRF leverages the existing volume rendering method (Mildenhall et al., 2020) which is naturally differentiable and can accurately render the RGB per light ray. ",
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+    {
+        "type": "text",
+        "text": "3 GRF ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 OVERVIEW ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Our GRF models the complex 3D scenes including the geometry and appearance as a neural function $f _ { \\mathbf { W } }$ where W represent learnable parameters. This function takes a set of $M$ images together with their camera poses and intrinsics $\\{ ( \\mathcal { T } _ { 1 } , \\pmb { \\xi } _ { 1 } , \\pmb { K } _ { 1 } ) \\cdot \\cdot \\cdot ( \\mathcal { T } _ { m } , \\pmb { \\xi } _ { m } , \\pmb { K } _ { m } ) \\cdot \\cdot \\cdot ( \\mathcal { T } _ { M } , \\pmb { \\xi } _ { M } , \\pmb { K } _ { M } ) \\} _ { \\{ \\pmb { \\xi } _ { 1 } , \\pmb { \\xi } _ { 1 } , \\pmb { K } _ { 1 } \\} }$ , which are sampled from a complex 3D scene, a query 3D point $p = \\{ x _ { p } , y _ { p } , z _ { p } \\}$ , and its query viewpoint $\\mathcal { V } _ { p } = \\{ x _ { p } ^ { v } , y _ { p } ^ { v } , z _ { p } ^ { v } \\}$ as input, and then predicts the RGB value $\\{ r _ { p } , g _ { p } , b _ { p } \\}$ and the volumetric density $d _ { p }$ of that query 3D point $p$ observed from the query viewpoint $\\gamma _ { p }$ . Formally, it is defined as below: ",
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/5d3c6c72c009754a82db30af933402975893bbbecaaa520ff95b6877dfab572f.jpg",
+        "text": "$$\n( r _ { p } , g _ { p } , b _ { p } , d _ { p } ) = f _ { \\mathbf { W } } \\left( \\Big \\{ ( \\mathcal { T } _ { 1 } , \\pmb { \\xi } _ { 1 } , K _ { 1 } ) \\cdot \\cdot \\cdot ( \\mathcal { T } _ { M } , \\pmb { \\xi } _ { M } , K _ { M } ) \\Big \\} , \\{ x _ { p } , y _ { p } , z _ { p } \\} , \\{ x _ { p } ^ { v } , y _ { p } ^ { v } , z _ { p } ^ { v } \\} \\right)\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Basically, this function is a general radiance field (GRF) which parameterizes arbitrary 3D scenes observed by the input $M$ images. In the meantime, it returns both the appearance and geometry, when being queried at any location $p$ from any viewpoint $\\nu$ in 3D space. ",
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+    },
+    {
+        "type": "text",
+        "text": "As illustrated in Figure 1, the proposed GRF firstly extracts general features for each light ray hitting through every pixel, and then reprojects those features back to the query 3D point $p$ . After that, the corresponding RGB value and volumetric density of $p$ are inferred from those features given a specific query viewpoint. This simple design of GRF follows the principle of classic multi-view geometry (Hartley & Zisserman, 2004), therefore driving the learned implicit representations to be multi-view consistent. In particular, for any query 3D point in space, its features are always the same when being viewing from different angles. Hence, its rendered RGBs and volumetric densities are more likely to be consistent from different viewpoints. To realize our neural function $f _ { \\mathbf { W } }$ , we introduce four components: 1) A feature extractor for every 2D pixel; 2) A reprojector to transform 2D features to 3D space; 3) An aggregator to obtain general features for a 3D point; and 4) A neural renderer to infer the appearance and geometry for that 3D point. All these components are connected and trained end-to-end with arbitrary scenes. ",
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+    {
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+        "img_path": "images/56d1944f4d23a0a0a15e2089b674d2311f3342f1b49f441d770342b7eecce36d.jpg",
+        "image_caption": [
+            "Figure 1: Overview of the proposed general radiance field. "
+        ],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "3.2 EXTRACTING GENERAL FEATURES FOR 2D PIXELS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The whole input set of observed 2D images together describes the geometry and appearance of a 3D scene, while each pixel of those images describes a specific point of that scene. This module aims to extract the general features of each pixel, so to learn the regional description and geometric patterns for each light ray. A naive approach is to directly use the raw rgb values as the pixel features. However, this is sub-optimal because the raw rgb values are sensitive to lighting conditions, environmental noise, etc. In order to learn more general and robust patterns for each pixel, we turn to use a more powerful encoder-decoder based convolutional neural network (CNN). As shown in Figure 2, our CNN module is designed with the follow two features: ",
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+    {
+        "type": "image",
+        "img_path": "images/b2e9605ba008f372de37a5ed751410ac325144fee0bb3ce35aceec07022d20bd.jpg",
+        "image_caption": [
+            "Figure 2: The CNN module to extract pixel features. "
+        ],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "• Instead of directly feeding raw RGB images into the CNN module, we stack (duplicate) the corresponding viewpoint, i.e., the xyz location of the camera, to each pixel of the image. This allows the learned pixel features to be explicitly aware of its relative position in the 3D scene space. Note that, we empirically find that stacking the additional camera rotation and intrinsics to each pixel does not noticeably improve the performance, especially for scenes centered at the origin. • We use skip connections between the encoder and decoder to preserve high frequency local features for each pixel, while optionally integrating a couple of fully connected (fc) layers in the middle of the CNN module to learn global features. The mixture of hierarchical features tends to to be more general and representative, effectively aiding the network in practice. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Details of the CNN module is presented in the appendix and all input images share the same CNN module. Note that, there are many ways to construct a pixel feature extractor, but identifying an optimal CNN module here is out the scope of this paper. ",
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+    {
+        "type": "text",
+        "text": "3.3 REPROJECTING 2D FEATURES TO 3D SPACE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Considering the extracted pixel features are a compact description of the light ray emitting from the camera center up to the 3D scene surface, we naturally reproject the pixel features back to the 3D space along the light ray. Since there are no depth scans paired with RGB images, it is impossible to determine which particular 3D surface point the pixel features belong to. In this module, we preliminarily regard the pixel features as the representation of every location along the light ray in 3D space. With this simple formulation, every 3D point can theoretically have a copy of its corresponding 2D pixel features from each 2D image. Formally, given a 3D point $p$ , an observed 2D view $\\mathcal { I } _ { m }$ together with the camera pose $\\xi _ { m }$ and the intrinsics $K _ { m }$ , the corresponding 2D pixel features $\\mathbf { F } _ { p } ^ { m }$ is retrieved by the reprojection operation below: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/93bd6c5f152edf53249bcebd35c1b2f68cdf40f352ed334374b867c5c5628be1.jpg",
+        "text": "$$\n\\mathbf { F } _ { p } ^ { m } = \\mathcal { P } \\Big ( \\{ \\boldsymbol { \\mathcal { T } } _ { k } , \\boldsymbol { \\xi } _ { m } , K _ { m } \\} , \\{ x _ { p } , y _ { p } , z _ { p } \\} , \\mathbf { I } _ { m } \\Big )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where the function $\\mathcal { P } ( )$ follows the principle of multi-view geometry (Hartley & Zisserman, 2004) and ${ \\mathbf I } _ { m }$ represents the image features extracted by the CNN module in Section 3.2. However, since the pixels of 2D images are discrete and bounded within a certain spatial size, while the 3D points are continuous in the scene space, after the 3D point $p$ is projected to the plane of image $\\mathcal { T } _ { m }$ , we apply two approximations to deal with the following issues. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "• If the point lies inside of the image, we simply select the nearest pixel and duplicate its features to the 3D point. Note that, more advanced techniques may be applied to address the discretization issue, such as linear interpolation or designing a kernel function. • If the point lies outside of the image, we assign a zero vector to the 3D point, which means there is no information observed. In fact, we empirically find that the nearest interpretation can also achieve good performance, but it is only applicable for relatively small-scale scenes. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Overall, the above simple reprojection operation explicitly retains the extracted 2D pixel features back to 3D space via the principle of geometry, and the module is naturally differentiable. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4 OBTAINING GENERAL FEATURES FOR 3D POINTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Given a query 3D point $p$ , it is able to retrieve a feature vector from each input image. However, given a set of input images, it is challenging to obtain a final feature vector for the point $p$ , because: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• The total number of input images for each 3D scene is variable and there is no order for images. As a consequence, the retrieved feature vectors are also unordered and have arbitrary size. • Since there are no depth scans paired with the input RGB images, it is unable to decide which feature vectors are the true descriptions of the query point due to visual occlusions. Ideally, these features can be aware of the relative distance to the query point and then selected automatically. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To tackle these critical issues, we formulate this problem as an attention aggregation process. In particular, given the query 3D point $p$ , its query viewpoint $\\nu _ { p }$ , and the set of retrieved pixel features $\\{ \\mathbf { F } _ { p } ^ { 1 } \\cdot \\cdot \\cdot \\mathbf { F } _ { p } ^ { m } \\cdot \\cdot \\cdot \\mathbf { F } _ { p } ^ { M } \\}$ : ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• For each retrieved feature vector $\\mathbf { F } _ { p } ^ { m }$ , we firstly use shared MLPs to integrate the information of query point $p$ , generating a new feature vector $\\hat { \\mathbf { F } } _ { p } ^ { m }$ which is aware of the relative distance to the query point $p$ . Formally, it is defined as: $\\hat { \\mathbf { F } } _ { p } ^ { m } = M L P s ( \\mathbf { F } _ { p } ^ { m } \\oplus [ x _ { p } , y _ { p } , z _ { p } ] ) , \\quad ( \\oplus \\mathrm { m e a n s ~ c o n c a t e n a t i o n ~ i n ~ n e u r a l ~ n e t s } )$ (3)   \n• After obtaining the new set of position-aware features $\\{ \\hat { \\mathbf { F } } _ { p } ^ { 1 } \\cdot \\cdot \\cdot \\hat { \\mathbf { F } } _ { p } ^ { m } \\cdot \\cdot \\cdot \\hat { \\mathbf { F } } _ { p } ^ { M } \\}$ , we use the existing attention aggregation methods such as AttSets (Yang et al., 2020) and Slot Attention (Locatello et al., 2020) to compute a unique feature vector $\\bar { \\mathbf { F } } _ { p }$ for the query 3D point $p$ . Basically, the attention mechanism learns a unique weight for all input features and then aggregates them together. According to the theoretical analysis in (Yang et al., 2020) and (Locatello et al., 2020), the selected attention mechanisms are permutation invariant with regard to the input set of feature vectors and can process an arbitrary number of elements. Formally, it is defined as: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/904fa8bc269de40581217d509910ed3b07f202a888ada9353fe5726b085967fe.jpg",
+        "text": "$$\n\\bar { \\mathbf { F } } _ { p } = \\mathcal { A } \\big ( \\hat { \\mathbf { F } } _ { p } ^ { 1 } \\cdot \\cdot \\cdot \\hat { \\mathbf { F } } _ { p } ^ { m } \\cdot \\cdot \\cdot \\hat { \\mathbf { F } } _ { p } ^ { M } \\big ) , \\quad \\mathrm { ( \\mathcal { A } ~ i s ~ a n ~ e x i s t i n g ~ a t t e n t i o n ~ f u n c t i o n ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Above all, for every query 3D point $p$ , its final features $\\bar { \\mathbf { F } } _ { p }$ explicitly preserve the general geometric patterns retrieved from the input 2D observations and are also made aware of the particular 3D point location within the scene space. This allows the 3D point features $\\bar { \\mathbf { F } } _ { p }$ to be general and representative for its own geometry and appearance. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.5 RENDERING 3D FEATURES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "With the learned features $\\bar { \\mathbf { F } } _ { p }$ for any query 3D point $p$ , this module aims to infer its volumetric density $d _ { p }$ in 3D space and the corresponding RGB values $\\{ r _ { p } , g _ { p } , b _ { p } \\}$ given the query viewpoint $\\nu _ { p }$ . In particular, we exactly follow the simple MLP-based design of NeRF (Mildenhall et al., 2020) in the rendering process. As illustrated in Equation 5, the volumetric density $d _ { p }$ is modeled as a function of the point features $\\bar { \\mathbf { F } } _ { p }$ only. This allows the estimated density $d _ { p }$ at the query point $p$ to remain unchanged under different query viewpoints, thus maintaining consistent geometry for each point. The RGB is modeled as a function of both point features and the query viewpoint. Details of these MLPs refers to NeRF (Mildenhall et al., 2020). ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/26d919dff65d750e3d751851267bb1bf28a0879c63c1fd693d6eb7ee186b7238.jpg",
+        "text": "$$\nd _ { p } = M L P s ( \\bar { \\mathbf { F } } _ { p } ) , \\qquad \\quad ( r _ { p } , g _ { p } , b _ { p } ) = M L P s \\Bigl ( \\bar { \\mathbf { F } } _ { p } , \\{ x _ { p } ^ { v } , y _ { p } ^ { v } , z _ { p } ^ { v } \\} \\Bigr )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To render the color of any query ray passing through the 3D scene, the classic volume rendering (Kajiya & Herzen, 1984) is applied and the integral is estimated by sampling a discrete set of 3D points along the query ray. In particular, we exactly follow the formulation of ray marching and the hierarchical sampling strategy in NeRF (Mildenhall et al., 2020). Eventually, novel 2D images can be directly synthesized from our GRF by querying light rays. This allows the entire network to be trainable only with a set of 2D images, without requiring 3D data. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.6 IMPLEMENTATION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The proposed four modules are connected and trained end-to-end. In the implementation, details of the CNN module and the attention module for different experiments are presented in the appendix. In the neural rendering module, all the designs of neural layers and other settings simply follow NeRF (Mildenhall et al., 2020). In our network, all the 3D locations and RGB values are processed by the positional encoding proposed by NeRF (Mildenhall et al., 2020). The L2 loss between the rendered RGB and the ground truth is used to optimize the whole neural network. All these details are presented in the appendix. ",
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+    {
+        "type": "text",
+        "text": "4 EVALUATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTS ON SHAPENETV2 DATASET ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Following the experimental settings of SRNs (Sitzmann et al., 2019), we firstly evaluate our GRF on the chair and car classes of ShapeNetv2. Particularly, the chair has 4612 objects for training, 662 for validation and 1317 for testing, while the car has 2151 objects for training, 352 for validation and 704 for testing. Each training object has randomly sampled 50 images with a resolution of $1 2 8 \\times 1 2 8$ pixels. We conduct the following three groups of experiments. ",
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+    {
+        "type": "text",
+        "text": "• Group 1: Novel-view synthesis of seen objects in the training split of the same category. In particular, we train two separate models on the training split of chairs and cars, and these two models are separately tested on the trained objects to generate completely novel views. During testing, the model is fed with 50 trained views and infers another 251 novel views for evaluation. • Group 2: Novel-view synthesis of unseen objects in the testing split of the same category. The trained two models in Group 1 are tested on novel objects of the same category. During testing, the model is fed with 2 novel views of each novel object, inferring 251 novel views for evaluation. • Group 3: Similar to Group 2, but only 1 novel view is fed into the model for novel view synthesis. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Basically, the experiments of Group 1, 2 and 3 are exactly the same as in SRNs (Sitzmann et al., 2019) which is also the state of the art on this ShapeNetv2 dataset. Table 1 compares the quantitative results of our GRF and four baselines. Note that, the recent NeRF and NSVF are not scalable to learn large number of scenes simultaneously because each scene is encoded into the network parameters. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Analysis. 1) Our method outperforms SRNs on 50-view reconstruction, demonstrating the strong capability of our GRF for 3D scene representation. Figure 3 shows the qualitative results. 2) Our GRF lags behind SRNs for novel scene reconstruction in both Group 2 and 3. Fundamentally, this is because our GRF solves a much harder problem than SRNs. In particular, our network directly infers the novel scene representation in a single forward pass, while SRNs needs to be retrained on all novel scenes to optimize the latent code. As a result, the experiments of Group 2 and 3 are significantly in favour of SRNs. 3) To further demonstrate the advantage of GRF over SRNs, we directly evaluate the trained SRNs model on unseen objects (of the same category) without retraining. For comparison, we also directly evaluate the trained GRF model on the same novel objects. Figure 4 shows the qualitative results. It can be seen that if not retrained, SRNs completely fails to reconstruct unseen car instances, but randomly generates similar cars from learned prior knowledge, primarily because its latent code has not been updated from the unseen objects. In contrast, by learning pixel local patterns, our GRF generalizes well to novel objects directly. This highlights the superiority of the simple formulation of our GRF. ",
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+    {
+        "type": "table",
+        "img_path": "images/e4fa34648ba5c9c90c1bdadb195d161c5ec17e80900f684eac0614a5458324ab.jpg",
+        "table_caption": [
+            "Table 1: Comparison of the PSNR (in dB) and SSIM of reconstructed images in the ShapeNetv2 dataset by our GRF, the deterministic GQN (Eslami et al., 2018), TCO (Tatarchenko et al., 2015), WRL (Worrall et al., 2017) and SRNs (Sitzmann et al., 2019). The higher the scores, the better the synthesized novel views. Note that, our GRF solves a harder problem than SRNs, because GRF infers the novel scene representation in a single forward pass, while SRNs cannot. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\"></td><td colspan=\"2\">50 Images (Group 1)</td><td colspan=\"2\">2 Images (Group 2)</td><td colspan=\"2\">1 Image (Group 3)</td></tr><tr><td>Chairs</td><td>Cars</td><td>Chairs</td><td>Cars</td><td>Chairs</td><td>Cars</td></tr><tr><td>TCO</td><td>24.31/ 0.92</td><td>20.38/0.83</td><td>21.33/ 0.88</td><td>18.41 /0.80</td><td>21.27 / 0.88</td><td>18.15/0.79</td></tr><tr><td>WRL</td><td>24.57 / 0.93</td><td>19.16 /0.82</td><td>22.28/0.90</td><td>17.20 /0.78</td><td>22.11/0.90</td><td>16.89 / 0.77</td></tr><tr><td>dGQN</td><td>22.72/0.90</td><td>19.61 /0.81</td><td>22.36 /0.89</td><td>18.79 /0.79</td><td>21.59 / 0.87</td><td>18.19 /0.78</td></tr><tr><td>SRNs</td><td>26.23 / 0.95</td><td>26.32 /0.94</td><td>24.48 / 0.92</td><td>22.94 / 0.88</td><td>22.89 / 0.91</td><td>20.72 / 0.85</td></tr><tr><td>GRF (Ours)</td><td>27.10 / 0.95</td><td>27.21 / 0.94</td><td>22.65/0.88</td><td>22.06 /0.86</td><td>21.25/0.86</td><td>19.84/ 0.81</td></tr></table>",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/2df62752b06cd043c2e56bd864bd2c55fce00ed1bd7adfe827095d12d01968ae.jpg",
+        "image_caption": [
+            "Figure 3: Qualitative results of SRNs and our GRF for 50-view reconstruction of chairs and cars. ",
+            "or on par with the state-of-the-art NSVF on all metrics. Figure 6 shows the qualitative results. "
+        ],
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+        "image_caption": [
+            "Figure 4: Qualitative results of SRNs (without retraining) and our GRF for 50-view reconstruction of unseen cars. SRNs fails to recover faithful shapes for unseen cars without retraining. "
+        ],
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+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.2 EXPERIMENTS ON SYNTHETIC-NERF DATASET ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We further evaluate our approach on the more complex dataset Synthetic-NeRF (Mildenhall et al., 2020). It consists of pathtraced images of 8 synthetic scenes with much more complicated geometry and realistic non-Lambertian materials. Each scene has 100 views for training and 200 novel views for testing. Each image has a high resolution of $8 0 0 \\times 8 0 0$ pixels. ",
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+    {
+        "type": "text",
+        "text": "Single-scene Learning. To demonstrate the representational capability of our GRF, we train a single model for each synthetic scene, following the same experimental settings of NSVF (Liu et al., 2020). Table 2 compares the quantitative results. Our method is better ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/414f5dd3af395f2566e75e951ed08981951767a1f8507c7e13c4426b34f8a5a7.jpg",
+        "table_caption": [
+            "Table 2: The average scores of PSNR, SSIM and LPIPS (Zhang et al., 2018) for our GRF, SRNs (Sitzmann et al., 2019), NV (Lombardi et al., 2019), LLFF (Mildenhall et al., 2019), NeRF (Mildenhall et al., 2020) and NSVF (Liu et al., 2020) on the Synthetic-NeRF dataset for single-scene learning. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td></tr><tr><td>SRNs</td><td>22.26</td><td>0.846</td><td>0.170</td></tr><tr><td>NV</td><td>26.05</td><td>0.893</td><td>0.160</td></tr><tr><td>LLFF</td><td>24.88</td><td>0.911</td><td>0.114</td></tr><tr><td>NeRF</td><td>31.01</td><td>0.947</td><td>0.081</td></tr><tr><td>NSVF</td><td>31.74</td><td>0.953</td><td>0.047</td></tr><tr><td>GRF(Ours)</td><td>32.06</td><td>0.960</td><td>0.048</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Novel-scene Generalization. To evaluate the generalization of GRF on complex scenes, we train a single model simultaneously from randomly selected 4 scenes of Synthetic-NeRF, i.e., Chair, Mic, Ship, and Hotdog, and then conduct the following two groups of experiments. ",
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+    {
+        "type": "text",
+        "text": "Group 1: The trained model is directly tested on the remaining 4 novel scenes of Synthetic-NeRF dataset, i.e., Drums, Lego, Materials, and Ficus. Fundamentally, this experiment is to evaluate whether the learned limited features can truly generalize to new scenarios. This is extremely challenging because there are only 4 training scenes and the overall shapes of the 4 novel scenes are dramatically different from the trained ones. ",
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+    {
+        "type": "text",
+        "text": "Group 2: The trained model is further finetuned on each of the four novel scenes with 100, 1k, and 10k iterations separately. In total, we obtain $4 \\times 3 = 1 2$ new models. For comparison, we also train NeRF on each of the four novel scenes with 100, 1k, 10k iterations from scratch. This group of experiments evaluates how the initially-learned general features of our GRF improve novel scene learning. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Analysis. Table 3 compares the quantitative results. We can see that: 1) In Group 1, our GRF can indeed generalize to novel scenes with complex and completely different overall geometries, demonstrating the effectiveness of learning point local features in GRF. As shown in Figure 5, the overall shape and appearance of Lego can be satisfactorily recovered, though it has never been seen before. 2) In Group 2, our GRF can quickly learn high-quality scene representations given a sparse number of training pix",
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+    {
+        "type": "table",
+        "img_path": "images/3080c6c65b006f8fbd396908b54525d24ac650e1d6e039623be0413789aa141b.jpg",
+        "table_caption": [
+            "Table 3: The average scores of PSNR, SSIM and LPIPS for GRF and NeRF on four novel scenes of Synthetic-NeRF in Group 1&2. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>PSNR↑ SSIM↑LPIPS↓</td></tr><tr><td>GRF (Group 1) 13.62 0.763 0.246</td></tr><tr><td>NeRF(Group 2, 100 iters) 15.15 0.752 0.359 NeRF(Group 2,1k iters) 19.81 0.809 0.228 NeRF(Group 2,10k iters) 23.35 0.875 0.137</td></tr><tr><td>GRF (Group 2, 100 iters) 19.69 0.835 0.169 GRF (Group 2, 1k iters) 22.00 0.876 0.128 GRF (Group 2,10k iters) 25.10 0.916 0.089</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "els, thanks to the initially learned general features. Compared with the NeRF trained from scratch, the learned GRF significantly improves novel scene learning, achieving much better results given the same amount of training signals of new scenes. Figure 5 qualitatively shows the generalization of GRF and more results are presented in appendix. ",
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+        "img_path": "images/2b33473f8cad83e6fc6ade9aac3c829b58988eea96b3c91c1071bdc0382ad409.jpg",
+        "image_caption": [
+            "Figure 5: Qualitative results of our GRF for novel scene generalization. "
+        ],
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+    {
+        "type": "text",
+        "text": "4.3 EXPERIMENTS ON REAL-WORLD COMPLEX SCENES ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluate the novel view synthesis on complex real-world scenes captured by cellphones. There are 8 scenes, 5 from LLFF (Mildenhall et al., 2019) and 3 from NeRF. Each scene has 20 to 62 images, 1/8 of which for testing. All images have high resolution of $1 0 0 8 \\times 7 5 6$ pixels. ",
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+    {
+        "type": "text",
+        "text": "Single-scene Learning. We train a single model for each real-world scene, following the same experimental settings of NeRF. Table 4 compares the quantitative results. Note that, the recent ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/ce266d353b51e877a8a168b6fc05c2de9bc86a65d823ee30f8ce43337a952611.jpg",
+        "table_caption": [
+            "Table 4: Comparison of the average PSNR, SSIM and LPIPS scores of our GRF, SRNs, LLFF and NeRF in the challenging real-world dataset for single-scene learning. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td></tr><tr><td>SRNs</td><td>22.84</td><td>0.668</td><td>0.378</td></tr><tr><td>LLFF</td><td>24.13</td><td>0.798</td><td>0.212</td></tr><tr><td>NeRF</td><td>26.50</td><td>0.811</td><td>0.250</td></tr><tr><td>GRF(Ours)</td><td>26.64</td><td>0.837</td><td>0.178</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "NSVF (Liu et al., 2020) is unable to process these forward-facing scenes because the predefined voxels cannot represent the unbounded 3D space. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Analysis. Our method surpasses the state of the art by large margins, especially over the SSIM and LPIPS metrics. Compared with PSNR which only measures the average per-pixel accuracy, the metrics SSIM and LPIPS favor high quality of photorealism, highlighting the superiority of our GRF to generate truly realistic images. Figure 6 shows the qualitative results. As highlighted by the red circles, our GRF can generate fine-grained and natural geometries in pixel level, while NeRF produces many artifacts. This clearly demonstrates our GRF indeed learns general and precise pixel features from the observed 2D images for 3D scene representation and rendering. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Compared with Synthetic-NeRF, this dataset is much harder because of the complicated geometries and the inaccuracy of estimated camera poses. Therefore, it is non-trivial to conduct real-world experiments for novel-scene generalization or large-scale scene reconstruction on ScanNet (Dai et al., 2017) or “Tanks and Temples” (Knapitsch et al., 2017). We leave these for future exploration. ",
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+        "page_idx": 7
+    },
+    {
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+        "img_path": "images/d2dd827b02e584e3f576cc7940a4fd39cb69b473fd797cdb28c8a8cf51466e19.jpg",
+        "image_caption": [
+            "NeRF GRF (Ours) Ground Truth Figure 6: Qualitative results on Synthetic-NeRF (1st row) and real-world datasets (2nd row). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.4 ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To evaluate the effectiveness of the key components of our GRF, we conduct 3 groups of ablation experiments on the car category in ShapeNetv2 dataset. In particular, we train on the entire training split then randomly select 500 objects from the training split for testing. During testing, the model infers 50 novel views for each of these 500 objects for evaluation. Basically, this setting is similar to the experiment of Group 1 in Section 4.1. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/d65a0d05a7af6ab59e853844ea4a4e8a65732bc36324d880ac1e76483adc362e.jpg",
+        "table_caption": [
+            "Table 5: The average scores of PSNR and SSIM for ablated GRF in the subset of car category. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>PSNR↑</td><td>SSIM↑</td></tr><tr><td>(1)Remove Input Viewpoints</td><td>20.13</td><td>0.807</td></tr><tr><td>(2) Remove Pixel Local Features</td><td>20.23</td><td>0.818</td></tr><tr><td>(3) Replace Attention by Maxpool</td><td>24.88</td><td>0.914</td></tr><tr><td>(4) The Full Model</td><td>27.16</td><td>0.942</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• Group 1: The viewpoints of the input images are removed from the CNN module. The CNN module is not explicitly aware of the relative position of the pixel features. • Group 2: The decoder of the CNN module is removed and each input image is encoded as a global feature vector. For any 3D query point which is rightly projected into the image boundary, that global feature vector is retrieved and reprojected to the query point. Fundamentally, this modified CNN module can be regarded as a hyper-network that learns a conditional embedding from input images and then feed it into NeRF, but it sacrifices the precise pixel local features. • Group 3: The advanced attention module is replaced by max-pooling to aggregate the pixel features, aiming to investigate how the visual occlusion can be better addressed using soft attention. • Group 4: The full model is trained and tested with the same settings for comparison. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Analysis. Table 5 compares the performance of all ablated models. We can see that: 1) The greatest impact is caused by the removal of image viewpoints from the CNN module and the lack of local pixel features to represent 3D points. It highlights that obtaining the position-aware and precise pixel features is crucial to represent 3D scenes from 2D images, while learning a simple hyper-network for NeRF cannot achieve comparable performance. 2) Using max-pooling to select the reprojected pixel features is sub-optimal to address the visual occlusions from different viewing angles. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To further investigate the attention module, the maximal attention scores are retrieved and visualized (detailed in appendix). It shows that the attention module can truly drive the network to focus on the visible pixel local features among the multiple intersected light rays. We also conduct additional ablation experiments (detailed in appendix), demonstrating that our GRF can indeed generalize well to novel scenes with severe visual occlusions and is robust to a variable number of input images. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            794,
+            318,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The proposed method models complex 3D scenes as a general radiance field. We demonstrated that it can learn general and robust 3D point features from a set of 2D observations. By using the principle of multi-view geometry to precisely map 2D pixel features back to 3D space and leveraging the attention mechanism to implicitly address the visual occlusions, our GRF can synthesize truly realistic 2D novel views. However, there are still limitations that lead to the future work. (1) More advanced CNN modules can be designed to learn better pixel features. (2) Depth scans can be integrated into the network to explicitly address visual occlusions. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1 DETAILS OF NETWORK ARCHITECTURE ",
+        "bbox": [
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+            133,
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+            148
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/3ac8d85f8a85bfd8277fb757a68ee73ad0999e9f05f89b1e6333da2f9edbaa85.jpg",
+        "table_caption": [
+            "Table 6: The CNN Module for Experiments on ShapeNetv2 Dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Type Size/Channels Activation Stride</td></tr><tr><td>Input: embedding of RGB and viewpoint</td></tr><tr><td>1 - 64 ReLU 2</td></tr><tr><td>Li: Conv 7 × 7 L2: Conv 3 × 3 128 ReLU 2</td></tr><tr><td>L3: Conv 3 × 3 256 ReLU 2</td></tr><tr><td></td></tr><tr><td>L4: Conv 3 × 3 512 ReLU 2</td></tr><tr><td>L5: Conv 4 × 4 128 ReLU 4</td></tr><tr><td>L6: Flatten /Tile - 1</td></tr><tr><td>L7: Concat (L6,L4) - 1</td></tr><tr><td>L8:Dilated Conv 3×3 256 ReLU 4</td></tr><tr><td>L8: Concat (L8,L3) 1 1 =</td></tr><tr><td>L9:Dilated Conv 3 × 3 128 ReLU 8</td></tr><tr><td>L9: Concat (L9,L2) = 一 =</td></tr><tr><td>L10: Dilated Conv 3 × 3 64 ReLU 16</td></tr><tr><td>L10: Concat (L10, L1) - ■ = L11: Dilated Conv 3 × 3 128 ReLU 32</td></tr></table>",
+        "bbox": [
+            246,
+            188,
+            748,
+            428
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/f3cfc5c0b697f7ff2d4d137d9755f6843fc3a68f617b71895b640e8384cb92d3.jpg",
+        "table_caption": [
+            "Table 7: The CNN Module for Experiments on Synthetic-NeRF Dataset. The average pooling is added to aggressively downsample the feature maps. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Type Size/Channels Activation Stride</td></tr><tr><td>Input: embedding of RGB and viewpoint - -</td></tr><tr><td>L1: Conv 7 × 7 64 ReLU 2</td></tr><tr><td>L2: Conv 3 × 3 128 ReLU 2</td></tr><tr><td>L3: Conv 3 × 3 256 ReLU 2</td></tr><tr><td>L4: Conv 3 × 3 512 ReLU 2</td></tr><tr><td>L4: AveragePooling 5 × 5</td></tr><tr><td>= =</td></tr><tr><td>L5: Conv 5 × 5 128 ReLU 5</td></tr><tr><td>L6:Flatten /Tile - 1</td></tr><tr><td>L7: Concat (L6,L4) 1 1</td></tr><tr><td>L8: Deconv 3× 3 256 ReLU 2</td></tr><tr><td>L8: Concat (L8, L3) 1 1</td></tr><tr><td>L9: Deonv 3 × 3 128 ReLU 2</td></tr><tr><td>L9: Concat (L9,L2) 1 1</td></tr><tr><td>L10: Deconv 3×3 64 ReLU 2</td></tr><tr><td>L10: Concat (L10,L1) - 1 L11: Deconv 3 × 3 128 ReLU 2</td></tr></table>",
+        "bbox": [
+            246,
+            486,
+            748,
+            739
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/15ad75e56b1cd73cb68236429d499cf4d8210d2b20664c25830133d3707753ff.jpg",
+        "table_caption": [
+            "Table 8: The CNN Module for Experiments on the real-world Dataset. The average pooling is added to aggressively downsample the feature maps. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Type Size/Channels Activation Stride</td></tr><tr><td>Input: embedding of RGB and viewpoint 1 1</td></tr><tr><td>L1: Conv 7 × 7 64 ReLU</td></tr><tr><td>22 L2: Conv 3 × 3 128 ReLU</td></tr><tr><td>L3: Conv 3 × 3 256 ReLU 2</td></tr><tr><td>L4: Conv 3 × 3 512 ReLU 2</td></tr><tr><td></td></tr><tr><td>L4: AveragePooling 8 × 8 1 =</td></tr><tr><td>L5: Conv 4 × 4 128 ReLU 4</td></tr><tr><td>L6: Flatten /Tile 1 1</td></tr><tr><td>L7: Concat (L6,L4) 1 1</td></tr><tr><td>L8: Deconv 3 × 3 256 ReLU 2</td></tr><tr><td>L8: Concat (L8,L3) - 1</td></tr><tr><td>L9: Deconv 3 × 3 128 ReLU</td></tr><tr><td>L9: Concat (L9,L2) 1 =</td></tr><tr><td>L10: Deconv 3×3 64 ReLU 2</td></tr><tr><td>L10: Concat (L10, L1) - =</td></tr><tr><td>L11: Deconv 3 × 3 128 ReLU 2</td></tr></table>",
+        "bbox": [
+            245,
+            140,
+            750,
+            392
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/5792f5022f202a1711b69c1ecc67365884b64b5fb9c2d196c0495c1ea99e0865.jpg",
+        "table_caption": [
+            "Table 9: The Attention Module, AttSets (Yang et al., 2020), for Experiments on the ShapeNetv2 Dataset. The simple AttSets is computationally efficient and we choose it to train the large-scale ShapeNetv2 dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Type</td><td>Size/Channels Activation</td><td></td></tr><tr><td>Input: Concat(K × 128,embedding of viewpoint)</td><td>=</td><td>=</td></tr><tr><td>L1: fc</td><td>256</td><td>ReLU</td></tr><tr><td>L2: fc</td><td>256</td><td>ReLU</td></tr><tr><td>L3: fc</td><td>256</td><td>ReLU</td></tr><tr><td>L4: fc</td><td>512</td><td>ReLU</td></tr><tr><td>L5: fc</td><td>512</td><td>ReLU</td></tr><tr><td>L6: softmax(L5)</td><td>1</td><td>1</td></tr><tr><td>L7: sum(L6*L5,axis=-2)</td><td>1</td><td>1</td></tr><tr><td>L8: fc</td><td>512</td><td>ReLU</td></tr></table>",
+        "bbox": [
+            240,
+            467,
+            754,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We use Slot Attention as the pixel feature aggregation module for experiments on the SyntheticNeRF and real-world datasets. In particular, we use two slots, two iterations, and the hidden size is 128. The final output two slots are flattened and a 256 dimensional vector is obtained. For details of Slot Attention refer to the paper (Locatello et al., 2020). ",
+        "bbox": [
+            174,
+            638,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Details of the neural rendering layers and the volume rendering can be found in NeRF (Mildenhall et al., 2020). We set the positional embedding length $L = 5$ for all inputs to the CNN module, except the rotation, which we convert to quaternion and embed at $L = 4$ . ",
+        "bbox": [
+            176,
+            702,
+            820,
+            743
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "During training, we feed the models between 2 and 6 views of each scene at each gradient step. We set the learning rate for the ShapeNetv2 models at 1e-4. We set the learning rate for leaves and orchids in the real-world dataset at 7e-5, and for the rest, we use 1e-4. For Synthetic-NeRF dataset, we use a learning rate of 1e-4. We use the Adam optimizer for all models, and train for $2 0 0 \\mathbf { k } { - } 3 0 0 \\mathbf { k }$ iterations. At each gradient step, we take 1000 rays for ShapeNetv2 with 32 coarse samples and 64 fine samples, and 800 rays for the real-world and Synthetic-NeRF datasets with 64 coarse samples and 192 fine samples. We train each model on a single Nvidia-V100 GPU with 32GB VRAM. ",
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "During testing for the ShapeNetv2 reconstruction from 50 images, we feed the model the 5 closest views to the desired novel view by cosine similarity from the 50 training images. During testing for single-scene reconstruction, we feed the model the 4 closest views by cosine similarity to the desired novel view. ",
+        "bbox": [
+            176,
+            854,
+            825,
+            910
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.2 DETAILS OF EXPERIMENTAL RESULTS ON THE SYNTHETIC-NERF DATASET ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            741,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/8a19c4d1e380f64c84286252d3506630858144ee54eca16f70dc58749599911e.jpg",
+        "table_caption": [
+            "Table 10: Comparison of the PSNR (in dB), SSIM and LPIPS (Zhang et al., 2018) scores of our GRF, SRNs (Sitzmann et al., 2019), NV (Lombardi et al., 2019), NeRF (Mildenhall et al., 2020) and NSVF (Liu et al., 2020) in the Synthetic-NeRF dataset for single-scene learning. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Chair</td><td>Drums</td><td>Lego</td><td>Mic</td><td>Materials</td><td>Ship</td><td>Hotdog</td><td>Ficus</td><td>Mean</td></tr><tr><td colspan=\"10\">PSNR↑</td></tr><tr><td>SRNs</td><td>26.96</td><td>17.18</td><td>20.85</td><td>26.85</td><td>18.09</td><td>20.60</td><td>26.81</td><td>20.73</td><td>22.26</td></tr><tr><td>NV</td><td>28.33</td><td>22.58</td><td>26.08</td><td>27.78</td><td>24.22</td><td>23.93</td><td>30.71</td><td>24.79</td><td>26.05</td></tr><tr><td>NeRF</td><td>33.00</td><td>25.01</td><td>32.54</td><td>32.91</td><td>29.62</td><td>28.65</td><td>36.18</td><td>30.13</td><td>31.01</td></tr><tr><td>NSVF</td><td>33.19</td><td>25.18</td><td>32.29</td><td>34.27</td><td>32.68</td><td>27.93</td><td>37.14</td><td>31.23</td><td>31.74</td></tr><tr><td>GRF(Ours)</td><td>34.51</td><td>25.83</td><td>32.92</td><td>33.94</td><td>30.91</td><td>30.12</td><td>37.47</td><td>30.75</td><td>32.06</td></tr><tr><td colspan=\"10\">SSIM↑</td></tr><tr><td>SRNs</td><td>0.910</td><td>0.766</td><td>0.809</td><td>0.947</td><td>0.808</td><td>0.757</td><td>0.923</td><td>0.849</td><td>0.846</td></tr><tr><td>NV</td><td>0.916</td><td>0.873</td><td>0.880</td><td>0.946</td><td>0.888</td><td>0.784</td><td>0.944</td><td>0.910</td><td>0.893</td></tr><tr><td>NeRF</td><td>0.967</td><td>0.925</td><td>0.961</td><td>0.980</td><td>0.949</td><td>0.856</td><td>0.974</td><td>0.964</td><td>0.947</td></tr><tr><td>NSVF</td><td>0.968</td><td>0.931</td><td>0.960</td><td>0.987</td><td>0.973</td><td>0.854</td><td>0.980</td><td>0.973</td><td>0.953</td></tr><tr><td>GRF(Ours)</td><td>0.981</td><td>0.937</td><td>0.967</td><td>0.987</td><td>0.963</td><td>0.891</td><td>0.983</td><td>0.969</td><td>0.960</td></tr><tr><td colspan=\"10\">LPIPS↓</td></tr><tr><td>SRNs</td><td>0.106</td><td>0.267</td><td>0.200</td><td>0.063</td><td>0.174</td><td>0.299</td><td>0.100</td><td>0.149</td><td>0.170</td></tr><tr><td>NV</td><td>0.109</td><td>0.214</td><td>0.175</td><td>0.107</td><td>0.130</td><td>0.276</td><td>0.109</td><td>0.162</td><td>0.160</td></tr><tr><td>NeRF</td><td>0.046</td><td>0.091</td><td>0.050</td><td>0.028</td><td>0.063</td><td>0.206</td><td>0.121</td><td>0.044</td><td>0.081</td></tr><tr><td>NSVF</td><td>0.043</td><td>0.069</td><td>0.029</td><td>0.010</td><td>0.021</td><td>0.162</td><td>0.025</td><td>0.017</td><td>0.047</td></tr><tr><td>GRF(Ours)</td><td>0.021</td><td>0.068</td><td>0.042</td><td>0.013</td><td>0.041</td><td>0.141</td><td>0.028</td><td>0.032</td><td>0.048</td></tr></table>",
+        "bbox": [
+            173,
+            185,
+            826,
+            474
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/7456beddc3b5748ce63369feae497d718a196676d033b0c2e1938a44b6190833.jpg",
+        "table_caption": [
+            "Table 11: The PSNR, SSIM and LPIPS scores of our GRF simultaneously trained on 4 scenes of the Synthetic-NeRF dataset for multi-scene learning. The scores of SRNs, NeRF and NSVF trained on single scenes are included for comparison. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Chair</td><td>Mic</td><td>Ship</td><td>Hotdog</td></tr><tr><td colspan=\"5\">PSNR↑</td></tr><tr><td>SRNs (Single-scene)</td><td>26.96</td><td>26.85</td><td>20.60</td><td>26.81</td></tr><tr><td>NeRF (Single-scene)</td><td>33.00</td><td>32.91</td><td>28.65</td><td>36.18</td></tr><tr><td>NSVF (Single-scene)</td><td>33.19</td><td>34.27</td><td>27.93</td><td>37.14</td></tr><tr><td>GRF (Multi-scene)</td><td>32.49</td><td>32.02</td><td>27.76</td><td>34.92</td></tr><tr><td colspan=\"5\">SSIM↑</td></tr><tr><td>SRNs (Single-scene)</td><td>0.910</td><td>0.947</td><td>0.757</td><td>0.923</td></tr><tr><td>NeRF (Single-scene)</td><td>0.967</td><td>0.980</td><td>0.856</td><td>0.974</td></tr><tr><td>NSVF (Single-scene)</td><td>0.968</td><td>0.987</td><td>0.854</td><td>0.980</td></tr><tr><td>GRF (Multi-scene)</td><td>0.971</td><td>0.982</td><td>0.866</td><td>0.975</td></tr><tr><td colspan=\"5\">LPIPS↓</td></tr><tr><td>SRNs (Single-scene)</td><td>0.106</td><td>0.063</td><td>0.299</td><td>0.100</td></tr><tr><td>NeRF (Single-scene)</td><td>0.046</td><td>0.028</td><td>0.206</td><td>0.121</td></tr><tr><td>NSVF (Single-scene)</td><td>0.043</td><td>0.010</td><td>0.162</td><td>0.025</td></tr><tr><td>GRF (Multi-scene)</td><td>0.032</td><td>0.019</td><td>0.167</td><td>0.040</td></tr></table>",
+        "bbox": [
+            173,
+            553,
+            826,
+            820
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/faa40c1a6f21d2a0d3d808f83a19ed8fcc279796274d22c7b01aac148f1c3535.jpg",
+        "table_caption": [
+            "Table 12: The PSNR, SSIM and LPIPS scores of our GRF and NeRF on four novel scenes of Synthetic-NeRF in Group 1&2 experiments. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"4\">Drums</td></tr><tr><td></td><td>Lego PSNR↑</td><td>Materials</td><td>Ficus</td></tr><tr><td colspan=\"2\"></td></tr><tr><td>GRF (Group 1) 13.23 13.53</td><td>12.26 15.47 15.42 15.72</td></tr><tr><td>NeRF( F(Group2,100 iters) 14.54 14.92</td><td>20.04 20.40 20.81</td></tr><tr><td>NeRF( (Group2,1k iters) 18.01</td><td>19.81 23.36 23.47</td></tr><tr><td>NeRF (Group 2,10k iters) 21.57 24.99</td><td>23.35</td></tr><tr><td>GRF (Group 2,100 iters) 18.70 20.24</td><td>18.81 21.03 19.69</td></tr><tr><td>GRF (Group2,1k iters) 20.49 23.64</td><td>21.87 22.02 22.00</td></tr><tr><td>GRF (Group2,10k iters) 23.11 27.07</td><td>25.11 25.11 25.10</td></tr><tr><td colspan=\"2\">SSIM↑</td></tr><tr><td>F (Group 1) 0.762</td><td>0.736 0.703 0.849</td></tr><tr><td>NeRF(Group 2,100 iters)</td><td>0.769 0.717 0.716</td></tr><tr><td>NeRF(Group 2,1k iters) 0.793 0.775</td><td>0.812 0.857 0.809 0.877 0.896 0.875</td></tr><tr><td>NeRF F(Group2,10k iters) 0.865</td><td>0.862</td></tr><tr><td>GRF(Group 2,100 iters) 0.822 0.813</td><td>0.829 0.878 0.835</td></tr><tr><td>GRF (Group2,1k iters) 0.856</td><td>0.877 0.878 0.894 0.876 0.916</td></tr><tr><td>GRF F(Group2,10k iters) 0.901</td><td>0.924 0.913 0.923</td></tr><tr><td colspan=\"2\">LPIPS↓</td></tr><tr><td></td><td>0.256 0.273 0.301 0.150</td></tr><tr><td>GRF (Group 1) NeRF(Group 2,100 iters)</td><td>0.332 0.395 0.314</td></tr><tr><td>NeRF(Group 2,1k iters) 0.254 0.264</td><td>0.393 0.359 0.229 0.164 0.228 0.137</td></tr><tr><td>NeRF (Group2,10k iters) 0.157</td><td>0.154 0.128 0.110</td></tr><tr><td>GRF (Group 2,100 iters) 0.196</td><td>0.203 0.159 0.117 0.169 0.128</td></tr><tr><td>GRF F(Group2,1k iters) 0.154 GRF (Group 2,10k iters) 0.104</td><td>0.138 0.123 0.097 0.090 0.090 0.071</td></tr></table>",
+        "bbox": [
+            171,
+            330,
+            825,
+            733
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.3 DETAILS OF EXPERIMENTAL RESULTS ON THE REAL-WORLD DATASET ",
+        "bbox": [
+            174,
+            103,
+            709,
+            118
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/13622607470041225f8d5395594024818146184a58c3007f10049788c300715c.jpg",
+        "table_caption": [
+            "Table 13: Comparison of the PSNR, SSIM and LPIPS scores of our GRF, SRNs (Sitzmann et al., 2019), LLFF (Mildenhall et al., 2019) and NeRF (Mildenhall et al., 2020) in the real-world dataset for single-scene learning. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Room</td><td>Fern</td><td>Leaves</td><td>Fortress</td><td>Orchids</td><td>Flower</td><td>T-Rex</td><td>Horns</td><td>Mean</td></tr><tr><td colspan=\"10\">PSNR↑</td></tr><tr><td>SRNs</td><td>27.29</td><td>21.37</td><td>18.24</td><td>26.63</td><td>17.37</td><td>24.63</td><td>22.87</td><td>24.33</td><td>22.84</td></tr><tr><td>LLFF</td><td>28.42</td><td>22.85</td><td>19.52</td><td>29.40</td><td>18.52</td><td>25.46</td><td>24.15</td><td>24.70</td><td>24.13</td></tr><tr><td>NeRF</td><td>32.70</td><td>25.17</td><td>20.92</td><td>31.16</td><td>20.36</td><td>27.40</td><td>26.80</td><td>27.45</td><td>26.50</td></tr><tr><td>GRF(Ours)</td><td>31.74</td><td>25.72</td><td>21.16</td><td>31.28</td><td>20.88</td><td>27.83</td><td>27.01</td><td>27.50</td><td>26.64</td></tr><tr><td colspan=\"10\">SSIM↑</td></tr><tr><td>SRNs</td><td>0.883</td><td>0.611</td><td>0.520</td><td>0.641</td><td>0.449</td><td>0.738</td><td>0.761</td><td>0.742</td><td>0.668</td></tr><tr><td>LLFF</td><td>0.932</td><td>0.753</td><td>0.697</td><td>0.872</td><td>0.588</td><td>0.844</td><td>0.857</td><td>0.840</td><td>0.798</td></tr><tr><td>NeRF</td><td>0.948</td><td>0.792</td><td>0.690</td><td>0.881</td><td>0.641</td><td>0.827</td><td>0.880</td><td>0.828</td><td>0.811</td></tr><tr><td>GRF(Ours)</td><td>0.951</td><td>0.827</td><td>0.727</td><td>0.898</td><td>0.667</td><td>0.852</td><td>0.901</td><td>0.873</td><td>0.837</td></tr><tr><td colspan=\"10\">LPIPS↓</td></tr><tr><td>SRNs</td><td>0.240</td><td>0.459</td><td>0.440</td><td>0.453</td><td>0.467</td><td>0.288</td><td>0.298</td><td>0.376</td><td>0.378</td></tr><tr><td>LLFF</td><td>0.155</td><td>0.247</td><td>0.216</td><td>0.173</td><td>0.313</td><td>0.174</td><td>0.222</td><td>0.193</td><td>0.212</td></tr><tr><td>NeRF</td><td>0.178</td><td>0.280</td><td>0.316</td><td>0.171</td><td>0.321</td><td>0.219</td><td>0.249</td><td>0.268</td><td>0.250</td></tr><tr><td>GRF(Ours)</td><td>0.104</td><td>0.191</td><td>0.238</td><td>0.127</td><td>0.275</td><td>0.176</td><td>0.146</td><td>0.169</td><td>0.178</td></tr></table>",
+        "bbox": [
+            173,
+            186,
+            825,
+            436
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "A.4 ANALYSIS OF ATTENTION MECHANISM(Generalizat ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            491,
+            117
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "The attention mechanism in our GRF aims to automatically select the correct pixel patch from multiple pixel patches where the light rays intersect at the same query 3D point in space. ",
+        "bbox": [
+            174,
+            128,
+            823,
+            160
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "In order to investigate how the attention mechanism learns to select the useful information, we retrieve the maximal attention score from the observed multiple pixel patches for analysis. Intuitively, the higher the attention score is assigned to a particular pixel patch, the more important that patch for inferring the novel pixel RGB. In particular, we conduct the following experiment using our GRF model trained on ShapeNetv2 Cars. In this case, the AttSets attention module is used (details are in Table 9). Given a query light ray, multiple 3D points are sampled to query the network. ",
+        "bbox": [
+            173,
+            167,
+            825,
+            261
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "We firstly try to find the 3D point which is near the surface according to the predicted volume density for points along the ray through a given pixel, if they exist. Otherwise, we ignore such pixels, making them white.   \nThen we compute the M feature vectors from the input M views for these surface points. Thirdly, the attention masks for those M feature vectors are computed. We identify the view whose sum of the attention mask along the feature axis is greatest as the main contributor for inferring the novel pixel RGB.   \nAfter querying light rays for each pixel, we obtain a rendered RGB image. At the same time, for each pixel of that image, we select the most important view from the M input views for the surfaceintersection point along the ray from the viewpoint through that pixel, according to the maximal attention score. Eventually, we obtain a Max Attention Map corresponding to the rendered RGB image. ",
+        "bbox": [
+            173,
+            267,
+            825,
+            453
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Figure 7 shows the qualitative results of the above experiment. In particular, we feed the three images (#1,#2,#3) of an unseen car into our GRF model which is well-trained on car category, and then render a new image (e.g., the 5th image in Figure 7). Note that, we carefully select the input 3 images and the rendered image with very large viewing baselines. In the mean time, we obtain and visualize the Max Attention Map corresponding to the rendered image. ",
+        "bbox": [
+            173,
+            460,
+            825,
+            537
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "For each pixel of the rendered image, we retrieve the input image pixel that has the highest attention score. Specifically, the rendered pixels with purple color correspond to the input image #1, the green pixels correspond to the input image #2, while the blue pixels correspond to the input image #3. ",
+        "bbox": [
+            174,
+            545,
+            825,
+            592
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Analysis. It can be seen that, when inferring a new image, the attention module of our GRF focuses on the most informative pixel patch from the multiple input pixel patches. In addition, it is able to truly deal with the visual occlusion. For example, when inferring the windshield of the car, the attention module focuses on the input image #2 where the windshield is visible, while ignoring the image #1 and #3 where the windshield is self-occluded. ",
+        "bbox": [
+            173,
+            598,
+            825,
+            676
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/20f93eecb76196ee8f164adc09b398e9f5dbb82c10a34c1453af75819e8f5736.jpg",
+        "image_caption": [
+            "Figure 7: Visualization of Max Attention Map from Multiple Input Images of a Novel Object for Inferring a Novel View. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            691,
+            825,
+            789
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "A.5 GENERALIZATION TO VISUAL OCCLUSIONS AND VARIABLE INPUT IMAGES ",
+        "bbox": [
+            174,
+            103,
+            740,
+            118
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "We carefully select the attention module, i.e., either AttSets or SlotAtt, to aggregate the features from an arbitrary number of input views. In order to evaluate how our GRF is able to generalize with a variable number of input views, especially when there is a very sparse number of views with severe visual occlusions, we conduct the following four groups of experiments. ",
+        "bbox": [
+            174,
+            128,
+            823,
+            190
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "• 1-view Reconstruction. We feed the a single image of a novel car into our GRF model which is well-trained on car category (trained with 5 images per object), and then render 9 new images from vastly different viewing angles. This is the extreme case where the majority of the object is self-occluded. ",
+        "bbox": [
+            174,
+            198,
+            825,
+            260
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "• 2-view Reconstruction. Similarly, we feed only two images of the novel car into the same model and render the same 9 novel views. In this case, more information is given to the network, but there are still many parts occluded. ",
+        "bbox": [
+            179,
+            261,
+            823,
+            305
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "• 5-view / 10-view Reconstruction. The same GRF model is fed with 5 and 10 views of the novel object, rendering the same set of new images. ",
+        "bbox": [
+            178,
+            306,
+            820,
+            338
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Analysis. Figure 8 shows the qualitative results. It can be seen that: 1) In the extreme case, i.e., 1-view reconstruction, our GRF is still able to recover the general 3D shape of the unseen object, including the visually occluded parts, primarily because our CNN model learns the hierarchical features including the high-level shapes. 2) Given more input views, the originally occluded parts tend to be observed from some viewing angles, and then these parts can be reconstructed better and better. This shows that our GRF is indeed able to effectively identify the corresponding useful pixel features for more accurately recovering shape and appearance. ",
+        "bbox": [
+            173,
+            344,
+            825,
+            453
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/452fb2908ba7c814b8da50727c0309cbd16427f8d4f7072fd5be8f665d791d99.jpg",
+        "image_caption": [
+            "Figure 8: Qualitative results of our GRF when being fed with a variable number of views of a novel object. The red circle highlights that the tail of the car is able to be recovered given more visual cues from more input images. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            470,
+            794,
+            861
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/5e585a9c6f55e0f762c826bccc3ad70554c0e9a9023da31e86706efd23ce1569.jpg",
+        "image_caption": [
+            "Figure 9: Qualitative results of SRNs and our GRF for 50-view reconstruction of chairs and cars. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            161,
+            193,
+            816,
+            820
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "image",
+        "img_path": "images/720b1b3f0a109fae40aade5f250bbe18fc1e9ff6bc9faf69eda08923c27ddadc.jpg",
+        "image_caption": [
+            "Figure 10: Qualitative results of our GRF for novel view depth and RGB estimation on the SyntheticNeRF dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            212,
+            803,
+            775
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "image",
+        "img_path": "images/cadef3a36c5e380d2f99cc5b827f2e4ca75d2c8175fcc3a245639772f579210d.jpg",
+        "image_caption": [
+            "Figure 11: Qualitative results of our GRF for novel view depth and RGB estimation on the SyntheticNeRF dataset. "
+        ],
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diff --git a/parse/train/cAvgPMAA3hb/cAvgPMAA3hb_middle.json b/parse/train/cAvgPMAA3hb/cAvgPMAA3hb_middle.json
new file mode 100644
index 0000000000000000000000000000000000000000..6239d1dc8aba30bda72ba8da60c4d3e2a1958f28
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+                                    "content": "tures for each pixel, while optionally integrating a couple of fully connected (fc) layers in the",
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+                                    "score": 1.0,
+                                    "content": "space along the light ray. Since there are no depth scans paired with RGB images, it is impossible",
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+                        },
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+                                    "content": "preliminarily regard the pixel features as the representation of every location along the light ray",
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+                                    "score": 1.0,
+                                    "content": "in 3D space. With this simple formulation, every 3D point can theoretically have a copy of its",
+                                    "type": "text"
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+                        },
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+                        },
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+                                    "content": "is no information observed. In fact, we empirically find that the nearest interpretation can also",
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+                                    "content": "achieve good performance, but it is only applicable for relatively small-scale scenes.",
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+                                    "content": "Overall, the above simple reprojection operation explicitly retains the extracted 2D pixel features",
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+                                    "content": ", it is able to retrieve a feature vector from each input image. However,",
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+                                    "score": 1.0,
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+                                    "content": "\\{ \\mathbf { F } _ { p } ^ { 1 } \\cdot \\cdot \\cdot \\mathbf { F } _ { p } ^ { m } \\cdot \\cdot \\cdot \\mathbf { F } _ { p } ^ { M } \\}",
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+                                    "score": 1.0,
+                                    "content": ". Formally, it is defined as:",
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+                                    "score": 0.8,
+                                    "content": "\\hat { \\mathbf { F } } _ { p } ^ { m } = M L P s ( \\mathbf { F } _ { p } ^ { m } \\oplus [ x _ { p } , y _ { p } , z _ { p } ] ) , \\quad ( \\oplus \\mathrm { m e a n s ~ c o n c a t e n a t i o n ~ i n ~ n e u r a l ~ n e t s } )",
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+                                    "content": "• After obtaining the new set of position-aware features",
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+                                    "score": 0.9,
+                                    "content": "\\{ \\hat { \\mathbf { F } } _ { p } ^ { 1 } \\cdot \\cdot \\cdot \\hat { \\mathbf { F } } _ { p } ^ { m } \\cdot \\cdot \\cdot \\hat { \\mathbf { F } } _ { p } ^ { M } \\}",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": ", we use the existing",
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+                                    "score": 1.0,
+                                    "content": "attention aggregation methods such as AttSets (Yang et al., 2020) and Slot Attention (Locatello",
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+                        },
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+                                    "score": 1.0,
+                                    "content": "et al., 2020) to compute a unique feature vector",
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+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": ". Basically, the atten-",
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+                                    "score": 1.0,
+                                    "content": "tion mechanism learns a unique weight for all input features and then aggregates them together.",
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+                                    "score": 1.0,
+                                    "content": "Given a query 3D point",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": ", it is able to retrieve a feature vector from each input image. However,",
+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": ". Formally, it is defined as:",
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+                                    "score": 0.8,
+                                    "content": "\\hat { \\mathbf { F } } _ { p } ^ { m } = M L P s ( \\mathbf { F } _ { p } ^ { m } \\oplus [ x _ { p } , y _ { p } , z _ { p } ] ) , \\quad ( \\oplus \\mathrm { m e a n s ~ c o n c a t e n a t i o n ~ i n ~ n e u r a l ~ n e t s } )",
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+                                    "content": "• After obtaining the new set of position-aware features",
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+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": ", we use the existing",
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+                        },
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+                                    "score": 1.0,
+                                    "content": "et al., 2020) to compute a unique feature vector",
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+                                    "type": "inline_equation"
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+                                    "score": 1.0,
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+                                    "score": 1.0,
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+                                    "content": "Following the experimental settings of SRNs (Sitzmann et al., 2019), we firstly evaluate our GRF on",
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+                                    "score": 1.0,
+                                    "content": "the chair and car classes of ShapeNetv2. Particularly, the chair has 4612 objects for training, 662 for",
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+                                    "content": "validation and 1317 for testing, while the car has 2151 objects for training, 352 for validation and",
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+                                    "score": 1.0,
+                                    "content": "704 for testing. Each training object has randomly sampled 50 images with a resolution of",
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+                                    "score": 0.88,
+                                    "content": "1 2 8 \\times 1 2 8",
+                                    "type": "inline_equation"
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+                            "index": 21
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+                                    "content": "pixels. We conduct the following three groups of experiments.",
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+                                    "score": 1.0,
+                                    "content": "• Group 1: Novel-view synthesis of seen objects in the training split of the same category. In",
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+                                    "score": 1.0,
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+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "testing, the model is fed with 50 trained views and infers another 251 novel views for evaluation.",
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+                                    "score": 1.0,
+                                    "content": "• Group 2: Novel-view synthesis of unseen objects in the testing split of the same category. The",
+                                    "type": "text"
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+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "trained two models in Group 1 are tested on novel objects of the same category. During testing,",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "the model is fed with 2 novel views of each novel object, inferring 251 novel views for evaluation.",
+                                    "type": "text"
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+                            "index": 29
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+                        {
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+                                    "score": 1.0,
+                                    "content": "• Group 3: Similar to Group 2, but only 1 novel view is fed into the model for novel view synthesis.",
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+                    ],
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+                                    "score": 1.0,
+                                    "content": "Basically, the experiments of Group 1, 2 and 3 are exactly the same as in SRNs (Sitzmann et al.,",
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+                                    "score": 1.0,
+                                    "content": "2019) which is also the state of the art on this ShapeNetv2 dataset. Table 1 compares the quantitative",
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+                                    "score": 1.0,
+                                    "content": "results of our GRF and four baselines. Note that, the recent NeRF and NSVF are not scalable to learn",
+                                    "type": "text"
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+                            "index": 33
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+                                    "score": 1.0,
+                                    "content": "Analysis. 1) Our method outperforms SRNs on 50-view reconstruction, demonstrating the strong",
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+                                    "content": "capability of our GRF for 3D scene representation. Figure 3 shows the qualitative results. 2) Our",
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+                                    "content": "is because our GRF solves a much harder problem than SRNs. In particular, our network directly",
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+                                    "score": 1.0,
+                                    "content": "on all novel scenes to optimize the latent code. As a result, the experiments of Group 2 and 3 are",
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+                                    "score": 1.0,
+                                    "content": "significantly in favour of SRNs. 3) To further demonstrate the advantage of GRF over SRNs, we",
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "directly evaluate the trained SRNs model on unseen objects (of the same category) without retrain-",
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+                                    "content": "Figure 4 shows the qualitative results. It can be seen that if not retrained, SRNs completely fails",
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+                                    "content": "4 EVALUATION",
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+                                    "content": "704 for testing. Each training object has randomly sampled 50 images with a resolution of",
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+                                    "score": 0.88,
+                                    "content": "1 2 8 \\times 1 2 8",
+                                    "type": "inline_equation"
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+                            "index": 21
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+                                    "content": "• Group 1: Novel-view synthesis of seen objects in the training split of the same category. In",
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+                        {
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+                                    "content": "trained two models in Group 1 are tested on novel objects of the same category. During testing,",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "• Group 3: Similar to Group 2, but only 1 novel view is fed into the model for novel view synthesis.",
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+                    ],
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+                                    "content": "Basically, the experiments of Group 1, 2 and 3 are exactly the same as in SRNs (Sitzmann et al.,",
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+                                    "score": 1.0,
+                                    "content": "2019) which is also the state of the art on this ShapeNetv2 dataset. Table 1 compares the quantitative",
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "results of our GRF and four baselines. Note that, the recent NeRF and NSVF are not scalable to learn",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 33
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+                                    "content": "large number of scenes simultaneously because each scene is encoded into the network parameters.",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "is because our GRF solves a much harder problem than SRNs. In particular, our network directly",
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+                            ],
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+                        {
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+                                    "score": 1.0,
+                                    "content": "infers the novel scene representation in a single forward pass, while SRNs needs to be retrained",
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+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "on all novel scenes to optimize the latent code. As a result, the experiments of Group 2 and 3 are",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 40
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "significantly in favour of SRNs. 3) To further demonstrate the advantage of GRF over SRNs, we",
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+                                }
+                            ],
+                            "index": 41
+                        },
+                        {
+                            "bbox": [
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+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "directly evaluate the trained SRNs model on unseen objects (of the same category) without retrain-",
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+                                    "score": 1.0,
+                                    "content": "ing. For comparison, we also directly evaluate the trained GRF model on the same novel objects.",
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "Figure 4 shows the qualitative results. It can be seen that if not retrained, SRNs completely fails",
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+                            ],
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+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "to reconstruct unseen car instances, but randomly generates similar cars from learned prior knowl-",
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+                                }
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+                                    "content": "and the inaccuracy of estimated camera poses. Therefore, it is non-trivial to conduct real-world ex-",
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+                                    "content": "4.4 ABLATION STUDY",
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+                                    "content": "• Group 4: The full model is trained and tested with the same settings for comparison.",
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+                                    "content": "Analysis. Table 5 compares the performance of all ablated models. We can see that: 1) The greatest",
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+                                    "content": "for inferring the novel pixel RGB. In particular, we conduct the following experiment using our GRF",
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+                                    "content": "model trained on ShapeNetv2 Cars. In this case, the AttSets attention module is used (details are in",
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+                                    "score": 1.0,
+                                    "content": "Table 9). Given a query light ray, multiple 3D points are sampled to query the network.",
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+                                    "content": "Then we compute the M feature vectors from the input M views for these surface points.",
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+                                    "content": "Thirdly, the attention masks for those M feature vectors are computed. We identify the view whose",
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+                                    "score": 1.0,
+                                    "content": "sum of the attention mask along the feature axis is greatest as the main contributor for inferring",
+                                    "type": "text"
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+                                    "score": 1.0,
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "After querying light rays for each pixel, we obtain a rendered RGB image. At the same time, for",
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+                            ],
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+                                    "score": 1.0,
+                                    "content": "each pixel of that image, we select the most important view from the M input views for the surface-",
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+                            ],
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+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "intersection point along the ray from the viewpoint through that pixel, according to the maximal",
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+                                    "score": 1.0,
+                                    "content": "attention score. Eventually, we obtain a Max Attention Map corresponding to the rendered RGB",
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+                        },
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+                                    "content": "Figure 7 shows the qualitative results of the above experiment. In particular, we feed the three im-",
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "then render a new image (e.g., the 5th image in Figure 7). Note that, we carefully select the input 3",
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+                            "index": 23
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+                            "bbox": [
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+                            "index": 24
+                        },
+                        {
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+                    ],
+                    "index": 23,
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+                                    "score": 1.0,
+                                    "content": "to truly deal with the visual occlusion. For example, when inferring the windshield of the car, the",
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+                                            "content": "Figure 7: Visualization of Max Attention Map from Multiple Input Images of a Novel Object for",
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+                                    "score": 1.0,
+                                    "content": "from an arbitrary number of input views. In order to evaluate how our GRF is able to generalize",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "with a variable number of input views, especially when there is a very sparse number of views with",
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+                                    "content": "severe visual occlusions, we conduct the following four groups of experiments.",
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+                                    "score": 1.0,
+                                    "content": "• 1-view Reconstruction. We feed the a single image of a novel car into our GRF model which",
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+                        },
+                        {
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+                        {
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+                                    "score": 1.0,
+                                    "content": "from vastly different viewing angles. This is the extreme case where the majority of the object is",
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+                                    "type": "text"
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+                                    "score": 1.0,
+                                    "content": "• 2-view Reconstruction. Similarly, we feed only two images of the novel car into the same model",
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+                            ],
+                            "index": 9
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "and render the same 9 novel views. In this case, more information is given to the network, but",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "there are still many parts occluded.",
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+                                    "score": 1.0,
+                                    "content": "• 5-view / 10-view Reconstruction. The same GRF model is fed with 5 and 10 views of the novel",
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+                                }
+                            ],
+                            "index": 12
+                        },
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new file mode 100644
index 0000000000000000000000000000000000000000..52f39de758e53334342c9f7fb2556d0bbc7bb561
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@@ -0,0 +1,495 @@
+# MOLLIFYING NETWORKS
+
+Caglar Gulcehre1, Marcin Moczulski2,∗, Francesco Visin3,∗ Yoshua Bengio1 1 University of Montreal, 2 University of Oxford, 3 Politecnico di Milano
+
+# ABSTRACT
+
+The optimization of deep neural networks can be more challenging than the traditional convex optimization problems due to highly non-convex nature of the loss function, e.g. it can involve pathological landscapes such as saddle-surfaces that can be difficult to escape from for algorithms based on simple gradient descent. In this paper, we attack the problem of optimization of highly non-convex neural networks objectives by starting with a smoothed – or mollified – objective function which becomes more complex as the training proceeds. Our proposition is inspired by the recent studies in continuation methods: similarly to curriculum methods, we begin by learning an easier (possibly convex) objective function and let it evolve during training until it eventually becomes the original, difficult to optimize objective function. The complexity of the mollified networks is controlled by a single hyperparameter that is annealed during training. We show improvements on various difficult optimization tasks and establish a relationship between recent works on continuation methods for neural networks and mollifiers.
+
+# 1 INTRODUCTION
+
+In the last few years, deep neural networks – i.e. convolutional networks (LeCun et al., 1989), LSTMs (Hochreiter & Schmidhuber, 1997a) or GRUs (Cho et al., 2014) – set the state of the art on a range of challenging tasks (Szegedy et al., 2014; Visin et al., 2015; Hinton et al., 2012; Sutskever et al., 2014; Bahdanau et al., 2014; Mnih et al., 2013; Silver et al., 2016). However when trained with variants of SGD (Bottou, 1998) deep networks can be difficult to optimize due to their highly non-linear and non-convex nature (Choromanska et al., 2014; Dauphin et al., 2014).
+
+A number of approaches were proposed to alleviate the difficulty of optimization: addressing the problem of the internal covariate shift with Batch Normalization (Ioffe & Szegedy, 2015), learning with a curriculum (Bengio et al., 2009), recently an approach to train RNNs with diffusion process (Mobahi, 2016), and graduated optimization (Hazan et al., 2015). The impact of noise injection on the behavior of modern deep learning methods has been explored by Neelakantan et al. (2015a). Hazan et al. (2015) have shown that injecting a particular noise and scheduling it carefully, can guarantee the convergence in $O ( 1 / \sigma ^ { 2 } \epsilon ^ { 2 } )$ steps for $\epsilon$ -optimal and $\sigma$ -nice functions. Similar to our work graduated optimization optimizes a smoothed objective function without performing expensive convolutions. Injecting noise to the activation functions and scheduling it have been recently shown to improve the performance on a wide variety of tasks (Gulcehre et al., 2016).
+
+We connect the ideas of curriculum learning and continuation methods with those arising from models with skip connections and using layers that compute near-identity transformations. Skip connections allow to train very deep residual and highway architectures (He et al., 2015; Srivastava et al., 2015) by skipping layers or block of layers. Similarly, it has been shown that stochastically changing the depth of a network during training (Huang et al., 2016b) does not prevent convergence and enables better generalization performance.
+
+We discuss the idea of mollification for neural networks – a form of differentiable smoothing of the loss function connected to noisy activations – which in our case can be interpreted as a form of adaptive noise injection which is controlled by a single hyperparameter. Inspired by Huang et al. (2016b), we use a hyperparameter to stochastically control the depth of our network. This allows us to start the optimization from a convex objective function (as long as the optimized criterion is convex, e.g. linear or logistic regression) and to slowly introduce more complexity into the model by annealing the hyperparameter, thus making the network deeper and increasingly non-linear.
+
+![](images/a43e573759ef754c90bf6fbf3615df4808ea481cc1b4c0fe1226a5cfec94d678.jpg)  
+Figure 1: A sequence of optimization problems of increasing complexity, where the first ones are easy to solve but only the last one corresponds to the actual problem of interest. It is possible to tackle the problems in order, starting each time at the solution of the previous one and tracking the local minima along the way.
+
+An important difference of our work compared to injecting noise to the gradients as it is explored in (Hazan et al., 2015; Neelakantan et al., 2015b) is that we inject the noise in the forward computation of the graph and we shape the cost function directly. As a result the cost function for the mollified network both at the test time and during the training are consistent and this makes the early-stopping much easier.
+
+# 2 MOLLIFYING OBJECTIVE FUNCTIONS
+
+# 2.1 CONTINUATION AND ANNEALING METHODS
+
+Continuation methods and simulated annealing provide a general strategy to reduce the impact of local minima and deal with non-convex, continuous, but not necessarily everywhere differentiable objective functions by smoothing the original objective function and gradually reducing the amount of smoothing during training (Allgower & Georg, 1980) (see Fig. 1).
+
+In machine learning, approaches based on curriculum learning (Bengio et al., 2009) are inspired by this principle and define a sequence of gradually more difficult training tasks (or training distributions) that eventually converge to the task of interest.
+
+In the context of stochastic gradient descent, we use a stochastic estimation of the gradient for the smoothed objective function. This is convenient because it may not be analytically feasible to compute the smoothed function, but a Monte-Carlo estimate can often be obtained easily.
+
+In this paper we construct a sequence of smoothed objective functions obtained with a form of mollification and we progressively optimize them. The training procedure iterates over the sequence of objective functions starting from the simpler ones – i.e. with a smoother loss surface – and moving towards more complex ones until the last, original, objective function is reached.1
+
+# 2.2 MOLLIFIERS AND WEAK GRADIENTS
+
+We smooth the loss function $\mathcal { L }$ , which is parametrized by $\boldsymbol \theta \in \mathbb { R } ^ { n }$ , by convolving it with another function $K ( \cdot )$ with stride $\tau \in \mathbb { R } ^ { n }$ :
+
+$$
+\mathscr { L } _ { K } ( \pmb { \theta } ) = ( \mathscr { L } * K ) ( \pmb { \theta } ) = \int _ { - \infty } ^ { + \infty } \mathscr { L } ( \pmb { \theta } - \pmb { \tau } ) K ( \pmb { \tau } ) d \pmb { \tau }
+$$
+
+Although there are many choices for the function $K ( \cdot )$ , we focus on those that satisfy the definition of a mollifier.
+
+A mollifier is an infinitely differentiable function that behaves like an approximate identity in the group of convolutions of integrable functions. If $K ( \cdot )$ is an infinitely differentiable function, that converges to the Dirac delta function when appropriately rescaled and for any integrable function $\mathcal { L }$ , then it is a mollifier:
+
+$$
+\mathcal { L } ( \pmb \theta ) = \operatorname* { l i m } _ { \epsilon  0 } \int \epsilon ^ { - n } K ( \pmb \tau / \epsilon ) \mathcal { L } ( \pmb \theta - \pmb \tau ) d \pmb \tau .
+$$
+
+If we choose $K ( \cdot )$ to be a mollifier and obtain the smoothed loss function $\mathcal { L } _ { K }$ as in Eqn. 1, we can take its gradient with respect to $\pmb \theta$ using directly the result from Evans (1998):
+
+$$
+\nabla _ { \pmb { \theta } } \mathcal { L } _ { K } ( \pmb { \theta } ) = \nabla _ { \pmb { \theta } } ( \mathcal { L } \ast K ) ( \pmb { \theta } ) = ( \mathcal { L } \ast \nabla K ) ( \pmb { \theta } ) .
+$$
+
+To relate the resulting gradient $\nabla _ { \pmb { \theta } } \mathcal { L } _ { K }$ to the gradient of the original function $\mathcal { L }$ , we introduce the notion of weak gradient, i.e. an extension to the idea of weak/distributional derivatives to functions with multidimensional arguments, such as loss functions of neural networks.
+
+For an integrable function $\mathcal { L }$ in space ${ \mathcal { L } } \in L ( [ a , b ] )$ , ${ \mathrm { ~  ~ g ~ } } \in \ L ( [ a , b ] ^ { n } )$ is a $n$ -dimensional weak gradient of $\mathcal { L }$ if it satisfies:
+
+$$
+\int _ { C } \mathbf { g } ( \pmb { \tau } ) K ( \pmb { \tau } ) d \pmb { \tau } = - \int _ { C } \mathcal { L } ( \pmb { \tau } ) \nabla K ( \pmb { \tau } ) d \pmb { \tau } ,
+$$
+
+where $K ( \tau )$ is an infinitely differentiable function vanishing at infinity, $C \in [ a , b ] ^ { n }$ and $\tau \in \mathbb { R } ^ { n }$
+
+As long as the chosen $K ( \cdot )$ fulfills the definition of a mollifier we can use Eqn. 3 and Eqn. $4 ^ { 2 }$ to rewrite the gradient as:
+
+$$
+\begin{array} { r l r } { \nabla _ { \pmb { \theta } } \mathcal { L } _ { K } ( \pmb { \theta } ) = ( \mathcal { L } \ast \nabla K ) ( \pmb { \theta } ) } & { } & { \mathrm { b y ~ E q n . ~ 3 } } \\ { = \displaystyle \int _ { C } \mathcal { L } ( \pmb { \theta } - \pmb { \tau } ^ { } ) \nabla K ( \pmb { \tau } ^ { } ) d \pmb { \tau } } \\ { = - \displaystyle \int _ { C } \mathbf { g } ( \pmb { \theta } - \pmb { \tau } ^ { } ) K ( \pmb { \tau } ^ { } ) d \pmb { \tau } } & { } & { \mathrm { b y ~ E q n . ~ 4 } } \end{array}
+$$
+
+For a differentiable almost everywhere function $\mathcal { L }$ , the weak gradient $g ( \pmb \theta )$ is equal to $\nabla _ { \boldsymbol { \theta } } \mathcal { L }$ almost everywhere. With a slight abuse of notation we can therefore write:
+
+$$
+\nabla _ { \pmb { \theta } } \mathcal { L } _ { K } ( \pmb { \theta } ) = - \int _ { C } \nabla _ { \pmb { \theta } } \mathcal { L } ( \pmb { \theta } - \pmb { \tau } ) K ( \pmb { \tau } ) d \pmb { \tau }
+$$
+
+# 2.3 GAUSSIAN MOLLIFIERS
+
+It is possible to use the standard Gaussian distribution $\mathcal { N } ( 0 , \bf { I } )$ as a mollifier $K ( \cdot )$ , as it satisfies the desired properties: it is infinitely differentiable, a sequence of properly rescaled Gaussian distributions converges to the Dirac delta function and it vanishes in infinity. With such a $K ( \cdot )$ the gradient becomes:
+
+$$
+\begin{array} { r l } {  { \nabla _ { \theta } \mathcal { L } _ { K = \mathcal { N } } ( \theta ) = - \int _ { C } \nabla _ { \theta } \mathcal { L } ( \theta - \tau ) p ( \tau ) d \tau } } \\ & { = \mathrm { E } _ { \tau } [ \nabla _ { \theta } \mathcal { L } ( \theta - \tau ) ] , \mathrm { w i t h } \ \tau \sim \mathcal { N } ( 0 , \ \mathbf { I } ) } \end{array}
+$$
+
+Exploiting the fact that a Gaussian distribution is a mollifier, we can focus on a sequence of mollifications indexed by scaling parameter $\epsilon$ introduced in Eqn. 2. A single element of this sequence takes the following form:
+
+$$
+\begin{array} { r l } & { \nabla _ { \boldsymbol { \theta } } \mathcal { L } _ { \mathcal { N } , \epsilon } ( \boldsymbol { \theta } ) = - \displaystyle \int _ { C } \nabla _ { \boldsymbol { \theta } } \mathcal { L } ( \boldsymbol { \theta } - \boldsymbol { \tau } ) \epsilon ^ { - 1 } p ( \boldsymbol { \tau } / \epsilon ) d \boldsymbol { \tau } } \\ & { ~ = \mathrm { E } _ { \boldsymbol { \tau } } [ \nabla _ { \boldsymbol { \theta } } \mathcal { L } ( \boldsymbol { \theta } - \boldsymbol { \tau } ) ] , \mathrm { w i t h } \boldsymbol { \tau } \sim \mathcal { N } ( 0 , \epsilon ^ { 2 } \mathbf { I } ) } \end{array}
+$$
+
+Replacing $\epsilon$ with $\sigma$ yields a sequence of mollifications indexed by $\sigma$
+
+$$
+\nabla _ { \pmb { \theta } } \mathcal { L } _ { \mathcal { N } , \sigma } ( \pmb { \theta } ) = \mathrm { E } _ { \pmb { \tau } } [ \nabla _ { \pmb { \theta } } \mathcal { L } ( \pmb { \theta } - \pmb { \tau } ) ] , \mathrm { w i t h } \ \pmb { \tau } \sim \mathcal { N } ( 0 , \ \sigma ^ { 2 } \mathbf { I } )
+$$
+
+with the following property (by Eqn. 2):
+
+$$
+\operatorname* { l i m } _ { \sigma \to 0 } \nabla _ { \pmb { \theta } } \mathcal { L } _ { \mathcal { N } , \sigma } ( \pmb { \theta } ) = \nabla _ { \pmb { \theta } } \mathcal { L } ( \pmb { \theta } )
+$$
+
+An intuitive interpretation of the result is that $\sigma$ determines the standard deviation of a mollifying Gaussian and is annealed in order to construct a sequence of gradually less "blurred" and closer approximations to $\mathcal { L }$ . This is consistent with the property that when $\sigma$ is annealed to zero we are optimizing the original function $\mathcal { L }$ .
+
+So far we obtained the mollified version $\mathcal { L } _ { K } ( \pmb { \theta } )$ of the cost function $\mathcal { L } ( \pmb { \theta } )$ by convolving it with a mollifier $K ( \pmb \theta )$ . The kernel $K ( \pmb \theta )$ corresponds to the average effect of injecting noise $\bar { \xi }$ sampled from standard Normal distribution. The amount of noise controls the amount of smoothing. Gradually reducing the noise during training is related to a form of simulated annealing (Kirkpatrick et al., 1983). Similarly to the analysis in Mobahi (2016), we can write a Monte-Carlo estimate of $\begin{array} { r } { \mathcal { L } _ { K } ( \pmb { \theta } ) = ( \mathcal { L } * K ) ( \pmb { \theta } ) \approx \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \mathcal { L } ( \pmb { \theta } - \pmb { \xi } ^ { ( i ) } ) } \end{array}$ . We provide the derivation and the gradient of this equation in Appendix A.
+
+The Monte-Carlo estimators of the mollifiers can be easily implemented with neural networks, where the layers typically have the form:
+
+$$
+\mathbf { h } ^ { l } = \mathbf { f } ( \mathbf { W } ^ { l } \mathbf { h } ^ { l - 1 } )
+$$
+
+with $\mathbf { h } ^ { l - 1 }$ a vector of activations from the previous layer in the hierarchy, $\mathbf { W } ^ { l }$ a matrix representing a linear transformation and f an element-wise non-linearity of choice.
+
+A mollification of such a layer can be formulated as:
+
+$$
+\mathbf { h } ^ { l } = \mathbf { f } ( ( \mathbf { W } ^ { l } - \boldsymbol { \xi } ^ { l } ) \mathbf { h } ^ { l - 1 } ) , \mathrm { w h e r e \ } \boldsymbol { \xi } ^ { l } \sim \mathcal { N } ( \mu , \boldsymbol { \sigma } ^ { 2 } )
+$$
+
+From Eqn. 16, it is easy to see that both weight noise methods proposed by Hinton $\&$ van Camp (1993) and Graves (2011) can be seen as a variation of Monte-Carlo estimate of mollifiers.
+
+# 2.4 GENERALIZED AND NOISY MOLLIFIERS
+
+We introduce a generalization of the concept of mollifiers that encompasses the approach we explored here and that is targeted during optimization via a continuation method using stochastic gradient descent.
+
+Definition 2.1. (Generalized Mollifier). A generalized mollifier is an operator, where $T _ { \sigma } ( f )$ defines a mapping between two functions, such that $T _ { \sigma } : f \to f ^ { * }$ .
+
+$$
+\operatorname* { l i m } _ { \sigma \to 0 } T _ { \sigma } f = f ,
+$$
+
+$$
+\begin{array} { r l } { { f ^ { 0 } = \displaystyle \operatorname* { l i m } _ { \sigma \to \infty } T _ { \sigma } f \quad } } & { { \mathrm { ~ i s ~ \ a n ~ i d e n t i t y ~ f u n c t i o n ~ } } } \\ { { \frac { \partial ( T _ { \sigma } f ) ( x ) } { \partial x } \quad } } & { { \mathrm { e x i s t s ~ } \forall x , \sigma > 0 } } \end{array}
+$$
+
+In addition, we consider noisy mollifiers which can be defined as an expected value of a stochastic function $\phi ( x , \xi )$ under some noise source $\xi$ with variance $\sigma$ :
+
+$$
+( T _ { \sigma } f ) ( x ) = E _ { \xi } [ \phi ( x , \xi _ { \sigma } ) ]
+$$
+
+Definition 2.2. (Noisy Mollifier). We call a stochastic function $\phi ( x , \xi _ { \sigma } )$ with input $x$ and noise $\xi$ a noisy mollifier if its expected value corresponds to the application of a generalized mollifier $T _ { \sigma }$ , as per Eqn. 20.
+
+The composition of two noisy mollifiers sharing the same $\sigma$ is also a noisy mollifier, since the three properties in the definition (Eqns. 17,18,19) are still satisfied. When $\sigma = 0$ no noise is injected and therefore the original function will be optimized. If $\sigma \to \infty$ instead, the function will become an identity function. Thus, for instance, if we mollify each layer of a feed-forward network except the output layer, when $\sigma \to \infty$ all the mollified layers will become identity function and the objective function of the network with respect to its inputs will be convex.
+
+Consequently, corrupting separately the activation function of each level of a deep neural network (but with a shared noise level $\sigma$ ) and annealing $\sigma$ yields a noisy mollifier for the objective function. This is related to the work of Mobahi (2016), who recently introduced a way of analytically smoothing of the non-linearities to help the training of recurrent networks. The differences of that approach from our algorithm is two-fold: we use a noisy mollifier (rather than an analytic smoothing of the network’s non-linearities) and we introduce (in the next section) a particular form of the noisy mollifier that empirically proved to work well.
+
+Shaping the cost function to define a sequence of costs that are progressing from easier to more difficult ones can be related to the reward shaping $\mathrm { N g }$ et al., 1999; $\bar { \bf N } \mathrm { \bf g }$ , 2003) algorithms. In our algorithm, we shape the cost and the model architecture itself, rather than rewards or the targets in order to make the optimization easier. In that sense, reward shaping can be considered to be more closer to curriculum learning.
+
+# 3 METHOD
+
+We propose an algorithm to mollify the cost of a neural network which also addresses an important drawback of the previously proposed noisy training procedures: as the noise gets larger, it can dominate the learning process and lead the algorithm to perform a random walk on the energy landscape of the objective function. Conversely in our algorithm, as the noise gets larger gradient descent minimizes a simpler (e.g. convex) but still meaningful objective function.
+
+We define the desired behavior of the network in the limit cases where the noise is very large or very small, and modify the model architecture accordingly. Specifically, during training we minimize a sequence of increasingly complex noisy objectives $\bar { \mathrm { L } } = ( \bar { \mathcal { L } } ^ { 1 } ( \pmb { \theta } ; \xi _ { \sigma _ { 1 } } ) , \bar { \mathcal { L } } ^ { 2 } ( \pmb { \theta } ; \xi _ { \sigma _ { 2 } } ) , \cdots , \bar { \mathcal { L } } ^ { k } ( \pmb { \theta } ; \xi _ { \sigma _ { k } } ) )$ that we obtain by annealing the scale (variance) of the noise $\sigma _ { i }$ . Let us note that our algorithm satisfies the fundamental properties of the generalized and noisy mollifiers that we introduced earlier.
+
+We use a noisy mollifier based on our definition in Section 2.4. Instead of convolving the objective function with a kernel:
+
+1. We start the training by optimizing a convex objective function that is obtained by configuring all the layers between the input and the last cost layer to compute an identity function, i.e., by skipping both the affine transformations and the blocks followed by nonlinearities. 2. During training, the magnitude of noise which is proportional to $p$ is annealed, allowing to gradually evolve from identity transformations to linear transformations between the layers. 3. Simultaneously, as we decrease the $p$ , the noisy mollification procedure allows the element-wise activation functions to gradually change from linear to be nonlinear.
+
+# 4 SIMPLIFYING THE OBJECTIVE FUNCTION FOR FEEDFORWARD NETWORKS
+
+For every unit of each layer, we either copy the activation (output) of the corresponding unit of the previous layer (the identity path in Figure 2) or output a noisy activation $\tilde { \mathbf { h } } ^ { l }$ of a non-linear transformation of it $\psi ( \mathbf { h } ^ { l - 1 } , \boldsymbol { \xi } ; \bar { \mathbf { W } } ^ { l } )$ , where $\boldsymbol { \xi }$ is noise, $\bar { \mathbf { W } } ^ { l }$ is a weight matrix applied on $\mathbf { h } ^ { l - 1 }$ and $\pi$ is a vector of binary decisions for each unit (the convolutional path in Figure 2):
+
+$$
+\begin{array} { c } { { \tilde { \mathbf { h } } ^ { l } = \psi ( \mathbf { h } ^ { l - 1 } , \pmb { \xi } ; \mathbf { W } ^ { l } ) } } \\ { { \phi ( \mathbf { h } ^ { l - 1 } , \pmb { \xi } , \pi ^ { l } ; \mathbf { W } ^ { l } ) = \pi ^ { l } \odot \mathbf { h } ^ { l - 1 } + ( 1 - \pi ^ { l } ) \odot \tilde { \mathbf { h } ^ { l } } } } \\ { { \mathbf { h } ^ { l } = \phi ( \mathbf { h } ^ { l - 1 } , \pmb { \xi } , \pi ^ { l } ; \mathbf { W } ^ { l } ) . } } \end{array}
+$$
+
+To decide which path to take, for each unit in the network, a binary stochastic decision is taken by drawing from a Bernoulli distribution with probability dependent on the decaying value of $p ^ { l }$ :
+
+$$
+\pi ^ { l } \sim \mathrm { B e r n o u l l i } ( p ^ { l } )
+$$
+
+If the number of hidden units of layer $l - 1$ and layer $l + 1$ is not the same, we can either zero-pad layer $l - 1$ before feeding it into the next layer or apply a linear projection to obtain the right dimensionality.
+
+For $p ^ { l } = 1$ , the layer computes the identity function leading to a convex objective. If $p ^ { l } = 0$ the layer computes the original non-linear transformation unfolding the full capacity of the model. We call the connections that are introduced this way as unitwise stochastic skip connections(USSC).
+
+In DropIn layers (Smith et al., 2016) a binary random vector is sampled from a Bernoulli distribution to decide whether to introduce skip a connection from the layer below $l - 1$ for each layer $l$ and they use this as a regularization. As opposed to the USSC, DropIn layers do not necessarily do not necessarily achieve a convex objective function as the DropIn ratio $( \hat { p ^ { l } } )$ increases.
+
+The pseudo-code for the mollified activations is reported in Algorithm 1.
+
+![](images/80eecd4ac991add0cb642b20c46377ca05f8af859e791101be6113b09d06839f.jpg)  
+Figure 2: Top: Stochastic depth. Bottom: mollifying network. The dashed line represents the optional residual connection. In the top path, the input is processed with a convolutional block followed by a noisy activation function, while in the bottom path the original activation of the layer $l - 1$ is propagated untouched. For each unit, one of the two paths in picked according to a binary stochastic decision $\pi$ .
+
+# Algorithm 1 Activation of a unit $i$ at layer $l$
+
+1: $x _ { i } \gets \mathbf { w } _ { i } ^ { \top } \mathbf { h } ^ { l - 1 } + b _ { i }$ . an affine transformation of hl−1   
+2: $\Delta _ { i } \gets \mathbf { u } ( x _ { i } ) - \mathbf { f } ( x _ { i } )$ $\mathsf { D } \Delta _ { i }$ is a measure of a saturation of a unit   
+3: $\sigma ( x _ { i } ) \gets ( \mathrm { s i g m o i d } ( a _ { i } \Delta _ { i } ) - 0 . 5 ) ^ { 2 }$ . std of the injected noise depends on $\Delta _ { i }$   
+4: $\xi _ { i } \sim \mathcal { N } ( 0 , 1 )$ . sampling the noise from a basic Normal distribution   
+5: $s _ { i } \gets p ^ { l } c \sigma ( x _ { i } ) | \xi _ { i } |$ . Half-Normal noise controlled by $\sigma ( x _ { i } )$ , const. $c$ and prob-ty $p ^ { l }$   
+6: $\psi ( x _ { i } , \xi _ { i } ) \gets \mathrm { s g n } ( \mathbf { u } ^ { * } ( x _ { i } ) ) \mathrm { m i n } ( | \mathbf { u } ^ { * } ( x _ { i } ) |$ , $| \mathbf { f } ^ { * } ( x _ { i } ) + \mathrm { s g n } ( \mathbf { u } ^ { * } ( x _ { i } ) ) | \hat { s _ { i } } | | ) + \bar { \mathbf { u } } ( 0 )$ $\triangleright$ noisy activation   
+7: $\pi _ { i } ^ { l } \sim { \bf B e r n o u l l i } ( p ^ { l } ) \qquad \triangleright p ^ { l }$ controls the variance of the noise AND the prob of skipping a unit   
+8: $\ddot { h } _ { i } ^ { l } = \psi ( x _ { i } , \xi _ { i } )$ . $\tilde { h } _ { i } ^ { l }$ is a noisy activation candidate   
+9: $\phi ( \mathbf { h } ^ { l - 1 } , \xi _ { i } , \pi _ { i } ^ { l } ; \mathbf { w } _ { i } ) = \pi _ { i } ^ { l } h _ { i } ^ { l - 1 } + ( 1 - \pi _ { i } ^ { l } ) \tilde { h } _ { i } ^ { l }$ . make a HARD decision between $h _ { i } ^ { l - 1 }$ and $\tilde { h } _ { i } ^ { l }$
+
+# 5 LINEARIZING THE NETWORK
+
+In Section 2, we show that convolving the objective function with a particular kernel can be approximated by adding noise to the activation function. This method may suffer from excessive random exploration when the noise is very large.
+
+We address this issue by bounding the element-wise activation function $\operatorname { f } ( \cdot )$ with its linear approximation when the variance of the noise is very large, after centering it at the origin. The resulting function $\mathrm { f } ^ { \ast } ( \cdot )$ is bounded and centered around the origin.
+
+Note that centering the sigmoid or hard-sigmoid will make them symmetric with respect to the origin. With a proper choice of the standard deviation $\sigma ( \mathbf { h } )$ , the noisy activation function becomes a linear function of the input when $\mathfrak { p }$ is large, as illustrated by Figure 10.
+
+Let $\mathbf { u } ^ { * } ( x ) = \mathbf { u } ( x ) - \mathbf { u } ( 0 )$ , where ${ \bf u } ( 0 )$ is the offset of the function from the origin, and $x _ { i }$ the $i$ -th dimension of an affine transformation of the output of the previous layer $\mathbf { h } ^ { l - 1 }$ : $x _ { i } = \mathbf { w } _ { i } ^ { \top } \mathbf { h } ^ { l - 1 } + b _ { i }$ . Then:
+
+$$
+\psi ( x _ { i } , \xi _ { i } ; \mathbf { w } _ { i } ) = \mathrm { s g n } ( \mathbf { u } ^ { * } ( x _ { i } ) ) \mathrm { m i n } ( | \mathbf { u } ^ { * } ( x _ { i } ) | , | \mathbf { f } ^ { * } ( x _ { i } ) + \mathrm { s g n } ( \mathbf { u } ^ { * } ( x _ { i } ) ) | s _ { i } | | ) + \mathbf { u } ( 0 )
+$$
+
+The noise is sampled from a Normal distribution with mean 0 and whose standard deviation depends on $c$ :
+
+$$
+s _ { i } \sim \mathcal { N } ( 0 , p c \sigma ( x _ { i } ) )
+$$
+
+# 5.1 LINEARIZING RELU ACTIVATION FUNCTION
+
+We have a simpler form of the equations to linearize ReLU (Nair & Hinton, 2010) activation function when $p ^ { l } \to \infty \quad$ . Instead of the complicated Eqn. 23. We can use a simpler equation as in Eqn. 26 to
+
+![](images/644f80b2c2d62505206241eaafd9d373597c1578b8fc6d71a55ea6c773a2d3d5.jpg)  
+Figure 3: The figures show how to evolve the model to make it closer to a linear network. Arrows denote the direction of the noise pushing the activation function towards the linear function. a) The quasi-convex envelope established by a $| \mathrm { { s i g m o i d } ( \cdot ) | }$ around $| 0 . 2 5 x |$ . b) A depiction of how the noise pushes the sigmoid to become a linear function.
+
+achieve the linearization of the activation function when we have a very large noise in the activation function:
+
+$$
+\begin{array} { c } { s _ { i } = \underset { \mathbf { \sigma } } { \mathrm { m i n i m u m } } ( | x _ { i } | , p \sigma ( x _ { i } ) | \boldsymbol { \xi } | ) } \\ { \psi ( x _ { i } , \xi _ { i } , \mathbf { w } _ { i } ) = \mathbf { f } ( x _ { i } ) - s _ { i } } \end{array}
+$$
+
+# 6 MOLLIFYING LSTMS AND GRUS
+
+In a similar vein it is possible to smooth the objective functions of LSTM and GRU networks by starting the optimization procedure with a simpler objective function such as optimizing a word2vec, BoW-LM or CRF objective function at the beginning of training and gradually increasing the difficulty of the optimization by increasing the capacity of the network.
+
+For GRUs we set the update gate to $\textstyle { \frac { 1 } { t } }$ – where $t$ is the time-step index – and reset the gate to 1 if the noise is very large, using Algorithm 1. Similarly for LSTMs, we can set the output gate to 1 and input gate to $\textstyle { \frac { 1 } { t } }$ and forget gate to $\textstyle { 1 - { \frac { 1 } { t } } }$ when the noise is very large. The output gate is 1 or close to 1 when the noise is very large. This way the LSTM will behave like a BOW model. In order to achieve this behavior, the activations $\psi ( x _ { t } , \dot { \xi _ { i } } )$ of the gates can be formulated as:
+
+$$
+\boldsymbol { \psi } ( x _ { t } ^ { l } , \xi ) = \mathbf { f } ( x _ { t } ^ { l } + p ^ { l } \sigma ( x ) | \xi | )
+$$
+
+By using a particular formulation of $\sigma ( x )$ that constraints it to be in expectation over $\xi$ when $p ^ { l } = 1$ , we can obtain a function for $\gamma \in \mathbb { R }$ within the range of $\operatorname { f } ( \cdot )$ that is discrete in expectation, but still per sample differentiable:
+
+$$
+\sigma ( x _ { t } ^ { l } ) = \frac { \mathrm { f } ^ { - 1 } ( \gamma ) - x _ { t } ^ { l } } { \mathrm { E } _ { \xi } [ | \xi | ] }
+$$
+
+We provide the derivation of Eqn. 28 in Appendix B. The gradient of the Eqn. 28 will be a Monte-Carlo approximation to the gradient of $\bar { \bf f } ( { \bf x } _ { t } ^ { l } )$ .
+
+# 7 ANNEALING SCHEDULES FOR $p$
+
+We used a different schedule for each layer of the network, such that the noise in the lower layers will anneal faster. This is similar to the linearly decaying probability of layers in Huang et al. (2016b).
+
+Exponential Decay In our experiments, we focused on using an annealing schedule similar to the inverse sigmoid rule in Bengio et al. (2015) with $p _ { t } ^ { l }$ ,
+
+$$
+p _ { t } ^ { l } = 1 - \mathrm { e } ^ { - \frac { k v _ { t } l } { t L } }
+$$
+
+with hyper-parameter model. We stop anne $k \geq 0$ at wh $t ^ { t h }$ update for the he expected de $l ^ { t h }$ $L$ is the number of layers reaches some threshold the. In $\begin{array} { r } { p _ { t } = \sum _ { i = 1 } ^ { L } p _ { t } ^ { l } } \end{array}$ $\delta$ our experiment we set $v _ { t }$ to be a moving average of the $\mathrm { l o s s } ^ { 3 }$ of the network, but for some of our experimnets that resulted unstable behaviors in the training and thus we have to fix $v _ { t }$ to 1. An advantage of using running average of loss for the $v _ { t }$ is because the behavior of the loss/optimization can directly influence the annealing behavior of the network. Because we will have:
+
+$$
+\operatorname* { l i m } _ { { \bf v } _ { t } \to \infty } p _ { t } ^ { l } = 1 \mathrm {  ~ \ a n d , ~ } \operatorname* { l i m } _ { { \bf v } _ { t } \to 0 } p _ { t } ^ { l } = 0 .
+$$
+
+This has a desirable property: when the training-loss is high, the noise injected into the system will be large as well. As a result, the model is encouraged to do more exploration, while if the model converges the noise injected into the system by the mollification procedure will be zero.
+
+Square-root Decay The square root decaying of the learning rate is very popular. In our experiments we have tried to decay the plt according to min(pmin, 1 − q $\begin{array} { r } { 1 - \sqrt { \frac { t } { N _ { e p o c h s } } } ) } \end{array}$ where $t$ is the index of the iteration/epoch over the data and $N _ { e p o c h s }$ corresponds to the total number of iterations that the model will do on the dataset. $p _ { m i n }$ is hyperparameter to define the lower bound on the $p _ { t } ^ { l }$ .
+
+Linear Decay We have also experimented with linearly decaying the $p _ { t } ^ { l }$ via $\begin{array} { r } { \operatorname* { m i n } ( p _ { m i n } , 1 - \frac { t } { N _ { e p o c h s } } ) } \end{array}$ We have compared the plot of different annealing methods described in this paper as in Figure 4.
+
+![](images/0edec83fe7d1c958a8488792b79a2fed28b69de56bc31b6fc444f2b0511e2401.jpg)  
+Figure 4: We compare different annealing schedules with respect to the time in y-axis(iterations).
+
+Furthermore, in our experiments we observe that training with noisy mollifiers can potentially be helpful for the generalization. This can be due to the noise induced to the backpropagation through the noisy mollification, that makes SGD more likely to converge to a flatter-minima (Hochreiter & Schmidhuber, 1997b) because the noise will help it escape from sharper local minima.
+
+# 8 EXPERIMENTS
+
+In this section we mainly focus on training of difficult to optimize models, in particular deep MLPs with sigmoid or tanh activation functions. The details of the experimental procedure is provided in Appendix C.
+
+# 8.1 EXPLORATORY STUDY ON MNIST
+
+We train a thin deep neural network on MNIST (LeCun & Cortes, 1998) dataset with 72 hidden layers and 100 hidden units. We train our model with Adam(Kingma & Ba, 2014) optimizer and fix the learning rate of all the models to $3 e { - 4 }$ . We have used the same learning-rate for all the models in order to factor out the possibility that a model converges faster, due to the fact of using a larger learning rate.
+
+Firstly, in Figure 5, we investigate the effect of using different annealing schedules. Exponential decay converges faster compared to linear decay and square root decay of $p$ . We find it to be very unstable to train our model with linear and square root decay in particular for large $c$ values. Thus we have to use smaller $c$ value (20 instead of 100) to be able to train the model with causing it to diverge.
+
+![](images/be0d19fd9d12424b4a91c95146d3758bf25debfc209b87d8462abca2c20cfb26.jpg)  
+Figure 5: We compare the training performance of different types of annealing methods used with mollification procedure to anneal the parameter of the mollification $p$ . Decaying the $p$ exponentially achieves both better training and validation performance.
+
+In Figure 6, we show the effect of using the noisy training procedure that we have introduced by sampling a mask from Bernoulli and Gaussian distributions versu using the deterministic approximation of this noisy procedure which we also use for the test time but during the training as well.
+
+![](images/4b8a3ac20b57ecaf26bfbb68e0f1deb19984fdd2b9a2d6ea11e3f0a10189a871.jpg)  
+Figure 6: We show the learning curves of the model where we don’t inject the noise during the training and instead use the deterministic approximation for the mollification during the training as well. The differerence in terms of speed of learning is very small.
+
+In Figure 7, we compare the results obtained for the model using mollification obtained with or without batch normalization and feed-forward residual networks. The mollified model performs very closely to the MLP trained with residual connections and the batchnorm. However, using residual connections and batch-norm does not seem to improve the results.
+
+We have tried to run experiment with the Monte-Carlo approximation of the mollification which is derived in Appendix A, however when we start with large noise and anneal the noise during the training, the model was very unstable and training was diverging. If we start with small noise and anneal the magnitude of the noise during the training, we could not observe any effect of it on the training.
+
+![](images/24f40f65392ee90e6386375192de3d2b7240bb6ad5b1529448614804c614f410.jpg)  
+Figure 7: We investigate the effect of using batch norm and residual connection for mollifcation and compare against to the network with residual connections and batch-norm. The effect of batch norm on this task for mollification seems to be very small and training convergence performance of the all the approaches are very close to each other.
+
+![](images/c41840e048fd609433dd3b430329544afc0d1b86c1b1c7b4544f1462c89c0e19.jpg)  
+Figure 8: The learning curves of a 6-layers MLP with sigmoid activation function on 40 bit parity task.
+
+Table 1: CIFAR10 deep convolutional neural network.   
+
+<table><tr><td></td><td>Test Accuracy</td></tr><tr><td>Stochastic Depth</td><td>93.25</td></tr><tr><td>Mollified Convnet</td><td>92.45</td></tr><tr><td>ResNet</td><td>91.78</td></tr></table>
+
+# 8.2 DEEP MLP EXPERIMENTS
+
+Deep Parity Experiments Training neural networks on a high-dimensional parity problem can be challenging (Graves, 2016; Kalchbrenner et al., 2015). We experiment on forty dimensional (bits) parity problem with 6-layer MLP using sigmoid activation function. All the models are initialized with Glorot initialization Glorot et al. (2011) and trained with SGD with momentum. We compare an MLP with residual connections using batch normalization and a mollified network with sigmoid activation function. As can be seen in Figure 8, the mollified network converges faster.
+
+Deep Pentomino Pentomino is a toy-image dataset where each image has 3 Pentomino blocks. The task is to predict whether if there is a different shape in the image or not (Gülçehre & Bengio, 2013). The best reported result on this task with MLPs is $6 8 . 1 5 \%$ accuracy (Gulcehre et al., 2014). The same model as ours trained without noisy activation function and vanilla residual connections scored $6 9 . 5 \%$ accuracy, while our mollified version scored $7 5 . 1 5 \%$ accuracy after 100 epochs of training on the $8 0 k$ dataset.
+
+CIFAR10 We experimented with deep convolutional neural networks of 110-layers with residual blocks and residual connections comparing our model against ResNet and Stochastic depth. We adapted the hyperparameters of the Stochastic depth network from Huang et al. (2016a) and we used the same hyperparameters for our algorithm. We report the training and validation curves of the three models in Figure 10 and the best test accuracy obtained early stopping on validation accuracy over 500 epochs in Table 1. Our model achieves better generalization than ResNet. Stochastic depth achieves better generalization, but it might be possible to combine both and obtain better results.
+
+![](images/345e69a1318bab42ba15b8295abfbd900d2a22fb372d4cfae106a2e0cb0e9af6.jpg)  
+Figure 9: The training curve of a bidirectionalRNN that predicts the embedding corresponding to a sequence of characters.
+
+Table 2: 3-layered LSTM network on word-level language modeling for PTB.   
+
+<table><tr><td></td><td>Test PPL</td></tr><tr><td>LSTM</td><td>119.4</td></tr><tr><td>Mollified LSTM</td><td>115.7</td></tr></table>
+
+# 9 LSTM EXPERIMENTS
+
+Predicting the Character Embeddings from Characters Learning the mapping from sequences of characters to the word-embeddings is a difficult problem. Thus one needs to use a highly non-linear function. We trained a word2vec model on Wikipedia with embeddings of size 500 (Mikolov et al., 2014) with a vocabulary of size 374557.
+
+LSTM Language Modeling We evaluate our model on LSTM language modeling. Our baseline model is a 3-layer stacked LSTM without any regularization. We observed that mollified model converges faster and achieves better results. We provide the results for PTB language modeling in Table 2.
+
+# 10 CONCLUSION
+
+We propose a novel method for training neural networks inspired by an idea of continuation, smoothing techniques and recent advances in non-convex optimization algorithms. The method makes learning easier by starting from a simpler model, solving a well-behaved problem, and gradually transitioning to a more complicated setting. We show improvements on very deep models, difficult to optimize tasks and compare with powerful techniques such as batch-normalization and residual connections. We also show that the mollification procedure improves the generalization performance of the model on two tasks.
+
+Our future work includes testing this method on large-scale language tasks that require long training time, e.g., machine translation and language modeling. Moreover, (Kaiser & Sutskever, 2015) observed that the training of Neural-GPU model can be improved significantly by using gradient noise which can be related to the smoothing of the loss surface, it would be interesting to try mollification on this model to see if the training of Neural GPU can be made easier by using mollification procedure.
+
+# ACKNOWLEDGEMENTS
+
+We thank Nicholas Ballas and Misha Denil for the valuable discussions and their feedback. We would like to also thank the developers of Theano 4, for developing such a powerful tool for scientific computing Theano Development Team (2016). We acknowledge the support of the following organizations for research funding and computing support: NSERC, Samsung, Calcul Québec, Compute Canada, the Canada Research Chairs and CIFAR.
+
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+# Appendix
+
+# A MONTE-CARLO ESTIMATE OF MOLLIFICATION
+
+${ \mathcal L } _ { K } ( \pmb \theta ) = ( { \mathcal L } * K ) ( \pmb \theta ) = \int _ { C } { \mathcal L } ( \pmb \theta - \xi ) K ( \xi ) d \xi$ which can be estimated by a Monte Carlo: $\approx \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \mathcal { L } ( \pmb { \theta } - \pmb { \xi } ^ { ( i ) } )$ , where $\xi ^ { ( i ) }$ is a realization of the noise random variable $\xi$ yielding $\frac { \partial \mathcal { L } _ { K } ( { \pmb \theta } ) } { \partial { \pmb \theta } }$ $\approx \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \frac { \partial \mathcal { L } ( \pmb { \theta } - \pmb { \xi } ^ { ( i ) } ) } { \partial \pmb { \theta } } .$
+
+Therefore introducing additive noise to the input of $\mathcal { L } ( \pmb \theta )$ is equivalent to mollification.
+
+# B DERIVATION OF THE NOISY ACTIVATIONS FOR THE GATING
+
+Assume that $z _ { t } ^ { l } = x _ { t } ^ { l } + p _ { t } ^ { l } \sigma ( x ) | \xi _ { t } ^ { l } |$ and $\mathrm { E } _ { \xi } [ \psi ( x _ { t } ^ { l } , \xi ) ] = t$ . Thus for all $z _ { t } ^ { l }$ ,
+
+$$
+\begin{array} { r l } & { \mathrm { E } _ { \xi } [ \psi ( x _ { t } ^ { l } , \xi _ { t } ^ { l } ) ] = \mathrm { E } _ { \xi } [ \mathrm { f } ( z _ { t } ^ { l } ) ] , } \\ & { \qquad t = \mathrm { E } _ { \xi } [ \mathrm { f } ( z _ { t } ^ { l } ) ] , \ \mathrm { a s s u m i n g ~ f ( \cdot ) ~ b e h a v e s ~ s i m i l a r ~ t o ~ a ~ l i n e a r ~ f u n c t i o n } ; } \\ & { \qquad \mathrm { E } _ { \xi } [ \mathrm { f } ( z _ { t } ^ { l } ) ] \approx \mathrm { f } ( \mathrm { E } _ { \xi } [ z _ { t } ^ { l } ] ) \ \mathrm { s i n c e ~ w e ~ u s e ~ h a r d - s i g m o i d ~ f o r ~ f ( \cdot ) ~ t h i s ~ w i l l ~ h o l d } . } \\ & { \qquad \mathrm { f } ^ { - 1 } ( t ) \approx \mathrm { E } _ { \xi } [ z _ { t } ^ { l } ] } \end{array}
+$$
+
+As in Eqn. 32, we can write the expectation of this equation as:
+
+$$
+\mathrm { f } ^ { - 1 } ( t ) \approx x _ { t } ^ { l } + p _ { t } ^ { l } \sigma ( x ) \mathrm { E } _ { \xi } [ \xi _ { t } ^ { l } ]
+$$
+
+Corollary, the value that $\sigma ( x _ { t } ^ { l } )$ should take in expectation for $p _ { t } ^ { l } = 1$ would be:
+
+$$
+\sigma ( x _ { t } ^ { l } ) \approx \frac { \mathrm { f } ^ { - 1 } ( t ) - x _ { t } ^ { l } } { \mathrm { E } _ { \xi } [ \xi _ { t } ^ { l } ] }
+$$
+
+In our experiments for $\operatorname { f } ( \cdot )$ we used the hard-sigmoid activation function. We used the following piecewise activation function in order to use it as $\mathrm { f } ^ { - 1 } ( x ) = 4 ( x - 0 . 5 )$ . During inference we use the expected value of random variables $\pi$ and $\xi$ .
+
+# C EXPERIMENTAL DETAILS
+
+# C.1 MNIST
+
+The weights of the models are initialized with Glorot & Bengio initialization Glorot et al. (2011). We use the learning rate of $4 e - 4$ along with RMSProp. We initialize $a _ { i }$ parameters of mollified activation function by sampling it from a uniform distribution, $\mathrm { { U } [ - 2 , 2 ] }$ . We used 100 hidden units at each layer with a minibatches of size 500.
+
+# C.2 PENTOMINO
+
+We train a 6−layer MLP with sigmoid activation function using SGD and momentum. We used 200 units per layer with sigmoid activation functions. We use a learning rate of $1 e - 3$ .
+
+# C.3 CIFAR10
+
+We use the same model with the same hyperparameters for both ResNet, mollified network and the stochastic depth. We borrowed the hyperparameters of the model from Huang et al. (2016a). Our mollified convnet model has residual connections coming from its layer below.
+
+![](images/8dbf86661abcd350698fec3b04fd3a0769dbc7ba6ca3d6a790e7f74289b8e903.jpg)  
+Figure 10: Training and validation losses over 500 epochs of a mollified convolutional network composed by 110-layers. We compare against ResNet and Stochastic depth.
+
+# C.4 PARITY
+
+The n-dimensional parity task is the task to figure out whether the sum of n-bits in a binary vector is even or odd. We use SGD with Nesterov momentum and initialize the weight matrices by using Glorot&Bengio initializationGlorot et al. (2011). For all models, we use the learning rate of $1 e - 3$ and momentum of 0.92. $a _ { i }$ is the parameters of mollified activation function are initialized by sampling from uniform distribution, $\mathsf { \bar { U } } [ - 2 , 2 ]$ .
+
+# C.5 LSTM LANGUAGE MODELING
+
+We trained 2-layered LSTM language models on PTB word-level. We used the models with the same hyperparameters as in Zaremba & Sutskever (2014). We used the same hyperparameters for both the mollified LSTM language model and the LSTM. We use hard-sigmoid activation function for both the LSTM and mollified LSTM language model. We use hard-sigmoid activation function for the gates of the LSTM.
+
+# C.6 PREDICTING THE CHARACTER EMBEDDINGS FROM CHARACTERS
+
+We use $1 0 k$ of these words as a validation and another $1 0 k$ word embeddings as test set. We train a bidirectional-LSTM on top of each sequence of characters for each word and on top of the representation of bidirectional LSTM, we use a 5-layered tanh-MLP to predict the word-embedding.
+
+We train our models using RMSProp and momentum with learning rate of $6 e - 4$ and momentum 0.92. The size of the minibatches, we used is 64. As seen in Figure 9, mollified LSTM network converges faster.
\ No newline at end of file
diff --git a/parse/train/r1G4z8cge/r1G4z8cge_content_list.json b/parse/train/r1G4z8cge/r1G4z8cge_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..3d525baf36d12c269cdfc1def044865f172e185d
--- /dev/null
+++ b/parse/train/r1G4z8cge/r1G4z8cge_content_list.json
@@ -0,0 +1,2470 @@
+[
+    {
+        "type": "text",
+        "text": "MOLLIFYING NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            99,
+            482,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Caglar Gulcehre1, Marcin Moczulski2,∗, Francesco Visin3,∗ Yoshua Bengio1 1 University of Montreal, 2 University of Oxford, 3 Politecnico di Milano ",
+        "bbox": [
+            184,
+            143,
+            720,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            209,
+            544,
+            224
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The optimization of deep neural networks can be more challenging than the traditional convex optimization problems due to highly non-convex nature of the loss function, e.g. it can involve pathological landscapes such as saddle-surfaces that can be difficult to escape from for algorithms based on simple gradient descent. In this paper, we attack the problem of optimization of highly non-convex neural networks objectives by starting with a smoothed – or mollified – objective function which becomes more complex as the training proceeds. Our proposition is inspired by the recent studies in continuation methods: similarly to curriculum methods, we begin by learning an easier (possibly convex) objective function and let it evolve during training until it eventually becomes the original, difficult to optimize objective function. The complexity of the mollified networks is controlled by a single hyperparameter that is annealed during training. We show improvements on various difficult optimization tasks and establish a relationship between recent works on continuation methods for neural networks and mollifiers. ",
+        "bbox": [
+            233,
+            239,
+            766,
+            424
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            469,
+            336,
+            484
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In the last few years, deep neural networks – i.e. convolutional networks (LeCun et al., 1989), LSTMs (Hochreiter & Schmidhuber, 1997a) or GRUs (Cho et al., 2014) – set the state of the art on a range of challenging tasks (Szegedy et al., 2014; Visin et al., 2015; Hinton et al., 2012; Sutskever et al., 2014; Bahdanau et al., 2014; Mnih et al., 2013; Silver et al., 2016). However when trained with variants of SGD (Bottou, 1998) deep networks can be difficult to optimize due to their highly non-linear and non-convex nature (Choromanska et al., 2014; Dauphin et al., 2014). ",
+        "bbox": [
+            174,
+            496,
+            825,
+            577
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A number of approaches were proposed to alleviate the difficulty of optimization: addressing the problem of the internal covariate shift with Batch Normalization (Ioffe & Szegedy, 2015), learning with a curriculum (Bengio et al., 2009), recently an approach to train RNNs with diffusion process (Mobahi, 2016), and graduated optimization (Hazan et al., 2015). The impact of noise injection on the behavior of modern deep learning methods has been explored by Neelakantan et al. (2015a). Hazan et al. (2015) have shown that injecting a particular noise and scheduling it carefully, can guarantee the convergence in $O ( 1 / \\sigma ^ { 2 } \\epsilon ^ { 2 } )$ steps for $\\epsilon$ -optimal and $\\sigma$ -nice functions. Similar to our work graduated optimization optimizes a smoothed objective function without performing expensive convolutions. Injecting noise to the activation functions and scheduling it have been recently shown to improve the performance on a wide variety of tasks (Gulcehre et al., 2016). ",
+        "bbox": [
+            174,
+            583,
+            825,
+            715
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We connect the ideas of curriculum learning and continuation methods with those arising from models with skip connections and using layers that compute near-identity transformations. Skip connections allow to train very deep residual and highway architectures (He et al., 2015; Srivastava et al., 2015) by skipping layers or block of layers. Similarly, it has been shown that stochastically changing the depth of a network during training (Huang et al., 2016b) does not prevent convergence and enables better generalization performance. ",
+        "bbox": [
+            174,
+            722,
+            825,
+            801
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We discuss the idea of mollification for neural networks – a form of differentiable smoothing of the loss function connected to noisy activations – which in our case can be interpreted as a form of adaptive noise injection which is controlled by a single hyperparameter. Inspired by Huang et al. (2016b), we use a hyperparameter to stochastically control the depth of our network. This allows us to start the optimization from a convex objective function (as long as the optimized criterion is convex, e.g. linear or logistic regression) and to slowly introduce more complexity into the model by annealing the hyperparameter, thus making the network deeper and increasingly non-linear. ",
+        "bbox": [
+            174,
+            808,
+            825,
+            901
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/a43e573759ef754c90bf6fbf3615df4808ea481cc1b4c0fe1226a5cfec94d678.jpg",
+        "image_caption": [
+            "Figure 1: A sequence of optimization problems of increasing complexity, where the first ones are easy to solve but only the last one corresponds to the actual problem of interest. It is possible to tackle the problems in order, starting each time at the solution of the previous one and tracking the local minima along the way. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            383,
+            93,
+            614,
+            204
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "An important difference of our work compared to injecting noise to the gradients as it is explored in (Hazan et al., 2015; Neelakantan et al., 2015b) is that we inject the noise in the forward computation of the graph and we shape the cost function directly. As a result the cost function for the mollified network both at the test time and during the training are consistent and this makes the early-stopping much easier. ",
+        "bbox": [
+            173,
+            295,
+            825,
+            363
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 MOLLIFYING OBJECTIVE FUNCTIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            382,
+            521,
+            398
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 CONTINUATION AND ANNEALING METHODS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            410,
+            524,
+            424
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Continuation methods and simulated annealing provide a general strategy to reduce the impact of local minima and deal with non-convex, continuous, but not necessarily everywhere differentiable objective functions by smoothing the original objective function and gradually reducing the amount of smoothing during training (Allgower & Georg, 1980) (see Fig. 1). ",
+        "bbox": [
+            173,
+            425,
+            825,
+            479
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In machine learning, approaches based on curriculum learning (Bengio et al., 2009) are inspired by this principle and define a sequence of gradually more difficult training tasks (or training distributions) that eventually converge to the task of interest. ",
+        "bbox": [
+            174,
+            484,
+            825,
+            526
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the context of stochastic gradient descent, we use a stochastic estimation of the gradient for the smoothed objective function. This is convenient because it may not be analytically feasible to compute the smoothed function, but a Monte-Carlo estimate can often be obtained easily. ",
+        "bbox": [
+            176,
+            532,
+            823,
+            571
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper we construct a sequence of smoothed objective functions obtained with a form of mollification and we progressively optimize them. The training procedure iterates over the sequence of objective functions starting from the simpler ones – i.e. with a smoother loss surface – and moving towards more complex ones until the last, original, objective function is reached.1 ",
+        "bbox": [
+            174,
+            579,
+            825,
+            635
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.2 MOLLIFIERS AND WEAK GRADIENTS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            651,
+            473,
+            665
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We smooth the loss function $\\mathcal { L }$ , which is parametrized by $\\boldsymbol \\theta \\in \\mathbb { R } ^ { n }$ , by convolving it with another function $K ( \\cdot )$ with stride $\\tau \\in \\mathbb { R } ^ { n }$ : ",
+        "bbox": [
+            176,
+            675,
+            823,
+            704
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a9bac6416796a497b88c3e4ce6ee927d26156c0e77a5af446fc6899bdbcd59c5.jpg",
+        "text": "$$\n\\mathscr { L } _ { K } ( \\pmb { \\theta } ) = ( \\mathscr { L } * K ) ( \\pmb { \\theta } ) = \\int _ { - \\infty } ^ { + \\infty } \\mathscr { L } ( \\pmb { \\theta } - \\pmb { \\tau } ) K ( \\pmb { \\tau } ) d \\pmb { \\tau }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            710,
+            666,
+            747
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Although there are many choices for the function $K ( \\cdot )$ , we focus on those that satisfy the definition of a mollifier. ",
+        "bbox": [
+            174,
+            762,
+            823,
+            790
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A mollifier is an infinitely differentiable function that behaves like an approximate identity in the group of convolutions of integrable functions. If $K ( \\cdot )$ is an infinitely differentiable function, that converges to the Dirac delta function when appropriately rescaled and for any integrable function $\\mathcal { L }$ , then it is a mollifier: ",
+        "bbox": [
+            173,
+            795,
+            825,
+            849
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6d7039927a5655136929072640ca21c54c0698a5482845f0a56f0ac661aa3fd4.jpg",
+        "text": "$$\n\\mathcal { L } ( \\pmb \\theta ) = \\operatorname* { l i m } _ { \\epsilon  0 } \\int \\epsilon ^ { - n } K ( \\pmb \\tau / \\epsilon ) \\mathcal { L } ( \\pmb \\theta - \\pmb \\tau ) d \\pmb \\tau .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            362,
+            856,
+            635,
+            888
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "If we choose $K ( \\cdot )$ to be a mollifier and obtain the smoothed loss function $\\mathcal { L } _ { K }$ as in Eqn. 1, we can take its gradient with respect to $\\pmb \\theta$ using directly the result from Evans (1998): ",
+        "bbox": [
+            171,
+            103,
+            823,
+            131
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9ca4455527ad8e48cb0dc887449843656638ce5fa51a93e426c6b1d342cc6429.jpg",
+        "text": "$$\n\\nabla _ { \\pmb { \\theta } } \\mathcal { L } _ { K } ( \\pmb { \\theta } ) = \\nabla _ { \\pmb { \\theta } } ( \\mathcal { L } \\ast K ) ( \\pmb { \\theta } ) = ( \\mathcal { L } \\ast \\nabla K ) ( \\pmb { \\theta } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            136,
+            653,
+            154
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To relate the resulting gradient $\\nabla _ { \\pmb { \\theta } } \\mathcal { L } _ { K }$ to the gradient of the original function $\\mathcal { L }$ , we introduce the notion of weak gradient, i.e. an extension to the idea of weak/distributional derivatives to functions with multidimensional arguments, such as loss functions of neural networks. ",
+        "bbox": [
+            176,
+            165,
+            825,
+            207
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For an integrable function $\\mathcal { L }$ in space ${ \\mathcal { L } } \\in L ( [ a , b ] )$ , ${ \\mathrm { ~  ~ g ~ } } \\in \\ L ( [ a , b ] ^ { n } )$ is a $n$ -dimensional weak gradient of $\\mathcal { L }$ if it satisfies: ",
+        "bbox": [
+            174,
+            212,
+            823,
+            241
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/60622f586baee3bc3539334cca7165100f6f0a5a632d956822958f2467eb800f.jpg",
+        "text": "$$\n\\int _ { C } \\mathbf { g } ( \\pmb { \\tau } ) K ( \\pmb { \\tau } ) d \\pmb { \\tau } = - \\int _ { C } \\mathcal { L } ( \\pmb { \\tau } ) \\nabla K ( \\pmb { \\tau } ) d \\pmb { \\tau } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            244,
+            640,
+            279
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $K ( \\tau )$ is an infinitely differentiable function vanishing at infinity, $C \\in [ a , b ] ^ { n }$ and $\\tau \\in \\mathbb { R } ^ { n }$ ",
+        "bbox": [
+            173,
+            284,
+            803,
+            299
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As long as the chosen $K ( \\cdot )$ fulfills the definition of a mollifier we can use Eqn. 3 and Eqn. $4 ^ { 2 }$ to rewrite the gradient as: ",
+        "bbox": [
+            169,
+            305,
+            823,
+            334
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ab966a70938a27a555b09f8f8bdc7ab717ee5da9253431febc7067405c41d1be.jpg",
+        "text": "$$\n\\begin{array} { r l r } { \\nabla _ { \\pmb { \\theta } } \\mathcal { L } _ { K } ( \\pmb { \\theta } ) = ( \\mathcal { L } \\ast \\nabla K ) ( \\pmb { \\theta } ) } & { } & { \\mathrm { b y ~ E q n . ~ 3 } } \\\\ { = \\displaystyle \\int _ { C } \\mathcal { L } ( \\pmb { \\theta } - \\pmb { \\tau } ^ { } ) \\nabla K ( \\pmb { \\tau } ^ { } ) d \\pmb { \\tau } } \\\\ { = - \\displaystyle \\int _ { C } \\mathbf { g } ( \\pmb { \\theta } - \\pmb { \\tau } ^ { } ) K ( \\pmb { \\tau } ^ { } ) d \\pmb { \\tau } } & { } & { \\mathrm { b y ~ E q n . ~ 4 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            337,
+            673,
+            426
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For a differentiable almost everywhere function $\\mathcal { L }$ , the weak gradient $g ( \\pmb \\theta )$ is equal to $\\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L }$ almost everywhere. With a slight abuse of notation we can therefore write: ",
+        "bbox": [
+            173,
+            436,
+            823,
+            465
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a9fdad77f9115695511a104d62d70da3586873f3308dc87bdb1dad68e4b4a2bd.jpg",
+        "text": "$$\n\\nabla _ { \\pmb { \\theta } } \\mathcal { L } _ { K } ( \\pmb { \\theta } ) = - \\int _ { C } \\nabla _ { \\pmb { \\theta } } \\mathcal { L } ( \\pmb { \\theta } - \\pmb { \\tau } ) K ( \\pmb { \\tau } ) d \\pmb { \\tau }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            361,
+            469,
+            637,
+            503
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 GAUSSIAN MOLLIFIERS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            517,
+            380,
+            531
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "It is possible to use the standard Gaussian distribution $\\mathcal { N } ( 0 , \\bf { I } )$ as a mollifier $K ( \\cdot )$ , as it satisfies the desired properties: it is infinitely differentiable, a sequence of properly rescaled Gaussian distributions converges to the Dirac delta function and it vanishes in infinity. With such a $K ( \\cdot )$ the gradient becomes: ",
+        "bbox": [
+            173,
+            541,
+            825,
+            597
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7aaf714baa513747c6fed1e6280acc13bffa5969bbaf78a60cbd51521a64a804.jpg",
+        "text": "$$\n\\begin{array} { r l } {  { \\nabla _ { \\theta } \\mathcal { L } _ { K = \\mathcal { N } } ( \\theta ) = - \\int _ { C } \\nabla _ { \\theta } \\mathcal { L } ( \\theta - \\tau ) p ( \\tau ) d \\tau } } \\\\ & { = \\mathrm { E } _ { \\tau } [ \\nabla _ { \\theta } \\mathcal { L } ( \\theta - \\tau ) ] , \\mathrm { w i t h } \\ \\tau \\sim \\mathcal { N } ( 0 , \\ \\mathbf { I } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            602,
+            684,
+            655
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Exploiting the fact that a Gaussian distribution is a mollifier, we can focus on a sequence of mollifications indexed by scaling parameter $\\epsilon$ introduced in Eqn. 2. A single element of this sequence takes the following form: ",
+        "bbox": [
+            174,
+            665,
+            826,
+            705
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9950b1b7bd20ac0fef703f794670413e1bf914a9c4f70040e2a7c0c3b5d68505.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { \\mathcal { N } , \\epsilon } ( \\boldsymbol { \\theta } ) = - \\displaystyle \\int _ { C } \\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } ( \\boldsymbol { \\theta } - \\boldsymbol { \\tau } ) \\epsilon ^ { - 1 } p ( \\boldsymbol { \\tau } / \\epsilon ) d \\boldsymbol { \\tau } } \\\\ & { ~ = \\mathrm { E } _ { \\boldsymbol { \\tau } } [ \\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } ( \\boldsymbol { \\theta } - \\boldsymbol { \\tau } ) ] , \\mathrm { w i t h } \\boldsymbol { \\tau } \\sim \\mathcal { N } ( 0 , \\epsilon ^ { 2 } \\mathbf { I } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            710,
+            684,
+            765
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Replacing $\\epsilon$ with $\\sigma$ yields a sequence of mollifications indexed by $\\sigma$ ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/37289e4b45dbdcb9cec31211bf54eada6503f4e6bfe11e4317f5fd524dee448a.jpg",
+        "text": "$$\n\\nabla _ { \\pmb { \\theta } } \\mathcal { L } _ { \\mathcal { N } , \\sigma } ( \\pmb { \\theta } ) = \\mathrm { E } _ { \\pmb { \\tau } } [ \\nabla _ { \\pmb { \\theta } } \\mathcal { L } ( \\pmb { \\theta } - \\pmb { \\tau } ) ] , \\mathrm { w i t h } \\ \\pmb { \\tau } \\sim \\mathcal { N } ( 0 , \\ \\sigma ^ { 2 } \\mathbf { I } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "with the following property (by Eqn. 2): ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3c7449568d79bb87b5729d188e6055dd46b393c8ef4407f517bed765d80d48c3.jpg",
+        "text": "$$\n\\operatorname* { l i m } _ { \\sigma \\to 0 } \\nabla _ { \\pmb { \\theta } } \\mathcal { L } _ { \\mathcal { N } , \\sigma } ( \\pmb { \\theta } ) = \\nabla _ { \\pmb { \\theta } } \\mathcal { L } ( \\pmb { \\theta } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "An intuitive interpretation of the result is that $\\sigma$ determines the standard deviation of a mollifying Gaussian and is annealed in order to construct a sequence of gradually less \"blurred\" and closer approximations to $\\mathcal { L }$ . This is consistent with the property that when $\\sigma$ is annealed to zero we are optimizing the original function $\\mathcal { L }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "So far we obtained the mollified version $\\mathcal { L } _ { K } ( \\pmb { \\theta } )$ of the cost function $\\mathcal { L } ( \\pmb { \\theta } )$ by convolving it with a mollifier $K ( \\pmb \\theta )$ . The kernel $K ( \\pmb \\theta )$ corresponds to the average effect of injecting noise $\\bar { \\xi }$ sampled from standard Normal distribution. The amount of noise controls the amount of smoothing. Gradually reducing the noise during training is related to a form of simulated annealing (Kirkpatrick et al., 1983). Similarly to the analysis in Mobahi (2016), we can write a Monte-Carlo estimate of $\\begin{array} { r } { \\mathcal { L } _ { K } ( \\pmb { \\theta } ) = ( \\mathcal { L } * K ) ( \\pmb { \\theta } ) \\approx \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\mathcal { L } ( \\pmb { \\theta } - \\pmb { \\xi } ^ { ( i ) } ) } \\end{array}$ . We provide the derivation and the gradient of this equation in Appendix A. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The Monte-Carlo estimators of the mollifiers can be easily implemented with neural networks, where the layers typically have the form: ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/050189de7e54b542177fd8538c98b7593e4d651e1001fe8adbca8f211bd00a87.jpg",
+        "text": "$$\n\\mathbf { h } ^ { l } = \\mathbf { f } ( \\mathbf { W } ^ { l } \\mathbf { h } ^ { l - 1 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            439,
+            266,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "with $\\mathbf { h } ^ { l - 1 }$ a vector of activations from the previous layer in the hierarchy, $\\mathbf { W } ^ { l }$ a matrix representing a linear transformation and f an element-wise non-linearity of choice. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A mollification of such a layer can be formulated as: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3467103ada77b6382f68c5afaac9af4dd31dec9653e774d1d9b33cabedf27408.jpg",
+        "text": "$$\n\\mathbf { h } ^ { l } = \\mathbf { f } ( ( \\mathbf { W } ^ { l } - \\boldsymbol { \\xi } ^ { l } ) \\mathbf { h } ^ { l - 1 } ) , \\mathrm { w h e r e \\ } \\boldsymbol { \\xi } ^ { l } \\sim \\mathcal { N } ( \\mu , \\boldsymbol { \\sigma } ^ { 2 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            347,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "From Eqn. 16, it is easy to see that both weight noise methods proposed by Hinton $\\&$ van Camp (1993) and Graves (2011) can be seen as a variation of Monte-Carlo estimate of mollifiers. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.4 GENERALIZED AND NOISY MOLLIFIERS ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We introduce a generalization of the concept of mollifiers that encompasses the approach we explored here and that is targeted during optimization via a continuation method using stochastic gradient descent. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 2.1. (Generalized Mollifier). A generalized mollifier is an operator, where $T _ { \\sigma } ( f )$ defines a mapping between two functions, such that $T _ { \\sigma } : f \\to f ^ { * }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f0c9f435739b6fad0bfe9a7785f0a75340c1eb0a978691f96934be1e7cbf6717.jpg",
+        "text": "$$\n\\operatorname* { l i m } _ { \\sigma \\to 0 } T _ { \\sigma } f = f ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8d81d610d83e3c4468314d45c09a25e522cbf3b2170425900cdf24b9ae72cbd0.jpg",
+        "text": "$$\n\\begin{array} { r l } { { f ^ { 0 } = \\displaystyle \\operatorname* { l i m } _ { \\sigma \\to \\infty } T _ { \\sigma } f \\quad } } & { { \\mathrm { ~ i s ~ \\ a n ~ i d e n t i t y ~ f u n c t i o n ~ } } } \\\\ { { \\frac { \\partial ( T _ { \\sigma } f ) ( x ) } { \\partial x } \\quad } } & { { \\mathrm { e x i s t s ~ } \\forall x , \\sigma > 0 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In addition, we consider noisy mollifiers which can be defined as an expected value of a stochastic function $\\phi ( x , \\xi )$ under some noise source $\\xi$ with variance $\\sigma$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ba9a406f9b3150a536dd8186a5a2abedfd676c09e938a7a9eeac6418b206e8aa.jpg",
+        "text": "$$\n( T _ { \\sigma } f ) ( x ) = E _ { \\xi } [ \\phi ( x , \\xi _ { \\sigma } ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 2.2. (Noisy Mollifier). We call a stochastic function $\\phi ( x , \\xi _ { \\sigma } )$ with input $x$ and noise $\\xi$ a noisy mollifier if its expected value corresponds to the application of a generalized mollifier $T _ { \\sigma }$ , as per Eqn. 20. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The composition of two noisy mollifiers sharing the same $\\sigma$ is also a noisy mollifier, since the three properties in the definition (Eqns. 17,18,19) are still satisfied. When $\\sigma = 0$ no noise is injected and therefore the original function will be optimized. If $\\sigma \\to \\infty$ instead, the function will become an identity function. Thus, for instance, if we mollify each layer of a feed-forward network except the output layer, when $\\sigma \\to \\infty$ all the mollified layers will become identity function and the objective function of the network with respect to its inputs will be convex. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Consequently, corrupting separately the activation function of each level of a deep neural network (but with a shared noise level $\\sigma$ ) and annealing $\\sigma$ yields a noisy mollifier for the objective function. This is related to the work of Mobahi (2016), who recently introduced a way of analytically smoothing of the non-linearities to help the training of recurrent networks. The differences of that approach from our algorithm is two-fold: we use a noisy mollifier (rather than an analytic smoothing of the network’s non-linearities) and we introduce (in the next section) a particular form of the noisy mollifier that empirically proved to work well. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Shaping the cost function to define a sequence of costs that are progressing from easier to more difficult ones can be related to the reward shaping $\\mathrm { N g }$ et al., 1999; $\\bar { \\bf N } \\mathrm { \\bf g }$ , 2003) algorithms. In our algorithm, we shape the cost and the model architecture itself, rather than rewards or the targets in order to make the optimization easier. In that sense, reward shaping can be considered to be more closer to curriculum learning. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            190,
+            282,
+            207
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We propose an algorithm to mollify the cost of a neural network which also addresses an important drawback of the previously proposed noisy training procedures: as the noise gets larger, it can dominate the learning process and lead the algorithm to perform a random walk on the energy landscape of the objective function. Conversely in our algorithm, as the noise gets larger gradient descent minimizes a simpler (e.g. convex) but still meaningful objective function. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We define the desired behavior of the network in the limit cases where the noise is very large or very small, and modify the model architecture accordingly. Specifically, during training we minimize a sequence of increasingly complex noisy objectives $\\bar { \\mathrm { L } } = ( \\bar { \\mathcal { L } } ^ { 1 } ( \\pmb { \\theta } ; \\xi _ { \\sigma _ { 1 } } ) , \\bar { \\mathcal { L } } ^ { 2 } ( \\pmb { \\theta } ; \\xi _ { \\sigma _ { 2 } } ) , \\cdots , \\bar { \\mathcal { L } } ^ { k } ( \\pmb { \\theta } ; \\xi _ { \\sigma _ { k } } ) )$ that we obtain by annealing the scale (variance) of the noise $\\sigma _ { i }$ . Let us note that our algorithm satisfies the fundamental properties of the generalized and noisy mollifiers that we introduced earlier. ",
+        "bbox": [
+            174,
+            295,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We use a noisy mollifier based on our definition in Section 2.4. Instead of convolving the objective function with a kernel: ",
+        "bbox": [
+            174,
+            369,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "1. We start the training by optimizing a convex objective function that is obtained by configuring all the layers between the input and the last cost layer to compute an identity function, i.e., by skipping both the affine transformations and the blocks followed by nonlinearities. 2. During training, the magnitude of noise which is proportional to $p$ is annealed, allowing to gradually evolve from identity transformations to linear transformations between the layers. 3. Simultaneously, as we decrease the $p$ , the noisy mollification procedure allows the element-wise activation functions to gradually change from linear to be nonlinear. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 SIMPLIFYING THE OBJECTIVE FUNCTION FOR FEEDFORWARD NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            523,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For every unit of each layer, we either copy the activation (output) of the corresponding unit of the previous layer (the identity path in Figure 2) or output a noisy activation $\\tilde { \\mathbf { h } } ^ { l }$ of a non-linear transformation of it $\\psi ( \\mathbf { h } ^ { l - 1 } , \\boldsymbol { \\xi } ; \\bar { \\mathbf { W } } ^ { l } )$ , where $\\boldsymbol { \\xi }$ is noise, $\\bar { \\mathbf { W } } ^ { l }$ is a weight matrix applied on $\\mathbf { h } ^ { l - 1 }$ and $\\pi$ is a vector of binary decisions for each unit (the convolutional path in Figure 2): ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f21a43fb99383f5278852be96b53c872f13a827906bda544c48a7da869cdc95d.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\tilde { \\mathbf { h } } ^ { l } = \\psi ( \\mathbf { h } ^ { l - 1 } , \\pmb { \\xi } ; \\mathbf { W } ^ { l } ) } } \\\\ { { \\phi ( \\mathbf { h } ^ { l - 1 } , \\pmb { \\xi } , \\pi ^ { l } ; \\mathbf { W } ^ { l } ) = \\pi ^ { l } \\odot \\mathbf { h } ^ { l - 1 } + ( 1 - \\pi ^ { l } ) \\odot \\tilde { \\mathbf { h } ^ { l } } } } \\\\ { { \\mathbf { h } ^ { l } = \\phi ( \\mathbf { h } ^ { l - 1 } , \\pmb { \\xi } , \\pi ^ { l } ; \\mathbf { W } ^ { l } ) . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To decide which path to take, for each unit in the network, a binary stochastic decision is taken by drawing from a Bernoulli distribution with probability dependent on the decaying value of $p ^ { l }$ : ",
+        "bbox": [
+            174,
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7d1bc2e0c9726684620e03af7856caaf1e0bee00ac936e8579f2a39c0be868b8.jpg",
+        "text": "$$\n\\pi ^ { l } \\sim \\mathrm { B e r n o u l l i } ( p ^ { l } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            431,
+            732,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "If the number of hidden units of layer $l - 1$ and layer $l + 1$ is not the same, we can either zero-pad layer $l - 1$ before feeding it into the next layer or apply a linear projection to obtain the right dimensionality. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For $p ^ { l } = 1$ , the layer computes the identity function leading to a convex objective. If $p ^ { l } = 0$ the layer computes the original non-linear transformation unfolding the full capacity of the model. We call the connections that are introduced this way as unitwise stochastic skip connections(USSC). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In DropIn layers (Smith et al., 2016) a binary random vector is sampled from a Bernoulli distribution to decide whether to introduce skip a connection from the layer below $l - 1$ for each layer $l$ and they use this as a regularization. As opposed to the USSC, DropIn layers do not necessarily do not necessarily achieve a convex objective function as the DropIn ratio $( \\hat { p ^ { l } } )$ increases. ",
+        "bbox": [
+            174,
+            847,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The pseudo-code for the mollified activations is reported in Algorithm 1. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/80eecd4ac991add0cb642b20c46377ca05f8af859e791101be6113b09d06839f.jpg",
+        "image_caption": [
+            "Figure 2: Top: Stochastic depth. Bottom: mollifying network. The dashed line represents the optional residual connection. In the top path, the input is processed with a convolutional block followed by a noisy activation function, while in the bottom path the original activation of the layer $l - 1$ is propagated untouched. For each unit, one of the two paths in picked according to a binary stochastic decision $\\pi$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Activation of a unit $i$ at layer $l$ ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            391,
+            464,
+            406
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "1: $x _ { i } \\gets \\mathbf { w } _ { i } ^ { \\top } \\mathbf { h } ^ { l - 1 } + b _ { i }$ . an affine transformation of hl−1   \n2: $\\Delta _ { i } \\gets \\mathbf { u } ( x _ { i } ) - \\mathbf { f } ( x _ { i } )$ $\\mathsf { D } \\Delta _ { i }$ is a measure of a saturation of a unit   \n3: $\\sigma ( x _ { i } ) \\gets ( \\mathrm { s i g m o i d } ( a _ { i } \\Delta _ { i } ) - 0 . 5 ) ^ { 2 }$ . std of the injected noise depends on $\\Delta _ { i }$   \n4: $\\xi _ { i } \\sim \\mathcal { N } ( 0 , 1 )$ . sampling the noise from a basic Normal distribution   \n5: $s _ { i } \\gets p ^ { l } c \\sigma ( x _ { i } ) | \\xi _ { i } |$ . Half-Normal noise controlled by $\\sigma ( x _ { i } )$ , const. $c$ and prob-ty $p ^ { l }$   \n6: $\\psi ( x _ { i } , \\xi _ { i } ) \\gets \\mathrm { s g n } ( \\mathbf { u } ^ { * } ( x _ { i } ) ) \\mathrm { m i n } ( | \\mathbf { u } ^ { * } ( x _ { i } ) |$ , $| \\mathbf { f } ^ { * } ( x _ { i } ) + \\mathrm { s g n } ( \\mathbf { u } ^ { * } ( x _ { i } ) ) | \\hat { s _ { i } } | | ) + \\bar { \\mathbf { u } } ( 0 )$ $\\triangleright$ noisy activation   \n7: $\\pi _ { i } ^ { l } \\sim { \\bf B e r n o u l l i } ( p ^ { l } ) \\qquad \\triangleright p ^ { l }$ controls the variance of the noise AND the prob of skipping a unit   \n8: $\\ddot { h } _ { i } ^ { l } = \\psi ( x _ { i } , \\xi _ { i } )$ . $\\tilde { h } _ { i } ^ { l }$ is a noisy activation candidate   \n9: $\\phi ( \\mathbf { h } ^ { l - 1 } , \\xi _ { i } , \\pi _ { i } ^ { l } ; \\mathbf { w } _ { i } ) = \\pi _ { i } ^ { l } h _ { i } ^ { l - 1 } + ( 1 - \\pi _ { i } ^ { l } ) \\tilde { h } _ { i } ^ { l }$ . make a HARD decision between $h _ { i } ^ { l - 1 }$ and $\\tilde { h } _ { i } ^ { l }$ ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 LINEARIZING THE NETWORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            574,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In Section 2, we show that convolving the objective function with a particular kernel can be approximated by adding noise to the activation function. This method may suffer from excessive random exploration when the noise is very large. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We address this issue by bounding the element-wise activation function $\\operatorname { f } ( \\cdot )$ with its linear approximation when the variance of the noise is very large, after centering it at the origin. The resulting function $\\mathrm { f } ^ { \\ast } ( \\cdot )$ is bounded and centered around the origin. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Note that centering the sigmoid or hard-sigmoid will make them symmetric with respect to the origin. With a proper choice of the standard deviation $\\sigma ( \\mathbf { h } )$ , the noisy activation function becomes a linear function of the input when $\\mathfrak { p }$ is large, as illustrated by Figure 10. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Let $\\mathbf { u } ^ { * } ( x ) = \\mathbf { u } ( x ) - \\mathbf { u } ( 0 )$ , where ${ \\bf u } ( 0 )$ is the offset of the function from the origin, and $x _ { i }$ the $i$ -th dimension of an affine transformation of the output of the previous layer $\\mathbf { h } ^ { l - 1 }$ : $x _ { i } = \\mathbf { w } _ { i } ^ { \\top } \\mathbf { h } ^ { l - 1 } + b _ { i }$ . Then: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d3ce9f3819bca81dc0de1cfc2bf008e295e1d224a2a62b8c68d791d67d3954c7.jpg",
+        "text": "$$\n\\psi ( x _ { i } , \\xi _ { i } ; \\mathbf { w } _ { i } ) = \\mathrm { s g n } ( \\mathbf { u } ^ { * } ( x _ { i } ) ) \\mathrm { m i n } ( | \\mathbf { u } ^ { * } ( x _ { i } ) | , | \\mathbf { f } ^ { * } ( x _ { i } ) + \\mathrm { s g n } ( \\mathbf { u } ^ { * } ( x _ { i } ) ) | s _ { i } | | ) + \\mathbf { u } ( 0 )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The noise is sampled from a Normal distribution with mean 0 and whose standard deviation depends on $c$ : ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aa8a9b68c9de09337719d7e42d807b4953173f578a18141b7d1cba2668e133e6.jpg",
+        "text": "$$\ns _ { i } \\sim \\mathcal { N } ( 0 , p c \\sigma ( x _ { i } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.1 LINEARIZING RELU ACTIVATION FUNCTION ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We have a simpler form of the equations to linearize ReLU (Nair & Hinton, 2010) activation function when $p ^ { l } \\to \\infty \\quad$ . Instead of the complicated Eqn. 23. We can use a simpler equation as in Eqn. 26 to ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/644f80b2c2d62505206241eaafd9d373597c1578b8fc6d71a55ea6c773a2d3d5.jpg",
+        "image_caption": [
+            "Figure 3: The figures show how to evolve the model to make it closer to a linear network. Arrows denote the direction of the noise pushing the activation function towards the linear function. a) The quasi-convex envelope established by a $| \\mathrm { { s i g m o i d } ( \\cdot ) | }$ around $| 0 . 2 5 x |$ . b) A depiction of how the noise pushes the sigmoid to become a linear function. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "achieve the linearization of the activation function when we have a very large noise in the activation function: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/008d8fbe2e5477a9e0c37b8b15192bbd2b6d56dfb8c0e07eb1e9d7de28cf304d.jpg",
+        "text": "$$\n\\begin{array} { c } { s _ { i } = \\underset { \\mathbf { \\sigma } } { \\mathrm { m i n i m u m } } ( | x _ { i } | , p \\sigma ( x _ { i } ) | \\boldsymbol { \\xi } | ) } \\\\ { \\psi ( x _ { i } , \\xi _ { i } , \\mathbf { w } _ { i } ) = \\mathbf { f } ( x _ { i } ) - s _ { i } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6 MOLLIFYING LSTMS AND GRUS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In a similar vein it is possible to smooth the objective functions of LSTM and GRU networks by starting the optimization procedure with a simpler objective function such as optimizing a word2vec, BoW-LM or CRF objective function at the beginning of training and gradually increasing the difficulty of the optimization by increasing the capacity of the network. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For GRUs we set the update gate to $\\textstyle { \\frac { 1 } { t } }$ – where $t$ is the time-step index – and reset the gate to 1 if the noise is very large, using Algorithm 1. Similarly for LSTMs, we can set the output gate to 1 and input gate to $\\textstyle { \\frac { 1 } { t } }$ and forget gate to $\\textstyle { 1 - { \\frac { 1 } { t } } }$ when the noise is very large. The output gate is 1 or close to 1 when the noise is very large. This way the LSTM will behave like a BOW model. In order to achieve this behavior, the activations $\\psi ( x _ { t } , \\dot { \\xi _ { i } } )$ of the gates can be formulated as: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b77f3e1ce6b4ee6007e9628c6b173d61d5ab3691adbebd286cbda48a4fed0ce7.jpg",
+        "text": "$$\n\\boldsymbol { \\psi } ( x _ { t } ^ { l } , \\xi ) = \\mathbf { f } ( x _ { t } ^ { l } + p ^ { l } \\sigma ( x ) | \\xi | )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "By using a particular formulation of $\\sigma ( x )$ that constraints it to be in expectation over $\\xi$ when $p ^ { l } = 1$ , we can obtain a function for $\\gamma \\in \\mathbb { R }$ within the range of $\\operatorname { f } ( \\cdot )$ that is discrete in expectation, but still per sample differentiable: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7462e8550aaec5c2e5686a1abb87a851f31e4c8c2cbd8bd761ef16c81cc03eb6.jpg",
+        "text": "$$\n\\sigma ( x _ { t } ^ { l } ) = \\frac { \\mathrm { f } ^ { - 1 } ( \\gamma ) - x _ { t } ^ { l } } { \\mathrm { E } _ { \\xi } [ | \\xi | ] }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            426,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We provide the derivation of Eqn. 28 in Appendix B. The gradient of the Eqn. 28 will be a Monte-Carlo approximation to the gradient of $\\bar { \\bf f } ( { \\bf x } _ { t } ^ { l } )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "7 ANNEALING SCHEDULES FOR $p$ ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We used a different schedule for each layer of the network, such that the noise in the lower layers will anneal faster. This is similar to the linearly decaying probability of layers in Huang et al. (2016b). ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Exponential Decay In our experiments, we focused on using an annealing schedule similar to the inverse sigmoid rule in Bengio et al. (2015) with $p _ { t } ^ { l }$ , ",
+        "bbox": [
+            173,
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/00e2dab64f5c89f51db46ef01b810fdff7d90c203c8ec182489fbefd5fe0064c.jpg",
+        "text": "$$\np _ { t } ^ { l } = 1 - \\mathrm { e } ^ { - \\frac { k v _ { t } l } { t L } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "with hyper-parameter model. We stop anne $k \\geq 0$ at wh $t ^ { t h }$ update for the he expected de $l ^ { t h }$ $L$ is the number of layers reaches some threshold the. In $\\begin{array} { r } { p _ { t } = \\sum _ { i = 1 } ^ { L } p _ { t } ^ { l } } \\end{array}$ $\\delta$ our experiment we set $v _ { t }$ to be a moving average of the $\\mathrm { l o s s } ^ { 3 }$ of the network, but for some of our experimnets that resulted unstable behaviors in the training and thus we have to fix $v _ { t }$ to 1. An advantage of using running average of loss for the $v _ { t }$ is because the behavior of the loss/optimization can directly influence the annealing behavior of the network. Because we will have: ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e11515286cdb82c18307c87f52abe95b1878a73f8e35bd5738690c359abb3f4b.jpg",
+        "text": "$$\n\\operatorname* { l i m } _ { { \\bf v } _ { t } \\to \\infty } p _ { t } ^ { l } = 1 \\mathrm {  ~ \\ a n d , ~ } \\operatorname* { l i m } _ { { \\bf v } _ { t } \\to 0 } p _ { t } ^ { l } = 0 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            189,
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+            215
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "This has a desirable property: when the training-loss is high, the noise injected into the system will be large as well. As a result, the model is encouraged to do more exploration, while if the model converges the noise injected into the system by the mollification procedure will be zero. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Square-root Decay The square root decaying of the learning rate is very popular. In our experiments we have tried to decay the plt according to min(pmin, 1 − q $\\begin{array} { r } { 1 - \\sqrt { \\frac { t } { N _ { e p o c h s } } } ) } \\end{array}$ where $t$ is the index of the iteration/epoch over the data and $N _ { e p o c h s }$ corresponds to the total number of iterations that the model will do on the dataset. $p _ { m i n }$ is hyperparameter to define the lower bound on the $p _ { t } ^ { l }$ . ",
+        "bbox": [
+            173,
+            270,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Linear Decay We have also experimented with linearly decaying the $p _ { t } ^ { l }$ via $\\begin{array} { r } { \\operatorname* { m i n } ( p _ { m i n } , 1 - \\frac { t } { N _ { e p o c h s } } ) } \\end{array}$ We have compared the plot of different annealing methods described in this paper as in Figure 4. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/0edec83fe7d1c958a8488792b79a2fed28b69de56bc31b6fc444f2b0511e2401.jpg",
+        "image_caption": [
+            "Figure 4: We compare different annealing schedules with respect to the time in y-axis(iterations). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            292,
+            420,
+            699,
+            645
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, in our experiments we observe that training with noisy mollifiers can potentially be helpful for the generalization. This can be due to the noise induced to the backpropagation through the noisy mollification, that makes SGD more likely to converge to a flatter-minima (Hochreiter & Schmidhuber, 1997b) because the noise will help it escape from sharper local minima. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "8 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            763,
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+            779
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this section we mainly focus on training of difficult to optimize models, in particular deep MLPs with sigmoid or tanh activation functions. The details of the experimental procedure is provided in Appendix C. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "8.1 EXPLORATORY STUDY ON MNIST ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            849,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We train a thin deep neural network on MNIST (LeCun & Cortes, 1998) dataset with 72 hidden layers and 100 hidden units. We train our model with Adam(Kingma & Ba, 2014) optimizer and fix the learning rate of all the models to $3 e { - 4 }$ . We have used the same learning-rate for all the models in order to factor out the possibility that a model converges faster, due to the fact of using a larger learning rate. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Firstly, in Figure 5, we investigate the effect of using different annealing schedules. Exponential decay converges faster compared to linear decay and square root decay of $p$ . We find it to be very unstable to train our model with linear and square root decay in particular for large $c$ values. Thus we have to use smaller $c$ value (20 instead of 100) to be able to train the model with causing it to diverge. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/be0d19fd9d12424b4a91c95146d3758bf25debfc209b87d8462abca2c20cfb26.jpg",
+        "image_caption": [
+            "Figure 5: We compare the training performance of different types of annealing methods used with mollification procedure to anneal the parameter of the mollification $p$ . Decaying the $p$ exponentially achieves both better training and validation performance. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In Figure 6, we show the effect of using the noisy training procedure that we have introduced by sampling a mask from Bernoulli and Gaussian distributions versu using the deterministic approximation of this noisy procedure which we also use for the test time but during the training as well. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/4b8a3ac20b57ecaf26bfbb68e0f1deb19984fdd2b9a2d6ea11e3f0a10189a871.jpg",
+        "image_caption": [
+            "Figure 6: We show the learning curves of the model where we don’t inject the noise during the training and instead use the deterministic approximation for the mollification during the training as well. The differerence in terms of speed of learning is very small. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In Figure 7, we compare the results obtained for the model using mollification obtained with or without batch normalization and feed-forward residual networks. The mollified model performs very closely to the MLP trained with residual connections and the batchnorm. However, using residual connections and batch-norm does not seem to improve the results. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have tried to run experiment with the Monte-Carlo approximation of the mollification which is derived in Appendix A, however when we start with large noise and anneal the noise during the training, the model was very unstable and training was diverging. If we start with small noise and anneal the magnitude of the noise during the training, we could not observe any effect of it on the training. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/24f40f65392ee90e6386375192de3d2b7240bb6ad5b1529448614804c614f410.jpg",
+        "image_caption": [
+            "Figure 7: We investigate the effect of using batch norm and residual connection for mollifcation and compare against to the network with residual connections and batch-norm. The effect of batch norm on this task for mollification seems to be very small and training convergence performance of the all the approaches are very close to each other. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            320,
+            111,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/c41840e048fd609433dd3b430329544afc0d1b86c1b1c7b4544f1462c89c0e19.jpg",
+        "image_caption": [
+            "Figure 8: The learning curves of a 6-layers MLP with sigmoid activation function on 40 bit parity task. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            243,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/f378c6167f9ea9fa01daa4b81cf06eab81db69264d83a592a7353cf57198fa76.jpg",
+        "table_caption": [
+            "Table 1: CIFAR10 deep convolutional neural network. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Test Accuracy</td></tr><tr><td>Stochastic Depth</td><td>93.25</td></tr><tr><td>Mollified Convnet</td><td>92.45</td></tr><tr><td>ResNet</td><td>91.78</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "8.2 DEEP MLP EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            604,
+            397,
+            617
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Deep Parity Experiments Training neural networks on a high-dimensional parity problem can be challenging (Graves, 2016; Kalchbrenner et al., 2015). We experiment on forty dimensional (bits) parity problem with 6-layer MLP using sigmoid activation function. All the models are initialized with Glorot initialization Glorot et al. (2011) and trained with SGD with momentum. We compare an MLP with residual connections using batch normalization and a mollified network with sigmoid activation function. As can be seen in Figure 8, the mollified network converges faster. ",
+        "bbox": [
+            174,
+            632,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Deep Pentomino Pentomino is a toy-image dataset where each image has 3 Pentomino blocks. The task is to predict whether if there is a different shape in the image or not (Gülçehre & Bengio, 2013). The best reported result on this task with MLPs is $6 8 . 1 5 \\%$ accuracy (Gulcehre et al., 2014). The same model as ours trained without noisy activation function and vanilla residual connections scored $6 9 . 5 \\%$ accuracy, while our mollified version scored $7 5 . 1 5 \\%$ accuracy after 100 epochs of training on the $8 0 k$ dataset. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "CIFAR10 We experimented with deep convolutional neural networks of 110-layers with residual blocks and residual connections comparing our model against ResNet and Stochastic depth. We adapted the hyperparameters of the Stochastic depth network from Huang et al. (2016a) and we used the same hyperparameters for our algorithm. We report the training and validation curves of the three models in Figure 10 and the best test accuracy obtained early stopping on validation accuracy over 500 epochs in Table 1. Our model achieves better generalization than ResNet. Stochastic depth achieves better generalization, but it might be possible to combine both and obtain better results. ",
+        "bbox": [
+            174,
+            830,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/345e69a1318bab42ba15b8295abfbd900d2a22fb372d4cfae106a2e0cb0e9af6.jpg",
+        "image_caption": [
+            "Figure 9: The training curve of a bidirectionalRNN that predicts the embedding corresponding to a sequence of characters. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            269,
+            112,
+            501,
+            238
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/81d61387d971eff68d8d3252462d7ec7a797292fc2c55e49e9d151dea12ae429.jpg",
+        "table_caption": [
+            "Table 2: 3-layered LSTM network on word-level language modeling for PTB. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Test PPL</td></tr><tr><td>LSTM</td><td>119.4</td></tr><tr><td>Mollified LSTM</td><td>115.7</td></tr></table>",
+        "bbox": [
+            566,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "9 LSTM EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            316,
+            387,
+            333
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Predicting the Character Embeddings from Characters Learning the mapping from sequences of characters to the word-embeddings is a difficult problem. Thus one needs to use a highly non-linear function. We trained a word2vec model on Wikipedia with embeddings of size 500 (Mikolov et al., 2014) with a vocabulary of size 374557. ",
+        "bbox": [
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+            348,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "LSTM Language Modeling We evaluate our model on LSTM language modeling. Our baseline model is a 3-layer stacked LSTM without any regularization. We observed that mollified model converges faster and achieves better results. We provide the results for PTB language modeling in Table 2. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "10 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We propose a novel method for training neural networks inspired by an idea of continuation, smoothing techniques and recent advances in non-convex optimization algorithms. The method makes learning easier by starting from a simpler model, solving a well-behaved problem, and gradually transitioning to a more complicated setting. We show improvements on very deep models, difficult to optimize tasks and compare with powerful techniques such as batch-normalization and residual connections. We also show that the mollification procedure improves the generalization performance of the model on two tasks. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Our future work includes testing this method on large-scale language tasks that require long training time, e.g., machine translation and language modeling. Moreover, (Kaiser & Sutskever, 2015) observed that the training of Neural-GPU model can be improved significantly by using gradient noise which can be related to the smoothing of the loss surface, it would be interesting to try mollification on this model to see if the training of Neural GPU can be made easier by using mollification procedure. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
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+    },
+    {
+        "type": "text",
+        "text": "We thank Nicholas Ballas and Misha Denil for the valuable discussions and their feedback. We would like to also thank the developers of Theano 4, for developing such a powerful tool for scientific computing Theano Development Team (2016). We acknowledge the support of the following organizations for research funding and computing support: NSERC, Samsung, Calcul Québec, Compute Canada, the Canada Research Chairs and CIFAR. ",
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+        "text": "Appendix ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "A MONTE-CARLO ESTIMATE OF MOLLIFICATION ",
+        "text_level": 1,
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+            174,
+            256,
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+    {
+        "type": "text",
+        "text": "${ \\mathcal L } _ { K } ( \\pmb \\theta ) = ( { \\mathcal L } * K ) ( \\pmb \\theta ) = \\int _ { C } { \\mathcal L } ( \\pmb \\theta - \\xi ) K ( \\xi ) d \\xi$ which can be estimated by a Monte Carlo: $\\approx \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\mathcal { L } ( \\pmb { \\theta } - \\pmb { \\xi } ^ { ( i ) } )$ , where $\\xi ^ { ( i ) }$ is a realization of the noise random variable $\\xi$ yielding $\\frac { \\partial \\mathcal { L } _ { K } ( { \\pmb \\theta } ) } { \\partial { \\pmb \\theta } }$ $\\approx \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\frac { \\partial \\mathcal { L } ( \\pmb { \\theta } - \\pmb { \\xi } ^ { ( i ) } ) } { \\partial \\pmb { \\theta } } .$ ",
+        "bbox": [
+            183,
+            284,
+            787,
+            438
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Therefore introducing additive noise to the input of $\\mathcal { L } ( \\pmb \\theta )$ is equivalent to mollification. ",
+        "bbox": [
+            173,
+            438,
+            740,
+            454
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B DERIVATION OF THE NOISY ACTIVATIONS FOR THE GATING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            470,
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+            488
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Assume that $z _ { t } ^ { l } = x _ { t } ^ { l } + p _ { t } ^ { l } \\sigma ( x ) | \\xi _ { t } ^ { l } |$ and $\\mathrm { E } _ { \\xi } [ \\psi ( x _ { t } ^ { l } , \\xi ) ] = t$ . Thus for all $z _ { t } ^ { l }$ , ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d81ce7f88b317f357b179064e036e23d39769074b19dcdf04d265f8ac162a209.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathrm { E } _ { \\xi } [ \\psi ( x _ { t } ^ { l } , \\xi _ { t } ^ { l } ) ] = \\mathrm { E } _ { \\xi } [ \\mathrm { f } ( z _ { t } ^ { l } ) ] , } \\\\ & { \\qquad t = \\mathrm { E } _ { \\xi } [ \\mathrm { f } ( z _ { t } ^ { l } ) ] , \\ \\mathrm { a s s u m i n g ~ f ( \\cdot ) ~ b e h a v e s ~ s i m i l a r ~ t o ~ a ~ l i n e a r ~ f u n c t i o n } ; } \\\\ & { \\qquad \\mathrm { E } _ { \\xi } [ \\mathrm { f } ( z _ { t } ^ { l } ) ] \\approx \\mathrm { f } ( \\mathrm { E } _ { \\xi } [ z _ { t } ^ { l } ] ) \\ \\mathrm { s i n c e ~ w e ~ u s e ~ h a r d - s i g m o i d ~ f o r ~ f ( \\cdot ) ~ t h i s ~ w i l l ~ h o l d } . } \\\\ & { \\qquad \\mathrm { f } ^ { - 1 } ( t ) \\approx \\mathrm { E } _ { \\xi } [ z _ { t } ^ { l } ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            245,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "As in Eqn. 32, we can write the expectation of this equation as: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bb4a7988b50191817819d708c72a878138289811bede4001d28e39ac2021dcb5.jpg",
+        "text": "$$\n\\mathrm { f } ^ { - 1 } ( t ) \\approx x _ { t } ^ { l } + p _ { t } ^ { l } \\sigma ( x ) \\mathrm { E } _ { \\xi } [ \\xi _ { t } ^ { l } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            662,
+            593,
+            681
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Corollary, the value that $\\sigma ( x _ { t } ^ { l } )$ should take in expectation for $p _ { t } ^ { l } = 1$ would be: ",
+        "bbox": [
+            173,
+            685,
+            687,
+            702
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/37f119cb5fff172401ae59c1cff8fb44f26310a702a76c6ba7c93fe86200e4a2.jpg",
+        "text": "$$\n\\sigma ( x _ { t } ^ { l } ) \\approx \\frac { \\mathrm { f } ^ { - 1 } ( t ) - x _ { t } ^ { l } } { \\mathrm { E } _ { \\xi } [ \\xi _ { t } ^ { l } ] }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            428,
+            704,
+            570,
+            741
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In our experiments for $\\operatorname { f } ( \\cdot )$ we used the hard-sigmoid activation function. We used the following piecewise activation function in order to use it as $\\mathrm { f } ^ { - 1 } ( x ) = 4 ( x - 0 . 5 )$ . During inference we use the expected value of random variables $\\pi$ and $\\xi$ . ",
+        "bbox": [
+            174,
+            752,
+            826,
+            796
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C EXPERIMENTAL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            814,
+            419,
+            830
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.1 MNIST ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            843,
+            272,
+            859
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The weights of the models are initialized with Glorot & Bengio initialization Glorot et al. (2011). We use the learning rate of $4 e - 4$ along with RMSProp. We initialize $a _ { i }$ parameters of mollified activation function by sampling it from a uniform distribution, $\\mathrm { { U } [ - 2 , 2 ] }$ . We used 100 hidden units at each layer with a minibatches of size 500. ",
+        "bbox": [
+            174,
+            869,
+            825,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.2 PENTOMINO ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            303,
+            117
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We train a 6−layer MLP with sigmoid activation function using SGD and momentum. We used 200 units per layer with sigmoid activation functions. We use a learning rate of $1 e - 3$ . ",
+        "bbox": [
+            174,
+            128,
+            823,
+            156
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C.3 CIFAR10 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            172,
+            287,
+            186
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We use the same model with the same hyperparameters for both ResNet, mollified network and the stochastic depth. We borrowed the hyperparameters of the model from Huang et al. (2016a). Our mollified convnet model has residual connections coming from its layer below. ",
+        "bbox": [
+            174,
+            198,
+            826,
+            239
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/8dbf86661abcd350698fec3b04fd3a0769dbc7ba6ca3d6a790e7f74289b8e903.jpg",
+        "image_caption": [
+            "Figure 10: Training and validation losses over 500 epochs of a mollified convolutional network composed by 110-layers. We compare against ResNet and Stochastic depth. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            230,
+            261,
+            766,
+            407
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C.4 PARITY ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            472,
+            269,
+            486
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The n-dimensional parity task is the task to figure out whether the sum of n-bits in a binary vector is even or odd. We use SGD with Nesterov momentum and initialize the weight matrices by using Glorot&Bengio initializationGlorot et al. (2011). For all models, we use the learning rate of $1 e - 3$ and momentum of 0.92. $a _ { i }$ is the parameters of mollified activation function are initialized by sampling from uniform distribution, $\\mathsf { \\bar { U } } [ - 2 , 2 ]$ . ",
+        "bbox": [
+            174,
+            497,
+            825,
+            565
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C.5 LSTM LANGUAGE MODELING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            580,
+            429,
+            594
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We trained 2-layered LSTM language models on PTB word-level. We used the models with the same hyperparameters as in Zaremba & Sutskever (2014). We used the same hyperparameters for both the mollified LSTM language model and the LSTM. We use hard-sigmoid activation function for both the LSTM and mollified LSTM language model. We use hard-sigmoid activation function for the gates of the LSTM. ",
+        "bbox": [
+            174,
+            606,
+            825,
+            672
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C.6 PREDICTING THE CHARACTER EMBEDDINGS FROM CHARACTERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            689,
+            671,
+            704
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We use $1 0 k$ of these words as a validation and another $1 0 k$ word embeddings as test set. We train a bidirectional-LSTM on top of each sequence of characters for each word and on top of the representation of bidirectional LSTM, we use a 5-layered tanh-MLP to predict the word-embedding. ",
+        "bbox": [
+            176,
+            714,
+            823,
+            755
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We train our models using RMSProp and momentum with learning rate of $6 e - 4$ and momentum 0.92. The size of the minibatches, we used is 64. As seen in Figure 9, mollified LSTM network converges faster. ",
+        "bbox": [
+            174,
+            762,
+            825,
+            803
+        ],
+        "page_idx": 14
+    }
+]
\ No newline at end of file
diff --git a/parse/train/r1G4z8cge/r1G4z8cge_middle.json b/parse/train/r1G4z8cge/r1G4z8cge_middle.json
new file mode 100644
index 0000000000000000000000000000000000000000..75f3d4d039d514c847e2d8a8a8b973b20cefb1bb
--- /dev/null
+++ b/parse/train/r1G4z8cge/r1G4z8cge_middle.json
@@ -0,0 +1,42075 @@
+{
+    "pdf_info": [
+        {
+            "preproc_blocks": [
+                {
+                    "type": "title",
+                    "bbox": [
+                        107,
+                        79,
+                        295,
+                        96
+                    ],
+                    "lines": [
+                        {
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "MOLLIFYING NETWORKS",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 0
+                        }
+                    ],
+                    "index": 0
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                    "lines": [
+                        {
+                            "bbox": [
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+                                    "bbox": [
+                                        114,
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+                                    ],
+                                    "score": 1.0,
+                                    "content": "Caglar Gulcehre1, Marcin Moczulski2,∗, Francesco Visin3,∗ Yoshua Bengio1",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 1
+                        },
+                        {
+                            "bbox": [
+                                111,
+                                125,
+                                408,
+                                139
+                            ],
+                            "spans": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "1 University of Montreal, 2 University of Oxford, 3 Politecnico di Milano",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 2
+                        }
+                    ],
+                    "index": 1.5
+                },
+                {
+                    "type": "title",
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+                                    "score": 1.0,
+                                    "content": "ABSTRACT",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 3
+                        }
+                    ],
+                    "index": 3
+                },
+                {
+                    "type": "text",
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+                    "lines": [
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+                                    "score": 1.0,
+                                    "content": "The optimization of deep neural networks can be more challenging than the",
+                                    "type": "text"
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+                            "index": 4
+                        },
+                        {
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+                                    "score": 1.0,
+                                    "content": "traditional convex optimization problems due to highly non-convex nature of the",
+                                    "type": "text"
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+                            "index": 5
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+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "loss function, e.g. it can involve pathological landscapes such as saddle-surfaces",
+                                    "type": "text"
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+                            ],
+                            "index": 6
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "that can be difficult to escape from for algorithms based on simple gradient descent.",
+                                    "type": "text"
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+                        },
+                        {
+                            "bbox": [
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+                                    "content": "In this paper, we attack the problem of optimization of highly non-convex neural",
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+                        {
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+                                    "score": 1.0,
+                                    "content": "networks objectives by starting with a smoothed – or mollified – objective function",
+                                    "type": "text"
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+                            "index": 9
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+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "which becomes more complex as the training proceeds. Our proposition is inspired",
+                                    "type": "text"
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+                            ],
+                            "index": 10
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "by the recent studies in continuation methods: similarly to curriculum methods,",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 11
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "we begin by learning an easier (possibly convex) objective function and let it",
+                                    "type": "text"
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+                            ],
+                            "index": 12
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "evolve during training until it eventually becomes the original, difficult to optimize",
+                                    "type": "text"
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+                            ],
+                            "index": 13
+                        },
+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "objective function. The complexity of the mollified networks is controlled by a",
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+                            "index": 14
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+                        {
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+                                    "score": 1.0,
+                                    "content": "single hyperparameter that is annealed during training. We show improvements",
+                                    "type": "text"
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+                            ],
+                            "index": 15
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "on various difficult optimization tasks and establish a relationship between recent",
+                                    "type": "text"
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+                            "index": 16
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+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "works on continuation methods for neural networks and mollifiers.",
+                                    "type": "text"
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+                        }
+                    ],
+                    "index": 10.5
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+                                    "score": 1.0,
+                                    "content": "1 INTRODUCTION",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 18
+                        }
+                    ],
+                    "index": 18
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                        393,
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+                    "lines": [
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+                                    "bbox": [
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+                                        393,
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+                                    "score": 1.0,
+                                    "content": "In the last few years, deep neural networks – i.e. convolutional networks (LeCun et al., 1989),",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 19
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "LSTMs (Hochreiter & Schmidhuber, 1997a) or GRUs (Cho et al., 2014) – set the state of the art on",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 20
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "a range of challenging tasks (Szegedy et al., 2014; Visin et al., 2015; Hinton et al., 2012; Sutskever",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 21
+                        },
+                        {
+                            "bbox": [
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+                                425,
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+                            "spans": [
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+                                    "bbox": [
+                                        105,
+                                        425,
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+                                        437
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+                                    "score": 1.0,
+                                    "content": "et al., 2014; Bahdanau et al., 2014; Mnih et al., 2013; Silver et al., 2016). However when trained",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 22
+                        },
+                        {
+                            "bbox": [
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+                                434,
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+                                448
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+                            "spans": [
+                                {
+                                    "bbox": [
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+                                        434,
+                                        505,
+                                        448
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+                                    "score": 1.0,
+                                    "content": "with variants of SGD (Bottou, 1998) deep networks can be difficult to optimize due to their highly",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 23
+                        },
+                        {
+                            "bbox": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "non-linear and non-convex nature (Choromanska et al., 2014; Dauphin et al., 2014).",
+                                    "type": "text"
+                                }
+                            ],
+                            "index": 24
+                        }
+                    ],
+                    "index": 21.5
+                },
+                {
+                    "type": "text",
+                    "bbox": [
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+                    "lines": [
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+                                    "bbox": [
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+                                    "score": 1.0,
+                                    "content": "A number of approaches were proposed to alleviate the difficulty of optimization: addressing",
+                                    "type": "text"
+                                }
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+                            "index": 25
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+                        {
+                            "bbox": [
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+                                    "score": 1.0,
+                                    "content": "smoothing of the non-linearities to help the training of recurrent networks. The differences of that",
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+                                    "score": 1.0,
+                                    "content": "et al., 1983). Similarly to the analysis in Mobahi (2016), we can write a Monte-Carlo estimate of",
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+                                    "content": "\\bar { \\mathrm { L } } = ( \\bar { \\mathcal { L } } ^ { 1 } ( \\pmb { \\theta } ; \\xi _ { \\sigma _ { 1 } } ) , \\bar { \\mathcal { L } } ^ { 2 } ( \\pmb { \\theta } ; \\xi _ { \\sigma _ { 2 } } ) , \\cdots , \\bar { \\mathcal { L } } ^ { k } ( \\pmb { \\theta } ; \\xi _ { \\sigma _ { k } } ) )",
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+                                    "score": 1.0,
+                                    "content": ". Let us note that our algorithm",
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+                                    "content": "\\bar { \\mathrm { L } } = ( \\bar { \\mathcal { L } } ^ { 1 } ( \\pmb { \\theta } ; \\xi _ { \\sigma _ { 1 } } ) , \\bar { \\mathcal { L } } ^ { 2 } ( \\pmb { \\theta } ; \\xi _ { \\sigma _ { 2 } } ) , \\cdots , \\bar { \\mathcal { L } } ^ { k } ( \\pmb { \\theta } ; \\xi _ { \\sigma _ { k } } ) )",
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+                                    "score": 1.0,
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+                                    "content": "4 SIMPLIFYING THE OBJECTIVE FUNCTION FOR FEEDFORWARD NETWORKS",
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+                                    "content": "Algorithm 1 Activation of a unit",
+                                    "type": "text"
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+                                    "content": "x _ { i } \\gets \\mathbf { w } _ { i } ^ { \\top } \\mathbf { h } ^ { l - 1 } + b _ { i }",
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+                                    "content": "\\Delta _ { i } \\gets \\mathbf { u } ( x _ { i } ) - \\mathbf { f } ( x _ { i } )",
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+                                    "content": "is a measure of a saturation of a unit",
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+                                    "score": 0.8,
+                                    "content": "\\sigma ( x _ { i } ) \\gets ( \\mathrm { s i g m o i d } ( a _ { i } \\Delta _ { i } ) - 0 . 5 ) ^ { 2 }",
+                                    "type": "inline_equation"
+                                },
+                                {
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+                                    "content": "\\xi _ { i } \\sim \\mathcal { N } ( 0 , 1 )",
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+                                {
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+                                    "score": 1.0,
+                                    "content": ". sampling the noise from a basic Normal distribution",
+                                    "type": "text"
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+                                    "content": "s _ { i } \\gets p ^ { l } c \\sigma ( x _ { i } ) | \\xi _ { i } |",
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+                                    "content": ". Half-Normal noise controlled by",
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+                                    "content": "\\pi _ { i } ^ { l } \\sim { \\bf B e r n o u l l i } ( p ^ { l } ) \\qquad \\triangleright p ^ { l }",
+                                    "type": "inline_equation"
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+                                    "content": "\\phi ( \\mathbf { h } ^ { l - 1 } , \\xi _ { i } , \\pi _ { i } ^ { l } ; \\mathbf { w } _ { i } ) = \\pi _ { i } ^ { l } h _ { i } ^ { l - 1 } + ( 1 - \\pi _ { i } ^ { l } ) \\tilde { h } _ { i } ^ { l }",
+                                    "type": "inline_equation"
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+                                    "score": 1.0,
+                                    "content": ". make a HARD decision between",
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+                                    "content": "5 LINEARIZING THE NETWORK",
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+                                    "score": 1.0,
+                                    "content": "function of the input when",
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+                                    "score": 1.0,
+                                    "content": ". sampling the noise from a basic Normal distribution",
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+                                    "content": "Deep Pentomino Pentomino is a toy-image dataset where each image has 3 Pentomino blocks.",
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+                                    "score": 1.0,
+                                    "content": "three models in Figure 10 and the best test accuracy obtained early stopping on validation accuracy",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
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+                                    "content": "LSTM Language Modeling We evaluate our model on LSTM language modeling. Our baseline",
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+                                    "score": 1.0,
+                                    "content": "of characters to the word-embeddings is a difficult problem. Thus one needs to use a highly non-linear",
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+                                    "score": 1.0,
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+                                    "score": 1.0,
+                                    "content": "LSTM Language Modeling We evaluate our model on LSTM language modeling. Our baseline",
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+                                    "score": 1.0,
+                                    "content": "verges faster and achieves better results. We provide the results for PTB language modeling in Table 2.",
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+                                    "content": "We propose a novel method for training neural networks inspired by an idea of continuation,",
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+                        },
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+                                    "content": "We thank Nicholas Ballas and Misha Denil for the valuable discussions and their feedback. We",
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+                                    "content": "computing Theano Development Team (2016). We acknowledge the support of the following",
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+                                    "content": "organizations for research funding and computing support: NSERC, Samsung, Calcul Québec,",
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+                                    "content": "REFERENCES",
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diff --git a/parse/train/r1G4z8cge/r1G4z8cge_model.json b/parse/train/r1G4z8cge/r1G4z8cge_model.json
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