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+# CANIFE: CRAFTING CANARIES FOR EMPIRICAL PRIVACY MEASUREMENT IN FEDERATED LEARNING
+
+Samuel Maddock*
+
+University of Warwick
+
+Alexandre Sablayrolles
+
+Meta AI
+
+Pierre Stock
+
+Meta AI
+
+# ABSTRACT
+
+Federated Learning (FL) is a setting for training machine learning models in distributed environments where the clients do not share their raw data but instead send model updates to a server. However, model updates can be subject to attacks and leak private information. Differential Privacy (DP) is a leading mitigation strategy which involves adding noise to clipped model updates, trading off performance for strong theoretical privacy guarantees. Previous work has shown that the threat model of DP is conservative and that the obtained guarantees may be vacuous or may overestimate information leakage in practice. In this paper, we aim to achieve a tighter measurement of the model exposure by considering a realistic threat model. We propose a novel method, CANIFE, that uses canaries—carefully crafted samples by a strong adversary to evaluate the empirical privacy of a training round. We apply this attack to vision models trained on CIFAR-10 and CelebA and to language models trained on Sent140 and Shakespeare. In particular, in realistic FL scenarios, we demonstrate that the empirical per-round epsilon obtained with CANIFE is  $4 - 5 \times$  lower than the theoretical bound.
+
+# 1 INTRODUCTION
+
+Federated Learning (FL) has recently become a popular paradigm for training machine learning models across a large number of clients, each holding local data samples (McMahan et al., 2017a). The primary driver of FL's adoption by the industry is its compatibility with the "privacy by design" principle, since the clients' raw data are not communicated to other parties during the training procedure (Kairouz et al., 2019; Huba et al., 2022; Xu et al., 2022). Instead, clients train the global model locally before sending back updates, which are aggregated by a central server. However, model updates, in their individual or aggregate form, leak information about the client local samples (Geiping et al., 2020; Gupta et al., 2022).
+
+Differential Privacy (DP) (Dwork et al., 2006; Abadi et al., 2016) is a standard mitigation to such privacy leakage. Its adaptation to the FL setting, DP-FEDAVG (McMahan et al., 2017b), provides user-level guarantees by adding Gaussian noise to the aggregated clipped model updates received by the server. In practice, training with strong privacy guarantees comes at the expense of model utility (Bassily et al., 2014; Kairouz et al., 2019), notwithstanding efforts to close this gap, either with public pre-training and partial model updates (Xu et al., 2022), accountants with better compositionality properties (Mironov, 2017) or DP variants such as DP-FTRL (Kairouz et al., 2021).
+
+Hence, it is common in practical deployments of DP-FL to train with a high privacy budget  $\varepsilon$  resulting in loose privacy guarantees (Ramaswamy et al., 2020). Such large privacy budgets often provide vacuous guarantees on the information leakage, for instance, against membership inference attacks (Mahloujifar et al., 2022). Encouragingly, recent work has shown that the information recovered in practice using state-of-the-art attacks is less than what theoretical bounds may allow (Nasr et al., 2021). This suggests that DP is conservative and that a tighter measurement of the model exposure may be achieved by considering more realistic threat models.
+
+In this paper, we propose to complement DP-FL training with a novel attack method, CANaries In Federated Environments (CANIFE), to measure empirical privacy under a realistic threat model. We assume that a rogue client wants to reconstruct data samples from the model updates. To make its job
+
+![](images/6cb5b47d0f222366b6ee4bc7f3860345db4df6254e614e43ebec85ab3092cfbc.jpg)  
+(a) Sent140 (2-layer LSTM)
+
+![](images/e5483ea9973e7d88a84d2936aa92cc7bdafd9721ad6e78a90fd18095222be449.jpg)  
+(b) CelebA (ResNet18)  
+Figure 1: Empirical privacy measurements over the course of FL training for LEAF benchmarks Sent140, CelebA and Shakespeare with  $\varepsilon \in \{10,30,50\}$ . We observe a notable gap between the theoretical  $\varepsilon$  obtained with DP-FEDSGD and the empirical  $\hat{\varepsilon}$  obtained with CANIFE.
+
+![](images/2bc1f53d72eb0e5c45ff52ce2fe7b638c8ea765563192b3aa30420197d5fc2d9.jpg)  
+(c) Shakespeare (2-layer LSTM)
+
+easier, this adversary is allowed to craft an outlier training sample, the canary. The training round proceeds normally, after which the rogue client performs a statistical test to detect the canary in the global noisy model update provided to the server by any secure aggregation protocol (see Figure 2). Finally, we translate the attack results into a per round measure of empirical privacy (Jagielski et al., 2020; Nasr et al., 2021) and propose a method using amplification by subsampling to compute the empirical privacy incurred during training as depicted in Figure 1 for standard FL benchmarks.
+
+Critically, our privacy attack is designed to approximate the worst-case data sample, not the worst-case update vector. The rogue client seeks to undermine the privacy guarantee by manipulating its input, which is consistent with FL environments using secure sandboxing to protect the integrity of the training process (Frey, 2021). We additionally model the server as the honest party, not allowing it to poison the global model in order to reconstruct training samples, in contrast with a recent line of work (Fowl et al., 2021; Boenisch et al., 2021; Wen et al., 2022; Fowl et al., 2022).
+
+In summary, our contributions are as follows:
+
+- We propose CANIFE (Section 3), a novel and practical privacy attack on FL that injects crafted canary samples. It augments the standard DP-FL training with a tight measure of the model's privacy exposure given a realistic yet conservative threat model.  
+- CANIFE is compatible with natural language and image modalities, lightweight and requires little representative data and computation to be effective. As a sanity check, we demonstrate that CANIFE tightly matches DP guarantees in a toy setup (Section 4.1) before exploring how it behaves in the federated setting (Section 4.2).  
+- Our work highlights the gap between the practical privacy leakage and the DP guarantees in various scenarios. For instance, on the CelebA benchmark, we obtain an empirical measure  $\hat{\varepsilon} \approx 6$  for a model trained with a formal privacy guarantee of  $\varepsilon = 50$ .
+
+# 2 BACKGROUND
+
+# 2.1 DIFFERENTIAL PRIVACY
+
+Differential Privacy (Dwork et al., 2006; Dwork & Roth, 2014) defines a standard notion of privacy that guarantees the output of an algorithm does not depend significantly on a single sample or user.
+
+Definition 1 (Differential Privacy). A randomised algorithm  $\mathcal{M}\colon \mathcal{D}\to \mathcal{R}$  satisfies  $(\varepsilon ,\delta)$ -differential privacy if for any two adjacent datasets  $D,D^{\prime}\in \mathcal{D}$  and any subset of outputs  $S\subseteq \mathcal{R}$ ,
+
+$$
+\mathbb {P} (\mathcal {M} (D) \in S) \leq e ^ {\varepsilon} \mathbb {P} (\mathcal {M} (D ^ {\prime}) \in S) + \delta .
+$$
+
+The privacy parameter  $\varepsilon$  is called the privacy budget and it determines an upper bound on the information an adversary can obtain from the output of an  $(\varepsilon, \delta)$ -DP algorithm. The parameter  $\delta$  defines the probability of failing to guarantee the differential privacy bound for any two adjacent datasets. In this work, we are interested in user-level differential privacy which takes  $D$  and  $D'$  to be adjacent if  $D'$  can be formed by adding or removing all samples associated with a single user from  $D$ .
+
+Standard DP results state that the privacy budget  $(\varepsilon, \delta)$  accumulates (roughly) in proportion to the square root of the number of iterations. Advanced privacy accountants leverage the uncertainty due to random sampling (Mironov, 2017; Wang et al., 2019b; Gopi et al., 2021). We use the Renyi Differential Privacy (RDP) accounting implemented in the Opacus library (Yousefpour et al., 2021).
+
+# 2.2 PRIVATE FEDERATED LEARNING
+
+A standard FL protocol, such as FEDAVG (McMahan et al., 2017a) computes a weighted average of the model updates from clients before performing a gradient descent step on the global model. Other variants exist to deal with common optimization problems in the federated setting such as convergence speed (Wang et al., 2019a), heterogeneity (Karimireddy et al., 2020a;b), reducing communication bandwidth (Alistarh et al., 2017) and adding momentum (Reddi et al., 2020). In this work, we focus on DP-FEDSGD, a private extension of FEDAVG (McMahan et al., 2017b). At each round, the selected clients compute their clipped model update  $u_{i}$  and transmit it to the server, which aggregates the model updates and adds Gaussian noise:
+
+$$
+\tilde {u} = \sum_ {i} u _ {i} + \mathcal {N} (0, \sigma^ {2} I _ {d}).
+$$
+
+In practice, federated algorithms rely on Secure Aggregation (SegAgg) protocols to aggregate each client update without revealing any individual  $u_{i}$  to the server or to other participants (Bonawitz et al., 2017; Bell et al., 2020). In a TEE-based SecAgg (Huba et al., 2022), a trusted execution environment (TEE) aggregates the individual model updates and calibrated DP noise before handing over the noisy model update  $\tilde{u}$  to the server. These specific models are orthogonal to our work and we assume from now on that the server and clients participate in a TEE-based SecAgg protocol. We discuss our threat model with regards to our attack in Section 3.1.
+
+# 2.3 ATTACKS & EMPIRICAL PRIVACY
+
+Centralized and FL Attacks. There is a vast body of literature on attacking models trained in the centralised setting. For example, membership inference attacks (MIA) attempt to distinguish whether a sample was present in the training set given only the trained model (Shokri et al., 2017; Sablayrolles et al., 2019). Others attacks consider the more difficult problem of reconstructing entire training samples from a trained model, often using (batch) gradient information (Yin et al., 2021; Jeon et al., 2021; Balle et al., 2022). Since model updates  $u_{i}$  are essentially just aggregated gradients, it is natural that FL updates may leak private information as well. Nasr et al. (2019) show that it is possible to perform both passive and active membership-inference attacks in the federated setting. Other works such as that of Fowl et al. (2021) and Wen et al. (2022) have designed attacks on model updates which allow for the reconstruction of training samples used in federated training. However, they assume that the server can poison the global model, whereas we assume an honest server.
+
+Canaries. The notion of canary samples usually refers to natural data samples used to measure memorization in large language models (Carlini et al., 2019; 2021; Thakkar et al., 2020; Shi et al., 2022). For instance, Parikh et al. (2022) propose to reconstruct canaries inserted in training data and Stock et al. (2022) insert canaries to track and quantify the information leakage when training a causal language model. In all prior work, the canary is either a sample from the training set or a handcrafted instance such as "My SSN is 123-45-6789". In contrast, CANIFE provides an explicit method for crafting canary samples that are as adversarial as possible within the given threat model (see Appendix E for canary samples) to obtain tight measurement of the model's exposure. Moreover, the proposed method applies to any training modality allowing to backpropagate in the sample space (in particular, pixels for images and tokens for natural language).
+
+Empirical Privacy. The proposed approach departs from existing attacks in the FL setup. For instance, Jayaraman & Evans (2019) and Jagielski et al. (2020) have derived empirical measures of privacy through attacks and often shown gaps between the empirically measured and theoretical privacy. More recently, Nasr et al. (2021) study a range of membership-inference attacks by varying adversary's powers. We argue that the threat model of many of these attacks is often too permissive relative to what a realistic adversary can achieve in the federated setting. For example, attacks in Nasr et al. (2021) assume knowledge of the other samples in the dataset.
+
+![](images/d526acba114ac48195b4cf0c7dbfd4b160dd3850ba2c92549715fe0b10e112fc.jpg)  
+Figure 2: Illustration of the proposed CANIFE method for one training round. The adversary is a rogue client that crafts a training sample resulting in an extremely out-of-distribution model update  $u_{c}$  and performs a membership inference attack on the (public) aggregated noisy model update  $\tilde{u}$  provided to the server by SecAgg. Attack results are converted to empirical privacy guarantees  $\hat{\varepsilon}$  for this round. We then compound these guarantees over the course over the training as in Section 3.4.
+
+# 3 METHODOLOGY
+
+The adversary wants to design a canary sample  $z$  to measure the empirical privacy of a given FL training round under a realistic threat model defined in Section 3.1. We view this problem as a membership inference game where a rogue client crafts a canary  $z$  specific to the global model at round  $r$  (Section 3.3) and uses this canary to produce an extremely out-of-distribution model update. After this, the server proceeds as usual, aggregating model updates via any secure aggregation primitive and adding noise. At the end of the round, the adversary attempts to detect the presence of the canary in the aggregated noisy model update  $\tilde{u}$  (Section 3.2) and computes a per-round empirical privacy measure. These per-round empirical privacy guarantees are compounded over the whole course of the training to give a final empirical privacy guarantee as explained in Section 3.4.
+
+# 3.1 THREAT MODEL
+
+We assume an honest-but-curious threat model where the clients and the server do not deviate from the training protocol (Bonawitz et al., 2017; Kairouz et al., 2019; Huba et al., 2022). In particular, the clients cannot directly manipulate gradients nor the final model updates from the local training and the server cannot modify the model maliciously to eavesdrop on the clients.
+
+To craft the canary for a given training round, we assume that the adversary — a rogue client — has access to (1) the current server-side public model and to (2) emulated mock clients. Mock clients are created using a public set of samples called the design pool whose distribution is similar to that of the clients. We show in Section 4 that the adversary is able to design a strong and robust canary sample even with a small amount of public data and under mild assumptions about the true data distribution. We also demonstrate that the computational resources required to design the canary are small, which means that even adversaries under resource constraints are able to launch such attacks.
+
+We argue such a threat model is realistic and discuss practical limitations that influence the adversary's success in Section 3.4: our goal is to design an attack that is as adversarial as possible (in order to derive a worst-case measure of empirical privacy) under a set of reasonable assumptions.
+
+# 3.2 CANARY DETECTION
+
+The adversary's objective is to craft a canary  $z$  such that the resulting model update is extremely out-distribution. More precisely, we require that the canary gradient  $\nabla_{\theta}\ell (z)$  is orthogonal to all individual clipped model updates  $u_{i}$  in the current round:  $\langle u_i,\nabla_\theta \ell (z)\rangle = 0$ . We demonstrate that this allows the adversary to detect the presence of the canary by separating two Gaussian distributions.
+
+The server trains a model parameterized by  $\theta \in \mathbb{R}^d$  with a certain level of DP noise  $\sigma$  across a population of clients each holding a local dataset  $D_{i}$ . Since the rogue client holds a dataset that
+
+only contains the canary  $D_{c} = \{z\}$ , its clipped model update will be proportional<sup>1</sup> to the canary gradient:  $u_{c} \propto \nabla_{\theta}\ell (z)$ . Recall that the aggregated private model update  $\tilde{u}$  is formed by the noisy sum of individual clipped model updates:  $\tilde{u} = \sum_{i}u_{i} + \mathcal{N}(0,\sigma^{2}I_{d})$ . Then, as a counter-factual, if the rogue client does not participate in the training round:
+
+$$
+\langle \tilde {u}, u _ {c} \rangle \propto \sum_ {i} \underbrace {\langle u _ {i} , \nabla_ {\theta} \ell (z) \rangle} _ {= 0 \text {b y d e s i g n}} + \underbrace {\langle \mathcal {N} (0 , \sigma^ {2} I _ {d}) , \nabla_ {\theta} \ell (z) \rangle} _ {= \mathcal {N} (0, \sigma^ {2} \| \nabla_ {\theta} \ell (z) \| ^ {2})}.
+$$
+
+Hence,  $\langle \tilde{u}, u_c \rangle$  follows a one-dimensional zero-mean Gaussian with variance  $\sigma_c^2$ , where  $\sigma_c$  accounts for the proportionality factor. Similarly, if the rogue client participates in the training round,  $\langle \tilde{u}, u_c \rangle$  follows a one-dimensional Gaussian with the same variance  $\sigma_c^2$  centered at  $\| u_c \|^2$ . Thus, the membership inference game is reduced to separating two Gaussian distributions centered at 0 and  $||u_c||^2$  respectively. Note that the result is unchanged up to a fixed scaling factor if the client updates  $u_i$  are weighted by their local dataset size as in (McMahan et al., 2017b). The testing approach is described in full in Algorithm 1 and involves computing the attack score  $s_r := \langle \tilde{u}, u_c \rangle$ . We derive a connection with the likelihood ratio test in Appendix A.
+
+# 3.3 CANARY DESIGN
+
+At a given round  $r$ , the rogue client is given the current server-side model parameterized by  $\theta$ . Given a set of heldout clipped model updates  $\{u_i\}$ , it creates the canary by minimizing:
+
+$$
+\mathcal {L} (z) = \sum_ {i} \left\langle u _ {i}, \nabla_ {\theta} \ell (z) \right\rangle^ {2} + \max  \left(C - | | \nabla_ {\theta} \ell (z) | |, 0\right) ^ {2}. \tag {1}
+$$
+
+Recall that  $\nabla_{\theta}\ell (z)$  denotes the gradient of the network's parameters with respect to its training loss  $\ell$  when fording the canary  $z$  through the network. The first loss term is designed to make the canary gradient  $\nabla_{\theta}\ell (z)$  orthogonal to the set of heldout model updates while the second term enforces the canary norm is not smaller than some constant, that we set to the clipping constant  $C$  of DP-FL. This ensures  $\| u_c\|^2 = C^2$  and for simplicity we fix  $C = 1$ . In Appendix D, we provide experiments that show that choosing the gradient norm constant too large (i.e., much larger than  $C$ ) has a detrimental effect on optimization.
+
+Using an automatic differentiation framework such as PyTorch (Paszke et al., 2019), we compute  $\nabla_z\mathcal{L}(z)$  and perform stochastic gradient descent directly in the sample space as described in Algorithm 1. (Recall that the model parameters  $\theta$  are fixed during the optimization procedure.) Computing  $\nabla_z\mathcal{L}(z)$  is straightforward for continuous data, such as images, as we can simply backpropagate in the pixel space. For language models, the problem is more complex as the input is a sequence of discrete tokens. Hence, we leverage the work of Guo et al. (2021), who use the Gumbel-Softmax distribution (Jang et al., 2016) to forge adversarial language examples. This allows them to use gradient-based methods by optimising a probability distribution over each token in a sequence.
+
+We investigate various methods to initialize the canary, including starting from a random training sample or random pixels or tokens. Depending on the task at hand, we might need to fix a target for the canary. For instance, for image classification, we need to assign the canary to a class in order to be able to compute  $\nabla_{\theta}\ell (z)$ . We investigate various canary initialization and target choice strategies experimentally in Section 4. We also investigate other optimization considerations for designing the canary such as slightly modifying the loss in Appendix D.
+
+Adversarial Examples. We can view the canary  $z$  as "adversarial" in the sense that it should be extremely out-of-distribution to get a worst-case measure of privacy. This is different from the unrelated notion of adversarial examples which typically constrain the changes of a sample to be imperceptible to humans (Biggio et al., 2013). In our setup, we do not impose this constraint as we wish to encompass the realistic worst-case of an extreme out-of-distribution sample.
+
+# 3.4 MEASURING EMPIRICAL PRIVACY
+
+The CANIFE attack is carried out over a certain number  $(n)$  of fake rounds where the server does not update its global model but where the pool of selected (regular) clients differs every time. Hence,
+
+Algorithm 1 CANIFE attack by a rogue client  
+Input: Design pool  $D_{\mathrm{pool}}$ , Design iterations  $T$ , Canary learning rate  $\beta$ , Global model  $\theta$   
+1: Form mock clients from the design pool  $D_{\mathrm{pool}}$   
+2: For each mock client  $i$ , compute the clipped model update  $u_i$   
+3: Initialise the canary  $z_0$   
+4: for  $t = 1, \dots, T$  do  
+5:  $\mathcal{L}(z_t) \gets \sum_i \langle u_i, C \cdot \nabla_\theta \ell(z_t) \rangle^2 + \max(C - ||\nabla_\theta \ell(z_t)||, 0)^2$   
+6: Compute  $\nabla_{z_t} \mathcal{L}(z_t)$   
+7:  $z_{t+1} \gets z_t - \beta \cdot \nabla_{z_t} \mathcal{L}(z_t)$   
+8: end for  
+9: Compute  $u_c$   
+10: return  $s_r \gets \langle \tilde{u}, u_c \rangle$   
+11:  
+    - Model update with  $D_c = \{(z_T, y_c)\}$   
+    -  $u_i$  is deemed public after the round has finished
+
+CANIFE has no influence on the model's final accuracy. The adversary crafts a single canary and inserts it into every fake round with probability  $1/2$ . Once the  $n$  attack scores are computed, the adversary deduces the performance of the attack at a calibrated threshold  $\gamma$ . Building on previous work (Jagielski et al., 2020; Nasr et al., 2021), we compute the empirical privacy guarantee  $\hat{\varepsilon}$  based on the False Positive Rate (FPR) and False Negative Rate (FNR) of the attack at the threshold  $\gamma$  as
+
+$$
+\hat {\varepsilon} = \max  \left(\log \frac {1 - \delta - \mathrm {F P R}}{\mathrm {F N R}}, \log \frac {1 - \delta - \mathrm {F N R}}{\mathrm {F P R}}\right).
+$$
+
+Our measure differs slightly from that of (Nasr et al., 2021) as our attack measures privacy at a single round of training. Thus,  $\hat{\varepsilon}$  is a per-round privacy measure which we denote as  $\hat{\varepsilon}_r$  and we compare this to a per-round theoretical  $\varepsilon_r$ . In order to deduce an overall empirical epsilon  $\hat{\varepsilon}$  we convert the per-round  $\hat{\varepsilon}_r$  into a noise estimate  $\hat{\sigma}_r$  under amplification by subsampling. We then compose for  $s$  steps with an RDP accountant, until we perform the next attack and update our  $\hat{\sigma}_r$ . To provide a worst-case measure we choose a threshold that maximises  $\hat{\varepsilon}$  under our attack. We note that  $n$  determines bounds on  $\hat{\varepsilon}_r$  and its CIs. With  $n = 100$ ,  $\hat{\varepsilon}_r$  is at most 3.89. Similarly, when computing  $\hat{\varepsilon}_r$ , we set  $\delta = 1/n$ . For further accounting details and the algorithm to compute  $\hat{\varepsilon}$  see Appendix C.
+
+Relationship to DP. Our threat model and canary design procedure have been chosen to ensure that the adversary is as strong as possible but still constrained by realistic assumptions. At least three factors of the design process restrict the adversary in practice:
+
+1. The adversary only has partial knowledge of the model updates in the form of an aggregated noisy sum  $\tilde{u}$ . Furthermore, when designing a canary, the adversary uses a design pool to mock clients. In our design, we have been conservative by assuming the adversary has access to heldout data that matches the federated distribution by constructing the design pool from the test set of our datasets. In practice, adversaries can form design pools from public datasets that match the task of the target model. See Appendix F for further discussion.  
+2. The optimization process induces two possible sources of error: the convergence itself and the generalization ability of the canaries to unseen model updates.  
+3. We calculate the maximum accuracy that can be derived from the attack and use the threshold that maximises  $\hat{\varepsilon}_r$ . This is conservative, as in practice an adversary would have to calibrate a threshold  $\gamma$  on a public heldout dataset. Furthermore, they would not be able to perform  $n$  fake rounds with the same canary.
+
+If these practical constraints were not imposed on an adversary, then the privacy measure we derive from our attack would essentially be tight with the DP upper bound. For example, if we could always design a perfect canary  $z$  that has gradient orthogonal to all possible model updates  $u_{i}$  then the DP noise  $\sigma$  completely determines whether we can distinguish if  $z$  is present in  $\tilde{u}$  or not. However, in practice, the assumptions listed above make the attack more difficult as we will see in Section 4.1.
+
+# 4 EXPERIMENTS
+
+In this section, we begin by highlighting the power of our attack on the simplest federated setting where participants only have a single sample (Section 4.1). We then show how our attack allows
+
+![](images/2ab43f9622b0806290edd5bbf93dc53cae1ca29c4d66c1a0fef73b687e157238.jpg)  
+(a)  $\sigma = 0$
+
+![](images/8fa3f4ca474b7721e28d9fa466803be5b1309365ae6f75034255882e9fe64896.jpg)  
+(b)  $\sigma = 0.423$  
+Figure 3: Experiments on CIFAR-10 on a toy setup where each client holds a single data sample. (a) Attack histograms without DP noise, showing that our attack completely succeeds. (b) Attack histograms with DP noise: as the DP noise  $\sigma$  increases, it becomes harder to separate both histograms. (c) Varying the design pool size and the number of participating clients. The  $y$ -axis shows the mean standard deviation of the histograms formed from the attack.
+
+![](images/456e3094dc4f52a55a52e5c784097c860aba65a7265a6c26b5134945b5268598.jpg)  
+(c)
+
+us to monitor privacy empirically over a federated training run (Section 4.2) before concluding with an ablation study that shows our canary is robust to various design choices (Section 4.3). We open-source the code for CANIFE design and test to reproduce our results<sup>2</sup>.
+
+Setup. We evaluate our attack on both image and language models. We utilise LEAF (Caldas et al., 2018) which provides benchmark federated datasets for simulating clients with non-IID data and a varying number of local samples. We study image classification on CIFAR10 (IID) (Krizhevsky et al., 2009) and CelebA (non-IID) (Liu et al., 2015). We train a simple Convolutional Neural Network (CNN) and a ResNet18 model. For our language tasks, we train an LSTM model on non-IID splits of Sent140 (Go et al., 2009) and Shakespeare (McMahan et al., 2017a). For more information on datasets, model architectures and training hyperparameters, see Appendix B. We train, design, and evaluate our attacks using the FLSim framework<sup>3</sup>. All experiments were run on a single A100 40GB GPU with model training taking at most a few hours. We discuss CPU benchmarks for canary design in Section 4.3. For canary design, we use the Adam optimizer (Kingma & Ba, 2014) with learning rate  $\beta = 1$  and fix  $C = 1$ . We form the CANIFE design pool from a LEAF test split, resulting in canaries designed on non-IID mock clients which approximates the training distribution. We have clients perform a single local epoch in all experiments, but see Appendix F for possible extensions to multiple local epochs. From now, we refer to "epoch" as one pass of the federated dataset (in expectation). For privacy accounting, we utilise the RDP accountant with subsampling (Mironov et al., 2019) implemented via the Opacus library (Yousefpour et al., 2021), sampling clients uniformly at each round (i.e., Poisson sampling), see Appendix C for more details.
+
+# 4.1 EXAMPLE ATTACK: CIFAR10
+
+We first investigate the simplest federated setting where each client holds a single sample. It follows that the clipped model update  $u_{i}$  is simply a scaled gradient of the client's single sample. This corresponds to the central setting where the number of clients per round is equivalent to a (central) batch size. In Figure 3, we train a ResNet18 model on CIFAR10 to  $60\%$  test accuracy and perform our attack. We design a single canary and have the server perform  $n = 100$  mock rounds where the model is frozen, with half having the canary client inserted and half without. We compute attack scores from these  $n = 100$  trials and form histograms.
+
+In Figure 3a, we present one histogram of the attack scores on a model that has 64 clients participating per round with no DP  $(\varepsilon = \infty)$ . We use a design pool of  $m = 512$  to design the canary. We observe that our attack is quite tight in this non-private setting. Since  $\sigma = 0$ , we hope that if the canary was designed well, the standard deviation of the histograms would also be close to 0. It turns out that the average standard deviation of the histograms is 0.006. Recall in Section 3.4, we discussed there is inherent error from both the optimization procedure and the fact the server
+
+![](images/2874e40d516f20002388b7a4c54da58eb5d927a03253811f235ac3cd4fa0983d.jpg)  
+(a) Sent140 (2-layer LSTM)
+
+![](images/d53122f55bc5f54e6db7610656a3f9c9ab8698054cbe6c714ee708eb2e46d25b.jpg)  
+(b) CelebA (ResNet18)  
+Figure 4: (a) and (b) Monitoring per-round empirical privacy on LEAF benchmarks. Models are trained to  $\varepsilon = 50$  with 100 clients per round. CIs for  $\hat{\varepsilon}_r$  are compared with the theoretical per-round epsilon  $\varepsilon_r$ , rounds with an upper CI of  $\infty$  do not have CIs displayed. (c) CANIFE empirical privacy incurred during the whole training, averaged over 5 independent runs (see Figure 1 for details).
+
+![](images/a631df7ce91f3c775d5c5ddcf86a04a3bcf950d389041f6f6425ace5c976b792.jpg)  
+(c) CANIFE empirical epsilon  $\hat{\varepsilon}$
+
+designs the canary on heldout data but in this case the error is small. In Figure 3b, we display another example attack, this time for a model trained under privacy with a final  $\varepsilon = 25$  corresponding to  $\sigma = 0.423$  with 64 clients per round. We observe the attack is still fairly tight as the standard deviation of the histograms (average 0.478) is close to that of the privacy noise  $\sigma$ .
+
+Finally, we explore how both the design pool size and the number of clients affect the standard deviation of the histograms (Figure 3c). We vary both the number of clients and the design pool size. We train a model without privacy for each combination of parameters until it reaches  $60\%$  train accuracy and then attack the model, plotting the average standard deviation of the histograms. We conclude with two observations. First, the attack is much tighter when there is a smaller number of clients participating per round. Second, the size of the design pool has a diminishing effect on reducing the standard deviation. We further explore these in Section 4.3.
+
+# 4.2 MONITORING EMPIRICAL PRIVACY
+
+One main goal of our attack is to provide a lightweight method for monitoring empirical privacy during federated training. We explore this in Figure 4, training ResNet18 on CelebA and a 2-layer LSTM model on Sent140 and Shakespeare. We train to a final  $\varepsilon = 50$  and achieve  $63.9\%$  test accuracy on Sent140,  $89.9\%$  on CelebA and  $44.8\%$  on Shakespeare. We carry out the CANIFE attack at a regular interval, freezing the model, designing a single canary, and performing  $n = 100$  attack trials with the designed canary before continuing training. This generates scores which are used to compute an empirical per-round privacy measure  $\hat{\varepsilon}_r$  as outlined in Section 3.4 (see also Appendix C.3 for further details). The canary sample  $z$  is initialised randomly and we explore how this affects optimization in Section 4.3. We note in practice that participants (and the server) would not want to waste rounds with a frozen model, see Appendix F for possible extensions.
+
+We monitor the optimization of the canary loss by computing the canary health which measures the percentage improvement of the final canary loss compared to the initial loss. A perfect health of 1 means the canary achieved minimal design loss. As an example, for Sent140, we find that the average canary health across 5 separate training runs is  $0.970 \pm 0.098$ . Excluding the first 1000 rounds from each run, the average canary health becomes  $0.990 \pm 0.01$ . Thus, after two model epochs, optimization stabilises and the initial canary loss is reduced by  $99\%$ .
+
+In Figure 4a, we display per-round privacy estimates  $\hat{\varepsilon}_r$  and their  $95\%$  confidence intervals (CIs) for Sent140 trained with  $\varepsilon = 50$  and compare to the (constant) theoretical per-round privacy  $\varepsilon_r$ . We observe  $\hat{\varepsilon}_r$  is initially small and grows to an almost constant level within a single epoch and stays there during training. This results in a  $4.5\times$  gap between the theoretical  $\varepsilon_r$  and  $\hat{\varepsilon}_r$  measured by our attack. We obtain similar  $4 - 5\times$  gaps for CelebA (ResNet18) in Figure 4b and Shakespeare in Appendix C.5. We compound these per-round estimates  $\hat{\varepsilon}_r$  to provide a cumulative measure of privacy  $\hat{\varepsilon}$  and display this in Figure 1 for models trained with  $\varepsilon \in \{10,30,50\}$ . Again, there is a significant gap between the final theoretical privacy and the measure derived under CANIFE. The final  $\hat{\varepsilon}$  averaged over 5 separate training runs for each tasks are shown in Figure 4c. We note significant gaps between tasks, most notably with Sent140. This is likely due to the small sample
+
+![](images/f83b8c33b662d1afd87dd5c0c01c3533fde2c2269d8d7e1c4876340cc2086eb2.jpg)  
+(a) Canary Initialisation
+
+![](images/2625d51d3b9ebafadbb53c2e2f783ba67624b86bd828e7da1a25097b6217b43d.jpg)  
+(b) Design Iterations: Varying  $T$  
+Figure 5: Ablations studies for CANIFE without addition of DP noise. (a) Canary initialization when varying the design pool size (CelebA, 4-layer CNN). (b) Number of design iterations required to produce robust and well-designed canaries (CelebA, 4-layer CNN). (c) Model vs attack accuracy across 5 runs (Sent140, 2-layer LSTM). Attack accuracy increases as model test accuracy increases.
+
+![](images/414641edb6ad767a355f9cf2b3f6011f3dfc187a410abf767a7d168cc5822b9c.jpg)  
+(c) Model test vs attack accuracy
+
+rate which provides significant privacy amplification. We also observe CANIFE estimates are stable across runs with relatively small standard deviation.
+
+# 4.3 ABLATION STUDY
+
+Canary Initialisation. In Figure 5a, we explore how initialising the canary affects the optimization. We consider two initialisation strategies: initialising the canary randomly or initialising the canary as a sample from the design pool (which is then excluded from the design). We observe no significant difference on the average standard deviation of the attack histograms.
+
+Design Pool Size. In Figure 5a, we vary the design pool size. We observe there is no significant effect on the average standard deviation of the attack histograms. This confirms what we observed in Figure 3c: the design pool size has diminishing impact on reducing the standard deviation.
+
+Design Iterations. In Figure 5b, we explore how the number of design iterations impacts the quality of the canary measured through the calibrated attack accuracy. We observe that just  $t = 1000$  iterations are needed to obtain  $95\%$  accuracy and that with  $t = 3000$  the attack improves to almost  $100\%$  accuracy, staying close to constant as  $t$  increases further. We additionally benchmarked the average CPU time on an M1 MacBook Air (2020) for a single design iteration. For Sent140, it takes an average of 0.06s per iteration and 0.23s for CelebA (ResNet18). For  $t = 2500$ , the design procedure took on average 165s and 591s respectively. Hence, the canary design process takes at most 10 minutes on a CPU and only a few minutes with a GPU. Thus our attack is lightweight, requiring only a few thousand design iterations to achieve near-optimal canaries.
+
+Model Accuracy. We conclude by noting the accuracy of the model and that of our attack are highly correlated. In Figure 5c, we plot both model test accuracy and calibrated attack accuracy across Sent140 trained without DP. We observe early in training, when model accuracy is low, that the attack accuracy is similarly low. Once the model converges to a sufficiently high test accuracy the attack accuracy is close to  $100\%$ . In experiments that require comparison across different models (e.g., Figure 3c) we checkpoint to a fixed test accuracy to avoid this confounding effect.
+
+# 5 CONCLUSION
+
+Motivated by the fact that DP is conservative, we consider a more realistic threat model to extract private information about client data when training a model with FL. We introduce CANIFE, a novel method to measure empirical privacy where a rogue client crafts a canary sample that results in an outlier model update. We argue that the difficulty of tracing this malicious model update in the aggregated noisy model update provides a tighter measurement of the model's privacy exposure. We hope this work can benefit practical deployments of DP-FL pipelines by complementing the theoretical bound with an arguably more realistic measure of the privacy leakage.
+
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+
+# A CONNECTIONS TO LIKELIHOOD RATIO TEST
+
+Let us assume that each model update follows a Gaussian distribution  $\mathcal{N}(\mu, \Sigma)$ . The sum of  $k$  model updates then either follows  $\mathcal{N}(k\mu, k\Sigma)$  (without the canary) or  $\mathcal{N}(k\mu + \nabla \ell(z), k\Sigma)$  (with the canary), recalling that  $u_c \propto \nabla \ell(z)$ . Then we have
+
+$$
+\begin{array}{l} p _ {0} (u) = \frac {1}{\sqrt {\det  (2 \pi k \Sigma)}} \exp \left(- (u - k \mu) ^ {T} (k \Sigma) ^ {- 1} (u - k \mu) / 2\right) \\ p _ {1} (u) = \frac {1}{\sqrt {\det  (2 \pi k \Sigma)}} \exp \left(- (u - (k \mu + \nabla \ell (z))) ^ {T} (k \Sigma) ^ {- 1} (u - (k \mu + \nabla \ell (z))) / 2\right) \\ = \frac {1}{\sqrt {\det (2 \pi k \Sigma)}} \exp \left(- ((u - k \mu) - \nabla \ell (z))) ^ {T} (k \Sigma) ^ {- 1} ((u - k \mu) - \nabla \ell (z)) / 2\right). \\ \end{array}
+$$
+
+We can write the (log) likelihood ratio as
+
+$$
+\begin{array}{l} \log \frac {p _ {1} (u)}{p _ {0} (u)} = \frac {1}{2} \left((u - k \mu) ^ {T} (k \Sigma) ^ {- 1} (u - k \mu) - ((u - k \mu) - \nabla \ell (z))\right) ^ {T} (k \Sigma) ^ {- 1} ((u - k \mu) - \nabla \ell (z)) \\ = \frac {1}{2} \left((u - k \mu) ^ {T} (k \Sigma) ^ {- 1} (u - k \mu) - ((u - k \mu) - \nabla \ell (z))) ^ {T} (k \Sigma) ^ {- 1} ((u - k \mu) - \nabla \ell (z))\right) \\ = \nabla \ell (z) ^ {T} (k \Sigma) ^ {- 1} (u - k \mu) - \frac {1}{2} \nabla \ell (z) ^ {T} (k \Sigma) ^ {- 1} \nabla \ell (z). \\ \end{array}
+$$
+
+In particular for the centers of the Gaussian with and without the canary,  $u \in \{k\mu, k\mu + \nabla \ell(z)\}$ ,
+
+$$
+\log \left(\frac {p _ {1} (u)}{p _ {0} (u)}\right) = \pm \frac {1}{2} \nabla \ell (z) ^ {T} (k \Sigma) ^ {- 1} \nabla \ell (z).
+$$
+
+Maximizing this term will thus help separate the two Gaussians. However, doing this directly is infeasible as it requires to form and invert the full covariance matrix  $\Sigma$  in very high dimensions. Instead, we propose to minimize  $z\mapsto (\nabla \ell (z)^T)\Sigma (\nabla \ell (z))$  as it is tractable and can be done with SGD. Note that for sample model updates  $\{u_i\}$  we can estimate the (uncentered) covariance matrix as  $\frac{1}{n}\sum_{i}u_{i}u_{i}^{T}$  and thus
+
+$$
+(\nabla \ell (z) ^ {T}) \Sigma (\nabla \ell (z)) \approx \frac {1}{n} \sum_ {i} \nabla \ell (z) ^ {T} (u _ {i} u _ {i} ^ {T}) \nabla \ell (z) = \frac {1}{n} \sum_ {i} \langle u _ {i}, \nabla \ell (z) \rangle^ {2}.
+$$
+
+Which, ignoring constants, is the first term of  $\mathcal{L}(z)$  defined in Equation 1. One could alternatively minimise  $\langle \nabla \ell (z),\hat{\mu}\rangle^2$  with  $\hat{\mu} = \frac{1}{n}\sum_{i}u_{i}$ . We explore the empirical differences in Appendix D.
+
+# B DATASETS & MODEL ARCHITECTURES
+
+Here we detail the training setup and model architectures for our experiments. In all experiments we train with DP-FEDSGD and without momentum. We use both client  $(\eta_C)$  and server  $(\eta_S)$  learning rates. We train without dropout in all model architectures. In more detail:
+
+- CIFAR10 is an image classification tasks with 10 classes (Krizhevsky et al., 2009). We train a ResNet18 model (He et al., 2016) on CIFAR10. We form an IID split of 50,000 train users and 10,000 test users where each user holds a single sample and thus has a local batch size of 1. We use a client learning rate of  $\eta_C = 0.01$  and server learning rate  $\eta_S = 1$ .  
+- CelebA is a binary image classification task (Liu et al., 2015). We train a ResNet18 model with 11, 177, 538 parameters and a simple Convolutional Neural Network (CNN) with four convolutional layers following that used by Caldas et al. (2018) with 29, 282 parameters. We use the standard non-IID LEAF split resulting in 8408 train users and 935 test users and a local batch size of 32. We train with a client learning rate of  $\eta_C = 0.899$  and a server learning rate of  $\eta_S = 0.0797$ .
+
+Algorithm 2 Measuring  $\hat{\varepsilon}$  
+Input: Number of rounds  $R$ , Privacy parameter  $\delta$ , Sampling rate  $q$ , Number of attack scores  $n$ , Attack frequency  $s$   
+1: for  $r = 1, \ldots, R$  do  
+2: if  $r \mod s = 0$  then  
+3: Freeze the model  $\theta_r$  and use Algorithm 1 to compute attack scores  $\{s_i\}_{i=1}^n$   
+4: Calculate FPR $_\gamma$  and FNR $_\gamma$  from  $\{s_i\}$  at each threshold  $\gamma \in \{s_1, \ldots, s_n\}$   
+5: Compute  $\hat{\varepsilon}_r \gets \max_\gamma \left( \log \frac{1 - \delta - \mathrm{FPR}_\gamma}{\mathrm{FNR}_\gamma}, \log \frac{1 - \delta - \mathrm{FNR}_\gamma}{\mathrm{FPR}_\gamma} \right)$   
+6: Compute  $\hat{\sigma}_r \gets \operatorname{GetNoise}(\hat{\varepsilon}_r, \delta)$   
+7:  $\hat{\sigma}_{r+i} \gets \hat{\sigma}_r$  for  $i \in [1, s)$   
+8: end if  
+9: end for  
+10: return  $\hat{\varepsilon} \gets \operatorname{GetPrivacy}(\{\hat{\sigma}_r\}; \delta, q)$   
+11: Compose each noise estimate under an RDP accountant with amplification by subsampling using sample rate  $q$ , see (Mironov et al., 2019)
+
+![](images/1c4aa205c7dba023ea99ee2ce147cf79ac90a9196d9f822cacc777b5b95dcf43.jpg)  
+(a) Shakespeare per-round privacy  $(\varepsilon = 50)$  
+Figure 6: Measuring privacy during Shakespeare training
+
+![](images/93369579ce3eb14578b086fb9873354cbe10d779c5d4e2b7c16d05faf15dfca4.jpg)  
+(b) Shakespeare per-round privacy  $(\varepsilon = \infty)$
+
+- Sent140 is a sentiment analysis (binary classification) task (Go et al., 2009). We train a 2-layer LSTM with 272, 102 parameters on Sent140 following the architecture of Caldas et al. (2018). We use standard non-IID LEAF splits resulting in 59, 214 train users and 39, 477 test users and a local batch size of 32. We train for 15 epochs, and for  $\varepsilon = 0$  achieve an average test accuracy of  $64.8\%$ . We train with a client learning rate of  $\eta_C = 5.75$  and a server learning rate of  $\eta_S = 0.244$ .  
+- Shakespeare is a next character prediction task with 47 classes (McMahan et al., 2017a). We train a similar LSTM model to Sent140, based on the architecture used by Caldas et al. (2018) with 819,920 parameters. We use standard non-IID LEAF splits with 1016 train users and 113 test users and a local batch size of 128. We train our models for 15 epochs resulting in an average final test accuracy of  $44.4\%$  for  $\varepsilon = 0$ . We use a client learning rate of  $\eta_{C} = 3$  and a server learning rate of  $\eta_{S} = 0.524$ .
+
+# C MEASURING EMPIRICAL PRIVACY
+
+# C.1 PRIVACY MEASURES AND ACCOUNTING
+
+Here we provide further details about privacy accounting and the different privacy measures that we analyse. In all experiments we use Rényi Differential Privacy (RDP) accounting with subsampling (Poisson sampling over the set of clients) to guarantee user-level DP. This is based on the DP-FEDAVG algorithm (McMahan et al., 2017b) with accounting implemented via the Opacus library (Yousefpour et al., 2021). More specifically, the accounting uses the RDP subsampling analysis derived from Mironov et al. (2019) and the RDP to  $(\varepsilon, \delta)$ -DP conversion from Balle et al. (2020).
+
+![](images/b7b9a303a52aec28a337d15c6d886d3f2893d7c9cadcb06d3744ae8ff637a4d6.jpg)  
+(a) Sent140 (2-layer LSTM)
+
+![](images/5d3ad6eafd12692343fdda2f58c03a01d15b1deeb2eda27c1c17f91165759fda.jpg)  
+(b) Shakespeare (2-layer LSTM)
+
+![](images/ac513e0e0f983c6d4857f459128591df5e6edf9b688eddd2eab6dd24fcc6ac00.jpg)  
+(c) CelebA (ResNet18)
+
+![](images/d8eda7ce42703fa043ff9ea12095329e06e1b2602f3dfcfacd5da6db806b89ce.jpg)  
+(a) Loss modifications  
+Figure 8: Further Ablations on CelebA; (a) CNN (b), (c); ResNet18
+
+![](images/2a05ff46c3d2fc73acc893d44b82d390d81bf96c002ceccb682f8112b86c558a.jpg)  
+(b) Varying number of clients per round: Attack accuracy
+
+![](images/687842a872660c6fb82adc978eba438951bf8c9433e470e89dd68e8ee75b97d0.jpg)  
+Figure 7: Per-round  $\hat{\varepsilon}_r$  estimates against attack accuracy. Estimates are computed over 5 training runs with a final theoretical  $\varepsilon = 50$ .  
+(c) Varying number of clients per round: Empirical  $\hat{\varepsilon}$
+
+In this work, we have exactly four different privacy measures. For the theoretical quantities we have the per-round guarantee  $\varepsilon_r$  and final privacy guarantee  $\varepsilon$  which are computed as follows:
+
+-  $\varepsilon_r$  - This is the per-round theoretical epsilon that is derived from the RDP accountant when the subsampling rate  $q$  is set to 1 and number of steps  $R = 1$ . This is a constant value (dependent on the noise multiplier  $\sigma$  and  $\delta$ ) and is the privacy guarantee of performing a single step of DP-FEDAVG.  
+-  $\varepsilon$ - This is the theoretical  $(\varepsilon, \delta)$ -DP guarantee of the model trained under (user-level) DP. We calculate this via an RDP accountant with subsampling where the sampling rate  $q$  is chosen to be the number of participating clients over the total number of clients in the population. Since we are using Gaussian noise this corresponds to the RDP analysis of the Subsampled Gaussian Mechanism (SGM), see (Mironov et al., 2019) for technical details.
+
+Similarly, we have analogous empirical measures  $\hat{\varepsilon}_r$  and  $\hat{\varepsilon}$  which are computed as follows:
+
+-  $\hat{\epsilon}_r$  - Since the attack used by CANIFE infers membership of the canary update at a particular round, the privacy measure derived from a set of CANIFE attack scores is a per-round measure. This is computed via the formula derived in Kairouz et al. (2015) i.e.,
+
+$$
+\hat {\varepsilon} _ {r} = \max  _ {\gamma} \left(\log \frac {1 - \delta - \mathrm {F P R} _ {\gamma}}{\mathrm {F N R} _ {\gamma}}, \log \frac {1 - \delta - \mathrm {F N R} _ {\gamma}}{\mathrm {F P R} _ {\gamma}}\right),
+$$
+
+where FPR, FNR are computed from the attack scores at round  $r$ . In our experiments we maximise  $\hat{\varepsilon}_r$  over the threshold  $\gamma$  to provide a worst-case measure. The quantity  $\hat{\varepsilon}_r$  is directly comparable to the theoretical  $\varepsilon_r$ . We also compute 95% confidence intervals (CIs) for  $\hat{\varepsilon}_r$  from the attack scores via the Clopper-Pearson method as in Nasr et al. (2021).
+
+-  $\hat{\varepsilon}$  - This is the empirical privacy measure of the model derived from CANIFE. One could apply basic composition to  $\hat{\varepsilon}_r$  over  $R$  rounds to obtain the empirical measure of  $R\hat{\varepsilon}_r$  but this results in suboptimal composition. Instead we compute  $\hat{\varepsilon}$  under the tighter composition
+
+<table><tr><td>Canary Sample after optimization</td></tr><tr><td>mRnt,,,,,&#x27;d,,,,R,,,,A,,,,,,,AA,,,,V2,E &gt;H &amp;3 4iYet I confess that often ere thisMday,x?h\nnPImh!v7DhNAd}I!H&#x27;;kXXI&#x27;PmP1Iert 6Fa th, nothing bu; aZ empty box, sir, wQich in my lord&#x27;s beealf I cote to e? rtat 3</td></tr></table>
+
+Table 1: Canary Samples on Shakespeare.
+
+of RDP with amplification by subsampling. To do so, we convert each  $\hat{\varepsilon}_r$  into an equivalent noise multiplier  $\hat{\sigma}_r$  and compound the noise over a number of rounds with the accountant. The quantity  $\hat{\varepsilon}$  is directly comparable to  $\varepsilon$ . See Section C.2 for more information.
+
+# C.2 ALGORITHM FOR  $\hat{\varepsilon}$
+
+In Section 3.4, we explained how we obtain a per-round privacy measurement  $\hat{\varepsilon}_r$  from CANIFE and compound this to form a global privacy estimate  $\hat{\varepsilon}$  over a training run. We detail this method in Algorithm 2. In order to compound our per-round estimates  $\hat{\varepsilon}_r$  from CANIFE, since we only attack the model every  $s$  rounds we assume that the noise estimate  $\hat{\sigma}_r$  remains constant between rounds  $r$  and  $r + s$  before we attack the model again and re-estimate the noise  $\hat{\sigma}_r$ .
+
+# C.3 DETAILS FOR MONITORING PRIVACY
+
+In Section 4.2, we present experiments using CANIFE to measure empirical privacy during the training run of federated models. We ran each training run five times and examples of these runs are displayed in Figure 4. Here we provide extra details of the training setup:
+
+CelebA. We train a ResNet18 model for 30 epochs and have 100 clients participate per training round. This results in 85 rounds per epoch. We freeze and attack the model every  $s = 40$  rounds resulting in 64 empirical privacy estimates  $(\hat{\varepsilon}_r)$  across training.  
+- Sent140. We train for 15 epochs and have 100 clients participate per training round. This results in 593 rounds per epoch and 8895 rounds in total. We freeze and attack the model every  $s = 100$  resulting in 90 empirical privacy measurements across training.  
+- Shakespeare. We train for 30 epochs and have 60 clients participate per training round. This results in 17 rounds per epoch and 510 training rounds in total. We freeze and attack the model every  $s = 8$  rounds resulting in 64 empirical privacy estimates across training.
+
+# C.4 RELATIONSHIP BETWEEN  $\hat{\varepsilon}_r$  AND ATTACK ACCURACY
+
+We display the relationship between the per-round measurement  $\hat{\varepsilon}_r$  and the accuracy of the attack for models with a final privacy of  $\varepsilon = 50$  in Figure 7. We also found a consistent relationship for  $\varepsilon \in \{10,30\}$ .
+
+# C.5 FURTHER EXPERIMENTS: SHAKESPECARE
+
+In Figure 4, we displayed example per-round estimates  $\hat{\varepsilon}_r$  across training runs for CelebA and Sent140. In Figure 6a we display a similar plot but for an example training run on Shakespeare. We also note that one can use CANIFE to measure  $\hat{\varepsilon}_r$  for models that are trained without privacy. In Figure 6b, we show an example training run on Shakespeare without DP ( $\varepsilon = \infty$ ). As mentioned in Section 3.4, the number of attack trials determines an upper bound on  $\hat{\varepsilon}_r$ . Here we can see that the empirical privacy measure is essentially constant throughout (non-private) training and consistently reaches these bounds. We note that  $\hat{\varepsilon}_r = \infty$  only once (when the maximum attack accuracy reaches  $100\%$ ).
+
+# D FURTHER ABLATION STUDIES
+
+Loss modification. As discussed in Appendix A, we can minimise the dot-product of the canary with the average model update ("Loss 1") or the covariance loss ("Loss 2") which we choose to
+
+![](images/d2d3005d052ee3e14c1c240db2cd4ad204ac48422c7d796525990acf46854eda.jpg)  
+(a) Attack accuracy
+
+![](images/ed7ec5f6727116f6b831b03d0fdbd60c87d567009dcd33ffe25d6e0f1a469e06.jpg)  
+(b) Canary health  
+Figure 9: Varying the canary gradient norm constant on CelebA (CNN)
+
+use in all experiments. In Figure 8a, we vary the design sample size and carry out 5 attacks with a checkpointed CNN model on CelebA while varying these losses. We observe that there is a clear difference in accuracy between the two losses, with Loss 2 having consistently high attack accuracy which is almost constant as we increase the sample size. We note that Loss 1 seems more sensitive to the total number of design samples, with increasing attack accuracy as the design sample size increases.
+
+Number of clients per round. In Figure 8b, we vary the number of clients per round and plot the average attack accuracy over 5 training runs. We note that the attack accuracy decreases slowly as the number of clients increases which is consistent with the observations made in Figure 3c. In Figure 8c, we plot the empirical estimate  $\hat{\varepsilon}$  during training. We find that although the average accuracy decreases, this does not have a significant effect on the final estimate  $\hat{\varepsilon}$  which decreases as the number of clients per round decreases.
+
+Canary gradient norm. To conclude, we investigate the effect of the canary gradient norm constant in the second term of the loss  $\mathcal{L}(z)$ . We fix the privacy clipping constant to be  $C = 1$  and vary the gradient norm constant on CelebA. We use a checkpointed CNN model trained without DP to  $70\%$  test accuracy. We attack the model 10 times and average the results. In Figure 9a, we display the average accuracy of the attack and in Figure 9b the average canary health as we vary the gradient norm constant. We observe that choosing the constant too large can significantly decrease the efficacy of the attack and that choosing the constant around  $C$  is enough to guarantee high accuracy and well-behaved optimization. We note that the canary health does not significantly decrease until the norm constant is chosen to be very large ( $>600$ ), yet the attack suffers a large drop in accuracy for constants  $>50$ . This implies that as you increase the constant, it is first possible to design a canary that has large gradient norm but at the expense of orthogonality to the other model updates, and as the constant gets very large, it becomes too difficult to optimise either term of  $\mathcal{L}(z)$ .
+
+# E CANARY SAMPLES
+
+We display in Table 1 some examples of designed canaries with CANIFE when initializing the crafting procedure with training sentences on Shakespeare.
+
+# F LIMITATIONS AND EXTENSIONS
+
+In our experiments we have made assumptions for CANIFE that may be limiting in specific practical FL scenarios. In this section, we highlight such limitations with various extensions for CANIFE.
+
+Multiple Local Epochs. Throughout Section 4, we have restricted clients to performing a single local epoch for simplicity. But in practice clients often perform multiple local epochs and this has shown to help improve the convergence of models trained in FL. CANIFE is not limited to methods
+
+with a single local epoch and in practice to audit methods like DP-FEDAVG with multiple local epochs there are two main solutions:
+
+1. Use CANIFE as is: There is nothing in the current formulation of CANIFE that does not apply to multiple local updates. Since in practice, the server will mock the adversary, they are free to choose how many local epochs the canary client performs. Thus, they can design a canary with a single local epoch to be orthogonal to clients who do multiple local epochs. We also believe this is most practical for an adversary since it is the easiest from the attackers viewpoint to design and optimize for.  
+2. Modify CANIFE loss: An alternative approach is to design a canary sample that has a model update (formed from multiple local epochs) that is orthogonal to all other model updates (which are also formed from multiple local epochs). To carry out such a design we can modify the CANIFE loss in equation 1 to include the canary model update  $u_{c}$  instead of the canary gradient  $\nabla_{\theta}\ell (z)$ . In order to calculate the gradient of such a loss it requires backpropagating through the multiple local updates and essentially "unrolling" the local SGD steps. This may be computationally burdensome, and so while it is possible to design such a canary sample, it may not be practically viable for any adversary (and/or server) depending on the model size and number of local epochs. Thus in practice, it may be simpler to audit the model via (1).
+
+Multiple Privacy Measurements. One current limitation of CANIFE is that it only produces a single privacy measurement  $(\hat{\varepsilon}_r)$  at a specific round. In practice, model auditors may want a more comprehensive empirical analysis of the model's privacy via multiple measurements, such as attacks that vary the threat model like that of Nasr et al. (2021). We believe CANIFE can be extended to support a "multiple measurement" approach with ease since there is some degree of freedom in the canary design. For example, by using different design pools, each one strengthening the adversary further (e.g. with more data and/or design pools that better approximate the federated distribution based on prior knowledge) and thus obtain a more holistic measure of the model's privacy. One can similarly vary the number of design epochs to simulate clients with limited computation. Designing multiple canaries per round under different constraints will generate a set of empirical epsilons that allow for more fine-grained statistics (for example, taking the maximum of the per-round empirical epsilons for a worst-case measure).
+
+Preventing Wasted FL Rounds. Another limitation of CANIFE is that in our experimental setup we freeze the model for a set number of rounds to compute attack scores of a designed canary. In practice, the server and clients would be unwilling to waste federated rounds to compute CANIFE scores. We believe CANIFE can be extended to support a more practical attack without wasting training rounds as follows:
+
+- Multiple Canaries: One alternative is that the server could design multiple canaries at a single round, and use these to obtain (multiple) attack scores (subject to compute limitations). This can help reduce the number of mock rounds being run.  
+- Running measurements: The previous approach may be computationally prohibitive depending on server resources and still requires frozen rounds. An alternative could be to maintain "running" attack measurements where we allow CANIFE to run alongside normal model training, letting the model change at each round. In this setup, the server can design a canary at each round, calculate attack scores and then proceed with updating the global model (without the canary inserted). This has minimal overhead to the server (who just needs to design the canary) and no additional overhead to clients (who just believe they are participating in a standard FL training round). The set of attack scores can be used to calculate empirical epsilons over various periods of training and one can change this period however they like. We emphasize that since our results (specifically Figure 4) show the per-round empirical measure is fairly stable after the first epoch or so of training, then the approach described here should still give stable results (even though the model would be changing at each round).
+
+Design Pool Assumptions. The design pool is an important component of our attack, and in order to present a conservative (worst-case) privacy measure in this work, we assume that the adversary
+
+has access to held-out data that approximates the federated training data well. To do this, we form the design pool from the test set of our datasets but in many practical scenarios this is not possible since the server may not know much about the (private) federated data. While this is a limitation of CANIFE, we believe that for many tasks it would be possible to form a design pool from public data. For example, if the adversary knows the exact task of the model (e.g. sentiment analysis) then the adversary can form a design pool from public datasets (e.g. Sent140) or even craft their own language data for canary design. We expect that this would be a reasonable proxy for the true federated dataset and would not significantly affect privacy measurement. In scenarios where the server has no prior knowledge, they could utilise private synthetic data generation methods or Federated Analytics (FA) to privately compute statistics about client data to guide the choice of design pool.

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+# CASR: GENERATING COMPLEX SEQUENCES WITH AUTOREGRESSIVE SELF-BOOST REFINEMENT
+
+Hongwei Han $^{1*}$  Mengyu Zhou $^{2\dagger}$  Shi Han $^{2}$  Xiu Li $^{1\dagger}$  Dongmei Zhang $^{2}$
+
+$^{1}$ Tsinghua Shenzhen International Graduate School, Tsinghua University
+
+2Microsoft Research
+
+hhw20@mails.tsinghua.edu.cn, li.xiu@sz.tsinghua.edu.cn
+
+{mezho, shihan, dongmeiz}@microsoft.com
+
+# ABSTRACT
+
+There are sequence generation tasks where the best order to generate the target sequence is not left-to-right. For example, an answer to the Sudoku game, a structured code like s-expression, and even a logical natural language answer where the analysis may be generated after the decision. We define the target sequences of those tasks as complex sequences. Obviously, a complex sequence should be constructed with multiple logical steps, and has dependencies among each part of itself (e.g. decisions depend on analyses). It's a great challenge for the classic left-to-right autoregressive generation system to generate complex sequences. Current approaches improve one-pass left-to-right generation on NLG tasks by generating different heuristic intermediate sequences in multiple stages. However, for complex sequences, the heuristic rules to break down them may hurt performance, and increase additional exposure bias. To tackle these challenges, we propose a PLM-friendly autoregressive self-boost refinement framework, CASR. When training, CASR inputs the predictions generated by the model itself at the previous refinement step (instead of those produced by heuristic rules). To find an optimal design, we also discuss model architecture, parameter efficiency and initialization strategy. By evaluating CASR on Sudoku, WebQSP, MTOP and KVRET through controlled experiments and empirical studies, we find that CASR produces high-quality outputs. CASR also improves Accuracy on Sudoku  $(70.93\% \rightarrow 97.28\%)$  and achieves state-of-the-art performance on KVRET with Micro F1 score  $(67.88\% \rightarrow 70.00\%)$ .
+
+# 1 INTRODUCTION
+
+![](images/1c07e647e8bf8c40967802e3c9c49e663c1dd951f56807a9a5b0bbad90dcdf99.jpg)  
+Figure 1: The Overview of CASR Framework.  $X$ ,  $Y$  and  $\hat{Y}$  denote the input, ground truth and prediction, respectively. The blue arrows show how we iteratively added back the previous-step prediction  $\hat{Y}^{t-1}$  to the input for generating refined output  $\hat{Y}^t$ .
+
+Sequence generation models are widely used in tasks related to natural, domain-specific and programming languages - E.g., question answering (Pandya & Bhatt, 2021), neural machine translation (Yang et al., 2020), speech recognition (Malik et al., 2021), automatic data analysis (Zhou et al., 2020), drug discovery (Kim et al., 2021), document summarization (Ma et al., 2020), code search and generation (Lee et al., 2021), etc.
+
+To achieve better performance on these tasks, recent works often adopt autoregressive (AR) models (Wu et al., 2016), especially the ones with one-pass L2R (left-to-right) token-by-token generation / decoding order. Many SOTA-performance generative PLMs (pre-trained language models) are one-pass L2R models, such as GPT (Radford et al., 2018), T5 (Raffel et al., 2020), Bart (Lewis et al., 2020), etc. Different from non-autoregressive (NAR) models (Gu et al., 2017) which assume independence among tokens, L2R models assume the conditional probability in the form of  $P(Y|X) = \prod_{i}P(y_{i}|X,y_{< i})$  which better captures the left-side dependencies that exist in most generation tasks. More variations of generation models are discussed in §2.1.
+
+However, for many sequence generation tasks, beyond the left-side dependencies, there are right-side dependencies in the answer sequence, which together lead to multi-hop dependency chains, making left-to-right not the best order for generation. We call these tasks as complex tasks, and the answer sequences of these tasks as complex sequences. Complex tasks (see more details in §2.2, including Sudoku (PARK), WebQSP (Yih et al., 2016), MTOP (Li et al., 2021), and KVRET (Eric et al., 2017), etc.) require better generation mechanism beyond one-pass L2R generation, since complex sequences are usually long, difficult, structured, or logical, which should be constructed with multiple logical steps.
+
+![](images/9d661c072cb4975d18441a64f13298496cf70cc3311d9c9972f0900ed217d4dd.jpg)  
+(a) A 9x9 Sodomoku Example. White cells denote blanks, and the green numbers in them denote the ground truth.  
+Figure 2: Examples of Selenium.
+
+![](images/79c429a70840e6de1629a8d5334cf11b59708e064b2e8408e6cee88300697d89.jpg)  
+(b) A 4x4 Sodomoku Example. The game solving order of a human is from "a" to "g", rather than row-by-row.
+
+Human beings solve a complex problem with respect to its intrinsic order. For example, the order to write hierarchical answers (such as s-
+
+Expression or SQL code) is usually bottom-up or top-down following the dependencies between components as discussed by Sun et al. (2020). The order to give an NL response is first analyses then decisions as discussed by CoT (Wei et al., 2022). The order to solve a puzzle (such as the example 4x4 Sudo game in Figure 2) is usually from easy parts to hard parts, because the hard parts become easier when the easy parts are correctly solved. (That is also verified in §5.1 where our CASR model learns to solve easy parts before hard ones.) Obviously, people give answers to different tasks in various orders with respect to all kinds of dependencies.
+
+Mimicking human behavior, some existing works design specific intermediate sequences to solve the dependency order challenge. E.g., templates (Hua & Wang, 2020) or heuristic rules (Zhang et al., 2018; Tan et al., 2021) are applied in autoregressive NL generation, allowing models to generate some parts (intermediate sequences) before the other parts in an answer via iterative refinement (rather than one-pass decoding). However, 1) it's really hard to design the best heuristic order and easy to miss intrinsic dependencies for some tasks, and we need expert knowledge or manual efforts to design specific heuristic orders for all different tasks. Besides, 2) when using teacher forcing strategy to parallel train all refinement iterations, additional exposure bias occurs.
+
+In this paper, CASR (Generating Complex Sequences with Auto-regressive Self-Boost Refinement) framework is proposed by us to: 1) decide intermediate sequences of complex answers for different tasks in a data-driven way, 2) avoid additional exposure bias. As shown in Figure 1 and will be discussed in §3, in CASR we design a model architecture (§3.2) to not only take in the original input  $X$ , but also the previous prediction  $\hat{Y}^{t - 1}$  in both training and inference. A special process (§3.1) is designed to train refinement models  $M^t$  for each step  $t = 0, 1, \dots, T - 1$ . To enhance the performance on downstream tasks, CASR models could be initialized with pre-trained language models (§3.3) such as T5 (Raffel et al., 2020), and even trained in a "Continue" way (§3.4) by initializing  $M^t$
+
+with  $M^{t-1}$ . For CASR models to be more efficient, we also explore the parameter-efficient model designs (SepEnc vs. UniEnc in §3.2), and tuning with parallel adapters (§3.3).
+
+We evaluate CASR (and several baselines) on WebQSP, MTOP, KVRET, and Suku in §4, achieving SOTA performance on KVRET. In detail, CASR improves F1 on WebQSP from 70.81 to 74.81, EM on MTOP from 78.64 to 81.92, Micro F1 on KVRET from 67.40 to  $70.00^{1}$ , and accuracy on Suku from 70.93 to 97.28. We find the optimal CASR design (§4.2) is "Fine-tuning" "SepEnc" with "Continue" strategy. We also do empirical studies on complexity (§5.1), attention map (§5.2), and visualize cases (§5.3). Then we find that, CASR benefits hard sequences more than easy ones, and CASR can indeed correct the wrong part of the previous prediction according to other parts it depends on (as shown in Figure 8).
+
+In summary, our major contributions are: First, we point out the challenge to generate complex sequences due to the existence of multi-hop dependency chains and conduct a comprehensive review on existing iterative refinement methods. Second, we propose an autoregressive self-boost refinement framework, CASR, to decide intermediate sequences of complex answers in a data-driven way. The code of CASR framework is open sourced in the repository at https://github.com/RalphHan/CASR. Third, we conduct experiments and empirical studies on four complex tasks to show CASR works and interpret how it works.
+
+# 2 RELATED WORK
+
+Table 1: The Objectives of Sequence Generation Methods.  
+
+<table><tr><td>Method</td><td>Objective</td></tr><tr><td>NAT (Gu et al., 2017)</td><td>ΠiP(yi|X)</td></tr><tr><td>INAT (Lee et al., 2018)</td><td>ΠiP(yi|t|X,ˆYt-1)</td></tr><tr><td>Levenshtein (Gu et al., 2019)</td><td>Imitate an expert policy to delete and insert</td></tr><tr><td>L2R (Wu et al., 2016)</td><td>ΠiP(yi|X,yi-1,...,y1)</td></tr><tr><td>XLNet (Yang et al., 2019)</td><td>ΠiP(yzi|X,yzi-1,...,yz1)</td></tr><tr><td>Bidirectional (Zhang et al., 2018)</td><td>ΠiP(yi|X,yi+1,...,yn) · ΠiP(yi|X,yi-1,...,y1,C^R2L)</td></tr><tr><td>Progressive (Tan et al., 2021)</td><td>ΠiP(yi|t|X,Yt-1,yi-1,...,yt)</td></tr><tr><td>Ours</td><td>ΠiP(yi|t|X,ˆYt-1,yi-1,...,y1t)</td></tr></table>
+
+Table 2: The Number of Samples in Train, Dev, and Test Splits of WebQSP, MTOP, KVRET, and Selenium.  
+
+<table><tr><td>Task</td><td>Train</td><td>Dev</td><td>Test</td></tr><tr><td>WebQSP</td><td>2673</td><td>309</td><td>1639</td></tr><tr><td>MTOP</td><td>15667</td><td>2235</td><td>4386</td></tr><tr><td>KVRET</td><td>6291</td><td>777</td><td>808</td></tr><tr><td>Sudoku</td><td>800K</td><td>100K</td><td>100K</td></tr></table>
+
+# 2.1 SEQUENCE GENERATION METHODS
+
+In recent years, many studies have been made on seq2seq generation. For input  $X$  and target  $Y$  sequences, the ultimate objective is to maximize  $P(Y|X)$ . As shown in Table 1, different methods formulates the objective differently. These methods can be divided into non-autoregressive (row 1-3) and autoregressive (row 4-7) ones. Highly related ones are run as baselines in Table 5.
+
+NAT (Gu et al., 2017) assumes that each token in sequence  $Y$  is mutually independent from each other, thus formulating  $P(Y|X)$  as  $\prod_{i} P(y_{i}|X)$ . To reduce the modeling bias in NAT, iterative refinement is applied on answers in INAT (Lee et al., 2018). Levenshtein Transformer (Gu et al., 2019) further breaks down each refinement step into deletion, insertion, and classification so as to allow the model to edit the generation in a non-autoregressive way. Despite their efficiency, non-autoregressive models perform relatively poor comparing to autoregressive ones (see §4.4).
+
+L2R (Wu et al., 2016) generation tries to decode from left to right, which is also adopted by GPT (Radford et al., 2018), basic transformer decoder (Vaswani et al., 2017), VAE decoder (Yu et al., 2020), and RNN decoder (Xu et al., 2015; Xia et al., 2017). XLNet (Yang et al., 2019) allows the model to generate with any given order  $Z$ , but cannot decide by itself the best order. Bidirectional decoder (Zhang et al., 2018) generates backward (R2L) then forward (L2R), which handles at most 2-step logic. It trains an additional R2L decoder from scratch, without leveraging generation capabilities of PLMs as in CASR (see §3.3).
+
+PAIR (Hua & Wang, 2020) is used in controlled text generation with a template (constructed from a set of provided key phrases and placeholders) as input. It fills and refines the placeholders in an auto-regressive way. The template, the sequence length and position of placeholders remain
+
+unchanged during refinement. Similarly, chain of thought prompting (Wei et al., 2022) constructs the intermediate process through manually provided templates. These two methods are not chosen as baselines in §4 because they both require expert designs for each specific task.
+
+Progressive generation (Tan et al., 2021) is proposed to remove the requirement of templates and the limitation of fixed length during refinement. It breaks down the vocabulary into multiple stages according to word importance (average of tfidf). Important words are generated first during early refinement steps, while other words (both important and unimportant) are generated later. Note that in Table 1,  $Y^{t-1}$  (row 7) denotes the intermediate sequence produced by the heuristics of progressive generation, but  $\hat{Y}^{t-1}$  (row 8) is the previous prediction generated by our CASR model (see §3).
+
+# 2.2 TASKS WITH COMPLEX ANSWERS
+
+The four complex-answer tasks referred in §1 will be further introduced in this section.
+
+WebQSP (Yih et al., 2016) is a classic dataset for KBQA(Knowledge Base Question Answering). The input consists of a knowledge graph and an NL query, and the output is an s-Expression which can be executed on the knowledge graph. The SOTA method of WebQSP  $(\mathrm{F}1 = 83.6\%)$  is ranking with bootstrapping negative samples (Ye et al., 2022).
+
+MTOP (Li et al., 2021) is a benchmark for comprehensive multilingual task-oriented semantic parsing. The input consists of a list of API calls and an NL query, and the output is a tree-based TOP Representation that can be executed.
+
+KVRET (Eric et al., 2017) is a benchmark for table conversation. The input consists of a table and an NL query, and the output is an NL response corresponding to the dialog. The SOTA method of MTOP  $(\mathrm{EM} = 86.78\%)$  and KVRET (Micro F1=67.88%) is multi-task (20+ tasks) prefix tuning with T5-3B as the backbone (Xie et al., 2022).
+
+Sudoku (PARK) is an open dataset on Kaggle. Its game target is to fill the blanks correctly with the constraint that any two numbers in the same row, column, and house shouldn't have the same value. We choose Sudoku as an intuitive toy task for better demonstration of ideas in this paper.
+
+Examples of WebQSP (Figure 5), MTOP (Figure 6) and KVRET (Figure 7) are shown in Appendix  $\S A$ . The number of samples in each split of the four tasks is shown in Table 2.
+
+# 3 CASR FRAMEWORK
+
+As shown in Figure 1 and introduced in §1 and §2.1, the key idea of CASR framework is to take the previous-step prediction  $\hat{Y}^{t-1}$  as part of the current-step  $(t)$  input for generating a refined output  $\hat{Y}^t$ . The iterative inference process in CASR follows the idea: As shown in Algorithm 1, the prediction at castep  $t$  is  $\hat{Y}^t (0 \leq t < T)$  generated by the corresponding CASR model  $M^t$ , where the input to  $M^0$  is the original input  $X$ , and the input to  $M^t$ $(t > 0)$  is  $(X, \hat{Y}^{t-1})$ . Following the notions in §2.1,  $X, Y$ , and  $\hat{Y}$  denote the input, the ground truth and the prediction, respectively. The iterative refinement process in CASR will take at most  $T$  casteps (CASR steps).
+
+The undetermined  $M^t$  models in the CASR inference process lead to more questions: 1) How to design the training process and objectives that match the inference process? 2) How to design CASR model architecture that leverages existing PLMs and takes the extra  $\hat{Y}^{t-1}$  as input? 3) When leveraging a large PLM, can we train and save CASR models  $M^t$  with less parameters? 4) Can the training and inference process be more effective by exploiting the relationships among CASR models  $M^t$  of each castep? In the following, we will discuss these problems one by one.
+
+# 3.1 TRAINING PROCESS
+
+As formulated in Algorithm 2, the training process is also iterative.
+
+At the 0-th castep, we train the model  $M^0$  normally on train set with the objective to maximize  $P(Y|X)$  (as discussed in §2.1). When the training is done, we use  $M^0$  to generate prediction  $\hat{Y}^0$  for each  $X$  in train and dev sets with beam searching (Ney et al., 1987).
+
+At the  $t$ -th castep  $(0 < t < T)$ , we train  $M^t$  to refine the last prediction  $\hat{Y}^{t-1}$ . The objective becomes maximizing  $P(Y|X, \hat{Y}^{t-1})$  on train set. After training, we generate  $\hat{Y}^t$  on all train and dev sets to compose the samples  $(X, Y, \hat{Y}^t)$  for the next castep  $t+1$ .
+
+After the training process,  $T$  versions of CASR model  $M^t (0 \leq t < T)$  are available for the inference process. Because the distribution of  $\hat{Y}^{t-1}$  could change with  $t$ , and different casteps may learn different refinement patterns, by default all the  $T$  versions are saved.
+
+In the following, we will discuss more details of the algorithms on  $M^t$  model architecture (§3.2), tuning (§3.3), and initialization (§3.4). More discussions on result selections are in Appendix §B.
+
+# 3.2 MODEL ARCHITECTURE
+
+![](images/3ecf40939c11cc94b5390a4cd1c49bc905faf421c8e66b68ecd68d87fb34d8b7.jpg)  
+AE and AD denote adapters of encoder and decoder. Blue and gray segments are the parameters to be trained and frozen, respectively.
+
+![](images/679e6e3fa484f6a15b2a73a7284960f40af82618a0a49b565e706c0dcfdad487.jpg)  
+Figure 4: Initialization Strategies.  
+Figure 3: Model Architectures and Parameter Efficiency Choices.
+
+The next problem after the training process is how to design a model architecture which could take  $(X, \hat{Y}^{t-1})$  as input and generate  $\hat{Y}^t$ . Starting from any Transformer encoder-decoder (Vaswani et al., 2017) based model (such as T5, or an untrained Transformer), in CASR we provide two modification approaches to take in  $\hat{Y}^{t-1}$ .
+
+As shown in Figure 3, SepEnc has two encoders to encode  $X$  and  $\hat{Y}^{t-1}$  separately (encode before concat), and UniEnc adopts only one encoder for both inputs (concat before encode). The output sequence length of SepEnc and UniEnc are both  $\mathrm{len}(X) + \mathrm{len}(\hat{Y}^{t-1})$ . Let  $H$  denote the encoder output. For SepEnc,  $H = \text{Concat}(\text{Encoder}_1(X), \text{Encoder}_2(\hat{Y}^{t-1}))$ , and for UniEnc,  $H = \text{Encoder}(\text{Concat}(X, \hat{Y}^{t-1}))$ .
+
+Both approaches have their own advantages: SepEnc naturally handles the distribution differences between  $X$  and  $\hat{Y}^{t-1}$ , while UniEnc requires fewer model parameters. UniEnc forces one encoder to handle two kinds of sequences with potentially different types of contents and lengths. As a tradeoff, SepEnc brings two times larger encoding part. And the decoder generates answers from  $H$ , just like the classic transformer decoder (Vaswani et al., 2017).
+
+# 3.3 PARAMETER EFFICIENCY
+
+Both SepEnc and UniEnc approaches in §3.2 could take a PLM (pre-trained language model, such as T5) as starting point for model parameter initialization (more details in §3.4). In this way, CASR could leverage the existing knowledge from the PLM to enhance performances on downstream tasks (e.g., the ones in §2.2).
+
+However, it's costly to fine-tune all the parameters from a large PLM and save all the parameters of all the  $T$  CASR models  $M^t$ . Thus, besides the fine-tuning approach, in CASR we also try parameter-efficient tuning. As shown in Figure 3, parallel adapters (He et al., 2021) could be added to every encoder and decoder. By adopting adapters, the parameters from the PLM are frozen, and only the parameters of the adapters are trained. In other words, one only needs to save the adapter (rather than the whole model) parameters for  $T$  times.
+
+# 3.4 INITIALIZATION STRATEGY
+
+In Algorithm 2, at the beginning of each castep, we initialize the parameter weights of  $M^t$  with a chosen strategy from {Restart, Continue}. Illustrations of each strategy are shown in Figure 4.
+
+"Restart" means initializing each  $M^t$  with the same PLM weights (or from scratch). For SepEnc architecture, Both  $Encoder_1$  and  $Encoder_2$  can be initialized with the parameter weights of the PLM encoder. "Continue" means initializing with the best checkpoint from the previous castep. In this way,  $M^t$  inherits the knowledge from  $M^{t-1}$  to avoid the potential cold start issue at castep  $t$ .
+
+In the "Restart" strategy, when comparing the late castep with the early castep, the improvement is only brought by refinement, rather than the extra continuous training steps (which exist in the "Continue" strategy). Note that, we design "Restart" mainly to decouple the effects of refinement from training steps, which is necessary for controlled experiments in §4.
+
+# 4 EXPERIMENTS
+
+In this section, we run controlled experiments to evaluate CASR designs (§3) on WebQSP (Yih et al., 2016), MTOP (Li et al., 2021), KVRET (Eric et al., 2017) and Sudo (PARK) (§2.2), and compare CASR with high-related baselines (§2.1).
+
+# 4.1 EXPERIMENT SETUP
+
+In this section, we explain how we design the controlled experiments and reproduce the high-related baselines. For more details about hyperparameter and resource consumption (number of parameters, training/inference time, etc.) please see Appendix §C.
+
+# 4.1.1 CONTROLLED EXPERIMENTS
+
+In Table 3, we apply CASR (with max castep  $T = 3$ ) on the 24-layer generative PLM T5-base (Raffel et al., 2020). Controlled experiments  $\mathrm{A} \sim \mathrm{E}$  are run to compare the variations of Parameter Efficiency ( $\S 3.3$ ), Initialization Strategy ( $\S 3.4$ ) and Model Architecture ( $\S 3.2$ ). The evaluation metrics (on test sets) are F1 score for WebQSP, template accuracy and exact match for MTOP, and BLEU and Micro F1 scores for KVRET. For each row, the
+
+Table 3: Controlled Experiments of CASR Framework on WebQSP, MTOP, and KVRET.  
+
+<table><tr><td rowspan="2">t</td><td colspan="3">CASR Variations</td><td>WebQSP</td><td colspan="2">MTOP</td><td colspan="2">KVRET</td><td rowspan="2">AVG</td></tr><tr><td>Param.E.</td><td>Init.</td><td>Arch.</td><td>F1</td><td>Acc.</td><td>EM</td><td>BLEU</td><td>Micro F1</td></tr><tr><td>A0</td><td>Fine-tune</td><td>Restart</td><td>SepEnc</td><td rowspan="4">70.81</td><td rowspan="4">82.49</td><td rowspan="4">78.64</td><td rowspan="4">18.33</td><td rowspan="4">67.40</td><td rowspan="4">63.53</td></tr><tr><td>B0</td><td>Fine-tune</td><td>Continue</td><td>SepEnc</td></tr><tr><td>C0</td><td>0</td><td>Restart</td><td>UniEnc</td></tr><tr><td>D0</td><td>Fine-tune</td><td>Continue</td><td>UniEnc</td></tr><tr><td>E0</td><td>Adapter</td><td>Restart</td><td>SepEnc</td><td>70.20</td><td>77.34</td><td>73.10</td><td>16.38</td><td>65.59</td><td>60.52</td></tr><tr><td>A1</td><td>Fine-tune</td><td>Restart</td><td>SepEnc</td><td>73.03</td><td>82.92</td><td>78.93</td><td>18.17</td><td>67.65</td><td>64.14</td></tr><tr><td>B1</td><td>Fine-tune</td><td>Continue</td><td>SepEnc</td><td>74.61</td><td>85.07</td><td>81.19</td><td>18.55</td><td>70.00*</td><td>65.88</td></tr><tr><td>C1</td><td>1</td><td>Restart</td><td>UniEnc</td><td>73.14</td><td>82.83</td><td>78.80</td><td>18.80</td><td>67.88</td><td>64.29</td></tr><tr><td>D1</td><td>Fine-tune</td><td>Continue</td><td>UniEnc</td><td>74.16</td><td>84.86</td><td>81.24</td><td>18.06</td><td>69.12</td><td>65.49</td></tr><tr><td>E1</td><td>Adapter</td><td>Restart</td><td>SepEnc</td><td>67.92</td><td>77.43</td><td>73.21</td><td>16.60</td><td>65.86</td><td>60.20</td></tr><tr><td>A2</td><td>Fine-tune</td><td>Restart</td><td>SepEnc</td><td>73.09</td><td>82.92</td><td>78.93</td><td>18.26</td><td>67.70</td><td>64.18</td></tr><tr><td>B2</td><td>Fine-tune</td><td>Continue</td><td>SepEnc</td><td>74.81</td><td>85.61</td><td>81.69</td><td>19.12</td><td>69.85</td><td>66.22</td></tr><tr><td>C2</td><td>2</td><td>Restart</td><td>UniEnc</td><td>73.14</td><td>82.83</td><td>78.77</td><td>18.78</td><td>68.06</td><td>64.32</td></tr><tr><td>D2</td><td>Fine-tune</td><td>Continue</td><td>UniEnc</td><td>74.32</td><td>85.66</td><td>81.92</td><td>18.34</td><td>68.80</td><td>65.81</td></tr><tr><td>E2</td><td>Adapter</td><td>Restart</td><td>SepEnc</td><td>68.13</td><td>77.45</td><td>73.23</td><td>16.63</td><td>65.66</td><td>60.22</td></tr></table>
+
+average (AVG) of all metrics (on test set) is calculated to provide an overall merged score.
+
+The three row blocks  $\overline{\mathrm{A0}}\sim \overline{\mathrm{E0}}$ $\overline{\mathrm{A1}}\sim \overline{\mathrm{E1}}$  and  $\overline{\mathrm{A2}}\sim \overline{\mathrm{E2}}$  correspond to  $M^0$ $M^1$  and  $M^2$  models at Castep  $t = 0,1,2$  , respectively. By definitions in  $\S 3$  , row  $\overline{\mathrm{A0}}\sim \overline{\mathrm{D0}}$  share the same result because initialization and architecture choices do not influence  $M^0$  models.
+
+Since CASR converges on Sudoku with larger  $T$ , we evaluate it separately. In Table 4, We train 12-layer encoder-decoder transformers from scratch with max castep  $T = 5$ . The metric is accuracy (\% correctly filled blanks). For a  $9 \times 9$  board, the input and output sequences of a Sudoku game are both 81-number serialized sequences. During training, we apply supervision only on blanks and apply constrained generation to
+
+Table 4: Selenium Testing Results.  
+
+<table><tr><td>t</td><td>Restart</td><td>Continue</td><td>Continue w/o Yt-1</td></tr><tr><td>0</td><td colspan="3">70.93</td></tr><tr><td>1</td><td>77.00</td><td>85.71</td><td>84.28</td></tr><tr><td>2</td><td>78.69</td><td>92.48</td><td>89.16</td></tr><tr><td>3</td><td>79.11</td><td>95.66</td><td>91.13</td></tr><tr><td>4</td><td>79.21</td><td>97.28</td><td>92.09</td></tr></table>
+
+make sure the model only predicts on blanks. And we fill the predictions back to the blanks of the input as the combination of  $X$  and  $\hat{Y}$ , at the beginning of each castep  $t > 0$ . In other words, we use a special UniEnc to encode the combination, with  $H = \text{Encoder}(C o m b i n e(X, \hat{Y}^{t-1}))$ . "Restart" and "Continue" correspond to the same choices as  $\mathbb{C}$  and  $\mathbb{D}$  in Table 3. "Continue w/o  $\hat{Y}^{t-1}$ " denotes that the previous predictions are not put back to the blanks of the input at each castep, which is equivalent to training the origin model  $T$  times steps as usual.
+
+# 4.1.2 BASELINE COMPARISONS
+
+In Table 5, we run some high-related baselines as introduced in §2.1. We run INAT and Levenshtein with the official code provided by faiseq², where self-attention layers of the encoder and the decoder are initialized with bert, leaving cross-attention layers of the decoder trained from scratch. To decouple the effect from PLMs and make a fairer comparison
+
+between NAR and AR methods, we add row  $\boxed{\mathrm{Xc}}$ , where the decoder of CASR is not initialized with T5 but from scratch (random weights), marked as “/Dec”. In row  $\boxed{\mathrm{L2}}$ , “CASR-L” denotes changing the backbone of row  $\boxed{\mathrm{B2}}$  from T5-base to T5-large. For fair comparisons, the original backbones in Bidirectional Decoder (GRU) and Progressive Generation (Bart) implementations are changed to T5-base (same as CASR), and set  $T = 3$  (same as CASR) for Progressive Generation.
+
+Table 5: Comparisons between CASR and Baselines.  
+"NAR" and "AR" denote non-autoregressive and autoregressive refinement.  
+“-.01-” denotes the metrics are lower than 0.01.  
+
+<table><tr><td rowspan="2" colspan="2"></td><td rowspan="2">Methods</td><td>WebQSP</td><td colspan="2">MTOP</td><td colspan="2">KVRET</td><td rowspan="2">AVG</td></tr><tr><td>F1</td><td>Acc.</td><td>EM</td><td>BLEU</td><td>Micro F1</td></tr><tr><td>Xa</td><td rowspan="2">NAR</td><td>INAT (Lee et al., 2018)</td><td colspan="5">—.01—</td><td>0.00</td></tr><tr><td>Xb</td><td>Levenshtein (Gu et al., 2019)</td><td>22.54</td><td colspan="4">—.01—</td><td>4.51</td></tr><tr><td>Xc</td><td rowspan="4">AR</td><td>CASR B / Dec</td><td>4.08</td><td>63.02</td><td>12.93</td><td>7.18</td><td>10.17</td><td>19.48</td></tr><tr><td>Xd</td><td>Bidirectional (Zhang et al., 2018)</td><td>68.43</td><td>61.33</td><td>57.84</td><td>8.98</td><td>54.77</td><td>50.27</td></tr><tr><td>Xe</td><td>Progressive (Tan et al., 2021)</td><td>72.05</td><td>80.80</td><td>77.04</td><td>15.49</td><td>64.54</td><td>61.98</td></tr><tr><td>B0</td><td>Finetune</td><td>70.81</td><td>82.49</td><td>78.64</td><td>18.33</td><td>67.40</td><td>63.53</td></tr><tr><td rowspan="2">B2</td><td rowspan="2">L2</td><td>CASR</td><td>74.81</td><td>85.61</td><td>81.69</td><td>19.12</td><td>69.85</td><td>66.22</td></tr><tr><td>CASR-L</td><td>77.99</td><td>87.73</td><td>84.54</td><td>18.14</td><td>68.80</td><td>67.44</td></tr></table>
+
+# 4.2 BEST CASR DESIGNS
+
+We achieve the SOTA performance on Micro F1 of KVRET, which is 70.00. The previous SOTA is 67.88 (Xie et al., 2022) based on T5-3B, and we beat it with a smaller T5-base.
+
+For the parameter initialization choices, "Continue" strategy brings better performance than "Restart". This is observed by comparing  $\mathbb{A}$  with  $\mathbb{B}$  rows,  $\mathbb{C}$  with  $\mathbb{D}$  rows in Table 3, and comparing "Restart" and "Continue" columns in Table 4.
+
+For the model architectures, by comparing A with C rows, B with D rows, we find that there is no great performance gap between SepEnc and UniEnc, and SepEnc is slightly better than UniEnc when combined with "Continue".
+
+For the parameter efficiency, by comparing  $\mathbf{E}$  with  $\mathbf{A}$  rows, we can find that adapter-tuning performs worse than fine-tuning. As the cost of freezing PLM parameters for efficiency, performance drops as expected when bringing adapter-tuning.
+
+# 4.3 ANALYSIS ON SELF-BOOST REFINEMENT
+
+As introduced in §3.4, we design "Restart" to decouple the effects of refinement from fitting the training set longer than  $t = 0$ . The variations with "Restart" strategy (A, C) and (E) in Table 3, "Restart" column in Table 4) demonstrates how self-boost refinement could improve the quality of generated sequences. Here the improvement is only brought by refinement since CASR models at all casteps are initialized with the same PLM/Scratch version and take the same training steps.
+
+In Table 4, comparing "Continue" with "Continue w/o  $\hat{Y}^{t-1}$ ", we find that self-boostly feeding  $\hat{Y}^{t-1}$  to the model is indeed helpful, which improves the accuracy from  $92.09\%$  to  $97.28\%$ .
+
+Interestingly, the performance gap between castep 0 and 1 is much larger than the gap between  $t$  to  $t + 1$  ( $t > 0$ ). One possible explanation is that later refinements have reached the upper bound.
+
+As we can see in Table 4, Selenium requires more refinement steps (5 in our case) for the gap to converge. This shows there are more logical steps required and more dependencies exist in the generated Selenium answers.
+
+# 4.4 COMPARISON WITH BASELINES
+
+In the NAR block of Table 5, we find the results of NAR methods are not on par with AR methods. With case study (Figure 10), we find the ability of NAR methods to learn syntax is poor. Comparing row  $\boxed{\mathrm{Xb}}$  and  $\boxed{\mathrm{Xc}}$ , we find the poorness of NAR methods doesn't come from the initialization of the decoder. Comparing row  $\boxed{\mathrm{Xd}}$ ,  $\boxed{\mathrm{Xe}}$ , and  $\boxed{\mathrm{B0}}$ , we find that those AR refinement methods using heuristic intermediate sequences are indeed harmful (worse than directly fine-tuning) for complex sequence generation. Compare row  $\boxed{\mathrm{B2}}$  and  $\boxed{\mathrm{L2}}$ , we find larger backbone (T5-base  $\rightarrow$  T5-large) is overall helpful (UnifiedSKG (Xie et al., 2022) observes the same phenomenon that T5-large is worse than T5-base on KVRET).
+
+In summary, following conclusions can be drawn for CASR:
+
+- Self-boost refinement leads to better outputs than vanilla AR (one-pass L2R) generation, NAR refinement generation, and heuristic-rule-based AR refinement generation.  
+- In general, the best combination is "Fine-tuning" "SepEnc" with "Continue" strategy.
+
+# 5 EMPIRICAL STUDIES
+
+Table 6: Result on Different Answer Length.  
+Produced by row  $\widehat{\mathbf{B}}$  in Table 3 (Finetune+Continue+SepEnc).  $\Delta_{t}$  denotes the improvement from castep  $t - 1$  to  $t$ .  
+
+<table><tr><td rowspan="2">t</td><td>Task</td><td colspan="2">WebQSP</td><td colspan="2">MTOP</td><td colspan="2">KVRET</td><td>AVG</td><td>Δt</td></tr><tr><td>Metric</td><td>F1</td><td>Acc</td><td>Match</td><td>Bleu</td><td>Micro F1</td><td></td><td></td><td></td></tr><tr><td rowspan="3">0</td><td>Short</td><td>78.80</td><td>85.23</td><td>83.65</td><td>14.05</td><td>65.91</td><td>65.53</td><td>-</td><td></td></tr><tr><td>Middle</td><td>83.07</td><td>84.83</td><td>80.51</td><td>21.93</td><td>71.16</td><td>68.30</td><td>-</td><td></td></tr><tr><td>Long</td><td>50.51</td><td>77.31</td><td>71.62</td><td>15.08</td><td>65.69</td><td>56.04</td><td>-</td><td></td></tr><tr><td rowspan="3">1</td><td>Short</td><td>80.12</td><td>87.21</td><td>85.50</td><td>12.17</td><td>65.93</td><td>66.19</td><td>0.66</td><td></td></tr><tr><td>Middle</td><td>86.41</td><td>87.19</td><td>83.01</td><td>19.81</td><td>71.51</td><td>69.59</td><td>1.29</td><td></td></tr><tr><td>Long</td><td>57.52</td><td>80.71</td><td>74.95</td><td>16.33</td><td>69.48</td><td>59.80</td><td>3.76</td><td></td></tr><tr><td rowspan="3">2</td><td>Short</td><td>80.12</td><td>87.96</td><td>86.25</td><td>12.32</td><td>64.52</td><td>66.23</td><td>0.05</td><td></td></tr><tr><td>Middle</td><td>86.41</td><td>87.53</td><td>83.34</td><td>19.77</td><td>71.79</td><td>69.77</td><td>0.18</td><td></td></tr><tr><td>Long</td><td>58.13</td><td>81.26</td><td>75.36</td><td>17.00</td><td>69.18</td><td>60.19</td><td>0.39</td><td></td></tr></table>
+
+Table 7: The average difficulty and ratio (sum to 1) of each correct-solving castep, produced by the setting of column "Continue" in Table 4.  
+
+<table><tr><td>t</td><td>AVG Difficulty</td><td>Ratio</td></tr><tr><td>0</td><td>82.80</td><td>63.54%</td></tr><tr><td>1</td><td>86.95</td><td>18.48%</td></tr><tr><td>2</td><td>88.60</td><td>8.50%</td></tr><tr><td>3</td><td>89.83</td><td>4.25%</td></tr><tr><td>4</td><td>90.86</td><td>2.51%</td></tr><tr><td>5 (failed)</td><td>92.21</td><td>2.72%</td></tr></table>
+
+In this section, we conduct empirical studies to interpret how CASR works.
+
+In §5.1, we verify that CASR brings more improvement on more complex sequence. Also, we find CASR model may implicitly learns dependency because it solves easy parts first during refinements.
+
+In §5.2, the cross-attention map between  $\hat{Y}^t$  and  $\hat{Y}^{t - 1}$  implies the informativeness and correctness of  $\hat{Y}^{t - 1}$ . Meanwhile, from a microscopic perspective, each row in the cross-attention map implies the foundation of changing a component, where the hot-attended parts can be considered as the cause of changes.
+
+In §5.3, through case studies we show the intermediate steps generated by CASR models.
+
+# 5.1 ANALYSIS ON COMPLEXITY
+
+CASR helps more on problems with more complexity. For WebQSP, MTOP, and KVRET, we assume that the complexity of a problem  $(X,Y)$  is positively correlated with the length of its answer. Thus, we divide each test dataset into "Short", "Middle" and "Long" equal splits according to  $\mathrm{len}(Y)$ , the length of ground truth answer. As we can see in Table 6, CASR brings the more improvement to the problems with "Long" answers than "Middle" and "Short" ones.
+
+Take Sodomu as example, we find that CASR solves easy blanks before difficult blanks during refinements. In late casteps, the generation benefits more from refinement by gradually handling
+
+complex dependencies. These are shown in Table 7 - As castep  $t$  increases, the average difficulty of remaining blanks increase. Here we formulate the difficulty $^3$  of a blank cell as: Difficulty =  $(r - 1) \cdot (c - 1) \cdot (h - 1)$  where  $r, c,$  and  $h$  denote the number of blanks in the same row, column, and house of the blank. Also, we can define the correct-solved castep for a blank: If a blank is correctly filled at castep  $t$  and remains the same for the rest casteps, then the blank is solved at castep  $t$  and  $t$  is the correct-solved castep for the blank. Otherwise, by default the correct-solved castep is set to  $T$ . The "Ratio" column in Table 7 means the percentage of blanks is correctly solved at castep  $t$ .
+
+# 5.2 ANALYSIS ON ATTENTION
+
+To measure to what extend does  $\hat{Y}^t$  attend to  $X$  or  $\hat{Y}^{t - 1}$  in cross-attention, we define the density. The cross attention map,  $A$ , is a tensor that displays how the tokens in  $\hat{Y}^t$  attend to  $H$ . Therefore, the size of  $A$  is [num-layers, num-heads,  $\mathrm{len}(\hat{Y}^t)$ ,  $\mathrm{len}(X) + \mathrm{len}(\hat{Y}^{t - 1})$ ]. We define the density to  $X$  and to  $\hat{Y}^{t - 1}$  as:  $D_X = A[...;len(X)]$ . mean() and  $D_{\hat{Y}^{t - 1}} = A[...;len(X)]:$ . mean(), which are the average attention weight among each layer, head, and token, for  $X$  and for  $\hat{Y}^{t - 1}$ .
+
+Similar to Table 6, We sort the samples according to their  $D_{\hat{Y}^{t-1}}$ , and divide a test dataset into sparse, middle, and dense splits, each holding 1/3 of the total test set. We evaluate the metrics of  $\hat{Y}^{t-1}$  on the three splits. We find the performance of  $\hat{Y}^{t-1}$  increases with  $D_{\hat{Y}^{t-1}}$ . Besides, we compute the average  $D_X$  and  $D_{\hat{Y}^{t-1}}$  over all samples in the test set, and find that  $D_{\hat{Y}^{t-1}}$  grows with  $t$ . Therefore,  $\hat{Y}^{t-1}$  becomes more and more informative thus be attended to by  $\hat{Y}^t$ . Please refer to Table 10 and Table 11 in Appendix §D for more details.
+
+Meanwhile, we draw top-5 positions in cross-attention map (filter out non-blank cells and self) for each changed cell in Figure 8. As we can see, the corrected cell usually attends to cells in the same row, column, or house.
+
+# 5.3 CASE STUDIES
+
+As examples, the intermediate CASR prediction of each castep are shown in Figure 9 of Appendix §D. We can find that as castep  $t$  increases, there is more green (keywords) and less red (mistakes) parts, which means  $\hat{Y}^t$  becomes closer to  $Y$ . In the WebQSP case, the wrong part (marked red) of  $\hat{Y}^0$  is deleted by late casteps. In the MTOP case, we find that CASR correctly generates the right part ("[SL:RECIPES INCLUDED INGR EDIENT dairy]") of the answer before the left part ("[SL:MET HOD_RECIPES recipe]"), which one-pass L2R decoding cannot achieve. For Sodomu, an interesting phenomenon occurs. Although the wrong predictions (marked red) become fewer with the growth of castep, some previous correct predictions are modified to wrong ones, and then modified back. This can be considered a kind of exploration.
+
+We also present the predictions from other experiments (basiines and CASR-L, as discussed in Table 5) in Figure 10, and we find it a challenge for NAR methods to generate complex sequences.
+
+# 6 CONCLUSION
+
+In this paper, CASR framework is proposed by us to generate Complex sequences with Autoregressive Self-Boost Refinement, which can decide intermediate sequences of complex answers in a data-driven way with no need for expert knowledge or manual efforts. Through controlled experiments, we find the optimal design in CASR is Fine-tuning SepEnc with Continue strategy. Also, the interpretability of CASR is enhanced via empirical studies on problem complexity and attention dependencies. CASR sheds more light on the sequence generation methods, especially on complex tasks (e.g. automatic data analysis and drug discovery), for future research.
+
+# ETHICS STATEMENT
+
+Datasets This work collects the public dataset for research purposes. We believe there is no privacy issue, because Selenium, WebQSP, MTOP, and KVRET are accessible to the public.  
+Models This work leverages T5-base and T5-large, which are pre-trained on C4, a clean corpus. Therefore, we can make sure that CASR will not produce discriminatory answers.  
+Computational Resources We train on 4 Tesla V100 GPUs, and the training and inference time consumption is show in §9, which is acceptable considering the performance gain.
+
+# REPRODUCIBILITY STATEMENT
+
+We describe experiment setup in §4.1 and provide the complete details such as hyperparameter (in §C.1) and resource consumption (in §C.2). please see §C. Meanwhile, we make our code open source in this repository https://github.com/RalphHan/CASR. These resources should be sufficient to reproduce results of the paper.
+
+# ACKNOWLEDGMENTS
+
+This research was partly supported by Shenzhen Stable Supporting Program (WDZC20200820200655001).
+
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+
+# A EXAMPLES
+
+UnifiedSKG (Xie et al., 2022) did a great job to formulate 21 structured knowledge grounding tasks into a unified seq2seq form, and we apply their input-output form of WebQSP, MTOP, and KVRET in CASR. And for Sodomu, we apply their original serialization method, which flattens the 9x9 game table row by row into an 81-dimensional number list. Examples of WebQSP (Figure 5), MTOP (Figure 6) and KVRET (Figure 7) are shown.
+
+# B BEST RESULT SELECTION
+
+The default strategy is to run all  $T$  casteps and take the Last output  $\hat{Y}^{T - 1}$  as the final prediction. However, one could argue that for a specific input  $X$ , the best prediction may occur early and more refinements may hurt. Following this we try to pick the  $\hat{Y}^t (0\leq t < T)$  with the highest prediction probability (given by  $M^t$ ):  $\hat{Y} = \hat{Y}^{t_{pick}}$  where  $t_{pick} = \arg\max_t P(\hat{Y}^t |X,\hat{Y}^{t - 1},M^t)$ . However, that doesn't always work and we leave it as furture work.
+
+# C EXPERIMENT DETAILS
+
+# C.1 HYPERPARAMETERS
+
+On WebQSP, MTOP and KVRET tasks, we initialize CASR models with T5-base (Raffel et al., 2020), a 24-layer generative PLM. We set max casteps  $T = 3$ . Max epoch  $E$  is set to 2K, 4K, and
+
+![](images/e6c2d619e252516e8cbb1e74c4bd9663afb89de880a087694ceddba7f27a897d.jpg)  
+Request: what school did sir ernest rutherford go to? Answer: (JOIN (R education.education.institution) (JOIN (R people.person.education) m.02m7r))
+
+![](images/c077605f69369c30ada58e7970c96a99c8ab3bb83acc36d24f76e1cdd885e29e.jpg)  
+Figure 5: An Example of WebQSP.  
+Request: does this recipe have? Answer: [IN:IS_true_RECIECs [SL:METHOD_RECIECs recipe] [SL:RECIPES_INDED_INGREDIENT dairy]  
+Figure 6: An Example of MTOP.
+
+![](images/14459e59c2eb6c48e034f5f51d5c0671fd70cd17850410db4bb2f1fb0aa6d2a9.jpg)  
+Request: where is the closest grocery store? Response: we are 4 miles away from whole_foods and from safeway: which one do you prefer?  
+Figure 7: An Example of KVRET.
+
+4K steps for fine-tuning WebQSP, MTOP and KVRET, respectively. For adapter-tuning, we double the epoch number, which is 4K, 8K, and 8K steps. For the three tasks, we set the batch-size to 128, learning-rate to 2e-5, max-input-length to 1024, max-generation-length to 128, beam-size to 4, and evaluate every 2K steps for checkpoint selection.
+
+For Sudoku, we train a 12-layer encoder-decoder transformer from scratch, with d-model=512, ffindim=2048, num-heads=8. We set max castep  $T = 5$  and max epoch  $E = 10\mathrm{K}$  steps. We set the batch-size to 1024, learning-rate to 2e-5, beam-size to 2, and evaluate every 2K steps for checkpoint selection.
+
+# C.2 RESOURCE CONSUMPTION
+
+Table 8: CASR Variations and Their Number of Parameters.  
+
+<table><tr><td colspan="5">A, E and D denote #parameters of adapter, encoder and decoder.</td></tr><tr><td></td><td>Tuning</td><td>Init.</td><td>Arch.</td><td>#Parameters</td></tr><tr><td>A</td><td>Fine-tune</td><td>Restart</td><td>SepEnc</td><td>T*(2E+D)</td></tr><tr><td>B</td><td>Fine-tune</td><td>Continue</td><td>SepEnc</td><td>T*(2E+D)</td></tr><tr><td>C</td><td>Fine-tune</td><td>Restart</td><td>UniEnc</td><td>T*(E+D)</td></tr><tr><td>D</td><td>Fine-tune</td><td>Continue</td><td>UniEnc</td><td>T*(E+D)</td></tr><tr><td>E</td><td>Adapter</td><td>Restart</td><td>SepEnc</td><td>T*A+(E+D)</td></tr></table>
+
+Table 9: Training and Inference Time Consumption (take WebQSP as an example)  
+
+<table><tr><td></td><td>Method</td><td>Training(h)</td><td>Inference(min)</td></tr><tr><td>Xa</td><td>INAT (Lee et al., 2018)</td><td>14</td><td>2</td></tr><tr><td>Xb</td><td>Levenshtein (Gu et al., 2019)</td><td>12</td><td>7</td></tr><tr><td>Xc</td><td>CASR B / Dec</td><td>8</td><td>23</td></tr><tr><td>Xd</td><td>Bidirectional (Zhang et al., 2018)</td><td>6</td><td>16</td></tr><tr><td>Xe</td><td>Progressive (Tan et al., 2021)</td><td>9</td><td>16</td></tr><tr><td>B0</td><td>Finetune</td><td>3</td><td>8</td></tr><tr><td>B2</td><td>CASR</td><td>10</td><td>25</td></tr><tr><td>L2</td><td>CASR-L</td><td>28</td><td>141</td></tr></table>
+
+Table 8 shows the total parameters of each controlled experiment to help the audience understand our methods. In row  $\mathbf{E}$ , the total parameters is  $\mathrm{T^{*}A + (E + D)}$  rather than  $\mathrm{T^{*}A + (2E + D)}$ , because  $\text{Encoder}_1$  and  $\text{Encoder}_2$  are initialized with the same PLM and frozen all the time.
+
+Table 9 shows the training and inference time consumption of CASR and baselines. Note that CASR-L runs slowly because we leverage deepspeed<sup>4</sup> to avoid OOM.
+
+We train on 4 Tesla V100 GPUs. It takes 10, 22, and 19 hours to fine-tune WebQSP, MTOP and KVRET, respectively, and 12, 25, and 21 hours to adapter-tune them. Adapter-tuning is intrinsically hard so it takes more training steps to tune fewer parameters comparing to fine-tuning. For Selenium, it takes 32 hours to train from scratch.
+
+# D DETAILED RESULTS OF EMPIRICAL STUDIES
+
+Table 10: Performance of last-step prediction  $\hat{Y}^1$  when castep  $t = 2$ , produced by the setting of row  $\widehat{\mathbf{B}}$  in Table 8, grouped by  $D_{\hat{Y}^1}$ , the density to  $\hat{Y}^1$ .  
+
+<table><tr><td>Task</td><td>WebQSP</td><td colspan="2">MTOP</td><td colspan="2">KVRET</td><td rowspan="2">AVG</td></tr><tr><td>Metric</td><td>F1</td><td>Acc</td><td>Match</td><td>Bleu</td><td>Micro F1</td></tr><tr><td>Sparse</td><td>51.88</td><td>78.52</td><td>72.72</td><td>16.70</td><td>64.17</td><td>56.80</td></tr><tr><td>Middle</td><td>85.84</td><td>87.53</td><td>83.17</td><td>20.18</td><td>75.09</td><td>70.36</td></tr><tr><td>Dense</td><td>85.78</td><td>89.08</td><td>87.63</td><td>20.26</td><td>85.80</td><td>73.71</td></tr></table>
+
+Table 11: The average density to input  $X$  and previous prediction  $\hat{Y}^{t - 1}$  of the setting of row  $\mathbf{B}$  in Table 8. Note that, when castep  $t = 0$ , the density to the previous prediction is 0, thus not listed.  
+
+<table><tr><td>t</td><td>Task</td><td>WebQSP</td><td>MTOP</td><td>KVRET</td><td>AVG</td></tr><tr><td rowspan="2">1</td><td>Input</td><td>0.10%</td><td>0.11%</td><td>0.40%</td><td>0.20%</td></tr><tr><td>Previous Prediction</td><td>0.34%</td><td>0.33%</td><td>0.84%</td><td>0.50%</td></tr><tr><td rowspan="2">2</td><td>Input</td><td>0.09%↓</td><td>0.10%↓</td><td>0.39%↓</td><td>0.19%↓</td></tr><tr><td>Previous Prediction</td><td>0.61%↑</td><td>0.43%↑</td><td>1.01%↑</td><td>0.68%↑</td></tr></table>
+
+![](images/e68a0fdc1ed65bec2737a8e89b0d3996cf80bce7960516c02853d9ce156ebefe.jpg)  
+t=0
+
+![](images/b125e68011985fc15192633541b13033c8fc57431a33c80871338250734fd40b.jpg)  
+t=1  
+Figure 8: The Dependencies between Casteps Discovered by CASR. Shown as purple ( $t = 1$  attends to  $t = 0$ ) and blue ( $t = 2$  attends to  $t = 1$ ) arrows.
+
+![](images/b099887832ef8996d1724934842bfbb68acb4d6f999eed6fa271118b1623c920.jpg)  
+$\mathbf{t} = 2$
+
+# D.1 AN EXAMPLE TO COMPUTE DIFFICULTY OF BLANKS
+
+In the game table Figure 2, for the top-left blank (the ground truth is 6 in green),  $r = 5$  (the ground truths of the row blanks are 6, 9, 4, 8, and 3 respectively),  $c = 6$  (the ground truths of the column blanks are 6, 4, 3, 9, 8, and 2), and  $h = 4$  (the ground truths of the house blanks are 6, 9, 7, and 4). Then, the difficulty to fill the top-left blank is  $(5 - 1)^{*}(6 - 1)^{*}(4 - 1) = 60$ .
+
+<table><tr><td>Task&amp;Input X</td><td colspan="3">CASR Predictions \(\hat{Y}^{\mathrm{t}}\)</td><td>Ground Truth Y</td><td></td></tr><tr><td>WebQSPinput:task: webqsp ; what school did sir ernest rutherford go to? ; structured knowledge: Ernest Rutherford: m.02m7r | m.068kst common-topic.notable_types m.01m9 | (omitted to save space) | m.018y0w base.act</td><td>t=0(AND (JOIN commontopic,nota ble_types m.01nf) JOIN(R education.educatio n.institution) JOIN(R people.person.education) m.02m7r)))</td><td>t=1(JOIN (R education.educatio n.institution) JOIN (R people.person.education) m.02m7r))</td><td>t=2(JOIN (R education.educatio n.institution) JOIN (R people.person.education) m.02m7r))</td><td>(JOIN (R education.educatio n.institution) (JOIN (R people.person.education) m.02m7r))</td><td></td></tr><tr><td>MTOPinput:task: mtop ; does this recipe have dairy ; structured knowledge: IN:GET:MESSAGE, WEATHER,... | (omitted to save space) | IN:FOLLOW: MUSIC</td><td>t=0|IN:IS_true_RECI PES[SL:RECIPIES_INCL UDED_INGREDIENT T dairy ]]</td><td>t=1|IN:IS_true_RECI PES[SL:METHOD_REC IPES recipe] [SL:RECIPIES_INCL UDED_INGREDIENT T dairy ]]</td><td>t=2|IN:IS_true_RECI PES[SL:METHOD_REC IPES recipe] [SL:RECIPIES_INCL UDED_INGREDIENT T dairy ]]</td><td>|IN:IS:true_RECIP ES[SL:METHOD_RECI PES recipe] [SL:RECIPIES_INCL UDED_INGREDIENT dairy ]]</td><td></td></tr><tr><td>KVRETinput:task: kvret ; where is the closest grocery Store ; structured knowledge: col : poi | poi_type | address | distance | traffic_info row 1 : pizza my heart | pizza restaurant | (omitted to save space) | 3 miles | moderate traffic ; context:</td><td>t=0whole_foods is 4 miles away at 819_alma_st.</td><td>t=1the closest grocery store is whole_foods at 819_alma_st.</td><td>t=2the closest grocery store is whole_foods which is 4 miles away at 819_alma_st.</td><td>we are 4 miles away from whole_foods and from safeway: which one do you prefer?</td><td></td></tr><tr><td>Sudoku</td><td>t=0</td><td>t=1</td><td>t=2,3,4</td><td>692487135183592746183592746574316928461753892329861574758249361915674283836925417247138659</td><td>692487135183592746574316928461753892329861574758249361915674283836925417247138659</td></tr></table>
+
+Figure 9: Case study for CASR (row  $\widehat{\mathbf{B}}$ ) on WebQSP, MTOP, KVRET, and Sudoku at different casteps. The first column shows how the input is serialized from Figure 5, Figure 6, and Figure 7. For WebQSP, MTOP, and KVRET, keywords are highlighted with green color, and red color denotes wrong predictions. For Sudoku, the white fields denote the blanks to be filled, the red numbers in them denote that the predictions are wrong, and the green ones denote the correct predictions. It's a coincidence that the Sudoku prediction for this sample remains the same for  $t = 2,3,4$ , so we draw them together.
+
+<table><tr><td>Method</td><td>WebQSP</td><td>MTOP</td><td>KVRET</td></tr><tr><td>INAT</td><td>(JOIN (R education.education.institution) (AND (JOIN education.educationdegree).02 (JOIN education.education.JO3.)www.w m.educationma02date)</td><td>and_IN: the _ ]ACT [ ] the for: ] |IN for</td><td>error iserror_ theerror_ at is is_error_ is_. . . would would is_is_error is is</td></tr><tr><td>Levenshtein</td><td>(JOIN (R education.education.institution) (JOIN (R people.person.education) m.03xsv3))</td><td>-</td><td>what is for you.</td></tr><tr><td>CASR /Dec</td><td>(JOIN (R education.education.institution) (JOIN (R people.person.education) m.02mjmr))</td><td>[IN:GETRECIPES [SL:RECIPES_DISH chicken ] ]</td><td>the nearest grocery Store is whole_foods at 819_alma_st. would you like directions there?</td></tr><tr><td>Bidirectional</td><td>(JOIN (R education.education.institution) (JOIN (R people.person.education) m.02m7r))</td><td>[IN:GETRECIPES [SL:RECIPES INCLUDE D_INGREDIENT dairy ] ]</td><td>safeway is 4 miles away.</td></tr><tr><td>Progressive</td><td>(AND (JOIN commontopic.notable_types m.01nf) (JOIN (R education.education.institution) (JOIN (R people.person.education) m.02m7r)))</td><td>[IN:IS:true_RECIPES [SL:RECIPES INCLUDE D_INGREDIENT dairy ] ]</td><td>the closest grocery store is safeway.</td></tr><tr><td>CASR-L</td><td>(JOIN (R education.education.institution) (JOIN (R people.person.education) m.02m7r))</td><td>[IN:IS:true_RECIPES [SL:RECIPES INCLUDE D_INGREDIENT dairy ] ]</td><td>the closest grocery store is whole_foods at 819 alma_st.</td></tr></table>
+
+Figure 10: Case study for other experiments (basiines and CASR-L).
+
+Algorithm 1 CASR Inference Process.  
+Input: max castep  $T$ ; input  $X$  (from test set); well-trained CASR models  $M^t$  ( $0 \leq t < T$ )  
+Output: the best prediction  $\hat{Y}$  for  $X$   
+1: for  $t$  in  $0, 1, \dots, T - 1$  do  
+2: /* argmax by beam searching */  
+3: if  $t = 0$  then  
+4:  $\hat{Y}^t \gets \text{argmax}_Y P(Y|X, M^t)$   
+5: else  
+6:  $\hat{Y}^t \gets \text{argmax}_Y P(Y|X, \hat{Y}^{t-1}, M^t)$   
+7: end if  
+8: end for  
+9: return  $\hat{Y}$
+
+Algorithm 2 CASR Training Process (§3.1).  
+Input: max castep  $T$ ; max epoch  $E$ ; all input  $X$  and ground truth  $Y$  in train  $(S_{t})$  and dev  $(S_{d})$  sets  
+Output: CASR models  $M^{t}$  ( $0 \leq t < T$ );  
+1: for  $t$  in  $0,1,\ldots,T-1$  do  
+2:  $\theta \gets$  parameter initialization (§3.4)  
+3: for  $e$  in  $0,1,\ldots,E-1$  do  
+4: /* argmax by updating  $\theta$  using train set */  
+5: if  $t = 0$  then  
+6:  $\theta \gets \text{argmax}_{\theta} \sum_{(X,Y) \in S_{t}} P(Y|X, \theta)$   
+7: else  
+8:  $\theta \gets \text{argmax}_{\theta} \sum_{(X,Y) \in S_{t}} P(Y|X, \hat{Y}^{t-1}, \theta)$   
+9: end if  
+10:  $M_{e}^{t} \gets \theta$   
+11: end for  
+12:  $M^{t} \gets$  checkpoint selection from  $M_{e}^{t}$  ( $0 \leq e < E$ ) on  $S_{d}$   
+13: /* predict with beam searching  $\forall X \in S_{t} \cup S_{d}$  using  $M^{t} \ast/$   
+14: if  $t = 0$  then  
+15:  $\hat{Y}^{t} \gets \text{argmax}_{Y} P(Y|X, M^{t})$   
+16: else  
+17:  $\hat{Y}^{t} \gets \text{argmax}_{Y} P(Y|X, \hat{Y}^{t-1}, M^{t})$   
+18: end if  
+19: end for  
+20: return  $M^{t}$  ( $0 \leq t < T$ )

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2023/CFlowNets_ Continuous Control with Generative Flow Networks/8b541aa9-1052-4ad9-9c1a-f789cead07db_origin.pdf

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2023/CFlowNets_ Continuous Control with Generative Flow Networks/full.md

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+# CFLOWNETS: CONTINUOUS CONTROL WITH GENERATIVE FLOW NETWORKS
+
+Yinchuan Li $^{1}$ , Shuang Luo $^{2,*}$ , Haozhi Wang $^{3}$ , Jianye Hao $^{1,3}$
+
+<sup>1</sup>Huawei Noah's Ark Lab, Beijing, China
+
+$^{2}$ Zhejiang University, Huangzhou, China
+
+$^{3}$ Tianjin University, Tianjin, China
+
+{liyinchuan, haojianye}@huawei.com, luoshuang@zju.edu.cn
+
+wanghaozhi@tju.edu.cn
+
+# ABSTRACT
+
+Generative flow networks (GFlowNets), as an emerging technique, can be used as an alternative to reinforcement learning for exploratory control tasks. GFlowNet aims to generate distribution proportional to the rewards over terminating states, and to sample different candidates in an active learning fashion. GFlowNets need to form a DAG and compute the flow matching loss by traversing the inflows and outflows of each node in the trajectory. No experiments have yet concluded that GFlowNets can be used to handle continuous tasks. In this paper, we propose generative continuous flow networks (CFlowNets) that can be applied to continuous control tasks. First, we present the theoretical formulation of CFlowNets. Then, a training framework for CFlowNets is proposed, including the action selection process, the flow approximation algorithm, and the continuous flow matching loss function. Afterward, we theoretically prove the error bound of the flow approximation. The error decreases rapidly as the number of flow samples increases. Finally, experimental results on continuous control tasks demonstrate the performance advantages of CFlowNets compared to many reinforcement learning methods, especially regarding exploration ability.
+
+# 1 INTRODUCTION
+
+As an emerging technology, generative flow networks (GFlowNets) (Bengio et al., 2021a;b) can make up for the shortcomings of reinforcement learning (Kaelbling et al., 1996; Sutton & Barto, 2018) on exploratory tasks. Specifically, based on the Bellman equation (Sutton & Barto, 2018), reinforcement learning is usually trained to maximize the expectation of future rewards; hence the learned policy is more inclined to sample action sequences with higher rewards. In contrast, the training goal of GFlowNets is to define a distribution proportional to the rewards over terminating states, i.e., the parent states of the final states, rather than generating a single high-reward action sequence (Bengio et al., 2021a). This is more like sampling different candidates in an active learning setting (Bengio et al., 2021b), thus better suited for exploration tasks.
+
+GFlowNets construct the state transitions of trajectories into a directed acyclic graph (DAG) structure. Each node in the graph structure corresponds to a different state, and actions correspond to transitions between different states, that is, an edge connecting different nodes in the graph. For discrete tasks, the number of nodes in this graph structure is limited, and each edge can only correspond to one discrete action. However, in real environments, the state and action spaces are continuous for many tasks, such as quadrupedal locomotion (Kohl & Stone, 2004), autonomous driving (Kiran et al., 2021; Shalev-Shwartz et al., 2016; Pan et al., 2017), or dexterous in-hand manipulation (Andrychowicz et al., 2020). Moreover, the reward distributions corresponding to these environments may be multimodal, requiring more diversity exploration. The needs of these environments closely match the strengths of GFlowNets. (Bengio et al., 2021b) proposes an idea for adapting GFlowNets to continuous tasks by replacing sums with integrals for continuous variables,
+
+and they suggest the use of integrable densities and detailed balance (DB) or trajectory balance (TB) Malkin et al. (2022) criterion to obtain tractable training objectives, which can avoid some integration operations. However, this idea has not been verified experimentally.
+
+In this paper, we propose generative Continuous Flow Networks, named CFlowNets for short, for continuous control tasks to generate policies that can be proportional to continuous reward functions. Applying GFlowNets to continuous control tasks is exceptionally challenging. In generative flow networks, the transition probability is defined as the ratio of action flow and state flow. For discrete state and action spaces, we can form a DAG and compute the state flow by traversing a node's incoming and outgoing flows. Conversely, it is impossible for continuous tasks to traverse all state-action pairs and corresponding rewards. To address this issue, we use important sampling to approximate the integrals over inflows and outflows in the flow-matching constraint, where we use a deep neural network to predict the parent nodes of each state in the sampled trajectory. The main contributions of this paper are summarized as the following:
+
+Main Contributions: 1) We extend the theoretical formulation and flow matching theorem of previous GFlowNets to continuous scenarios. Based on this, a loss function for training CFlowNets is presented; 2) We propose an efficient way to sample actions with probabilities approximately proportional to the output of the flow network, and propose a flow sampling approach to approximate continuous inflows and outflows, which allows us to construct a continuous flow matching loss; 3) We theoretically analyze the error bound between sampled flows and inflows/outflows, and the tail becomes minor as the number of flow samples increases; 4) We conduct experiments based on continuous control tasks to demonstrate that CFlowNets can outperform current state-of-the-art RL algorithms, especially in terms of exploration capabilities. To the best of our knowledge, our work is the first to empirically demonstrate the effectiveness of flow networks on continuous control tasks. The codes are available at http://gitee.com/mindsight/models/tree/master/research/gflownets/cflownets
+
+# 2 PRELIMINARIES
+
+# 2.1 MARKOV DECISION PROCESS
+
+A stochastic, discrete-time and sequential decision task can be described as a Markov Decision Process (MDP), which is canonically formulated by the tuple:
+
+$$
+M = \langle \mathcal {S}, \mathcal {A}, P, R, \gamma \rangle . \tag {1}
+$$
+
+In the process,  $s \in S$  represents the state space of the environment. At each time step, agent receives a state  $s$  and selects an action  $a$  on the action space  $\mathcal{A}$ . This results in a transition to the next state  $s'$  according to the state transition function  $P(s'|s, a): \mathcal{S} \times \mathcal{A} \times \mathcal{S} \to [0,1]$ . Then the agent gets the reward  $r$  based on the reward function  $R(s, a): \mathcal{S} \times \mathcal{A} \to \mathbb{R}$ . A stochastic policy  $\pi$  maps each state to a distribution over actions  $\pi(\cdot|s)$  and gives the probability  $\pi(a|s)$  of choosing action  $a$  in state  $s$ . The agent interacts with the environment by executing the policy  $\pi$  and obtaining the admissible trajectories  $\{(s_t, a_t, r_t, s_{t+1})\}_{t=1}^n$ , where  $n$  is the trajectory length. The goal of an agent is to maximize the discounted return  $\mathbb{E}_{s_{0:n}, a_{0:n}}[\sum_{t=0}^{\infty} \gamma^t r_t | s_0 = s, a_0 = a, \pi]$ , where  $\mathbb{E}$  is the expectation over the distribution of the trajectories and  $\gamma \in [0,1)$  is the discount factor.
+
+# 2.2 GENERATIVE FLOW NETWORK
+
+GFlowNet sees the MDP as a flow network. Define  $s' = T(s, a)$  and  $F(s)$  as the node's transition and the total flow going through  $s$ . Define an edge/action flow  $F(s, a) = F(s \to s')$  as the flow through an edge  $s \to s'$ . The training process of vanilla GFlowNets needs to sum the flow of parents and children through nodes (states), which depends on the discrete state space and discrete action space. The framework is optimized by the following flow consistency equations:
+
+$$
+\sum_ {s, a: T (s, a) = s ^ {\prime}} F (s, a) = R \left(s ^ {\prime}\right) + \sum_ {a ^ {\prime} \in \mathcal {A} \left(s ^ {\prime}\right)} F \left(s ^ {\prime}, a ^ {\prime}\right), \tag {2}
+$$
+
+which means that for any node  $s$ , the incoming flow equals the outgoing flow, which is the total flow  $F(s)$  of node  $s$ .
+
+# 3 CFLOWNETS: THEORETICAL FORMULATION
+
+Considering a continuous task with tuple  $(\mathcal{S},\mathcal{A})$ , where  $\mathcal{S}$  denotes the continuous state space and  $\mathcal{A}$  denotes the continuous action space. Define a trajectory  $\tau = (s_1,\dots,s_n)$  in this continuous task as a sequence sampled elements of  $\mathcal{S}$  such that every transition  $a_{t}:s_{t}\to s_{t + 1}\in \mathcal{A}$ . Further, we define an acyclic trajectory  $\tau = (s_1,\dots,s_n)$  as a trajectory satisfies the acyclic constraint:  $\forall s_m\in \tau ,s_k\in \tau ,m\neq k$ , we have  $s_m\neq s_k$ . Denote  $s_0$  and  $s_f$  respectively as the initial state and the final state related with the continuous task  $(\mathcal{S},\mathcal{A})$ , we define the complete trajectory as any sampled acyclic trajectory from  $(\mathcal{S},\mathcal{A})$  starting in  $s_0$  and ending in  $s_f$ . Correspondingly, a transition  $s\rightarrow s_f$  into the final state is defined as the terminating transition, and  $F(s\rightarrow s_f)$  is a terminating flow.
+
+A trajectory flow  $F(\tau): \tau \mapsto \mathbb{R}^{+}$  is defined as any nonnegative function defined on the set of complete trajectories  $\tau$ . For each trajectory  $\tau$ , the associated flow  $F(\tau)$  contains the number of particles (Bengio et al., 2021b) sharing the same path  $\tau$ . In addition, the tuple  $(S, A, F)$  is called a continuous flow network. Let  $T(s, a) = s'$  indicate an action  $a$  that could make a transition from state  $s$  to attain  $s'$ . Then we make the following assumptions.
+
+Assumption 1. Assume that the continuous take  $(\mathcal{S},\mathcal{A})$  is an "acyclic" task, which means that arbitrarily sampled trajectories  $\tau$  are acyclic, i.e.,
+
+$$
+s _ {i} \neq s _ {j}, \forall s _ {i}, s _ {j} \in \tau = (s _ {0}, \dots , s _ {n}), i \neq j.
+$$
+
+Assumption 2. Assume the flow function  $F(s, a)$  is Lipschitz continuous, i.e.,
+
+$$
+\left| F (s, a) - F \left(s, a ^ {\prime}\right) \right| \leq L \| a - a ^ {\prime} \|, a, a ^ {\prime} \in \mathcal {A}, \tag {3}
+$$
+
+$$
+\left| F (s, a) - F \left(s ^ {\prime}, a\right) \right| \leq L \| s - s ^ {\prime} \|, s, s ^ {\prime} \in \mathcal {S}, \tag {4}
+$$
+
+where  $L$  is a constant.
+
+Assumption 3. Assume that for any state pair  $(s_t, s_{t+1})$ , there is a unique action  $a_t$  such that  $T(s_t, a_t) = s_{t+1}$ , i.e., taking action  $a_t$  in  $s_t$  is the only way to get to  $s_{t+1}$ . Hence we can define  $s_t \coloneqq g(s_{t+1}, a_t)$ , where  $g(\cdot)$  is a transition function. And assume actions are the translation actions.
+
+The necessity and rationality of Assumptions 1-3 are analyzed in the appendix. Under Assumption 1, we define the parent set  $\mathcal{P}(s_t)$  of a state  $s_t$  as the set that contains all of the direct parents of  $s_t$  that could make a direct transition to  $s_t$ , i.e.,  $\mathcal{P}(s_t) = \{s \in S : T(s, a \in \mathcal{A}) = s_t\}$ . Similarly, define the child set  $\mathcal{C}(s_t)$  of a state  $s_t$  as the set contains all of the direct children of  $s_t$  that could make a direct transition from  $s_t$ , i.e.,  $\mathcal{C}(s_t) = \{s \in S : T(s_t, a \in \mathcal{A}) = s\}$ . Then, we have the following continuous flow definitions, where Assumptions 2-3 make these integrals integrable and meaningful.
+
+Definition 1 (Continuous State Flow). The continuous state flow  $F(s): \mathcal{S} \mapsto \mathbb{R}$  is the integral of the complete trajectory flows passing through the state:
+
+$$
+F (s) = \int_ {\tau : s \in \tau} F (\tau) \mathrm {d} \tau . \tag {5}
+$$
+
+Definition 2 (Continuous Inflows). For any state  $s_t$ , its inflows are the integral of flows that can reach state  $s_t$ , i.e.,
+
+$$
+\int_ {s \in \mathcal {P} \left(s _ {t}\right)} F (s \rightarrow s _ {t}) \mathrm {d} s = \int_ {s: T (s, a) = s _ {t}} F (s, a) \mathrm {d} s = F \left(s _ {t}\right) = \int_ {a: T (s, a) = s _ {t}} F (s, a) \mathrm {d} a, \tag {6}
+$$
+
+where  $a:s\to s_t$  and  $s = g(s_{t},a)$  since Assumption 3 holds.
+
+Definition 3 (Continuous Outflows). For any state  $s_t$ , the outflows are the integral of flows passing through state  $s_t$  with all possible actions  $a \in \mathcal{A}$ , i.e.,
+
+$$
+\int_ {s \in \mathcal {C} \left(s _ {t}\right)} F \left(s _ {t} \rightarrow s\right) \mathrm {d} s = F \left(s _ {t}\right) = \int_ {a \in \mathcal {A}} F \left(s _ {t}, a\right) \mathrm {d} a. \tag {7}
+$$
+
+Based on the above definitions, we can define the transition probability  $P(s \to s'|s)$  of edge  $s \to s'$  as a special case of conditional probability introduced in Bengio et al. (2021b). In particular, the forward transition probability is given by
+
+$$
+P _ {F} \left(s _ {t + 1} \mid s _ {t}\right) := P \left(s _ {t} \rightarrow s _ {t + 1} \mid s _ {t}\right) = \frac {F \left(s _ {t} \rightarrow s _ {t + 1}\right)}{F \left(s _ {t}\right)}. \tag {8}
+$$
+
+Similarly, the backwards transition probability is given by
+
+$$
+P _ {B} \left(s _ {t} \mid s _ {t + 1}\right) := P \left(s _ {t} \rightarrow s _ {t + 1} \mid s _ {t + 1}\right) = \frac {F \left(s _ {t} \rightarrow s _ {t + 1}\right)}{F \left(s _ {t + 1}\right)}. \tag {9}
+$$
+
+For any trajectory sampled from a continuous task  $(\mathcal{S},\mathcal{A})$  , we have
+
+$$
+\forall \tau = \left(s _ {1}, \dots , s _ {n}\right), P _ {F} (\tau) := \prod_ {t = 1} ^ {n - 1} P _ {F} \left(s _ {t + 1} \mid s _ {t}\right) \tag {10}
+$$
+
+$$
+\forall \tau = \left(s _ {1}, \dots , s _ {n}\right), P _ {B} (\tau) := \prod_ {t = 1} ^ {n - 1} P _ {B} \left(s _ {t} \mid s _ {t + 1}\right), \tag {11}
+$$
+
+and we further have
+
+$$
+\forall s \in \mathcal {S} \backslash \left\{s _ {f} \right\}, \int_ {s ^ {\prime} \in \mathcal {C} (s)} P _ {F} \left(s ^ {\prime} \mid s\right) d s ^ {\prime} = 1 \text {a n d} \forall s \in \mathcal {S} \backslash \left\{s _ {0} \right\}, \int_ {s ^ {\prime} \in \mathcal {P} (s)} P _ {B} \left(s ^ {\prime} \mid s\right) d s ^ {\prime} = 1. \tag {12}
+$$
+
+Given any trajectory  $\tau = (s_0, \dots, s_n, s)$  that starts in  $s_0$  and ends in  $s$ , a Markovian flow (Bengio et al., 2021b) is defined as the flow that satisfies
+
+$$
+P (s \rightarrow s ^ {\prime} | \tau) = P (s \rightarrow s ^ {\prime} | s) = P _ {F} (s ^ {\prime} | s),
+$$
+
+and the corresponding flow network  $(\mathcal{S},\mathcal{A},F)$  is called a Markovian flow network (Bengio et al., 2021b). Then, we present Theorem 1 proved in the appendix B.1, which is an extension of Proposition 19 in Bengio et al. (2021b) to continuous scenarios.
+
+Theorem 1 (Continuous Flow Matching Condition). Consider a non-negative function  $\hat{F}(s, a)$  taking a state  $s \in S$  and an action  $a \in \mathcal{A}$  as inputs. Then we have  $\hat{F}$  corresponds to a flow if and only if the following continuous flow matching conditions are satisfied:
+
+$$
+\forall s ^ {\prime} > s _ {0}, \hat {F} \left(s ^ {\prime}\right) = \int_ {s \in \mathcal {P} \left(s ^ {\prime}\right)} \hat {F} \left(s \rightarrow s ^ {\prime}\right) \mathrm {d} s = \int_ {s: T (s, a) = s ^ {\prime}} \hat {F} \left(s, a: s \rightarrow s ^ {\prime}\right) \mathrm {d} s \tag {13}
+$$
+
+$$
+\forall s ^ {\prime} <   s _ {f}, \hat {F} (s ^ {\prime}) = \int_ {s ^ {\prime \prime} \in \mathcal {C} (s ^ {\prime})} \hat {F} (s ^ {\prime} \to s ^ {\prime \prime}) \mathrm {d} s ^ {\prime \prime} = \int_ {a \in \mathcal {A}} \hat {F} (s ^ {\prime}, a) \mathrm {d} a.
+$$
+
+Furthermore,  $\hat{F}$  uniquely defines a Markovian flow  $F$  matching  $\hat{F}$  such that
+
+$$
+F (\tau) = \frac {\prod_ {t = 1} ^ {n + 1} \hat {F} \left(s _ {t - 1} \rightarrow s _ {t}\right)}{\prod_ {t = 1} ^ {n} \hat {F} \left(s _ {t}\right)}. \tag {14}
+$$
+
+Theorem 1 means that as long as any non-negative function satisfies the flow matching conditions, a unique flow is determined. Therefore, for sparse reward environments, i.e.,  $R(s) = 0$ ,  $\forall s \neq s_f$ , we can obtain the target flow by training a flow network that satisfies the flow matching conditions. Such learning machines are called CFlowNets, and we have the following continuous loss function:
+
+$$
+\mathcal {L} (\tau) = \sum_ {s _ {t} = s _ {1}} ^ {s _ {f}} \left(\int_ {s _ {t - 1} \in \mathcal {P} (s _ {t})} F (s _ {t - 1} \rightarrow s _ {t}) \mathrm {d} s _ {t - 1} - R (s _ {t}) - \int_ {s _ {t + 1} \in \mathcal {C} (s _ {t})} F (s _ {t} \rightarrow s _ {t + 1}) \mathrm {d} s _ {t + 1}\right) ^ {2}.
+$$
+
+However, obviously, the above continuous loss function cannot be directly applied in practice. Next, we propose a method to approximate the continuous loss function based on the sampled trajectories to obtain the flow model.
+
+# 4 CFLOWNETS: TRAINING FRAMEWORK
+
+For continuous tasks, it is usually difficult to access all state-action pairs to calculate continuous inflows and outflows. In the following, we propose the CFlowNets training framework to address this problem, which includes an action sampling process, a flow matching approximation process. Then, CFlowNets can be trained based on an approximate flow matching loss function.
+
+![](images/dbeebd9532d4778eb520aa3e0c8d242ea750b0743d9579cea4e9a6bba2b998c1.jpg)  
+Figure 1: Overall framework of CFlowNets. Left: During the environment interaction phase, we sample actions to update states with probabilities proportional to the reward according to CFlowNet. Middle: We randomly sample actions to approximately calculate the inflows and outflows, where a DNN is used to estimate the parent states. Right: Continuous flow matching loss is used to train the CFlowNet based on making inflows equal to outflows or reward.
+
+# 4.1 OVERALL FRAMEWORK
+
+The overview framework of CFlowNets is shown in Figure 1, including the environment interaction, flow sampling, and training procedures. During the environment interaction phase (Left part of Figure 1), we sample an action probability buffer based on the forward-propagation of CFlowNets. We name this process the action selection procedure, as detailed in Section 4.2. After acquiring the action, the agent can interact with the environment to update the state, and this process repeats several steps until the complete trajectory is sampled. Once a buffer of complete trajectories is available, we randomly sample  $K$  actions and compute the child states to approximately calculate the outflows. For the inflows, we use these sampled actions together with the current state as the input to the deep neural network  $G$  to estimate the parent states. Based on these, we can approximately determine the inflows. We name this process the flow matching approximation procedure (Middle part of Figure 1), as detailed in Section 4.3. Finally, based on the approximate inflows and outflows, we can train a CFlowNet based on the continuous flow matching loss function (Right part of Figure 1), as details in Section 4.4. The pseudocode is provided in Appendix C.
+
+# 4.2 ACTION SELECTION PROCEDURE
+
+Starting from an empty set, CFlowNets aim to obtain complete trajectories  $\tau = (s_0,s_1,\dots ,s_f)\in \mathcal{T}$  by iteratively sampling  $a_{t}\sim \pi (a_{t}|s_{t}) = \frac{F(s_{t},a_{t})}{F(s_{t})}$  with tuple  $\{(s_t,a_t,r_t,s_{t + 1})\}_{t = 0}^f$  . However, it is difficult to sample trajectories strictly according to the corresponding probability of  $a_{t}$  , since the actions are continuous, we cannot get the exact action probability distribution function based on the flow network  $F(s_{t},a_{t})$  . To solve this problem, at each state  $s_t$  , we first uniformly sample  $M$  actions from  $\mathcal{A}$  and generate an action probability buffer  $\mathcal{P} = \{F(s_t,a_i)\}_{i = 1}^M$  , which is used as an approximation of action probability distributions. Then we sample an action from  $\mathcal{P}$  according to the corresponding probabilities of all actions. Obviously, actions with larger  $F(s_{t},a_{i})$  will be sampled with higher probability. In this way, we approximately sample actions from a continuous distribution according to their corresponding probabilities.
+
+Remark 1. After the training process, for tasks that require a larger reward, we can sample actions with the maximum flow output in  $\mathcal{P}$  during the test process to obtain a relatively higher reward. How the output of the flow model is used is flexible, and we can adjust it for different tasks.
+
+# 4.3 FLOW MATCHING APPROXIMATION
+
+Once a batch of trajectories  $\mathcal{B}$  is available, to satisfy flow conditions, we require that for any node  $s_t$ , the inflows  $\int_{a:T(s,a) = s_t} F(s,a)\mathrm{d}a$  equals the outflows  $\int_{a\in \mathcal{A}} F(s_t,a)\mathrm{d}a$ , which is the total flow
+
+$F(s_{t})$  of node  $s_t$ . However, obviously, we cannot directly calculate the continuous inflows and outflows to complete the flow matching condition. An intuitive idea is to discretize the inflows and outflows based on a reasonable approximation and match the discretized flows. To do this, we sample  $K$  actions independently and uniformly from the continuous action space  $\mathcal{A}$  and calculate corresponding  $F(s_{t},a_{k}), k = 1,\dots,K$  as the outflows, i.e., we use the following approximation:
+
+$$
+\int_ {a \in \mathcal {A}} F \left(s _ {t}, a\right) \mathrm {d} a \approx \frac {\mu (\mathcal {A})}{K} \sum_ {k = 1} ^ {K} F \left(s _ {t}, a _ {k}\right), \tag {15}
+$$
+
+where  $\mu (\mathcal{A})$  denotes the measure of the continuous action space  $\mathcal{A}$
+
+By contrast, an approximation of inflow is more difficult since we should find the parent states first. To solve this problem, we construct a deep neural network  $G$  (named "retrieval" neural network) parameterized by  $\phi$  with  $(s_{t + 1},a_t)$  as the input while  $s_t$  as the output, and train this network based on  $\mathcal{B}$  with the MSE loss. That is, we want use  $G$  to fit function  $g(\cdot)$ . The network  $G$  is usually easy to train since we consider tasks satisfy Assumption 3, and we can obtain a high-precision network  $G$  through simple pre-training. As the training progresses, we can also occasionally update  $G$  based on the sampled trajectories to ensure accuracy. Then, the inflows can be calculated approximately:
+
+$$
+\int_ {a: T (g \left(s _ {t}, a\right), a) = s _ {t}} F (g \left(s _ {t}, a\right), a) \mathrm {d} a \approx \frac {\mu (\mathcal {A})}{K} \sum_ {k = 1} ^ {K} F \left(G _ {\phi} \left(s _ {t}, a _ {k}\right), a _ {k}\right). \tag {16}
+$$
+
+Next, by assuming that the flow function  $F(s,a)$  is Lipschitz continuous in Assumption 2, we could provide a non-asymptotic analysis for the error between the sample inflows/outflows and the true inflows/outflows. Theorem 2 establishes the error bound between the sample outflows (resp. inflows) and the actual outflows (resp. inflows) in the tail form and shows that the tail is decreasing exponentially. Furthermore, the tail gets much smaller with the increase of  $K$ , which means the sample outflows (resp. inflows) are a good estimation of the actual outflows (resp. inflows).
+
+Theorem 2. Let  $\{a_k\}_{k=1}^K$  be sampled independently and uniformly from the continuous action space  $\mathcal{A}$ . Assume  $G_{\phi^*}$  can optimally output the actual state  $s_t$  with  $(s_{t+1}, a_t)$ . For any bounded continuous action  $a \in \mathcal{A}$  and any state  $s_t \in S$ , we have
+
+$$
+\mathbb {P} \left(\left| \frac {\mu (\mathcal {A})}{K} \sum_ {k = 1} ^ {K} F \left(s _ {t}, a _ {k}\right) - \int_ {a \in \mathcal {A}} F \left(s _ {t}, a\right) \mathrm {d} a \right| \geq t\right) \leq 2 \exp \left(- \frac {K t ^ {2}}{2 \left(L \mu (\mathcal {A}) \operatorname {d i a m} (\mathcal {A})\right) ^ {2}}\right) \tag {17}
+$$
+
+and
+
+$$
+\begin{array}{l} \mathbb {P} \left(\left| \frac {\mu (\mathcal {A})}{K} \sum_ {k = 1} ^ {K} F \left(G _ {\phi^ {*}} \left(s _ {t}, a _ {k}\right), a _ {k}\right) - \int_ {a: T (s, a) = s _ {t}} F (s, a) \mathrm {d} a \right| \geq t\right) \\ \leq 2 \exp \left(- \frac {K t ^ {2}}{2 \left(L \mu (\mathcal {A}) \left(\operatorname {d i a m} (\mathcal {A}) + \operatorname {d i a m} (\mathcal {S})\right)\right) ^ {2}}\right), \tag {18} \\ \end{array}
+$$
+
+where  $L$  is the Lipschitz constant,  $\mathrm{diam}(\mathcal{A})$  denotes the diameter of the action space  $\mathcal{A}$  and  $\mathrm{diam}(S)$  denotes the diameter of the state space  $S$ .
+
+# 4.4 LOSS FUNCTION
+
+Based on (15) and (16), the continuous loss function can be approximated by
+
+$$
+\mathcal {L} _ {\theta} (\tau) = \sum_ {s _ {t} = s _ {1}} ^ {s _ {f}} \left[ \sum_ {k = 1} ^ {K} F _ {\theta} \left(G _ {\phi} \left(s _ {t}, a _ {k}\right), a _ {k}\right) - \lambda R \left(s _ {t}\right) - \sum_ {k = 1} ^ {K} F _ {\theta} \left(s _ {t}, a _ {k}\right) \right] ^ {2}, \tag {19}
+$$
+
+where  $\theta$  is the parameter of the flow network  $F(\cdot)$  and  $\lambda = K / \mu (\mathcal{A})$ . Note that in many tasks we cannot obtain exact  $\mu (\mathcal{A})$ . For such tasks, we can directly set  $\lambda$  to 1, and then adjust the reward shaping to ensure the convergence of the algorithm<sup>1</sup>.
+
+It is noteworthy that the magnitude of the state flow at different locations in the trajectory may not match. For example, the initial node flow is likely to be larger than the ending node flow. To solve this problem, inspired the log-scale loss introduced in GFlowNets (Bengio et al., 2021a), we can modify (19) into:
+
+$$
+\begin{array}{l} \mathcal {L} _ {\theta} (\tau) = \sum_ {s _ {t} = s _ {1}} ^ {s _ {f}} \left\{\log \left[ \epsilon + \sum_ {k = 1} ^ {K} \exp F _ {\theta} ^ {\log} (G _ {\phi} (s _ {t}, a _ {k}), a _ {k}) \right] \right. \\ \left. - \log \left[ \epsilon + \lambda R (s _ {t}) + \sum_ {k = 1} ^ {K} \exp F _ {\theta} ^ {\log} \left(s _ {t}, a _ {k}\right) \right] \right\} ^ {2}, \tag {20} \\ \end{array}
+$$
+
+where  $\epsilon$  is a hyper-parameter that helps to trade off small versus large flows and helps avoid the numerical problem of taking the logarithm of tiny flows. Note that Theorem 2 cannot be used to guarantee the unbiasedness of (20) because  $\log \mathbb{E}(x)\neq \mathbb{E}\log (x)$ . But experiments show that this approximation works well.
+
+# 5 RELATED WORKS
+
+Generative Flow Networks. Generative flow networks are proposed to enhance exploration capabilities by generating a distribution proportional to the rewards over terminating states (Bengio et al., 2021b;a). Since the network only samples actions based on the distribution of the corresponding rewards, rather than focusing only on actions that maximize rewards such as reinforcement learning, it can perform well on tasks with more diverse reward distributions, and has been successfully applied to molecule generation (Bengio et al., 2021a; Malkin et al., 2022; Jain et al., 2022), discrete probabilistic modeling (Zhang et al., 2022b), structure learning (Deleu et al., 2022), causal discovery Li et al. (2022) and graph neural network Li et al. (2023). The connection between deep generative models and GFlowNets is discussed in Zhang et al. (2022a) through the lens of Markov trajectory learning. In Bengio et al. (2021b), an idea is proposed for adapting GFlowNets to continuous tasks by replacing sums with integrals for continuous variables. Malkin et al. (2022) and Bengio et al. (2021b) propose detailed balance (DB) and trajectory balance (TB) objectives, which use parametric forward and backward policies in the objective function. These new objective functions do not require evaluating the flow model on multiple parents of a state, which is more efficient, especially for high-dimensional environments. Malkin et al. (2022) and Bengio et al. (2021b) mentioned that these objective functions can also be used in continuous scenarios by replacing the policy likelihoods in the objective with probability densities. A possible disadvantage is that it is not easy to estimate  $P_{F}$  and  $\bar{P}_B$  in a continuous environment, since the state space is much larger than in a discrete scenario, and a small error in modeling probability densities can greatly affect the final performance. How to combine DB and TB with CFlowNets will be a worthy future work.
+
+Continuous Reinforcement Learning. Policy gradient algorithms are widely used for reinforcement learning problems with continuous action spaces. The deterministic policy gradient (DPG) (Silver et al., 2014) algorithm is an actor-critic (Grondman et al., 2012; Rosenstein et al., 2004) method that uses an estimate of the learned value  $Q(s, a)$  to train a deterministic policy  $\mu: S \to \mathcal{A}$  parameterized by  $\theta^{\mu}$ . Compared with CFlowNets, the policy is updated by applying the chain rule to the expected return  $J$  from the start distribution with respect to the policy parameters:
+
+$$
+\begin{array}{l} \nabla_ {\theta^ {\mu}} J \approx \mathbb {E} _ {\mathcal {D}} \left[ \left. \nabla_ {\theta^ {\mu}} Q (s, a | \theta^ {Q}) \right| _ {a = \mu (s | \theta^ {\mu})} \right] \tag {21} \\ = \mathbb {E} _ {\mathcal {D}} \left[ \nabla_ {a} Q (s, a | \theta^ {Q}) \big | _ {a = \mu (s _ {t})} \nabla_ {\theta^ {\mu}} \mu (s | \theta^ {\mu}) \right], \\ \end{array}
+$$
+
+where  $\mathcal{D}$  is the replay buffer. The policy aims to maximize the expectation of future rewards, which are estimated by  $Q$ -learning. In this setting, the trajectories generated by the policy may be relatively homogeneous. However, the training goal of CFlowNets is to define a distribution proportional to the rewards over terminating states, resulting in more diverse trajectories that are beneficial for exploring the environment.
+
+Later, deep DPG (DDPG) (Lillicrap et al., 2015) improves DPG and has good sample efficiency but suffers from extreme brittleness and hyperparameter sensitivity. Therefore, it is difficult to extend
+
+DDPG to complex, high-dimensional tasks. To improve DDPG, twin delayed DDPG (TD3) (Fujimoto et al., 2018) adopts an actor-critic framework and considers the interaction between value update and function approximation error and in the policy. There are also some policy gradient (Sutton et al., 1999; Kohl & Stone, 2004; Khadka & Tumer, 2018) based algorithms that can be adapted for continuous tasks, such as proximal policy optimization (PPO) (Schulman et al., 2017) algorithms, asynchronous advantage actor-critic (A3C) (Stooke & Abbeel, 2018), and importance weighted actor-learner architecture (IMPALA) (Espeholt et al., 2018). PPO has the benefits of trust region policy optimization (Schulman et al., 2015), enabling multiple batches of data to be updated together. Therefore, it is simpler to implement, more general, and has lower sample complexity. Recently, phasic policy gradient (PPG) (Cobbe et al., 2021) is proposed to decouple the training between policy and value function while keeping their feature sharing, and PPG optimizes each objective with an appropriate level of sample reuse to improve sample efficiency. Most of these improved policy gradient methods can be classified as aiming at maximizing reward, so none of them are better suited for exploration tasks than CFlowNets.
+
+Furthermore, some maximum entropy (Pitis et al., 2020; Haarnoja et al., 2018a; Hazan et al., 2019; Yarats et al., 2021) based reinforcement learning algorithms can also be adapted for continuous tasks, such as soft actor-critic (SAC) (Haarnoja et al., 2018b). By maximizing the expected reward and entropy, the actor network of SAC can successfully complete tasks while acting as randomly as possible. The difference between CFlowNets and SAC is: 1) SAC selects actions by a Gaussian policy, which is less expressive than using a general unnormalized action p.d.f.  $F(s,a)$ ; 2) In the general case, SAC learns to be proportional to the long-term return, which generates the trajectory distribution satisfying  $p(\tau) \propto R(\tau)$  with  $R(\tau)$  is the return of  $\tau$ . CFlowNets considers all possible trajectories that lead to a terminal state  $s_f$ , and learn the policy to generate  $s_f$  with  $p(s_f) \propto R(s_f)$ .
+
+# 6 EXPERIMENTS
+
+To demonstrate the effectiveness of the proposed CFlowNets, we conduct experiments on several continuous control tasks with sparse rewards, including Point-Robot-Sparse, Reacher-Goal-Sparse, and Swimmer-Sparse. The visualization of these environments is shown in Figures 7, 8 and 9. Then we compare CFlowNets with a few state-of-the-art baseline RL algorithms, such as DDPG (Lillicrap et al., 2015), TD3 (Fujimoto et al., 2018), PPO (Schulman et al., 2017), and SAC (Haarnoja et al., 2018b). More implementation details are provided in Appendix D.
+
+Figure 2 illustrates the distributions of learned policies for CFlowNets and RL algorithms. All curves are max-min normalized. The gray curve is the ground truth of reward distribution generated by the agent's different actions when it goes to coordinates (7, 7), which indicates that the optimal action here is to go right or up. The red curve shows the flow network output of CFlowNets under different actions, indicating that CFlowNets have an excellent fitting ability to the reward. In contrast, other reinforcement learning algorithms have difficulty fitting the actual reward distribution well.
+
+![](images/88c4082c66c2b69f513c148e0eee5f6a51191cbee92d5ac54161ccf43b5dd9a4.jpg)  
+Figure 2: Reward distributions on Point-Robot-Sparse Task.
+
+Figures 3(a)-(c) show the number of valid-distinctive trajectories explored as training progresses in Point-Robot-Sparse,
+
+Reacher-Goal-Sparse, and Swimmer-Sparse environment, respectively. After a certain number of training epochs, 10000 trajectories are collected. A valid-distinctive trajectory is defined as a reward above a threshold  $\delta_r$  while the MSE between the trajectory and other trajectories is greater than another threshold  $\delta_{\mathrm{mse}}$ . That is, if the returns of both trajectories are high, but the two are close and the MSE is small, we consider it only one valid-distinctive exploration.  $\delta_r$  in Point-Robot-Sparse, Reacher-Goal-Sparse, and Swimmer-Sparse is set as 0.5, -0.2, 5.0, respectively.  $\delta_{\mathrm{mse}}$  in Point-Robot-Sparse, Reacher-Goal-Sparse, and Swimmer-Sparse is set as 0.02, 4.0, 1.0, respectively. As can be seen from the figure, DDPG, TD3 and PPO have the worst exploration ability, only one valid-distinctive trajectory is generated. SAC explores better at the beginning of training, and decreases as the training progresses and gradually converges. In contrast, the exploration ability of CFlowNets is very outstanding, the number of trajectories explored far exceeds other algorithms, and the exploration ability has been stable as the training progresses.
+
+![](images/6f6e7470a9c8508839cd0f13b9b9c4f9d66d53845b5ec125ca7c96832d24c564.jpg)  
+(a) Point-Robot-Sparse
+
+![](images/baaf85c5c3b815b4dad862a13ff0776c4a0e408b07a94a6654b9249823d2656e.jpg)  
+(b) Reacher-Goal-Sparse
+
+![](images/a7c38625307c88fa0d095cfd567793dc1da1e5f40a00fe7e94e3c7b44749cf98.jpg)  
+(c) Swimmer-Sparse
+
+![](images/78090654d04d1dc9e884e475ebda0099cd2641adf98197d5e88eda4525c6ba67.jpg)  
+(d) Point-Robot-Sparse
+
+![](images/fdc90788f3f3975cd4e33621f4310a9dd2c926f3f7da9b405f2a20fe9ebd7c69.jpg)  
+(e) Reacher-Goal-Sparse
+
+![](images/72c967ccca20de5c931c9d6a7be7002f95cff0410cabcc4e02fab99b7c3ba97a.jpg)  
+(f) Swimmer-Sparse  
+Figure 3: Comparison results of CFlowNets, DDPG, TD3, SAC and PPO on Point-Robot-Sparse, Reacher-Goal-Sparse, and Swimmer-Sparse tasks. Top: Number of valid-distinctive trajectories generated under 10000 explorations. Bottom: The average reward of different methods.
+
+Figures 3(d)-(f) indicate the rewards during the training process in Point-Robot-Sparse, Reacher-Goal-Sparse, and Swimmer-Sparse environment, respectively. The shaded region represents  $95\%$  confidence interval across 5 runs. Figure 3(d) and Figure 3(e) show that CFlowNets has the fastest and more stable upward trend, and the final reward is ahead of that of other algorithms by a large margin. In contrast, CFlowNets do not perform as well as other algorithms in Figure 3(f). Since the rewards in Point-Robot-Sparse and Reacher-Goal-Sparse are more evenly distributed, so these two tasks are more inclined to exploration. CFlowNets has better exploration ability and hence can converge stably. As for Swimmer-Sparse, its reward distribution is relatively steep, and sampling near the maximum reward can achieve faster convergence. It is reasonable for CFlowNets to perform worse than RL on this task in terms of reward. However, in this environment, CFN can still maintain a good exploration ability.
+
+# 7 CONCLUSION
+
+In this paper, we propose generative continuous flow networks to enhance exploration in continuous control tasks. The theoretical formulation of CFlowNets is first presented. Then, a training framework for CFlowNets is proposed, including the action selection process, the flow approximation algorithm, and the continuous flow matching loss function. Theoretical analysis shows that the error of the flow approximation decreases rapidly as the number of flow samples increases. Experimental results on continuous control tasks illustrate the performance advantages of CFlowNets compared to many reinforcement learning methods. Especially in the exploration ability, the effect of CFlowNets far exceeds other state-of-the-art reinforcement learning algorithms.
+
+Limitations: Similar to GFlowNets, CFlowNets aims to sample actions according to the flow network, rather than selecting actions with maximizing rewards. Therefore, CFlowNets are more suitable for exploration-biased tasks. It does not perform as well as reinforcement learning on tasks that aim to maximize reward. Of course, the purpose of CFlowNets is not to completely replace reinforcement learning, but as a supplement to reinforcement learning, giving a new option for continuous control tasks. Future work: Future work will be how to combine CFlowNets with DB and TB objective functions to improve training efficiency.
+
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+David Silver, Guy Lever, Nicolas Heess, Thomas Degris, Daan Wierstra, and Martin Riedmiller. Deterministic policy gradient algorithms. In International conference on machine learning, pp. 387-395. PMLR, 2014.  
+Adam Stooke and Pieter Abbeel. Accelerated methods for deep reinforcement learning. arXiv preprint arXiv:1803.02811, 2018.  
+Richard S Sutton and Andrew G Barto. Reinforcement learning: An introduction. MIT press, 2018.  
+Richard S Sutton, David McAllester, Satinder Singh, and Yishay Mansour. Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems, 12, 1999.  
+Roman Vershynin. High-dimensional probability: An introduction with applications in data science, volume 47. Cambridge university press, 2018.  
+Denis Yarats, Rob Fergus, Alessandro Lazaric, and Lerrel Pinto. Reinforcement learning with prototypical representations. In International Conference on Machine Learning, pp. 11920-11931. PMLR, 2021.
+
+Dinghuai Zhang, Ricky TQ Chen, Nikolay Malkin, and Yoshua Bengio. Unifying generative models with gflownets. arXiv preprint arXiv:2209.02606, 2022a.  
+Dinghuai Zhang, Nikolay Malkin, Zhen Liu, Alexandra Volokhova, Aaron Courville, and Yoshua Bengio. Generative flow networks for discrete probabilistic modeling. arXiv preprint arXiv:2202.01361, 2022b.
+
+# A DISCUSSIONS
+
+# A.1 WHY IS ASSUMPTION 1 NECESSARY AND REASONABLE?
+
+Necessity: For most environments, it is difficult to generate cycles when sampling a trajectory in a continuous space. Since  $\forall t$ ,  $\mu(\{s_0, \dots, s_t\}) = 0$  and  $\mu(\mathcal{A}) = \mu(\mathcal{A} \setminus \{s_0, \dots, s_t\})$ , that is, the probability of  $s_{t+1} \in \{s_0, \dots, s_t\}$  is very small. However, cycles often arise when certain environments have some special constraints. For example, a simple pendulum task (see Figure 4), the action is to control the pendulum to rotate from the previous position to the next position at a certain angle. For this task, it is difficult for a pendulum to rotate to exactly the same position in continuous space. However, if a wall is added to the task, the pendulum can easily go to the same position (see Figure 5), i.e., a cycle will occur. Therefore, we still need to add an acyclic assumption to make the theory and performance of CFlowNets guaranteed.
+
+Rationality: This assumption is reasonable because for many continuous environments it is difficult to form cycles in trajectories without special constraints. Even for tasks prone to form cycles, we can directly add time steps in the state space to satisfy this assumption.
+
+![](images/41cdffcc2ee6ccd6f146d03fc8760e1f44c900c905b678ffd5ca66c54ba9f0a8.jpg)  
+Figure 4: Pendulum. It is difficult for the state to be completely consistent in this continuous space.
+
+![](images/3c9de3916077bde3496570b6b47f8c0cb4fc1c1a3caf747df3f6b74a453cc353.jpg)  
+Figure 5: Pendulum-with-Wall. The state becomes consistent when reaching the wall.
+
+![](images/2a721ec52490e11f2d40cf480f02f1a83370c7f1e2335c6f4b6381e3b52d9123.jpg)  
+(a) for  $(s,a,a^{\prime})$
+
+![](images/d502dfa97e3a0e0d5032daea54122973896be40c64601f81342a5dbe0ac9366c.jpg)  
+(b) for  $(s,s^{\prime},a)$  
+Figure 6: Accumulated maximum Lipschitz constant of flow network  $F(s, a)$ .
+
+# A.2 WHY IS ASSUMPTION 2 NECESSARY AND REASONABLE?
+
+Necessity: This assumption is mainly used to guarantee the existence of flow-related integrals, and to ensure that Theorem 2 holds.
+
+Rationality: We justify this assumption based on simulations. As shown in Figure 6, we calculate  $\frac{|F(s,a) - F(s,a')|}{\|a - a'\|}$  and  $\frac{|F(s,a) - F(s',a)|}{\|s - s'\|}$  of each sample tuple  $(s,a,a')$  and  $(s,s',a)$  to analysis the Lipschitz constant, respectively. Their accumulated maximum Lipschitz constants are shown in Figures 6 (a) and (b), respectively. Clearly, there exists a finite Lipschitz constant for our flow network.
+
+In addition, Lipschitz continuous is a common assumption of neural networks, just some quick examples: Du et al. (2019); Jacot et al. (2018); Allen-Zhu et al. (2019); Alistarh et al. (2018) all use this assumption to prove the convergence of algorithms.
+
+# A.3 WHY IS ASSUMPTION 3 NECESSARY AND REASONABLE?
+
+Necessity: This assumption is used in Definition 2 and enables the retrieval neural network to fit the function  $g(s, a)$ . While there is a one-to-one correspondence between most environment state transitions and actions, there are still some special cases where, given a state pair  $(s, s')$ , there can be an infinite number of actions. For example: for Pendulum-with-Wall in Figure 5, after reaching the wall, continuing to increase the action will not continue to change the state  $s'$ . In addition, a special case of the translation action could be  $T(s, a) = s + a$  or using the special linear group, such that Definition 2 and 3 hold. The translation action is used to ensure that there is no Jacobian term in the continuous flow definition.
+
+Rationality: This assumption is a property of many environments and therefore reasonable. For environments that do not satisfy this assumption, we can try to satisfy this assumption by modifying the state to add more information. For example, we can add the duration of the action to the state space of Pendulum-with-Wall task in Figure 5. Even if the action increases after reaching the wall, the position information will not be changed, but the duration will increase, so that the state transition and the action will correspond one-to-one. The worst case is that we cannot change the environment to satisfy the assumption. At this time, we mainly need to solve the problem that the output of the retrieval neural network  $G$  cannot be multiple when the input is fixed. One of our conjectures is that maybe we can alleviate this problem by adding some small random noise to the input, but this idea has not been tested.
+
+# B PROOFS
+
+# B.1 PROOF OF THEOREM 1
+
+Theorem 1. (Continuous Flow Matching Condition). Consider a non-negative function  $\hat{F}(s, a)$  taking a state  $s \in S$  and an action  $a \in \mathcal{A}$  as inputs. Then we have  $\hat{F}$  corresponds to a flow if and only if the following continuous flow matching conditions are satisfied:
+
+$$
+\begin{array}{l} \forall s ^ {\prime} > s _ {0}, \hat {F} (s ^ {\prime}) = \int_ {s \in \mathcal {P} (s ^ {\prime})} \hat {F} (s \to s ^ {\prime}) \mathrm {d} s = \int_ {s: T (s, a) = s ^ {\prime}} \hat {F} (s, a: s \to s ^ {\prime}) \mathrm {d} s \\ \forall s ^ {\prime} <   s _ {f}, \hat {F} (s ^ {\prime}) = \int_ {s ^ {\prime \prime} \in \mathcal {C} (s ^ {\prime})} \hat {F} (s ^ {\prime} \rightarrow s ^ {\prime \prime}) \mathrm {d} s ^ {\prime \prime} = \int_ {a \in \mathcal {A}} \hat {F} (s ^ {\prime}, a) \mathrm {d} a. \\ \end{array}
+$$
+
+Furthermore,  $\hat{F}$  uniquely defines a Markovian flow  $F$  matching  $\hat{F}$  such that
+
+$$
+F (\tau) = \frac {\prod_ {t = 1} ^ {n + 1} \hat {F} \left(s _ {t - 1} \rightarrow s _ {t}\right)}{\prod_ {t = 1} ^ {n} \hat {F} \left(s _ {t}\right)}. \tag {23}
+$$
+
+Proof. The proof is an extension of that of Proposition 19 in Bengio et al. (2021b) to the continuous case. We first prove the necessity part of the proof. Given a flow network, for non-initial and non-final nodes on a trajectory, the set of complete trajectories passing through state  $s'$  is the union of the sets of trajectories going through  $s \to s'$  for all  $s \in \mathcal{P}(s')$ , and also is the union of the sets of trajectories going through  $s' \to s''$  for all  $s'' \in \mathcal{C}(s')$ , i.e.,
+
+$$
+\{\tau \in \mathcal {T}: s ^ {\prime} \in \tau \} = \bigcup_ {s \in \mathcal {P} (s ^ {\prime})} \{\tau \in \mathcal {T}: s \to s ^ {\prime} \in \tau \} = \bigcup_ {s ^ {\prime \prime} \in \mathcal {C} (s ^ {\prime})} \{\tau \in \mathcal {T}: s ^ {\prime} \to s ^ {\prime \prime} \in \tau \}.
+$$
+
+Then we have
+
+$$
+F (s ^ {\prime}) = \int_ {\tau : s ^ {\prime} \in \tau} F (\tau) \mathrm {d} \tau = \int_ {s \in \mathcal {P} (s ^ {\prime})} \int_ {\tau : s \to s ^ {\prime} \in \tau} F (\tau) \mathrm {d} \tau \mathrm {d} s = \int_ {s \in \mathcal {P} (s ^ {\prime})} F (s \to s ^ {\prime}) \mathrm {d} s,
+$$
+
+and
+
+$$
+F (s ^ {\prime}) = \int_ {\tau : s ^ {\prime} \in \tau} F (\tau) \mathrm {d} \tau = \int_ {s ^ {\prime \prime} \in \mathcal {C} (s ^ {\prime})} \int_ {\tau : s ^ {\prime} \to s ^ {\prime \prime} \in \tau} F (\tau) \mathrm {d} \tau \mathrm {d} s ^ {\prime \prime} = \int_ {s ^ {\prime \prime} \in \mathcal {C} (s ^ {\prime})} F (s ^ {\prime} \to s ^ {\prime \prime}) \mathrm {d} s ^ {\prime \prime}.
+$$
+
+Then we finish the necessity part. Next we show sufficiency. Let  $\hat{Z} = \hat{F}(s_0)$  be the partition function and  $\hat{P}_F$  be the forward probability function, then there exists a unique Markovian flow  $F$  with forward transition probability function  $P_F = \hat{P}_F$  and partition function  $Z$  according to Proposition 18 in Bengio et al. (2021b), and such that
+
+$$
+F (\tau) = \hat {Z} \prod_ {t = 1} ^ {n + 1} \hat {P} _ {F} \left(s _ {t} \mid s _ {t - 1}\right) = \frac {\prod_ {t = 1} ^ {n + 1} \hat {F} \left(s _ {t - 1} \rightarrow s _ {t}\right)}{\prod_ {t = 1} ^ {n} \hat {F} \left(s _ {t}\right)}, \tag {24}
+$$
+
+where  $s_{n+1} = s_f$ . In addition, according to Lemma 1, we have
+
+$$
+\int_ {\tau \in \mathcal {T} _ {0, s}} P _ {B} (\tau) \mathrm {d} \tau = \int_ {\tau \in \mathcal {T} _ {0, s}} \prod_ {s _ {t} \to s _ {t + 1} \in \tau} P _ {B} (s _ {t} | s _ {t + 1}) \mathrm {d} \tau = 1.
+$$
+
+Lemma 1. Considering a continuous task  $(S, \mathcal{A})$ , where we have the transition probabilities defined in (8) and (9). Define  $\mathcal{T}_{s,f}$  and  $\mathcal{T}_{0,s}$  as the set of trajectories sampled from a continuous task starting in  $s$  and ending in  $s_f$ ; and starting in  $s_0$  and ending in  $s$ , respectively. Then we have
+
+$$
+\forall s \in \mathcal {S} \backslash \{s _ {f} \}, \int_ {\tau \in \mathcal {T} _ {s, f}} P _ {F} (\tau) \mathrm {d} \tau = 1 \tag {25}
+$$
+
+$$
+\forall s \in \mathcal {S} \backslash \{s _ {0} \}, \int_ {\tau \in \mathcal {T} _ {0, s}} P _ {B} (\tau) \mathrm {d} \tau = 1. \tag {26}
+$$
+
+Thus, we have for  $s' \neq s_0$ :
+
+$$
+\begin{array}{l} F(s^{\prime}) = \hat{Z}\int_{\tau \in \mathcal{T}_{0,s^{\prime}}}\prod_{(s_{t}\to s_{t + 1})\in \tau}\hat{P}_{F}(s_{t + 1}|s_{t})\mathrm{d}\tau \\ = \hat {Z} \frac {\hat {F} \left(s ^ {\prime}\right)}{\hat {F} \left(s _ {0}\right)} \int_ {\tau \in \mathcal {T} _ {0, s ^ {\prime}}} \prod_ {\left(s _ {t} \rightarrow s _ {t + 1}\right) \in \tau} \hat {P} _ {B} \left(s _ {t} \mid s _ {t + 1}\right) \mathrm {d} \tau = \hat {F} \left(s ^ {\prime}\right). \tag {27} \\ \end{array}
+$$
+
+Combine (27) with  $P_F = \hat{P}_F$  yields  $\forall s \to s' \in \mathcal{A}$ ,  $F(s \to s') = \hat{F}(s \to s')$ . Finally, according to Proposition 16 in Bengio et al. (2021b), for any Markovian flow  $F'$  matching  $\hat{F}$  on states and edges, we have  $F'(\tau) = F(\tau)$ , which shows the uniqueness property. Then we complete the proof.
+
+# B.2 PROOF OF THEOREM 2
+
+Theorem 2. Let  $\{a_k\}_{k=1}^K$  be sampled independently and uniformly from the continuous action space  $\mathcal{A}$ . Assume  $G_{\phi^*}$  can optimally output the actual state  $s_t$  with  $(s_{t+1}, a_t)$ . For any bounded continuous action  $a \in \mathcal{A}$  and any state  $s_t \in S$ , we have
+
+$$
+\mathbb {P} \left(\left| \frac {\mu (\mathcal {A})}{K} \sum_ {k = 1} ^ {K} F \left(s _ {t}, a _ {k}\right) - \int_ {a \in \mathcal {A}} F \left(s _ {t}, a\right) \mathrm {d} a \right| \geq t\right) \leq 2 \exp \left(- \frac {K t ^ {2}}{2 \left(L \mu (\mathcal {A}) \operatorname {d i a m} (\mathcal {A})\right) ^ {2}}\right) \tag {28}
+$$
+
+and
+
+$$
+\begin{array}{l} \mathbb {P} \left(\left| \frac {\mu (\mathcal {A})}{K} \sum_ {k = 1} ^ {K} F \left(G _ {\phi^ {*}} \left(s _ {t}, a _ {k}\right), a _ {k}\right) - \int_ {a: T (s, a) = s _ {t}} F (s, a) \mathrm {d} a \right| \geq t\right) \\ \leq 2 \exp \left(- \frac {K t ^ {2}}{2 \left(L \mu (\mathcal {A}) \left(\operatorname {d i a m} (\mathcal {A}) + \operatorname {d i a m} (\mathcal {S})\right)\right) ^ {2}}\right), \tag {29} \\ \end{array}
+$$
+
+where  $L$  is the Lipschitz constant,  $\mathrm{diam}(\mathcal{A})$  denotes the diameter of the action space  $\mathcal{A}$  and  $\mathrm{diam}(S)$  denotes the diameter of the state space  $S$ .
+
+Proof. First, we show that the expectation of sample outflow is the true outflow and the expectation of sample inflow is the true inflow in Lemma 2.
+
+Lemma 2. Let  $\{a_k\}_{k=1}^K$  be sampled independently and uniformly from the continuous action space  $\mathcal{A}$ . Assume  $G_{\phi^*}$  can optimally output the actual state  $s_t$  with  $(s_{t+1}, a_t)$ . Then for any state  $s_t \in \mathcal{S}$ , we have
+
+$$
+\mathbb {E} \left[ \frac {\mu (\mathcal {A})}{K} \sum_ {k = 1} ^ {K} F \left(s _ {t}, a _ {k}\right) \right] = \int_ {a \in \mathcal {A}} F \left(s _ {t}, a\right) \mathrm {d} a \tag {30}
+$$
+
+and
+
+$$
+\mathbb {E} \left[ \frac {\mu (\mathcal {A})}{K} \sum_ {k = 1} ^ {K} F \left(G _ {\phi^ {*}} \left(s _ {t}, a _ {k}\right), a _ {k}\right) \right] = \int_ {a: T (s, a) = s _ {t}} F (s, a) \mathrm {d} a, \tag {31}
+$$
+
+where  $s = g(s_{t},a)$
+
+Then, define the following terms:
+
+$$
+\Gamma_ {k} = \frac {\mu (\mathcal {A})}{K} F \left(s _ {t}, a _ {k}\right) - \frac {1}{K} \int_ {a \in \mathcal {A}} F \left(s _ {t}, a\right) \mathrm {d} a = \frac {1}{K} \int_ {a \in \mathcal {A}} \left[ F \left(s _ {t}, a _ {k}\right) - F \left(s _ {t}, a\right) \right] \mathrm {d} a \tag {32}
+$$
+
+and
+
+$$
+\begin{array}{l} \Lambda_ {k} = \frac {\mu (\mathcal {A})}{K} F \left(G _ {\phi^ {\star}} \left(s _ {t}, a _ {k}\right), a _ {k}\right) - \frac {1}{K} \int_ {a: T (s, a) = s _ {t}} F (s, a) \mathrm {d} a (33) \\ = \frac {1}{K} \int_ {a: T (s, a) = s _ {t}} \left[ F \left(G _ {\phi^ {\star}} \left(s _ {t}, a _ {k}\right), a _ {k}\right) - F (s, a) \right] \mathrm {d} a, (34) \\ \end{array}
+$$
+
+where  $s = g(s_{t},a)$ .
+
+Note that the variables  $\{\Gamma_k\}_{k=1}^K$  are independent and  $\mathbb{E}[\Gamma_k] = 0, k = 1, \ldots, K$  according to Lemma 2. So the following equations hold
+
+$$
+\mathbb {P} \left(\left| \frac {\mu (\mathcal {A})}{K} \sum_ {k = 1} ^ {K} F \left(s _ {t}, a _ {k}\right) - \int_ {a \in \mathcal {A}} F \left(s _ {t}, a\right) \mathrm {d} a \right| \geq t\right) = \mathbb {P} \left(\left| \sum_ {k = 1} ^ {K} \Gamma_ {k} \right| \geq t\right) \tag {35}
+$$
+
+and
+
+$$
+\mathbb {P} \left(\left| \frac {\mu (\mathcal {A})}{K} \sum_ {k = 1} ^ {K} F \left(G _ {\phi^ {\star}} \left(s _ {t}, a _ {k}\right), a _ {k}\right) - \int_ {a: T (s, a) = s _ {t}} F (s, a) d a \right| \geq t\right) = \mathbb {P} \left(\left| \sum_ {k = 1} ^ {K} \Lambda_ {k} \right| \geq t\right). \tag {36}
+$$
+
+Since  $F(s, a)$  is a Lipschitz function, we have
+
+$$
+\begin{array}{l} \left| \Gamma_ {k} \right| \leq \frac {1}{K} \int_ {a \in \mathcal {A}} \left| F (s _ {t}, a _ {k}) - F (s _ {t}, a) \right| d a \\ \leq \frac {L}{K} \int_ {a \in \mathcal {A}} | | a _ {k} - a | | \mathrm {d} a \leq \frac {L \mu (\mathcal {A}) \operatorname {d i a m} (\mathcal {A})}{K}. \tag {37} \\ \end{array}
+$$
+
+Together with Assumption 3, that is, for any pair of  $(s,a)$  satisfying  $T(s,a) = s_t$ ,  $a$  is unique if we fix  $s$ , we have
+
+$$
+\begin{array}{l} \left| \Lambda_ {k} \right| \leq \frac {1}{K} \int_ {a: T (s, a) = s _ {t}} \left| F \left(G _ {\phi^ {\star}} \left(s _ {t}, a _ {k}\right), a _ {k}\right) - F (s, a) \right| d a \\ \leq \frac {1}{K} \int_ {a: T (s, a) = s _ {t}} | F \left(G _ {\phi^ {\star}} \left(s _ {t}, a _ {k}\right), a _ {k}\right) - F (s, a _ {k}) + F (s, a _ {k}) - F (s, a) | \mathrm {d} a \\ \leq \frac {1}{K} \int_ {a: T (s, a) = s _ {t}} L | | G _ {\phi^ {*}} \left(s _ {t}, a _ {k}\right) - s | | + L | | a _ {k} - a | | \mathrm {d} a \\ \leq \frac {L \mu (\mathcal {A}) \left(\operatorname {d i a m} (\mathcal {A}) + \operatorname {d i a m} (\mathcal {S})\right)}{K}. \tag {38} \\ \end{array}
+$$
+
+Lemma 3 (Hoeffding's inequality, Vershynin (2018)). Let  $x_{1}, \ldots, x_{K}$  be independent random variables. Assume the variables  $\{x_{k}\}_{k=1}^{K}$  are bounded in the interval  $[T_{l}, T_{r}]$ . Then for any  $t > 0$ , we have
+
+$$
+\mathbb {P} \left(\left| \sum_ {k = 1} ^ {K} \left(x _ {k} - \mathbb {E} x _ {k}\right) \right| \geq t\right) \leq 2 \exp \left(- \frac {2 t ^ {2}}{K \left(T _ {r} - T _ {l}\right) ^ {2}}\right). \tag {39}
+$$
+
+Incorporating  $T_r = \frac{L}{K} \mu(\mathcal{A}) \mathrm{diam}(\mathcal{A})$  and  $T_l = -\frac{L}{K} \mu(\mathcal{A}) \mathrm{diam}(\mathcal{A})$  in Lemma 3 with (37), and incorporating  $T_r = \frac{L}{K} \mu(\mathcal{A})(\mathrm{diam}(\mathcal{A}) + \mathrm{diam}(\mathcal{S}))$  and  $T_l = -\frac{L}{K} \mu(\mathcal{A})(\mathrm{diam}(\mathcal{A}) + \mathrm{diam}(\mathcal{S}))$  in Lemma 3 with (38), we complete the proof.
+
+# B.3 PROOF OF LEMMA 1
+
+Lemma 1. Considering a continuous task  $(S, \mathcal{A})$ , where we have the transition probabilities defined in (8) and (9). Define  $\mathcal{T}_{s,f}$  and  $\mathcal{T}_{0,s}$  as the set of trajectories sampled from a continuous task starting in  $s$  and ending in  $s_f$ ; and starting in  $s_0$  and ending in  $s$ , respectively. Then we have
+
+$$
+\forall s \in \mathcal {S} \backslash \{s _ {f} \}, \int_ {\tau \in \mathcal {T} _ {s, f}} P _ {F} (\tau) \mathrm {d} \tau = 1 \tag {40}
+$$
+
+$$
+\forall s \in \mathcal {S} \backslash \left\{s _ {0} \right\}, \int_ {\tau \in \mathcal {T} _ {0, s}} P _ {B} (\tau) \mathrm {d} \tau = 1. \tag {41}
+$$
+
+Proof. We show by strong induction that (40) holds, mainly following the proof of Lemma 5 in Bengio et al. (2021b), and then extending to (41) is trivial. Define  $d$  as the maximum trajectory length in  $\mathcal{T}_{s,f}, s \neq s_f$ , we have:
+
+Base cases: If  $d = 1$ , then
+
+$$
+\int_ {\tau \in \mathcal {T} _ {s, f}} P _ {F} (\tau) \mathrm {d} \tau = P _ {F} (s \rightarrow s _ {f}) = 1
+$$
+
+holds by noting  $\mathcal{T}_{s,f} = \{(s\to s_f)\}$
+
+Induction steps: Consider  $d > 1$ , by noting (12) we have
+
+$$
+\begin{array}{l} \int_ {\tau \in \mathcal {T} _ {s, f}} P _ {F} (\tau) \mathrm {d} \tau = \int_ {s ^ {\prime} \in \mathcal {C} (s)} \int_ {\tau \in \mathcal {T} _ {s \rightarrow s ^ {\prime}, f}} P _ {F} (\tau) \mathrm {d} \tau \mathrm {d} s ^ {\prime} (42) \\ = \int_ {s ^ {\prime} \in \mathcal {C} (s)} \int_ {\tau \in \mathcal {T} _ {s ^ {\prime}, f}} P _ {F} \left(s ^ {\prime} \mid s\right) P _ {F} (\tau) \mathrm {d} \tau \mathrm {d} s ^ {\prime} (43) \\ = \int_ {s ^ {\prime} \in \mathcal {C} (s)} P _ {F} \left(s ^ {\prime} \mid s\right) \mathrm {d} s ^ {\prime} \int_ {\tau \in \mathcal {T} _ {s ^ {\prime}, f}} P _ {F} (\tau) \mathrm {d} \tau = 1, (44) \\ \end{array}
+$$
+
+where the last equality follows by the induction hypotheses.
+
+# B.4 PROOF OF LEMMA 2
+
+Lemma 2. Let  $\{a_k\}_{k=1}^K$  be sampled independently and uniformly from the continuous action space  $\mathcal{A}$ . Assume  $G_{\phi^*}$  can optimally output the actual state  $s_t$  with  $(s_{t+1}, a_t)$ . Then for any state  $s_t \in S$ , we have
+
+$$
+\mathbb {E} \left[ \frac {\mu (\mathcal {A})}{K} \sum_ {k = 1} ^ {K} F \left(s _ {t}, a _ {k}\right) \right] = \int_ {a \in \mathcal {A}} F \left(s _ {t}, a\right) \mathrm {d} a \tag {45}
+$$
+
+and
+
+$$
+\mathbb {E} \left[ \frac {\mu (\mathcal {A})}{K} \sum_ {k = 1} ^ {K} F \left(G _ {\phi^ {*}} \left(s _ {t}, a _ {k}\right), a _ {k}\right) \right] = \int_ {a: T (s, a) = s _ {t}} F (s, a) \mathrm {d} s, \tag {46}
+$$
+
+where  $s = g(s_{t},a)$ .
+
+Proof. Since  $\{a_k\}_{k=1}^K$  is sampled independently and uniformly from the continuous action space  $\mathcal{A}$ , then we have
+
+$$
+\mathbb {E} \left[ F \left(s _ {t}, a _ {k}\right) \right] = \frac {1}{\mu (\mathcal {A})} \int_ {a \in \mathcal {A}} F \left(s _ {t}, a\right) \mathrm {d} a. \tag {47}
+$$
+
+Therefore, we obtain
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \frac {\mu (\mathcal {A})}{K} \sum_ {k = 1} ^ {K} F \left(s _ {t}, a _ {k}\right) \right] = \frac {\mu (\mathcal {A})}{K} \sum_ {k = 1} ^ {K} \mathbb {E} [ F \left(s _ {t}, a _ {k}\right) ] (48) \\ = \int_ {a \in \mathcal {A}} F \left(s _ {t}, a\right) \mathrm {d} a. (49) \\ \end{array}
+$$
+
+Since Assumption 3 holds, for any pair of  $(s,a)$  satisfying  $T(s,a) = s_t$ ,  $a$  is unique if we fix  $s$ , we have
+
+$$
+\mathbb {E} \left[ F \left(G _ {\phi^ {\star}} \left(s _ {t}, a _ {k}\right), a _ {k}\right) \right] = \frac {1}{\mu (\mathcal {A})} \int_ {a: T (s, a) = s _ {t}} F (s, a) \mathrm {d} a,
+$$
+
+where  $s = g(s_{t},a)$ .
+
+Therefore, we get
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \frac {\mu (\mathcal {A})}{K} \sum_ {k = 1} ^ {K} F \left(G _ {\phi^ {\star}} \left(s _ {t}, a _ {k}\right), a _ {k}\right) \right] = \frac {\mu (\mathcal {A})}{K} \sum_ {k = 1} ^ {K} \mathbb {E} \left[ F \left(G _ {\phi^ {\star}} \left(s _ {t}, a _ {k}\right), a _ {k}\right) \right] \\ = \int_ {a: T (s, a) = s _ {t}} F (s, a) \mathrm {d} a. \\ \end{array}
+$$
+
+Then we complete the proof.
+
+![](images/774b8e19cd985421967fe99beecd98536c52b89ed59309c612faa36afb181a37.jpg)
+
+# C PSEUDOCODE OF CFLOWNETS
+
+For clarity, we show pseudocode for CFlowNets in Algorithm 1.
+
+Algorithm 1 Generative Continuous Flow Networks (CFlowNets) Algorithm  
+Initialize: Flow network  $\theta$ ; a pretrained retrieval network  $G_{\phi}$ ; and empty buffer  $\mathcal{D}$  and  $\mathcal{P}$  
+1: repeat  
+2: Set  $t = 0$ ,  $s = s_0$   
+3: while  $s \neq \text{terminal}$  and  $t < T$  do  
+4: Uniformly sample  $M$  actions  $\{a_i\}_{i=1}^M$  from action space  $\mathcal{A}$   
+5: Compute edge flow  $F_{\theta}(s_t, a_i)$  for each  $a_i \in \{a_i\}_{i=1}^M$  to generate  $\mathcal{P}$   
+6: Sample  $a_t \sim \mathcal{P}$  and execute  $a_t$  in the environment to obtain  $r_{t+1}$  and  $s_{t+1}$   
+7:  $t = t + 1$   
+8: end while  
+9: Store episodes  $\{(s_t, a_t, r_t, s_{t+1})\}_{t=1}^T$  in replay buffer  $\mathcal{D}$   
+10: [Optional] Fine-tuning retrieval network  $G_{\phi}$  based on  $\mathcal{D}$   
+11: Sample a random minibatch  $\mathcal{B}$  of episodes from  $\mathcal{D}$   
+12: Uniformly sample  $K$  actions  $\{a_k\}_{k=1}^K$  from action space  $\mathcal{A}$  for each state in  $\mathcal{B}$   
+13: Compute parent states according to  $\{G_{\phi}(s, a_k)\}_{k=1}^K$  for each state in  $\mathcal{B}$   
+14: Inflows:  
+15: log[ε + ∑k=1K exp  $F_{\theta}^{\log}(G_{\phi}(s_t, a_k), a_k)$ ]  
+16: Outflows or reward:  
+17: log[ε + λR(s_t) + ∑k=1K exp  $F_{\theta}^{\log}(s_t, a_k)$ ]  
+18: Update flow network  $F_{\theta}$  according to (20)  
+19: until convergence
+
+![](images/d2c9b2dcab20d71172ecb40fde386a29367d5d89e3f853dbd7dbd6730a780a23.jpg)  
+(a)
+
+![](images/ff57c77a22320b3c43f5d95b65529838fdc6060deb22175a3f9ad661961b9442.jpg)  
+(b)
+
+![](images/297aa317069fb863e2830e075b7c9895625b6802914a0c5ca817aa7019c51e39.jpg)  
+(c)
+
+![](images/7e83c4a396c445609de23139ef43cff2f739f6c1b63cad95c6daab3721f5d5cc.jpg)  
+(d)
+
+![](images/1b2bcd58d1a2a21a82e22c890802eacf5a1f78e7dcabe03869c6b881d613f785.jpg)  
+Figure 7: Visualization of Point-Robot-Sparse task.  
+(a)  
+Figure 8: Visualization of Reacher-Goal-Sparse task.
+
+![](images/85f3d68482d751b59f6e58917a5a12da6d0d0488ff4019316218daeb1b6db02a.jpg)  
+(b)
+
+![](images/b2816f36a439813496941ae7cf88f122496071cbdc19dbef00294376308d55f0.jpg)  
+(c)
+
+![](images/5c0654fed6a50c1da0ea4d6934908464183b189fdcb93114b8592b065fc587f8.jpg)  
+(d)
+
+![](images/28290f4abecea995f535086559e5449d8ef4ebe4172cf92d673b23b5983500e5.jpg)  
+(a)
+
+![](images/c6df41727222d09eb4dc41469f68673f5cf4bf36947f030da1d65b6c2873d505.jpg)  
+(b)  
+Figure 9: Visualization of Swimmer-Sparse task.
+
+![](images/13a2d16ff75260f4dfc49de1362f42e4211de7287452a2936ce7c661c1827af8.jpg)  
+(c)
+
+![](images/1098430c1c181103195fe4cbf3dba30bfa9a7697d15c5d8628aa00fc6f0afc05.jpg)  
+(d)
+
+# D ADDITIONAL EXPERIMENTS
+
+# D.1 VISUALIZATION OF ENVIRONMENT
+
+As shown in Figures 7, 8 and 9, we provide the visualization of Point-Robot-Sparse, Reacher-Goal-Sparse, and Swimmer-Sparse tasks. In Point-Robot-Sparse, the goal of the agent is to navigate two different goals. The agent starts at the starting coordinate  $(0,0)$  and moves towards the target coordinate one step at a time. The environment has two target coordinates  $(5,10)$  and  $(10,5)$  with a maximum episode length of 12, and the environment returns a reward only when the last step is reached. Rewards are issued by measuring the distance between the agent's current position and the target node, and the closer the distance, the greater the reward. Each time the agent can take a step from any angle to the upper right.
+
+Both Reacher-Goal-Sparse and Swimmer-Sparse are adapted from OpenAI Gym's MuJoCo environment. In the Reacher-Goal-Sparse, "Reacher" is a two-jointed robotic arm. The goal of the agent is to reach a randomly generated target by moving the robots end effector. Figure 8 shows the movement process of the robotic arm. By adjusting the torque applied at the hinge joint, the end effector can gradually approach the target. In the Swimmer-Spars, the "swimmer" is suspended in a
+
+two-dimensional pool, and the goal is to move as fast as possible towards the right or left. Figure 9 shows the shape change process of the robot during motion. By taking the action that applies torque on the rotors and using the fluids friction, the robot can swim faster. We set the maximum number of steps to 50 for these two environments. For Reacher-Goal-Sparse, when the last step is reached, the environment returns a reward that measures how far the agent is from the randomly generated target. The closer the agent is to the target, the greater the reward. For Swimmer-Sparse, the farther to the left or right from the starting point, the greater the reward returned.
+
+# D.2 ADDITIONAL ANALYSIS
+
+Figure 10 shows that the average reward and reward distribution of different algorithms on the Point-Robot-OneGoal-Sparse task, where an agent needs to navigate to a specific location. Figure 10 (a) indicates that CFLowNets can obtain the highest average return compared to other RL-based algorithms. In Figure 10 (b), all algorithms are able to fit the reward distribution well under the one goal setting, while CFLowNets can achieve better. Note that RL algorithms can also learn the reward distribution in this task, since maximizing the reward is the optimal policy in the case of a single objective, and the policy is not difficult to learn.
+
+In Figure 11, we provide the action reward distribution of different algorithms with 2e4 total timesteps on Point-Robot-Sparse with Point (4,8), Point (8,4) and Point (7,7), respectively. Note that unlike Figure 2, where the total number of timesteps is 1e5, here we show the result with 2e4 total timesteps since we found DDPG is overfit after 1e5 timesteps in this task. Therefore we show the results without overfitting for a fairer comparison. We can see that no matter at which point, the policy of CFlowNets can better match the real reward distribution. For example, at points (4,8) and (8,4), CFlowNet tends to choose actions that guide the agent towards (5, 10) and (10, 5), respectively. For a location between two goals (point (7,7)), there are two directions that allow the agent to reach goals with high rewards. In contrast, the policy learned by RL algorithms can only occasion-
+
+![](images/29856523316f7df79c772efd5b4de42a50970b2472210c68ae4cf8da0414d375.jpg)  
+(a) Point-Robot-OneGoal-Sparse
+
+![](images/0a6032b3e4dcc44a24baef27c6d652b6236bd04cd00f75356abb677ade79625f.jpg)  
+(b) Point-Robot-OneGoal-Sparse
+
+![](images/9a5855b2513563d0fe1a3986aae58224c29f34b4ac0dfa01abf1473456f54e50.jpg)  
+Figure 10: The average reward and reward distributions of CFlowNets, DDPG, TD3, SAC and PPO on Point-Robot-OneGoal-Sparse task.  
+(a) Point (4,8)  
+Figure 11: The reward distributions of different points on Point-Robot-Sparse task.
+
+![](images/e1e30151c90cf47443c3ab916b7356267271261bc075339924bb276a48d69a48.jpg)  
+(b) Point (8,4)
+
+![](images/0877ac26547be0f7600fba725fd1b33b591c74c99db25bb8339bc71198e362ad.jpg)  
+(c) Point (7,7)
+
+![](images/17f2fc6243050e65599e9dd4ba01cdeda79b73b04e6e72445b21e7eb946eeb8b.jpg)  
+(a) DDPG
+
+![](images/8731fae76cb562f779c00d42e56b7e3f13450333843a99f9d8bc99744e1cfcbe.jpg)  
+(b) TD3
+
+![](images/e46390f0f4f41dd5e426aa8b47d3938ca911f9ae973c8edc730cc5a26ac82d48.jpg)  
+(c) SAC  
+Figure 12: Sampled trajectories on Point-Robot-OneGoal-Sparse task.
+
+![](images/49a99962d8e95cd2b90ffa451e3f68be07c06d471658f2ba9d69a3ee549fcc62.jpg)  
+(d) PPO
+
+![](images/fa0623e8e565b4bb97780615ff3936814364dbf487124d1caa43f591340f8a95.jpg)  
+(e) CFlowNets
+
+![](images/8216ea1cced88583389bde567efa192a8ac9167f02c397e109a22505dcaf5829.jpg)  
+(a) DDPG  
+Figure 13: Sampled trajectories on Point-Robot-Sparse task.
+
+![](images/9ec6089ca796334e9cfddeb1bf55c6d00b34a123c04ab64e936b170fde4258e0.jpg)  
+(b) TD3
+
+![](images/39cd42e7c0b7ca8e545570f6be6a613d59517f24cb37b799a492fff8bbfade36.jpg)  
+(c) SAC
+
+![](images/620c85d895429b5fe8548fd2118061cf94cf1e96da853098ffd3d17a5e981c27.jpg)  
+(d) PPO
+
+![](images/b7e0e9909e78e99298bc65d341ae1650f0a40ad33e0e9908181b4ac6a1fd7051.jpg)  
+(e) CFlowNets
+
+ally match the true reward distribution of a certain point, and cannot stably match every point. This also shows that the policies learned by RL algorithms is relatively simple. CFlowNets learn more diverse policies for agents to reach different goals with high rewards, while other methods usually find one goal instead of all potentially high reward locations.
+
+Figure 12 and Figure 13 show the results of trajectories visualization produced by different algorithms. In the Point-Robot-OneGoal-Sparse task, the trajectories of DDPG, TD3, and PPO are single, while SAC can select actions from the policy probability distribution, so different trajectories can be obtained. In contrast, CFlowNets found more diverse trajectories and also found the highest reward goal (thickened red trajectory), which means that CFlowNets can better explore the region near the goal. In the Point-Robot-Sparse task, the RL-based algorithms seek only one goal. However, CFlowNets can find all goals.
+
+![](images/399304a68d2c1fb79d1cb09e42809f63a0edca4c893f6210399ff74b0d0a8ef5.jpg)  
+Figure 14: Reward distributions on Point-Robot-Sparse Task.
+
+It is worth noting that in Figure 13 (e), the density of CFlowNets sampling trajectories is not as dense as in
+
+Figure 12 (e) near the maximum reward. Rather, it is denser on the diagonal. This is because in most positions, the action probability of choosing to go up and to the right is relatively high, so it is easier to go to the diagonal direction in combination. In addition, the reward on the line between two goals is not small. When sampling according to the output of the flow model as a probability, many trajectories themselves are more likely to reach the diagonal. Figure 14 shows the true reward distribution of Point-Robot-Sparse, where the reward is higher in the area near two goals and the line between two goals.
+
+# D.3 EXPERIMENT RESULTS ON HIGHWAY-PARKING-Sparse
+
+We evaluate the performance of CFlowNets on Highway-Parking-Sparse, which is an ego-vehicle control task. As shown in Figure 15, the goal is to make the ego-vehicle park in a given space with the appropriate orientation by adjusting its controller. The dimension of the vehicle observation is 18, consisting of the distance between the vehicle and parking, the vehicle speed, the triangular heading information, the goal the agent should attempt to achieve, and the goal that it currently
+
+![](images/ef884080d26bd224bbcf57b5b1ca0347667cfba0476b7df2dbf3a7a218f27443.jpg)  
+(a)
+
+![](images/4bbfbd08bb70eb058bbb3870bd67868c81c30ef32ce27a7f4f63691b399d3334.jpg)  
+(b)
+
+![](images/f2d2e881c443d9bf95d9ddeaa0c44271fb0308d42a8d7aa71a4c3f11f632d7d4.jpg)  
+(c)
+
+![](images/c2cae67dde3e98cb340c79e368905aafe51f2c4398852f74653e5cc36c1b8f61.jpg)  
+(d)
+
+![](images/9c14d1c69b657a2d3011a74b9b58a8d7044b5f52dbdad35f06131be67403e0c4.jpg)  
+(a) Average Reward
+
+![](images/cb0b6badbffad835ee8097c9c15737edaf4518e6e104c6ef93aed477a3d28fb2.jpg)  
+Figure 15: Visualization of Highway-Parking-Sparse task.  
+(b) Number of Distinctive Trajectories  
+Figure 16: The average reward and number of valid-distinctive trajectories generated under 10000 explorations of CFlowNets, DDPG, TD3, and SAC on Highway-Parking-Sparse.
+
+achieves. The action space includes control over the throttle and steering angle, and the reward function is set as the distance between the ego-vehicle and parking. Figure 16 shows the average reward and the number of valid-distinctive trajectories explored as training progresses of different algorithms, which illustrates that the performance of CFlowNets is more promising than other RL-based algorithms. Even for higher-dimensional continuous tasks, CFlowNets have very competitive reward results (outperforming DDPG, TD3, and SAC), while achieving much better exploration performance than RL-based algorithms.
+
+# D.4 BASELINES
+
+We compare our proposed CFlowNets to the following baselines:
+
+- Deep Deterministic Policy Gradient (DDPG) (Lillicrap et al., 2015). https://github.com/sfujim/TD3/blob/master/DDPG.py  
+- Twin Delayed Deep Deterministic Policy Gradient (TD3) (Fujimoto et al., 2018). https://github.com/sfujim/TD3  
+- Soft Actor-Critic (SAC) (Haarnoja et al., 2018b). https://github.com/denisyarats/pytorch_sac/  
+- Proximal Policy Optimization (PPO) (Schulman et al., 2017). https://github.com/ DLR-RM/stable-baselines3/blob/master/stable_baselines3/ppo/ ppo.py
+
+# D.5 HYPER-PARAMETER
+
+We provide the hyper-parameters of all compared methods under different environments in Table 1, Table 2, Table 3, Table 4, and Table 5.
+
+As for "Total Timesteps", "Start Traning Timestep", "Max Episode Length", "Actor Network Hidden Layers", "Critic Network Hidden Layers", "Optimizer", "Learning Rate", and "Discount Fac
+
+tor", we set them the same for all algorithms for a fair comparison. As for these specific parameters for baseline algorithms, we remain them the same as those in the original code to achieve good performance. As for these specific parameters of our CFlowNets, we set the number of sample flows to 100 and the action probability buffer size to 1000 to tradeoff the performance and computational load. Note that CFlowNets dose not require as large a replay buffer size as other RL algorithms, since the exploration ability of CFlowNets is better than that of others. And a good policy can already be learned from a small replay buffer. This is also an advantage of CFlowNets compared to RL based algorithms.
+
+Table 1: Hyper-parameters of CFlowNets under different environments.  
+
+<table><tr><td></td><td>Point-Robot-Sparse</td><td>Reacher-Goal-Sparse</td><td>Swimmer-Sparse</td></tr><tr><td>Total Timesteps</td><td>100,000</td><td>100,000</td><td>100,000</td></tr><tr><td>Start Traning Timestep</td><td>4,000</td><td>7,500</td><td>7,500</td></tr><tr><td>Max Episode Length</td><td>12</td><td>50</td><td>50</td></tr><tr><td>Flow Network Hidden Layers</td><td>[256,256]</td><td>[256,256]</td><td>[256,256]</td></tr><tr><td>Retrieval Network Hidden Layers</td><td>[256,256,256]</td><td>[256,256,256]</td><td>[256,256,256]</td></tr><tr><td>Optimizer</td><td>Adam</td><td>Adam</td><td>Adam</td></tr><tr><td>Learning Rate</td><td>0.0003</td><td>0.0003</td><td>0.0003</td></tr><tr><td>Batchsize</td><td>128</td><td>128</td><td>128</td></tr><tr><td>Number of Sample Flows</td><td>100</td><td>100</td><td>100</td></tr><tr><td>Action Probability Buffer Size</td><td>1,000</td><td>1,0000</td><td>10,000</td></tr><tr><td>Replay Buffer Size</td><td>8,000</td><td>2,000</td><td>2,000</td></tr><tr><td>ε</td><td>1.0</td><td>1.0</td><td>1.0</td></tr></table>
+
+Table 2: Hyper-parameter of DDPG under different environments.  
+
+<table><tr><td></td><td>Point-Robot-Sparse</td><td>Reacher-Goal-Sparse</td><td>Swimmer-Sparse</td></tr><tr><td>Total Timesteps</td><td>100,000</td><td>100,000</td><td>100,000</td></tr><tr><td>Start Traning Timestep</td><td>4,000</td><td>7,500</td><td>7,500</td></tr><tr><td>Max Episode Length</td><td>12</td><td>50</td><td>50</td></tr><tr><td>Actor Network Hidden Layers</td><td>[256,256]</td><td>[256,256]</td><td>[256,256]</td></tr><tr><td>Critic Network Hidden Layers</td><td>[256,256]</td><td>[256,256]</td><td>[256,256]</td></tr><tr><td>Optimizer</td><td>Adam</td><td>Adam</td><td>Adam</td></tr><tr><td>Learning Rate</td><td>0.0003</td><td>0.0003</td><td>0.0003</td></tr><tr><td>Batchsize</td><td>256</td><td>256</td><td>256</td></tr><tr><td>Discount Factor</td><td>0.99</td><td>0.99</td><td>0.99</td></tr><tr><td>Replay Buffer Size</td><td>100,000</td><td>100,000</td><td>100,000</td></tr><tr><td>Target Network Update Rate</td><td>0.005</td><td>0.005</td><td>0.005</td></tr></table>
+
+Table 3: Hyper-parameter of TD3 under different environments.  
+
+<table><tr><td></td><td>Point-Robot-Sparse</td><td>Reacher-Goal-Sparse</td><td>Swimmer-Sparse</td></tr><tr><td>Total Timesteps</td><td>100,000</td><td>100,000</td><td>100,000</td></tr><tr><td>Start Traning Timestep</td><td>4,000</td><td>7,500</td><td>7,500</td></tr><tr><td>Max Episode Length</td><td>12</td><td>50</td><td>50</td></tr><tr><td>Actor Network Hidden Layers</td><td>[256,256]</td><td>[256,256]</td><td>[256,256]</td></tr><tr><td>Critic Network Hidden Layers</td><td>[256,256]</td><td>[256,256]</td><td>[256,256]</td></tr><tr><td>Optimizer</td><td>Adam</td><td>Adam</td><td>Adam</td></tr><tr><td>Learning Rate</td><td>0.0003</td><td>0.0003</td><td>0.0003</td></tr><tr><td>Batchsize</td><td>128</td><td>128</td><td>128</td></tr><tr><td>Discount Factor</td><td>0.99</td><td>0.99</td><td>0.99</td></tr><tr><td>Replay Buffer Size</td><td>100,000</td><td>100,000</td><td>100,000</td></tr><tr><td>Gaussian Exploration Noise</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Target Network Update Rate</td><td>0.005</td><td>0.005</td><td>0.005</td></tr></table>
+
+Table 4: Hyper-parameter of SAC under different environments.  
+
+<table><tr><td></td><td>Point-Robot-Sparse</td><td>Reacher-Goal-Sparse</td><td>Swimmer-Sparse</td></tr><tr><td>Total Timesteps</td><td>100,000</td><td>100,000</td><td>100,000</td></tr><tr><td>Start Traning Timestep</td><td>4,000</td><td>7,500</td><td>7,500</td></tr><tr><td>Max Episode Length</td><td>12</td><td>50</td><td>50</td></tr><tr><td>Actor Network Hidden Layers</td><td>[256,256]</td><td>[256,256]</td><td>[256,256]</td></tr><tr><td>Critic Network Hidden Layers</td><td>[256,256]</td><td>[256,256]</td><td>[256,256]</td></tr><tr><td>Optimizer</td><td>Adam</td><td>Adam</td><td>Adam</td></tr><tr><td>Learning Rate</td><td>0.0003</td><td>0.0003</td><td>0.0003</td></tr><tr><td>Batchsize</td><td>1024</td><td>1024</td><td>1024</td></tr><tr><td>Discount Factor</td><td>0.99</td><td>0.99</td><td>0.99</td></tr><tr><td>Replay Buffer Size</td><td>100,000</td><td>100,000</td><td>100,000</td></tr><tr><td>Target Update Interval</td><td>1</td><td>1</td><td>1</td></tr></table>
+
+Table 5: Hyper-parameter of PPO under different environments.  
+
+<table><tr><td></td><td>Point-Robot-Sparse</td><td>Reacher-Goal-Sparse</td><td>Swimmer-Sparse</td></tr><tr><td>Total Timesteps</td><td>100,000</td><td>100,000</td><td>100,000</td></tr><tr><td>Max Episode Length</td><td>12</td><td>50</td><td>50</td></tr><tr><td>Policy Network Hidden Layers</td><td>[64,64]</td><td>[64,64]</td><td>[64,64]</td></tr><tr><td>Value Network Hidden Layers</td><td>[64,64]</td><td>[64,64]</td><td>[64,64]</td></tr><tr><td>Optimizer</td><td>Adam</td><td>Adam</td><td>Adam</td></tr><tr><td>Learning Rate</td><td>0.0003</td><td>0.0003</td><td>0.0003</td></tr><tr><td>Batchsize</td><td>64</td><td>64</td><td>64</td></tr><tr><td>Discount Factor</td><td>0.99</td><td>0.99</td><td>0.99</td></tr><tr><td>GAE Parameter</td><td>0.95</td><td>0.95</td><td>0.95</td></tr><tr><td>Timesteps per Update</td><td>2048</td><td>2048</td><td>2048</td></tr><tr><td>Number of Epochs</td><td>10</td><td>10</td><td>10</td></tr><tr><td>Clipping Parameter</td><td>0.2</td><td>0.2</td><td>0.2</td></tr><tr><td>Value Loss Coefficient</td><td>0.5</td><td>0.5</td><td>0.5</td></tr></table>

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2023/CLARE_ Conservative Model-Based Reward Learning for Offline Inverse Reinforcement Learning/7c474f8a-4a8b-4a06-8c5a-6f6fae7e739f_origin.pdf

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2023/CLARE_ Conservative Model-Based Reward Learning for Offline Inverse Reinforcement Learning/full.md

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+# CLARE: CONSERVATIVE MODEL-BASED REWARD LEARNING FOR OFFLINE INVERSE REINFORCEMENT LEARNING
+
+Sheng Yue $^{1}$ , Guanbo Wang $^{2}$ , Wei Shao $^{3}$ , Zhaofeng Zhang $^{4}$ , Sen Lin $^{5}$ , Ju Ren $^{1,6\dagger}$ , Junshan Zhang $^{3}$
+
+$^{1}$ Tsinghua University,  $^{2}$ Tongji University,  $^{3}$ University of California, Davis,
+
+<sup>4</sup>Arizona State University, <sup>5</sup>Ohio State University, <sup>6</sup>Zhongguancun Laboratory
+
+# ABSTRACT
+
+This work aims to tackle a major challenge in offline Inverse Reinforcement Learning (IRL), namely the reward extrapolation error, where the learned reward function may fail to explain the task correctly and misguide the agent in unseen environments due to the intrinsic covariate shift. Leveraging both expert data and lower-quality diverse data, we devise a principled algorithm (namely CLARE) that solves offline IRL efficiently via integrating "conservatism" into a learned reward function and utilizing an estimated dynamics model. Our theoretical analysis provides an upper bound on the return gap between the learned policy and the expert policy, based on which we characterize the impact of covariate shift by examining subtle two-tier tradeoffs between the "exploitation" (on both expert and diverse data) and "exploration" (on the estimated dynamics model). We show that CLARE can provably alleviate the reward extrapolation error by striking the right "exploitation-exploration" balance therein. Extensive experiments corroborate the significant performance gains of CLARE over existing state-of-the-art algorithms on MuJoCo continuous control tasks (especially with a small offline dataset), and the learned reward is highly instructive for further learning (source code).
+
+# 1 INTRODUCTION
+
+The primary objective of Inverse Reinforcement Learning (IRL) is to learn a reward function from demonstrations (Arora & Doshi, 2021; Russell, 1998). In general, conventional IRL methods rely on extensive online trials and errors that can be costly or require a fully known transition model (Abbeel & Ng, 2004; Ratliff et al., 2006; Ziebart et al., 2008; Syed & Schapire, 2007; Boularias et al., 2011; Osa et al., 2018), struggling to scale in many real-world applications. To tackle this problem, this paper studies offline IRL, with focus on learning from a previously collected dataset without online interaction with the environment. Offline IRL holds tremendous promise for safety-sensitive applications where manually identifying an appropriate reward is difficult but historical datasets of human demonstrations are readily available (e.g., in healthcare, autonomous driving, robotics, etc.). In particular, since the learned reward function is a succinct representation of an expert's intention, it is useful for policy learning (e.g., in offline Imitation Learning (IL) (Chan & van der Schaar, 2021)) as well as a number of broader applications (e.g., task description (Ng et al., 2000) and transfer learning (Herman et al., 2016)).
+
+This work aims to address a major challenge in offline IRL, namely the reward extrapolation error, where the learned reward function may fail to correctly explain the task and misguide the agent in unseen environments. This issue results from the partial coverage of states in the restricted expert demonstrations (i.e., covariate shift) as well as the high-dimensional and expressive function approximation for the reward. It is further exacerbated due to no reinforcement signal for supervision and the intrinsic reward ambiguity therein. In fact, similar challenges related to the extrapolation
+
+error in the value function have been widely observed in offline (forward) RL, e.g., in Kumar et al. (2020); Yu et al. (2020; 2021). Unfortunately, to the best of our knowledge, this challenge remains not well understood in offline IRL, albeit there is some recent progress (Zolna et al., 2020; Garg et al., 2021; Chan & van der Schaar, 2021). Thus motivated, the key question this paper seeks to answer is: "How to devise offline IRL algorithms that can ameliorate the reward extrapolation error effectively?"
+
+We answer this question by introducing a principled offline IRL algorithm, named conservative model-based reward learning (CLARE), leveraging not only (limited) higher-quality expert data but also (potentially abundant) lower-quality diverse data to enhance the coverage of the state-action space for combating covariate shift. CLARE addresses the above-mentioned challenge by appropriately integrating conservatism into the learned reward to alleviate the possible misguidance in out-of-distribution states, and improves the reward generalization ability by utilizing a learned dynamics model. More specifically, CLARE iterates between conservative reward updating and safe policy improvement, and the reward function is updated via improving its values on weighted expert and diverse state-actions while in turn cautiously penalizing those generated from model rollouts. As a result, it can encapsulate the expert intention while conservatively evaluating out-of-distribution state-actions, which in turn encourages the policy to visit data-supported states and follow expert behaviors and hence achieves safe policy search.
+
+![](images/fdc070c28aff280859a2a1d209fa962f80c99826afc63109eead50e6921a498d.jpg)  
+Figure 1: An illustration of the two-tier tradeoffs in CLARE.
+
+Technically, there are highly nontrivial two-tier tradeoffs that CLARE has to delicately calibrate: "balanced exploitation" of the expert and diverse data, and "exploration" of the estimated model.2 As illustrated in Fig. 1, The first tradeoff arises because CLARE relies on both exploiting expert demonstrations to infer the reward and exploiting diverse data to handle the covariate shift caused by the insufficient state-action coverage of limited demonstration data. At a higher level, CLARE needs to judiciously explore the estimated model to escape the offline data manifold for better generalization. To this end, we first introduce the new pointwise weight parameters for offline data points (state-action pairs) to capture the subtle two-tier exploitation-exploration tradeoffs. Then, we rigorously quantify its impact on the performance by providing an upper bound on the return gap between the learned policy and the expert policy. Based on the theoretical quantification, we derive the optimal weight parameters whereby CLARE can strike the balance appropriately to minimize the return gap. Our findings reveal that the reward function obtained by CLARE can effectively capture the expert intention and provably ameliorate the extrapolation error in offline IRL.
+
+Finally, extensive experiments are carried out to compare CLARE with state-of-the-art offline IRL and offline IL algorithms on MuJoCo continuous control tasks. Our results demonstrate that even using small offline datasets, CLARE obtains significant performance gains over existing algorithms in continuous, high-dimensional environments. We also show that the learned reward function can explain the expert behaviors well and is highly instructive for further learning.
+
+# 2 PRELIMINARIES
+
+Markov decision process (MDP) can be specified by tuple  $M \doteq \langle S, \mathcal{A}, T, R, \mu, \gamma \rangle$ , consisting of state space  $S$ , action space  $\mathcal{A}$ , transition function  $T: S \times \mathcal{A} \to \mathcal{P}(S)$ , reward function  $R: S \times \mathcal{A} \to \mathbb{R}$ , initial state distribution  $\mu: S \to [0,1]$ , and discount factor  $\gamma \in (0,1)$ . A stationary stochastic policy maps states to distributions over actions as  $\pi: S \to \mathcal{P}(\mathcal{A})$ . We define the normalized state-action occupancy measure (abbreviated as occupancy measure) of policy  $\pi$
+
+under transition dynamics  $T$  as  $\rho^{\pi}(s,a)\doteq (1 - \gamma)\sum_{h = 0}^{\infty}\gamma^{h}\operatorname *{Pr}(s_{h} = s|T,\pi ,\mu)\pi (a|s)$ . The objective of reinforcement learning (RL) can be expressed as maximizing expected cumulative rewards:  $\max_{\pi \in \Pi}J(\pi)\doteq \mathbb{E}_{s,a\sim \rho^{\pi}}[R(s,a)]$ , where  $\Pi$  is the set of all stationary stochastic policies that take actions in  $\mathcal{A}$  given states in  $S$ .<sup>3</sup>
+
+Maximum entropy IRL (MaxEnt IRL) aims to learn the reward function from expert demonstrations and reason about the stochasticity therein (Ziebart et al., 2008; Ho & Ermon, 2016). Based on demonstrations sampled from expert policy  $\pi^{E}$ , the MaxEnt IRL problem is given by
+
+$$
+\min  _ {r \in \mathcal {R}} \left(\max  _ {\pi \in \Pi} \alpha H (\pi) + \mathbb {E} _ {s, a \sim \rho^ {\pi}} [ r (s, a) ]\right) - \mathbb {E} _ {s, a \sim \rho^ {E}} [ r (s, a) ] + \psi (r), \tag {1}
+$$
+
+with  $H(\pi) \doteq -\iint \rho^{\pi}(s, a) \log \pi(a|s) \, \mathrm{d}s \, \mathrm{d}a$  being the  $\gamma$ -discounted causal entropy,  $\mathcal{R}$  a family of reward functions,  $\alpha \geq 0$  the weight parameter, and  $\psi: \mathbb{R}^{S \times \mathcal{A}} \to \mathbb{R} \cup \{\infty\}$  a convex reward regularizer Fu et al. (2018); Qureshi et al. (2018). Problem (1) looks for a reward function assigning higher rewards to the expert policy and lower rewards to other policies, along with the best policy under the learned reward function. Although enjoying strong theoretical justification and achieving great performance in many applications, MaxEnt IRL has to solve a forward RL problem in the inner loop that involves extensive online interactions with the environment.
+
+Offline IRL is the setting where the algorithm is neither allowed to interact with the environment nor provided reinforcement signals. It only has access to static dataset  $\mathcal{D} = \mathcal{D}_E \cup \mathcal{D}_B$  consisting of expert dataset  $\mathcal{D}_E \doteq \{(s_i, a_i, s_i')\}_{i=1}^{D_E}$  and diverse dataset  $\mathcal{D}_B \doteq \{(s_i, a_i, s_i')\}_{i=1}^{D_B}$  collected by expert policy  $\pi^E$  and behavior policy  $\pi^B$ , respectively. The goal of offline IRL is to infer a reward function capable of explaining the expert's preferences from the given dataset.
+
+# 3 CLARE: CONSERVATIVE MODEL-BASED REWARD LEARNING
+
+A naive solution for offline IRL is to retrofit MaxEnt IRL to the offline setting via estimating a dynamics model using offline data (e.g., in Tanwani & Billard (2013); Herman et al. (2016)). Unfortunately, it has been reported that this naive paradigm often suffers from unsatisfactory performance in high-dimensional and continuous environments Jarrett et al. (2020). The underlying reasons for this issue include: (1) the dependence on full knowledge of the reward feature function, and (2) the lack of effective mechanisms to tackle the reward extrapolation error caused by covariate shift (as stated in Section 1). Nevertheless, we believe that utilizing a learned dynamics model is beneficial because it is expected to provide broader generalization by learning on additional model-generated synthetic data (Yu et al., 2020; 2021; Lin et al., 2021). With this insight, this work focuses on the model-based offline IRL method that is robust to covariate shift while enjoying the model's generalization ability.
+
+As illustrated in Fig. 1, there are two-tier subtle tradeoffs that need to be carefully balanced between exploiting the offline data and exploring model-based synthetic data. On one hand, the higher-quality expert demonstrations are exploited to infer the intention and abstract the reward function therein, while the lower-quality diverse data is exploited to enrich data support. On the other hand, it is essential to prudently explore the estimated dynamics model to improve the generalization capability while mitigating overfitting errors in inaccurate regions. To this end, we devise conservative model-based reward learning (CLARE) based on MaxEnt IRL, where the new pointwise weight parameters are introduced for each offline state-action pair to capture the tradeoffs subtly. We elaborate further in what follows.
+
+As outlined below, CLARE iterates between (I) conservative reward updating and (II) safe policy improvement, under a dynamics model (denoted by  $\hat{T}$ ) learned from offline dataset.
+
+(I) Conservative reward updating. Given current policy  $\pi$ , dynamics model  $\hat{T}$ , and offline datasets  $\mathcal{D}_E$  and  $\mathcal{D}_B$ , CLARE updates reward function  $r$  based on the following loss:
+
+$$
+L (r | \pi) \doteq \underbrace {Z _ {\beta} \mathbb {E} _ {s , a \sim \tilde {\rho} ^ {\pi}} [ r (s , a) ]} _ {\text {p e n a l i z e d o n m o d e l r o l l o u t s}} - \underbrace {\mathbb {E} _ {s , a \sim \tilde {\rho} ^ {E}} [ r (s , a) ]} _ {\text {i n c r e a s e d o n e x p e r t d a t a}} - \underbrace {\mathbb {E} _ {s , a \sim \tilde {\rho} ^ {D}} [ \beta (s , a) r (s , a) ]} _ {\text {w e i g h t i n g e x p e r t a n d d i v e r s e d a t a}} + \underbrace {Z _ {\beta} \psi (r)} _ {\text {r e g u l a r i z e r}}, \tag {2}
+$$
+
+where  $\tilde{\rho}^D (s,a)\doteq (|\mathcal{D}_E(s,a)| + |\mathcal{D}_B(s,a)|) / (D_E + D_B)$  is the empirical distribution of  $(s,a)$  in the union dataset  $\mathcal{D} = \mathcal{D}_E\cup \mathcal{D}_B$  and  $\tilde{\rho}^{E}\doteq |\mathcal{D}_{E}(s,a)| / D_{E}$  is that for expert dataset  $\mathcal{D}_E;\hat{\rho}^\pi$  is the occupancy measure when rolling out  $\pi$  with dynamics model  $\widehat{T}$ ; and  $\psi$  denotes a convex regularizer mentioned above. One key step is to add an additional term weighting the reward of each offline state-action by  $\beta (s,a)$ , which is a "fine-grained control" for the exploitation of the offline data. For the data deserving more exploitation (e.g., expert behaviors with sufficient data support), we can set a relatively large  $\beta (s,a)$ ; otherwise, we decrease its value. Besides, it can also control the exploration of the model subtly (consider that if we set all  $\beta (s,a) = 0$ , Eq. (2) reduces to MaxEnt IRL, enabling the agent to explore the model without restrictions). Here,  $Z_{\beta}\doteq 1 + \mathbb{E}_{s',a'\sim \tilde{\rho}^D}[\beta (s',a')]$  is a normalization term. The new ingredients beyond MaxEnt IRL are highlighted in blue.
+
+Observe that in Eq. (2), by decreasing the reward loss, CLARE pushes up the reward on good off-line state-action that characterized by larger  $\beta(s, a)$ , while pushing down the reward on potentially out-of-distribution ones that generated from model rollouts. This is similar to COMBO (Yu et al., 2021) in spirit, a state-of-the-art offline forward RL algorithm, and results in a conservative reward function. It can encourage the policy to cautiously exploring the state-actions beyond offline data manifold, thus capable of mitigating the misguidance issue and guiding safe policy search. In Section 4, we will derive a closed-form optimal  $\beta(s, a)$  that enables CLARE to achieve a proper exploration-exploitation trade-off by minimizing a return gap from the expert policy.
+
+(II) Safe policy improvement. Given updated reward function  $r$ , the policy is improved by solving
+
+$$
+\max  _ {\pi \in \Pi} L (\pi | r) \doteq Z _ {\beta} \mathbb {E} _ {s, a \sim \hat {\rho} ^ {\pi}} [ r (s, a) ] + \alpha \widehat {H} (\pi), \tag {3}
+$$
+
+where  $\alpha \geq 0$  is a weight parameter, and  $\widehat{H} (\pi)\doteq -\iint \hat{\rho}^{\pi}(s,a)\log \pi (a|s)\mathrm{d}s\mathrm{d}a$  is the  $\gamma$ -discounted causal entropy induced by the policy and learned dynamics model. Due to the embedded expert intention and conservatism in the reward function, the policy is updated safely by carrying out conservative model-based exploration. One can use any well-established MaxEnt RL approach to solve this problem by simulating with model  $\widehat{T}$  and reward function  $r$ . It is worth noting that for Problem (3) in this step, the practical implementation of CLARE works well with a small number of updates in each iteration (see Sections 5 and 6).
+
+# 4 THEORETICAL ANALYSIS OF CLARE
+
+In this section, we focus on answering the following question: "How to set  $\beta(s,a)$  for each offline state-action pair to strike the two-tier exploitation-exploration balance appropriately?" To this end, we first quantify the impact of the tradeoffs via bounding the return gap between the learned policy and expert policy. Then, we derive the optimal weight parameters to minimize this gap. All the detailed proofs can be found in Appendix B. Notably, this section works with finite state and action spaces, but our algorithms and experiments run in high-dimensional and continuous environments.
+
+# 4.1 CONVERGENCE ANALYSIS
+
+We first characterize the policy learned by CLARE, in terms of  $\beta(s,a)$  and empirical distributions  $\tilde{\rho}^E$  and  $\tilde{\rho}^D$ . Before proceeding, it is easy to see CLARE is iteratively solving the min-max problem:
+
+$$
+\min  _ {r \in \mathcal {R}} \max  _ {\pi \in \Pi} \underbrace {\alpha \widehat {H} (\pi) + Z _ {\beta} \mathbb {E} _ {\hat {\rho} ^ {\pi}} [ r (s , a) ] - \mathbb {E} _ {\tilde {\rho} ^ {D}} [ \beta (s , a) r (s , a) ] - \mathbb {E} _ {\tilde {\rho} ^ {E}} [ r (s , a) ] + Z _ {\beta} \psi (r)} _ {\doteq L (\pi , r)}. \tag {4}
+$$
+
+For dynamics  $T$ , define the set of occupancy measures satisfying Bellman flow constraints as
+
+$$
+\mathcal {C} _ {T} \doteq \left\{\rho \in \mathbb {R} ^ {| S | | A |}: \rho \geq 0 \text {a n d} \sum_ {a} \rho (s, a) = \mu (s) + \gamma \sum_ {s ^ {\prime}, a} T \left(s \mid s ^ {\prime}, a\right) \rho \left(s ^ {\prime}, a\right) \forall s \in \mathcal {S} \right\}. \tag {5}
+$$
+
+We first provide the following results for switching between policies and occupancy measures, which allow us to use  $\pi_{\rho}$  to denote the unique policy for occupancy measure  $\rho$ .
+
+Lemma 4.1 (Theorem 2 in Syed et al. (2008)). If  $\rho \in \mathcal{C}_T$ , then  $\rho$  is the occupancy measure for stationary policy  $\pi_{\rho}(a|s) \doteq \rho(s, a) / \sum_{a'} \rho(s, a')$ , and  $\pi_{\rho}$  is the only stationary policy with occupancy measure  $\rho$ .
+
+Lemma 4.2 (Lemma 3.2 in Ho & Ermon (2016)). Denote  $\bar{H}(\rho) \doteq -\sum_{s,a} \rho(s,a) \log \frac{\rho(s,a)}{\sum_{a'} \rho(s,a')}$ . Then,  $\bar{H}$  is strictly concave, and for all  $\pi \in \Pi$  and  $\rho \in \mathcal{C}_T$ ,  $H(\pi) = \bar{H}(\rho^\pi)$  and  $\bar{H}(\rho) = H(\pi_\rho)$  hold true, where  $\pi_\rho(a|s) \doteq \rho(s,a) / \sum_{a'} \rho(s,a')$ .
+
+Based on Lemma 4.1 and Lemma 4.2, we have the follow results on the learned policy.
+
+Theorem 4.1. Assume that  $\beta(s, a) \geq -\tilde{\rho}^E(s, a) / \tilde{\rho}^D(s, a)$  holds for  $(s, a) \in \mathcal{D}$ . For Problem (4), the following relationship holds:
+
+$$
+\min  _ {r \in \mathcal {R}} \max  _ {\pi \in \Pi} L (\pi , r) = \max  _ {\hat {\rho} \in \mathcal {C} _ {\hat {T}}} \alpha \bar {H} (\hat {\rho}) - Z _ {\beta} D _ {\psi} \left(\hat {\rho}, \frac {\tilde {\rho} ^ {E} + \beta \tilde {\rho} ^ {D}}{Z _ {\beta}}\right), \tag {6}
+$$
+
+with  $D_{\psi}(\rho_1,\rho_2)\dot{=}\psi^* (\rho_2 - \rho_1)$  , where  $\psi^{*}$  is the convex conjugate of  $\psi$
+
+Notably, by selecting appropriate forms of reward regularizers  $\psi$ ,  $D_{\psi}$  can belong to a wide-range of statistical distances. For example, if  $\psi(r) = \alpha r^2$ , then  $D_{\psi}(\rho_1, \rho_2) = \frac{1}{4\alpha} \chi^2(\rho_1, \rho_2)$ ; if  $\psi$  restricts  $r \in [-R^{\max}, R^{\max}]$ , then  $D_{\psi}(\rho_1, \rho_2) = 2R^{\max} D_{\mathrm{TV}}(\rho_1, \rho_2)$  (Garg et al., 2021). Theorem 4.1 implies that CLARE implicitly seeks a policy under  $\hat{T}$  whose occupancy measure stays close to an interpolation of the empirical distributions of expert dataset  $\mathcal{D}_E$  and union offline dataset  $\mathcal{D}$ . The interpolation reveals that CLARE is trying to trade off the exploration of the model and exploitation of offline data by selecting proper weight parameters  $\beta(s, a)$ . For example, if  $\beta(s, a) = 0$  for all  $(s, a) \in \mathcal{D}$ , CLARE will completely follow the occupancy measure of the (empirical) expert policy by explore the model freely. In contrast, if  $\beta(s, a)$  increases with  $\tilde{\rho}^D(s, a)$ , the learned policy will look for richer data support.
+
+Remarks. Looking deeper into Eq. (6), the target occupancy measure can be expressed equivalently as  $\frac{(1 + \beta D_E / D)\tilde{\rho}^E + (\beta D_S / D)\tilde{\rho}^B}{Z_\beta}$ , after rearranging terms in the above interpolation. As a result, CLARE also subtly balances the exploitation between the expert and diverse datasets to extract potentially valuable information in the sub-optimal data.
+
+# 4.2 STRIKING THE RIGHT EXPLORATION-EXPLOITATION BALANCE
+
+Next, we show how to set  $\beta(s, a)$  properly to achieve the right two-tier balance.
+
+Recall that  $J(\pi) \doteq \mathbb{E}_{s,a \sim \rho^{\pi}}[R(s,a)]$  is the return achieved by policy  $\pi$ . The next result provides a upper bound on the return gap between  $J(\pi)$  and  $J(\pi^{E})$ , which hinges on the intrinsic trade-offs.
+
+Theorem 4.2. Suppose  $|R(s, a)| \leq 1$  for any  $s \in S, a \in \mathcal{A}$ . For any stationary policy  $\pi$ , let  $\hat{\rho}^{\pi}$  denote the occupancy measure of  $\pi$  under estimated model  $\hat{T}$ . We have that
+
+$$
+J \left(\pi^ {E}\right) - J (\pi) \leq C \cdot \mathbb {E} _ {s, a \sim \hat {\rho} ^ {\pi}} \left[ D _ {\mathrm {T V}} \left(T (\cdot | s, a), \widehat {T} (\cdot | s, a)\right) \right] + 2 \left(D _ {\mathrm {T V}} \left(\hat {\rho} ^ {\pi}, \tilde {\rho} ^ {E}\right) + D _ {\mathrm {T V}} \left(\tilde {\rho} ^ {E}, \rho^ {E}\right)\right), \tag {7}
+$$
+
+where  $C \doteq \frac{2\gamma}{1 - \gamma}$ , and  $\rho^{E}$  is the occupancy measure of expert policy  $\pi^{E}$  under true dynamics  $T$ .
+
+Remarks. Theorem 4.2 indicates that a good policy learned from the estimated model not only follows the expert behaviors but also keeps in the "safe region" of the learned model, i.e., visiting the state-actions with less model estimation inaccuracy. Under the concentration assumption, the following holds with probability greater than  $1 - \delta$ :
+
+$$
+J (\pi^ {E}) - J (\pi) \leq \underbrace {\mathbb {E} _ {s , a \sim \hat {\rho} ^ {\pi}} \left[ \frac {C C _ {\delta}}{\sqrt {| \mathcal {D} _ {E} (s , a) | + | \mathcal {D} _ {B} (s , a) |}} \right]} _ {\mathrm {(a)}} + 2 \underbrace {D _ {\mathrm {T V}} (\hat {\rho} ^ {\pi} , \tilde {\rho} ^ {E})} _ {\mathrm {(b)}} + 2 \underbrace {D _ {\mathrm {T V}} (\tilde {\rho} ^ {E} , \rho^ {E})} _ {\mathrm {(c)}},
+$$
+
+where  $\mathcal{D}(s,a)\doteq \{(s',a')\in \mathcal{D}:s' = s,a' = a\}$ . It aligns well with the aforementioned exploration-exploitation balance: 1) Term (a) captures the exploitation of offline data support; 2) Term (b) captures the exploitation of expert data and the exploration of the model (recall that  $\hat{\rho}^{\pi}$  is the occupancy measure of rolling out  $\pi$  with  $\widehat{T}$ ); and 3) Term (c) captures the distributional shift in offline learning. Importantly, the result in Theorem 4.2 connects the true return of a policy with its occupancy measure on the learned model. This gives us a criteria to evaluate the performance of a policy from offline. Define  $c(s,a)\doteq C\cdot D_{\mathrm{TV}}(T(\cdot |s,a),\widehat{T} (\cdot |s,a))$  and  $c^{\mathrm{min}}\doteq \min_{s,a}c(s,a)$ . Subsequently, we derive the policy that minimizes the RHS of Eq. (7).
+
+# Algorithm 1: Conservative model-based reward learning (CLARE)
+
+Input: expert data  $\mathcal{D}_E$ , diverse data  $\mathcal{D}_B$ , bar  $u$ , learning rate  $\eta$ , policy regularizer weight  $\lambda$  Learn dynamics model  $\hat{T}$  represented by an ensemble of neural networks using all offline data; Set weight  $\beta(s, a)$  for each offline state-action tuple  $(s, a) \in \mathcal{D}_E \cup \mathcal{D}_B$  by Eq. (11); Initialize the policy  $\pi_\theta$  and reward function  $r_\phi$  parameterized by  $\theta$  and  $\phi$  respectively; while not done do
+
+(Safe policy improvement) Run a MaxEnt RL algorithm for some steps with model  $\widehat{T}$  and current reward function  $r_{\phi}$  to update policy  $\pi_{\theta}$ , based on  $L(\pi_{\theta}|r_{\phi}) - \lambda D_{\mathrm{KL}}(\pi^{b}||\pi_{\theta})$ ;
+
+(Conservative reward updating) Update  $r_{\phi}$  by  $\phi \gets \phi -\eta \nabla_{\phi}L(r_{\phi}|\pi_{\theta})$  for a few steps;
+
+end
+
+Theorem 4.3. Under the same conditions as in Theorem 4.2, the optimal occupancy measure minimizing the upper bound of Eq. (7) is given as follows:
+
+$$
+\hat {\rho} ^ {*} (s, a) = \left\{ \begin{array}{l l} \tilde {\rho} ^ {E} (s, a) + \Delta_ {\rho}, & i f c (s, a) \leq c ^ {\min }, \\ 0, & i f c (s, a) > c ^ {\min } + 2, \\ \tilde {\rho} ^ {E} (s, a), & o t h e r w i s e. \end{array} \right. \tag {8}
+$$
+
+where  $\Delta_{\rho} \doteq \frac{\sum_{s',a'} \mathbf{1}[c(s',a') - c^{\min} > 2] \cdot \tilde{\rho}^{E}(s',a')}{|\mathcal{N}_{\min}|}$  and  $\mathcal{N}_{\min} \doteq \{(s,a) \in \mathcal{D} : c(s,a) \leq c^{\min}\}$ .
+
+As shown in Theorem 4.3, the "optimal" policy leaned on model  $\widehat{T}$  conservatively explores the model by avoiding the visit of risky state-actions. Meantime, it cleverly exploits the accurate region, such that it does not deviate large from the expert. Now, we are ready to derive the optimal values of the weight parameters.
+
+Corollary 4.1. Suppose that when  $\tilde{\rho}^D (s,a) = 0$ ,  $c(s,a) > c^{\min}$  holds for each  $(s,a)\in S\times A$ . Under the same condition as in Theorem 4.3, if  $\beta (s,a)$  are set as
+
+$$
+\beta^ {*} (s, a) = \left\{ \begin{array}{l l} \frac {\Delta_ {\rho}}{\tilde {\rho} ^ {D} (s , a)}, & \text {i f} c (s, a) \leq c ^ {\min } \text {a n d} \tilde {\rho} ^ {D} (s, a) > 0, \\ - \frac {\tilde {\rho} ^ {E} (s , a)}{\tilde {\rho} ^ {D} (s , a)}, & \text {i f} c (s, a) > c ^ {\min } + 2 \text {a n d} \tilde {\rho} ^ {D} (s, a) > 0, \\ 0, & \text {o t h e r w i s e}, \end{array} \right. \tag {9}
+$$
+
+then it follows that
+
+$$
+\min  _ {r \in \mathcal {R}} \max  _ {\pi \in \Pi} L (\pi , r) = \max  _ {\pi} \alpha \bar {H} \left(\hat {\rho} ^ {\pi}\right) - Z _ {\beta} D _ {\psi} \left(\hat {\rho} ^ {\pi}, \hat {\rho} ^ {*}\right). \tag {10}
+$$
+
+Corollary 4.1 provides the value of  $\beta(s, a)$  for each  $(s, a) \in \mathcal{D}$  such that the learned reward function can guide the policy to minimize the return gap in Eq. (7). It indicates that the right exploitation-exploration trade-off can be provably balanced via setting the weight parameters properly. In particular,  $\beta^{*}$  assigns positive weight to the offline state-action with accurate model estimation and negative weight to that with large model error. It enables CLARE to learn a conservative reward function that pessimistically evaluates the out-of-distribution states and actions, capable of ameliorating the extrapolation error in unseen environments. However, the optimal weights require the model error,  $c(s, a)$ , which is typically hard to obtain (especially in high-dimensional and continuous spaces). Section 5 will solve this problem by extending this result with the aid of the model ensembles and uncertainty quantification techniques.
+
+# 5 PRACTICAL IMPLEMENTATION
+
+Learning dynamics models. Following the state-of-the-art model-based methods (Yu et al., 2020; 2021), we model the transition dynamics by an ensemble of neural networks, each of which outputs a Gaussian distribution over next states, i.e.,  $\{\widehat{T}_i(s'|s,a) = \mathcal{N}(\mu_i(s,a),\Sigma_i(s,a))\}_{i=1}^N$ .
+
+Weights in continuous environments. The ideas of achieving CLARE in continuous environments are 1) to approximately see the offline data as sampled from a large discrete space, and 2)
+
+to use an uncertainty quantification technique for quantifying the model error. Specifically, because state-action pairs are basically different from each other in this setting, we let  $\tilde{\rho}^D (s,a) = 1 / D$  and  $\tilde{\rho}^{E}(s,a) = 1 / D_{E}$ , and employ the uncertainty estimator,  $c(s,a) = \max_{i\in [N]}\| \Sigma_i(s,a)\| _F$ , proposed in Yu et al. (2020) for model error evaluation. Guided by the analytical results in Corollary 4.1, we compute the weights for each  $(s,a)\in \mathcal{D}$  via slight relaxation as follows:
+
+$$
+\beta (s, a) = \left\{ \begin{array}{l l} \frac {N ^ {\prime \prime} D}{N ^ {\prime} D _ {E}}, & \text {i f} c (s, a) \leq u, \\ - \frac {D}{D _ {E}} \cdot \mathbf {1} [ (s, a) \in \mathcal {D} _ {E} ], & \text {i f} c (s, a) > u, \\ 0, & \text {o t h e r w i s e}, \end{array} \right. \tag {11}
+$$
+
+where  $N' \doteq \sum_{(s,a) \in \mathcal{D}} \mathbf{1}[c(s,a) \leq u]$  and  $N'' \doteq \sum_{(s,a) \in \mathcal{D}_E} \mathbf{1}[c(s,a) > u]$ . Here, coefficient  $u$  is a user-chosen hyper-parameter for controlling the conservatism level of CLARE. If one wants the learned policy to be trained more conservatively on offline data support,  $u$  should be small; otherwise,  $u$  can be chose to be large for better exploration.
+
+Reward and policy regularizers. In the experiments, we use  $\psi (r) = r^2$  as the reward regularizer. Additionally, when updating the policy, we use a KL divergence as a regularizer with empirical behavior policy  $\pi^b$  induced by a subset of the offline dataset,  $\mathcal{D}'\subset \mathcal{D}$ , as follows:
+
+$$
+D _ {\mathrm {K L}} (\pi^ {b} \| \pi) \doteq \mathbb {E} _ {s \in \mathcal {D} ^ {\prime}} \left[ \mathbb {E} _ {a \sim \pi^ {b} (\cdot | s)} \left[ \log \pi^ {b} (a | s) \right] - \mathbb {E} _ {a \sim \pi^ {b} (\cdot | s)} \left[ \log \pi (a | s) \right] \right],
+$$
+
+where  $\pi^b (a|s) = \frac{\sum_{(s',a')\in\mathcal{D}'}\mathbf{1}[s' = s,a' = a]}{\sum_{(s',a')\in\mathcal{D}'}\mathbf{1}[s' = s]}$  if  $(s,a)\in \mathcal{D}'$ , and  $\pi^b (a|s) = 0$  otherwise. It can be implemented by adding  $-\mathbb{E}_{s,a\sim \mathcal{D}'}[\log \pi (a|s)]$  to the actor loss. The intuition is to encourage the actor to perform in support of the real data for accelerating safe policy improvement. While this regularization lacks theoretical guarantees, we empirically find that it can indeed speed up the training.
+
+Practical algorithm design. The pseudocode of CLARE is depicted in Algorithm 1. The policy improvement phase can be implemented by the standard implementation of SAC (Haarnoja et al., 2018) with a change of the additional policy regularizer. We elaborate more details in the Appendix A.
+
+# 6 EXPERIMENTS
+
+Next, we use experimental studies to evaluate CLARE and answer the following key questions: (1) How does CLARE perform on the standard offline RL benchmarks in comparison to existing state-of-the-art algorithms? (2) How does CLARE perform given different dataset sizes? (3) How does the "conservatism level",  $u$ , affect the performance? (4) How fast does CLARE converge? (5) Can the learned reward function effectively explain the expert intention?
+
+To answer these questions, we compare CLARE with the following existing offline IRL methods on the D4RL benchmark (Fu et al., 2020): 1) IQ-LEARN (Garg et al., 2021), a state-of-the-art model-free offline IRL algorithm; 2) AVRIL (Chan & van der Schaar, 2021), another recent model-free offline IRL method; 3) EDM (Jarrett et al., 2020), a state-of-the-art offline IL approach; and 4) Behavior Cloning (BC). To demonstrate the poor performance of the naive approach using a simple combination of IRL with model-based offline forward RL (MORL) method, we also consider a baseline algorithm, namely MOMAX, by directly using COMBO (Yu et al., 2021) in the inner loop of MaxEnt IRL. We present the results on continuous control tasks (including Half-Cheetah, Walker2d, Hopper, and Ant) consisting of three data qualities (random, medium, and expert). Experimental set-up and hyperparameters are described in detailed in Appendix A.
+
+![](images/8eae25be81751b04bc3473c6c4744ae366b1d6546de003df857565512bab3cf0.jpg)  
+Figure 2: CLARE against other algorithms on all tasks over different dataset sizes consisting of expert and medium data equally.
+
+![](images/d185fc766755a795440e7d383ab52dedcb16692adbd5792aed8555713e70d4ce.jpg)
+
+![](images/338bfbf4af02982090f7b4d563fa7d61573a97583d7a89093449f9fdbb3cc7e3.jpg)
+
+![](images/01e6bbac9ff41d611eb16b08a036b9591c65018a0e4ab7efb69ce271fe3ec8b2.jpg)
+
+Table 1: Results on D4RL datasets. For each task, the experiments are carried out with three different data combinations: 1) 10k expert tuples, 2) 5k expert and 5k medium tuples, and 3) 5k expert and 5k random tuples. The data scores below for 1), 2), and 3) correspond to expert, medium, and random data, respectively. We tune IQ-LEARN, EDM, and AVRIL based on their publicly available source code. Results are averaged over 7 random seeds. The highest score across all algorithms is bold.  
+
+<table><tr><td>Dataset type</td><td>Environment</td><td>Data score</td><td>CLARE</td><td>BC</td><td>IQ-LEARN</td><td>EDM</td><td>AVRIL</td><td>MOMAX</td></tr><tr><td rowspan="4">Exp. &amp; Rand.</td><td>Walker2d</td><td>1.9</td><td>2873.8</td><td>17.8</td><td>256.9</td><td>165.5</td><td>100.9</td><td>-525.4</td></tr><tr><td>Hopper</td><td>18.4</td><td>1891.5</td><td>110.2</td><td>523.6</td><td>178.8</td><td>178.3</td><td>0.7</td></tr><tr><td>Ant</td><td>-64.4</td><td>1960.0</td><td>-427.6</td><td>-247.2</td><td>-3000.9</td><td>1000.1</td><td>113.8</td></tr><tr><td>Half-Cheetah</td><td>-505.1</td><td>1113.7</td><td>-86.7</td><td>123.9</td><td>-346.7</td><td>-1093.5</td><td>-11.0</td></tr><tr><td rowspan="4">Exp. &amp; Med.</td><td>Walker2d</td><td>3496.3</td><td>3613.4</td><td>1674.2</td><td>1676.8</td><td>175.7</td><td>184.0</td><td>19.6</td></tr><tr><td>Hopper</td><td>1422.7</td><td>2135.0</td><td>947.0</td><td>2049.8</td><td>194.4</td><td>183.7</td><td>27.6</td></tr><tr><td>Ant</td><td>3969.0</td><td>3879.4</td><td>2146.0</td><td>222.2</td><td>-3001.5</td><td>1001.0</td><td>-33.2</td></tr><tr><td>Half-Cheetah</td><td>4667.8</td><td>4888.6</td><td>2375.0</td><td>2957.7</td><td>-298.3</td><td>-1195.6</td><td>-0.2</td></tr><tr><td rowspan="4">Exp.</td><td>Walker2d</td><td>5010.4</td><td>4990.5</td><td>1665.7</td><td>2445.4</td><td>189.7</td><td>194.1</td><td>23.2</td></tr><tr><td>Hopper</td><td>3603.2</td><td>2604.5</td><td>1436.1</td><td>2854.4</td><td>192.5</td><td>183.9</td><td>34.5</td></tr><tr><td>Ant</td><td>5172.8</td><td>3940.3</td><td>1797.9</td><td>375.4</td><td>-3000.6</td><td>1000.2</td><td>48.1</td></tr><tr><td>Half-Cheetah</td><td>10748.7</td><td>4975.1</td><td>242.4</td><td>3750.5</td><td>-299.5</td><td>-619.0</td><td>-0.4</td></tr></table>
+
+![](images/1c1bd5aa5fcf6f324c10fa33b04aa399c2c04f9e0b936093f33903307d56cb26.jpg)  
+(a) Impact of  $u$
+
+![](images/0d5f12f1c38bf8c2d04f1d275c70954577e78786204349a78fb40cec5e732e56.jpg)  
+(b) Convergence speed.
+
+![](images/9c293cbf33ccd5cfd3c087bfc85e1317b7fb8fe98d732c33cd12dcfc8513589d.jpg)  
+(c) Convergence speed.  
+Figure 3: Performance of CLARE. 1) Impact of  $u$ : Figure 3(a) shows the impact of user-chosen parameter  $u$  on the performance using 10k expert tuples. 2) Convergence speed: Figures 3(c) and 3(b) show the convergence of CLARE using 10k expert and 10k medium tuples. In each iteration, CLARE carries out policy improvement by total 10k gradient updates (total 500 epochs with 20 gradient steps per epoch) for the actor and critic networks using SAC. 3) Recovered reward: Figure 3(d) shows the result of training SAC via replacing the underlying reward by the one learned from CLARE.
+
+![](images/5516d2acd83f6024d225eaa8b2772f8a825de2c2400f05af3da607b37fb72bdd.jpg)  
+(d) Recovered reward.
+
+Results on MuJoCo control. To answer the first question and validate the effectiveness of the learned reward, we evaluate CLARE on different tasks using limited state-action tuples sampled from D4RL datasets. The ranges of standard deviations of the results in Exp. & Rand., Exp. & Med. and Exp. are 156.4-280.5, 15.7-127.8 and 42.4-89.5, respectively. As shown in Table 6, CLARE yields the best performance by a significant margin on almost all datasets, especially with low-quality data thereof. It demonstrates that the reward function learned by CLARE can effectively guide offline policy search while exploiting the useful knowledge in the diverse data.
+
+Results under different dataset sizes. To answer the second question, we vary the total numbers of state-action tuples from 2k to 100k and present the results on different tasks in Figure 2. CLARE reaches expert performance on each task with sufficient data. Albeit with very limited data, CLARE also achieves strong performance over existing algorithms, revealing its great sample efficiency.
+
+Results under different  $u$ . To answer the third question, we normalize the uncertainty measure to [0, 1] and vary  $u$  from 0.1 to 1.0. Due to Eq. (11), a smaller  $u$  corresponds to a more conservative CLARE. As illustrated in Figure 3(a), the performance becomes better with the decrease of  $u$  value. It validates the importance of the embedded conservatism in alleviating the extrapolation error. We empirically find that the performance with respect to  $u$  varies in different tasks. Thus, we treat it as a hyper-parameter to tune In practice.
+
+Convergence speed. To answer the fourth question, we present the results on the convergence speed of CLARE in Figure 3(b), revealing its great learning efficiency. It showcases that CLARE converges in 5 iterations with totally less than  $50\mathrm{k}$  gradient steps.
+
+Recovered reward function. To answer the last question, we evaluate the learned reward function by transferring it to the real environment. As demonstrated in Figure 3(c), the reward function is highly instructive for online learning. It implies that it can effectively reduce the reward extrapolation error and represent the task preferences well. Surprisingly, compared to the true reward function, the policy trained via the learned one performs more stably. The reason is that the learned one incorporates conservatism and thus is capable of penalizing risks and guide safe policy search.
+
+# 7 RELATED WORK
+
+Offline IRL. To side-step the expensive online environmental interactions in classic IRL, offline IRL aims to infer a reward function and recover the expert policy only from a static dataset with no access to the environment. Klein et al. (2011) extend the classic apprenticeship learning (i.e., Abbeel & Ng (2004)) to batch and off-policy cases by introducing a temporal difference method, namely LSTD-  $\mu$  , to compute the feature expectations therein. Klein et al. (2012) further introduce a linearly parameterized score function-based multi-class classification algorithm to output reward function based on an estimate of expert feature expectation. Herman et al. (2016) present a gradient-based solution that simultaneously estimates the feature weights and parameters of the transition model by taking into account the bias of the demonstrations. Lee et al. (2019) propose Deep Successor Feature Networks (DSFN) that estimates feature expectations in an off-policy setting. However, the assumption of full knowledge of the reward feature functions in Klein et al. (2011); Herman et al. (2016); Lee et al. (2019); Jain et al. (2019); Pirotta & Restelli (2016); Ramponi et al. (2020) is often unrealistic, because the choice of features is problem-dependent and can become a very hard task for complex problems (Arora & Doshi, 2021; Piot et al., 2014). To address this problem, Piot et al. (2014) propose a non-parametric algorithm, called RCAL, using boosting method to minimize directly the criterion without the step of choosing features. Konyushkova et al. (2020) propose two semi-supervised learning algorithms that learn a reward function from limited human reward annotations. Zolna et al. (2020) further propose ORIL that can learn from both expert demonstrations and a large unlabeled set of experiences without human annotations. Chan & van der Schaar (2021) use a variational method to jointly learn an approximate posterior distribution over the reward and policy. Garg et al. (2021) propose an off-policy IRL approach, namely IQ-Learn, implicitly representing both reward and policy via a learned soft Q-function. Nevertheless, these methods primarily concentrate on offline policy learning with learning reward function being an intermediate step. Due to the intrinsic covariate shift, these methods may suffer from severe reward extrapolation error, leading to misguidance in unseen environments and low learning efficiency.
+
+Offline IL. Akin to offline IRL, offline imitation learning (offline IL) deals with training an agent to directly mimic the actions of a demonstrator in an entirely offline fashion. Behavioral cloning (BC (Ross & Bagnell, 2010)) is indeed an intrinsically offline solution, but it fails to exploit precious dynamics information. To tackle this issue, several recent works propose dynamics-aware offline IL approaches, e.g., Kostrikov et al. (2019); Jarrett et al. (2020); Chang et al. (2021); Swamy et al. (2021). In contrast to directly mimicking the expert as done in offline IL, offline IRL explicitly learns the expert's reward function from offline datasets, which can take into account the temporal structure and inform what the expert wishes to achieve, rather than simply what they are reacting to. It enables agents to understand and generalize these "intentions" when encountering similar environments and therefore makes offline IRL more robust (Lee et al., 2019). In addition, the learned reward function can succinctly explain the expert's objective, which is also useful in a number of broader applications (e.g., task description Ng et al. (2000) and transfer learning Herman et al. (2016)).
+
+# 8 CONCLUSION
+
+This paper introduces a new offline IRL algorithm (namely CLARE) to approaching the reward extrapolation error (caused by covariate shift) via incorporating conservatism into a learned reward function and utilizing an estimated dynamics model. Our theoretical analysis characterizes the impact of covariate shift by quantifying a subtle two-tier exploitation-exploration tradeoffs, and we show that CLARE can provably alleviate the reward extrapolation error by striking the right trade-offs therein. Extensive experiments corroborate that CLARE outperforms existing methods in continuous, high-dimensional environments by a significant margin, and the learned reward function represents the task preferences well.
+
+# ACKNOWLEDGMENTS
+
+This research was supported in part by the National Natural Science Foundation of China under Grant No. 62122095, 62072472, and U19A2067, by NSF Grants CNS-2203239, CNS-2203412, and RINGS-2148253, and by a grant from the Guoqiang Institute, Tsinghua University.
+
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+
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+
+# A EXPERIMENTAL DETAILS
+
+In this section, we present necessary experimental details for reproducibility.
+
+# A.1 PRACTICAL IMPLEMENTATION DETAILS
+
+In the experiment, our implementation is built upon the open source framework of offline RL algorithms, provided at: https://github.com/polixir/OfflineRL, including data sampling, policy testing, dynamics model structure, etc. The implementation of SAC in the policy improvement uses the open source code available at: https://github.com/pranz24/pytorch-soft-actor-critic (under the MIT License). Additionally, the expert and diverse state-action pairs are sampled at random from the D4RL dataset provided at: https://github.com/rail-berkeley/d4rl (under the Apache License 2.0).
+
+Model learning. Following the same line as in Yu et al. (2020; 2021), we model the transition dynamics by an ensemble of probabilistic neural networks, each of which takes the current state and action as input and outputs a Gaussian distribution over next states, i.e.,  $\{\widehat{T}_i(s'|s,a) = \mathcal{N}(\mu_i(s,a),\Sigma_i(s,a))\}_{i=1}^N$ . Using offline state-action pairs, 7 models are trained independently via maximum likelihood, each of which is represented as by a 4-layer feedforward neural network with 256 hidden units. The best 5 models are picked based on the validation prediction error on a held-out set. During model rollouts, one model will be selected randomly from the ensemble.
+
+Policy improvement. We represent both critic and actor as a 2-layer feedforward neural network with 256 hidden units and Swish activation functions. In each iteration, we update the critic and actor networks using SAC (Haarnoja et al., 2018) for 500 epochs (each has 20 gradient updates). As described in Section 5, we use a KL divergence with the behavior policy to accelerate inner-loop policy search. It is implemented by adding  $-\mathbb{E}_{s,a\sim \mathcal{D}'}[\log \pi (a|s)]$  ( $\mathcal{D}' \in \mathcal{D}$ ) to the actor loss. An instantiation of the policy improvement can be found in Algorithm 2.
+
+Algorithm 2: Safe policy improvement  
+Input: offline dataset  $\mathcal{D}$  , policy regularizer weight  $\lambda$  , rollout horizon  $H$  , rollout batchsize  $B$  the number of epochs  $E$  , dynamics ensemble  $\{\widehat{T_i}\}_{i = 1}^N$  , reward function  $r_{\phi}$  , policy  $\pi_{\theta}$  Initialize model buffer  $\mathcal{D}_{\mathrm{model}}\gets \emptyset$  .   
+for epoch  $= 1$  to  $E$  do   
+for  $b = 1$  to  $B$  in parallel do Sample state  $s_1$  from  $\mathcal{D}$  as the initial state of the rollout; for  $h = 1$  to  $H$  do Sample action  $a_{h}\sim \pi_{\theta}(\cdot |s_{h})$  Randomly pick dynamics  $\widehat{T}$  from  $\{\widehat{T_i}\}_{i = 1}^N$  and sample  $s_{h + 1}\sim \widehat{T} (\cdot |s_h,a_h)$  Compute  $r_h\gets r_\phi (s_h,a_h)$  Add sample  $(s_h,a_h,r_h,s_{h + 1})$  to  $\mathcal{D}_{\mathrm{model}}$  end   
+end Sample batches from  $\mathcal{D}_{\mathrm{model}}$  and use SAC to update policy  $\pi_{\theta}$  with  $-\mathbb{E}_{s,a\sim \mathcal{D}'}[\log \pi_{\theta}(a|s)]$ $(\mathcal{D}'\in \mathcal{D})$  added on the policy loss;   
+end
+
+Reward updating. We represent the reward function as a 4-layer feedforward neural network with 256 hidden units and Swish activate functions. In each iteration, the reward function is updated by 5 gradient steps with stepsize  $5 \times 10^{-5}$ , based on the following practical reward loss:
+
+$$
+\begin{array}{l} L \left(r _ {\phi}\right) \doteq Z _ {\beta} \mathbb {E} _ {\mathcal {D} _ {\text {r e p l a y}}} \left[ r _ {\phi} (s, a) \right] + Z _ {\beta} \mathbb {E} _ {s, a \sim \mathcal {D} \cup \mathcal {D} _ {\text {r e p l a y}}} \left[ r _ {\phi} (s, a) ^ {2} \right] \\ - \mathbb {E} _ {s, a \sim \mathcal {D} _ {E}} [ r _ {\phi} (s, a) ] - \mathbb {E} _ {s, a \sim \mathcal {D}} [ \beta (s, a) r _ {\phi} (s, a) ]. \tag {12} \\ \end{array}
+$$
+
+We use replay buffer  $\mathcal{D}_{\mathrm{replay}}$  across iterations to save the simulated data for training stability. An instantiation of reward updating is shown in Algorithm 3.
+
+Practical algorithm. Based on Algorithms 2 and 3, a detailed CLARE algorithm is outlined in Algorithm 4.
+
+Algorithm 3: Conservative reward updating  
+Input: expert data  $\mathcal{D}_E$  , diverse data  $\mathcal{D}_B$  , replay buffer  $\mathcal{D}_{\mathrm{replay}}$  , model buffer  $\mathcal{D}_{\mathrm{model}}$  , reward function  $r_{\phi}$  , learning rate  $\eta$  , the number of steps  $T$    
+Update replay buffer  $\mathcal{D}_{\mathrm{replay}}\gets \mathcal{D}_{\mathrm{replay}}\cup \mathcal{D}_{\mathrm{model}}$  .   
+for  $t = 1$  to  $T$  do Update the parameters of reward function  $r_{\phi}$  by  $\phi \leftarrow \phi -\eta \nabla L(r_{\phi})$    
+end
+
+Algorithm 4: Conservative model-based reward learning (CLARE)  
+Input: expert data  $\mathcal{D}_E$  , diverse data  $\mathcal{D}_B$  , bar  $u$  , learning rate  $\eta$  , policy regularizer weight  $\lambda$  Learn dynamics model  $\hat{T}$  represented by an ensemble of neural networks using all offline data; Set weight  $\beta (s,a)$  for each offline state-action tuple  $(s,a)\in \mathcal{D}_E\cup \mathcal{D}_B$  by Eq. (11); Initialize the policy  $\pi_{\theta}$  and reward function  $r_{\phi}$  parameterized by  $\theta$  and  $\phi$  respectively; Initialize replay buffer  $\mathcal{D}_{\mathrm{replay}}\gets \emptyset$  while not done do (Safe policy improvement) Run Algorithm 2 to update policy  $\pi_{\theta}$  and get model buffer  $\mathcal{D}_{\mathrm{model}}$  (Conservative reward updating) Run Algorithm 3 to update reward function  $r_{\phi}$  end
+
+# A.2 HYPERPARAMETERS
+
+We summarize the hyperparameters used in the evaluation as follows.
+
+Conservatism level  $u$ . For all tasks, we normalize the uncertainty measure to [0, 1] and test  $u$  from set  $\{0.4, 0.6, 0.8\}$ . The result is shown in Table 3. In each experiment, we select the  $u$  value that achieves the maximum corresponding score.
+
+Learning rates. For all experiments, the reward learning rate is  $\eta = 5\times 10^{-5}$ . Our empirical studies indicate that a relatively small reward learning rate leads to more stable training. Additionally, the learning rates for actor and critic are both  $3\times 10^{-4}$ , and that for dynamics model is  $10^{-3}$ .
+
+Policy regularization. For all experiments, the policy regularization weight is  $\lambda = 0.25$ .
+
+The additional hyperparameters are listed in Table A.2.
+
+Table 2: Hyperparameters for CLARE. Instead of  $u$ , the hyperparameters used in the evaluation are identical across different tasks (Half-Cheetah, Walker2d, Hopper, and Ant).  
+
+<table><tr><td>Hyperparameter</td><td>Value</td></tr><tr><td>Reward learning rate (η)</td><td>5 × 10-5</td></tr><tr><td>Rollout batchsize (B)</td><td>5000</td></tr><tr><td>Rollout horizon (H)</td><td>5</td></tr><tr><td>Policy regularization weight (λ)</td><td>0.25</td></tr><tr><td>Discount factor (γ)</td><td>0.99</td></tr><tr><td># steps per reward updating (T)</td><td>5</td></tr><tr><td># epochs (E)</td><td>500</td></tr><tr><td># steps per epoch</td><td>20</td></tr><tr><td>Actor learning rate</td><td>3 × 10-4</td></tr><tr><td>Critic learning rate</td><td>3 × 10-4</td></tr></table>
+
+# A.3 MORE EXPERIMENTAL RESULTS
+
+We further evaluate CLARE by answering the following two questions: 1) Can CLARE exploit the useful information from diverse datasets? 2) How does CLARE perform compared to the simple
+
+Table 3: Performance under different  $u$  values. We tune  $u$  from set  $\{ {0.4},{0.6},{0.8}\}$  . For each MuJoCo task, the experiments are carried out with three data combinations: 1)  ${10}\mathrm{k}$  expert state-action tuples, 2)  $5\mathrm{k}$  expert and  $5\mathrm{k}$  medium state-action tuples,and 3)  $5\mathrm{k}$  expert and  $5\mathrm{k}$  random state-action tuples. The highest score across different  $u$  is bold.  
+
+<table><tr><td>Dataset type</td><td>Environment</td><td>u = 0.4</td><td>u = 0.6</td><td>u = 0.8</td></tr><tr><td rowspan="4">Exp. &amp; Rand.</td><td>Walker2d</td><td>2896.93</td><td>2989.79</td><td>1083.17</td></tr><tr><td>Hopper</td><td>1187.22</td><td>1841.15</td><td>1508.09</td></tr><tr><td>Ant</td><td>2047.98</td><td>1496.09</td><td>1337.01</td></tr><tr><td>Half-Cheetah</td><td>453.03</td><td>1118.58</td><td>849.42</td></tr><tr><td rowspan="4">Exp. &amp; Med.</td><td>Walker2d</td><td>3334.55</td><td>3680.78</td><td>3275.21</td></tr><tr><td>Hopper</td><td>1722.44</td><td>2107.90</td><td>1963.59</td></tr><tr><td>Ant</td><td>3568.49</td><td>3805.64</td><td>2635.30</td></tr><tr><td>Half-Cheetah</td><td>4955.23</td><td>4349.88</td><td>4000.17</td></tr><tr><td rowspan="4">Exp.</td><td>Walker2d</td><td>4674.52</td><td>4958.04</td><td>4742.20</td></tr><tr><td>Hopper</td><td>1954.04</td><td>2605.82</td><td>2328.22</td></tr><tr><td>Ant</td><td>2747.10</td><td>3925.90</td><td>3330.48</td></tr><tr><td>Half-Cheetah</td><td>5050.05</td><td>4942.20</td><td>4542.45</td></tr></table>
+
+combination of MORL and (online) IRL methods? 3) What is the impact of reward weighting? 4) What is the impact of expert sample sizes?
+
+Exploitation on diverse data. Table 4 shows the results under different data combinations. By using additional medium data, the performance can be improved over that only using 5k expert tuples. The underlying rationale is: 1) The diverse datasets contain some good state-actions; 2) the diverse data support enables CLARE to safely generalize to the states beyond expert data manifold.
+
+Table 4: Impact of diverse data. For each MuJoCo task, the experiments are carried out with three data combinations: 1) 10k expert state-action tuples, 2) 5k expert state-action tuples, 3) 5k expert and 5k medium state-action tuples, and 4) 5k expert and 5k random state-action tuples.  
+
+<table><tr><td>Task</td><td>Exp. (5k) &amp; Rand. (5k)</td><td>Exp. (5k) &amp; Med. (5k)</td><td>Exp. (5k)</td><td>Exp. (10k)</td></tr><tr><td>Walker2d</td><td>2973.88</td><td>3613.49</td><td>2858.29</td><td>4990.57</td></tr><tr><td>Hopper</td><td>1891.55</td><td>2135.07</td><td>1885.76</td><td>2604.59</td></tr><tr><td>Ant</td><td>1960.05</td><td>3879.48</td><td>1978.08</td><td>3940.30</td></tr><tr><td>Half-Cheetah</td><td>1113.75</td><td>4888.64</td><td>1714.30</td><td>4975.17</td></tr></table>
+
+Expert sample sizes. Table 5 shows the average returns (over 5 random seeds) under different expert sample sizes with the fixed number of medium data (50k). It can corroborate our analytical results that with a relatively sufficient data coverage of the empirical expert behaviors, the performance is dominated by the expert sample size (combining Theorem 4.2, Theorem 4.3 and Corollary 4.1).
+
+Table 5: Results under different expert sample sizes.  
+
+<table><tr><td>Dataset</td><td>2k</td><td>5k</td><td>10k</td><td>20k</td><td>50k</td><td>100k</td></tr><tr><td>Half-Cheetah</td><td>4753.9</td><td>4978.2</td><td>5206.5</td><td>7865.5</td><td>10930.1</td><td>11121.9</td></tr><tr><td>Hopper</td><td>1989.9</td><td>2273.4</td><td>2507.8</td><td>2991.8</td><td>3571.1</td><td>3566.4</td></tr><tr><td>Walker</td><td>3439.9</td><td>3632.2</td><td>4753.3</td><td>4982.4</td><td>4977.8</td><td>4991.3</td></tr><tr><td>Ant</td><td>3375.4</td><td>3866.5</td><td>3968.9</td><td>4385.7</td><td>4797.5</td><td>4910.6</td></tr></table>
+
+Comparison to MOMAX. To demonstrate the poor performance of the naive approach using a simple combination of IRL with model-based offline forward RL (MORL) method, we design a baseline directly using a state-of-the-art MORL method, COMBO (Yu et al., 2021), in the inner loop of MaxEnt IRL (Eq. (1)), called MOMAX. As shown in Figure 4, MOMAX does not work well in
+
+![](images/83af61630e0592e61fd6e7562d42604f67d97be0d39694737609c26f324c4f7f.jpg)  
+Figure 4: Comparison to MOMAX. Each experiment uses 10k expert and 10k medium state-actions.
+
+![](images/3ac664152dee7afd1d0a50b0eca7d6b7a5922e2319292086a3fa340baf78d893.jpg)
+
+![](images/f8691e116b979fc79094f386be64067a7d21e75cfd8630481a420ac28603f49b.jpg)
+
+![](images/5c5a062bb3b8a584ef86fa5dc7b9d301557943488e6db98a2a90a403473bf91d.jpg)
+
+![](images/1c135a38502a7dce604c3d1c9a216a687818006439f4e7aa32d551c55a0adcef.jpg)  
+Figure 5: Ablation study of reward weighting. Each experiment uses 5k expert and 50k medium state-action samples from the D4RL benchmark.
+
+![](images/107cd5608a302a11649158e7a8527ac5a24c43b63560c20e1bbdf07613697a59.jpg)
+
+![](images/61e37249f0e445d65075b7236acf63daf4a8e98733302bc533e8174f98d1b95a.jpg)
+
+![](images/fc67768109d16e6b51da2163f074fdb8ea083e857d68c92e596e48f11759ec7b.jpg)
+
+these continuous control tasks. It reveals the challenges of repurposing the online IRL methods in the offline IRL setting.
+
+Ablation study of reward weighting. Fig. 5 shows the impact of reward weighting on performance. CLARE basically reduces to MaxEnt IRL with no reward weighting and thus can not deal with the extrapolation error effectively in offline learning. Fig. 5 also demonstrates that the conservative reward function can stabilize the training.
+
+# A.4 COMPUTATIONAL COMPLEXITY
+
+We implement the code in PyTorch 1.11.0 on a server with a 32-Cores AMD Ryzen Threadripper PRO 3975WX and a Intel GeForce RTX 3090 Ti. For all tasks, CLARE converges in one hour (around 5-10 iterations with total 50k-100k gradient steps).
+
+# B PROOFS
+
+In this section, we provide detailed proofs of main results in Section 4.
+
+# B.1 PROOF OF THEOREM 4.1
+
+This proof is built on that for Ho & Ermon (2016, Proposition 3.1).
+
+First, it follows from Eq. (4) that
+
+$$
+\begin{array}{l} L (\pi , r) = \alpha \widehat {H} (\pi) + Z _ {\beta} \mathbb {E} _ {\hat {\rho} ^ {\pi}} [ r (s, a) ] - \mathbb {E} _ {\tilde {\rho} ^ {D}} [ \beta (s, a) r (s, a) ] - \mathbb {E} _ {\tilde {\rho} ^ {E}} [ r (s, a) ] + Z _ {\beta} \psi (r) \\ = \alpha \widehat {H} (\pi) + \sum_ {s, a} \left(Z _ {\beta} \hat {\rho} ^ {\pi} (s, a) - \tilde {\rho} ^ {D} (s, a) \beta (s, a) - \tilde {\rho} ^ {E} (s, a)\right) r (s, a) + Z _ {\beta} \psi (r) \\ = \alpha \widehat {H} (\pi) + Z _ {\beta} \sum_ {s, a} \left(\hat {\rho} ^ {\pi} (s, a) - \frac {\tilde {\rho} ^ {D} (s , a) \beta (s , a) + \tilde {\rho} ^ {E} (s , a)}{Z _ {\beta}}\right) r (s, a) + Z _ {\beta} \psi (r) \\ \end{array}
+$$
+
+$$
+\begin{array}{l} = \alpha \widehat {H} (\pi) + Z _ {\beta} \left(\mathbb {E} _ {\hat {\rho} ^ {\pi}} [ r (s, a) ] - \mathbb {E} _ {\tilde {\rho} ^ {I}} [ r (s, a) ] + \psi (r)\right). \\ (\text {d e n o t i n g} \tilde {\rho} ^ {I} (s, a) \doteq \frac {\tilde {\rho} ^ {D} (s , a) \beta (s , a) + \tilde {\rho} ^ {E} (s , a)}{Z _ {\beta}}) \\ \end{array}
+$$
+
+where the last equality holds due to
+
+$$
+\begin{array}{l} Z _ {\beta} = 1 + \mathbb {E} _ {s, a \sim \hat {\rho} ^ {D}} [ \beta (s, a) ] \\ = \sum_ {s, a} \tilde {\rho} ^ {E} (s, a) + \tilde {\rho} ^ {D} (s, a) \beta (s, a) \\ \geq 0. \quad (\text {f r o m} \beta (s, a) \geq - \tilde {\rho} ^ {E} (s, a) / \tilde {\rho} ^ {D} (s, a) \text {f o r} (s, a) \in \mathcal {D}) \\ \end{array}
+$$
+
+Thanks to Lemma 4.1, there exists a one-to-one correspondence between  $\Pi$  and  $\mathcal{C}_{\widehat{T}}$ . Thus, we can rewrite
+
+$$
+\min  _ {r \in \mathcal {R}} \max  _ {\pi \in \Pi} L (\pi , r) = \min  _ {r \in \mathcal {R}} \max  _ {\hat {\rho} \in \mathcal {C} _ {\hat {T}}} \underbrace {\alpha \widehat {H} (\pi) + Z _ {\beta} \left(\mathbb {E} _ {\hat {\rho}} [ r (s , a) ] - \mathbb {E} _ {\tilde {\rho} ^ {I}} [ r (s , a) ] + \psi (r)\right)} _ {\doteq \bar {L} (\hat {\rho}, r)}. \tag {13}
+$$
+
+It is easy to see that  $\mathcal{R}$  is compact and convex. Besides, from the proof of Ho & Ermon (2016, Proposition 3.1),  $\mathcal{C}_{\hat{T}}$  is also a compact and convex set. Accordingly, based on the concavity of  $\bar{H}$  (Lemma 4.2), the minimax theorem holds (Du & Pardalos, 2013), and hence we have
+
+$$
+\begin{array}{l} \min _ {r \in \mathcal {R}} \max _ {\hat {\rho} \in \mathcal {C} _ {\hat {T}}} \bar {L} (\hat {\rho}, r) = \max _ {\hat {\rho} \in \mathcal {C} _ {\hat {T}}} \min _ {r \in \mathcal {R}} \bar {L} (\hat {\rho}, r) \\ = \max  _ {\hat {\rho} \in \mathcal {C} _ {\hat {T}}} \alpha \bar {H} (\hat {\rho}) + Z _ {\beta} \left(\min  _ {r \in \mathcal {R}} \mathbb {E} _ {\hat {\rho}} [ r (s, a) ] - \mathbb {E} _ {\tilde {\rho} ^ {I}} [ r (s, a) ] + \psi (r)\right) \\ = \max  _ {\hat {\rho} \in \mathcal {C} _ {\hat {T}}} \alpha \bar {H} (\hat {\rho}) + Z _ {\beta} \psi^ {*} \left(\tilde {\rho} ^ {I} - \tilde {\rho}\right) \quad (f r o m \\ = \max  _ {\tilde {\rho} \in \mathcal {C} _ {\hat {T}}} \alpha \bar {H} (\hat {\rho}) + Z _ {\beta} D _ {\psi} (\tilde {\rho}, \tilde {\rho} ^ {I}). \tag {14} \\ \end{array}
+$$
+
+Additionally, denote  $r^*$  and  $\hat{\rho}^*$  as
+
+$$
+r ^ {*} \in \arg \operatorname * {m i n} _ {r \in \mathcal {R}} \max  _ {\hat {\rho} \in \mathcal {C} _ {\hat {T}}} \bar {L} (\hat {\rho}, r), \quad \hat {\rho} ^ {*} \in \arg \max  _ {\hat {\rho} \in \mathcal {C} _ {\hat {T}}} \alpha \bar {H} (\hat {\rho}) + Z _ {\beta} D _ {\psi} (\tilde {\rho}, \tilde {\rho} ^ {I}). \tag {15}
+$$
+
+Due to Eq. (14),  $(r^{*},\hat{\rho}^{*})$  is a saddle point of  $\bar{L}$ , and thus  $\hat{\rho}^{*}\in \arg \max_{\hat{\rho}\in \mathcal{C}_{\hat{T}}}\bar{L} (\hat{\rho},r^{*})$ . By Lemma 4.1, it is easy to see that policy  $\pi^{*}$  (that corresponds to  $\hat{\rho}^*$ ) satisfies  $\pi^{*}\in \arg \max_{\pi \in \Pi}L(\pi ,r^{*})$ , thereby completing the proof.
+
+# B.2 PROOF OF THEOREM 4.2
+
+We present the following two lemmas before our main result.
+
+Lemma B.1. Denoting  $p_1(x,y) = q_1(x)q_1(y|x)$  and  $p_2(x,y) = q_2(x)q_2(y|x)$  as two joint distributions over finite spaces, we can bound the total variation distance (TVD) between  $p_1$  and  $p_2$  as
+
+$$
+D _ {\mathrm {T V}} \left(p _ {1}, p _ {2}\right) \leq E _ {x \sim q _ {1} (x)} \left[ D _ {\mathrm {T V}} \left(q _ {1} (\cdot | x), q _ {2} (\cdot | x)\right) \right] + D _ {\mathrm {T V}} \left(q _ {1}, q _ {2}\right). \tag {16}
+$$
+
+Proof. The proof is straight-forward:
+
+$$
+\begin{array}{l} D _ {\mathrm {T V}} (p _ {1}, p _ {2}) = \frac {1}{2} \sum_ {x, y} \left| p _ {1} (x, y) - p _ {2} (x, y) \right| \\ = \frac {1}{2} \sum_ {x, y} \left| q _ {1} (x) q _ {1} (y | x) - q _ {2} (x) q _ {2} (y | x) \right| \\ = \frac {1}{2} \sum_ {x, y} \left| q _ {1} (x) q _ {1} (y | x) - q _ {1} (x) q _ {2} (y | x) + q _ {1} (x) q _ {2} (y | x) - q _ {2} (x) q _ {2} (y | x) \right| \\ \leq \frac {1}{2} \sum_ {x, y} q _ {1} (x) \left| q _ {1} (y | x) - q _ {2} (y | x) \right| + \frac {1}{2} \sum_ {x, y} q _ {2} (y | x) \left| q _ {1} (x) - q _ {2} (x) \right| \\ \end{array}
+$$
+
+$$
+\begin{array}{l} = \sum_ {x} q _ {1} (x) \cdot \frac {1}{2} \sum_ {y} | q _ {1} (y | x) - q _ {2} (y | x) | + \frac {1}{2} \sum_ {x} | q _ {1} (x) - q _ {2} (x) | \sum_ {y} q _ {2} (y | x) \\ = E _ {x \sim q _ {1} (x)} \left[ D _ {\mathrm {T V}} \left(q _ {1} (y | x), q _ {2} (y | x)\right) \right] + D _ {\mathrm {T V}} \left(q _ {1}, q _ {2}\right), \tag {17} \\ \end{array}
+$$
+
+where the last equality is obtained due to  $\sum_{y}q_{2}(y|x) = 1$
+
+Lemma B.2. Suppose that we have two Markov chain transition distributions  $T_{1}(s^{\prime}|s)$  and  $T_{2}(s^{\prime}|s)$ , and the initial state distributions are the same. Then, for each  $h \in [1,2,\ldots)$ , the TVD of state marginals in time step  $h$  is bounded as
+
+$$
+D _ {\mathrm {T V}} \left(p _ {1} ^ {h}, p _ {2} ^ {h}\right) \leq \sum_ {h ^ {\prime} = 0} ^ {h - 1} E _ {s \sim p _ {2} ^ {h ^ {\prime}}} \left[ D _ {\mathrm {T V}} \left(T _ {1} (\cdot | s), T _ {2} (\cdot | s)\right) \right], \tag {18}
+$$
+
+where  $p_i^h (s)\doteq \operatorname *{Pr}(s_h = s\mid T_i,\mu)$  for  $i = 1,2$
+
+Proof. First, we have
+
+$$
+\begin{array}{l} \left| p _ {1} ^ {h} (s) - p _ {2} ^ {h} (s) \right| \\ = \left| \sum_ {s ^ {\prime}} T _ {1} (s | s ^ {\prime}) p _ {1} ^ {h - 1} \left(s ^ {\prime}\right) - \sum_ {s ^ {\prime}} T _ {2} (s | s ^ {\prime}) p _ {2} ^ {h - 1} \left(s ^ {\prime}\right) \right| \\ \leq \sum_ {s ^ {\prime}} \left| T _ {1} (s | s ^ {\prime}) p _ {1} ^ {h - 1} (s ^ {\prime}) - T _ {2} (s | s ^ {\prime}) p _ {2} ^ {h - 1} (s ^ {\prime}) \right| \\ = \sum_ {s ^ {\prime}} \left| T _ {1} (s | s ^ {\prime}) p _ {1} ^ {h - 1} (s ^ {\prime}) - T _ {1} (s | s ^ {\prime}) p _ {2} ^ {h - 1} (s ^ {\prime}) + T _ {1} (s | s ^ {\prime}) p _ {2} ^ {h - 1} (s ^ {\prime}) - T _ {2} (s | s ^ {\prime}) p _ {2} ^ {h - 1} (s ^ {\prime}) \right| \\ \leq \sum_ {s ^ {\prime}} \left(T _ {1} (s | s ^ {\prime}) \left| p _ {1} ^ {h - 1} (s ^ {\prime}) - p _ {2} ^ {h - 1} (s ^ {\prime}) \right| + p _ {2} ^ {h - 1} (s ^ {\prime}) \left| T _ {1} (s | s ^ {\prime}) - T _ {2} (s | s ^ {\prime}) \right|\right) \\ = \sum_ {s ^ {\prime}} T _ {1} \left(s \mid s ^ {\prime}\right) \left| p _ {1} ^ {h - 1} \left(s ^ {\prime}\right) - p _ {2} ^ {h - 1} \left(s ^ {\prime}\right) \right| + E _ {s ^ {\prime} \sim p _ {2} ^ {h - 1}} \left[ \left| T _ {1} \left(s \mid s ^ {\prime}\right) - T _ {2} \left(s \mid s ^ {\prime}\right) \right| \right]. \tag {19} \\ \end{array}
+$$
+
+Thus, we can write
+
+$$
+\begin{array}{l} D _ {\mathrm {T V}} (p _ {1} ^ {h}, p _ {2} ^ {h}) \\ = \frac {1}{2} \sum_ {s} \left| p _ {1} ^ {h} (s) - p _ {2} ^ {h} (s) \right| \\ \leq \frac {1}{2} \sum_ {s} E _ {s ^ {\prime} \sim p _ {2} ^ {h - 1}} \left[ \left| T _ {1} (s | s ^ {\prime}) - T _ {2} (s | s ^ {\prime}) \right| \right] + \frac {1}{2} \sum_ {s} \sum_ {s ^ {\prime}} T _ {1} (s | s ^ {\prime}) \left| p _ {1} ^ {h - 1} (s ^ {\prime}) - p _ {2} ^ {h - 1} (s ^ {\prime}) \right| (usingEq.(19)) \\ = \frac {1}{2} \sum_ {s} E _ {s ^ {\prime} \sim p _ {2} ^ {h - 1}} \left[ \left| T _ {1} (s | s ^ {\prime}) - T _ {2} (s | s ^ {\prime}) \right| \right] + \frac {1}{2} \sum_ {s ^ {\prime}} \left| p _ {1} ^ {h - 1} (s ^ {\prime}) - p _ {2} ^ {h - 1} (s ^ {\prime}) \right| \sum_ {s} T _ {1} (s | s ^ {\prime}) \\ = E _ {s ^ {\prime} \sim p _ {2} ^ {h - 1}} \left[ \frac {1}{2} \sum_ {s} \left| T _ {1} (s | s ^ {\prime}) - T _ {2} (s | s ^ {\prime}) \right| \right] + \frac {1}{2} \sum_ {s ^ {\prime}} \left| p _ {1} ^ {h - 1} (s ^ {\prime}) - p _ {2} ^ {h - 1} (s ^ {\prime}) \right| \\ (\text {u s i n g} \sum_ {s} T _ {1} (s | s ^ {\prime}) = 1) \\ = E _ {s ^ {\prime} \sim p _ {2} ^ {h - 1}} \left[ D _ {\mathrm {T V}} \left(T _ {1} (\cdot | s ^ {\prime}), T _ {2} (\cdot | s ^ {\prime})\right) \right] + D _ {\mathrm {T V}} \left(p _ {1} ^ {h - 1}, p _ {2} ^ {h - 1}\right) (20) \\ \leq \sum_ {h ^ {\prime} = 0} ^ {h - 1} E _ {s \sim p _ {2} ^ {h ^ {\prime}}} \left[ D _ {\mathrm {T V}} \left(T _ {1} (\cdot | s), T _ {2} (\cdot | s)\right) \right] + D _ {\mathrm {T V}} \left(p _ {1} ^ {0}, p _ {2} ^ {0}\right) \quad \text {(i t e r a t i v e l y u s i n g E q . (2 0))} \\ = \sum_ {h ^ {\prime} = 0} ^ {h - 1} E _ {s \sim p _ {2} ^ {h ^ {\prime}}} \left[ D _ {\mathrm {T V}} \left(T _ {1} (\cdot | s), T _ {2} (\cdot | s)\right) \right], \quad \text {(d u e t o s a m e i n i t i a l s t a t e d i s t r i b u t i o n s)} \\ \end{array}
+$$
+
+which completes the proof.
+
+Observe that Lemma B.1 bounds the TVD of a joint distribution by the TVDs of its corresponding conditional and marginal distributions, and that Lemma B.2 bounds the difference of two MDPs' state visitations in each time step by the cumulative dynamics differences. Next, we provide the following lemma that bounds the difference between the expert's and learned policy's occupancy measures from above.
+
+Lemma B.3. For each  $\hat{\rho} \in \mathcal{C}_{\widehat{T}}$ , denote  $\hat{\pi}$  as its corresponding stationary policy, i.e.,  $\hat{\pi} \doteq \hat{\rho}(s,a)/\sum_{a'} \hat{\rho}(s,a')$ , and  $\hat{\rho}^{\hat{\pi}}$  denote the occupancy measure of  $\hat{\pi}$  under true transition dynamics  $T$ . Then, the following holds:
+
+$$
+D _ {\mathrm {T V}} \left(\rho^ {\hat {\pi}}, \rho^ {E}\right) \leq \frac {\gamma}{1 - \gamma} \mathbb {E} _ {s, a \sim \hat {\rho}} \left[ D _ {\mathrm {T V}} \left(T (\cdot | s, a), \widehat {T} (\cdot | s, a)\right) \right] + D _ {\mathrm {T V}} (\hat {\rho}, \tilde {\rho} ^ {E}) + D _ {\mathrm {T V}} \left(\tilde {\rho} ^ {E}, \rho^ {E}\right), \tag {21}
+$$
+
+where  $\rho^{E}$  is the occupancy measure of expert policy  $\pi^{E}$  under the true transition dynamics.
+
+Proof. For conciseness, let  $\rho_1 \doteq \rho^{\hat{\pi}}$  and  $\rho_2 \doteq \hat{\rho}$ . Using the triangle inequality, it is easy to see that
+
+$$
+D _ {\mathrm {T V}} \left(\rho_ {1}, \rho^ {E}\right) \leq D _ {\mathrm {T V}} \left(\rho_ {1}, \rho_ {2}\right) + D _ {\mathrm {T V}} \left(\rho_ {2}, \tilde {\rho} ^ {E}\right) + D _ {\mathrm {T V}} \left(\tilde {\rho} ^ {E}, \rho^ {E}\right), \tag {22}
+$$
+
+where  $\tilde{\rho}^E$  is the empirical occupancy measure of expert policy  $\pi^E$ . To bound  $D_{\mathrm{TV}}(\rho_1,\rho_2)$ , denoting  $p_1^h (s,a)\doteq \operatorname *{Pr}(s_h = s,a_h = a\mid T,\hat{\pi},\mu)$  and  $p_2^h (s,a)\doteq \operatorname *{Pr}(s_h = s,a_h = a\mid \widehat{T},\hat{\pi},\mu)$  (the difference between them is marked in red), we can write
+
+(using the definition of occupancy measure in Section 2)
+
+$$
+\begin{array}{l} D _ {\mathrm {T V}} (\rho_ {1}, \rho_ {2}) = \frac {1}{2} \sum_ {s, a} \left| \rho_ {1} (s, a) - \rho_ {2} (s, a) \right| \\ = \frac {1}{2} \sum_ {s, a} \left| (1 - \gamma) \sum_ {h = 0} ^ {\infty} \gamma^ {h} p _ {1} ^ {h} (s, a) - (1 - \gamma) \sum_ {h = 0} ^ {\infty} \gamma^ {h} p _ {2} ^ {h} (s, a) \right| \\ = \frac {1 - \gamma}{2} \sum_ {s, a} \left| \sum_ {h = 0} ^ {\infty} \gamma^ {h} \left(p _ {1} ^ {h} (s, a) - p _ {2} ^ {h} (s, a)\right) \right| \\ \leq \frac {1 - \gamma}{2} \sum_ {h = 0} ^ {\infty} \sum_ {s, a} \gamma^ {h} \left| p _ {1} ^ {h} (s, a) - p _ {2} ^ {h} (s, a) \right| \\ = (1 - \gamma) \sum_ {h = 0} ^ {\infty} \gamma^ {h} \cdot \frac {1}{2} \sum_ {s, a} \left| p _ {1} ^ {h} (s, a) - p _ {2} ^ {h} (s, a) \right| \\ = (1 - \gamma) \sum_ {h = 0} ^ {\infty} \gamma^ {h} \cdot D _ {\mathrm {T V}} \left(p _ {1} ^ {h} (s, a), p _ {2} ^ {h} (s, a)\right) \\ \leq (1 - \gamma) \sum_ {h = 0} ^ {\infty} \gamma^ {h} D _ {\mathrm {T V}} \left(p _ {1} ^ {h} (s), p _ {2} ^ {h} (s)\right), \tag {23} \\ \end{array}
+$$
+
+where  $p_1^h(s) \doteq \operatorname{Pr}(s_h = s \mid T, \hat{\pi}, \mu)$ ,  $p_2^h(s) \doteq \operatorname{Pr}(s_h = s \mid \widehat{T}, \hat{\pi}, \mu)$ , and the last inequation holds due to Lemma B.1 (note that  $p_1^h(s, a) = p_1^h(s) \hat{\pi}(a|s)$  and  $p_2^h(s, a) = p_2^h(s) \hat{\pi}(a|s)$ ). Denote  $T_1(s'|s) \doteq \sum_a T(s', a|s)$  and  $T_2(s'|s) \doteq \sum_a \widehat{T}(s', a|s)$ , where we slightly overload notations using  $T(s', a|s) \doteq \hat{\pi}(a|s) T(s'|s, a)$  and  $\widehat{T}(s', a|s) \doteq \hat{\pi}(a|s) \widehat{T}(s'|s, a)$ . We obtain
+
+$$
+\begin{array}{l} D _ {\mathrm {T V}} \left(T _ {1} (\cdot | s), T _ {2} (\cdot | s)\right) = \frac {1}{2} \sum_ {s ^ {\prime}} \left| T _ {1} (s ^ {\prime} | s) - T _ {2} (s ^ {\prime} | s) \right| \\ = \frac {1}{2} \sum_ {s ^ {\prime}} \left| \sum_ {a} T (s ^ {\prime}, a | s) - \widehat {T} (s ^ {\prime}, a | s) \right| \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \leq \frac {1}{2} \sum_ {s ^ {\prime}, a} \left| T (s ^ {\prime}, a | s) - \widehat {T} (s ^ {\prime}, a | s) \right| \\ = D _ {\mathrm {T V}} \left(T \left(s ^ {\prime}, a | s\right), \widehat {T} \left(s ^ {\prime}, a | s\right)\right) \\ \leq E _ {a \sim \hat {\pi} (a | s)} \left[ D _ {\mathrm {T V}} \left(T (\cdot | s, a), \hat {T} (\cdot | s, a)\right) \right]. \\ \end{array}
+$$
+
+(seeing  $\hat{\pi}(a|s)$  as  $q_1(x), q_2(x), T(s'|s, a)$  as  $q_1(y|x)$ , and  $\widehat{T}(s'|s, a)$  as  $q_2(y|x)$ , and then using Lemma B.1)
+
+Based on that, the following holds:
+
+$$
+D _ {\mathrm {T V}} \left(\rho_ {1}, \rho_ {2}\right) \leq (1 - \gamma) \sum_ {h = 0} ^ {\infty} \gamma^ {h} D _ {\mathrm {T V}} \left(p _ {1} ^ {h} (s), p _ {2} ^ {h} (s)\right) \tag {fromEq.(23)}
+$$
+
+$$
+\leq (1 - \gamma) \sum_ {h = 1} ^ {\infty} \gamma^ {h} \sum_ {h ^ {\prime} = 0} ^ {h - 1} E _ {s \sim p _ {2} ^ {h ^ {\prime}} (s)} \left[ D _ {\mathrm {T V}} \left(T _ {1} (\cdot | s), T _ {2} (\cdot | s)\right) \right]
+$$
+
+(using fact  $D_{\mathrm{TV}}(p_1^0 (s),p_2^0 (s)) = D_{\mathrm{TV}}(\mu ,\mu) = 0$  and Lemma B.2)
+
+$$
+\leq (1 - \gamma) \sum_ {h = 1} ^ {\infty} \gamma^ {h} \sum_ {h ^ {\prime} = 0} ^ {h - 1} E _ {s \sim p _ {2} ^ {h ^ {\prime}} (s)} \left[ E _ {a \sim \hat {\pi} (a | s)} \left[ D _ {\mathrm {T V}} \left(T (\cdot | s, a), \widehat {T} (\cdot | s, a)\right) \right] \right]
+$$
+
+(using the above result)
+
+$$
+= (1 - \gamma) \sum_ {h = 1} ^ {\infty} \gamma^ {h} \sum_ {h ^ {\prime} = 0} ^ {h - 1} \mathbb {E} _ {s, a \sim p _ {2} ^ {h ^ {\prime}} (s, a)} \left[ D _ {\mathrm {T V}} \left(T (\cdot | s, a), \widehat {T} (\cdot | s, a)\right) \right]
+$$
+
+noting that  $p_2^h (s,a) = p_2^h (s)\hat{\pi} (a|s))$
+
+$$
+= (1 - \gamma) \sum_ {s, a} D _ {\mathrm {T V}} \left(T (\cdot | s, a), \widehat {T} (\cdot | s, a)\right) \sum_ {h = 1} ^ {\infty} \gamma^ {h} \sum_ {h ^ {\prime} = 0} ^ {h - 1} p _ {2} ^ {h ^ {\prime}} (s, a)
+$$
+
+(expanding the expectation and rearranging terms)
+
+$$
+= \gamma (1 - \gamma) \sum_ {s, a} D _ {\mathrm {T V}} \left(T (\cdot | s, a), \widehat {T} (\cdot | s, a)\right) \underbrace {\sum_ {h = 1} ^ {\infty} \sum_ {h ^ {\prime} = 0} ^ {h - 1} \gamma^ {h - 1} p _ {2} ^ {h ^ {\prime}} (s , a)} _ {\dot {=} A}
+$$
+
+$$
+= \gamma (1 - \gamma) \sum_ {s, a} D _ {\mathrm {T V}} \left(T (\cdot | s, a), \widehat {T} (\cdot | s, a)\right) \underbrace {\sum_ {h = 1} ^ {\infty} \gamma^ {h - 1} \sum_ {h ^ {\prime} = 0} ^ {\infty} \gamma^ {h ^ {\prime}} p _ {2} ^ {h ^ {\prime}} (s , a)} _ {\doteq B}
+$$
+
+$(B$  is derived by rearranging the terms in  $A$
+
+$$
+= \gamma (1 - \gamma) \sum_ {s, a} D _ {\mathrm {T V}} \left(T (\cdot | s, a), \widehat {T} (\cdot | s, a)\right) \sum_ {h = 1} ^ {\infty} \gamma^ {h - 1} \frac {\rho_ {2} (s , a)}{1 - \gamma}
+$$
+
+noting that  $\rho_{2}(s,a) = (1 - \gamma)\sum_{h^{\prime} = 0}^{\infty}\gamma^{h^{\prime}}p_{2}^{h^{\prime}}(s,a))$
+
+$$
+\begin{array}{l} = \sum_ {s, a} D _ {\mathrm {T V}} \left(T (\cdot | s, a), \widehat {T} (\cdot | s, a)\right) \sum_ {h = 1} ^ {\infty} \gamma^ {h} \rho_ {2} (s, a) \\ = \sum_ {s, a} \rho_ {2} (s, a) D _ {\mathrm {T V}} \left(T (\cdot | s, a), \widehat {T} (\cdot | s, a)\right) \sum_ {h = 1} ^ {\infty} \gamma^ {h} \\ = \frac {\gamma}{1 - \gamma} \mathbb {E} _ {s, a \sim \rho_ {2}} \left[ D _ {\mathrm {T V}} \left(T (\cdot | s, a), \hat {T} (\cdot | s, a)\right) \right]. \tag {24} \\ \end{array}
+$$
+
+Substituting Eq. (24) in Eq. (22) gives the desired result.
+
+![](images/480269a69064a8b35fc1c0b9e17c5fad4c141554cb99c0af0966d3963f58f1a1.jpg)
+
+Denoting  $\rho^{\pi}$  as the occupancy measure of  $\pi$  under underlying dynamics model  $T$ , We can write
+
+$$
+J \left(\pi^ {E}\right) - J \left(\rho^ {\pi}\right) = \sum_ {s, a} \rho^ {E} (s, a) R (s, a) - \sum_ {s, a} \rho^ {\pi} (s, a) R (s, a) \quad \text {(f r o m t h e d e f i n i t i o n)}
+$$
+
+$$
+\begin{array}{l} = \sum_ {s, a} \left(\rho^ {E} (s, a) - \rho^ {\pi} (s, a)\right) R (s, a) \\ \leq \sum_ {s, a} \left| \rho^ {E} (s, a) - \rho^ {\pi} (s, a) \right| \quad (\text {d u e t o} | R (s, a) | \leq 1) \\ = 2 D _ {\mathrm {T V}} \left(\rho^ {\pi}, \rho^ {E}\right). \tag {25} \\ \end{array}
+$$
+
+Then, based on Lemma B.3, the desired result in Theorem 4.2 can be obtained by combining Eq. (25) with Eq. (21).
+
+# B.3 PROOF OF THEOREM 4.3
+
+Recall that  $c(s, a) = C \cdot D_{\mathrm{TV}}(T(\cdot | s, a), \widehat{T}(\cdot | s, a))$ . We define
+
+$$
+\begin{array}{l} f (\rho) \doteq \mathbb {E} _ {s, a \sim \rho} [ c (s, a) ] + 2 D _ {\mathrm {T V}} (\rho , \tilde {\rho} ^ {E}) \\ = \sum_ {s, a} c (s, a) \rho (s, a) + \left| \rho (s, a) - \tilde {\rho} ^ {E} (s, a) \right|. \tag {26} \\ \end{array}
+$$
+
+Thanks to Lemma 4.1, minimizing the RHS of Eq. (7) is equivalent to the following problem:
+
+$$
+\min  _ {\rho \in \mathcal {P} (\mathcal {S} \times \mathcal {A})} f (\rho). \tag {27}
+$$
+
+Let  $\delta(s, a) \doteq \rho(s, a) - \tilde{\rho}^E(s, a)$ . Then, Problem (27) can be transformed to the following one:
+
+$$
+\min  _ {\delta} \sum_ {s, a} c (s, a) \delta (s, a) + | \delta (s, a) | \tag {28}
+$$
+
+$$
+\text {s . t .} \sum_ {s, a} \delta (s, a) = 0 \tag {29}
+$$
+
+$$
+\delta (s, a) \geq - \tilde {\rho} ^ {E} (s, a) \quad s \in \mathcal {S}, a \in \mathcal {A}. \tag {30}
+$$
+
+For conciseness, we rewrite Problem (28)-(30) as the following form:
+
+$$
+\min  _ {\delta} g (\delta) \doteq \sum_ {i = 1} ^ {n} c _ {i} \delta_ {i} + | \delta_ {i} | \tag {31}
+$$
+
+$$
+\text {s . t .} \sum_ {i = 1} ^ {n} \delta_ {i} = 0 \tag {32}
+$$
+
+$$
+\delta_ {i} \geq - \tilde {\rho} _ {i} ^ {E} \quad i \in [ n ] \tag {33}
+$$
+
+where  $i$  corresponds to a state-action pair,  $n\dot{=} |\mathcal{S}|\cdot |\mathcal{A}|$ $[n]\doteq \{1,2,\dots ,n\}$  , and  $\delta \doteq \{\delta_i:i\in [n]\}$
+
+Due to Eq. (32) and Eq. (33),  $[n]$  can be divided into two disjoint sets,  $\mathcal{N}_1(\delta) \doteq \{i \in [n] : \delta_i > 0\}$  and  $\mathcal{N}_2(\delta) \doteq \{i \in [n] : \delta_i \leq 0\}$  ( $\mathcal{N}_1(\delta) = \emptyset$  iff all  $\delta_i = 0$ ). Thus, we can write
+
+$$
+g (\delta) = \sum_ {i \in \mathcal {N} _ {1} (\delta)} (c _ {i} + 1) \delta_ {i} + \sum_ {j \in \mathcal {N} _ {2} (\delta)} (c _ {j} - 1) \delta_ {j}. \tag {34}
+$$
+
+For any  $\delta$  meeting Constraints (32) and (33), we denote  $\delta'$  (which should be  $\delta_{\mathcal{N}_1}'$  if written in full) satisfying  $\delta_j' = -\mathbf{1}[c_j - c_1^{\min} > 2] \cdot \tilde{\rho}_j^E$  for all  $j \in \mathcal{N}_2(\delta)$ ,  $\delta_i' = 0$  for all  $i \in \mathcal{N}_1(\delta) \setminus \mathcal{N}_{\min}$ , and  $\delta_i' = \sum_{j \in \mathcal{N}_2(\delta)} \mathbf{1}[c_j - c_1^{\min} > 2] \cdot \tilde{\rho}_j^E / |\mathcal{N}_{\min}(\delta)|$  for all  $i \in \mathcal{N}_{\min}(\delta)$ , where  $\mathcal{N}_{\min}(\delta) \doteq \{i \in \mathcal{N}_1(\delta) : i \in \arg \min_{i' \in \mathcal{N}_1(\delta)} c_{i'}\}$  and  $c_1^{\min} \doteq \min_{i \in \mathcal{N}_1(\delta)} c_i$ . Then, we have
+
+$$
+\begin{array}{l} g \left(\delta^ {\prime}\right) = \sum_ {i \in \mathcal {N} _ {1} (\delta)} \left(c _ {i} + 1\right) \delta_ {i} ^ {\prime} + \sum_ {j \in \mathcal {N} _ {2} (\delta)} \left(c _ {j} - 1\right) \delta_ {j} ^ {\prime} \\ = \sum_ {i \in \mathcal {N} _ {\min } (\delta)} (c _ {i} + 1) \delta_ {i} ^ {\prime} + \sum_ {i ^ {\prime} \in \mathcal {N} _ {1} (\delta) \backslash \mathcal {N} _ {\min } (\delta)} (c _ {i ^ {\prime}} + 1) \delta_ {i ^ {\prime}} ^ {\prime} + \sum_ {j \in \mathcal {N} _ {2} (\delta)} (c _ {j} - 1) \delta_ {j} ^ {\prime} \\ = \sum_ {i \in \mathcal {N} _ {\min } (\delta)} (c _ {i} + 1) \delta_ {i} ^ {\prime} + \sum_ {i ^ {\prime} \in \mathcal {N} _ {1} (\delta) \backslash \mathcal {N} _ {\min } (\delta)} (c _ {i ^ {\prime}} + 1) \delta_ {i ^ {\prime}} ^ {\prime} - \sum_ {j \in \mathcal {N} _ {2} (\delta)} \mathbf {1} [ c _ {j} - c _ {1} ^ {\min } > 2 ] \cdot (c _ {j} - 1) \tilde {\rho} _ {j} ^ {E} \\ \end{array}
+$$
+
+$$
+= (c _ {1} ^ {\min } + 1) \sum_ {j \in \mathcal {N} _ {2} (\delta)} \mathbf {1} [ c _ {j} - c _ {1} ^ {\min } > 2 ] \cdot \tilde {\rho} _ {j} ^ {E} - \sum_ {j \in \mathcal {N} _ {2} (\delta)} \mathbf {1} [ c _ {j} - c _ {1} ^ {\min } > 2 ] \cdot (c _ {j} - 1) \cdot \tilde {\rho} _ {j} ^ {E}
+$$
+
+(due to  $\delta_{i'}' = 0$ )
+
+$$
+= \sum_ {j \in \mathcal {N} _ {2} (\delta)} \mathbf {1} \left[ c _ {j} - c _ {1} ^ {\min } > 2 \right] \cdot \left(c _ {1} ^ {\min } - c _ {j} + 2\right) \cdot \tilde {\rho} _ {j} ^ {E}. \tag {35}
+$$
+
+Regarding  $g(\delta)$ , the following holds:
+
+$$
+\begin{array}{l} g (\delta) = \sum_ {i \in \mathcal {N} _ {1} (\delta)} (c _ {i} + 1) \delta_ {i} + \sum_ {j \in \mathcal {N} _ {2} (\delta)} (c _ {j} - 1) \delta_ {j} \\ \geq \sum_ {i \in \mathcal {N} _ {1} (\delta)} \left(c _ {1} ^ {\min } + 1\right) \delta_ {i} + \sum_ {j \in \mathcal {N} _ {2} (\delta)} \mathbf {1} \left[ c _ {j} - c _ {1} ^ {\min } > 2 \right] \cdot \left(c _ {j} - 1\right) \delta_ {j} \\ + \sum_ {j \in \mathcal {N} _ {2} (\delta)} \mathbf {1} \left[ c _ {j} - c _ {1} ^ {\min } \leq 2 \right] \cdot (c _ {j} - 1) \delta_ {j} \\ = - \sum_ {j \in \mathcal {N} _ {2} (\delta)} \mathbf {1} \left[ c _ {j} - c _ {1} ^ {\min } > 2 \right] \cdot \left(c _ {1} ^ {\min } + 1\right) \delta_ {j} + \sum_ {j \in \mathcal {N} _ {2} (\delta)} \mathbf {1} \left[ c _ {j} - c _ {1} ^ {\min } > 2 \right] \cdot \left(c _ {j} - 1\right) \delta_ {j} \\ + \left(c _ {1} ^ {\min } + 1\right) \left(\sum_ {i \in \mathcal {N} _ {1} (\delta)} \delta_ {i} + \sum_ {j \in \mathcal {N} _ {2} (\delta)} \mathbf {1} \left[ c _ {j} - c _ {1} ^ {\min } > 2 \right] \cdot \delta_ {j}\right) \\ + \sum_ {j \in \mathcal {N} _ {2} (\delta)} \mathbf {1} \left[ c _ {j} - c _ {1} ^ {\min } \leq 2 \right] \cdot (c _ {j} - 1) \delta_ {j} \\ \end{array}
+$$
+
+(adding and subtracting  $-\sum_{j \in \mathcal{N}_2(\delta)} \mathbf{1}[c_j - c_1^{\min} > 2] \cdot (c_1^{\min} + 1)\delta_j)$
+
+$$
+\begin{array}{l} = \sum_ {j \in \mathcal {N} _ {2} (\delta)} \mathbf {1} \left[ c _ {j} - c _ {1} ^ {\min } > 2 \right] \cdot \left(c _ {j} - c _ {1} ^ {\min } - 2\right) \delta_ {j} \\ + \left(c _ {1} ^ {\min } + 1\right) \left(\sum_ {i \in \mathcal {N} _ {1} (\delta)} \delta_ {i} + \sum_ {j \in \mathcal {N} _ {2} (\delta)} \mathbf {1} \left[ c _ {j} - c _ {1} ^ {\min } > 2 \right] \cdot \delta_ {j}\right) \\ + \sum_ {j \in \mathcal {N} _ {2} (\delta)} \mathbf {1} \left[ c _ {j} - c _ {1} ^ {\min } \leq 2 \right] \cdot (c _ {j} - 1) \delta_ {j} \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \geq \sum_ {j \in \mathcal {N} _ {2} (\delta)} \mathbf {1} \left[ c _ {j} - c _ {1} ^ {\min } > 2 \right] \cdot \left(c _ {1} ^ {\min } - c _ {j} + 2\right) \tilde {\rho} _ {j} ^ {E} \\ + \left(c _ {1} ^ {\min } + 1\right) \left(\sum_ {i \in \mathcal {N} _ {1} (\delta)} \delta_ {i} + \sum_ {j \in \mathcal {N} _ {2} (\delta)} \mathbf {1} \left[ c _ {j} - c _ {1} ^ {\min } > 2 \right] \cdot \delta_ {j}\right) \\ + \sum_ {j \in \mathcal {N} _ {2} (\delta)} \mathbf {1} \left[ c _ {j} - c _ {1} ^ {\min } \leq 2 \right] \cdot \left(c ^ {\min } + 1\right) \delta_ {j} \quad \text {(n o t i n g t h a t} \delta_ {j} \leq 0) \\ = \sum_ {j \in \mathcal {N} _ {2} (\delta)} \mathbf {1} \left[ c _ {j} - c _ {1} ^ {\min } > 2 \right] \cdot \left(c _ {1} ^ {\min } - c _ {j} + 2\right) \tilde {\rho} _ {j} ^ {E} \\ + \left(c _ {1} ^ {\min } + 1\right) \left(\sum_ {i \in \mathcal {N} _ {1} (\delta)} \delta_ {i} + \sum_ {j \in \mathcal {N} _ {2} (\delta)} \left(\mathbf {1} \left[ c _ {j} - c _ {1} ^ {\min } > 2 \right] + \mathbf {1} \left[ c _ {j} - c _ {1} ^ {\min } \leq 2 \right]\right) \cdot \delta_ {j}\right) \\ = \sum_ {j \in \mathcal {N} _ {2} (\delta)} \mathbf {1} \left[ c _ {j} - c _ {1} ^ {\min } > 2 \right] \cdot \left(c _ {1} ^ {\min } - c _ {j} + 2\right) \tilde {\rho} _ {j} ^ {E} + \left(c _ {1} ^ {\min } + 1\right) \underbrace {\left(\sum_ {i \in \mathcal {N} _ {1} (\delta)} \delta_ {i} + \sum_ {j \in \mathcal {N} _ {2} (\delta)} \delta_ {j}\right)} _ {\text {(d u e t o C o n s t r a i n t (3 2))}} \\ = g \left(\delta^ {\prime}\right). \quad (\text {d u e t o E q .} (3 5)) \\ \end{array}
+$$
+
+Denoting  $\mathcal{G} \doteq \{\delta \in \mathbb{R}^{|\mathcal{S}| \cdot |\mathcal{A}|}$  s.t. (32) and (33), we have the following fact:
+
+$$
+\delta_ {\mathcal {N} _ {1}} ^ {\prime} = \underset {\delta \in \mathcal {G} (\mathcal {N} _ {1})} {\arg \min } g (\delta), \tag {36}
+$$
+
+where  $\mathcal{G}(\mathcal{N}_1) \doteq \{\delta \in \mathcal{G} : \mathcal{N}_1(\delta) = \mathcal{N}_1$  and  $\mathcal{N}_2(\delta) = [n] \backslash \mathcal{N}_1\}$ . Due to Eq. (35), we have
+
+$$
+\min  _ {\delta \in \mathcal {G}} g (\delta) = \min  _ {\mathcal {N} _ {1} \subset [ n ]} g \left(\delta_ {\mathcal {N} _ {1}} ^ {\prime}\right) = \min  _ {\mathcal {N} _ {1} \subset [ n ]} \sum_ {j \in [ n ] \backslash \mathcal {N} _ {1}} \mathbf {1} \left[ c _ {j} - c _ {1} ^ {\min } > 2 \right] \cdot \left(c _ {1} ^ {\min } - c _ {j} + 2\right) \cdot \tilde {\rho} _ {j} ^ {E}. \tag {37}
+$$
+
+Let  $c^{\min} \doteq \min_{i \in [n]} c_i$  and  $\mathcal{N}_1^* \doteq \{i \in [n] : c_i = c^{\min}\}$ . The following fact is true:
+
+$$
+\begin{array}{l} g \left(\delta_ {\mathcal {N} _ {1}} ^ {\prime}\right) - g \left(\delta_ {\mathcal {N} _ {1} ^ {*}} ^ {\prime}\right) = \sum_ {j \in [ n ] \backslash \mathcal {N} _ {1}} \mathbf {1} \left[ c _ {j} - c _ {1} ^ {\min } > 2 \right] \cdot \left(c _ {1} ^ {\min } - c _ {j} + 2\right) \\ - \sum_ {j ^ {\prime} \in [ n ] \backslash \mathcal {N} _ {1} ^ {*}} \mathbf {1} \left[ c _ {j ^ {\prime}} - c ^ {\min } > 2 \right] \cdot \left(c ^ {\min } - c _ {j ^ {\prime}} + 2\right) \\ \geq \sum_ {j \in [ n ] \backslash \mathcal {N} _ {1}} \mathbf {1} \left[ c _ {j} - c _ {1} ^ {\min } > 2 \right] \cdot \left(c ^ {\min } - c _ {j} + 2\right) \\ - \sum_ {j ^ {\prime} \in [ n ] \backslash \mathcal {N} _ {1} ^ {*}} \mathbf {1} \left[ c _ {j ^ {\prime}} - c ^ {\min } > 2 \right] \cdot \left(c ^ {\min } - c _ {j ^ {\prime}} + 2\right) \quad (\text {d u e t o} c ^ {\min } \leq c _ {1} ^ {\min }) \\ \geq 0, \tag {38} \\ \end{array}
+$$
+
+where the last inequality holds because  $\{j\in [n] / \mathcal{N}_1:c_j - c_1^{\min} > 2\}$  is a subset of  $\{j^{\prime}\in [n] / \mathcal{N}_{1}^{*}:c_{j^{\prime}} - c^{\mathrm{min}} > 2\}$ . Thus,  $\delta^{*}\doteq \delta_{\mathcal{N}_{1}^{*}}^{\prime} = \min_{\delta \in \mathcal{G}}g(\delta)$ , and we can express  $\delta^{*}$  as
+
+$$
+\delta^ {*} (s, a) = \left\{ \begin{array}{l l} \frac {\sum_ {s ^ {\prime} , a ^ {\prime}} \mathbf {1} \left[ c \left(s ^ {\prime} , a ^ {\prime}\right) - c ^ {\min } > 2 \right] \cdot \tilde {\rho} ^ {E} \left(s ^ {\prime} , a ^ {\prime}\right)}{| \mathcal {N} _ {\min } |}, & \text {i f} c (s, a) \leq c ^ {\min } \\ - \tilde {\rho} ^ {E} (s, a), & \text {i f} c (s, a) > c ^ {\min } + 2 \\ 0, & \text {o t h e r w i s e} \end{array} \right. \tag {39}
+$$
+
+where  $\mathcal{N}_{\mathrm{min}} = \mathcal{N}_1^*$ . Due to  $\delta(s, a) + \tilde{\rho}^E(s, a) = \rho(s, a)$ , we obtain the optimal solution of Problem (27) as follows:
+
+$$
+\rho^ {*} (s, a) = \left\{ \begin{array}{l l} \frac {\sum_ {s ^ {\prime} , a ^ {\prime}} \mathbf {1} \left[ c \left(s ^ {\prime} , a ^ {\prime}\right) - c ^ {\min } > 2 \right] \cdot \tilde {\rho} ^ {E} \left(s ^ {\prime} , a ^ {\prime}\right)}{| \mathcal {N} _ {\min } |} + \tilde {\rho} ^ {E} (s, a), & \text {i f} c (s, a) \leq c ^ {\min } \\ 0, & \text {i f} c (s, a) > c ^ {\min } + 2 \\ \tilde {\rho} ^ {E} (s, a), & \text {o t h e r w i s e} \end{array} \right. \tag {40}
+$$
+
+thereby completing the proof.
+
+# B.4 PROOF OF COROLLARY 4.1
+
+Because  $c(s, a) > c^{\min}$  when  $\tilde{\rho}^{D}(s, a) = 0$ , if  $\tilde{\rho}^{D}(s, a) = 0$ , then  $\rho^{*}(s, a) = 0$  holds. The desired result can be easily obtained by seeing  $\beta^{*}(s, a)\tilde{\rho}^{D}(s, a)$  as  $\delta^{*}(s, a)$  in the proof of Theorem 4.3.
+
+# B.5 MINIMIZING A CHI-SQUARED DIVERGENCE
+
+The  $f$ -divergence between two distributions  $\rho_{1}$  and  $\rho_{2}$  is defined as
+
+$$
+D _ {f} \left(\rho_ {1} \| \rho_ {2}\right) = \mathbb {E} _ {\rho_ {2}} \left[ f \left(\frac {\rho_ {1}}{\rho_ {2}}\right) \right] = \sup  _ {g} \mathbb {E} _ {X \sim \rho_ {1}} [ g (X) ] - \mathbb {E} _ {X \sim \rho_ {2}} \left[ f ^ {*} (g (X)) \right] \tag {41}
+$$
+
+where  $f^{*}$  is the convex conjugate. The  $\chi^2$ -divergence is the  $f$ -divergence with  $f(x) = (x - 1)^2$  and  $f^{*}(y) = \frac{y^{2}}{4} + y$ , i.e.,
+
+$$
+\chi^ {2} \left(\rho_ {1}, \rho_ {2}\right) = \sup  _ {g} E _ {X \sim \rho_ {1}} [ g (X) ] - E _ {X \sim \rho_ {2}} \left[ \frac {g (X) ^ {2}}{4} + g (X) \right] \tag {42}
+$$
+
+By interpreting  $g = -r$  and  $X = (s, a)$ , the following holds:
+
+$$
+\chi^ {2} \left(\rho_ {1}, \rho_ {2}\right) = \sup  _ {r} E _ {(s, a) \sim \rho_ {2}} [ r (s, a) ] - E _ {(s, a) \sim \rho_ {1}} [ r (s, a) ] - \frac {1}{4} E _ {(s, a) \sim \rho_ {2}} [ r (s, a) ^ {2} ] \tag {43}
+$$
+
+Thus, using a convex reward regularizer  $\psi (r) = \frac{r^2}{4\delta}$  enables CLARE to minimize a  $\chi^2$ -divergence between the target policy and learned policy, i.e.,  $\max_{\hat{\rho}\in \mathcal{C}_{\hat{T}}}\alpha \bar{H} (\hat{\rho}) - Z_{\beta}\delta \cdot \chi^{2}(\hat{\rho},\hat{\rho}^{*})$

2023/CLARE_ Conservative Model-Based Reward Learning for Offline Inverse Reinforcement Learning/images.zip

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2023/CLIP-ViP_ Adapting Pre-trained Image-Text Model to Video-Language Alignment/300040d0-80cf-4593-ab2e-edbe3e46a15e_content_list.json

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+    {
+        "type": "text",
+        "text": "CLIP-VIP: ADAPTING PRE-TRAINED IMAGE-TEXT MODEL TO VIDEO-LANGUAGE ALIGNMENT",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Hongwei Xue $^{1*}$ , Yuchong Sun $^{2*}$ , Bei Liu $^{3\\dagger}$ , Jianlong Fu $^{3\\dagger}$ , Ruihua Song $^{2}$ , Houqiang Li $^{1}$ , Jiebo Luo $^{4}$",
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+        "type": "list",
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+        "list_items": [
+            "<sup>1</sup>University of Science and Technology of China, Hefei, China,",
+            "$^{2}$ Renmin University of China, Beijing, China,",
+            "<sup>3</sup>Microsoft Research, Beijing, China,",
+            "<sup>4</sup>University of Rochester, Rochester, NY"
+        ],
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+    {
+        "type": "text",
+        "text": "ABSTRACT",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Pre-trained image-text models, like CLIP, have demonstrated the strong power of vision-language representation learned from a large scale of web-collected image-text data. In light of the well-learned visual features, there are works that transfer image representation to the video domain and achieve good results. However, adapting image-text pre-trained models to video-text pre-training (i.e., post-pretraining) has not demonstrated a significant advantage yet. In this paper, we tackle this challenge by raising and addressing two questions: 1) what are the factors hindering post-pretraining CLIP from improving performance on video-text tasks, and 2) how to mitigate the impact of these factors. Through a series of comparative experiments and analyses, we find that the data scale and domain gap between language sources have large impacts. By these observations, we propose an Omnisource Cross-modal Learning method equipped with a Video Proxy mechanism on the basis of CLIP, namely CLIP-ViP. Extensive results show that our approach improves the performance of CLIP on video-text retrieval by a large margin. Our model achieves state-of-the-art results on a variety of datasets, including MSR-VTT, DiDeMo, LSMDC, and ActivityNet. We release our code and pre-trained CLIP-ViP models at https://github.com/microsoft/XPretrain/tree/main/CLIP-ViP.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In the past few years, vision-language pre-training has achieved great success on cross-modal representation learning from a large scale of web-crawled data (Radford et al., 2021; Jia et al., 2021; Li et al., 2021; Wang et al., 2021b; Zellers et al., 2021; 2022; Bain et al., 2021). Among them, image-text pre-trained models (Radford et al., 2021; Jia et al., 2021) have shown powerful capability for various downstream tasks, including visual understanding (Gu et al., 2021; Wang et al., 2021a; Rao et al., 2022), image-text generation (Patashnik et al., 2021; Mokady et al., 2021) and so on (Guzhov et al., 2022; Zhang et al., 2022). In light of the well-learned and enriched visual representation, some works directly adapt image-text pre-trained models to video-text downstream tasks without further pre-training on video data (Luo et al., 2021; Fang et al., 2021; Gorti et al., 2022; Zhao et al., 2022), while still outperforming models pre-trained on video data (Xu et al., 2021b; Bain et al., 2021).",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Utilizing an existing powerful image-text pre-trained model for further video-text pre-training (i.e., post-pretraining) is able to reduce the required training cost by making good use of the knowledge learned from images. However, adapting image-text pre-trained models to video-text data for postpretraining has not demonstrated a significant advantage yet, and thus is still under-explored. A preliminary study is conducted by CLIP4Clip (Luo et al., 2021) which adopts MeanPooling by averaging multiple frame features based on the CLIP model on a subset of Howto100M (Miech et al., 2019). While the improvement over directly using the image-text pre-trained model is marginal for",
+        "bbox": [
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2023",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "page_footnote",
+        "text": "*Equal contributor. This work was performed when Hongwei Xue and Yuchong Sun were visiting Microsoft Research as research interns.",
+        "bbox": [
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+    },
+    {
+        "type": "page_footnote",
+        "text": "† Corresponding authors.",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_number",
+        "text": "1",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "either zero-shot or fine-tuning settings. In this paper, we aim to explore how to effectively adapt the image-text pre-trained model (e.g., CLIP) to video-language representation learning for video-text tasks (e.g., text-to-video retrieval).",
+        "bbox": [
+            169,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To unleash the power of video data to adapt image-text pre-trained models for post-pretraining, we conduct several preliminary experiments to figure out the challenges that hinder post-pretraining. First, we explore post-pretraining an image-text pre-trained model (i.e., CLIP) with MeanPooling on video-text datasets with different scales, including WebVid-2.5M (Bain et al., 2021) and HDVILA-100M (Xue et al., 2022). The result shows that the scale of data is critical for video-text post-pretraining. Data on a small scale makes the model easy to over-fit the new data while the knowledge learned from image-text is suppressed and the performance is reduced. Second, we investigate the language domain gap between pre-training data and downstream data. By calculating the Normalized Mutual Information (NMI) on clusters of text features, we find that there is a large domain gap between subtitles that are used in large-scale video-text pre-training data and descriptive texts in downstream tasks.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To mitigate the impact of the above factors, we propose CLIP-ViP to adapt the pre-trained image-text model CLIP for video-text pre-training. First, we introduce auxiliary captions that have a smaller language domain gap with downstream data into existing large-scale video-text data. Instead of using a video captioning model which may cause data leakage by training on the same dataset with video-text downstream tasks, and considering a better visual captioning capability, we adopt an image captioning model to generate an auxiliary caption of middle frame in each video. In order to adapt a Transformer-based vision encoder to process both images and videos with minimal modification, we then propose video proxy tokens and design a proxy-guided video attention mechanism for the Vision Transformer (ViT). Specifically, during attention computation in each block, video proxy tokens can interact with all tokens, while patch tokens only interact with video proxy tokens and patch tokens within the same frame. Our vision encoder only increases negligible parameters and calculations compared to the vanilla Vision Transformer while increasing the generality and extendability. To facilitate cross-modal representation learning from both caption-frame and video-subtitle data types at the same time, we propose an Omnisource Cross-modal Learning (OCL) method for pre-training and study a series of variants to find the best fusion strategy.",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Our experimental results show that our approach improves the performance of CLIP on text-to-video retrieval tasks by a large margin. We also conduct ablation studies to verify the effectiveness of each part in our approach. Our contributions are summarized as follows: (1) We are one of the first to explore factors that hinder video post-pretraining on pre-trained image-text models; (2) We propose CLIP-ViP that can effectively leverage image-text pre-trained model for post-pretraining; (3) We conduct extensive experiments to verify the effectiveness of our method. Our model outperforms the state-of-the-art results by a large margin on four widely-used benchmarks.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Vision-Language Pre-Training End-to-end models (Lei et al., 2021; Xue et al., 2022; Zellers et al., 2021; Fu et al., 2021; Huang et al., 2020; 2021b; Xue et al., 2021; Li et al., 2021; Kim et al., 2021; Huang et al., 2021a; Sun et al., 2022) for vision-language pre-training are replacing the traditional approach using pre-extracted visual features by off-the-shelf models (Sun et al., 2019; Xu et al., 2021b; Zhu & Yang, 2020; Li et al., 2020b;a; Chen et al., 2020). Training end-to-end models on large-scale web-collected data also gradually demonstrates the big advantages (Radford et al., 2021; Jia et al., 2021; Xue et al., 2022; Zellers et al., 2021; 2022). Unlike images that have alt-texts, large-scale video datasets suitable for pre-training usually use subtitles as text sources (Miech et al., 2019; Xue et al., 2022). Subtitles are much noisier than alt-texts, according to (Miech et al., 2019), typical examples of incoherence include the content producer asking viewers to subscribe to their channel, talking about something unrelated to the video, or describing something before or after it happens. Bain et al. collect a video dataset WebVid (Bain et al., 2021) with textual description annotations. Their texts are well aligned with the video and avoid suffering from ASR errors. However, the vast majority of WebVid videos are sourced from a stock footage website, so scaling up is under limitation. The video-subtitle data is more easily accessible on the web and thus suitable for scaling up. In this paper, we investigate the unfavorable factors of video-subtitle data and explore how to mitigate the impact of these factors.",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2023",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "2",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/076dac7c87df262effc168cb0293de331d94f61edad53411a3ec3cabbec3f770.jpg",
+        "image_caption": [
+            "Figure 1: The curve of finetuning results during post-pretraining. The x-axis indicates the percentage of training steps. The y-axis indicates average value of Recall@1, 5 and 10. [Best viewed in color]"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/5de8e054660fbfafcd8c1d7a33bff1c3cdbf9907729d5df98b046c96bd5fe7bb.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>NMI Score</td><td>MSR-VTT</td><td>DiDeMo</td><td>Mean</td></tr><tr><td>HD-VILAsub</td><td>0.831</td><td>0.684</td><td>0.758</td></tr><tr><td>HD-VILAcap</td><td>0.317</td><td>0.621</td><td>0.469</td></tr><tr><td>WebVid</td><td>0.420</td><td>0.488</td><td>0.454</td></tr><tr><td>COCO</td><td>0.373</td><td>0.758</td><td>0.566</td></tr><tr><td>CC12M</td><td>0.445</td><td>0.673</td><td>0.559</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Normalized Mutual Information (NMI) score of language features extracted on series of data and downstream tasks. Larger value indicates larger domain gap.",
+        "bbox": [
+            526,
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+            268
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Pre-trained Models for Video-Text Retrieval The great success of the CLIP has demonstrated its unprecedented power on varies downstream tasks, including vision understanding (Gu et al., 2021; Wang et al., 2021a; Rao et al., 2022), image-text generation (Patashnik et al., 2021; Mokady et al., 2021) and so on (Guzhov et al., 2022; Zhang et al., 2022). By contrastive learning on large-scale image-text pairs, CLIP learns enriched visual concepts for images. Recently, some works directly transfer CLIP to video-text retrieval without further pretraining on video data (post-pretraining) (Luo et al., 2021; Fang et al., 2021; Gorti et al., 2022; Zhao et al., 2022; Wang et al., 2022c). Their work takes the performance of video-text retrieval to a new level, outperforming existing models pre-trained on video data (Xu et al., 2021b; Bain et al., 2021; Xue et al., 2022; Ge et al., 2022; Wang et al., 2022a). They transfer CLIP from views of feature aggregation (Luo et al., 2021; Zhao et al., 2022; Fang et al., 2021; Gorti et al., 2022) or representation alignment (Fang et al., 2021; Gorti et al., 2022; Wang et al., 2022c). In parallel with these works, we study post-pretraining with video data on top of CLIP in an effective way and our model can be combined with other approaches effectively.",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 PRELIMINARY ANALYSIS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            492,
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+            507
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we first study the impact of the data scale for adapting image-text pre-training to video-text post-pretraining, and then investigate how the language domain gap affects the adaption.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.1 POST-PRETRAINING WITH DIFFERENT DATA SCALES",
+        "text_level": 1,
+        "bbox": [
+            171,
+            558,
+            578,
+            571
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To study the effectiveness of different data scales, we use the CLIP-ViT-B/32 model (Radford et al., 2021) as the base image-text pre-trained model and adopt MeanPooling for video adaption like CLIP4Clip (Luo et al., 2021) by averaging multiple frame features as video feature. Two open-domain video-text datasets are used: WebVid-2.5M (Bain et al., 2021) with 2.5 million pairs and HD-VILA-100M (Xue et al., 2022) with 100M pairs. We also adopt a subset of HD-VILA-100M containing random  $10\\%$  data (namely HD-VILA-10M) as a middle setting. We run the same number of steps on all settings, equivalent to one epoch on HD-VILA-100M. We uniformly sample 12 frames from each video and apply the same hyper-parameters as described in Section 5 for all settings.",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "During post-pretraining, we evaluate the pre-trained models by fine-tuning on MSR-VTT text-to-video retrieval task. Figure 1 shows the performance trend. We observe an overfitting phenomenon that continuous post-pretraining leads to a performance drop. And the drop is more significant for smaller data (e.g., WebVid-2.5M and HD-VILA-10M). As CLIP is pre-trained on 400 million image-text pairs, further training on small data makes the model tend to overfit the new data while the implicit knowledge learned from the image-text pairs is degrading. As a consequence, the performance will drop, even worse than using CLIP directly. Thus we adopt HD-VILA-100M due to its large scale and diverse category.",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 LANGUAGE DOMAIN GAP WITH DOWNSTREAM DATA",
+        "text_level": 1,
+        "bbox": [
+            171,
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+            834
+        ],
+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "It is intuitive that pre-training on data with the same domain as downstream data can benefit downstream tasks. For most video-text tasks like video-text retrieval, texts are descriptive sentences of videos (i.e., captions). While for HD-VILA-100M, which we will use for pre-training, the texts are auto-transcribed subtitles and they indicate very different relevance to visual information compared to descriptive texts. Meanwhile, auto-transcribed subtitles suffer from irrelevance, misalignment, and ASR errors (Miech et al., 2019). To better explore the language domain gap between pre",
+        "bbox": [
+            169,
+            840,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2023",
+        "bbox": [
+            173,
+            32,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/c09f6fd3093317bdc4c09b3ca3bdc291d612b36973f3ca5ac07b210a9dbcf999.jpg",
+        "image_caption": [
+            "Figure 2: The framework of CLIP-ViP with a text encoder and a vision encoder. Taken features  $V$ ,  $F$ ,  $S$ ,  $C$  of videos, frames, subtitles, captions as input, a series of Omnisource cross-modal learning variants are studied to explore better representation learning losses: (a)  $\\mathcal{L}_{V \\leftrightarrow S} + \\mathcal{L}_{F \\leftrightarrow C}$ ; (b)  $\\mathcal{L}_{V \\leftrightarrow S} + \\mathcal{L}_{V \\leftrightarrow C}$ ; (c)  $\\mathcal{L}_{V \\leftrightarrow S} + \\mathcal{L}_{V \\leftrightarrow C} + \\mathcal{L}_{F \\leftrightarrow C}$ ; (d)  $\\mathcal{L}_{V \\leftrightarrow S, C} + \\mathcal{L}_{F \\leftrightarrow C}$ . In the vision encoder, Video proxy tokens and the ViP-guided attention mechanism is proposed to transfer CLIP into the video domain. [Best viewed in color]"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "training data and downstream data, we measure the inconsistency by calculating the dissimilarity between their language features. For downstream language data, we choose two typical video-text retrieval datasets: MSR-VTT (Xu et al., 2016) and DiDeMo (Anne Hendricks et al., 2017). For pretraining language, we select four types: video subtitles of HD-VILA-100M (HD-VILAsub), video captions of WebVid-2.5M, image captions of MS-COCO (Lin et al., 2014), and web-collected alttexts of Conceptual Caption 12M (Changpinyo et al., 2021). In addition, we analyze auto-generated captions of HD-VILA-100M (HD-VILAscap), which will be introduced in Section 4.",
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+    {
+        "type": "text",
+        "text": "We use a Transformer Encoder initialized from CLIP (Radford et al., 2021) to extract text features. To quantify the domain gap of languages between pre-training and downstream data, we first mix their text features and then use K-means to get two clusters. Then we calculate the Normalized Mutual Information (NMI) between cluster labels and ground-truth labels of pre-training or downstream. A larger NMI value means that the two types of features are easy to be distinguished, thus there is a larger domain gap. For each comparison, we randomly sample 1000 texts from each type of data for 10 times and adopt the average of 10 results. We report the results in Table 1. Comparing the values of all pre-training data types, we find that the NMI score between HD-VILAsub and downstream data is much larger than others, especially for MSR-VTT downstream dataset. This indicates that direct training with subtitles may introduce inconsistency with downstream tasks.",
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+    {
+        "type": "text",
+        "text": "4 APPROACH",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we will introduce the proposed CLIP-ViP video pre-training framework. To bridge language domain gaps between image and video datasets, we first introduce an in-domain auxiliary data generation method. Then, we propose a novel Video Proxy mechanism to enable the Vision Transformer (ViT) model for both image and video encoding. We further present an Omnisource Cross-modal Learning (OCL) method which can jointly learn cross-modal representation from video-text and image-text pairs.",
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+    {
+        "type": "text",
+        "text": "4.1 IN-DOMAIN AUXILIARY DATA GENERATION",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Motivated by the analysis in Section 3, we introduce auxiliary captions into large-scale video-subtitle data to reduce the language domain gap between pre-training and downstream data. We adopt an",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2023",
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+    },
+    {
+        "type": "page_number",
+        "text": "4",
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+    {
+        "type": "text",
+        "text": "image captioning model for two reasons. 1) Most SOTA video captioning models are trained with video-text datasets (e.g., MSR-VTT, ActivityNet) which are also used for downstream tasks. We avoid data leakage to perform pre-training agnostic to downstream data. 2) The performance of existing video captioning models lags far behind that of images. Thus, we choose a powerful image captioning model OFA-Caption (Wang et al., 2022b) to generate one caption for the middle frame of each video in HD-VILA-100M. We use the default setting of the OFA-Caption model. As a result, we generate 100M sentences with a max length of 16 words. This method can be applied to any video data and we will release the generated captions to facilitate future research.",
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+    {
+        "type": "text",
+        "text": "4.2 VIDEO PROXY MECHANISM",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Since video is an ordered sequence of frames, it is critical to learn the frame aggregation and temporality when transferring to the video domain. Meanwhile, to keep the high generality and extendability of the Vision Transformer (ViT) backbone, we aim to find a simple but effective way to transfer ViT to enable both image and video encoding with minimal modifications. Given a video containing  $T$  frames:  $\\{f_1, f_2, \\dots, f_T\\}$ , we follow CLIP to divide each frame into N patches:  $\\{f_t^1, f_t^2, \\dots, f_t^N \\mid t \\in [1, T]\\}$ . Then we add spatio-temporal positional embedding to each flattened 2D patches:",
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+    {
+        "type": "equation",
+        "text": "\n$$\ng \\left(f _ {t} ^ {n}\\right) = \\operatorname {L i n e a r} \\left(f _ {t} ^ {n}\\right) + \\operatorname {P o s} _ {s} (n) + \\operatorname {P o s} _ {t} (t), \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $Linear(*)$  is a linear layer,  $Pos_{s}(n)$  and  $Pos_{t}(t)$  is the learnable spatial and temporal positional embedding, respectively. The whole video can be divided into  $T \\times N$  patch tokens.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To model spatial information from multi-frames, one simple way is directly feeding all tokens into CLIP's vision encoder and conducting attention across all tokens. However, this method introduces significant conflicts with CLIP. As CLIP is pre-trained on image and text pairs, it has difficulty handling interactions of tokens between images/frames during training. We also verify it by experiments as Full Attention setting in Table 2. Instead, we introduce a Video Proxy token to act as a proxy that helps each local patch perceive video-level temporal information.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Before feeding into CLIP, we concatenate patch tokens with a set of learnable parameters called video proxy tokens:  $\\mathcal{P} = \\{p_1,p_2,\\dots,p_M\\}$ , where  $M$  is the number of video proxy tokens. Then all  $T\\times N + M$  tokens will be fed into the ViT of CLIP. The output of the first video proxy token will be regarded as the video's representation. We also design a proxy-guided attention mechanism for the vanilla ViT. In the attention score calculation of each block, video proxy tokens attend to all tokens, while patch tokens only attend to tokens in the same frame plus video proxy tokens. This mechanism can be formulated as attention mask  $\\mathcal{M}_{\\mathrm{ViP}}$ :",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {M} _ {\\mathrm {V i P}} (q, k) = \\left\\{ \\begin{array}{l l} 1 & \\text {i f} q \\in \\mathcal {P} \\text {o r} k \\in \\mathcal {P} \\text {o r} (q, k) \\text {i n t h e s a m e f r a m e}, \\\\ 0 & \\text {o t h e r w i s e}, \\end{array} \\right. \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where  $q$  and  $k$  is the key and query in attention calculation. Patch tokens can obtain global information from video proxy tokens while reducing inconsistencies with the original CLIP's calculation. Our experiment in Section 5 demonstrates the superiority of this mechanism.",
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+    {
+        "type": "text",
+        "text": "For the input type of the image/frame, we use linear interpolation to get a middle temporal positional embedding, then treat the image/frame as a special single-frame video. This method enables joint training on both videos and images in the same batch, as our proxy-guided attention mechanism reduces the difference in calculations between video and image.",
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+    {
+        "type": "text",
+        "text": "4.3 OMNISOURCE CROSS-MODAL LEARNING",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To learn rich video-language alignment from video-subtitle pairs and reduce the language domain gap with downstream data by corresponding auxiliary frame-caption pairs, we study joint CrossModal Learning on the omnisor input. Following most works of learning multimodal alignment on dual encoders (Radford et al., 2021; Xue et al., 2022; Li et al., 2021; Luo et al., 2020; Xu et al., 2021b; Luo et al., 2021), we use info-NCE loss to perform contrastive learning. There are two formats of visual source: video sequences and single frames, and two types of text source: subtitles and captions in our work. We denote them by  $V$ ,  $F$ ,  $S$ , and  $C$  respectively. We define a source-wise info-NCE loss by:",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {v 2 t} = - \\frac {1}{B} \\sum_ {i = 1} ^ {B} \\log \\frac {e ^ {v _ {i} ^ {\\top} t _ {i} / \\tau}}{\\sum_ {j = 1} ^ {B} e ^ {v _ {i} ^ {\\top} t _ {j} / \\tau}}, \\quad \\mathcal {L} _ {t 2 v} = - \\frac {1}{B} \\sum_ {i = 1} ^ {B} \\log \\frac {e ^ {t _ {i} ^ {\\top} v _ {i} / \\tau}}{\\sum_ {j = 1} ^ {B} e ^ {t _ {i} ^ {\\top} v _ {j} / \\tau}} \\tag {3}\n$$\n",
+        "text_format": "latex",
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+        "page_idx": 4
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2023",
+        "bbox": [
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+            32,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/cdec6a191547f01e6e9b212e2e23ba1435242a3660ced1a8659b552d594d8e69.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>R@1 ↑</td><td>R@5 ↑</td><td>R@10 ↑</td><td>Mean ↑</td></tr><tr><td>MeanPool</td><td>43.4</td><td>70.9</td><td>81.1</td><td>65.1</td></tr><tr><td>SeqTransformer</td><td>44.6</td><td>71.2</td><td>81.8</td><td>65.9</td></tr><tr><td>Full Attention</td><td>42.8</td><td>70.1</td><td>80.3</td><td>64.4</td></tr><tr><td>2 Video Proxy Tokens</td><td>45.8</td><td>71.3</td><td>81.7</td><td>66.3</td></tr><tr><td>4 Video Proxy Tokens</td><td>46.5</td><td>72.1</td><td>82.5</td><td>67.0</td></tr><tr><td>8 Video Proxy Tokens</td><td>45.7</td><td>72.7</td><td>81.7</td><td>66.7</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/7a92300a296dc3dbb417a103e8a3b36a093fb82e2f6dbbbf6a93c240c9afde86.jpg",
+        "table_caption": [
+            "Table 2: MSR-VTT text-to-video retrieval results of finetuning CLIP by different settings. Mean  $\\uparrow$  indicates the average value of Recall@1, 5, and 10. All results are based on CLIP-ViT-B/32."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"4\">MSR-VTT Retrieval</td><td colspan=\"4\">DiDeMo Retrieval</td></tr><tr><td>R@1 ↑</td><td>R@5 ↑</td><td>R@10 ↑</td><td>Mean ↑</td><td>R@1 ↑</td><td>R@5 ↑</td><td>R@10 ↑</td><td>Mean ↑</td></tr><tr><td>CLIP-MeanPool</td><td>43.4</td><td>70.9</td><td>81.1</td><td>65.1</td><td>40.6</td><td>67.5</td><td>77.2</td><td>61.8</td></tr><tr><td>CLIP-ViP</td><td>46.5</td><td>72.1</td><td>82.5</td><td>67.0</td><td>40.6</td><td>70.4</td><td>79.3</td><td>63.4</td></tr><tr><td>+ LV↔S</td><td>47.7</td><td>72.1</td><td>82.4</td><td>67.4</td><td>44.6</td><td>73.9</td><td>81.9</td><td>66.8</td></tr><tr><td>+ LV↔S + LF↔C</td><td>49.3</td><td>74.8</td><td>83.8</td><td>69.3</td><td>48.4</td><td>74.5</td><td>84.4</td><td>69.1</td></tr><tr><td>+ LV↔S + LV↔C</td><td>49.6</td><td>74.2</td><td>84.0</td><td>69.3</td><td>48.5</td><td>76.6</td><td>83.6</td><td>69.5</td></tr><tr><td>+ LV↔S + LV↔C + LF↔C</td><td>49.6</td><td>74.5</td><td>83.8</td><td>69.3</td><td>48.5</td><td>76.8</td><td>84.1</td><td>69.8</td></tr><tr><td>+ LV↔S,C + LF↔C</td><td>49.6</td><td>74.5</td><td>84.8</td><td>69.6</td><td>48.2</td><td>76.7</td><td>84.4</td><td>69.8</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Ablation study of different losses. We report text-to-video results of models finetuned on MSR-VTT and DiDeMo. Mean  $\\uparrow$  means the average of Recall@1, 5 and 10. All results are based on CLIP-ViT-B/32. All post-pretraining steps are equivalent to one epoch on HD-VILA-100M.",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where  $v_{i}$  and  $t_j$  are the normalized embeddings of  $i$ -th visual feature in  $X \\in \\{V, F\\}$  and  $j$ -th text feature in  $Y \\in \\{S, C\\}$  in a batch of size  $B$ .  $\\tau$  is a learnable temperature. The overall alignment loss  $\\mathcal{L}_{X \\leftrightarrow Y}$  is the average of  $\\mathcal{L}_{v2t}$  and  $\\mathcal{L}_{t2v}$ . For example,  $\\mathcal{L}_{V \\leftrightarrow S}$  represents info-NCE loss within video-subtitle pairs in a batch, which pulls aligned pairs together in embedding space while pushing apart misaligned pairs.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We study the reasonable variants of OCL: (a)  $\\mathcal{L}_{V\\leftrightarrow S} + \\mathcal{L}_{F\\leftrightarrow C}$ : Simple combination of two source-wise losses on video-subtitle and frame-caption pairs; (b)  $\\mathcal{L}_{V\\leftrightarrow S} + \\mathcal{L}_{V\\leftrightarrow C}$ : As there is also content correlation between videos and its middle-frame captions, we explore to add a loss on videocaption pairs to baseline loss  $\\mathcal{L}_{V\\leftrightarrow S}$ ; (c)  $\\mathcal{L}_{V\\leftrightarrow S} + \\mathcal{L}_{V\\leftrightarrow C} + \\mathcal{L}_{F\\leftrightarrow C}$ : Combination of (a) and (c); (d)  $\\mathcal{L}_{V\\leftrightarrow S,C} + \\mathcal{L}_{F\\leftrightarrow C}$ : A video corresponds to both a subtitle and auxiliary caption. Compare to (c), the numbers of negative pairs in  $\\mathcal{L}_{v2t}$  can be expanded. The  $\\mathcal{L}_{v2t}$  in  $\\mathcal{L}_{V\\leftrightarrow S,C}$  is rewritten as:",
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {v 2 t} = - \\frac {1}{2 B} \\sum_ {i = 1} ^ {B} \\left(\\log \\frac {e ^ {v _ {i} ^ {\\top} s _ {i} / \\tau}}{\\sum_ {j = 1} ^ {B} e ^ {v _ {i} ^ {\\top} s _ {j} / \\tau} + e ^ {v _ {i} ^ {\\top} c _ {j \\neq i} / \\tau}} + \\log \\frac {e ^ {v _ {i} ^ {\\top} c _ {i} / \\tau}}{\\sum_ {j = 1} ^ {B} e ^ {v _ {i} ^ {\\top} c _ {j} / \\tau} + e ^ {v _ {i} ^ {\\top} s _ {j \\neq i} / \\tau}}\\right), \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where  $s_i \\in S$  and  $c_i \\in C$ . The  $\\mathcal{L}_{t2v}$  in  $\\mathcal{L}_{V \\leftrightarrow S, C}$  is equal to (c). We compare all variants with the baseline  $\\mathcal{L}_{V \\leftrightarrow S}$  and report results in Section 5.",
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+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENT",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "5.1 EXPERIMENTAL DETAILS",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Video-Text Post-Pretraining. To pre-train the proposed CLIP-ViP model, we uniformly sample 12 frames and resize all frames to  $224 \\times 224$  from video clips with an average length of 13.4 seconds. The sampled frames can well cover the semantics conveyed from a video clip. For text, we adopt the CLIP's tokenizer to split a sentence into word tokens with a max length of 70. We use AdamW optimizer (Loshchilov & Hutter, 2019), and empirically set an initial learning rate as 5e-6 and a fixed weight decay as 5e-2. For the learning rate schedule, we adopt a cosine decay with a warm-up strategy. We train our model with 32 NVIDIA Tesla V100 GPUs in a batch size of 1024. The contrastive similarity is calculated on gathered features from all GPUs. We set training steps to one epoch on HD-VILA-100M for all ablation studies and three epochs for the full setting.",
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+    },
+    {
+        "type": "text",
+        "text": "Fine-tuning Training. To better adapt CLIP-ViP to downstream tasks, we reuse most hyperparameters of post-pretraining in fine-tuning with some exceptions. 1) Batch size: we fine-tune",
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+        "page_idx": 5
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2023",
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+    {
+        "type": "page_number",
+        "text": "6",
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+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Post-pretrain Data</td><td colspan=\"4\">MSR-VTT Retrieval</td><td colspan=\"4\">DiDeMo Retrieval</td></tr><tr><td>R@1↑</td><td>R@5↑</td><td>R@10↑</td><td>Mean ↑</td><td>R@1↑</td><td>R@5↑</td><td>R@10↑</td><td>Mean ↑</td></tr><tr><td>w/o Post-pretrain</td><td>46.5</td><td>72.1</td><td>82.5</td><td>67.0</td><td>40.6</td><td>70.4</td><td>79.3</td><td>63.4</td></tr><tr><td>HD-VILAsub</td><td>47.7</td><td>72.1</td><td>82.4</td><td>67.4</td><td>44.6</td><td>73.9</td><td>81.9</td><td>66.8</td></tr><tr><td>HD-VILAcap</td><td>45.9</td><td>73.0</td><td>81.8</td><td>66.9</td><td>44.9</td><td>74.4</td><td>82.3</td><td>67.2</td></tr><tr><td>HD-VILAsub+cap</td><td>49.6</td><td>74.5</td><td>84.8</td><td>69.6</td><td>48.2</td><td>76.7</td><td>84.4</td><td>69.8</td></tr><tr><td>ImageCaption</td><td>45.6</td><td>70.7</td><td>81.1</td><td>65.8</td><td>43.7</td><td>69.5</td><td>77.9</td><td>63.7</td></tr><tr><td>HD-VILAsub + ImageCaption</td><td>49.1</td><td>73.1</td><td>83.5</td><td>68.6</td><td>47.0</td><td>75.3</td><td>84.1</td><td>68.8</td></tr></table>",
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+        "type": "text",
+        "text": "Table 4: Ablation study of post-pretrain data. We report text-to-video results of models finetuned on MSR-VTT and DiDeMo. Mean  $\\uparrow$  indicates an average of Recall@1, 5 and 10. For all results, the model is designed with 4 video proxy tokens and pre-trained on CLIP-ViT-B/32. All post-pretraining steps are equivalent to one epoch on HD-VILA-100M.",
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+        "type": "text",
+        "text": "our model with a batch size of 128 for all downstream tasks for a fair comparison. 2) Learning rate and weight decay: we empirically set them to 1e-6 and 0.2, respectively. 3) Number of epochs: due to the various scales of downstream datasets, we set epoch numbers to 5, 20, 10, and 20 for MSR-VTT, DiDeMo, LSMDC, and ActivityNet, respectively. 4) Frame number: for a fair comparison, we set frame number to 12 except for ActivityNet Captions (set to 32) as its videos are much longer (180 seconds on average). Note that the hyper-parameters of downstream training are the same in all settings in the ablation study.",
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+        "text": "Downstream Datasets. To evaluate performances of video pre-training models, we conduct text-to-video retrieval experiments on four typical datasets. (a) MSR-VTT (Xu et al., 2016) contains 10K YouTube videos with 200K descriptions. We follow previous works (Yu et al., 2018; Liu et al., 2019) to train models on 9K videos, and report results on the 1K-A test set. (b) DiDeMo (Anne Hendricks et al., 2017) consists of 10K Flickr videos annotated with 40K sentences. We follow (Liu et al., 2019; Zhang et al., 2018) to evaluate paragraph-to-video retrieval and concatenate all descriptions of a video as one query. (c) LSMDC (Rohrbach et al., 2016) consists of 118,081 video clips sourced from 202 movies with one caption corresponding to each clip. Evaluation is conducted on a test set of 1,000 videos from movies disjoint from the train and validation sets. (d) ActivityNet Captions (Krishna et al., 2017a) contains 20K YouTube videos annotated with 100K sentences. We follow the paragraph-to-video retrieval setting (Zhang et al., 2018; Liu et al., 2019) to train models on 10K videos and report results on the val1 set with 4.9K videos.",
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+    {
+        "type": "text",
+        "text": "5.2 ABLATION STUDIES",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Video Proxy Mechanism. For the vision encoder, we evaluate our proposed Video Proxy (ViP) mechanism with different numbers of proxies and compare it with different model structures (i.e. MeanPool, SeqTransformer, Full Attention) by fine-tuning the same pre-trained model on MSR-VTT retrieval task. MeanPool simply takes the average of frame features as the representation of the whole video. For SeqTransformer, we follow the seqTransf type in CLIP4Clip (Luo et al., 2021) and the residual connection in their implementation. Full Attention setting takes all patch tokens as the input of the vision encoder and attention is conducted across all tokens. All models are initialized with CLIP-ViT-B/32. The results are shown in Table 2. Compared to the MeanPool baseline which completely disregards temporality, SeqTransformer improves the average Recall@1,5,10 by  $0.8\\%$ . Full Attention type leads to a significant performance drop and we observe a worse initial status and slower convergence than other settings during the experiment. This is consistent with our analysis in Section 4.2 that directly using CLIP for all patches' attention computation will decrease the advantage of CLIP. In our method, different numbers of video proxy tokens all result in significant performance gain on R@1 (e.g.,  $3.1\\%$  by 4 proxies), while only increasing negligible parameters: 3K compared to 86M of ViT backbone. Compared with other settings, our methods in all settings have the most improvement which indicates that our proposed video proxy mechanism can effectively leverage the image-text pre-trained model for video-text post-pretraining.",
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+    {
+        "type": "text",
+        "text": "Omnisource Cross-modal Learning. To verify the effectiveness of the proposed Omnisource Cross-modal Learning (OCL) and compare its variants, we set a post-pretraining and fine-tuning pipeline and adopt the same hyper-parameters for all experiments.  $\\mathcal{L}_{V\\leftrightarrow S}$  is the baseline contrastive",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2023",
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+    },
+    {
+        "type": "page_number",
+        "text": "7",
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+    {
+        "type": "table",
+        "img_path": "images/9f8a64848520e3dd04e3d952d5c6775edf390d9c8081f649d686b720aff40a9e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"4\">MSR-VTT Retrieval</td><td colspan=\"4\">ActivityNet Captions Retrieval</td></tr><tr><td>R@1↑</td><td>R@5↑</td><td>R@10↑</td><td>Mean↑</td><td>R@1↑</td><td>R@5↑</td><td>R@10↑</td><td>Mean↑</td></tr><tr><td>ClipBERT (Lei et al., 2021)</td><td>22.0</td><td>46.8</td><td>59.9</td><td>6.0</td><td>21.3</td><td>49.0</td><td>63.5</td><td>6.0</td></tr><tr><td>VLM (Xu et al., 2021a)</td><td>28.1</td><td>55.5</td><td>67.4</td><td>4.0</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>MMT (Gabeur et al., 2020)</td><td>26.6</td><td>57.1</td><td>69.6</td><td>4.0</td><td>28.7</td><td>61.4</td><td>-</td><td>3.3</td></tr><tr><td>Support Set (Patrick et al., 2021)</td><td>30.1</td><td>58.5</td><td>69.3</td><td>3.0</td><td>29.2</td><td>61.6</td><td>-</td><td>3.0</td></tr><tr><td>Frozen (Bain et al., 2021)</td><td>31.0</td><td>59.5</td><td>70.5</td><td>3.0</td><td>28.8</td><td>60.9</td><td>-</td><td>3.0</td></tr><tr><td>VideoCLIP (Xu et al., 2021b)</td><td>30.9</td><td>55.4</td><td>66.8</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>HD-VILA (Xue et al., 2022)</td><td>35.6</td><td>65.3</td><td>78.0</td><td>3.0</td><td>28.5</td><td>57.4</td><td>-</td><td>4.0</td></tr><tr><td>Florence (Yuan et al., 2021)</td><td>37.6</td><td>63.8</td><td>72.6</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>All-in-One (Wang et al., 2022a)</td><td>37.9</td><td>68.1</td><td>77.1</td><td>-</td><td>22.4</td><td>53.7</td><td>67.7</td><td>5.0</td></tr><tr><td>BridgeFormer (Ge et al., 2022)</td><td>37.6</td><td>64.8</td><td>75.1</td><td>3.0</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>CLIP-ViT-B/32:</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>CLIP4Clip (Luo et al., 2021)</td><td>44.5</td><td>71.4</td><td>81.6</td><td>2.0</td><td>40.5</td><td>72.4</td><td>-</td><td>2.0</td></tr><tr><td>CenterCLIP (Zhao et al., 2022)</td><td>44.2</td><td>71.6</td><td>82.1</td><td>2.0</td><td>43.9</td><td>74.6</td><td>85.8</td><td>2.0</td></tr><tr><td>XPool (Gorti et al., 2022)</td><td>46.9</td><td>72.8</td><td>82.2</td><td>2.0</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>CLIP2Video (Fang et al., 2021)</td><td>45.6</td><td>72.6</td><td>81.7</td><td>2.0</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>CLIP2Video†(Bogolin et al., 2022)</td><td>47.2</td><td>73.0</td><td>83.0</td><td>2.0</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>CLIP2TV (Gao et al., 2021)</td><td>46.1</td><td>72.5</td><td>82.9</td><td>2.0</td><td>44.1</td><td>75.2</td><td>-</td><td>2.0</td></tr><tr><td>DRL (Wang et al., 2022c)</td><td>47.4</td><td>74.6</td><td>83.8</td><td>2.0</td><td>44.2</td><td>74.5</td><td>86.1</td><td>2.0</td></tr><tr><td>CAMoE* (Cheng et al., 2021)</td><td>47.3</td><td>74.2</td><td>84.5</td><td>2.0</td><td>51.0</td><td>77.7</td><td>-</td><td>-</td></tr><tr><td>Ours</td><td>50.1</td><td>74.8</td><td>84.6</td><td>1.0</td><td>51.1</td><td>78.4</td><td>88.3</td><td>1.0</td></tr><tr><td>Ours*</td><td>55.9</td><td>77.0</td><td>86.8</td><td>1.0</td><td>59.1</td><td>83.9</td><td>91.3</td><td>1.0</td></tr><tr><td>CLIP-ViT-B/16:</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>CenterCLIP (Zhao et al., 2022)</td><td>48.4</td><td>73.8</td><td>82.0</td><td>2.0</td><td>46.2</td><td>77.0</td><td>87.6</td><td>2.0</td></tr><tr><td>CLIP2TV (Gao et al., 2021)</td><td>49.3</td><td>74.7</td><td>83.6</td><td>2.0</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DRL (Wang et al., 2022c)</td><td>50.2</td><td>76.5</td><td>84.7</td><td>1.0</td><td>46.2</td><td>77.3</td><td>88.2</td><td>2.0</td></tr><tr><td>DRL†(Wang et al., 2022c)</td><td>53.3</td><td>80.3</td><td>87.6</td><td>1.0</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Ours</td><td>54.2</td><td>77.2</td><td>84.8</td><td>1.0</td><td>53.4</td><td>81.4</td><td>90.0</td><td>1.0</td></tr><tr><td>Ours*</td><td>57.7</td><td>80.5</td><td>88.2</td><td>1.0</td><td>61.4</td><td>85.7</td><td>92.6</td><td>1.0</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 5: Comparison with SOTA models in MSR-VTT (Xu et al., 2016) and ActivityNet (Krishna et al., 2017a) text-to-video retrieval tasks. * and † respectively denotes that the method uses DSL (Cheng et al., 2021) and QB-Norm (Bogolin et al., 2022) as post-processing operations.",
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+    {
+        "type": "text",
+        "text": "loss on video-subtitle pairs. After introducing auxiliary captions, we study four variants of OCL Loss: (a)  $\\mathcal{L}_{V\\leftrightarrow S} + \\mathcal{L}_{F\\leftrightarrow C}$ ; (b)  $\\mathcal{L}_{V\\leftrightarrow S} + \\mathcal{L}_{V\\leftrightarrow C}$ ; (c)  $\\mathcal{L}_{V\\leftrightarrow S} + \\mathcal{L}_{V\\leftrightarrow C} + \\mathcal{L}_{F\\leftrightarrow C}$ ; (d)  $\\mathcal{L}_{V\\leftrightarrow S,C} + \\mathcal{L}_{F\\leftrightarrow C}$  as explained in Section 4.3. We pre-train models with each loss function for only one epoch due to the costly training, then finetune on two video-text retrieval datasets: MSR-VTT and DiDeMo. We compare the results with CLIP-MeanPool and CLIP using the proposed Video Proxy mechanism without post-pretraining (i.e., CLIP-ViP). The results are listed in Table 3. On MSR-VTT dataset, we find that  $\\mathcal{L}_{V\\leftrightarrow S}$  brings very little improvement:  $0.4\\%$  on average of Recall@1, 5, 10. This is due to the large domain gap between MSR-VTT and post-pretraining data. Combined with auxiliary captions, four variants of OCL loss all bring significant improvements: over  $3\\%$  on Recall@1 and over  $2.3\\%$  on average of Recall@1, 5, 10. On DiDeMo dataset, based on the improvement brought by  $\\mathcal{L}_{V\\leftrightarrow S}$ , OCL further improve the results by a large margin:  $8\\%$  on average of Recall@1, 5, 10. Finally,  $\\mathcal{L}_{V\\leftrightarrow S,C} + \\mathcal{L}_{F\\leftrightarrow C}$  performs best which is applied as our final setting.",
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+        "text": "Auxiliary Data. In this part, we ablate the contribution of large-scale noisy data and auxiliary data. For uni-source, we use video-subtitle pairs and video-caption data for post-pretraining by vanilla contrastive loss. For data combination, we apply OCL under  $\\mathcal{L}_{V\\leftrightarrow S,C} + \\mathcal{L}_{F\\leftrightarrow C}$  setting to post-pretrain on the combined data. From Table 4, Omnisource post-pretraining results are much better than two uni-source results. On MSR-VTT, both uni-source post-pretraining show limited improvement:  $67.4\\%$  and  $66.9\\%$  compared with  $67.0\\%$ . While the Omnisource post-pretraining brings a significant improvement of  $2.6\\%$ . On DiDeMo, the benefit of data combination is also obvious, with nearly double the improvements brought by uni-source. These results show that the auxiliary data together with our designed joint learning method can effectively adapt the image-text pre-trained model to video-text post-pretraining.",
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+    {
+        "type": "text",
+        "text": "As the generation of auxiliary captions is based on OFA-Caption (Wang et al., 2022b), a powerful image-text pre-trained model, we also explore only including existing data in post-pretraining. We choose image-text pairs of several widely-adopted datasets: MS-COCO, Visual Genome (VG) (Krishna et al., 2017b), Flickr-30K (Young et al., 2014), SBU (Ordonez et al., 2011), CC3M (Sharma et al., 2018) and CC12M as our auxiliary data (namely ImageCaption). To ablate the contribution of these data, we add experiments of post-pretraining on ImageCaption alone and HD-VILA-100M",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2023",
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+    },
+    {
+        "type": "page_number",
+        "text": "8",
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+    {
+        "type": "table",
+        "img_path": "images/0a9ac59db49e10044cecde9217dd2c9ba51eed05979fa930dc4ac5219c04d327.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"4\">DiDeMo Retrieval</td><td colspan=\"4\">LSMDC Retrieval</td></tr><tr><td>R@1↑</td><td>R@5↑</td><td>R@10↑</td><td>Mean ↑</td><td>R@1↑</td><td>R@5↑</td><td>R@10↑</td><td>Mean ↑</td></tr><tr><td>MMT (Gabeur et al., 2020)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>12.9</td><td>29.9</td><td>40.1</td><td>19.3</td></tr><tr><td>ClipBERT (Lei et al., 2021)</td><td>20.4</td><td>48.0</td><td>60.8</td><td>6.0</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Frozen (Bain et al., 2021)</td><td>31.0</td><td>59.8</td><td>72.4</td><td>3.0</td><td>15.0</td><td>30.8</td><td>40.3</td><td>20.0</td></tr><tr><td>HD-VILA (Xue et al., 2022)</td><td>28.8</td><td>57.4</td><td>69.1</td><td>4.0</td><td>17.4</td><td>34.1</td><td>44.1</td><td>15.0</td></tr><tr><td>All-in-One (Wang et al., 2022a)</td><td>32.7</td><td>61.4</td><td>73.5</td><td>3.0</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>BridgeFormer (Ge et al., 2022)</td><td>37.0</td><td>62.2</td><td>73.9</td><td>3.0</td><td>17.9</td><td>35.4</td><td>44.5</td><td>15.0</td></tr><tr><td>CLIP-ViT-B/32:</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>CLIP4Clip (Luo et al., 2021)</td><td>43.4</td><td>70.2</td><td>80.6</td><td>2.0</td><td>21.6</td><td>41.8</td><td>49.8</td><td>11.0</td></tr><tr><td>CenterCLIP (Zhao et al., 2022)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>21.7</td><td>39.8</td><td>49.8</td><td>11.0</td></tr><tr><td>XPool (Gorti et al., 2022)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>22.7</td><td>42.6</td><td>51.2</td><td>10.0</td></tr><tr><td>CLIP2TV (Gao et al., 2021)</td><td>45.5</td><td>69.7</td><td>80.6</td><td>2.0</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DRL (Wang et al., 2022c)</td><td>47.9</td><td>73.8</td><td>82.7</td><td>2.0</td><td>24.9</td><td>45.7</td><td>55.3</td><td>7.0</td></tr><tr><td>CAMoE* (Cheng et al., 2021)</td><td>43.8</td><td>71.4</td><td>-</td><td>-</td><td>25.9</td><td>46.1</td><td>53.7</td><td>-</td></tr><tr><td>Ours</td><td>48.6</td><td>77.1</td><td>84.4</td><td>2.0</td><td>25.6</td><td>45.3</td><td>54.4</td><td>8.0</td></tr><tr><td>Ours*</td><td>53.8</td><td>79.6</td><td>86.5</td><td>1.0</td><td>26.0</td><td>46.4</td><td>54.9</td><td>8.0</td></tr><tr><td>CLIP-ViT-B/16:</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>CLIP4Clip by (Zhao et al., 2022)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>24.1</td><td>45.0</td><td>55.1</td><td>8.0</td></tr><tr><td>CenterCLIP (Zhao et al., 2022)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>24.2</td><td>46.2</td><td>55.9</td><td>8.0</td></tr><tr><td>DRL (Wang et al., 2022c)</td><td>49.0</td><td>76.5</td><td>84.5</td><td>2.0</td><td>26.5</td><td>47.6</td><td>56.8</td><td>7.0</td></tr><tr><td>Ours</td><td>50.5</td><td>78.4</td><td>87.1</td><td>1.0</td><td>29.4</td><td>50.6</td><td>59.0</td><td>5.0</td></tr><tr><td>Ours*</td><td>55.3</td><td>82.0</td><td>89.3</td><td>1.0</td><td>30.7</td><td>51.4</td><td>60.6</td><td>5.0</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Table 6: Comparison with SOTA models in DiDeMo (Anne Hendricks et al., 2017) and LSMDC (Rohrbach et al., 2016) text-to-video retrieval tasks. * denotes using post-processing DSL (Cheng et al., 2021).",
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+    {
+        "type": "text",
+        "text": "combined with ImageCaption. From Table 4, post-pretraining on ImageCaption alone results in performance degradation on MSR-VTT and marginal improvement on DiDeMo. In contrast, ImageCaption yields significant performance gains on both datasets when used as auxiliary data for HD-VILA-100M. This further illustrates the importance of the combination of large-scale noisy data and auxiliary data.",
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+    {
+        "type": "text",
+        "text": "5.3 COMPARISON TO STATE-OF-THE-ART MODELS",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We compare our model under full setting (in three epochs) with state-of-the-art works on the text-to-video retrieval task. The results of fine-tuning on four datasets (i.e., MSR-VTT, DiDeMo, ActivityNet Captions, LSMDC) are shown in Table 5 and 6, respectively. We clarify the backbone for CLIP-based works. We only add results with DSL (Cheng et al., 2021) to make fair comparison with some methods using post-processing operations e.g., DSL (Cheng et al., 2021) or QB-Norm (Bogolin et al., 2022). Our model achieves the best results on all datasets in both CLIP-ViT-B/32 and CLIP-ViT-B/16. Note that some existing methods are also applicable on top of our models as our modification to the CLIP model is minimal. Note that even without post-processing (e.g., DSL), our results still surpass methods using post-processing operations on most datasets. Besides, adding DSL will greatly improve the performance of our model since our model has good bidirectional vision-language correspondence. The good results on the ActivityNet Captions dataset also indicate that our model can generalize well to long videos. Overall, the improvements on different datasets demonstrate the superiority of the video-language representation learned by our CLIP-ViP model.",
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+        "type": "text",
+        "text": "6 CONCLUSION",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this paper, we study further pre-training (post-pretraining) image-text models like CLIP on large-scale video data. We first conduct a preliminary analysis to reveal the factors hindering video postpretraining. Motivated by findings, we propose CLIP-ViP which includes an Omnisource Cross-modal Learning method and a Video Proxy mechanism. The Video Proxy mechanism can better model videos containing temporal information while reducing conflicts with the pre-trained CLIP model. The Omnisource Cross-modal Learning alleviates the problem caused by the domain gap between video-subtitle and downstream data. Extensive results show that our approach improves the performance of CLIP on video-text retrieval by a large margin and also achieves new state-of-the-art results on four widely-used video-language benchmarks.",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2023",
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+        "type": "page_number",
+        "text": "9",
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+    {
+        "type": "text",
+        "text": "REFERENCES",
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+        "text": "Published as a conference paper at ICLR 2023",
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+        ],
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+    {
+        "type": "text",
+        "text": "A MORE DOWNSTREAM TASKS",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We add various suitable downstream tasks to verify that our model can be generalizable to other tasks. Specifically, we add four zero-shot evaluations on video action recognition and videolanguage multi-choice tasks:",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "HMDB51 Action Recognition HMDB51 (Kuehne et al., 2011) is a video dataset containing realistic videos from various sources, including movies and web videos. There are 51 action categories (such as \"jump\", \"kiss\" and \"laugh\"). We evaluate our model by zero-shot classification on HMDB51. To adapt to our model, we formulate video classification as video-text alignment, using text prompts \"a person [label]\". The prediction is made on top-1 similarity between query video and prompted texts. We directly test our model without finetuning to report zero-shot accuracy results.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "SomethingSomethingV2 (SSv2) Action Recognition. SomethingSomethingV2 (Goyal et al., 2017) dataset is a large collection of labeled video clips that show humans performing pre-defined basic actions with everyday objects. It requires models to develop fine-grained understanding of basic actions that occur in the physical world. To adapt to our model, we design a multi-choice task on SSv2. The model has to choose the ground truth annotation among all categories. We directly test our model without finetuning to report zero-shot accuracy results.",
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+    },
+    {
+        "type": "text",
+        "text": "MSR-VTT Multi-Choice (MC). MSR-VTT MC is a benchmark build on MSR-VTT video dataset. Given a video query and five descriptive sentences, MSR-VTT MC requires the model to choose a single best answer in the candidates. We directly test our model without finetuning to report zero-shot accuracy results.",
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "LSMDC Multi-Choice (MC). LSMDC MC is a benchmark build on movie data. Similar with MSR-VTT MC, it requires the model to choose a single best answer from 5 candidates. We report zero-shot accuracy on 10000 test videos.",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "From the results in Table 7, we can find that post pretraining CLIP on subtitle alone will lead to a performance drop on most tasks: HMDB51 SSv2 and MSR-VTT MC. This phenomenon is in consistency with our assumption as subtitles usually have unique form. By our final setting, CLIP-ViP",
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2023",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "page_number",
+        "text": "13",
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+    {
+        "type": "table",
+        "img_path": "images/cacc276b412663d4f75c6316634ef61377721f6ff9df99e11f97a7066ba885f9.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>HMDB51</td><td>SSv2</td><td>MSR-VTT MC</td><td>LSMDC MC</td></tr><tr><td>CLIP-MeanPool</td><td>40.3</td><td>25.8</td><td>89.9</td><td>64.3</td></tr><tr><td>CLIP-ViP w/o post-pretraining</td><td>41.1</td><td>26.1</td><td>89.4</td><td>63.5</td></tr><tr><td>CLIP-ViP w/ video-subtitle</td><td>36.6</td><td>28.0</td><td>86.6</td><td>67.0</td></tr><tr><td>CLIP-ViP full</td><td>44.8</td><td>33.0</td><td>90.3</td><td>67.6</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "achieves better results than CLIP on all evaluations, especially on SSv2 which requires more understanding on temporality (Wang et al., 2022d). These results demonstrate that CLIP-ViP outperforms CLIP on various video benchmarks and further show the generalization of our model CLIP-ViP.",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B EFFICIENCY OF POST-PRETRAINING",
+        "text_level": 1,
+        "bbox": [
+            171,
+            329,
+            508,
+            345
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In light of the good image representation of CLIP, our post-pretraining significantly reduces the training cost, Compared with HD-VILA (Xue et al., 2022) which trains video-language representation from scratch, CLIP-ViP outperforms HD-VILA by a large margin with about 1/30 training time. To make more fair comparison with training from scratch. We conduct an experiment of training from scratch with the same setting and training time (one epoch). Then we finetune on MSR-VTT Retrieval to evaluate the pre-trained model.",
+        "bbox": [
+            169,
+            359,
+            826,
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+    },
+    {
+        "type": "table",
+        "img_path": "images/2eae2f4233d5df8a53a5f76fd49e81abab981e1d603c5d2a09d2f603d558fa12.jpg",
+        "table_caption": [
+            "Table 7: Evaluation on more downstream tasks. We report zero-shot accuracy  $(\\%)$  on HMDB51, SSv2, MSRVTT-MC, and LSMDC MC. All models are base size with patch size of 32. Details of post-pretraining are the same as in Section 5.2."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>MSR-VTT R@1</td><td>MSR-VTT R@5</td><td>MSR-VTT R@10</td></tr><tr><td>CLIP-ViP (from scratch)</td><td>28.5</td><td>55.2</td><td>68.6</td></tr><tr><td>CLIP-ViP</td><td>49.6</td><td>74.5</td><td>84.8</td></tr></table>",
+        "bbox": [
+            207,
+            455,
+            790,
+            515
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table 8: Comparison with training CLIP-ViP from scratch. We report finetuning results on MSR-VTT Retrieval. All models are base size with patch size of 32. Details of post-pretraining are the same as in Section 5.2.",
+        "bbox": [
+            169,
+            523,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "From the results in Table 8, we can see that there is a huge performance gap. Without basing on a CLIP, CLIP-VIP can not leverage the rich knowledge entailed in CLIP, thus leading to a much lower training efficiency.",
+        "bbox": [
+            169,
+            583,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2023",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 13
+    },
+    {
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+        "text": "14",
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+    }
+]

2023/CLIP-ViP_ Adapting Pre-trained Image-Text Model to Video-Language Alignment/300040d0-80cf-4593-ab2e-edbe3e46a15e_origin.pdf

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2023/CLIP-ViP_ Adapting Pre-trained Image-Text Model to Video-Language Alignment/full.md

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+# CLIP-VIP: ADAPTING PRE-TRAINED IMAGE-TEXT MODEL TO VIDEO-LANGUAGE ALIGNMENT
+
+Hongwei Xue $^{1*}$ , Yuchong Sun $^{2*}$ , Bei Liu $^{3\dagger}$ , Jianlong Fu $^{3\dagger}$ , Ruihua Song $^{2}$ , Houqiang Li $^{1}$ , Jiebo Luo $^{4}$
+
+<sup>1</sup>University of Science and Technology of China, Hefei, China,  
+$^{2}$ Renmin University of China, Beijing, China,  
+<sup>3</sup>Microsoft Research, Beijing, China,  
+<sup>4</sup>University of Rochester, Rochester, NY
+
+# ABSTRACT
+
+Pre-trained image-text models, like CLIP, have demonstrated the strong power of vision-language representation learned from a large scale of web-collected image-text data. In light of the well-learned visual features, there are works that transfer image representation to the video domain and achieve good results. However, adapting image-text pre-trained models to video-text pre-training (i.e., post-pretraining) has not demonstrated a significant advantage yet. In this paper, we tackle this challenge by raising and addressing two questions: 1) what are the factors hindering post-pretraining CLIP from improving performance on video-text tasks, and 2) how to mitigate the impact of these factors. Through a series of comparative experiments and analyses, we find that the data scale and domain gap between language sources have large impacts. By these observations, we propose an Omnisource Cross-modal Learning method equipped with a Video Proxy mechanism on the basis of CLIP, namely CLIP-ViP. Extensive results show that our approach improves the performance of CLIP on video-text retrieval by a large margin. Our model achieves state-of-the-art results on a variety of datasets, including MSR-VTT, DiDeMo, LSMDC, and ActivityNet. We release our code and pre-trained CLIP-ViP models at https://github.com/microsoft/XPretrain/tree/main/CLIP-ViP.
+
+# 1 INTRODUCTION
+
+In the past few years, vision-language pre-training has achieved great success on cross-modal representation learning from a large scale of web-crawled data (Radford et al., 2021; Jia et al., 2021; Li et al., 2021; Wang et al., 2021b; Zellers et al., 2021; 2022; Bain et al., 2021). Among them, image-text pre-trained models (Radford et al., 2021; Jia et al., 2021) have shown powerful capability for various downstream tasks, including visual understanding (Gu et al., 2021; Wang et al., 2021a; Rao et al., 2022), image-text generation (Patashnik et al., 2021; Mokady et al., 2021) and so on (Guzhov et al., 2022; Zhang et al., 2022). In light of the well-learned and enriched visual representation, some works directly adapt image-text pre-trained models to video-text downstream tasks without further pre-training on video data (Luo et al., 2021; Fang et al., 2021; Gorti et al., 2022; Zhao et al., 2022), while still outperforming models pre-trained on video data (Xu et al., 2021b; Bain et al., 2021).
+
+Utilizing an existing powerful image-text pre-trained model for further video-text pre-training (i.e., post-pretraining) is able to reduce the required training cost by making good use of the knowledge learned from images. However, adapting image-text pre-trained models to video-text data for postpretraining has not demonstrated a significant advantage yet, and thus is still under-explored. A preliminary study is conducted by CLIP4Clip (Luo et al., 2021) which adopts MeanPooling by averaging multiple frame features based on the CLIP model on a subset of Howto100M (Miech et al., 2019). While the improvement over directly using the image-text pre-trained model is marginal for
+
+either zero-shot or fine-tuning settings. In this paper, we aim to explore how to effectively adapt the image-text pre-trained model (e.g., CLIP) to video-language representation learning for video-text tasks (e.g., text-to-video retrieval).
+
+To unleash the power of video data to adapt image-text pre-trained models for post-pretraining, we conduct several preliminary experiments to figure out the challenges that hinder post-pretraining. First, we explore post-pretraining an image-text pre-trained model (i.e., CLIP) with MeanPooling on video-text datasets with different scales, including WebVid-2.5M (Bain et al., 2021) and HDVILA-100M (Xue et al., 2022). The result shows that the scale of data is critical for video-text post-pretraining. Data on a small scale makes the model easy to over-fit the new data while the knowledge learned from image-text is suppressed and the performance is reduced. Second, we investigate the language domain gap between pre-training data and downstream data. By calculating the Normalized Mutual Information (NMI) on clusters of text features, we find that there is a large domain gap between subtitles that are used in large-scale video-text pre-training data and descriptive texts in downstream tasks.
+
+To mitigate the impact of the above factors, we propose CLIP-ViP to adapt the pre-trained image-text model CLIP for video-text pre-training. First, we introduce auxiliary captions that have a smaller language domain gap with downstream data into existing large-scale video-text data. Instead of using a video captioning model which may cause data leakage by training on the same dataset with video-text downstream tasks, and considering a better visual captioning capability, we adopt an image captioning model to generate an auxiliary caption of middle frame in each video. In order to adapt a Transformer-based vision encoder to process both images and videos with minimal modification, we then propose video proxy tokens and design a proxy-guided video attention mechanism for the Vision Transformer (ViT). Specifically, during attention computation in each block, video proxy tokens can interact with all tokens, while patch tokens only interact with video proxy tokens and patch tokens within the same frame. Our vision encoder only increases negligible parameters and calculations compared to the vanilla Vision Transformer while increasing the generality and extendability. To facilitate cross-modal representation learning from both caption-frame and video-subtitle data types at the same time, we propose an Omnisource Cross-modal Learning (OCL) method for pre-training and study a series of variants to find the best fusion strategy.
+
+Our experimental results show that our approach improves the performance of CLIP on text-to-video retrieval tasks by a large margin. We also conduct ablation studies to verify the effectiveness of each part in our approach. Our contributions are summarized as follows: (1) We are one of the first to explore factors that hinder video post-pretraining on pre-trained image-text models; (2) We propose CLIP-ViP that can effectively leverage image-text pre-trained model for post-pretraining; (3) We conduct extensive experiments to verify the effectiveness of our method. Our model outperforms the state-of-the-art results by a large margin on four widely-used benchmarks.
+
+# 2 RELATED WORK
+
+Vision-Language Pre-Training End-to-end models (Lei et al., 2021; Xue et al., 2022; Zellers et al., 2021; Fu et al., 2021; Huang et al., 2020; 2021b; Xue et al., 2021; Li et al., 2021; Kim et al., 2021; Huang et al., 2021a; Sun et al., 2022) for vision-language pre-training are replacing the traditional approach using pre-extracted visual features by off-the-shelf models (Sun et al., 2019; Xu et al., 2021b; Zhu & Yang, 2020; Li et al., 2020b;a; Chen et al., 2020). Training end-to-end models on large-scale web-collected data also gradually demonstrates the big advantages (Radford et al., 2021; Jia et al., 2021; Xue et al., 2022; Zellers et al., 2021; 2022). Unlike images that have alt-texts, large-scale video datasets suitable for pre-training usually use subtitles as text sources (Miech et al., 2019; Xue et al., 2022). Subtitles are much noisier than alt-texts, according to (Miech et al., 2019), typical examples of incoherence include the content producer asking viewers to subscribe to their channel, talking about something unrelated to the video, or describing something before or after it happens. Bain et al. collect a video dataset WebVid (Bain et al., 2021) with textual description annotations. Their texts are well aligned with the video and avoid suffering from ASR errors. However, the vast majority of WebVid videos are sourced from a stock footage website, so scaling up is under limitation. The video-subtitle data is more easily accessible on the web and thus suitable for scaling up. In this paper, we investigate the unfavorable factors of video-subtitle data and explore how to mitigate the impact of these factors.
+
+![](images/076dac7c87df262effc168cb0293de331d94f61edad53411a3ec3cabbec3f770.jpg)  
+Figure 1: The curve of finetuning results during post-pretraining. The x-axis indicates the percentage of training steps. The y-axis indicates average value of Recall@1, 5 and 10. [Best viewed in color]
+
+<table><tr><td>NMI Score</td><td>MSR-VTT</td><td>DiDeMo</td><td>Mean</td></tr><tr><td>HD-VILAsub</td><td>0.831</td><td>0.684</td><td>0.758</td></tr><tr><td>HD-VILAcap</td><td>0.317</td><td>0.621</td><td>0.469</td></tr><tr><td>WebVid</td><td>0.420</td><td>0.488</td><td>0.454</td></tr><tr><td>COCO</td><td>0.373</td><td>0.758</td><td>0.566</td></tr><tr><td>CC12M</td><td>0.445</td><td>0.673</td><td>0.559</td></tr></table>
+
+Table 1: Normalized Mutual Information (NMI) score of language features extracted on series of data and downstream tasks. Larger value indicates larger domain gap.
+
+Pre-trained Models for Video-Text Retrieval The great success of the CLIP has demonstrated its unprecedented power on varies downstream tasks, including vision understanding (Gu et al., 2021; Wang et al., 2021a; Rao et al., 2022), image-text generation (Patashnik et al., 2021; Mokady et al., 2021) and so on (Guzhov et al., 2022; Zhang et al., 2022). By contrastive learning on large-scale image-text pairs, CLIP learns enriched visual concepts for images. Recently, some works directly transfer CLIP to video-text retrieval without further pretraining on video data (post-pretraining) (Luo et al., 2021; Fang et al., 2021; Gorti et al., 2022; Zhao et al., 2022; Wang et al., 2022c). Their work takes the performance of video-text retrieval to a new level, outperforming existing models pre-trained on video data (Xu et al., 2021b; Bain et al., 2021; Xue et al., 2022; Ge et al., 2022; Wang et al., 2022a). They transfer CLIP from views of feature aggregation (Luo et al., 2021; Zhao et al., 2022; Fang et al., 2021; Gorti et al., 2022) or representation alignment (Fang et al., 2021; Gorti et al., 2022; Wang et al., 2022c). In parallel with these works, we study post-pretraining with video data on top of CLIP in an effective way and our model can be combined with other approaches effectively.
+
+# 3 PRELIMINARY ANALYSIS
+
+In this section, we first study the impact of the data scale for adapting image-text pre-training to video-text post-pretraining, and then investigate how the language domain gap affects the adaption.
+
+# 3.1 POST-PRETRAINING WITH DIFFERENT DATA SCALES
+
+To study the effectiveness of different data scales, we use the CLIP-ViT-B/32 model (Radford et al., 2021) as the base image-text pre-trained model and adopt MeanPooling for video adaption like CLIP4Clip (Luo et al., 2021) by averaging multiple frame features as video feature. Two open-domain video-text datasets are used: WebVid-2.5M (Bain et al., 2021) with 2.5 million pairs and HD-VILA-100M (Xue et al., 2022) with 100M pairs. We also adopt a subset of HD-VILA-100M containing random  $10\%$  data (namely HD-VILA-10M) as a middle setting. We run the same number of steps on all settings, equivalent to one epoch on HD-VILA-100M. We uniformly sample 12 frames from each video and apply the same hyper-parameters as described in Section 5 for all settings.
+
+During post-pretraining, we evaluate the pre-trained models by fine-tuning on MSR-VTT text-to-video retrieval task. Figure 1 shows the performance trend. We observe an overfitting phenomenon that continuous post-pretraining leads to a performance drop. And the drop is more significant for smaller data (e.g., WebVid-2.5M and HD-VILA-10M). As CLIP is pre-trained on 400 million image-text pairs, further training on small data makes the model tend to overfit the new data while the implicit knowledge learned from the image-text pairs is degrading. As a consequence, the performance will drop, even worse than using CLIP directly. Thus we adopt HD-VILA-100M due to its large scale and diverse category.
+
+# 3.2 LANGUAGE DOMAIN GAP WITH DOWNSTREAM DATA
+
+It is intuitive that pre-training on data with the same domain as downstream data can benefit downstream tasks. For most video-text tasks like video-text retrieval, texts are descriptive sentences of videos (i.e., captions). While for HD-VILA-100M, which we will use for pre-training, the texts are auto-transcribed subtitles and they indicate very different relevance to visual information compared to descriptive texts. Meanwhile, auto-transcribed subtitles suffer from irrelevance, misalignment, and ASR errors (Miech et al., 2019). To better explore the language domain gap between pre
+
+![](images/c09f6fd3093317bdc4c09b3ca3bdc291d612b36973f3ca5ac07b210a9dbcf999.jpg)  
+Figure 2: The framework of CLIP-ViP with a text encoder and a vision encoder. Taken features  $V$ ,  $F$ ,  $S$ ,  $C$  of videos, frames, subtitles, captions as input, a series of Omnisource cross-modal learning variants are studied to explore better representation learning losses: (a)  $\mathcal{L}_{V \leftrightarrow S} + \mathcal{L}_{F \leftrightarrow C}$ ; (b)  $\mathcal{L}_{V \leftrightarrow S} + \mathcal{L}_{V \leftrightarrow C}$ ; (c)  $\mathcal{L}_{V \leftrightarrow S} + \mathcal{L}_{V \leftrightarrow C} + \mathcal{L}_{F \leftrightarrow C}$ ; (d)  $\mathcal{L}_{V \leftrightarrow S, C} + \mathcal{L}_{F \leftrightarrow C}$ . In the vision encoder, Video proxy tokens and the ViP-guided attention mechanism is proposed to transfer CLIP into the video domain. [Best viewed in color]
+
+training data and downstream data, we measure the inconsistency by calculating the dissimilarity between their language features. For downstream language data, we choose two typical video-text retrieval datasets: MSR-VTT (Xu et al., 2016) and DiDeMo (Anne Hendricks et al., 2017). For pretraining language, we select four types: video subtitles of HD-VILA-100M (HD-VILAsub), video captions of WebVid-2.5M, image captions of MS-COCO (Lin et al., 2014), and web-collected alttexts of Conceptual Caption 12M (Changpinyo et al., 2021). In addition, we analyze auto-generated captions of HD-VILA-100M (HD-VILAscap), which will be introduced in Section 4.
+
+We use a Transformer Encoder initialized from CLIP (Radford et al., 2021) to extract text features. To quantify the domain gap of languages between pre-training and downstream data, we first mix their text features and then use K-means to get two clusters. Then we calculate the Normalized Mutual Information (NMI) between cluster labels and ground-truth labels of pre-training or downstream. A larger NMI value means that the two types of features are easy to be distinguished, thus there is a larger domain gap. For each comparison, we randomly sample 1000 texts from each type of data for 10 times and adopt the average of 10 results. We report the results in Table 1. Comparing the values of all pre-training data types, we find that the NMI score between HD-VILAsub and downstream data is much larger than others, especially for MSR-VTT downstream dataset. This indicates that direct training with subtitles may introduce inconsistency with downstream tasks.
+
+# 4 APPROACH
+
+In this section, we will introduce the proposed CLIP-ViP video pre-training framework. To bridge language domain gaps between image and video datasets, we first introduce an in-domain auxiliary data generation method. Then, we propose a novel Video Proxy mechanism to enable the Vision Transformer (ViT) model for both image and video encoding. We further present an Omnisource Cross-modal Learning (OCL) method which can jointly learn cross-modal representation from video-text and image-text pairs.
+
+# 4.1 IN-DOMAIN AUXILIARY DATA GENERATION
+
+Motivated by the analysis in Section 3, we introduce auxiliary captions into large-scale video-subtitle data to reduce the language domain gap between pre-training and downstream data. We adopt an
+
+image captioning model for two reasons. 1) Most SOTA video captioning models are trained with video-text datasets (e.g., MSR-VTT, ActivityNet) which are also used for downstream tasks. We avoid data leakage to perform pre-training agnostic to downstream data. 2) The performance of existing video captioning models lags far behind that of images. Thus, we choose a powerful image captioning model OFA-Caption (Wang et al., 2022b) to generate one caption for the middle frame of each video in HD-VILA-100M. We use the default setting of the OFA-Caption model. As a result, we generate 100M sentences with a max length of 16 words. This method can be applied to any video data and we will release the generated captions to facilitate future research.
+
+# 4.2 VIDEO PROXY MECHANISM
+
+Since video is an ordered sequence of frames, it is critical to learn the frame aggregation and temporality when transferring to the video domain. Meanwhile, to keep the high generality and extendability of the Vision Transformer (ViT) backbone, we aim to find a simple but effective way to transfer ViT to enable both image and video encoding with minimal modifications. Given a video containing  $T$  frames:  $\{f_1, f_2, \dots, f_T\}$ , we follow CLIP to divide each frame into N patches:  $\{f_t^1, f_t^2, \dots, f_t^N \mid t \in [1, T]\}$ . Then we add spatio-temporal positional embedding to each flattened 2D patches:
+
+$$
+g \left(f _ {t} ^ {n}\right) = \operatorname {L i n e a r} \left(f _ {t} ^ {n}\right) + \operatorname {P o s} _ {s} (n) + \operatorname {P o s} _ {t} (t), \tag {1}
+$$
+
+where  $Linear(*)$  is a linear layer,  $Pos_{s}(n)$  and  $Pos_{t}(t)$  is the learnable spatial and temporal positional embedding, respectively. The whole video can be divided into  $T \times N$  patch tokens.
+
+To model spatial information from multi-frames, one simple way is directly feeding all tokens into CLIP's vision encoder and conducting attention across all tokens. However, this method introduces significant conflicts with CLIP. As CLIP is pre-trained on image and text pairs, it has difficulty handling interactions of tokens between images/frames during training. We also verify it by experiments as Full Attention setting in Table 2. Instead, we introduce a Video Proxy token to act as a proxy that helps each local patch perceive video-level temporal information.
+
+Before feeding into CLIP, we concatenate patch tokens with a set of learnable parameters called video proxy tokens:  $\mathcal{P} = \{p_1,p_2,\dots,p_M\}$ , where  $M$  is the number of video proxy tokens. Then all  $T\times N + M$  tokens will be fed into the ViT of CLIP. The output of the first video proxy token will be regarded as the video's representation. We also design a proxy-guided attention mechanism for the vanilla ViT. In the attention score calculation of each block, video proxy tokens attend to all tokens, while patch tokens only attend to tokens in the same frame plus video proxy tokens. This mechanism can be formulated as attention mask  $\mathcal{M}_{\mathrm{ViP}}$ :
+
+$$
+\mathcal {M} _ {\mathrm {V i P}} (q, k) = \left\{ \begin{array}{l l} 1 & \text {i f} q \in \mathcal {P} \text {o r} k \in \mathcal {P} \text {o r} (q, k) \text {i n t h e s a m e f r a m e}, \\ 0 & \text {o t h e r w i s e}, \end{array} \right. \tag {2}
+$$
+
+where  $q$  and  $k$  is the key and query in attention calculation. Patch tokens can obtain global information from video proxy tokens while reducing inconsistencies with the original CLIP's calculation. Our experiment in Section 5 demonstrates the superiority of this mechanism.
+
+For the input type of the image/frame, we use linear interpolation to get a middle temporal positional embedding, then treat the image/frame as a special single-frame video. This method enables joint training on both videos and images in the same batch, as our proxy-guided attention mechanism reduces the difference in calculations between video and image.
+
+# 4.3 OMNISOURCE CROSS-MODAL LEARNING
+
+To learn rich video-language alignment from video-subtitle pairs and reduce the language domain gap with downstream data by corresponding auxiliary frame-caption pairs, we study joint CrossModal Learning on the omnisor input. Following most works of learning multimodal alignment on dual encoders (Radford et al., 2021; Xue et al., 2022; Li et al., 2021; Luo et al., 2020; Xu et al., 2021b; Luo et al., 2021), we use info-NCE loss to perform contrastive learning. There are two formats of visual source: video sequences and single frames, and two types of text source: subtitles and captions in our work. We denote them by  $V$ ,  $F$ ,  $S$ , and  $C$  respectively. We define a source-wise info-NCE loss by:
+
+$$
+\mathcal {L} _ {v 2 t} = - \frac {1}{B} \sum_ {i = 1} ^ {B} \log \frac {e ^ {v _ {i} ^ {\top} t _ {i} / \tau}}{\sum_ {j = 1} ^ {B} e ^ {v _ {i} ^ {\top} t _ {j} / \tau}}, \quad \mathcal {L} _ {t 2 v} = - \frac {1}{B} \sum_ {i = 1} ^ {B} \log \frac {e ^ {t _ {i} ^ {\top} v _ {i} / \tau}}{\sum_ {j = 1} ^ {B} e ^ {t _ {i} ^ {\top} v _ {j} / \tau}} \tag {3}
+$$
+
+<table><tr><td>Model</td><td>R@1 ↑</td><td>R@5 ↑</td><td>R@10 ↑</td><td>Mean ↑</td></tr><tr><td>MeanPool</td><td>43.4</td><td>70.9</td><td>81.1</td><td>65.1</td></tr><tr><td>SeqTransformer</td><td>44.6</td><td>71.2</td><td>81.8</td><td>65.9</td></tr><tr><td>Full Attention</td><td>42.8</td><td>70.1</td><td>80.3</td><td>64.4</td></tr><tr><td>2 Video Proxy Tokens</td><td>45.8</td><td>71.3</td><td>81.7</td><td>66.3</td></tr><tr><td>4 Video Proxy Tokens</td><td>46.5</td><td>72.1</td><td>82.5</td><td>67.0</td></tr><tr><td>8 Video Proxy Tokens</td><td>45.7</td><td>72.7</td><td>81.7</td><td>66.7</td></tr></table>
+
+Table 2: MSR-VTT text-to-video retrieval results of finetuning CLIP by different settings. Mean  $\uparrow$  indicates the average value of Recall@1, 5, and 10. All results are based on CLIP-ViT-B/32.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="4">MSR-VTT Retrieval</td><td colspan="4">DiDeMo Retrieval</td></tr><tr><td>R@1 ↑</td><td>R@5 ↑</td><td>R@10 ↑</td><td>Mean ↑</td><td>R@1 ↑</td><td>R@5 ↑</td><td>R@10 ↑</td><td>Mean ↑</td></tr><tr><td>CLIP-MeanPool</td><td>43.4</td><td>70.9</td><td>81.1</td><td>65.1</td><td>40.6</td><td>67.5</td><td>77.2</td><td>61.8</td></tr><tr><td>CLIP-ViP</td><td>46.5</td><td>72.1</td><td>82.5</td><td>67.0</td><td>40.6</td><td>70.4</td><td>79.3</td><td>63.4</td></tr><tr><td>+ LV↔S</td><td>47.7</td><td>72.1</td><td>82.4</td><td>67.4</td><td>44.6</td><td>73.9</td><td>81.9</td><td>66.8</td></tr><tr><td>+ LV↔S + LF↔C</td><td>49.3</td><td>74.8</td><td>83.8</td><td>69.3</td><td>48.4</td><td>74.5</td><td>84.4</td><td>69.1</td></tr><tr><td>+ LV↔S + LV↔C</td><td>49.6</td><td>74.2</td><td>84.0</td><td>69.3</td><td>48.5</td><td>76.6</td><td>83.6</td><td>69.5</td></tr><tr><td>+ LV↔S + LV↔C + LF↔C</td><td>49.6</td><td>74.5</td><td>83.8</td><td>69.3</td><td>48.5</td><td>76.8</td><td>84.1</td><td>69.8</td></tr><tr><td>+ LV↔S,C + LF↔C</td><td>49.6</td><td>74.5</td><td>84.8</td><td>69.6</td><td>48.2</td><td>76.7</td><td>84.4</td><td>69.8</td></tr></table>
+
+Table 3: Ablation study of different losses. We report text-to-video results of models finetuned on MSR-VTT and DiDeMo. Mean  $\uparrow$  means the average of Recall@1, 5 and 10. All results are based on CLIP-ViT-B/32. All post-pretraining steps are equivalent to one epoch on HD-VILA-100M.
+
+where  $v_{i}$  and  $t_j$  are the normalized embeddings of  $i$ -th visual feature in  $X \in \{V, F\}$  and  $j$ -th text feature in  $Y \in \{S, C\}$  in a batch of size  $B$ .  $\tau$  is a learnable temperature. The overall alignment loss  $\mathcal{L}_{X \leftrightarrow Y}$  is the average of  $\mathcal{L}_{v2t}$  and  $\mathcal{L}_{t2v}$ . For example,  $\mathcal{L}_{V \leftrightarrow S}$  represents info-NCE loss within video-subtitle pairs in a batch, which pulls aligned pairs together in embedding space while pushing apart misaligned pairs.
+
+We study the reasonable variants of OCL: (a)  $\mathcal{L}_{V\leftrightarrow S} + \mathcal{L}_{F\leftrightarrow C}$ : Simple combination of two source-wise losses on video-subtitle and frame-caption pairs; (b)  $\mathcal{L}_{V\leftrightarrow S} + \mathcal{L}_{V\leftrightarrow C}$ : As there is also content correlation between videos and its middle-frame captions, we explore to add a loss on videocaption pairs to baseline loss  $\mathcal{L}_{V\leftrightarrow S}$ ; (c)  $\mathcal{L}_{V\leftrightarrow S} + \mathcal{L}_{V\leftrightarrow C} + \mathcal{L}_{F\leftrightarrow C}$ : Combination of (a) and (c); (d)  $\mathcal{L}_{V\leftrightarrow S,C} + \mathcal{L}_{F\leftrightarrow C}$ : A video corresponds to both a subtitle and auxiliary caption. Compare to (c), the numbers of negative pairs in  $\mathcal{L}_{v2t}$  can be expanded. The  $\mathcal{L}_{v2t}$  in  $\mathcal{L}_{V\leftrightarrow S,C}$  is rewritten as:
+
+$$
+\mathcal {L} _ {v 2 t} = - \frac {1}{2 B} \sum_ {i = 1} ^ {B} \left(\log \frac {e ^ {v _ {i} ^ {\top} s _ {i} / \tau}}{\sum_ {j = 1} ^ {B} e ^ {v _ {i} ^ {\top} s _ {j} / \tau} + e ^ {v _ {i} ^ {\top} c _ {j \neq i} / \tau}} + \log \frac {e ^ {v _ {i} ^ {\top} c _ {i} / \tau}}{\sum_ {j = 1} ^ {B} e ^ {v _ {i} ^ {\top} c _ {j} / \tau} + e ^ {v _ {i} ^ {\top} s _ {j \neq i} / \tau}}\right), \tag {4}
+$$
+
+where  $s_i \in S$  and  $c_i \in C$ . The  $\mathcal{L}_{t2v}$  in  $\mathcal{L}_{V \leftrightarrow S, C}$  is equal to (c). We compare all variants with the baseline  $\mathcal{L}_{V \leftrightarrow S}$  and report results in Section 5.
+
+# 5 EXPERIMENT
+
+# 5.1 EXPERIMENTAL DETAILS
+
+Video-Text Post-Pretraining. To pre-train the proposed CLIP-ViP model, we uniformly sample 12 frames and resize all frames to  $224 \times 224$  from video clips with an average length of 13.4 seconds. The sampled frames can well cover the semantics conveyed from a video clip. For text, we adopt the CLIP's tokenizer to split a sentence into word tokens with a max length of 70. We use AdamW optimizer (Loshchilov & Hutter, 2019), and empirically set an initial learning rate as 5e-6 and a fixed weight decay as 5e-2. For the learning rate schedule, we adopt a cosine decay with a warm-up strategy. We train our model with 32 NVIDIA Tesla V100 GPUs in a batch size of 1024. The contrastive similarity is calculated on gathered features from all GPUs. We set training steps to one epoch on HD-VILA-100M for all ablation studies and three epochs for the full setting.
+
+Fine-tuning Training. To better adapt CLIP-ViP to downstream tasks, we reuse most hyperparameters of post-pretraining in fine-tuning with some exceptions. 1) Batch size: we fine-tune
+
+<table><tr><td rowspan="2">Post-pretrain Data</td><td colspan="4">MSR-VTT Retrieval</td><td colspan="4">DiDeMo Retrieval</td></tr><tr><td>R@1↑</td><td>R@5↑</td><td>R@10↑</td><td>Mean ↑</td><td>R@1↑</td><td>R@5↑</td><td>R@10↑</td><td>Mean ↑</td></tr><tr><td>w/o Post-pretrain</td><td>46.5</td><td>72.1</td><td>82.5</td><td>67.0</td><td>40.6</td><td>70.4</td><td>79.3</td><td>63.4</td></tr><tr><td>HD-VILAsub</td><td>47.7</td><td>72.1</td><td>82.4</td><td>67.4</td><td>44.6</td><td>73.9</td><td>81.9</td><td>66.8</td></tr><tr><td>HD-VILAcap</td><td>45.9</td><td>73.0</td><td>81.8</td><td>66.9</td><td>44.9</td><td>74.4</td><td>82.3</td><td>67.2</td></tr><tr><td>HD-VILAsub+cap</td><td>49.6</td><td>74.5</td><td>84.8</td><td>69.6</td><td>48.2</td><td>76.7</td><td>84.4</td><td>69.8</td></tr><tr><td>ImageCaption</td><td>45.6</td><td>70.7</td><td>81.1</td><td>65.8</td><td>43.7</td><td>69.5</td><td>77.9</td><td>63.7</td></tr><tr><td>HD-VILAsub + ImageCaption</td><td>49.1</td><td>73.1</td><td>83.5</td><td>68.6</td><td>47.0</td><td>75.3</td><td>84.1</td><td>68.8</td></tr></table>
+
+Table 4: Ablation study of post-pretrain data. We report text-to-video results of models finetuned on MSR-VTT and DiDeMo. Mean  $\uparrow$  indicates an average of Recall@1, 5 and 10. For all results, the model is designed with 4 video proxy tokens and pre-trained on CLIP-ViT-B/32. All post-pretraining steps are equivalent to one epoch on HD-VILA-100M.
+
+our model with a batch size of 128 for all downstream tasks for a fair comparison. 2) Learning rate and weight decay: we empirically set them to 1e-6 and 0.2, respectively. 3) Number of epochs: due to the various scales of downstream datasets, we set epoch numbers to 5, 20, 10, and 20 for MSR-VTT, DiDeMo, LSMDC, and ActivityNet, respectively. 4) Frame number: for a fair comparison, we set frame number to 12 except for ActivityNet Captions (set to 32) as its videos are much longer (180 seconds on average). Note that the hyper-parameters of downstream training are the same in all settings in the ablation study.
+
+Downstream Datasets. To evaluate performances of video pre-training models, we conduct text-to-video retrieval experiments on four typical datasets. (a) MSR-VTT (Xu et al., 2016) contains 10K YouTube videos with 200K descriptions. We follow previous works (Yu et al., 2018; Liu et al., 2019) to train models on 9K videos, and report results on the 1K-A test set. (b) DiDeMo (Anne Hendricks et al., 2017) consists of 10K Flickr videos annotated with 40K sentences. We follow (Liu et al., 2019; Zhang et al., 2018) to evaluate paragraph-to-video retrieval and concatenate all descriptions of a video as one query. (c) LSMDC (Rohrbach et al., 2016) consists of 118,081 video clips sourced from 202 movies with one caption corresponding to each clip. Evaluation is conducted on a test set of 1,000 videos from movies disjoint from the train and validation sets. (d) ActivityNet Captions (Krishna et al., 2017a) contains 20K YouTube videos annotated with 100K sentences. We follow the paragraph-to-video retrieval setting (Zhang et al., 2018; Liu et al., 2019) to train models on 10K videos and report results on the val1 set with 4.9K videos.
+
+# 5.2 ABLATION STUDIES
+
+Video Proxy Mechanism. For the vision encoder, we evaluate our proposed Video Proxy (ViP) mechanism with different numbers of proxies and compare it with different model structures (i.e. MeanPool, SeqTransformer, Full Attention) by fine-tuning the same pre-trained model on MSR-VTT retrieval task. MeanPool simply takes the average of frame features as the representation of the whole video. For SeqTransformer, we follow the seqTransf type in CLIP4Clip (Luo et al., 2021) and the residual connection in their implementation. Full Attention setting takes all patch tokens as the input of the vision encoder and attention is conducted across all tokens. All models are initialized with CLIP-ViT-B/32. The results are shown in Table 2. Compared to the MeanPool baseline which completely disregards temporality, SeqTransformer improves the average Recall@1,5,10 by  $0.8\%$ . Full Attention type leads to a significant performance drop and we observe a worse initial status and slower convergence than other settings during the experiment. This is consistent with our analysis in Section 4.2 that directly using CLIP for all patches' attention computation will decrease the advantage of CLIP. In our method, different numbers of video proxy tokens all result in significant performance gain on R@1 (e.g.,  $3.1\%$  by 4 proxies), while only increasing negligible parameters: 3K compared to 86M of ViT backbone. Compared with other settings, our methods in all settings have the most improvement which indicates that our proposed video proxy mechanism can effectively leverage the image-text pre-trained model for video-text post-pretraining.
+
+Omnisource Cross-modal Learning. To verify the effectiveness of the proposed Omnisource Cross-modal Learning (OCL) and compare its variants, we set a post-pretraining and fine-tuning pipeline and adopt the same hyper-parameters for all experiments.  $\mathcal{L}_{V\leftrightarrow S}$  is the baseline contrastive
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">MSR-VTT Retrieval</td><td colspan="4">ActivityNet Captions Retrieval</td></tr><tr><td>R@1↑</td><td>R@5↑</td><td>R@10↑</td><td>Mean↑</td><td>R@1↑</td><td>R@5↑</td><td>R@10↑</td><td>Mean↑</td></tr><tr><td>ClipBERT (Lei et al., 2021)</td><td>22.0</td><td>46.8</td><td>59.9</td><td>6.0</td><td>21.3</td><td>49.0</td><td>63.5</td><td>6.0</td></tr><tr><td>VLM (Xu et al., 2021a)</td><td>28.1</td><td>55.5</td><td>67.4</td><td>4.0</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>MMT (Gabeur et al., 2020)</td><td>26.6</td><td>57.1</td><td>69.6</td><td>4.0</td><td>28.7</td><td>61.4</td><td>-</td><td>3.3</td></tr><tr><td>Support Set (Patrick et al., 2021)</td><td>30.1</td><td>58.5</td><td>69.3</td><td>3.0</td><td>29.2</td><td>61.6</td><td>-</td><td>3.0</td></tr><tr><td>Frozen (Bain et al., 2021)</td><td>31.0</td><td>59.5</td><td>70.5</td><td>3.0</td><td>28.8</td><td>60.9</td><td>-</td><td>3.0</td></tr><tr><td>VideoCLIP (Xu et al., 2021b)</td><td>30.9</td><td>55.4</td><td>66.8</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>HD-VILA (Xue et al., 2022)</td><td>35.6</td><td>65.3</td><td>78.0</td><td>3.0</td><td>28.5</td><td>57.4</td><td>-</td><td>4.0</td></tr><tr><td>Florence (Yuan et al., 2021)</td><td>37.6</td><td>63.8</td><td>72.6</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>All-in-One (Wang et al., 2022a)</td><td>37.9</td><td>68.1</td><td>77.1</td><td>-</td><td>22.4</td><td>53.7</td><td>67.7</td><td>5.0</td></tr><tr><td>BridgeFormer (Ge et al., 2022)</td><td>37.6</td><td>64.8</td><td>75.1</td><td>3.0</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>CLIP-ViT-B/32:</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>CLIP4Clip (Luo et al., 2021)</td><td>44.5</td><td>71.4</td><td>81.6</td><td>2.0</td><td>40.5</td><td>72.4</td><td>-</td><td>2.0</td></tr><tr><td>CenterCLIP (Zhao et al., 2022)</td><td>44.2</td><td>71.6</td><td>82.1</td><td>2.0</td><td>43.9</td><td>74.6</td><td>85.8</td><td>2.0</td></tr><tr><td>XPool (Gorti et al., 2022)</td><td>46.9</td><td>72.8</td><td>82.2</td><td>2.0</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>CLIP2Video (Fang et al., 2021)</td><td>45.6</td><td>72.6</td><td>81.7</td><td>2.0</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>CLIP2Video†(Bogolin et al., 2022)</td><td>47.2</td><td>73.0</td><td>83.0</td><td>2.0</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>CLIP2TV (Gao et al., 2021)</td><td>46.1</td><td>72.5</td><td>82.9</td><td>2.0</td><td>44.1</td><td>75.2</td><td>-</td><td>2.0</td></tr><tr><td>DRL (Wang et al., 2022c)</td><td>47.4</td><td>74.6</td><td>83.8</td><td>2.0</td><td>44.2</td><td>74.5</td><td>86.1</td><td>2.0</td></tr><tr><td>CAMoE* (Cheng et al., 2021)</td><td>47.3</td><td>74.2</td><td>84.5</td><td>2.0</td><td>51.0</td><td>77.7</td><td>-</td><td>-</td></tr><tr><td>Ours</td><td>50.1</td><td>74.8</td><td>84.6</td><td>1.0</td><td>51.1</td><td>78.4</td><td>88.3</td><td>1.0</td></tr><tr><td>Ours*</td><td>55.9</td><td>77.0</td><td>86.8</td><td>1.0</td><td>59.1</td><td>83.9</td><td>91.3</td><td>1.0</td></tr><tr><td>CLIP-ViT-B/16:</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>CenterCLIP (Zhao et al., 2022)</td><td>48.4</td><td>73.8</td><td>82.0</td><td>2.0</td><td>46.2</td><td>77.0</td><td>87.6</td><td>2.0</td></tr><tr><td>CLIP2TV (Gao et al., 2021)</td><td>49.3</td><td>74.7</td><td>83.6</td><td>2.0</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DRL (Wang et al., 2022c)</td><td>50.2</td><td>76.5</td><td>84.7</td><td>1.0</td><td>46.2</td><td>77.3</td><td>88.2</td><td>2.0</td></tr><tr><td>DRL†(Wang et al., 2022c)</td><td>53.3</td><td>80.3</td><td>87.6</td><td>1.0</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Ours</td><td>54.2</td><td>77.2</td><td>84.8</td><td>1.0</td><td>53.4</td><td>81.4</td><td>90.0</td><td>1.0</td></tr><tr><td>Ours*</td><td>57.7</td><td>80.5</td><td>88.2</td><td>1.0</td><td>61.4</td><td>85.7</td><td>92.6</td><td>1.0</td></tr></table>
+
+Table 5: Comparison with SOTA models in MSR-VTT (Xu et al., 2016) and ActivityNet (Krishna et al., 2017a) text-to-video retrieval tasks. * and † respectively denotes that the method uses DSL (Cheng et al., 2021) and QB-Norm (Bogolin et al., 2022) as post-processing operations.
+
+loss on video-subtitle pairs. After introducing auxiliary captions, we study four variants of OCL Loss: (a)  $\mathcal{L}_{V\leftrightarrow S} + \mathcal{L}_{F\leftrightarrow C}$ ; (b)  $\mathcal{L}_{V\leftrightarrow S} + \mathcal{L}_{V\leftrightarrow C}$ ; (c)  $\mathcal{L}_{V\leftrightarrow S} + \mathcal{L}_{V\leftrightarrow C} + \mathcal{L}_{F\leftrightarrow C}$ ; (d)  $\mathcal{L}_{V\leftrightarrow S,C} + \mathcal{L}_{F\leftrightarrow C}$  as explained in Section 4.3. We pre-train models with each loss function for only one epoch due to the costly training, then finetune on two video-text retrieval datasets: MSR-VTT and DiDeMo. We compare the results with CLIP-MeanPool and CLIP using the proposed Video Proxy mechanism without post-pretraining (i.e., CLIP-ViP). The results are listed in Table 3. On MSR-VTT dataset, we find that  $\mathcal{L}_{V\leftrightarrow S}$  brings very little improvement:  $0.4\%$  on average of Recall@1, 5, 10. This is due to the large domain gap between MSR-VTT and post-pretraining data. Combined with auxiliary captions, four variants of OCL loss all bring significant improvements: over  $3\%$  on Recall@1 and over  $2.3\%$  on average of Recall@1, 5, 10. On DiDeMo dataset, based on the improvement brought by  $\mathcal{L}_{V\leftrightarrow S}$ , OCL further improve the results by a large margin:  $8\%$  on average of Recall@1, 5, 10. Finally,  $\mathcal{L}_{V\leftrightarrow S,C} + \mathcal{L}_{F\leftrightarrow C}$  performs best which is applied as our final setting.
+
+Auxiliary Data. In this part, we ablate the contribution of large-scale noisy data and auxiliary data. For uni-source, we use video-subtitle pairs and video-caption data for post-pretraining by vanilla contrastive loss. For data combination, we apply OCL under  $\mathcal{L}_{V\leftrightarrow S,C} + \mathcal{L}_{F\leftrightarrow C}$  setting to post-pretrain on the combined data. From Table 4, Omnisource post-pretraining results are much better than two uni-source results. On MSR-VTT, both uni-source post-pretraining show limited improvement:  $67.4\%$  and  $66.9\%$  compared with  $67.0\%$ . While the Omnisource post-pretraining brings a significant improvement of  $2.6\%$ . On DiDeMo, the benefit of data combination is also obvious, with nearly double the improvements brought by uni-source. These results show that the auxiliary data together with our designed joint learning method can effectively adapt the image-text pre-trained model to video-text post-pretraining.
+
+As the generation of auxiliary captions is based on OFA-Caption (Wang et al., 2022b), a powerful image-text pre-trained model, we also explore only including existing data in post-pretraining. We choose image-text pairs of several widely-adopted datasets: MS-COCO, Visual Genome (VG) (Krishna et al., 2017b), Flickr-30K (Young et al., 2014), SBU (Ordonez et al., 2011), CC3M (Sharma et al., 2018) and CC12M as our auxiliary data (namely ImageCaption). To ablate the contribution of these data, we add experiments of post-pretraining on ImageCaption alone and HD-VILA-100M
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">DiDeMo Retrieval</td><td colspan="4">LSMDC Retrieval</td></tr><tr><td>R@1↑</td><td>R@5↑</td><td>R@10↑</td><td>Mean ↑</td><td>R@1↑</td><td>R@5↑</td><td>R@10↑</td><td>Mean ↑</td></tr><tr><td>MMT (Gabeur et al., 2020)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>12.9</td><td>29.9</td><td>40.1</td><td>19.3</td></tr><tr><td>ClipBERT (Lei et al., 2021)</td><td>20.4</td><td>48.0</td><td>60.8</td><td>6.0</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Frozen (Bain et al., 2021)</td><td>31.0</td><td>59.8</td><td>72.4</td><td>3.0</td><td>15.0</td><td>30.8</td><td>40.3</td><td>20.0</td></tr><tr><td>HD-VILA (Xue et al., 2022)</td><td>28.8</td><td>57.4</td><td>69.1</td><td>4.0</td><td>17.4</td><td>34.1</td><td>44.1</td><td>15.0</td></tr><tr><td>All-in-One (Wang et al., 2022a)</td><td>32.7</td><td>61.4</td><td>73.5</td><td>3.0</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>BridgeFormer (Ge et al., 2022)</td><td>37.0</td><td>62.2</td><td>73.9</td><td>3.0</td><td>17.9</td><td>35.4</td><td>44.5</td><td>15.0</td></tr><tr><td>CLIP-ViT-B/32:</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>CLIP4Clip (Luo et al., 2021)</td><td>43.4</td><td>70.2</td><td>80.6</td><td>2.0</td><td>21.6</td><td>41.8</td><td>49.8</td><td>11.0</td></tr><tr><td>CenterCLIP (Zhao et al., 2022)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>21.7</td><td>39.8</td><td>49.8</td><td>11.0</td></tr><tr><td>XPool (Gorti et al., 2022)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>22.7</td><td>42.6</td><td>51.2</td><td>10.0</td></tr><tr><td>CLIP2TV (Gao et al., 2021)</td><td>45.5</td><td>69.7</td><td>80.6</td><td>2.0</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DRL (Wang et al., 2022c)</td><td>47.9</td><td>73.8</td><td>82.7</td><td>2.0</td><td>24.9</td><td>45.7</td><td>55.3</td><td>7.0</td></tr><tr><td>CAMoE* (Cheng et al., 2021)</td><td>43.8</td><td>71.4</td><td>-</td><td>-</td><td>25.9</td><td>46.1</td><td>53.7</td><td>-</td></tr><tr><td>Ours</td><td>48.6</td><td>77.1</td><td>84.4</td><td>2.0</td><td>25.6</td><td>45.3</td><td>54.4</td><td>8.0</td></tr><tr><td>Ours*</td><td>53.8</td><td>79.6</td><td>86.5</td><td>1.0</td><td>26.0</td><td>46.4</td><td>54.9</td><td>8.0</td></tr><tr><td>CLIP-ViT-B/16:</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>CLIP4Clip by (Zhao et al., 2022)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>24.1</td><td>45.0</td><td>55.1</td><td>8.0</td></tr><tr><td>CenterCLIP (Zhao et al., 2022)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>24.2</td><td>46.2</td><td>55.9</td><td>8.0</td></tr><tr><td>DRL (Wang et al., 2022c)</td><td>49.0</td><td>76.5</td><td>84.5</td><td>2.0</td><td>26.5</td><td>47.6</td><td>56.8</td><td>7.0</td></tr><tr><td>Ours</td><td>50.5</td><td>78.4</td><td>87.1</td><td>1.0</td><td>29.4</td><td>50.6</td><td>59.0</td><td>5.0</td></tr><tr><td>Ours*</td><td>55.3</td><td>82.0</td><td>89.3</td><td>1.0</td><td>30.7</td><td>51.4</td><td>60.6</td><td>5.0</td></tr></table>
+
+Table 6: Comparison with SOTA models in DiDeMo (Anne Hendricks et al., 2017) and LSMDC (Rohrbach et al., 2016) text-to-video retrieval tasks. * denotes using post-processing DSL (Cheng et al., 2021).
+
+combined with ImageCaption. From Table 4, post-pretraining on ImageCaption alone results in performance degradation on MSR-VTT and marginal improvement on DiDeMo. In contrast, ImageCaption yields significant performance gains on both datasets when used as auxiliary data for HD-VILA-100M. This further illustrates the importance of the combination of large-scale noisy data and auxiliary data.
+
+# 5.3 COMPARISON TO STATE-OF-THE-ART MODELS
+
+We compare our model under full setting (in three epochs) with state-of-the-art works on the text-to-video retrieval task. The results of fine-tuning on four datasets (i.e., MSR-VTT, DiDeMo, ActivityNet Captions, LSMDC) are shown in Table 5 and 6, respectively. We clarify the backbone for CLIP-based works. We only add results with DSL (Cheng et al., 2021) to make fair comparison with some methods using post-processing operations e.g., DSL (Cheng et al., 2021) or QB-Norm (Bogolin et al., 2022). Our model achieves the best results on all datasets in both CLIP-ViT-B/32 and CLIP-ViT-B/16. Note that some existing methods are also applicable on top of our models as our modification to the CLIP model is minimal. Note that even without post-processing (e.g., DSL), our results still surpass methods using post-processing operations on most datasets. Besides, adding DSL will greatly improve the performance of our model since our model has good bidirectional vision-language correspondence. The good results on the ActivityNet Captions dataset also indicate that our model can generalize well to long videos. Overall, the improvements on different datasets demonstrate the superiority of the video-language representation learned by our CLIP-ViP model.
+
+# 6 CONCLUSION
+
+In this paper, we study further pre-training (post-pretraining) image-text models like CLIP on large-scale video data. We first conduct a preliminary analysis to reveal the factors hindering video postpretraining. Motivated by findings, we propose CLIP-ViP which includes an Omnisource Cross-modal Learning method and a Video Proxy mechanism. The Video Proxy mechanism can better model videos containing temporal information while reducing conflicts with the pre-trained CLIP model. The Omnisource Cross-modal Learning alleviates the problem caused by the domain gap between video-subtitle and downstream data. Extensive results show that our approach improves the performance of CLIP on video-text retrieval by a large margin and also achieves new state-of-the-art results on four widely-used video-language benchmarks.
+
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+
+# A MORE DOWNSTREAM TASKS
+
+We add various suitable downstream tasks to verify that our model can be generalizable to other tasks. Specifically, we add four zero-shot evaluations on video action recognition and videolanguage multi-choice tasks:
+
+HMDB51 Action Recognition HMDB51 (Kuehne et al., 2011) is a video dataset containing realistic videos from various sources, including movies and web videos. There are 51 action categories (such as "jump", "kiss" and "laugh"). We evaluate our model by zero-shot classification on HMDB51. To adapt to our model, we formulate video classification as video-text alignment, using text prompts "a person [label]". The prediction is made on top-1 similarity between query video and prompted texts. We directly test our model without finetuning to report zero-shot accuracy results.
+
+SomethingSomethingV2 (SSv2) Action Recognition. SomethingSomethingV2 (Goyal et al., 2017) dataset is a large collection of labeled video clips that show humans performing pre-defined basic actions with everyday objects. It requires models to develop fine-grained understanding of basic actions that occur in the physical world. To adapt to our model, we design a multi-choice task on SSv2. The model has to choose the ground truth annotation among all categories. We directly test our model without finetuning to report zero-shot accuracy results.
+
+MSR-VTT Multi-Choice (MC). MSR-VTT MC is a benchmark build on MSR-VTT video dataset. Given a video query and five descriptive sentences, MSR-VTT MC requires the model to choose a single best answer in the candidates. We directly test our model without finetuning to report zero-shot accuracy results.
+
+LSMDC Multi-Choice (MC). LSMDC MC is a benchmark build on movie data. Similar with MSR-VTT MC, it requires the model to choose a single best answer from 5 candidates. We report zero-shot accuracy on 10000 test videos.
+
+From the results in Table 7, we can find that post pretraining CLIP on subtitle alone will lead to a performance drop on most tasks: HMDB51 SSv2 and MSR-VTT MC. This phenomenon is in consistency with our assumption as subtitles usually have unique form. By our final setting, CLIP-ViP
+
+<table><tr><td>Model</td><td>HMDB51</td><td>SSv2</td><td>MSR-VTT MC</td><td>LSMDC MC</td></tr><tr><td>CLIP-MeanPool</td><td>40.3</td><td>25.8</td><td>89.9</td><td>64.3</td></tr><tr><td>CLIP-ViP w/o post-pretraining</td><td>41.1</td><td>26.1</td><td>89.4</td><td>63.5</td></tr><tr><td>CLIP-ViP w/ video-subtitle</td><td>36.6</td><td>28.0</td><td>86.6</td><td>67.0</td></tr><tr><td>CLIP-ViP full</td><td>44.8</td><td>33.0</td><td>90.3</td><td>67.6</td></tr></table>
+
+achieves better results than CLIP on all evaluations, especially on SSv2 which requires more understanding on temporality (Wang et al., 2022d). These results demonstrate that CLIP-ViP outperforms CLIP on various video benchmarks and further show the generalization of our model CLIP-ViP.
+
+# B EFFICIENCY OF POST-PRETRAINING
+
+In light of the good image representation of CLIP, our post-pretraining significantly reduces the training cost, Compared with HD-VILA (Xue et al., 2022) which trains video-language representation from scratch, CLIP-ViP outperforms HD-VILA by a large margin with about 1/30 training time. To make more fair comparison with training from scratch. We conduct an experiment of training from scratch with the same setting and training time (one epoch). Then we finetune on MSR-VTT Retrieval to evaluate the pre-trained model.
+
+Table 7: Evaluation on more downstream tasks. We report zero-shot accuracy  $(\%)$  on HMDB51, SSv2, MSRVTT-MC, and LSMDC MC. All models are base size with patch size of 32. Details of post-pretraining are the same as in Section 5.2.  
+
+<table><tr><td>Model</td><td>MSR-VTT R@1</td><td>MSR-VTT R@5</td><td>MSR-VTT R@10</td></tr><tr><td>CLIP-ViP (from scratch)</td><td>28.5</td><td>55.2</td><td>68.6</td></tr><tr><td>CLIP-ViP</td><td>49.6</td><td>74.5</td><td>84.8</td></tr></table>
+
+Table 8: Comparison with training CLIP-ViP from scratch. We report finetuning results on MSR-VTT Retrieval. All models are base size with patch size of 32. Details of post-pretraining are the same as in Section 5.2.
+
+From the results in Table 8, we can see that there is a huge performance gap. Without basing on a CLIP, CLIP-VIP can not leverage the rich knowledge entailed in CLIP, thus leading to a much lower training efficiency.

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2023/CLIPSep_ Learning Text-queried Sound Separation with Noisy Unlabeled Videos/5bdac351-fe7e-44aa-a3e2-6a95a971f2f7_origin.pdf

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2023/CLIPSep_ Learning Text-queried Sound Separation with Noisy Unlabeled Videos/full.md

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+# CLIPSEP: LEARNING TEXT-QUERYIED SOUND SEPARATION WITH NOISY UNLABELLED VIDEOS
+
+Hao-Wen Dong $^{1,2*}$  Naoya Takahashi $^{1\dagger}$  Yuki Mitsufuji $^{1}$   
+Julian McAuley $^{2}$  Taylor Berg-Kirkpatrick $^{2}$
+
+$^{1}$ Sony Group Corporation  $^{2}$ University of California San Diego hwdong@ucsd.edu, {Naoya.Takahashi,Yuhki.Mitsufuji}@sony.com, {jmcauley,tberg}@ucsd.edu
+
+# ABSTRACT
+
+Recent years have seen progress beyond domain-specific sound separation for speech or music towards universal sound separation for arbitrary sounds. Prior work on universal sound separation has investigated separating a target sound out of an audio mixture given a text query. Such text-queried sound separation systems provide a natural and scalable interface for specifying arbitrary target sounds. However, supervised text-queried sound separation systems require costly labeled audio-text pairs for training. Moreover, the audio provided in existing datasets is often recorded in a controlled environment, causing a considerable generalization gap to noisy audio in the wild. In this work, we aim to approach text-queried universal sound separation by using only unlabeled data. We propose to leverage the visual modality as a bridge to learn the desired audio-textual correspondence. The proposed CLIPSep model first encodes the input query into a query vector using the contrastive language-image pretraining (CLIP) model, and the query vector is then used to condition an audio separation model to separate out the target sound. While the model is trained on image-audio pairs extracted from unlabeled videos, at test time we can instead query the model with text inputs in a zero-shot setting, thanks to the joint language-image embedding learned by the CLIP model. Further, videos in the wild often contain off-screen sounds and background noise that may hinder the model from learning the desired audio-textual correspondence. To address this problem, we further propose an approach called noise invariant training for training a query-based sound separation model on noisy data. Experimental results show that the proposed models successfully learn text-queried universal sound separation using only noisy unlabeled videos, even achieving competitive performance against a supervised model in some settings.
+
+# 1 INTRODUCTION
+
+Humans can focus on to a specific sound in the environment and describe it using language. Such abilities are learned using multiple modalities—auditory for selective listening, vision for learning the concepts of sounding objects, and language for describing the objects or scenes for communication. In machine listening, selective listening is often cast as the problem of sound separation, which aims to separate sound sources from an audio mixture (Cherry, 1953; Bach & Jordan, 2005). While text queries offer a natural interface for humans to specify the target sound to separate from a mixture (Liu et al., 2022; Kilgour et al., 2022), training a text-queried sound separation model in a supervised manner requires labeled audio-text paired data of single-source recordings of a vast number of sound types, which can be costly to acquire. Moreover, such isolated sounds are often recorded in controlled environments and have a considerable domain gap to recordings in the wild, which usually contain arbitrary noise and reverberations. In contrast, humans often leverage the visual modality to assist learning the sounds of various objects (Baillargeon, 2002). For instance, by observing a dog barking, a human can associate the sound with the dog, and can separately learn that the animal is called a "dog." Further, such learning is possible even if the sound is observed in a noisy environment, e.g.,
+
+when a car is passing by or someone is talking nearby, where humans can still associate the barking sound solely with the dog. Prior work in psychophysics also suggests the intertwined cognition of vision and hearing (Sekuler et al., 1997; Shimojo & Shams, 2001; Rahne et al., 2007).
+
+Motivated by this observation, we aim to tackle text-queried sound separation using only unlabeled videos in the wild. We propose a text-queried sound separation model called CLIPSep that leverages abundant unlabeled video data resources by utilizing the contrastive image-language pretraining (CLIP) (Radford et al., 2021) model to bridge the audio and text modalities. As illustrated in Figure 1, during training, the image feature extracted from a video frame by the CLIP-image encoder is used to condition a sound separation model, and the model is trained to separate the sound that corresponds to the image query in a self-supervised setting. Thanks to the properties of the CLIP model, which projects corresponding text and images to close embeddings, at test time we instead use the text feature obtained by the CLIP-text encoder from a text query in a zero-shot setting.
+
+However, such zero-shot modality transfer can be challenging when we use videos in the wild for training as they often contain off-screen sounds and voice overs that can lead to undesired audio-visual associations. To address this problem, we propose the noise invariant training (NIT), where query-based separation heads and permutation invariant separation heads jointly estimate the noisy target sounds. We validate in our experiments that the proposed noise invariant training reduces
+
+![](images/ed16478f7f0fb9ebf7dc78ca3013e090b54588052c959230d573ec37d4b53c43.jpg)  
+Figure 1: An illustration of modality transfer.
+
+the zero-shot modality transfer gap when the model is trained on a noisy dataset, sometimes achieving competitive results against a fully supervised text-queried sound separation system.
+
+Our contributions can be summarized as follows: 1) We propose the first text-queried universal sound separation model that can be trained on unlabeled videos. 2) We propose a new approach called noise invariant training for training a query-based sound separation model on noisy data in the wild. Audio samples can be found on an our demo website. For reproducibility, all source code, hyperparameters and pretrained models are available at: https://github.com/sony/CLIPSeq.
+
+# 2 RELATED WORK
+
+Universal sound separation Much prior work on sound separation focuses on separating sounds for a specific domain such as speech (Wang & Chen, 2018) or music (Takahashi & Mitsufuji, 2021; Mitsufuji et al., 2021). Recent advances in domain specific sound separation lead several attempts to generalize to arbitrary sound classes. Kavalerov et al. (2019) reported successful results on separating arbitrary sounds with a fixed number of sources by adopting the permutation invariant training (PIT) (Yu et al., 2017), which was originally proposed for speech separation. While this approach does not require labeled data for training, a post-selection process is required as we cannot not tell what sounds are included in each separated result. Follow-up work (Ochiai et al., 2020; Kong et al., 2020) addressed this issue by conditioning the separation model with a class label to specify the target sound in a supervised setting. However, these approaches still require labeled data for training, and the interface for selecting the target class becomes cumbersome when we need a large number of classes to handle open-domain data. Wisdom et al. (2020) later proposed an unsupervised method called mixture invariant training (MixIT) for learning sound separation on noisy data. MixIT is designed to separate all sources at a time and also requires a post-selection process such as using a pre-trained sound classifier (Scott et al., 2021), which requires labeled data for training, to identify the target sounds. We summarize and compare related work in Table 1.
+
+Query-based sound separation Visual information has been used for selecting the target sound in speech (Ephrat et al., 2019; Afouras et al., 2020), music (Zhao et al., 2018; 2019; Tian et al., 2021) and universal sounds (Owens & Efros, 2018; Gao et al., 2018; Rouditchenko et al., 2019). While many image-queried sound separation approaches require clean video data that contains isolated sources, Tzinis et al. (2021) introduced an unsupervised method called AudioScope for separating on-screen sounds using noisy videos based on the MixIT model. While image queries can serve as a
+
+<table><tr><td>Method</td><td>Query type</td><td>Unlabeled data</td><td>Noisy data</td></tr><tr><td>USS (Kavalerov et al., 2019)</td><td>×</td><td>✓</td><td></td></tr><tr><td>MixIT (Wisdom et al., 2020)</td><td>×</td><td>✓</td><td>✓</td></tr><tr><td>Universal Sound Selector (Ochiai et al., 2020)</td><td>Label</td><td></td><td></td></tr><tr><td>USS-Label (Kong et al., 2020)</td><td>Label</td><td></td><td>(✓)</td></tr><tr><td>PixelPlayer (Zhao et al., 2018)</td><td>Image</td><td>✓</td><td>(✓)</td></tr><tr><td>AudioScope (Tzinis et al., 2021)</td><td>Image</td><td>✓</td><td>✓</td></tr><tr><td>SoundFilter (Gfeller et al., 2021)</td><td>Audio</td><td>✓</td><td></td></tr><tr><td>Zero-shot audio separation (Chen et al., 2022)</td><td>Audio</td><td></td><td>(✓)</td></tr><tr><td>Text-Quried (Liu et al., 2022)</td><td>Text</td><td></td><td></td></tr><tr><td>Text/Audio-Quried (Kilgour et al., 2022)</td><td>Text / Audio</td><td></td><td></td></tr><tr><td>CLIPSep (ours)</td><td>Text / Image</td><td>✓</td><td></td></tr><tr><td>CLIPSep-NIT (ours)</td><td>Text / Image</td><td>✓</td><td>✓</td></tr></table>
+
+Table 1: Comparisons of related work in sound separation.  $(\checkmark)$  indicates that the problem of noisy training data is partially addressed. CLIPSeq-NIT denotes the proposed CLIPSeq model trained with the noise invariant training. To the best of our knowledge, no previous work has attempted the problem of label-free text-queried sound separation.
+
+natural interface for specifying the target sound in certain use cases, images of target sounds become unavailable in low-light conditions and for sounds from out-of-screen objects.
+
+Another line of research uses the audio modality to query acoustically similar sounds. Chen et al. (2022) showed that such approach can generalize to unseen sounds. Later, Gfeller et al. (2021) cropped two disjoint segments from single recording and used them as a query-target pair to train a sound separation model, assuming both segments contain the same sound source. However, in many cases, it is impractical to prepare a reference audio sample for the desired sound as the query.
+
+Most recently, text-queried sound separation has been studied as it provides a natural and scalable interface for specifying arbitrary target sounds as compared to systems that use a fixed set of class labels. Liu et al. (2022) employed a pretrained language model to encode the text query, and condition the model to separate the corresponding sounds. Kilgour et al. (2022) proposed a model that accepts audio or text queries in a hybrid manner. These approaches, however, require labeled text-audio paired data for training. Different from prior work, our goal is to learn text-queried sound separation for arbitrary sound without labeled data, specifically using unlabeled noisy videos in the wild.
+
+Contrastive language-image-audio pretraining The CLIP model (Radford et al., 2021) has been used as a pretraining of joint embedding spaces among text, image and audio modalities for downstream tasks such as audio classification (Wu et al., 2022; Guzhov et al., 2022) and sound guided image manipulation (Lee et al., 2022). Pretraining is done either in a supervised manner using labels (Guzhov et al., 2022; Lee et al., 2022) or in a self-supervised manner by training an additional audio encoder to map input audio to the pretrained CLIP embedding space (Wu et al., 2022). In contrast, we explore the zero-shot modality transfer capability of the CLIP model by freezing the pre-trained CLIP model and directly optimizing the rest of the model for the target sound separation task.
+
+# 3 METHOD
+
+# 3.1 CLIPSEP—LEARNING TEXT-QUIERIED SOUND SEPARATION WITHOUT LASELED DATA
+
+In this section, we propose the CLIPSep model for text-queried sound separation without using labeled data. We base the CLIPSep model on Sound-of-Pixels (SOP) (Zhao et al., 2018) and replace the video analysis network of the SOP model. As illustrated in Figure 2, during training, the model takes as inputs an audio mixture  $\pmb{x} = \sum_{i=1}^{n} s_i$ , where  $s_1, \ldots, s_n$  are the  $n$  audio tracks, along with their corresponding images  $y_1, \ldots, y_n$  extracted from the videos. We first transform the audio mixture  $\pmb{x}$  into a magnitude spectrogram  $X$  and pass the spectrogram through an audio U-Net (Ronneberger et al., 2015; Jansson et al., 2017) to produce  $k$  ( $\geq n$ ) intermediate masks  $\tilde{M}_1, \ldots, \tilde{M}_k$ . On the other stream, each image is encoded by the pretrained CLIP model (Radford et al., 2021) into an embedding  $e_i \in \mathbb{R}^{512}$ . The CLIP embedding  $e_i$  will further be projected to a query vector
+
+![](images/97716ae9774adb0d3c6e52204b3377652b8913291b5dc35f297ad74ecd304472.jpg)  
+Figure 2: An illustration of the proposed CLIPSeq model for  $n = 2$ . During training, we mix audio from two videos and train the model to separate each audio source given the corresponding video frame as the query. At test time, we instead use a text query in the form of "a photo of [user input query]" to query the sound separation model. Thanks to the properties of the pretrained CLIP model, the query vectors we obtain for the image and text queries are expected to be close.
+
+$\pmb{q}_i \in \mathbb{R}^k$  by a projection layer, which is expected to extract only audio-relevant information from  $\pmb{e}_i$ .<sup>2</sup> Finally, the query vector  $\pmb{q}_i$  will be used to mix the intermediate masks into the final predicted masks  $\hat{M}_i = \sum_{j=1}^{k} \sigma(w_{ij} q_{ij} \tilde{M}_j + b_i)$ , where  $\pmb{w}_i \in \mathbb{R}^k$  is a learnable scale vector,  $b_i \in \mathbb{R}$  a learnable bias, and  $\sigma(\cdot)$  the sigmoid function. Now, suppose  $M_i$  is the ground truth mask for source  $\pmb{s}_i$ . The training objective of the model is the sum of the weighted binary cross entropy losses for each source:
+
+$$
+\mathcal {L} _ {C L I P S e p} = \sum_ {i = 1} ^ {n} W B C E (M _ {i}, \hat {M} _ {i}) = \sum_ {i = 1} ^ {n} X \odot \left(- M _ {i} \log \hat {M} _ {i} - (1 - M _ {i}) \log \left(1 - \hat {M} _ {i}\right)\right). \tag {1}
+$$
+
+At test time, thanks to the joint image-text embedding offered by the CLIP model, we feed a text query instead of an image to the query model to obtain the query vector and separate the target sounds accordingly (see Appendix A for an illustration). As suggested by Radford et al. (2021), we prefix the text query into the form of "a photo of [user input query]" to reduce the generalization gap. $^3$
+
+# 3.2 NOISE INVARIANT TRAINING—HANDLING NOISY DATA IN THE WILD
+
+While the CLIPSeq model can separate sounds given image or text queries, it assumes that the sources are clean and contain few query-irrelevant sounds. However, this assumption does not hold for videos in the wild as many of them contain out-of-screen sounds and various background noises. Inspired by the mixture invariant training (MixIT) proposed by Wisdom et al. (2020), we further propose the noise invariant training (NIT) to tackle the challenge of training with noisy data. As illustrated in Figure 3, we introduce  $n$  additional permutation invariant heads called noise heads to the CLIPSeq model, where the masks predicted by these heads are interchangeable during loss computation. Specifically, we introduce  $n$  additional projection layers, and each of them takes as input the sum of all query vectors produced by the query heads (i.e.,  $\sum_{i=1}^{n} q_i$ ) and produce a vector that is later used to mix the intermediate masks into the predicted noise mask. In principle, the query masks produced by the query vectors are expected to extract query-relevant sounds due to their stronger correlations to their corresponding queries, while the interchangeable noise masks should 'soak up' other sounds.
+
+![](images/def34b0ec30f86e05d6dd7bee0612035b1f3bf8dc2293582c4f952b906092787.jpg)  
+Figure 3: An illustration of the proposed CLIPSeq-NIT model for  $n = 2$ . Similar to CLIPSeq, we train the model to separate each audio source given the corresponding query image during training and switch to using a text query at test time. The two predicted noise masks are interchangeable for loss computation during training, and they are discarded at test time (grayed out paths).
+
+Mathematically, let  $M_1^Q, \ldots, M_n^Q$  be the predicted query masks and  $M_1^N, \ldots, M_n^N$  be the predicted noise masks. Then, the noise invariant loss is defined as:
+
+$$
+\mathcal {L} _ {N I T} = \min  _ {(j _ {1}, \dots , j _ {n}) \in \Sigma_ {n}} \sum_ {i = 1} ^ {n} W B C E \left(M _ {i}, \min  \left(1, \hat {M} _ {i} ^ {Q} + \hat {M} _ {j _ {i}} ^ {N}\right)\right), \tag {2}
+$$
+
+where  $\Sigma_{n}$  denotes the set of all permutations of  $\{1,\dots ,n\}$ .<sup>4</sup> Take  $n = 2$  for example.<sup>5</sup> We consider the two possible ways for combining the query heads and the noise heads:
+
+(Arrangement 1)  $\hat{M}_1 = \min \left(1, \hat{M}_1^Q + \hat{M}_1^N\right), \quad \hat{M}_2 = \min \left(1, \hat{M}_2^Q + \hat{M}_2^N\right),$  (3)
+
+(Arrangement 2)  $\hat{M}_1' = \min \left(1, \hat{M}_1^Q + \hat{M}_2^N\right), \quad \hat{M}_2' = \min \left(1, \hat{M}_2^Q + \hat{M}_1^N\right)$ . (4)
+
+Then, the noise invariant loss is defined as the smallest loss achievable:
+
+$$
+\mathcal {L} _ {N I T} ^ {(2)} = \min  \left(W B C E \left(M _ {1}, \hat {M} _ {1}\right) + W B C E \left(M _ {2}, \hat {M} _ {2}\right), W B C E \left(M _ {1}, \hat {M} _ {1} ^ {\prime}\right) + W B C E \left(M _ {2}, \hat {M} _ {2} ^ {\prime}\right)\right). \tag {5}
+$$
+
+Once the model is trained, we discard the noise heads and use only the query heads for inference (see Appendix A for an illustration). Unlike the MixIT model (Wisdom et al., 2020), our proposed noise invariant training still allows us to specify the target sound by an input query, and it does not require any post-selection process as we only use the query heads during inference.
+
+In practice, we find that the model tends to assign part of the target sounds to the noise heads as these heads can freely enjoy the optimal permutation to minimize the loss. Hence, we further introduce a regularization term to penalize producing high activations on the noise masks:
+
+$$
+\mathcal {L} _ {R E G} = \max  \left(0, \sum_ {i = 1} ^ {n} \operatorname {m e a n} \left(\hat {M} _ {i} ^ {N}\right) - \gamma\right), \tag {6}
+$$
+
+where  $\gamma \in [0,n]$  is a hyperparameter that we will refer to as the noise regularization level. The proposed regularization has no effect when the sum of the means of all the noise masks is lower than a predefined threshold  $\gamma$ , while having a linearly growing penalty when the sum is higher than  $\gamma$ . Finally, the training objective of the CLIPSeq-NIT model is a weighted sum of the noise invariant loss and regularization term:  $\mathcal{L}_{CLIPsep-NIT} = \mathcal{L}_{NIT} + \lambda \mathcal{L}_{REG}$ , where  $\lambda \in \mathbb{R}$  is a weight hyperparameter. We set  $\lambda = 0.1$  for all experiments, which we find work well across different settings.
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Unlabeled data</td><td rowspan="2">Post-proc. free</td><td colspan="2">Query type</td><td colspan="2">SDR [dB]</td></tr><tr><td>Training</td><td>Test</td><td>Mean</td><td>Median</td></tr><tr><td>Mixture</td><td>-</td><td>-</td><td>-</td><td>-</td><td>0.00 ± 0.89</td><td>0.00</td></tr><tr><td colspan="7">Text-queried models</td></tr><tr><td>CLIPSeq</td><td>✓</td><td>✓</td><td>Image</td><td>Text</td><td>5.49 ± 0.72</td><td>4.97</td></tr><tr><td>CLIPSeq-Text</td><td></td><td>✓</td><td>Text</td><td>Text</td><td>7.91 ± 0.81</td><td>7.46</td></tr><tr><td>CLIPSeq-Hybrid</td><td></td><td>✓</td><td>Text + Image</td><td>Text</td><td>8.36 ± 0.83</td><td>8.72</td></tr><tr><td colspan="7">Image-queried models</td></tr><tr><td>SOP (Zhao et al., 2018)</td><td>✓</td><td>✓</td><td>Image</td><td>Image</td><td>6.59 ± 0.85</td><td>6.22</td></tr><tr><td>CLIPSeq</td><td>✓</td><td>✓</td><td>Image</td><td>Image</td><td>7.03 ± 0.70</td><td>5.85</td></tr><tr><td>CLIPSeq-Text</td><td></td><td>✓</td><td>Text</td><td>Image</td><td>6.25 ± 0.72</td><td>6.19</td></tr><tr><td>CLIPSeq-Hybrid</td><td></td><td>✓</td><td>Text + Image</td><td>Image</td><td>8.06 ± 0.79</td><td>8.01</td></tr><tr><td colspan="7">Nonqueried models</td></tr><tr><td>LabelSep</td><td></td><td>✓</td><td>Label</td><td>Label</td><td>8.18 ± 0.80</td><td>7.82</td></tr><tr><td>PIT (Yu et al., 2017)</td><td>✓</td><td></td><td>×</td><td>×</td><td>8.68 ± 0.76</td><td>7.67</td></tr></table>
+
+Table 2: Results on the MUSIC dataset. Standard errors are reported in the mean SDR column. Bold values indicate the largest SDR achieved per group.
+
+# 4 EXPERIMENTS
+
+We base our implementations on the code provided by Zhao et al. (2018) (https://github.com/hangzhaomit/Sound-of-Pixels). Implementation details can be found in Appendix C.
+
+# 4.1 EXPERIMENTS ON CLEAN DATA
+
+We first evaluate the proposed CLIPSep model without the noise invariant training on musical instrument sound separation task using the MUSIC dataset, as done in (Zhao et al., 2018). This experiment is designed to focus on evaluating the quality of the learned query vectors and the zero-shot modality transferability of the CLIPSep model on a small, clean dataset rather than showing its ability to separate arbitrary sounds. The MUSIC dataset is a collection of 536 video recordings of people playing a musical instrument out of 11 instrument classes. Since no existing work has trained a text-queried sound separation model using only unlabeled data to our knowledge, we compare the proposed CLIPSep model with two baselines that serve as upper bounds—the PIT model (Yu et al., 2017, see Appendix D for an illustration) and a version of the CLIPSep model where the query model is replaced by learnable embeddings for the labels, which we will refer to as the LabelSep model. In addition, we also include the SOP model (Zhao et al., 2018) to investigate the quality of the query vectors as the CLIPSep and SOP models share the same network architecture except the query model.
+
+We report the results in Table 2. Our proposed CLIPSep model achieves a mean signal-to-distortion ratio (SDR) (Vincent et al., 2006) of  $5.49\mathrm{dB}$  and a median SDR of  $4.97\mathrm{dB}$  using text queries in a zero-shot modality transfer setting. When using image queries, the performance of the CLIPSep model is comparable to that of the SOP model. This indicates that the CLIP embeddings are as informative as those produced by the SOP model. The performance difference between the CLIPSep model using text and image queries at test time indicates the zero-shot modality transfer gap. We observe  $1.54\mathrm{dB}$  and  $0.88\mathrm{dB}$  differences on the mean and median SDRs, respectively. Moreover,
+
+![](images/c8902041efae76b0e9ccd8d803736f97a530702b6eb75e062c30a27ee694f9b7.jpg)  
+Figure 4: Mean SDR and standard errors of the models trained and tested on different modalities.
+
+we also report in Table 2 and Figure 4 the performance of the CLIPSeq models trained on different modalities to investigate their modality transferability in different settings. We notice that when we train the CLIPSeq model using text queries, dubbed as CLIPSeq-Text, the mean SDR using text queries increases to  $7.91\mathrm{dB}$ . However, when we test this model using image queries, we observe a  $1.66\mathrm{dB}$  difference on the mean SDR as compared to that using text queries, which is close to
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Unlabeled data</td><td rowspan="2">Post-proc. free</td><td colspan="2">MUSIC+</td><td colspan="2">VGGSound-Clean+</td></tr><tr><td>Mean SDR</td><td>Median SDR</td><td>Mean SDR</td><td>Median SDR</td></tr><tr><td>Mixture</td><td>-</td><td>-</td><td>4.49 ± 1.41</td><td>2.04</td><td>-0.77 ± 1.31</td><td>-0.84</td></tr><tr><td colspan="7">Text-queried models</td></tr><tr><td>CLIPSeq</td><td>✓</td><td>✓</td><td>9.71 ± 1.21</td><td>8.73</td><td>2.76 ± 1.00</td><td>3.95</td></tr><tr><td>CLIPSeq-NIT</td><td>✓</td><td>✓</td><td>10.27 ± 1.04</td><td>10.02</td><td>3.05 ± 0.73</td><td>3.26</td></tr><tr><td>BERTSep</td><td></td><td>✓</td><td>4.67 ± 0.44</td><td>4.41</td><td>5.09 ± 0.80</td><td>5.49</td></tr><tr><td>CLIPSeq-Text</td><td></td><td>✓</td><td>10.73 ± 0.99</td><td>9.93</td><td>5.49 ± 0.82</td><td>5.06</td></tr><tr><td colspan="7">Image-queried models</td></tr><tr><td>SOP (Zhao et al., 2018)</td><td>✓</td><td>✓</td><td>11.44 ± 1.18</td><td>11.18</td><td>2.99 ± 0.84</td><td>3.89</td></tr><tr><td>CLIPSeq</td><td>✓</td><td>✓</td><td>12.20 ± 1.17</td><td>12.42</td><td>5.46 ± 0.79</td><td>5.35</td></tr><tr><td>CLIPSeq-NIT</td><td>✓</td><td>✓</td><td>11.28 ± 1.08</td><td>10.83</td><td>4.84 ± 0.66</td><td>3.57</td></tr><tr><td>CLIPSeq-Text</td><td></td><td>✓</td><td>9.89 ± 1.04</td><td>8.09</td><td>2.45 ± 0.70</td><td>1.74</td></tr><tr><td colspan="7">Nonqueried models</td></tr><tr><td>PIT (Yu et al., 2017)</td><td>✓</td><td></td><td>12.24 ± 1.20</td><td>12.53</td><td>5.73 ± 0.79</td><td>4.97</td></tr><tr><td>LabelSep</td><td></td><td>✓</td><td>-</td><td>-</td><td>5.55 ± 0.81</td><td>5.29</td></tr></table>
+
+Table 3: Results of the MUSIC+ and VGGSound-Clean+ evaluations (see Section 4.2). Standard errors are reported in the mean SDR [dB] columns. Bold values indicate the largest SDR achieved per group. We use  $\gamma = 0.25$  for CLIPSeq-NIT. Note that the LabelSep model does not work on the MUSIC dataset due to the different label taxonomies of the MUSIC and VGGSound datasets.
+
+the mean SDR difference we observe for the model trained with image queries. Finally, we train a CLIPSeq model using both text and image queries in alternation, dubbed as CLIPSeq-Hybrid. We see that it leads to the best test performance for both text and image modalities, and there is only a mean SDR difference of  $0.30\mathrm{dB}$  between using text and image queries. As a reference, the LabelSep model trained with labeled data performs worse than the CLIPSeq-Hybrid model using text queries. Further, the PIT model achieves a mean SDR of  $8.68\mathrm{dB}$  and a median SDR of  $7.67\mathrm{dB}$ , but it requires post-processing to figure out the correct assignments.
+
+# 4.2 EXPERIMENTS ON NOISY DATA
+
+Next, we evaluate the proposed method on a large-scale dataset aiming at universal sound separation. We use the VGGSound dataset (Chen et al., 2020), a large-scale audio-visual dataset containing more than 190,000 10-second videos in the wild out of more than 300 classes. We find that the audio in the VGGSound dataset is often noisy and contains off-screen sounds and background noise. Although we train the models on such noisy data, it is not suitable to use the noisy data as targets for evaluation because it fails to provide reliable results. For example, if the target sound labeled as "dog barking" also contains human speech, separating only the dog barking sound provides a lower SDR value than separating the mixture of dog barking sound and human speech even though the text query is "dog barking". (Note that we use the labels only for evaluation but not for training.) To avoid this issue, we consider the following two evaluation settings:
+
+- MUSIC+: Samples in the MUSIC dataset are used as clean targets and mixed with a sample in the VGGSound dataset as an interference. The separation quality is evaluated on the clean target from the MUSIC dataset. As we do not use the MUSIC dataset for training, this can be considered as zero-shot transfer to a new data domain containing unseen sounds (Radford et al., 2019; Brown et al., 2020). To avoid the unexpected overlap of the target sound types in the MUSIC and VGGSound datasets caused by the label mismatch, we exclude all the musical instrument playing videos from the VGGSound dataset in this setting.  
+- VGGSound-Clean+: We manually collect 100 clean samples that contain distinct target sounds from the VGGSound test set, which we will refer to as VGGSound-Clean. We mix an audio sample in VGGSound-Clean with another in the test set of VGGSound. Similarly, we consider the VGGSound audio as an interference sound added to the relatively cleaner VGGSound-Clean audio and evaluate the separation quality on the VGGSound-Clean stem.
+
+![](images/99f6f812a44102ac641f388f6620699193328aa74cfc22658e4057806a929992.jpg)  
+(a) Input mixture
+
+![](images/c118a66b42e2dfdded800c15962364f63d2f54fac7ecdd051f78dd328e0bfe40.jpg)  
+(b) Ground truth
+
+![](images/9d9ad3fcfb04a0b213a4002fe25e615157dbcdce9379a6fbb57db1ebbc0177e4.jpg)  
+(c) Interference  
+Figure 5: Example results of the proposed CLIPSeq-NIT model with  $\gamma = 0.25$  on the MUSIC $^+$  dataset. We mix the an audio sample ("violin" in this example) in the MUSIC dataset with an interference audio sample ("people sobbing" in this example) in the VGGSound dataset to create an artificial mixture. (b) and (c) show the reconstructed signals using the ground truth ideal binary masks. The spectrograms are shown in the log frequency scale. We observe that the proposed model successfully separates the desired sounds (i.e., (d)) from query-irrelevant noise (i.e., (e) and (f)).
+
+![](images/a67d4508400d1b7265b594a16d5fdc3944814fe333b9898dd10a2a9eea71a41c.jpg)  
+(d) Prediction
+
+![](images/220a23892ea406bbfaa69445b586652c2226be316b5f60906082f358f1aeaba2.jpg)  
+(e) Noise head 1
+
+![](images/d43a9a5675b46c978f338b6572d1ebe4a647e4238c54658b7c394930a2240715.jpg)  
+(f) Noise head 2
+
+Table 3 shows the evaluation results. First, CLIPSep successfully learns text-queried sound separation even with noisy unlabeled data, achieving 5.22 dB and 3.53 dB SDR improvements over the mixture on  $\mathrm{MUSIC^{+}}$  and VGGSound-Clean $^{+}$ , respectively. By comparing CLIPSep and CLIPSep-NIT, we observe that NIT improves the mean SDRs in both settings. Moreover, on  $\mathrm{MUSIC^{+}}$ , CLIPSep-NIT's performance matches that of CLIPSep-Text, which utilizes labels for training, achieving only a 0.46 dB lower mean SDR and even a 0.05 dB higher median SDR. This result suggests that the proposed self-supervised text-queried sound separation method can learn separation capability competitive with the fully supervised model in some target sounds. In contrast, there is still a gap between them on VGGSound-Clean $^{+}$ , possibly because the videos of non-music-instrument objects are more noisy in both audio and visual domains, thus resulting in a more challenging zero-shot modality transfer. This hypothesis is also supported by the higher zero-shot modality transfer gap (mean SDR difference of image- and text-queried mode) of 1.79 dB on VGGSound-Clean $^{+}$  than that of 1.01 dB on  $\mathrm{MUSIC^{+}}$  for CLIPSep-NIT. In addition, we consider another baseline model that replaces the CLIP model in CLIPSep with a BERT encoder (Devlin et al., 2019), which we call BERTSep. Interestingly, although BERTSep performs similarly to CLIPSep-Text on VGGSound-Clean $^{+}$ , the performance of BERTSep is significantly lower than that of CLIPSep-Text on  $\mathrm{MUSIC^{+}}$ , indicating that BERTSep fails to generalize to unseen text queries. We hypothesize that the CLIP text embedding captures the timbral similarity of musical instruments better than the BERT embedding do, because the CLIP model is aware of the visual similarity between musical instruments during training. Moreover, it is interesting to see that CLIPSep outperforms CLIPSep-NIT when an image query is used at test time (domain-matched condition), possibly because images contain richer context information such as objects nearby and backgrounds than labels, and the models can use such information to better separate the target sound. While CLIPSep has to fully utilize such information, CLIPSep-NIT can use the noise heads to model sounds that are less relevant to the image query. Since we remove the noise heads from CLIPSep-NIT during the evaluation, it can rely less on such information from the image, thus improving the zero-shot modality transferability. Figure 5 shows an example of the separation results on  $\mathrm{MUSIC^{+}}$  (see Figures 12 to 15 for more examples). We observe that the two noise heads contain mostly background noise. Audio samples can be found on our demo website. $^{1}$
+
+# 4.3 EXAMINING THE EFFECTS OF THE NOISE REGULARIZATION LEVEL  $\gamma$
+
+In this experiment, we examine the effects of the noise regularization level  $\gamma$  in Equation (6) by changing the value from 0 to 1. As we can see from Figure 6 (a) and (b), CLIPSep-NIT with  $\gamma = 0.25$  achieves the highest SDR on both evaluation settings. This suggests that the optimal  $\gamma$  value is not sensitive to the evaluation dataset. Further, we also report in Figure 6 (c) the total mean noise head activation,  $\sum_{i=1}^{n} \mathrm{mean}(\hat{M}_i^N)$ , on the validation set. As  $\hat{M}_i^N$  is the mask estimate for the noise, the total mean noise head activation value indicates to what extent signals are assigned to the noise head. We observe that the proposed regularizer successfully keeps the total mean noise head activation close to the desired level,  $\gamma$ , for  $\gamma \leq 0.5$ . Interestingly, the total mean noise head activation is still around 0.5 when  $\gamma = 1.0$ , suggesting that the model inherently tries to use both the query-heads and the noise heads to predict the noisy target sounds. Moreover, while we discard the noise heads during evaluation in our experiments, keeping the noise heads can lead to a higher SDR as shown in
+
+![](images/952a2207c94d4e4edfd55039aebe40c1b11aaa5fbdb752c13c775857dbd8d555.jpg)  
+Figure 6: Effects of the noise regularization level  $\gamma$  for the proposed CLIPSeq-NIT model—mean SDR for the (a) MUSIC $^{+}$  and (b) VGGSound-Clean $^{+}$  evaluations, and (c) the total mean noise head activation,  $\sum_{i=1}^{n} \text{mean}(\hat{M}_i^N)$ , on the validation set. The shaded areas show standard errors.
+
+![](images/8b355bc2bbdfc8d8400d704f0b1a3acf389dadda9730e58e45bb77ec5073cd4b.jpg)
+
+![](images/8a75135f83299e94e95d0579702f72cd5cb36cf5cb49e3175d6cd43272ee3fb6.jpg)
+
+Figure 6 (a) and (b), which can be helpful in certain use cases where a post-processing procedure similar to the PIT model (Yu et al., 2017) is acceptable.
+
+# 5 DISCUSSIONS
+
+For the experiments presented in this paper, we work on labeled datasets so that we can evaluate the performance of the proposed models. However, our proposed models do not require any labeled data for training, and can thus be trained on larger unlabeled video collections in the wild. Moreover, we observe that the proposed model shows the capability of combing multiple queries, e.g., "a photo of [query A] and [query B]", to extract multiple target sounds, and we report the results on the demo website. This offers a more natural user interface against having to separate each target sound and mix them via an additional post-processing step. We also show in Appendix G that our proposed model is robust to different text queries and can extract the desired sounds.
+
+In our experiments, we often observe a modality transfer gap greater than 1 dB difference of SDR. A future research direction is to explore different approaches to reduce the modality transfer gap. For example, the CLIP model is pretrained on a different dataset, and thus finetuning the CLIP model on the target dataset can help improve the underlying modality transferability within the CLIP model. Further, while the proposed noise invariant training is shown to improve the training on noisy data and reduce the modality transfer gap, it still requires a sufficient audio-visual correspondence for training video. In other words, if the audio and images are irrelevant in most videos, the model will struggle to learn the correspondence between the query and target sound. In practice, we find that the data in the VGGSound dataset often contains off-screen sounds and the labels sometimes correspond to only part of the video content. Hence, filtering on the training data to enhance its audio-visual correspondence can also help reduce the modality transfer gap. This can be achieved by self-supervised audio-visual correspondence prediction (Arandjelović & Zisserman, 2017a;b) or temporal synchronization (Korbar et al., 2018; Owens & Efros, 2018).
+
+Another future direction is to explore the semi-supervised setting where a small subset of labeled data can be used to improve the modality transferability. We can also consider the proposed method as a pretraining on unlabeled data for other separation tasks in the low-resource regime. We include in Appendix H a preliminary experiment in this aspect using the ESC-50 dataset (Piczak, 2015).
+
+# 6 CONCLUSION
+
+In this work, we have presented a novel text-queried universal sound separation model that can be trained on noisy unlabeled videos. In this end, we have proposed to use the contrastive image-language pretraining to bridge the audio and text modalities, and proposed the noise invariant training for training a query-based sound separation model on noisy data. We have shown that the proposed models can learn to separate an arbitrary sound specified by a text query out of a mixture, even achieving competitive performance against a fully supervised model in some settings. We believe our proposed approach closes the gap between the ways humans and machines learn to focus on a sound in a mixture, namely, the multi-modal self-supervised learning paradigm of humans against the supervised learning paradigm adopted by existing label-based machine learning approaches.
+
+# ACKNOWLEDGEMENTS
+
+We would like to thank Stefan Uhlich, Giorgio Fabbro and Woosung Choi for their helpful comments during the preparation of this manuscript. We also thank Mayank Kumar Singh for supporting the setup of the subjective test in Appendix F. Hao-Wen thank J. Yang and Family Foundation and Taiwan Ministry of Education for supporting his PhD study.
+
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+
+# A INFERENCE PIPELINE OF CLIPSEP AND CLIPSEP-NIT
+
+Figure 7 illustrates the inference pipeline for the proposed CLIPSeq and CLIPSeq-NIT models. For the CLIPSeq-NIT model, we discard the noise heads at test time. The masked spectrogram is combined with the input phase and converted to the waveform by the inverse short-time Fourier transform (STFT).
+
+# B QUERY ENSEMBLING
+
+Radford et al. (2021) suggest that using a prompt template in the form of "a photo of [user input query]" helps bridge the distribution gap between text queries used for zero-shot image classification and text in the training dataset for the CLIP model. They further show that the ensemble of various prompt templates improve the generalizability. Motivated by this observation, we adopt a similar idea and use several query templates at test time (see Table 4). These query templates are heuristically chosen to handle the noisy images extracted from videos.
+
+![](images/f12a658b9bef817bfe4e5b83e2deb1af259e6d6e434f37d0428375149921a8a8.jpg)  
+Figure 7: Inference pipeline of the proposed CLIPSeq and CLIPSeq-NIT models. Note that the mixture spectrogram and the masks are shown in log frequency scale, while the masked spectrogram is shown in linear frequency scale.
+
+<table><tr><td>Query template</td><td>Example query for the label “dog barking”</td></tr><tr><td>a photo of [user input query]</td><td>a photo of dog barking</td></tr><tr><td>a photo of the small [user input query]</td><td>a photo of the small dog barking</td></tr><tr><td>a low resolution photo of a [user input query]</td><td>a low resolution photo of a dog barking</td></tr><tr><td>a photo of many [user input query]</td><td>a photo of many dog barking</td></tr></table>
+
+Table 4: Query templates used in our experiments.
+
+# C IMPLEMENTATION DETAILS
+
+We implement the audio model as a 7-layer U-Net (Ronneberger et al., 2015). We use  $k = 32$ . We use binary masks as the ground truth masks during training while using the raw, real-valued masks for evaluation. We train all the models for 200,000 steps with a batch size of 32. We use the Adam optimizer (Kingma & Ba, 2015) with  $\beta_{1} = 0.9$ ,  $\beta_{2} = 0.999$  and  $\epsilon = 10^{-8}$ . In addition, we clip the norm of the gradients to 1.0 (Zhang et al., 2020). We adopt the following learning rate schedule with a warm-up—the learning rate starts from 0 and grows to 0.001 after 5,000 steps, and then it linearly drops to 0.0001 at 100,000 steps and keeps this value thereafter. We validate the model every 10,000 steps using image queries as we do not assume labeled data is available for the validation set. We use a sampling rate of  $16,000\mathrm{Hz}$  and work on audio clips of length 65,535 samples ( $\approx$  4 seconds). During training, we randomly sample a center frame from a video and extract three frames (images) with 1-sec intervals and 4-sec audio around the center frame. During inference, for image-queried models, we extract three frames with 1-sec intervals around the center of the test clip. For the spectrogram computation, we use a filter length of 1024, a hop length of 256 and a window size of 1024 in the short-time Fourier transform (STFT). We resize images extracted from video to a size of 224-by-224 pixels. For the CLIPSeq-Hybrid model, we alternatively train the model with text and image queries, i.e., one batch with all image queries and next with all text queries, and so on. We implement all the models using the PyTorch library (Paszke et al., 2019). We compute the signal-to-distortion ratio (SDR) using museval (Stöter et al., 2018).
+
+In our preliminary experiments, we also tried directly predicting the final mask by conditioning the audio model on the query vector. We applied this modification for both SOP and CLIPSep models, however, we observe that this architecture is prone to overfitting. We hypothesize that this is because the audio model is powerful enough to remember the subtle clues in the query vector, which hinder the generalization to a new sound and query. In contrast, the proposed architecture first predicts over-determined masks and then combines them on the basis of the query vector, which avoids the overfitting problem due to the simple fusion step.
+
+![](images/19041cb7e86d64c8e63eebac33664f365c6df22cee6fbcef7e7b98689328d49d.jpg)  
+Figure 8: An illustration of the PIT model for  $n = 2$ . The two predicted masks are interchangeable during the loss computation. Since the two predicted masks are interchangeable, the PIT model requires an additional post-selection step to obtain the target sound.
+
+# D PERMUTATION INVARIANT TRAINING
+
+Figure 8 illustrates the permutation invariant training (PIT) model (Yu et al., 2017). The permutation invariant loss is defined as follows for  $n = 2$ .
+
+$$
+\mathcal {L} _ {P I T} = \min  \left(W B C E \left(M _ {1}, \hat {M} _ {1}\right) + W B C E \left(M _ {2}, \hat {M} _ {2}\right), W B C E \left(M _ {1}, \hat {M} _ {2}\right) + W B C E \left(M _ {2}, \hat {M} _ {1}\right)\right), \tag {7}
+$$
+
+where  $\hat{M}_1$  and  $\hat{M}_2$  are the predicted masks. Note that the PIT model requires an additional postselection step to obtain the target sound.
+
+# E QUALITATIVE EXAMPLE RESULTS
+
+We show in Figures 12 to 15 some example results. More results and audio samples can be found at https://sony.github.io/CLIPSeq/.
+
+# F SUBJECTIVE EVALUATION
+
+We conduct a subjective test to evaluate whether the SDR results aligned with perceptual quality. As done in the Sound of Pixel (Zhao et al., 2018), separated audio samples are randomly presented to evaluators, and the following question is asked: "Which sound do you hear? 1. A, 2. B, 3. Both, or 4. None of them". Here A and B are replaced by labels of their mixture sources, e.g. A=accordion, B=engine accelerating. Ten samples (including naturally occurring mixture) are evaluated for each model and 16 evaluators have participated in the evaluation. Table 5 shows the percentages of samples which are correctly identified the target sound class (Correct), which are incorrectly identified the target sound sources (Wrong), which are selected as both sounds are audible (Both), and which are selected as neither of the sounds are audible (None). The results indicate that the evaluators more often choose the correct sound source for CLIPSep-NIT  $(83.8\%)$  than CLIPSep  $(66.3\%)$  with text queries. Notably, CLIPSep-NIT with text-query obtained a higher correct score than that with image-query, which matches the training mode. This is probably because image queries often contain information about backgrounds and environments, hence, some noise and off-screen sounds are also suggested by the image-queries and leak to the query head. In contrast, text-queries purely contain the information of target sounds, thus, the query head more aggressively extract the target sounds.
+
+# G ROBUSTNESS TO DIFFERENT QUERIES
+
+To examine the model's robustness to different queries, we take the same input mixture and query the model with different text queries. We use the CLIPSeq-NIT model on the MUSIC $^+$  dataset and
+
+<table><tr><td>Model</td><td>Query type</td><td>Correct [%]</td><td>Wrong [%]</td><td>Both [%]</td><td>None [%]</td></tr><tr><td>Mixture</td><td>-</td><td>17.5</td><td>10.0</td><td>72.5</td><td>0.0</td></tr><tr><td rowspan="2">CLIPSeq</td><td>Image</td><td>70.6</td><td>0.0</td><td>29.4</td><td>0.0</td></tr><tr><td>Text</td><td>66.3</td><td>3.8</td><td>30.0</td><td>0.0</td></tr><tr><td rowspan="2">CLIPSeq-NIT</td><td>Image</td><td>68.8</td><td>1.9</td><td>28.1</td><td>1.3</td></tr><tr><td>Text</td><td>83.8</td><td>0.6</td><td>15.0</td><td>0.6</td></tr></table>
+
+Table 5: Subjective test results.  
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Post-processing free</td><td colspan="2">Dataset</td><td colspan="2">SDR</td></tr><tr><td>Training</td><td>Finetuning</td><td>Mean</td><td>Median</td></tr><tr><td>Mixture</td><td>-</td><td>-</td><td>-</td><td>0.00 ± 0.44</td><td>0.00</td></tr><tr><td>PIT</td><td>×</td><td>VGGSound</td><td>-</td><td>4.90 ± 0.26</td><td>2.44</td></tr><tr><td rowspan="3">CLIPSep</td><td>✓</td><td>VGGSound</td><td>-</td><td>1.07 ± 0.28</td><td>2.34</td></tr><tr><td>✓</td><td>ESC-50</td><td>-</td><td>5.18 ± 0.26</td><td>5.09</td></tr><tr><td>✓</td><td>VGGSound</td><td>ESC-50</td><td>6.73 ± 0.26</td><td>5.89</td></tr></table>
+
+Table 6: Results on the ESC-50 dataset. Standard errors are reported in the mean SDR column.
+
+report in Figure 16 the results. We see that the model is robust to different text queries and can extract the desired sounds. Audio samples can be found at https://sony.github.io/CLIPSeq/.
+
+# H FINETUNING EXPERIMENTS ON THE ESC-50 DATASET
+
+In this experiment, we aim to examine the possibilities of having a clean dataset for further finetuning. We consider the ESC-50 dataset (Piczak, 2015), a collection of 2,000 high-quality environmental audio recordings, as the clean dataset here.<sup>6</sup> We report the experimental results in Table 6. We can see that the model pretrained on VGGSound does not generalize well to the ESC-50 dataset as the ESC-50 contains much cleaner sounds, i.e., without query-irrelevant sounds and background noise. Further, if we train the CLIPSeq model from scratch on the ESC-50 dataset, it can only achieve a mean SDR of 5.18 dB and a median SDR of 5.09 dB. However, if we take the model pretrained on the VGGSound dataset and finetune it on the ESC-50 dataset, it can achieve a mean SDR of 6.73 dB and a median SDR of 4.89 dB, resulting in an improvement of 1.55 dB on the mean SDR.
+
+# I TRAINING BEHAVIORS
+
+We present in Figure 9 the training and validation losses along the training progress. Please note that we only show the results obtained using text queries for reference but do not use them for choosing the best model. We also evaluate the intermediate checkpoints every 10,000 steps and present in Figure 10 the test SDR along the training progress. In addition, for the CLIPSeq-NIT model, we visualize in Figure 11 the total mean noise head activation,  $\sum_{i=1}^{n}\mathrm{mean}(\hat{M}_i^N)$ , along the training progress. We can see that the total mean noise head activation stays around the desired level for  $\gamma = 0.1, 0.25$ . For  $\gamma = 0.5$  and the unregularized version, the total mean noise head activation converges to a similar value around 0.55.
+
+![](images/9135c5c549135ac9f116dd5d2629e7cae0a1a3102cceb5707c65db72fc4cbfaa.jpg)  
+(a) CLIPSeq
+
+![](images/d8a39bea48a26001e1c2313645c57123e84ac175add47f29a61f3de635f08d6f.jpg)  
+(b) PIT
+
+![](images/98eb756440afeb67dd15ba3b8c82e7345eb5c280967dc25dc4c496aa4d74c2f8.jpg)  
+(c) CLIPSeq-NIT (unregularized)
+
+![](images/0ad99211aa189c135cacd2406975839e1f91079683bed67cff025ef05406968f.jpg)  
+(d) CLIPSeq-NIT  $(\gamma = 0.5)$
+
+![](images/fc62abb8864a8e57ac60d48eb9d239c5a9f4f6e6bcbb6af9d3d3376328a20d9e.jpg)  
+(e) CLIPSeq-NIT  $(\gamma = 0.25)$
+
+![](images/2c074988484ed5743e427e0e23a368969272c9edfca250570c3d1087b46cde25.jpg)  
+(f) CLIPSeq-NIT  $(\gamma = 0.1)$  
+Figure 9: Training and validation losses along the training progress on the VGGSound dataset. We also include the losses computed using text queries instead of image queries. The y-axes are intentionally set to the same range for easy comparison. Note that we do not use the validation results obtained with text queries for choosing the best model.
+
+![](images/eca4bf7e698ebc00558364e52ce115ce55f121e52b99d03385ecb6d3ee7dfec4.jpg)  
+(a) CLIPSeq
+
+![](images/38b3dccdb701b1337b6ab8afde39f930aac09144d786fdf48e2e7e2fd547bd72.jpg)  
+(b) PIT
+
+![](images/bbc2bcd3c820832d0fb46c558977d80faef6f0ab5050c2df3815ca47fc7f9734.jpg)  
+(c) CLIPSeq-NIT (unregularized)
+
+![](images/c233b47649cc214bc35fe8aa0e13e6f7d8b713b59f4372b232b7669848f656f3.jpg)  
+(d) CLIPSeq-NIT  $(\gamma = 0.5)$
+
+![](images/d90b72e5c117c3c9abb215b87cc45acd80e88bc605079d249dfe34564f65896c.jpg)  
+(e) CLIPSeq-NIT  $(\gamma = 0.25)$  
+Figure 10: Test SDR along the training progress on the VGGSound-Clean dataset. The y-axes are intentionally set to the same range for easy comparison.
+
+![](images/92873e8dceccc78ca71e125b3885681b4112b8f515ecc91ac050fba7b97fe307.jpg)  
+(f) CLIPSeq-NIT  $(\gamma = 0.1)$
+
+![](images/4c26c58734cf05821919675c4e04ca0843ae222d961b9a488d54d8952ed6794c.jpg)  
+Figure 11: Total mean noise head activation,  $\sum_{i=1}^{n} \text{mean}(\hat{M}_i^N)$ , on the validation set for the CLIPSEP-NIT models along the training progress.
+
+![](images/1e601eb9a4fc56b28812b896f74547fc5d4a96185b36016c3d7f75bd55dfe0cc.jpg)  
+Mixture
+
+![](images/4271d1a4d82d5cb5167541332b73ba449cec6aa2fc71f9ba9b89f6506889e187.jpg)  
+Ground truth
+
+![](images/0cb8db016729c3d47eb20d4347364fd9bcc1ba129deb77bd3d96aa37362f682f.jpg)  
+Ground truth (Interference)
+
+![](images/d034f97491796c11d96a46dd32107cfcc1d9c32fee3fb5de60538fd0ff57eb5e.jpg)
+
+![](images/05252022b00768381aa60bb03672edb951a42e2e76dce762d9a20b0a9775b61b.jpg)
+
+![](images/01c8434d3656a243c0693199c8713f92e2af6ba53e370b84704eaff47048a8ce.jpg)  
+CLIPSeq
+
+![](images/65da869ee43e9329930af48d503211b5b994816024c032ef52cd72e4cd4d062f.jpg)  
+CLIPSeq-NIT  $(\gamma = 0.25)$
+
+![](images/6d3c399d98782a9218cf907242c9277eedee9924ce810387296a1eaa2b550d5d.jpg)  
+PIT (Yu et al., 2017)
+
+![](images/619e5eb0bfd1ae9a408b7398cf5ff692db6c4ca86024156d0c45da90cbacbe72.jpg)
+
+![](images/50b877597f7885966f10a58fbe0cb8f6487cd09a4abc39008b146cf11b47fe87.jpg)
+
+![](images/8b35c6064424ae9bacd60cdf7cd2f318a3ddb2cace306823f39327584bf4199c.jpg)
+
+![](images/92bdedb19be1202694c9d931e554b42782e695eec81c51fe0291d8478f07f493.jpg)  
+Noise head 1 (CLIPSeq-NIT)
+
+![](images/fe521ee99965448f26691633a69f32cf78ecb3c7f19c5b8946a5a1dea223254b.jpg)  
+Noise head 2 (CLIPSeq-NIT)
+
+![](images/adfdee1e60bcd9577d59993cfa643d068c393d83c727c6f83785ac1d82cd7082.jpg)  
+Figure 12: Example results on the MUSIC $^+$  dataset. Target source—"violin"; interference—"people sobbing"; query—"violin". The spectrograms and masks are shown in the log and linear frequency scales, respectively.
+
+![](images/f2ae33182be9f847236116aa42f6e642c442b4f8ab7aafc8fdd5fd08c609dfcf.jpg)
+
+![](images/2ac1012da043c977db905a2c60916049b862d795382472696eef28989bef2db8.jpg)  
+Mixture
+
+![](images/aa5501b5b4fd46d11c767b6a8c2428c3d628e81eb59a6805121df25d367dd40f.jpg)  
+Ground truth
+
+![](images/212bb49ef8cec9d6c1e1a175e521bdeca746cc461292f4d17473b81e63cb7199.jpg)
+
+![](images/205bb96d72cc219fb2fa9ebdd0835d6624356ffa854172e3071c9853edee3d4c.jpg)  
+Ground truth (Interference)
+
+![](images/3e6a79cba201d3056b4b41a2ad47882a38e592ac8137264ac632f448662ab6a6.jpg)
+
+![](images/276595f6753d57a519c3d1dffe4a0701a028a8a6fb3df7c6cc7755032032fa77.jpg)  
+CLIPSep
+
+![](images/1f25bc5f124ed36ea0e0a4e3673298b85913e8ef0c8f647973c15fa2748d82a0.jpg)  
+CLIPSeq-NIT  $(\gamma = 0.25)$
+
+![](images/9755d0ad12306e466bb1ac3286c0799123e2baa630f7e4ad9aa209b04935bbb1.jpg)
+
+![](images/ae1f71b91196d0ef9f55cb6654c38853a99c46c313bed9daf3d2da35261706de.jpg)
+
+![](images/170be811ec5076f3672d7248a6fb7137721f22b7ec3f4126371aeb9b1f1021e3.jpg)  
+PIT (Yu et al., 2017)
+
+![](images/23c962befd428c6c596f7d296d2c45182c7a95219e52c2641eacde63932f6a77.jpg)
+
+![](images/78b8fce702588960906287ade0221f8b65d1c4d57c246b4d3f8d093003fca776.jpg)  
+Noise head 1 CLIPSeq-NIT
+
+![](images/4283f67fbdf4b514a90089c4cb1aeb6910337b7d894aa52e1200e6411e8d6fea.jpg)  
+Noise head 2 (CLIPSeq-NIT)
+
+![](images/f985abfbed0051a806c71f699b2c8ae54d10b2982a504841b68cff8d82b31131.jpg)  
+Figure 13: Example results on the MUSIC $^+$  dataset. Target source—"acoustic guitar"; interference—"cheetah chirrup", query—"acoustic guitar". The spectrograms and masks are shown in the log and linear frequency scales, respectively.
+
+![](images/c07fc63bf5ab7a6b0727f9ebe3ef0e893539bf15e43c7b29e44cf6981123e2aa.jpg)
+
+![](images/ca1badaad152aa8633f7f633ae6ebf62672cc60815c1368dbbd6949d1c18cd8f.jpg)
+
+![](images/b91b10efd1e5254b4bd37a5254fafbf40d98d25a7b23c26ed2bfd61222ae4edd.jpg)
+
+![](images/3466ee0406b7ef653ef599c41fa249ee199302e48cffb8f3e4c61fd506ce471d.jpg)
+
+![](images/272a966c4fb0e3810629d9aba27b7c52dc0f77387008a2c4b766f482953c7fa9.jpg)
+
+![](images/28d18f441e0d38e1f04ca729853a052a2fa2274dfd445eae3a4ba7efa6c0c376.jpg)
+
+![](images/519e2b179cc2d9f56e437825c013e3d2c1972e53cbcbe031076023fd7f8c63e5.jpg)  
+Figure 14: Example results on the VGGSound-Clean $^+$  dataset. Target source—"cat growling"; interference—"railroad car train wagon"; query—"cat growling". The spectrograms and masks are shown in the log and linear frequency scales, respectively.
+
+![](images/c0cbb737ce51610f477c5ab8f1c34c59ea5df90dae537bdaf7072f7505dc9318.jpg)
+
+![](images/725334ebfaed33d78814bf3093a332fabf02a9918e2e017a33c735ab077f08a3.jpg)
+
+![](images/d30551131f6675860cbff9eb3bbfa07cc00d094a2726d434a257083977e330a8.jpg)
+
+![](images/adf5bdbe0e8ea579e3787b9a9e8cdcb427f7fb2f68bfad1b97b2f68bbd701d39.jpg)
+
+![](images/8afdacb40baa6d2afb60780a75c86d22fc303cc34f42fbb0f86e3d48562a2aea.jpg)
+
+![](images/c645c71b22bd182ef41bc97ef4684e96a9602018360f05a208bd847926b87476.jpg)
+
+![](images/acb7a0422c9373a8e926fca1053e2e57ef0ecfd15d9513a72c6f21fed57cdec7.jpg)
+
+![](images/28c18b8ab4de7862aac1e83ac2150c58bff5d216707dce0113620b3846c53c1a.jpg)
+
+![](images/36b3654496c7f5e5d9d443a3c32ba950babb16cf0048e91be43c94c554fe07a4.jpg)  
+Figure 15: Example results on the VGGSound-Clean+ dataset. Target source—"electric grinder grinding"; interference—"vehicle horn car horn honking"; query—"electric grinder grinding". The spectrograms and masks are shown in the log and linear frequency scales, respectively. Note that the PIT model requires a post-selection step to get the correct source. Without the post-selection step, the PIT model returns the right source in only a  $50\%$  chance.
+
+![](images/f63278600a65a8dbb249cf46983a13821a8a67a5e2da816351b243ef69ac9530.jpg)
+
+![](images/f2867a54ac6e7c1b9848361e2a97e20e61f2876b1515e5028c6a14187d09a5fb.jpg)
+
+![](images/4683f2825f90c3d9ccb7b25522c5f86813ba01d5ccbb6db2639423874c8c332b.jpg)
+
+![](images/a0d066f423e2f0673e62420482d0afca96b33cdd9b2edd991d21425196ac573c.jpg)
+
+![](images/416e81375e5b547a6a098656168740bf57fe1fe3b22be2af2fdec507cdac78c1.jpg)
+
+![](images/d65ac132608ca936f3ab389e24a0a6fa81a798ab7ed8aa15e02f2fc68add3993.jpg)
+
+![](images/074aa94c7d5b124f28da9a6cd04b8215b153e44839e3cc0251fae237a6c88c84.jpg)
+
+![](images/ffbf4aa7d7b7f96976833c68c9411fb851137bfa0917a2c4a029398a4a967cf0.jpg)  
+Figure 16: Query robustness experiment on the MUSIC $^+$  dataset. Target source—“acoustic guitar”; interference—“cheetah chirrup”. The spectrograms are shown in the log frequency scale.
+
+![](images/29d328fd9202ca018f203abf384f1c3cde1b3288bccae02fe54b1d0775578f85.jpg)  
+Prediction (Query: "a man is playing acoustic guitar in a room")

2023/CLIPSep_ Learning Text-queried Sound Separation with Noisy Unlabeled Videos/images.zip

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2023/CO3_ Cooperative Unsupervised 3D Representation Learning for Autonomous Driving/00ffb332-642a-4192-9865-4b968dcad722_origin.pdf

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2023/CO3_ Cooperative Unsupervised 3D Representation Learning for Autonomous Driving/full.md

@@ -0,0 +1,542 @@
+# CO3: COOPERATIVE UNSUPERVISED 3D REPRESENTATION LEARNING FOR AUTONOMOUS DRIVING
+
+Runjian Chen  $^{1}$ , Yao Mu  $^{1}$ , Runsen Xu  $^{4}$ , Wenqi Shao  $^{2*}$ , Chenhan Jiang  $^{5}$ , Hang Xu  $^{3}$ , Yu Qiao  $^{2}$ , Zhenguo Li  $^{3}$ , Ping Luo  $^{1*}$
+
+```txt
+{rjchen, muyao}@connect.hku.hk pluo.lhi@gmail.com {shaowenqi, qiaoyu}@pjlab.org.cn li.zhenguo@huawei.com xbjxh@live.com runsenxu@connect.cuhk.edu.hk cjiangao@connect.hkust.hk 1 The University of Hong Kong   
+2 Shanghai AI Laboratory   
+3 Huawei Noah's Ark Lab   
+4 The Chinese University of Hong Kong   
+5 Hong Kong University of Science and Technology
+```
+
+# ABSTRACT
+
+Unsupervised contrastive learning for indoor-scene point clouds has achieved great successes. However, unsupervised representation learning on outdoor-scene point clouds remains challenging because previous methods need to reconstruct the whole scene and capture partial views for the contrastive objective. This is infeasible in outdoor scenes with moving objects, obstacles, and sensors. In this paper, we propose CO3, namely Cooperative Contrastive Learning and Contextual Shape Prediction, to learn 3D representation for outdoor-scene point clouds in an unsupervised manner. CO3 has several merits compared to existing methods. (1) It utilizes LiDAR point clouds from vehicle-side and infrastructure-side to build views that differ enough but meanwhile maintain common semantic information for contrastive learning, which are more appropriate than views built by previous methods. (2) Alongside the contrastive objective, we propose contextual shape prediction to bring more task-relevant information for unsupervised 3D point cloud representation learning and we also provide a theoretical analysis for this pretraining goal. (3) As compared to previous methods, representation learned by CO3 is able to be transferred to different outdoor scene datasets collected by different type of LiDAR sensors. (4) CO3 improves current state-of-the-art methods on both Once, KITTI and NuScenes datasets by up to  $2.58\mathrm{mAP}$  in 3D object detection task and  $3.54\mathrm{mIoU}$  in LiDAR semantic segmentation task. Codes and models will be released here. We believe CO3 will facilitate understanding LiDAR point clouds in outdoor scene.
+
+# 1 INTRODUCTION
+
+LiDAR is an important sensor for autonomous driving in outdoor environments and both of the machine learning and computer vision communities have shown strong interest on perception tasks on LiDAR point clouds, including 3D object detection, segmentation and tracking. Up to now, randomly initializing and directly training from scratch on detailed annotated data still dominates this field. On the contrary, recent research efforts (He et al., 2020; Tian et al., 2019; Caron et al., 2020; Grill et al., 2020; Wang et al., 2021) in image domain focus on unsupervised representation learning with contrastive objective on different views built from different augmentation of images. They pre-train the 2D backbone with a large-scale dataset like ImageNet (Deng et al., 2009) in an unsupervised manner and use the pre-trained backbone to initialize downstream neural networks on different datasets and tasks, which achieve significant performance improvement in 2D object detection and semantic segmentation (Girshick et al., 2014; Lin et al., 2017; Ren et al., 2015). Inspired by these
+
+![](images/df8d61b5a4110d5a02934ef60b444cc73c3f15fbf9d4aeff46dda84c8a942019.jpg)  
+(a) Indoor Scene
+
+![](images/696c1e5d322506e3dd805347b7a4bdbbc61df57ebb003c968566cccec39d5897.jpg)
+
+![](images/97247d12b6b47ec0fd162df6e739268048e07ae6186f21ae91705b3d08112e9c.jpg)
+
+![](images/64965e5d7de0286c447d00bc9e1886290600db0187d25121f5111e28b415c9a6.jpg)
+
+![](images/e6c18c571c8371e2c344e6bceae86366fdb2469bdaa093c23c42b8ea5bba9eaa.jpg)
+
+![](images/376dd5d75858230b405ac47ea1c4602a7b5d6bd721d814666a5f645c5b3accf2.jpg)
+
+![](images/d20aa2fedf91c59a7114188fd42022c922bc8130a0bb4eb36fe2cea81c4fd55b.jpg)  
+(c) Our CO3: Cooperative View from Infrastructure and Vehicle
+
+![](images/a1ff914341f2ac0aa265921ee0099bbb019a6ee05a2358935aac31e2cd2081b1.jpg)  
+(b) Previous Methods in Outdoor Scene
+
+![](images/f482a290c86f9c7de651a49666014c52fe005f17b34676e2b2401df035e9d72f.jpg)  
+Figure 1: Example views built by different methods in contrastive learning, including (a) previous indoor-scene methods (b) previous outdoor-scene methods and (c) the proposed CO3. Compared to previous methods, CO3 can build two views that differ a lot and share adequate common semantics.
+
+successes, we explore unsupervised representation learning for outdoor scene point clouds to learn general representations for different architectures on various downstream datasets and tasks.
+
+In the past decade, learning 3D representation from unlabelled data has achieved great success in indoor-scene point clouds. PointContrast (Xie et al., 2020) is the pioneering work and proposes to reconstruct the whole indoor scenes, collect partial point clouds from two different poses and utilize them as two views in contrastive learning to learn dense (point-level or voxel-level) representation. More recent works such as (Hou et al., 2021) and (Liu et al., 2020) also need the reconstruction and this naturally assumes that the environment is static. Fig. 1 (a) shows an example of views in PointContrast (Xie et al., 2020). We can see that the views differ a lot because they are captured from different poses but meanwhile, they still contain enough common semantic information such as the same sofa and table. These are demonstrated important properties of views in contrastive learning in (Tian et al., 2020).
+
+However, outdoor scenes are dynamic and large-scale, making it impossible to reconstruct the whole scenes for building views. Thus, methods in (Xie et al., 2020; Hou et al., 2021; Liu et al., 2020) cannot be directly transferred but there exists two possible alternatives to build views. The first idea, embraced by (Liang et al., 2021; Yin et al., 2022), is to apply data augmentation to single frame of point cloud and treat the original and augmented versions as different views, which are indicated by the first and second pictures in Fig. 1 (b). However, all the augmentation of point clouds, including random drop, rotation and scaling, can be implemented in a linear transformation and views constructed in this way do not differ enough. The second idea is to consider point clouds at different timestamps as different views, represented by (Huang et al., 2021). Yet the moving objects would make it hard to find correct correspondence for contrastive learning. See the first and third pictures in Fig. 1 (b), while the autonomous vehicle is waiting at the crossing, other cars and pedestrians are moving around. The autonomous vehicle has no idea about how they move and is not able to find correct correspondence (common semantics). Due to these limitations, it is still challenging when transferring the pre-trained 3D encoders to datasets collected by different LiDAR sensors. Could we find better views to learn general representations for outdoor-scene LiDAR point clouds?
+
+In this paper, we propose COoperative COntrastive Learning and COntextual Shape Prediction, namely CO3, to explore the potential of utilizing vehicle-infrastructure cooperation dataset for building adequate views in unsupervised 3D representation learning. As shown in (c) in Fig. 1, a recently released infrastructure-vehicle-cooperation dataset called DAIR-V2X (Yu et al., 2022) is utilized to learn general 3D representations. Point clouds from both vehicle and infrastructure sides are captured at the same timestamp thus views share adequate common semantic. Meanwhile infrastructure-side and vehicle-side point clouds differ a lot. These properties make views constructed in this way appropriate in contrastive learning. Besides, as proposed in (Wang et al., 2022), representations learned by pure contrastive learning lack task-relevant information. Thus we further add a pre-training goal
+
+![](images/6fc2fe03076fe4914a303cc9482f432d23e993c8dbf5fc3e4ea00f9230c710a4.jpg)
+
+![](images/04bb15fef09977b65a68985ea8a192f41e3dff36f1ce247957d935a74f13a859.jpg)  
+Figure 2: (a) shows the unsupervised 3D representation learning pipeline. (b) presents the performance changes after pre-training with different methods. Our CO3 achieves consistent improvement.
+
+called contextual shape prediction to reconstruct local point distribution, encouraging our proposed CO3 to capture more task-relevant information.
+
+Our contributions can be summarized as follows. (1) CO3 is proposed to utilize the vehicleinfrastructure-cooperation dataset to build adequate views for unsupervised contrastive 3D representation learning on outdoor-scene point clouds. (2) A shape-context prediction task is proposed alongside to inject task-relevant information, which is beneficial for downstream 3D detection and LiDAR semantic segmentation tasks. (3) The learned 3D representations is generic enough to be well transferred to datasets collected by different LiDAR sensors. (4) Extensive experiments demonstrate the effectiveness of CO3. For example, on 3D object detection task, CO3 improves Second, CenterPoints on Once dataset by 1.07 and  $2.58\mathrm{mAPs}$  respectively. As for LiDAR semantic segmentation task, CO3 improves Cylinder3D on NuScenes dataset by  $3.54\mathrm{mIoUs}$ .
+
+# 2 RELATED WORKS
+
+3D Perception Tasks. 3D object detection aims to predict 3D boundary boxes for different objects in the LiDAR point clouds. Current 3D detectors can be divided into three main streams due to the 3D backbones they use: (1) point-based methods (Shi et al., 2019; Chen et al., 2017; Yang et al., 2018) use point-based 3D backbone. (2) voxel-based methods (Zhou & Tuzel, 2018; Lang et al., 2019; Su et al., 2015; Shi et al., 2020b; Yin et al., 2021; Fan et al., 2021) generally transform point cloud into voxel grids and process them using 3D volumetric convolutions. (3) point-voxel-combined methods (Shi et al., 2020a; 2021; Deng et al., 2021) utilize features from both (1) and (2). LiDAR Semantic Segmentation aims to predict per-point label for LiDAR point clouds. Cylinder3D (Zhu et al., 2021) and PVKD (Hou et al., 2022) are current SOTA methods for LiDAR Semantic Segmentation.
+
+Unsupervised 3D Representation Learning. As shown in (a) of Fig. 2, unsupervised 3D representation learning aims to pre-train only for one time and downstream to different architectures on various datasets and tasks to achieve performance gain. PointContrast (Xie et al., 2020) is the pioneering work for unsupervised contrastive learning on indoor-scene point clouds, which relies on the reconstructed point clouds for constructing adequate views. To extend their ideas to outdoor-scene point clouds, GCC-3D and ProposalContrast (Liang et al., 2021; Yin et al., 2022) augment single frame of point cloud to build views and STRL (Huang et al., 2021) utilizes point clouds at different timestamps as views for contrastive learning. Alongside contrastive learning, authors in (Hu et al., 2021) propose to use safe space prediction as a pretext task for self-supervised representation learning. However, as discussed in Sec. 1, previous works fail to build suitable views and their learned representations are unable to transfer to datasets collected by different LiDAR sensors. In this paper, we propose to use point clouds from vehicle and infrastructure to construct views in contrastive learning and also extend the idea of shape context to introduce task-relevant information with a local-distribution prediction goal. The performance change after initialized with different methods are shown in (b) of Fig. 2. CO3 generally improve the performance and achieve more performance gains than other pre-training methods for different detectors on different datasets.
+
+# 3 METHODS
+
+In this section, we introduce the proposed CO3 for unsupervised representation learning on LiDAR point clouds in outdoor scenes. As detailed in Fig. 3, CO3 has two pre-training objectives: (a) a
+
+![](images/44fc0c7ba6eaa1a3c0bb76811845d504ad9f427c70cd512a1c7212eb81fa663d.jpg)  
+Figure 3: The pipeline of CO3. With vehicle-side and fusion point clouds as inputs, we first process them with the 3D backbone and propose two pre-training objectives: (a) Cooperative Contrastive Loss (b) Contextual Shape Prediction Loss
+
+cooperative contrastive learning goal on dense (point-level or voxel-level) representations between vehicle-side and fusion point clouds, which provides adequate views for contrastive learning. (b) a contextual shape prediction loss to bring in more task-relevant information. To start with, we discuss the problem formulation and overall pipeline in Sec. 3.1. Then we respectively introduce the cooperative contrastive objective and contextual shape prediction goal in Sec. 3.2 and Sec. 3.3.
+
+# 3.1 PROBLEM FORMULATION AND PIPELINE
+
+Notation. To begin with, we denote LiDAR point clouds  $\mathbf{P} \in \mathbb{R}^{N \times (3 + d)}$  as the concatenation of the  $xyz$ -coordinate  $\mathbf{C} \in \mathbb{R}^{N \times 3}$  and the point features  $\mathbf{F} \in \mathbb{R}^{N \times d}$ , which leads to  $\mathbf{P} = [\mathbf{C}, \mathbf{F}]$ . Here  $N$  denotes the number of points (or voxels) and  $d$  is the number of point feature channels. For raw LiDAR point clouds,  $d = 1$  represents the intensity of each point. Moreover, we use the subscripts (e.g. 'v' and 'i') to indicate point clouds from different sources. For example,  $\mathbf{P}_{\mathrm{v}} = [\mathbf{C}_{\mathrm{v}}, \mathbf{F}_{\mathrm{v}}] \in \mathbb{R}^{N_{\mathrm{v}} \times (3 + d)}$  and  $\mathbf{P}_{\mathrm{i}} = [\mathbf{C}_{\mathrm{i}}, \mathbf{F}_{\mathrm{i}}] \in \mathbb{R}^{N_{\mathrm{i}} \times (3 + d)}$  respectively represent  $N_{\mathrm{v}}$  and  $N_{\mathrm{i}}$  points (or voxels) in vehicle and infrastructure sides. Besides, each pair of vehicle-side and infrastructure-side point clouds is associated with a transformation  $\mathcal{T}$  mapping infrastructure-side coordinate to that of vehicle-side.
+
+Pre-processing and Encoding. For cooperative learning, we need to first align the point clouds from both sides in the same coordinate. Hence, we transform the infrastructure-side point clouds to  $\mathbf{P}_{\mathrm{i}}^{\prime} = [\mathbf{C}_{\mathrm{i}}^{\prime},\mathbf{F}_{\mathrm{i}}]$  where  $\mathbf{C}_{\mathrm{i}}^{\prime} = \mathcal{T}(\mathbf{C}_{\mathrm{i}})$ . However, we empirically find that the contrastive learning built upon vehicle-side point cloud  $\mathbf{P}_{\mathrm{v}}$  and transformed infrastructure-side point clouds  $\mathbf{P}_{\mathrm{i}}^{\prime}$  only achieve marginal performance gains than training from scratch on downstream ONCE dataset (0.53 mAPs vs 2.58 mAPs, also see Table 7 in Appendix D). We believe this stems from the sparsity of LiDAR point clouds, which sometimes make it difficult to find good positive pairs to perform contrastive learning. To mitigate this problem, we concatenate the LiDAR point clouds from both sides to fusion point clouds  $\mathbf{P}_{\mathrm{f}} = [\mathbf{P}_{\mathrm{v}},\mathbf{P}_{\mathrm{i}}^{\prime}]\in \mathbb{R}^{(N_{\mathrm{v}} + N_{\mathrm{i}})\times (3 + d)}$ , which is used as the contrastive view of vehicle-side point clouds  $\mathbf{P}_{\mathrm{v}}$  as shown in Fig.3. By default of notations, we can express the coordinate and feature of fusion point clouds as  $\mathbf{C}_{\mathrm{f}} = [\mathbf{C}_{\mathrm{v}},\mathbf{C}_{\mathrm{i}}^{\prime}]$  and  $\mathbf{F}_{\mathrm{f}} = [\mathbf{F}_{\mathrm{v}},\mathbf{F}_{\mathrm{i}}]$ , respectively. Then  $\mathbf{P}_{\mathrm{v}}$  and  $\mathbf{P}_{\mathrm{f}}$  are embedded by the 3D encoder  $f^{\mathrm{enc}}$  to obtain their 3D representations. We use subscript 'v/f' to indicate that the same operation is applied respectively on both point clouds.
+
+$$
+\hat {\mathbf {P}} _ {\mathrm {v} / \mathrm {f}} = f ^ {\mathrm {e n c}} \left(\mathbf {P} _ {\mathrm {v} / \mathrm {f}}\right) \tag {1}
+$$
+
+where  $\hat{\mathbf{P}}_{\mathrm{v / f}}\in \mathbb{R}^{\hat{N}_{\mathrm{v / f}}\times (3 + \hat{d})}$  indicates the vehicle point cloud and fusion point cloud respectively. Here we use  $\hat{N}_{\mathrm{v / f}}$  to denote the number of points after encoding because pooling operation often exists in 3D encoders (Graham et al., 2018) and changes the number of voxels/points. Moreover,  $\hat{d}$  is the number of feature channels after encoding.
+
+Loss Function. To guide the 3D encoder to learn good representations in an unsupervised manner, our proposed CO3 consists of a cooperative contrastive loss  $\mathcal{L}_{\mathrm{CO}_2}$  and a contextual shape prediction loss  $\mathcal{L}_{\mathrm{CSP}}$ . The overall loss function is given by:
+
+$$
+\mathcal {L} = \frac {1}{| \mathcal {P} _ {\mathrm {v} / \mathrm {f}} |} \sum_ {\mathbf {P} _ {\mathrm {v} / \mathrm {f}} \in \{\mathcal {P} _ {\mathrm {v} / \mathrm {f}} \}} \mathcal {L} _ {\mathrm {C O} _ {2}} \left\{\hat {\mathbf {P}} _ {\mathrm {v}}, \hat {\mathbf {P}} _ {\mathrm {f}} \right\} + w \times \mathcal {L} _ {\mathrm {C S P}} \left\{\hat {\mathbf {P}} _ {\mathrm {v}}, \hat {\mathbf {P}} _ {\mathrm {f}}, \mathbf {C} _ {\mathrm {f}} \right\} \tag {2}
+$$
+
+where  $\mathcal{P}_{\mathrm{v / f}}$  denote a batch of vehicle and fusion point clouds and  $|\mathcal{P}_{\mathrm{v / f}}|$  indicates the batch size.  $\mathcal{L}_{\mathrm{CO_2}}$  applies contrastive learning on the encoded vehicle and fusion pointclouds. Meanwhile,  $\mathcal{L}_{\mathrm{CSP}}$  introduces more task-relevant information into  $f^{\mathrm{enc}}$  by using the encoded features to predict contextual shape obtained by the coordinate of fusion point clouds  $\mathbf{C}_{\mathrm{f}}$ .  $w$  is a weight to balance the losses.
+
+# 3.2 COOPERATIVE CONTRASTIVE OBJECTIVE
+
+Unsupervised contrastive learning has been demonstrated successful in image domain (He et al., 2020; Tian et al., 2019) and indoor-scene point clouds (Xie et al., 2020). However, when it turns to outdoor-scene LiDAR point clouds, building adequate views, which share common semantics while differing enough, for contrastive learning is difficult. To tackle this challenge, we utilize a recently released vehicle-infrastructure-cooperation dataset called DAIR-V2X (Yu et al., 2022) and use vehicle-side point clouds and fusion point clouds as views for contrastive representation learning. Views built in this way differ a lot because they are captured at different positions and they share enough information because they are captured at the same timestamp. More details about view building in contrastive learning can be found in Appendix A.
+
+Contrastive Head. Following BYOL (Grill et al., 2020), we construct contrastive head of cooperative contrastive objective by a Multi-Layer-Perceptron (MLP) layer and a  $\ell_2$ -normalization. Specifically, the embedded features of vehicle and fusion point clouds,  $\hat{\mathbf{F}}_{\mathrm{v}}$  and  $\hat{\mathbf{F}}_{\mathrm{f}}$ , are first projected by a MLP layer denoted as  $\mathrm{MLP}_1$ . Then an  $\ell_2$ -normalization is applied along feature dimension. This process is described below, where  $\mathbf{Z}_{\mathrm{v/f}} \in \mathbb{R}^{\hat{N}_{\mathrm{v/f}} \times d_z}$  and  $d_z$  is the output feature dimension.
+
+$$
+\hat {\mathbf {Z}} _ {\mathrm {v} / \mathrm {f}} = \operatorname {M L P} _ {1} \left(\hat {\mathbf {F}} _ {\mathrm {v} / \mathrm {f}}\right), \quad \mathbf {Z} _ {\mathrm {v} / \mathrm {f}} = \hat {\mathbf {Z}} _ {\mathrm {v} / \mathrm {f}} / \| \hat {\mathbf {Z}} _ {\mathrm {v} / \mathrm {f}} \| _ {2} \tag {3}
+$$
+
+Cooperative Contrastive Loss. For cooperative contrastive learning, we obtain positive pairs by exploring the correspondence between coordinates  $\hat{\mathbf{C}}_{\mathrm{v}}$  and  $\hat{\mathbf{C}}_{\mathrm{f}}$ . In detail, we uniformly sample  $N_{1}$  points from the vehicle point cloud and find their corresponding points in the fusion point cloud to form  $N_{1}$  pairs of features  $(\{z_{\mathrm{v / f}}^{n}\}_{n = 1}^{N_{1}})$  from  $\mathbf{Z}_{\mathrm{v / f}}$  for contrastive learning, where  $z_{\mathrm{v / f}}^{n}\in \mathbb{R}^{1\times d_{z}}$ . We treat corresponding points (or voxels) as positive pairs and otherwise negative pairs for contrastive learning. Example pairs are shown in Fig. 3. The final loss function is formulated below,
+
+$$
+\mathcal {L} _ {\mathrm {C O} _ {2}} = \frac {1}{N _ {1}} \sum_ {n = 1} ^ {N _ {1}} - \log \left(\frac {\exp \left(z _ {\mathrm {v}} ^ {n} \cdot z _ {\mathrm {f}} ^ {n} / \tau\right)}{\sum_ {i = 1} ^ {N _ {1}} \exp \left(z _ {\mathrm {v}} ^ {n} \cdot z _ {\mathrm {f}} ^ {i} / \tau\right)}\right) \tag {4}
+$$
+
+where  $\tau$  is the temperature parameter. The bottom right box in Fig. 3 shows examples for this loss. As ground points only contain background information that is not related to perception tasks, we mark those points with height value lower than a threshold  $z_{\mathrm{thd}}$  as ground points and filter them out when sampling.
+
+# 3.3 CONTEXTUAL SHAPE PREDICTION
+
+CO3 aims to learn representations applicable to various downstream datasets. However, as demonstrated in (Wang et al., 2022), pure contrastive loss in Eqn. (4) would result in the lack of task-relevant information in the learned representations, making it hard to generalize across different architectures and datasets. Meanwhile, an additional reconstruction objective could increase the mutual information between the representations and the input views, which would bring in task-relevant information. Please refer to detailed explanations about this in Appendix B. For outdoor-scene point clouds, it is difficult to reconstruct the whole scene with point/voxel-level representations. To mitigate this issue, we propose to reconstruct the neighborhood of each point/voxel with its representation.
+
+Local Distribution. Shape context has been demonstrated as useful point feature descriptor in previous works (Hou et al., 2021; Belongie et al., 2002; Kortgen et al., 2003; Xie et al., 2018). Fig. 4 shows two examples of shape context with 8 bins (the number of bins can be changed) around the query point, which
+
+![](images/f28a54b7ed2b8a6aa136e8ec99d190c26aa467ec2b8ab03a1ecc05d700651033.jpg)  
+Figure 4: Two examples of shape context.
+
+is marked as a larger black point. Previous works use the number of points in each bin as the descriptor for the query point. However, it can be difficult for the neural net to learn to regress the exact number of points in each bin. Thus, we propose to use the representation to predict a local distribution built upon shape context. In practice, we divide the neighborhood of each point in fusion point clouds into  $N_{\mathrm{bin}} = 32$  bins along  $xyz$ -plane with  $R_1 = 0.5m$  and  $R_2 = 4m$ . Then we compute the "raw" shape context  $Q' \in \mathbb{R}^{N_{\mathrm{f}} \times N_{\mathrm{bin}}}$ , which describe the number of points in each bin around every fusion point. Next, we apply  $\ell_2$ -normalization to  $Q'$  and a consecutive softmax to obtain the ground-truth shape context  $Q \in \mathbb{R}^{N_{\mathrm{f}} \times N_{\mathrm{bin}}}$  as described below, where  $Q_{i,*} \in \mathbb{R}^{1 \times N_{\mathrm{bin}}}$  describes the ground-truth local distribution for  $i$ -th point.
+
+$$
+Q _ {i, *} = \operatorname {s o f t m a x} \left(Q _ {i, *} ^ {\prime} / \left\| Q _ {i, *} ^ {\prime} \right\| _ {2}\right) \tag {5}
+$$
+
+Prediction Loss. We use the encoded features of both vehicle and fusion point clouds, i.e.  $\hat{\mathbf{F}}_{\mathrm{v / f}}$ , to predict the local distribution. To be specific,  $\hat{\mathbf{F}}_{\mathrm{v / f}}$  is first passed through a MLP layer  $\mathrm{MLP}_2$  and softmax operation is applied on the projected features to get the predicted local distribution.
+
+$$
+P _ {\mathrm {v / f}} = \operatorname {s o f t m a x} \left(\mathrm {M L P} _ {2} \left(\hat {\mathbf {F}} _ {\mathrm {v / f}}\right)\right) \tag {6}
+$$
+
+where  $P_{\mathrm{v / f}} \in \mathbb{R}^{\hat{N}_{\mathrm{v / f}} \times N_{\mathrm{bin}}}$ . Then we uniformly sample  $N_2$  predictions  $\left(\{p_{\mathrm{v / f}}^n\}_{n=1}^{N_2}\right)$  from  $P_{\mathrm{v / f}}$  and find their corresponding "ground truth" in  $Q$  by coordinate correspondence, where we have  $\{q_{\mathrm{v / f}}^n\}_{n=1}^{N_2}$ . The dimensions of each prediction and "ground truth" are  $p_{\mathrm{v / f}}^n \in \mathbb{R}^{1 \times N_{bin}}$  and  $q_{\mathrm{v / f}}^n \in \mathbb{R}^{1 \times N_{bin}}$ . Finally, the prediction loss with KL-divergence is given by,
+
+$$
+\mathcal {L} _ {\mathrm {C S P}} = \frac {1}{N _ {2}} \sum_ {n = 1} ^ {N _ {2}} \sum_ {m = 1} ^ {N _ {\text {b i n}}} \left(p _ {\mathrm {v}} ^ {n, m} \log \frac {p _ {\mathrm {v}} ^ {n , m}}{q _ {\mathrm {v}} ^ {n , m}} + p _ {\mathrm {f}} ^ {n, m} \log \frac {p _ {\mathrm {f}} ^ {n , m}}{q _ {\mathrm {f}} ^ {n , m}}\right) \tag {7}
+$$
+
+# 4 EXPERIMENTS
+
+The goal for unsupervised representation learning is to learn general representation that can benefit different downstream architectures on different downstream datasets and tasks. In this section, we design experiments to answer the question whether CO3 learns such representation as compared to previous methods. We first provide experiment setups in Sec. 4.1 and then discuss main results in Sec. 4.2. We also conduct ablation study and qualitative visualization in Sec. 4.3 and 4.4.
+
+# 4.1 EXPERIMENT SETUP
+
+Pre-training Dataset. We utilize the recently released vehicle-infrastructure-cooperation dataset called DAIR-V2X (Yu et al., 2022) to pre-train the 3D encoder. DAIR-V2X is the first real-world autonomous dataset for vehicle-infrastructure-cooperative task. The LiDAR sensor at vehicle-side is 40-beam while a 120-beam LiDAR is utilized at infrastructure-side. There are 38845 LiDAR frames (10084 in vehicle-side and 22325 in infrastructure-side) for cooperative-detection task. The dataset contains around 7000 synchronized cooperative samples in total.
+
+Implementation Details of CO3. We use Sparse-Convolution as the 3D encoder which is a 3D convolutional network because it is widely used as 3D encoders in current state-of-the-art methods (Zhou & Tuzel, 2018; Yin et al., 2021; Shi et al., 2020a). We set the number of feature channels  $d^{enc} = 64$ , the temperature parameter in contrastive learning  $\tau = 0.07$ , the dimension of common feature space of vehicle-side and fusion point clouds  $d_{z} = 256$  and the sample number in cooperative contrastive loss  $N_{1} = 2048$ . For contextual shape prediction, we set the number of bins  $N_{bin} = 32$ , the sample number  $N_{2} = 2048$  and the weighting constant  $w = 10$ . The threshold for ground point filtering is  $z_{\mathrm{thd}} = 1.6\mathrm{m}$ . We empirically find that freezing the parameters of  $\mathrm{MLP}_2$  brings better results in detection task thus we fix them.
+
+Baselines. We implement STRL (Huang et al., 2021) and use the official code of ProposalContrast (Yin et al., 2022) as baselines. Besides, as proposed in (Mao et al., 2021), several methods in image domain and indoor-scene point clouds can be transferred to outdoor scene point clouds, including Swav (Caron et al., 2020), Deep Cluster (Caron et al., 2018), BYOL (Grill et al., 2020) and Point Contrast (Xie et al., 2020). Note that in order to make fair comparisons, all the pre-training methods are pre-trained on DAIR-V2X. Thus there might exist number discrepancy between the results and previous benchmarks.
+
+<table><tr><td rowspan="2">Init.</td><td colspan="3">Second</td><td colspan="3">PV-RCNN</td><td colspan="3">CenterPoint</td></tr><tr><td>Overall</td><td>0-30m</td><td>30-50m</td><td>Overall</td><td>0-30m</td><td>30-50m</td><td>Overall</td><td>0-30m</td><td>30-50m</td></tr><tr><td>Rand</td><td>52.21</td><td>60.71</td><td>47.31</td><td>54.55</td><td>63.05</td><td>50.00</td><td>55.92</td><td>66.39</td><td>50.16</td></tr><tr><td>Swav</td><td>53.03</td><td>61.38</td><td>48.34</td><td>54.89</td><td>63.41</td><td>50.31</td><td>57.00</td><td>67.54</td><td>50.60</td></tr><tr><td>Deep Cluster</td><td>52.30</td><td>61.03</td><td>47.00</td><td>54.91</td><td>63.84</td><td>50.29</td><td>57.65</td><td>68.05</td><td>51.04</td></tr><tr><td>BYOL</td><td>45.24</td><td>53.84</td><td>40.22</td><td>49.41</td><td>58.04</td><td>45.19</td><td>52.17</td><td>63.53</td><td>45.00</td></tr><tr><td>PointContrast</td><td>47.64</td><td>56.91</td><td>42.54</td><td>50.49</td><td>58.94</td><td>46.30</td><td>54.17</td><td>65.57</td><td>46.53</td></tr><tr><td>ProposalContrast</td><td>50.70</td><td>59.42</td><td>46.09</td><td>52.79</td><td>60.97</td><td>48.91</td><td>56.02</td><td>66.26</td><td>48.70</td></tr><tr><td>STRL</td><td>51.57</td><td>59.80</td><td>46.59</td><td>54.25</td><td>63.03</td><td>49.31</td><td>57.44</td><td>67.90</td><td>50.47</td></tr><tr><td>Ours</td><td>53.28+1.07</td><td>62.16</td><td>49.33</td><td>55.17+0.61</td><td>63.94</td><td>50.29</td><td>58.50+2.58</td><td>69.09</td><td>51.51</td></tr></table>
+
+Table 1: Results of 3D object detection on Once dataset (Mao et al., 2021). We conduct experiments on 3 different detectors: Second (Zhou & Tuzel, 2018) (short as Sec.), PV-RCNN (Shi et al., 2021) (short as PV) and CenterPoint (Yin et al., 2021) (short as Cen.) and 8 different initialization methods including random (short as Rand, i.e. training from scratch), Swav (Caron et al., 2020), Deep Cluster (short as D. Cl.) (Caron et al., 2018), BYOL (Grill et al., 2020), Point Contrast (short as P.C.) (Xie et al., 2020), GCC-3D (Liang et al., 2021) and STRL (Huang et al., 2021). Results are mAPs in %. "0-30m" and "30-50m" respectively indicate results for objects in 0 to 30 meters and 30 to 50 meters. The "Overall" metric highlighted in red is the overall mAP, which serves as major metric for comparisons. We use bold font for the best overall mAP of each detector for better understanding.
+
+Downstream Tasks. Two downstream tasks are selected for evaluation: 3D object detection and LiDAR semantic segmentation. 3D object detection task takes raw 3D point clouds as inputs and aims to output 3D boundary boxes of different objects in the scene. LiDAR semantic segmentation assign each 3D point a category label, including Car, Pedestrian, Bicycle, Truck and so on.
+
+3D Object Detection. We select two downstream datasets: Once (Mao et al., 2021) and KITTI (Geiger et al., 2012). Once has  $15\mathrm{k}$  fully annotated frames of LiDAR scans. A 40-beam LiDAR is used to collect the point cloud data. We adopt common practice, including point cloud range and voxel size, in their public code repository. mAPs (mean accurate precisions) in different ranges and overall mAP are presented. KITTI is a widely used self-driving dataset, where point clouds are collected by LiDAR with 64 beams. It contains around 7k samples for training and another 7k for evaluation. All the results are evaluated by mAPs with three difficulty levels: Easy, Moderate and Hard. We select several current state-of-the-art methods implemented in the public repository of Once dataset (Mao et al., 2021) $^{1}$  and OpenPCDet $^{2}$  to evaluate the quality of representations learned by CO3, including Second (Zhou & Tuzel, 2018), CenterPoint (Yin et al., 2021) and PV-RCNN (Shi et al., 2020a).
+
+LiDAR Semantic Segmentation. We select NuScenes (Caesar et al., 2020) as downstream dataset. There are 1000 scenes each of which lasts 20 seconds in NuScenes. It is collected with a 32-beam LiDAR sensor and the total number of frames is 40,000. We select Cylinder3D (Zhu et al., 2021) as downstream architecture. The full training is time-consuming (at least 4 days for one training) and we use a 1/8 training schedule setting to evaluate whether CO3 is able to speed up training. We follow the conventional evaluation metrics. mAPs for detailed categories and mean intersection-over-union (mIoU) for overall evaluation are presented. Per-class mIoU is first computed as  $\mathrm{mIoU}_i = \frac{\mathrm{TP}_i}{\mathrm{TP}_i + \mathrm{FP}_i + \mathrm{FN}_i}$ , where  $\mathrm{TP}_i$ ,  $\mathrm{FP}_i$  and  $\mathrm{FN}_i$  respectively represent true positive, false positive and false negative for class  $i$ . We then average over classes and get the final mIoU.
+
+# 4.2 MAIN RESULTS
+
+Once Detection. As shown in Table 1, when initialized by CO3, all the three detectors achieve the best performance on the overall mAPs, which we value the most, and CenterPoint (Yin et al., 2021) achieves the highest overall mAP (58.50) with 2.58 improvement. The improvement on PV-RCNN (Shi et al., 2021) is 0.62 in mAP (similar lower improvement with other pre-training methods) because PV-RCNN (Shi et al., 2021) has both the point-based and voxel-based 3D backbones, among which CO3 only pre-trains the voxel-based branch. It can be found that other baselines are not able to learn general representation for different architectures. For example, STRL achieves +1.52 mAPs improvements on CenterPoint but degrades the performance of Second and PV-RCNN. On the contrary, CO3 achieve consistent improvement over different detectors.
+
+KITTI Detection. As shown in Table 2, when initialized by CO3, PV-RCNN (Shi et al., 2021) achieves the best performance on Easy and Hard (+1.19) level and third place on Moderate level.
+
+<table><tr><td rowspan="2">Initialization</td><td colspan="3">Second</td><td colspan="3">PV-RCNN</td></tr><tr><td>Easy</td><td>Moderate</td><td>Hard</td><td>Easy</td><td>Moderate</td><td>Hard</td></tr><tr><td>Random</td><td>73.29</td><td>63.16</td><td>60.34</td><td>78.54</td><td>67.23</td><td>63.68</td></tr><tr><td>Swav</td><td>73.23</td><td>64.05</td><td>60.90</td><td>78.43</td><td>67.91</td><td>64.60</td></tr><tr><td>Deep Cluster</td><td>73.19</td><td>63.37</td><td>60.08</td><td>77.05</td><td>67.06</td><td>64.50</td></tr><tr><td>BYOL</td><td>71.05</td><td>60.39</td><td>56.98</td><td>77.96</td><td>67.50</td><td>64.42</td></tr><tr><td>PointContrast</td><td>72.67</td><td>62.74</td><td>59.21</td><td>77.62</td><td>67.79</td><td>63.31</td></tr><tr><td>ProposalContrast</td><td>74.23</td><td>64.22</td><td>60.88</td><td>77.28</td><td>66.47</td><td>63.22</td></tr><tr><td>STRL</td><td>73.95</td><td>63.90</td><td>60.90</td><td>77.10</td><td>66.21</td><td>62.90</td></tr><tr><td>Ours</td><td>74.40+1.11</td><td>64.38+1.22</td><td>60.90+0.56</td><td>78.84+0.30</td><td>67.75+0.52</td><td>64.87+1.09</td></tr></table>
+
+Table 2: Results of 3D object detection on KITTI dataset (Geiger et al., 2012). Results are mAPs in %. "Easy", "Moderate" and "Hard" respectively indicate difficulty levels defined in KITTI dataset. We use bold font for the best mAP of each detector in each difficulty level for better understanding.  
+
+<table><tr><td>Initialization</td><td>mIoU</td><td>Car</td><td>Truck</td><td>Con. Veh.</td><td>Ped.</td><td>Trailer</td><td>Bic.</td><td>M.C.</td><td>S.W.</td><td>Terrain</td><td>Veg.</td></tr><tr><td>Random</td><td>63.34</td><td>84.32</td><td>70.25</td><td>28.29</td><td>64.46</td><td>41.35</td><td>0.00</td><td>60.46</td><td>69.76</td><td>71.21</td><td>83.32</td></tr><tr><td>P.C.</td><td>64.31</td><td>84.70</td><td>74.36</td><td>30.81</td><td>63.52</td><td>47.13</td><td>10.16</td><td>55.55</td><td>70.38</td><td>71.43</td><td>84.65</td></tr><tr><td>STRL</td><td>64.71</td><td>84.66</td><td>76.65</td><td>27.30</td><td>63.29</td><td>52.76</td><td>12.79</td><td>60.11</td><td>70.27</td><td>71.70</td><td>84.60</td></tr><tr><td>Ours</td><td>66.88+3.54</td><td>85.52</td><td>77.00</td><td>36.00</td><td>66.93</td><td>53.13</td><td>19.51</td><td>70.65</td><td>70.40</td><td>72.43</td><td>84.88</td></tr></table>
+
+Table 3: LiDAR semantic segmentation on NuScenes (Caesar et al., 2020). Results are mIoU and mAPs in  $\%$  . We shows details results in each category. "Con. Veh.", "Ped.", "Bic.", "M.C.", "S. W." and "Veg." are abbreviations respectively for Construction Vehicle, Pedestrian, Bicycle, MotoCycle, Sidewalk and Vegetation. We use bold font for the best results in each column for better understanding.  
+
+<table><tr><td>Init.</td><td>Overall</td><td>Vehicle</td><td>Pedestrian</td><td>Cyclist</td></tr><tr><td>Random</td><td>55.92</td><td>62.85</td><td>45.52</td><td>59.39</td></tr><tr><td>Sup.</td><td>57.50</td><td>63.86</td><td>46.96</td><td>61.68</td></tr><tr><td>CO3</td><td>58.50</td><td>64.60</td><td>48.83</td><td>62.17</td></tr></table>
+
+Table 4: Comparison to supervised initialization  
+
+<table><tr><td>Init.</td><td>Overall</td><td>Vehicle</td><td>Pedestrian</td><td>Cyclist</td></tr><tr><td>From Scratch</td><td>55.92</td><td>62.85</td><td>45.52</td><td>59.39</td></tr><tr><td>CO3 w. ground</td><td>57.37</td><td>63.19</td><td>48.16</td><td>60.76</td></tr><tr><td>CO3</td><td>58.50</td><td>64.60</td><td>48.83</td><td>62.17</td></tr></table>
+
+Table 5: Ablation study on filtering out ground points.
+
+Meanwhile, when Second (Zhou & Tuzel, 2018) is equipped with CO3, it achieves the highest mAPs on Easy level (+1.11), Moderate level (+1.22) and Hard level (+0.56). The lower gains on the KITTI dataset (Geiger et al., 2012) stem from the smaller number of training samples (half of that in Once (Mao et al., 2021)), which makes the detectors easily reach their capacity and improvement is hard to achieve. Consistent results across different initialization methods demonstrate this.
+
+NuScene Semantic Segmentation. As shown in Table 3, CO3 improve Cylinder3D by 3.54 in mIoU and also achieves the best performance among the four initialization methods. Meanwhile, when initialized by CO3, Cylinder3D achieves the best mAPs among all the categories. On truck and construction vehicle, CO3 improve the performance of random initialization by 6.75 and 7.71 mAPs, which is very important in autonomous driving because correct segmentation of different vehicles can help benefit control and avoid accidents.
+
+Comparison to Supervised Pre-training. We find backbone pre-trained on 3D detection task from the official codebase of DAIR-V2X dataset and use it to initialize CenterPoint, after which we fine-tune it on Once dataset. Results are shown in Table 4. It can be found that although supervised pre-training achieve improvement as compared to training from scratch, CO3 leads to the best results in all categories and the overall mAP. This is because supervised pre-training overfit to DAIR-V2X dataset, making the improvement lower than CO3 in 3D detection task on Once dataset.
+
+Overall Evaluation. To summarize, CO3 achieves consistent performance gains over different architectures on different tasks (3D object detection and LiDAR semantic segmentation) and datasets (Once, KITTI and NuScenes) when pre-trained only on DAIR-V2X dataset. In comparison, other baseline pre-training methods only occasionally improve the performance and sometimes even bring degradation. These demonstrate CO3 learns general 3D representations.
+
+# 4.3 ABLATION STUDY
+
+The influence of ground points in pre-training. We first conduct ablation experiments on filtering out ground points. We use CO3 to pre-train 3D backbone without filtering out ground points and downstream it to 3D object detection with CenterPoint on Once. Results are shown in Table 5. When pre-training without filtering out ground points, the performance of CO3 drop in each category and
+
+<table><tr><td rowspan="2">Init.</td><td colspan="4">Once (CenterPoint)</td><td colspan="4">KITTI (Second)</td></tr><tr><td>Overall</td><td>Vehicle</td><td>Pedestrian</td><td>Cyclist</td><td>Overall</td><td>Vehicle</td><td>Pedestrian</td><td>Cyclist</td></tr><tr><td>Random</td><td>55.92</td><td>62.85</td><td>45.52</td><td>59.39</td><td>63.16</td><td>77.45</td><td>48.71</td><td>63.32</td></tr><tr><td>Contextual Shape Prediction Only</td><td>57.30</td><td>62.86</td><td>49.17</td><td>59.86</td><td>63.36</td><td>77.75</td><td>49.16</td><td>63.18</td></tr><tr><td>Cooperative Contrastive Only</td><td>57.53</td><td>63.39</td><td>48.14</td><td>61.05</td><td>63.41</td><td>77.40</td><td>47.78</td><td>65.06</td></tr><tr><td>CO3</td><td>58.50</td><td>64.50</td><td>48.83</td><td>62.17</td><td>64.38</td><td>77.95</td><td>49.59</td><td>65.60</td></tr></table>
+
+Table 6: Results of ablation study on Once (Mao et al., 2021) and KITTI (Geiger et al., 2012). We use CenterPoint (Yin et al., 2021) on Once and Second (Zhou & Tuzel, 2018) on KITTI. Results are mAPs in %. For Once, results are average across different ranges. For KITTI, results are all in moderate level. We highlight the best performance in each column for better understanding.
+
+![](images/71706bf79c2b7ace89ee5a3da8c654ba97ec94b00daa9d9848c7562451714792.jpg)  
+Pre-trained by CO3
+
+![](images/bd48d2ff41011f71907a3cddd94cf98d78bf924320927f8a357923b9fcb6bf12.jpg)
+
+![](images/b555a8ade0dbd18c0161f3ee52b6ada3ff7b18340ae231e2df8943619099febc.jpg)
+
+![](images/76b37745f4621ec138c6c02cbee2c7a22d839557f44cb19c3f68c623a088f41e.jpg)  
+Pre-trained by PointContrast
+
+![](images/dd6e1596872882af5ad3f0b66ee5effcdb03142da12662210202979ba6ac9a8a.jpg)  
+Pre-trained by CO3
+
+![](images/70cc22c85df5f7c14c513c1e2b655e0f73397c84248db0d6200950d2b7f63bf2.jpg)  
+(a) Case 1
+
+![](images/be2c7560bd8e991343d4130ad644d50a85148d49e07b6cda1cb550489f7b3fda.jpg)  
+(c) Case 3
+
+![](images/16069799b5328f5ef60c7e7ad79bff774101bac5a8901d9fb9c8cdb415987987.jpg)  
+Pre-trained by PointContrast
+
+![](images/3b9d32a9d50ead4f22c0c7f783f7762204e5d783c09ba87ddd6fc73991f756b7.jpg)  
+Pre-trained by CO3
+
+![](images/a9d8dae00e7c80e60344a4393db5687a328c6744fab041ef0e5affd5774aa077.jpg)
+
+![](images/c0c98a3dc8467e3c8e058fda41b4c2eb561e0b58666ee7135fef68e6f9c7180d.jpg)
+
+![](images/bb36962531a0f931c0619be4f8fa1fa4150a860038323689de7dadc23692da96.jpg)  
+Pre-trained by PointContrast
+
+![](images/534fd241fd504a530f9c185d4609ddbe8de9d3665779e88b2ae5e60097cf4213.jpg)  
+Pre-trained by CO3  
+Figure 5: Visualization for detection results. Green boxes are predicted ones and red boxes are the ground truth.
+
+![](images/1aa19214b660e496fcc0de60b255d2d950b84fc19934062b63b28685589c0828.jpg)  
+(b) Case 2  
+(d) Case 4
+
+![](images/25e5eb731408899dbeed81e87e5ec3ddb585b9ddf09777fecb62644f70eab00c.jpg)  
+Pre-trained by PointContrast
+
+the overall mAP. This demonstrate the effectiveness of filtering out ground points because ground points contain background information that is not useful for 3D perception tasks.
+
+The effect of each component in CO3. We conduct ablation experiments to analyze the effectiveness of different components. We respectively pre-train the 3D encoder with cooperative contrastive objective and contextual shape prediction objective. As shown in Table 6, it can be found that each objective alone can achieve slight improvement, which demonstrates the effectiveness of either part. Besides, when pre-trained by CO3, we achieve the best performance on the overall mAPs. A more detailed discussion on the ablation study is provided in Appendix G
+
+# 4.4 QUALITATIVE EXPERIMENT
+
+We use the 3D backbone pre-trained by CO3 and PointContrast to initialize CenterPoint and train it on Once dataset. Then we visualize the detection results in Fig. 5, where predicted boxes are marked as green and the ground truth boxes are red. In Case 1 and 2, when the detector is initialized by CO3, the detection results are more correct in headings as shown in the zoom-in area. Correct heading prediction is important especially for control in autonomous driving. In Case 3 and 4, it can be found that CO3 helps the CenterPoint detect object with only a few points captured by the LiDAR sensor and meanwhile, PointContrast initialization fails to detect them. This is also essential in autonomous driving because detection failure can sometimes lead to disaster.
+
+# 5 CONCLUSION AND FUTURE WORK
+
+In this paper, we propose CO3, namely Cooperative Contrastive Learning and Contextual Shape Prediction, for unsupervised 3D representation learning in outdoor scenes. The recently released vehicle-infrastructure-cooperation dataset DAIR-V2X is utilized to build views for cooperative contrastive learning. Meanwhile the contextual shape prediction objective provides task-relevant information for the 3D encoders. Our experiments demonstrate that the representation learned by CO3 can be transferred to various architectures and different downstream datasets and tasks to achieve performance gain. Currently the size of the real cooperation dataset is relatively small and it will be interesting if larger cooperative datasets can be collected for pre-training in the future.
+
+# ACKNOWLEDGMENTS
+
+Ping Luo is supported by the General Research Fund of HK No.27208720, No.17212120, and No.17200622.
+
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+
+# A BACKGROUND ABOUT VIEW BUILDING IN CONTRASTIVE LEARNING
+
+In this section, we discuss how to build proper views in contrastive learning. Firstly, we introduce the formulation of contrastive learning and some important properties of good views in contrastive learning as proposed in (Tian et al., 2020). Then we discuss view building for LiDAR point clouds.
+
+View Building Contrastive Learning. Unsupervised representation learning aims to pre-train 2D/3D backbones on a dataset without labels, which can be transferred to downstream datasets and tasks to achieve performance improvement over training from scratch (random initialization). Recently, unsupervised contrastive learning achieves great success in image domain (He et al., 2020; Tian et al., 2019; Caron et al., 2020; Grill et al., 2020; Wang et al., 2021). Given a batch of images  $\mathcal{X}$  as inputs, these works first apply two kinds of random augmentations for each image  $x^{n} \in \mathcal{X}$  ( $n = 1, 2, \dots, N$  where  $N$  is the number of images in the batch) to get augmented images  $x_{1}^{n}$  and  $x_{2}^{n}$ , which are called different views of  $x^{n}$ . The main objective of contrastive learning is to pull together the representations of views of the same image in the feature space while pushing away representations of different images, as indicated in the equation below:
+
+$$
+L _ {\text {c o n}} = \frac {1}{N} \sum_ {n = 1} ^ {N} - \log \left(\frac {\exp \left(z _ {1} ^ {n} \cdot z _ {2} ^ {n} / \tau\right)}{\sum_ {i = 1} ^ {N} \exp \left(z _ {1} ^ {n} \cdot z _ {2} ^ {i} / \tau\right)}\right) \quad \text {w i t h} \tag {8}
+$$
+
+$$
+z _ {1, 2} ^ {n} = f _ {\mathrm {I}} ^ {\mathrm {e n c}} \left(x _ {1, 2} ^ {n}\right) \quad n = 1, 2, \dots , N
+$$
+
+where  $f_{\mathrm{I}}^{\mathrm{enc}}$  is the 2D backbone used to extract representations.  $z_{1,2}^{n}$  is the encoded representations for views  $x_{1,2}^{n}$ . To apply contrastive loss in the first line in Eqn. (8), views of the same image are considered as positive pairs and other pairs are negative ones. The numerator indicates the similarity of positive pairs while the denominator sums up the positive similarity and the sum of similarity of negative pairs.  $\tau$  is the temperature parameter. Minimizing this loss equals to maximize the similarity of positive pairs and minimize the similarity of negative pairs.
+
+Authors in (Tian et al., 2020) discuss what property views should have to benefit contrastive learning via Information Theory and propose that mutual information (Shannon, 2001; Kreer, 1957; Wikipedia contributors, 2022) of views  $I(x_1^n;x_2^n)$ , can indicate the quality of the learned representations. Mutual information formally quantifies "how much information about one random variable we can obtain when observing the other one". Experiments on images suggest that there exists a "sweet spot" for  $I(x_{1}^{n};x_{2}^{n})$  where the pre-trained backbone can achieve the most significant performance improvement in downstream tasks. This means the mutual information between views can neither be too low (sharing little semantics) nor too high (differing little). Further experiments in (Tian et al., 2020) indicate that the mutual information of different augmented images is high and reducing  $I(x_{1}^{n};x_{2}^{n})$  by applying stronger augmentations is effective. The performance on downstream tasks increases at the beginning and then decreases when the augmentations are too strong.
+
+Views Building for LiDAR Point Clouds. As discussed in the main paper, it is impossible for us to reconstruct the whole outdoor-scene for constrative learning, which is demonstrated useful in indoor-scene (Xie et al., 2020; Hou et al., 2021; Liu et al., 2020). And there exists two alternatives to build views for outdoor-scene LiDAR point clouds. The first one (Liang et al., 2021; Yin et al., 2022) is to apply data augmentation to single frame of point cloud and treat the original and augmented versions as different views, which is similar to what previous works do in image domain. However, the augmentations in image domain are highly non-linear while all the augmentation of point clouds, including random drop, rotation and scaling, can be implemented in a linear transformation. As claimed in (Tian et al., 2020), the highly non-linear augmentations on images already bring high mutual information between views. Thus views of LiDAR point clouds built in this way would have higher mutual information, which is not adequate for learning representations. The second intuitive idea to build views is to utilize point clouds at different timestamps, embraced by (Huang et al., 2021). However, outdoor-scenes are dynamic and the autonomous driving vehicle has no idea about how other objects (cars, pedestrians, etc.) move. Thus observing one view (timestamp  $t$ ) bring in little information about the other one (timestamp  $t + 10$  for example), meaning that  $I(x_1^n; x_2^n)$  can be extremely low and this can be harmful to the learned representations. Due to these limitations, pretrained 3D encoders in (Liang et al., 2021; Huang et al., 2021) cannot achieve noticeable improvement when transferring to datasets collected by different LiDAR sensors. Thus, in this paper, we propose to utilize the vehicle-infrastructure-cooperation dataset (Yu et al., 2022), which capture the same scene from different view-point at the same timestamp, for contrastive representation learning. Views built
+
+with this dataset neither share too much information (captured from different view-points) nor share too little information (captured at the same time, easy to find correspondence), which is adequate for contrastive learning.
+
+# B RECONSTRUCTION OBJECTIVE FOR TASK-RELEVANT INFORMATION
+
+In this section, we borrow the ideas from (Wang et al., 2022) to explain why pure contrastive learning bring less improvements as shown in Table 3 in the main paper. Firstly, we give the definition of sufficient representation and minimal sufficient representation in contrastive learning. Then, we present analysis on image classification problem as downstream task and we refer readers to (Wang et al., 2022) for other downstream tasks. Finally, we propose our pre-training objective for LiDAR point clouds.
+
+Sufficient Representation and Minimal Sufficient Representation. Sufficient Representation  $z_{1,\mathrm{suf}}^n$  of  $x_1^n$  contains all the information that is shared by  $x_1^n$  and  $x_2^n$ , which means  $z_{1,\mathrm{suf}}^n$  can be used to express common semantics shared by these two views. Among all the sufficient representations for  $x_2^n$ , minimal sufficient representation  $z_{1,\min}^n$  contains the least information about  $x_1^n$ . The learned representation in contrastive learning is sufficient and almost minimal. Assuming that  $\mathcal{Z}_{1,\mathrm{suf}}$  is the set of all possible sufficient representations of view  $x_1^n$ , we can define these two concepts as belows,
+
+Definition 1.  $z_{1,suf}^n$  of view  $x_1^n$  is sufficient for  $x_2^n$  if and only if  $I(z_{1,suf}^n, x_2^n) = I(x_1^n, x_2^n)$ .
+
+Definition 2.  $z_{1,\min}^n \in \mathcal{Z}_{1,suf}$  of view  $x_1^n$  is minimal sufficient if and only if  $I(z_{1,\min}^n, x_1^n) \leq I(z_{1,suf}^n, x_1^n), \forall z_{1,suf}^n \in \mathcal{Z}_{1,suf}$ .
+
+Theorem. (1)  $z_{1,suf}$  provides more information about the downstream task  $T$  than  $z_{1,\min}$ . (2) The upper bound of error rates in downstream tasks using minimal sufficient representations are higher than that of sufficient representations. That is,
+
+$$
+\begin{array}{l} I (z _ {1, \text {s u f}}, T) \geq I (z _ {1, \min }, T) \\ \sup  \left\{P _ {\text {s u f}} ^ {e} \right\} \leq \sup  \left\{P _ {\min } ^ {e} \right\} \\ \end{array}
+$$
+
+This gap stems from the missing task-relevant information in  $z_{1,\min}$ . To prevent this problem, authors in (Wang et al., 2022) propose to add a reconstruction objective (reconstruct  $x_{1}$  using  $z_{1}$ ) alongside the contrastive loss to increase  $I(z_{1},x_{1})$ , which indirectly increases  $I(z_{1},T|x_{2})$  and brings improvement in downstream classification problem over pure contrastive learning.
+
+Proof for Image Classification Problem. We denote the downstream classification task as  $T$  and the task-relevant information in minimal sufficient representation  $z_{1,\min}$  can be described as below:
+
+$$
+\begin{array}{l} I \left(z _ {1, \text {s u f}}, T\right) = I \left(z _ {1, \min }, T\right) + \left[ I \left(x _ {1}, T \mid z _ {1, \min }\right) - I \left(x _ {1}, T \mid z _ {1, \text {s u f}}\right) \right] \\ \geq I \left(z _ {1, \min }, T\right) \\ \end{array}
+$$
+
+(10)
+
+To begin with,  $z_{1,\mathrm{suf}}$  and  $z_{1,\mathrm{min}}$  are sufficient representations and they contain two parts of information: shared information between  $x_{1}$  and  $x_{2}$ , and extra information about  $x_{1}$ . Thus the mutual information  $I(z_{1,\mathrm{suf}},T)$  can be decomposed into  $I(z_{1,\mathrm{min}},T)$  and  $[I(x_1,T|z_{1,\mathrm{min}}) - I(x_1,T|z_{1,\mathrm{suf}})]$ , where  $I(x_{1},T|z_{1,\mathrm{min}})$  indicates the information about  $T$  we can obtain by observing  $x_{1}$  when  $z_{1,\mathrm{min}}$  is known. As  $z_{1,\mathrm{suf}}$  contains more information about  $x_{1}$  than  $z_{1,\mathrm{min}}$ , the second term is larger than zero and the right-hand-side of the first line in Eqn. (10) is larger than  $I(z_{1,\mathrm{min}},T)$ . This indicates that  $z_{1,\mathrm{suf}}$  contains more task-relevant information than  $z_{1,\mathrm{min}}$  and thus would have better performance in  $T$ . Then we consider using Bayes error rate  $P^{e}$  (Fukunaga, 2013), which is the lower-bound of achievable error for the classifier, to analyze performance of  $z_{1,\mathrm{suf}}$  and  $z_{1,\mathrm{min}}$  on downstream classification problem. We have
+
+$$
+\begin{array}{l} P _ {\text {s u f}} ^ {e} \leq 1 - \exp \left[ - H (T) + I \left(x _ {1}, x _ {2}, T\right) + I \left(z _ {1, \text {s u f}}, T \mid x _ {2}\right) \right] \\ P _ {\min } ^ {e} \leq 1 - \exp \left[ - H (T) + I \left(x _ {1}, x _ {2}, T\right) \right] \\ \end{array}
+$$
+
+(11)
+
+where  $H(T)$  is the entropy of the task. Since  $1 - \exp[-(H(T) + I(x_1, x_2, T) + I(z_{1,\mathrm{surf}}, T|x_2))] \leq 1 - \exp[-H(T) + I(x_1, x_2, T)]$ , the upper bound of Bayes error rate of minimal sufficient representation is larger than that of sufficient representations. This indicates that ideally  $z_{1,\mathrm{surf}}$  can achieve better performance than  $z_{1,\mathrm{min}}$  in classification problem.
+
+Contextual Shape Prediction Objective. As it is impossible to reconstruct the whole scene point cloud with point-level or voxel-level representations. We propose an additional pre-training objective to predict distribution of local neighborhood using point/voxel-level representation. We use shape context to describe the local neighborhood distribution of a point/voxel, which has been demonstrated as a useful local distribution descriptor in previous works (Hou et al., 2021; Belongie et al., 2002; Kortgen et al., 2003; Xie et al., 2018). As demonstrated in our ablation study (Table 3 in main paper), this additional pre-training objective brings more significant performance improvement over pre-trained by pure contrastive loss. We also provide python-style code for computing shape context as followings
+
+Algorithm 1 Implementation of Contextual Shape Computation in Python Style.  
+```python
+class Contextual_Shape(object):
+    def __init__(self, r1=0.125, r2=2, nbins_xy=2, nbins_zy=2):
+        self.r1 = r1
+        self.r2 = r2
+        self.nbins_xy = nbins_xy
+        self.nbins_zy = nbins_zy
+        self.partitionions = nbins_xy * nbins_zy * 2
+def pdist_batch(rel_trans):
+    D2 = torch.sum(rel_trans.pow(2), 3)
+    return torch.sqrt(D2 + 1e-7)
+def compute_rel_trans_batch(A, B):
+    return A unsqueeze(1) - B unsqueeze(2)
+def hash_batch(A, B, seed):
+    mask = (A >= 0) & (B >= 0)
+    C = torch.zeros_like(A) - 1
+    C[mask] = A[mask] * seed + B[mask]
+    return C
+def computeAngles_batch(rel_trans):
+    angles_xy = torch atan2(rel_trans[., :, :, 1], rel_trans[., :, :, 0])
+    angles_xy = torch.fmod(angles_xy + 2 * math.pi, 2 * math.pi)
+    angles_zy = torch atan2(rel_trans[., :, :, 1], rel_trans[., :, :, 2])
+    angles_zy = torch.fmod(angles_zy + 2 * math.pi, math.pi)
+    return angles_xy, angles_zy
+def compute_partitions_batch(self, xyz_batch):
+    rel_trans_batch = ShapeContext.compute_rel_trans_batch(xyz_batch,
+                     xyz_batch)
+    # compute angles from different points to the query one
+    angles_xy_batch, angles_zy_batch = ShapeContext.computeAngles_batch(
+        rel_trans_batch)
+    angles_xy_bins_batch = torch.floor(angles_xy_batch / (2 * math.pi /
+            self.nbins_xy))
+    angles_zy_batch = torch.floor(angles_zy_batch / (math.pi / self.
+            nbins_zy))
+    angles_bins_batch = ShapeContext.hash_batch(angles_xy_bins_batch,
+                        angles_zybins_batch, self.nbins_zy)
+    # compute distances between different points and the query one
+    distance_matrix_batch = ShapeContext.pdist_batch(rel_trans_batch)
+    dist_bins_batch = torch.zeros_like(angles_bins_batch) - 1
+    # generate partitions for each points
+    mask_batch = (distance_matrix_batch >= self.r1) & (
+        distance_matrix_batch < self.r2)
+    dist_bins_batch[mask_batch] = 0
+    mask_batch = distance_matrix_batch >= self.r2
+    dist_bins_batch[mask_batch] = 1
+    bins_batch = ShapeContext.hash_batch(dist_bins_batch,
+                     angles_bins_batch, self.nbins_xy * self.nbins_zy)
+    return bins_batch
+```
+
+# C DATASETS DETAILS
+
+In this section, we introduce details about different datasets used in the main paper for evaluation and also two more datasets in additional experiments.
+
+DAIR-V2X. DAIR-V2X (Yu et al., 2022) is the first real-world autonomous dataset for vehicleinfrastructure-cooperative detection task. It covers various scenes, including cities and highways, and different whether condition including sunny, rainy and foggy days. A virtual world coordinate is used to align the vehicle LiDAR coordinate and infrastructure LiDAR coordinate. There are 38845 LiDAR frames (10084 in vehicle-side and 22325 in infrastructure-side) for cooperative-detection task. The dataset contains around 7000 synchronized cooperative samples in total and we utilize them to pre-train 3D encoder in an unsupervised manner via the proposed CO3. The LiDAR sensor at vehicle-side is 40-beam while a 120-beam LiDAR is utilized at infrastructure-side.
+
+Once. Once (Mao et al., 2021) is a large-scale autonomous dataset for evaluating self-supervised methods with 1 Million LiDAR frames and only 15k fully annotated frames with 3 classes (Vehicle, Pedestrian, Cyclist). A 40-beam LiDAR is used in (Mao et al., 2021) to collect the point cloud data. We adopt common practice, including point cloud range and voxel size, in their public code repository<sup>3</sup>. As for the evaluation metrics, IoU thresholds 0.7, 0.3, 0.5 are respectively adopted for vehicle, pedestrian, cyclist. Then 50 score thresholds with the recall rates ranging from 0.02 to 1.00 (step size if 0.02) are computed and the 50 corresponding values are used to draw a PR curve, resulting in the final mAPs (mean accurate precisions) for each category. We also further average over the three categories and compute an 'Overall' mAP for evaluations.
+
+KITTI. KITTI (Geiger et al., 2012) is a widely used self-driving dataset, where point clouds are collected by LiDAR with 64 beams. It contains around 7k samples for training and another 7k for evaluation. For point cloud range and voxel size, we adopt common practice in current popular codebase like MMDetection3D $^4$  and OpenPCDet $^5$ . All the results are evaluated by mAPs with three difficulty levels: Easy, Moderate and Hard. These three results are further average and an 'Overall' mAP is generated for comparisons.
+
+# D ADDITIONAL EXPERIMENT RESULTS
+
+# D.1 INFRASTRUCTURE AS VIEW IN CONTRASTIVE LEARNING
+
+We also conduct ablation study on the fusion view. Instead of using fusion point clouds as view in contrastive learning, we directly use infrastructure side point cloud as another view. Results are shown in Table 7. It can be found that if we directly use infrastructure-side point cloud for contrastive learning, the performance improvement is very marginal. This stems from the sparsity of LiDAR point clouds, which sometimes make it difficult to find good positive pairs to perform contrastive learning.
+
+<table><tr><td>Init.</td><td>Overall</td><td>Vehicle</td><td>Pedestrian</td><td>Cyclist</td></tr><tr><td>Random</td><td>55.92</td><td>62.85</td><td>45.52</td><td>59.39</td></tr><tr><td>Inf-view</td><td>56.45</td><td>62.71</td><td>46.63</td><td>60.02</td></tr><tr><td>CO3</td><td>58.50</td><td>64.60</td><td>48.83</td><td>62.17</td></tr></table>
+
+Table 7: Results of directly using infrastructure-side point cloud as view in contrastive learning.
+
+# E IMPLEMENTATION DETAILS
+
+In this section, we introduce some details about implementation in both pre-training stage and fine-tuning stage. Common settings of pre-training and fine-tuning are listed in Table 8 and we discuss other settings that vary in different detectors later.
+
+<table><tr><td>Configuration</td><td>Pre-training</td><td>KITTI</td><td>Once</td></tr><tr><td>optimizer</td><td>AdamW</td><td>Adam</td><td>Adam</td></tr><tr><td>base learning rate</td><td>0.0001</td><td>0.003</td><td>0.003</td></tr><tr><td>weight decay</td><td>0.01</td><td>0.01</td><td>0.01</td></tr><tr><td>batch size</td><td>16</td><td>-</td><td>-</td></tr><tr><td>learning rate schedule</td><td>cyclic</td><td>cyclic</td><td>cyclic</td></tr><tr><td>GPU numbers</td><td>8</td><td>4</td><td>4</td></tr><tr><td>training epochs</td><td>20</td><td>80</td><td>80</td></tr></table>
+
+Table 8: Details about implementations. "Pre-training" means settings in  $DAIR-V2X$  (Yu et al., 2022). "KITTI" and "Once" respectively indicate settings for detection tasks in  $KITTI$  (Geiger et al., 2012) and  $Once$  (Mao et al., 2021). We list all common settings and discuss those vary in different detectors below, which are marked as “-” in this table.
+
+Other Pre-training Settings. To accelerate the pre-training process, we utilize the "repeated dataset" in MMDetection3D and the schedule is set to 10 epochs, which equals to 20 epochs without "repeated dataset". Thus the number 20 for training epochs in Table 8 is tilt.
+
+Other Fine-tuning Settings. Batch size settings for different detectors on different datasets are shown in Table 9. We use different types of GPUs, different number of GPUs and different version of PyTorch (Paszke et al., 2019) as compared to those used in the codebases, which may lead to degrading when training from scratch. Thus these parameters are tuned based on the original settings from the codebases to make the performance of training from scratch match or even surpass the results they published.
+
+<table><tr><td>Detectors</td><td>KITTI</td><td>Once</td></tr><tr><td>Second</td><td>48</td><td>48</td></tr><tr><td>PV-RCNN</td><td>16</td><td>48</td></tr><tr><td>CenterPoint</td><td>-</td><td>32</td></tr></table>
+
+Table 9: Details about batchsize settings for different detectors on different datasets. “-” means there is no configuration for the detector on the exact dataset in the codebase or we do not conduct the downstream experiments.
+
+F TABLE OF NOTATION  
+
+<table><tr><td>Variable</td><td>Description</td><td>Note</td></tr><tr><td>Pv/f</td><td>Raw vehicle/fusion LiDAR points.</td><td></td></tr><tr><td>Pv/f</td><td>Point/Voxel-level features after embedded by the 3D backbone.</td><td>Indicating same process applied to vehicle/fusion point cloud.</td></tr><tr><td>Zv/f</td><td>Projected point/voxel-level features in the common feature space.</td><td></td></tr><tr><td>Zv/f</td><td>Normalized point/voxel-level features in the common feature space.</td><td>An L-2 normalization is applied on the feature dimension of Zv/f.zn/vfis sampled point/voxel feature for contrastive learning.</td></tr><tr><td>N1</td><td>The number of samples for contrastive learning.</td><td></td></tr><tr><td>τ</td><td>Temperature parameter in contrastive learning.</td><td></td></tr><tr><td>Q&#x27;</td><td>A matrix in dimension RNf×Nbin indicating the number of points in each bin in the neighborhood of each fusion point.</td><td></td></tr><tr><td>Q</td><td>Apply ℓ2-normalization and softmax to Q&#x27; then we get Q ∈ RNf×Nbin, which describes a local geometry distribution in the neighborhood of each fusion point.</td><td>qn/vfis sampled “ground truth” distribution from Q</td></tr><tr><td>P</td><td>Predicted local geometry distribution from Penc v/f.</td><td>pn/vfis sampled distribution prediction from P.</td></tr><tr><td>N2</td><td>The number of samples for contextual shape prediction.</td><td></td></tr></table>
+
+Table 10: Detailed description of variables
+
+# G DISCUSSION ABOUT THE ABLATION STUDY
+
+![](images/f86c08614c3de9be67216bd7cc32d993c53670c89faa0475d40b7b7458a913cc.jpg)
+
+![](images/74a29a241cdfbf34cbb7a1c43ceb246003cfbd272abcb951f391cbbb630d0f6f.jpg)  
+(a) Different Cases of Pedestrian Class
+
+![](images/2b485c1b5bceb36c11c5b63cc04d8aedb916a8259a9ea5ebcaaa641f644cab08.jpg)
+
+![](images/95dcd8c6247784ab9594ca8abdb692742bdf1d040fe8cad4f1bf350281fe7bc0.jpg)
+
+![](images/a7d482ad234f7d46338d14a439f0915e80fb4232d1243a7329f41545ca3f2f2d.jpg)
+
+![](images/8a2cb38339e766eb44945cfb802a64f2a7a6ab9db5ee36d86e2f7de3c9dc085c.jpg)
+
+![](images/543eb2a8b9f0cfae80cf46bea02ebc45641f420773acabbbb5387fa29a84fccd.jpg)  
+Figure 6: Example samples of pedestrian and cyclist. LiDAR points are shown in red. It can be found that pedestrian category has consistent shape while the shape of cyclist varies across time and identities.
+
+![](images/ce4cb5048ee71c98eb910b48c106c69cfab6b6b789503d6c6681da034c950da9.jpg)  
+(b) Different Cases of Cyclist Class
+
+![](images/5f8cfa1df152e5dc41e5da3946c07e29bc233ae270023dc40474acb28ef9a74b.jpg)
+
+![](images/e3567fa6558795c02cf215b5caa52851d841818f041d5b969703fcf9ac401d82.jpg)
+
+![](images/b5d6f1c76ad4673dc4827dc5e4e0de8485bece30c5859eb2b76d07db850f42e7.jpg)
+
+It can be found in Table 6 that using contextual shape prediction loss only brings more improvement on Pedestrian class while cooperative contrastive loss introduce more gains on the Cyclist class. In this section, we provide a discussion on this phenomenon.
+
+First of all, the "ground truth" shape context is computed with point clouds and the contextual shape prediction goal is to predict the local point distribution with voxel-level representations. This
+
+enables the voxel-level representation to predict the structure inside a voxel. Meanwhile, cooperative contrastive only focuses on contrast different voxel representation. For pedestrian class, there are usually only one or two voxels for one pedestrian. Thus, with cooperative contrastive loss only, the representation fail to recognize the inner structure of the voxel and thus cannot improve upon the pedestrian category. On the contrary, when we pre-train the encoder with contextual shape prediction loss only, the learned representation is able to express the structure inside the voxel and this helps with the downstream detection on the pedestrian category.
+
+Second, as the contextual shape prediction goal aims to capture local shape distribution, it fails to learn good representation with varying shape. Compared to pedestrians whose shape is always cylinder-like, cyclists with their bicycles are usually captured with different poses, leading to different shapes. A visualization of different cases of the two categories are provided in Fig. 6. Thus the representation fails to learn knowledge when predicting varying shape of the same semantic. When we look at the cyclist category, it can be found that pre-training with contextual shape loss only brings little improvement.
+
+Besides, as discussed in Appendix B, the two pre-training objectives are complementary. Pure cooperative contrastive learning makes the representation minimal sufficient, which lacks of task-relevant information. The contextual shape prediction loss brings more task-relevant information by increasing the mutual information between the representations and the inputs. Thus combining them leads to better performance.
+
+# H MORE EXPERIMENTS.
+
+# H.1 ABLATION EXPERIMENTS ON SHAPE CONTEXT.
+
+We conduct a ablation experiment on the contextual shape prediction loss. Previous works (Belongie et al., 2002; Kortgen et al., 2003) use the number of points in each bin as the feature of the query point. However, we think it is difficult for the network to directly regress the exact number of points in the bins. Thus we propose to construction a local distribution and use the representation to predict it. In this part, we conduct ablation study on this design. We use the exact number of points as prediction goal and keep other parts of our method the same. The experiment results are shown in Table 11 and it can be found that directly transfer the idea of shape context only brings little improvement (0.94mAP) as compared to the proposed CO3 (2.58 mAP). Thus our insight can also be transferred to indoor scene point clouds.
+
+<table><tr><td>Initialization</td><td>Overall mAP</td><td>Vehicle</td><td>Pedestrian</td><td>Cyclist</td></tr><tr><td>Random</td><td>55.92</td><td>62.85</td><td>45.52</td><td>59.39</td></tr><tr><td>CO3 with exact number of points prediction</td><td>56.86 (+0.94)</td><td>62.79</td><td>47.63</td><td>60.15</td></tr><tr><td>CO3</td><td>58.50 (+2.58)</td><td>64.60</td><td>48.83</td><td>62.17</td></tr></table>
+
+Table 11: Ablation experiments on shape context.
+
+# H.2 EXPERIMENTS COMPARED TO SAFETY SPACE FORECASTING.
+
+We also conduct experiments where freespace forecasting (Hu et al., 2021) is used as pre-training goal. The pre-training setting is kept the same as all the other initialization methods. The results are shown in Table 12. It can be found that pre-training with safety space forecasting (Hu et al., 2021) brings minor performance gain. We think this might stems from the objective of the pre-training goal. The pre-training goal of (Hu et al., 2021) is to predict freespace for safe driving and their downstream task is motion planning. However, 3D perception task requires semantic information and simply predicting freespace does not help to distinguish the representation of different objects. This is why pre-training with freespace forecasting bring lower performance improvement on 3D perception task.
+
+# H.3 PARAMETER SENSITIVITY EXPERIMENTS
+
+In this part, we conduct parameter sensitivity experiments on temperature parameter and radius in contextual shape prediction loss.
+
+<table><tr><td>Initialization</td><td>Overall mAP</td><td>Vehicle</td><td>Pedestrian</td><td>Cyclist</td></tr><tr><td>From Scratch</td><td>55.92</td><td>62.85</td><td>45.52</td><td>59.39</td></tr><tr><td>Freespace forecasting</td><td>56.18 (+0.26)</td><td>63.33</td><td>46.19</td><td>59.01</td></tr><tr><td>CO3</td><td>58.50 (+2.58)</td><td>64.60</td><td>48.83</td><td>62.17</td></tr></table>
+
+We first fix all the other parameters in the main paper and change  $R1$  and  $R2$  in the contextual shape prediction loss. We pre-train the 3D encoder on the DAIR-V2X dataset and downstream it to Once dataset with CenterPoint as detector. Results are shown in Table 13. It can be found that for the same R2, an increasing R1 brings more improvement on the downstream detection tasks. This might stem from that a larger inner radius can help capture more neighborhood information, making the contextual shape prediction goal more meaningful. It can also be found that when  $R2$  is relatively small, for example  $R2 = 3.5\mathrm{m}$  and  $R1 = 1\mathrm{m}$ , the performance drops. This might stem from the relatively small R2 which might fail to capture the local shape distribution of a larger object like cars or trucks. Also, we did not search these parameters before and we surprisingly find that using  $R1 = 1.5\mathrm{m}$  and  $R2 = 4\mathrm{m}$  brings the best performance (58.86 mAP).
+
+Table 12: Experiment results on freespace forecasting pre-training.  
+
+<table><tr><td>R1/m
+R2/m</td><td>0.5</td><td>1.0</td><td>1.5</td><td>2.0</td></tr><tr><td>3.5</td><td>57.55</td><td>57.31</td><td>58.00</td><td>58.42</td></tr><tr><td>4.0</td><td>57.53</td><td>58.50</td><td>58.86</td><td>58.48</td></tr></table>
+
+In the experiment on temperature, all other parameters are kept the same as those in the main experiment and we only change the temperature in the cooperative contrastive loss. We pre-train the 3D backbone on DAIR-V2X dataset with different parameter settings and fine-tune it on Once dataset with CenterPoint. The results are shown in Table 14. It can be found that as the temperature increases to 0.15, the downstream performance drop to  $57.34\mathrm{mAP}$ . This is because a higher temperature brings smaller gap between negative pairs and this makes the representations hard to separate different objects. Also, as we keep decreasing the temperature, the overall downstream performance drop to  $57.74\mathrm{mAP}$ . This is because too small temperature push representation of different objects too far away but ignore the similar semantic meaning for the same category. For example, it will make representation of two different cars far away and this will harm the performance of downstream detection task. It can also be found that using adequate temperature brings comparable performance  $(58.50\mathrm{mAP}$  vs  $58.10\mathrm{mAP})$
+
+Table 13: Results on parameter sensitivity experiments on temperature.  
+
+<table><tr><td>Temperature</td><td>0.02</td><td>0.07</td><td>0.1</td><td>0.15</td></tr><tr><td>Overall mAP</td><td>57.74</td><td>58.50</td><td>58.10</td><td>57.34</td></tr></table>
+
+Table 14: Results on parameter sensitivity experiments on radius in contextual shape prediction loss.
+
+# I DISCUSSION ON THE INFLUENCE TO V2X COMMUNITY.
+
+The V2X community is developing rapidly and the original motivation of V2X setting is to alleviate occlusion and long-range sensing problems (Yu et al., 2022). There are several V2X settings including vehicle-to-infrastructure and vehicle-to-vehicle.
+
+In this paper, we propose to use cooperation dataset for 3D unsupervised representation learning and achieve performance improvement on perception task using vehicle-side point clouds only. Although our experiments are conducted on vehicle-infrastructure dataset, our method can also be applied on other V2X settings without any label. As labeling is the most intensive and time-consuming part in the collection of cooperation dataset, we believe larger scale of unlabeled cooperation dataset will be collected in the future for unsupervised 3D representation learning to introduce more performance gain. Also, it is expensive and difficult to deploy V2X settings everywhere. In our work, we pretrain on such cooperation dataset and achieve improvements on downstream tasks with vehicle-side
+
+point clouds as inputs, which is a new exciting finding about the V2X research. We believe the promising results will encourage more attempts in cooperative unsupervised representation learning and accelerate the development of the V2X community.

2023/CO3_ Cooperative Unsupervised 3D Representation Learning for Autonomous Driving/images.zip

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