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+# TO LEARN EFFECTIVE FEATURES: UNDERSTANDING THE TASK-SPECIFIC ADAPTATION OF MAML
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Meta learning, an effective way for learning unseen tasks with few samples, is an important research area in machine learning. Model Agnostic MetaLearning (MAML) (Finn et al. (2017)) is one of the most well-known gradientbased meta learning algorithms, that learns the meta-initialization through the inner and outer optimization loop. The inner loop is to perform fast adaptation in several gradient update steps with the support datapoints, while the outer loop to generalize the updated model to the query datapoints. Recently, it has been argued that instead of rapid learning and adaptation, the learned meta-initialization through MAML has already absorbed the high-quality features prior, where the task-specific head at training facilitates the feature learning. In this work, we investigate the impact of the task-specific adaptation of MAML and discuss the general formula for other gradient-based and metric-based meta-learning approaches. From our analysis, we further devise the Random Decision Planes (RDP) algorithm to find a suitable linear classifier without any gradient descent step and the Meta Contrastive Learning (MCL) algorithm to exploit the inter-samples relationship instead of the expensive inner-loop adaptation. We conduct sufficient experiments on various datasets to explore our proposed algorithms.
+
+# 1 INTRODUCTION
+
+Few-shot learning, aiming to learn from few labelled examples, is a great challenge for modern machine learning systems. Meta learning, an effective way for tracking this challenge, enables the model to learn general knowledge across a distribution of tasks. Various ideas of meta learning have been proposed to address the few-shot problems. Gradient-based meta learning (Finn et al. (2017); Nichol et al. (2018)) learns the meta-parameters that can be quickly adapted to new tasks by few gradient descent steps. Metric-based meta learning (Koch et al. (2015); Vinyals et al. (2016); Snell et al. (2017)) proposes to learn a metric space by comparing different datapoints. Memorybased meta learning (Santoro et al. (2016)) can rapidly assimilate new data and leverage the stored information to make predictions.
+
+Model Agnostic Meta-Learning (MAML) (Finn et al. (2017)) is one of the most well-known gradient-based meta learning algorithms, that learns the meta-initialization parameters through the inner optimization loop and the outer optimization loop. For a given task, the inner loop is to perform fast adaptation in several gradient descent steps with the support datapoints, while the outer loop to generalize the updated model to the query datapoints. With the learned meta-initialization, the model can be quickly adapted to the unseen tasks with few labelled samples. Following the MAML algorithm, many significant variants (Finn et al. (2018); Rusu et al. (2018); Oreshkin et al. (2018); Bertinetto et al. (2018); Lee et al. (2019b)) are studied under the few-shot setting.
+
+To understand how the MAML works, Raghu et al. (2019) conduct a series of experiments and claim that rather than rapid learning and adaptation, the learned meta-initialization has already absorbed the high-quality features prior, thus the representations after fine-tuning are almost the same for the coming unseen tasks. Also, the task specific head of MAML at training facilitates the learning of better features. In this paper, we further design more representative experiments and present a formal argument to explain the importance of the task specific adaptation. Actually, the multi-step taskspecific adaptation, making the body and head have similar classification capabilities, can provide better gradient descent direction for the features learning of body. We also notice that for both the gradient-based methods (e.g. MAML (Finn et al. (2017)), MetaOptNet (Lee et al. (2019b))) and metric-based methods (e.g. Prototypical Networks (Snell et al. (2017))) that attempt to learn a taskspecific head using the support datapoints, the adaptation is a common mode for features learning of body but varied in different methods.
+
+Based on our analysis, we first propose a new training paradigm to find a decision plane (linear classifier) for guidance with no gradient descent step during the inner loop and get more supporting conclusions. Moreover, we devise another training paradigm that removes the inner loop and trains the model with only the query datapoints. Specifically, inspired by contrastive representation learning (Oord et al. (2018); Chen et al. (2020); He et al. (2020)), we exploit the inter-samples relationship of query set to find a guidance for the body across different tasks. This meta contrastive learning algorithm even achieves competitive results comparable to some state-of-the-art methods. In total, our contributions can be listed as follows:
+
+1. We present sufficient experiments and formal argument to explore the impact of the taskspecific adaptation for body features learning and discuss the general formula for other gradient-based and metric-based meta-learning approaches.   
+2. We devise a training algorithm to obtain a decision plane with no gradient descent step during the inner loop, named as Random Decision Planes (RDP), and get more supporting conclusions.   
+3. Unlike prior gradient-based methods, we propose the Meta Contrastive Learning (MCL) algorithm to exploit the inter-samples relations instead of training a task-specific head during the inner loop. Even without the task-specific adaptation for guidance, our algorithm still achieve better results with even less computation costs.   
+4. We empirically shows the effectiveness of the proposed algorithm with different backbones on four benchmark datasets: miniImageNet (Vinyals et al. (2016)), tieredImageNet (Ren et al. (2018)), CIFAR-FS (Bertinetto et al. (2018)) and FC100 (Oreshkin et al. (2018)).
+
+# 2 RELATED WORKS
+
+MAML (Finn et al. (2017)) is a highly influential gradient-based meta learning algorithm for fewshot learning. The amazing experiment results on several public few-shot datasets have proved its effectiveness. Following the core idea of MAML, there are numerous works to handle the data insufficiency problem in few-shot learning. Some works (Oreshkin et al. (2018); Vuorio et al. (2019)) introduce the task-dependent representations via conditioning the feature extractor on the specific task to improve the performance. Sun et al. (2019) also employ the meta-learned scaling and shifting parameters for transferring from another large-scale dataset. Others (Grant et al. (2018); Finn et al. (2018); Lee et al. (2019a)) study this problem from the perspective of Bayesian approach. Unlike prior methods, we provide two training paradigms, one with no gradient descent step during the inner loop and another removing the inner loop and exploiting the inter-sample relations for training.
+
+Recent works also explore the key factors that makes the meta-learned model perform better than others at few-shot tasks. Chen et al. (2019) discovers that a deeper backbone has a large effect on the success of meta learning algorithm, while Goldblum et al. (2020) finds that the meta learning tends to cluster object classes more tightly in feature space for those methods that fix the backbone during the inner loop (Bertinetto et al. (2018); Rusu et al. (2018)). A very recent work (Raghu et al. (2019)) argues that the meta-trained model can be applied to new task due to the high-quality features prior learned by the meta-initialized parameters rather than rapid learning. In this paper, we further study the impact of the task-specific adaptation for feature learning. Based on the analysis, we devise two algorithms, Random Decision Planes (RDP) and Meta Contrastive Learning (MCL) requiring less computation cost but still with competitive performance.
+
+# 3 MODEL-AGNOSTIC META LEARNING (MAML)
+
+The MAML aims to learn the meta-initialized parameters $\theta$ for the coming unseen tasks through the inner optimization loop and the outer optimization loop. Under the $N$ -way- $K$ -shot setting, for a task $T _ { b }$ sampled from the task distribution $P ( T )$ , we have a support set of $N \times K$ examples $T _ { b } ^ { s }$ and a query set $T _ { b } ^ { q }$ , where $N$ is the number of sampled class and $K$ is the number of instances for each class. During the inner loop, with the support set $T _ { b } ^ { s }$ , we perform fast adaptation in several gradient descent steps and obtain the task-specific parameters $\theta _ { T _ { b } } ^ { t }$ where $t$ is the number of gradient descent steps, given by:
+
+Table 1: The evaluation results of 5-way-K-shot learning for methods with different training regimes on the MiniImageNet and TieredImageNet datasets.   
+
+<table><tr><td>Method</td><td>MiniImageNet-5-way</td><td>TieredImageNet-5-way</td></tr><tr><td>Multi-Head(1)</td><td>38.66 ± 0.34</td><td>31.78 ± 0.37</td></tr><tr><td>Multi-Task(1)</td><td>40.14 ± 0.38</td><td>33.62 ± 0.38</td></tr><tr><td>MAML (2017)(1)</td><td>49.31 ± 0.40</td><td>52.45 ± 0.48</td></tr><tr><td>ANIL (Almost No Inner Loop)(1)</td><td>50.23 ± 0.42</td><td>52.69 ± 0.47</td></tr><tr><td>BOHI (Body Outer loop, Head Inner Loop)(1)</td><td>50.61 ± 0.43</td><td>53.60 ± 0.48</td></tr><tr><td>Multi-Head(5)</td><td>48.99 ± 0.33</td><td>41.48 ± 0.38</td></tr><tr><td>Multi-Task(5)</td><td>50.82 ± 0.35</td><td>44.94 ± 0.39</td></tr><tr><td>MAML (2017)(5)</td><td>64.77 ± 0.36</td><td>67.66 ± 0.42</td></tr><tr><td>ANIL (Almost No Inner Loop)(5)</td><td>65.98 ± 0.38</td><td>67.44 ± 0.43</td></tr><tr><td>BOHI (Body Outer loop,Head Inner Loop)(5)</td><td>66.14 ± 0.37</td><td>68.39 ± 0.42</td></tr></table>
+
+$$
+\theta _ { T _ { b } } ^ { t } = \theta _ { T _ { b } } ^ { t - 1 } - \alpha \nabla _ { \theta _ { T _ { b } } ^ { t - 1 } } \mathcal { L } _ { T _ { b } ^ { s } } ( \theta _ { T _ { b } } ^ { t - 1 } )
+$$
+
+where $\alpha$ is the step size for inner loop and $\mathcal { L } _ { T _ { b } ^ { s } } ( \theta _ { T _ { b } } ^ { t - 1 } )$ denoted as the loss on the support set $T _ { b } ^ { s }$ after $t - 1$ steps. With the query set $T _ { b } ^ { q }$ b , we compute the meta loss on the task-specific parameters $\theta _ { T _ { b } } ^ { t }$ and backward to update the meta-initialized parameters $\theta$ , given by
+
+$$
+\theta = \theta - \beta \nabla _ { \theta } \frac { 1 } { B } \sum _ { b = 1 } ^ { B } \mathcal { L } _ { T _ { b } ^ { q } } ( \theta _ { T _ { b } } ^ { t } )
+$$
+
+where $\beta$ is the learning rate and $B$ is the number of sampled tasks in a batch.
+
+# 4 IMPACT OF TASK-SPECIFIC ADAPTATION
+
+# 4.1 THE MULTI-STEP TASK-SPECIFIC ADAPTATION IS IMPORTANT.
+
+To explore the effectiveness of MAML, Raghu et al. (2019) have conducted sufficient experiments, indicating that the network body (the representation layers) has already absorbed the high-quality features prior. During meta-testing, instead of fine tuning on the network head (the classifier), simply building the prototypes with the support set can achieve comparable performance to MAML. Raghu et al. (2019) also shows that the task specificity of head at training can facilitate feature learning and ensure good representation learning in the network body. In our work, we show that besides the task specificity of head, the multi-step adaptation is also essential, and further study the role of network body and head during meta-training. We devise several methods using different training regimes: (1) Multi-Task, where all the tasks simply share one common head and the model is trained in a traditional way without inner loop adaptation; (2) Multi-Head, where different tasks are equipped with different heads for task specificity and the model is trained in a traditional way without inner loop adaptation; (3) Almost No Inner Loop (ANIL), where the network body is fixed during the inner loop; (4) Body Outer Loop, Head Inner Loop (BOHI), where the network body is updated only by the outer loop and the head is adapted only during the inner loop, making the head’s meta-initialized parameters unchanged. More algorithms’ details can be found in Appendix B, and implementation details can be found in Appendix C.1.
+
+Following Raghu et al. (2019), we employ the cosine similarities between prototypes and the query datapoints to evaluate the quality of features learned. As Table 1 shows, even equipped with taskspecific head, the Multi-Head training still performs worse than the standard MAML algorithm by a large margin, indicating the multi-step adaptation of MAML is helpful for features learning. The results of Multi-Head and Multi-Task show the importance of multi-step task-specific adaptation.
+
+![](images/323b27e20fba56523cde9743ed75492f6ca7f5fff7291b139b0b1cd66eb3f61e.jpg)  
+Figure 1: The adaptation of the random initialized model for the sampled tasks in different steps.
+
+![](images/a40e1540dcd28c9760c7e7608a38024c2c13f4c874dc9ed191732136043309bf.jpg)  
+Figure 2: The adaptation after 5,000 iterations for the sampled tasks in different steps .
+
+As the results shown in Table 1, the ANIL training remains effective comparable to the standard MAML algorithm, indicating that the task-specific adaptation of network body is unnecessary to learn good features. More interestingly, the BOHI training that keeps the meta-initialization of head unchanged even performs better than MAML, further demonstrating that good features learning depends on the multi-step task-specific adaptation of head during inner loop more than updating the meta-initialization of head in outer loop. Also, the ANIL and BOHI have similar performance, indicating that compared with learned prior knowledge in head, the inner loop adaptation, as a guidance, contributes more to the features learning. More experimental results can be found in Appendix C.2.
+
+# .2 WHY IS MULTI-STEP TASK-SPECIFIC ADAPTATION IMPORTANT?
+
+Having observed that the MAML algorithm outperforms the Multi-Task training by a large margin and the multi-step task-specific adaptation is important for features learning, we extend our analysis to explore the reason why the inner loop adaptation is essential for MAML at different stages of meta training. Specifically, we freeze the initialized MAML model and model at 5,000 iterations, sample validation tasks from the task distribution, and record the test accuracy of model in different inner loop steps. Both the body accuracy based on prototypes construction and head accuracy based on fine-tuning are given in Figure 1 and Figure 2, where “Task ID” stands for different tasks. As the results shows, at different stages of meta training, the head accuracy increases significantly in the first few adaptation steps since the model has learnt the correspondence between sample and label. However, at the beginning of training, there is only a small improvement on the body accuracy after first adaptation step. In Figure 2, as the model converges, the body accuracy even decreases in the first few adaptation steps. In the following steps, with the task-specific adaptation of head, the network body then learns better representations, further demonstrating that the multistep task-specific adaptation, making the body and head have similar classification capabilities, can be regarded as a guidance to provide better gradient descent direction for the feature learning of body.
+
+Algorithm 1 The Random Decision Planes (RDP) Algorithm for N-way-K-shot learning   
+
+<table><tr><td>Algorithm1 The Random Decision Planes (RDP) Algorithm for N-way-K-shot learning Input: Network Body fe,Learning Rate β, Task Distribution P(T) while not done do Sample a batch of tasks {Tb}b=1, where Tb ~ P(T) forb∈{1,..,B} do</td></tr><tr><td>for each sample x in {TTdo</td></tr><tr><td>z = |lfe(x)ll end for define CrossEntropyLoss(H,D) as the cross entropy loss</td></tr><tr><td>on the features representations set D with head H. W*= argmin CrossEntropyLoss(W,{(z,)))</td></tr><tr><td>WEP Lb = CrossEntropyLoss(W*,{(z&#x27;,y)}K)</td></tr><tr><td>end for 0=θ-βVθB∑b=1Lb B end while</td></tr></table>
+
+To understand this intuitive argument better, we consider a sample $( { \pmb x } , y )$ for few-shot classification where the cross entropy loss is employed, formulated as:
+
+$$
+\mathcal { L } _ { c } = - \mathrm { l o g } ( \frac { \mathrm { e x p } ( w _ { y } ^ { \top } h ) } { \sum _ { k } \mathrm { e x p } ( w _ { k } ^ { \top } h ) } ) = - w _ { y } ^ { \top } h + \mathrm { l o g } ( \sum _ { k } \mathrm { e x p } ( w _ { k } ^ { \top } h ) )
+$$
+
+where $\{ \pmb { w } _ { 1 } , \pmb { w } _ { 2 } , . . . , \pmb { w } _ { k } \}$ is the weights of the classifier head, $^ { h }$ is the body representation of $_ { \textbf { \em x } }$ . The gradients of loss $\mathcal { L } _ { c }$ with respect to the body representation $^ { h }$ are denoted by,
+
+$$
+\frac { \partial \mathcal { L } _ { c } } { \partial \pmb { h } } = - \pmb { w } _ { y } + \frac { \sum _ { k } \pmb { w } _ { k } \mathrm { e x p } ( \pmb { w } _ { k } ^ { \top } \pmb { h } ) } { \sum _ { k } \mathrm { e x p } ( \pmb { w } _ { k } ^ { \top } \pmb { h } ) } = - \pmb { w } _ { y } + \bar { \pmb { w } }
+$$
+
+where $\bar { \pmb w }$ is exactly the weighted average of the weights $\{ \pmb { w } _ { 1 } , \pmb { w } _ { 2 } , . . . , \pmb { w } _ { k } \}$ . As shown in Equation 4, a reasonable direction for the network body to minimize the target loss $\mathcal { L } _ { c }$ is to make the representation $^ { h }$ closer to the corresponding class weight ${ \pmb w } _ { y }$ , given by $\pmb { h } = \pmb { h } + \lambda ( \pmb { w } _ { y } - \pmb { \bar { w } } )$ . As the model converges, in the first few adaptation steps, there is a significant margin between the performance of head and body, and the classifier weights contain little knowledge about correspondence between samples and labels and differences between different classes. With the low-performance head, this updating rule for body may lead to a decline in the quality of features, which also explains why the simpler BOHI, ANIL even performs better than MAML in Table 1. After several adaptation steps during the inner loop, the body then receives the useful guidance for features learning from the taskspecific head since ${ \pmb w } _ { y }$ can better express its corresponding class. The formulation above shows that the multi-step task-specific adaptation, making the body and head have similar classification capabilities, can provide better gradient descent direction for the features learning of body.
+
+# 4.3 TASK-SPECIFIC ADAPTATION IN OTHER META-LEARNING ALGORITHMS
+
+Having noticed that the multi-step task-specific adaptation of MAML, which promotes the performance of head, can facilitate the features learning of body. It works similarly for other gradientbased methods that use end-to-end fine-tuning, such as Reptile (Nichol et al. (2018)). In the case of meta-learning methods that fix the network body and only update the head during the inner loop, such as MetaOptNet (Lee et al. (2019b)) and R2-D2 (Bertinetto et al. (2018)), the convex optimization of head also aims to provide a classifier with better classification capabilities. For metric-based methods, such as Prototypical Networks (Snell et al. (2017)), the adaptation of head is actually conducted through the nearest neighbor algorithm. In conclusion, the adaptation is a common mode but varied in different methods. These meta-learning algorithms reveal a general formula that the inner loop is for building a task-specific head that matches the classification capabilities of body and the outer loop for task-independent features learning.
+
+Table 2: The evaluation results of 5-way-K-shot learning for the standard MAML and Random Decision Planes (RDP) with different backbones.   
+
+<table><tr><td>Method</td><td>Backbone</td><td>MiniImageNet-5-way</td><td> TieredImageNet-5-way</td><td>FC100-5-way</td></tr><tr><td>MAML (2017)(1)</td><td>Conv4</td><td>49.31 ± 0.40</td><td>52.45 ± 0.48</td><td>34.75 ± 0.39</td></tr><tr><td>RDP(1)</td><td>Conv4</td><td>46.12 ± 0.38</td><td>47.63 ± 0.44</td><td>36.63 ± 0.38</td></tr><tr><td>RDP(1)</td><td>ResNet12</td><td>51.16 ± 0.43</td><td>51.37 ± 0.46</td><td>37.54 ± 0.40</td></tr><tr><td>MAML (2017)(5)</td><td>Conv4</td><td>64.77 ± 0.36</td><td>67.66 ± 0.42</td><td>43.90 ± 0.38</td></tr><tr><td>RDP(5)</td><td>Conv4</td><td>63.34 ± 0.36</td><td>65.19 ± 0.42</td><td>49.46 ± 0.39</td></tr><tr><td>RDP(5)</td><td>ResNet12</td><td>65.72 ± 0.36</td><td>66.31 ± 0.41</td><td>50.29 ± 0.38</td></tr></table>
+
+![](images/97034f91d46151f9006f5649e7a8354abc351edc92e686aa641234f661b23922.jpg)  
+Figure 3: The effect of the number of decision planes on the miniImageNet and FC100 datasets.
+
+# 5 THE RANDOM DECISION PLANES ALGORITHM
+
+As discussed above, the multi-step adaptation based on gradient descent during the inner loop aims to provide guidance for features learning of body. From this consideration, we suppose that if a suitable linear classifier is given, the feature learning can be facilitated even without gradient descent during the inner loop. From this consideration, we devise such an algorithm named Random Decision Planes (RDP), where a classifier is chosen from a predefined set $\mathcal { P }$ according to the target loss on the support set. The predefined set of classifier $\mathcal { P }$ consists of $n _ { p }$ different orthonormal matrices that are generated through the Gram-Schmidt method from random matrices. During the inner loop, without gradient descent, we directly choose a most suitable classifier as the network head which minimizes the cross entropy loss on the support set. In the outer loop, we compute the loss based on the chosen head and run backward to update the network body. A formal description of RDP is presented in Algorithm 1. The implementation details can be found in Appendix C.1.
+
+The overall evaluation results on three datasets are presented in Table 2. Note that we also remove the head and construct the prototypes from the body network $f _ { \theta }$ for predictions during meta-testing. The proposed RDP algorithm performs comparably to the standard MAML method on three datasets, especially on the FC100 dataset. Without any task-specific adaptation for the network body, a best performing classifier chosen from a set of randomly generated subspaces can also be a guidance to facilitate the features learning, further suggesting that a head with better classification capabilities, is key factor to learn good representations even if the chosen approximate head performs worse than a gradient-based head, and the main purpose of task-specific adaptation is to adjust the lowperformance head for features learning of body.
+
+# Algorithm 2 The Meta Contrastive Learning (MCL) Algorithm for N-way learning
+
+<table><tr><td>Input: Network Body fo, Projection Layer gφ,Learning Rate β, Constant T,Task Distribution P(T) while not done do Sample a batch of tasks {Tb}B=1, where Tb ~ P(T)</td></tr><tr><td>for b ∈ {1,...,B} do</td></tr><tr><td>and y2k-1= y2k where k ∈ {1,.,N}). for i ∈ {1,...,2N} do</td></tr><tr><td>zi=gΦ(fe(x)） end for</td></tr><tr><td>for i ∈{1.,..., 2N} and j ∈{1,...,2N} do Si,j= zzj/(zil|lzjl)</td></tr><tr><td>end for define l(i,j)=-log( exp(si,j/T) (∑11xp(s/</td></tr><tr><td>Lb=2∑_1[l（2k -1,2k)+ (2k,2k -1)] N end for θ=0-βθB∑B=1Lb JB</td></tr></table>
+
+Also, we conduct experiments to explore the impact of the number of decision planes. Results are shown in Figure 3 on two datasets. With a small set of decision planes, it can be more difficult to find a suitable head to guide the features learning, while with enough decision planes, the performance then reaches the upper limit.
+
+# 6 THE META CONTRASTIVE LEARNING ALGORITHM
+
+We have already seen that the multi-step task-specific adaptation to improve the classifier head can essentially facilitate the features learning of body. In total, prior gradient-based methods based on the cross-entropy loss proposes to learn the correspondence between samples and assigned labels for different tasks, thus requiring the task-specific adaptation for the classifier head during inner loop. Since the task-specific head also serves for features learning of body, we wonder if we can remove the inner loop or adaptation, and make full use of the labels information in other way to be a guidance for features learning. From this consideration and inspired by recent works (Chen et al. (2020); He et al. (2020)) about self-supervised contrastive learning, we further devise the Meta Contrastive Learning (MCL) algorithm that directly removes the inner loop and exploits the inter-sample relationship with only the query set.
+
+Specifically, rather than using cross entropy loss for task-specific adaptation, we simply impose that normalized representations from the same class are closer together than representations from different classes. For $N$ -way few-shot learning, we sample two examples per class to build the query set. Next, for a given anchor example, the meta contrastive loss pulls it closer to the point of same class while pushes the anchor farther away from the negative examples of other classes. Following Chen et al. (2020), we also employ a small neural network projection layer that maps the body features to the space where contrastive loss is applied. A formal description of MCL is presented in Algorithm 2. The implementation details can be found in Appendix C.1.
+
+During meta-testing, we discard the projection layer $g _ { \phi }$ and construct the prototypes from the body network $f _ { \theta }$ for predictions. The overall evaluation results on the MiniImageNet, TieredImageNet and FC100 datasets are presented in Table 3. Note that TADAM (Oreshkin et al. (2018)) employs a extra task embedding network (TEN) block to predict element-wise scale and shift vectors, and MetaOptNet (Lee et al. (2019b)) proposes to learn a linear support vector machine (SVM) as classifier head during the inner loop. Unlike those methods, our MCL method is arguably simpler. By exploiting the relationship between different samples, we are able to remove the inner loop which contains a complex adaptation process, and devise a contrastive loss to train the network body directly. As the results shows, our method outperforms almost previous well-designed methods and also achieves results comparable to MetaOptNet. More experimental results and time-efficiency analysis can be found in Appendix C.2 and C.3.
+
+Table 3: The evaluation results of 5-way-K-shot learning for the Meta Contrastive Learning (MCL) and other baselines with different backbones.   
+
+<table><tr><td>Method</td><td>Backbone</td><td>MiniImageNet-5-way</td><td>TieredImageNet-5-way</td><td>FC100-5-way</td></tr><tr><td>MAML (2017)(1)</td><td>Conv4</td><td>49.31 ± 0.40</td><td>52.45 ± 0.48</td><td>34.75 ± 0.39</td></tr><tr><td>MCL(1)</td><td>Conv4</td><td>50.73 ± 0.43</td><td>53.12 ± 0.48</td><td>37.73 ± 0.38</td></tr><tr><td>TADAM (2018)(1)</td><td>ResNet12</td><td>58.50 ± 0.30</td><td></td><td>40.10 ± 0.40</td></tr><tr><td>MetaOptNet (2019b)(1)</td><td>ResNet12</td><td>62.64 ± 0.61</td><td>65.99 ± 0.72</td><td>41.10 ± 0.60</td></tr><tr><td>MCL(1)</td><td>ResNet12</td><td>62.14 ± 0.43</td><td>65.98 ± 0.50</td><td>41.38 ± 0.40</td></tr><tr><td>MAML (2017)(5)</td><td>Conv4</td><td>64.77 ± 0.36</td><td>67.66 ± 0.42</td><td>43.90 ± 0.38</td></tr><tr><td>MCL(5)</td><td>Conv4</td><td>66.25 ± 0.36</td><td>69.31 ± 0.41</td><td>51.29 ± 0.39</td></tr><tr><td>TADAM (2018)(5)</td><td>ResNet12</td><td>76.70 ± 0.30</td><td></td><td>56.10 ± 0.40</td></tr><tr><td>MetaOptNet (2019b)(5)</td><td>ResNet12</td><td>78.63 ± 0.46</td><td>81.56 ± 0.53</td><td>55.50 ± 0.60</td></tr><tr><td>MCL(5)</td><td>ResNet12</td><td>78.34 ± 0.33</td><td>81.09 ± 0.37</td><td>56.64 ± 0.39</td></tr></table>
+
+![](images/ba3d9472b7185a0c80973c9148357d7d8caf73f0516c3efb9d5c9089fce9606d.jpg)  
+Figure 4: The effect of output dimension of $g _ { \phi }$ on the MiniImageNet dataset.
+
+We also study the impact of the projection layer $g _ { \phi }$ . Figure 4 shows the evaluation results with different output dimensions. Note that “None” means that there is no projection layer for loss computation. As the results show, for a deeper ResNet12 backbone, the projection layer facilitates the features learning a lot ( $56 \%$ for 5-shot, ${ > } 5 \%$ for 1-shot). We conjecture that the projection layer is trained to extract task-specific information useful for the contrastive loss, while the body representations $^ { h }$ learns more general information. More analysis can be found in Appendix C.4.
+
+# 7 CONCLUSION
+
+In this paper, based on the hypothesis that feature reuse is the dominant factor for the success of MAML algorithm, we further study the impact of task-specific adaptation and devise several training regimes including BOHI, Multi-Head and so on. Also, we provide a more formal argument from the perspective of gradient descent optimization. Based on analysis above, we find that the multistep task-specific adaptation, making the body and head have similar classification capabilities, can provide better gradient descent direction for the features learning of body. We further connect our results to other meta-learning algorithm, showing the adaptation is a common mode but varied in different methods. From our consideration, we devise the RDP algorithm where a suitable linear classifier is chosen without gradient descent and get more supporting conclusions. We also build the
+
+MCL algorithm that removes the inner loop and exploit the inter-sample relationship, and achieve results comparable to some state-of-the-art methods.
+
+# REFERENCES
+
+Luca Bertinetto, Joao F Henriques, Philip HS Torr, and Andrea Vedaldi. Meta-learning with differentiable closed-form solvers. arXiv preprint arXiv:1805.08136, 2018.
+
+Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. arXiv preprint arXiv:2002.05709, 2020.
+
+Wei-Yu Chen, Yen-Cheng Liu, Zsolt Kira, Yu-Chiang Frank Wang, and Jia-Bin Huang. A closer look at few-shot classification. arXiv preprint arXiv:1904.04232, 2019.
+
+Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation of deep networks. arXiv preprint arXiv:1703.03400, 2017.
+
+Chelsea Finn, Kelvin Xu, and Sergey Levine. Probabilistic model-agnostic meta-learning. In Advances in Neural Information Processing Systems, pp. 9516–9527, 2018.
+
+Spyros Gidaris and Nikos Komodakis. Dynamic few-shot visual learning without forgetting. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4367– 4375, 2018.
+
+Micah Goldblum, Steven Reich, Liam Fowl, Renkun Ni, Valeriia Cherepanova, and Tom Goldstein. Unraveling meta-learning: Understanding feature representations for few-shot tasks. arXiv preprint arXiv:2002.06753, 2020.
+
+Erin Grant, Chelsea Finn, Sergey Levine, Trevor Darrell, and Thomas Griffiths. Recasting gradientbased meta-learning as hierarchical bayes. arXiv preprint arXiv:1801.08930, 2018.
+
+Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual representation learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 9729–9738, 2020.
+
+Gregory Koch, Richard Zemel, and Ruslan Salakhutdinov. Siamese neural networks for one-shot image recognition. In ICML deep learning workshop, volume 2. Lille, 2015.
+
+Alex Krizhevsky, Vinod Nair, and Geoffrey Hinton. Cifar-10 (canadian institute for advanced research). URL http://www. cs. toronto. edu/kriz/cifar. html, 5, 2010.
+
+Hae Beom Lee, Hayeon Lee, Donghyun Na, Saehoon Kim, Minseop Park, Eunho Yang, and Sung Ju Hwang. Learning to balance: Bayesian meta-learning for imbalanced and out-of-distribution tasks. arXiv preprint arXiv:1905.12917, 2019a.
+
+Kwonjoon Lee, Subhransu Maji, Avinash Ravichandran, and Stefano Soatto. Meta-learning with differentiable convex optimization. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 10657–10665, 2019b.
+
+Alex Nichol, Joshua Achiam, and John Schulman. On first-order meta-learning algorithms. arXiv preprint arXiv:1803.02999, 2018.
+
+Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. arXiv preprint arXiv:1807.03748, 2018.
+
+Boris Oreshkin, Pau Rodr´ıguez Lopez, and Alexandre Lacoste. Tadam: Task dependent adaptive ´ metric for improved few-shot learning. In Advances in Neural Information Processing Systems, pp. 721–731, 2018.
+
+Siyuan Qiao, Chenxi Liu, Wei Shen, and Alan L Yuille. Few-shot image recognition by predicting parameters from activations. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 7229–7238, 2018.
+
+Aniruddh Raghu, Maithra Raghu, Samy Bengio, and Oriol Vinyals. Rapid learning or feature reuse? towards understanding the effectiveness of maml. arXiv preprint arXiv:1909.09157, 2019.
+
+Sachin Ravi and Hugo Larochelle. Optimization as a model for few-shot learning. 2016.
+
+Mengye Ren, Eleni Triantafillou, Sachin Ravi, Jake Snell, Kevin Swersky, Joshua B Tenenbaum, Hugo Larochelle, and Richard S Zemel. Meta-learning for semi-supervised few-shot classification. arXiv preprint arXiv:1803.00676, 2018.
+
+Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al. Imagenet large scale visual recognition challenge. International journal of computer vision, 115(3):211–252, 2015.
+
+Andrei A Rusu, Dushyant Rao, Jakub Sygnowski, Oriol Vinyals, Razvan Pascanu, Simon Osindero, and Raia Hadsell. Meta-learning with latent embedding optimization. arXiv preprint arXiv:1807.05960, 2018.
+
+Adam Santoro, Sergey Bartunov, Matthew Botvinick, Daan Wierstra, and Timothy Lillicrap. Metalearning with memory-augmented neural networks. In International conference on machine learning, pp. 1842–1850, 2016.
+
+Jake Snell, Kevin Swersky, and Richard Zemel. Prototypical networks for few-shot learning. In Advances in neural information processing systems, pp. 4077–4087, 2017.
+
+Qianru Sun, Yaoyao Liu, Tat-Seng Chua, and Bernt Schiele. Meta-transfer learning for few-shot learning. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 403–412, 2019.
+
+Oriol Vinyals, Charles Blundell, Timothy Lillicrap, Daan Wierstra, et al. Matching networks for one shot learning. In Advances in neural information processing systems, pp. 3630–3638, 2016.
+
+Risto Vuorio, Shao-Hua Sun, Hexiang Hu, and Joseph J Lim. Multimodal model-agnostic metalearning via task-aware modulation. In Advances in Neural Information Processing Systems, pp. 1–12, 2019.
+
+# A FEW-SHOT IMAGE CLASSIFICATION DATASETS
+
+In this section, we introduce four benchmark datasets often used for few-shot image classification: the miniImageNet (Vinyals et al. (2016)), tieredImageNet (Ren et al. (2018)), CIFAR-FS (Bertinetto et al. (2018)) and FC100 (Oreshkin et al. (2018)).
+
+The miniImageNet (Vinyals et al. (2016)) dataset is standard benchmark for few-shot image classification, comprises 100 classes randomly chosen from the original ImageNet (Russakovsky et al. (2015)) dataset, where 64 classes is used for meta-training, 16 classes for meta-validation and 20 classes for meta-testing. Each class contains 600 images of size $8 4 \times 8 4$ . Since the original class splits are unavailable, we use the commonly-used split proposed in Ravi & Larochelle (2016).
+
+The tieredImageNet (Ren et al. (2018)) dataset is another larger subset of ImageNet (Russakovsky et al. (2015)). This dataset contains 608 classes that are grouped into 34 high-level categories, where 20 categories (351 classes) are used for meta-training, 6 categories (97 classes) for meta-validation and 8 categories(160 classes) for meta-testing. All images are also size of $8 4 \times 8 4$ .
+
+The CIFAR-FS (Bertinetto et al. (2018)) dataset is a few-shot image classification benchmark, consisting of all 100 classes from CIFAR-100 (Krizhevsky et al. (2010)). These classes are randomly split into 64, 16, and 20 separately for meta-training, meta-validation and meta-testing. Each class contains 600 images of size $3 2 \times 3 2$ .
+
+The FC100 (Oreshkin et al. (2018)) dataset is another benchmark derived from CIFAR100 (Krizhevsky et al. (2010)). This dataset comprises 100 classes that are grouped into 20 highlevel categories, where 12 categories (60 classes) are used for meta-training, 4 categories (20 classes) for meta-validation and 4 categories (20 classes) for meta-testing. Each class contains 600 images of size $3 2 \times 3 2$ .
+
+Table 4: The evaluation results of 5-way-K-shot learning on the MiniImageNet dataset.   
+
+<table><tr><td>Method</td><td>Conv4, K=1</td><td>Conv4, K=5</td><td>ResNet12, K=1</td><td>ResNet12, K=5</td></tr><tr><td>MAML (Finn et al. (2017))</td><td>49.31 ± 0.40</td><td>64.77 ± 0.36</td><td>57.45 ± 0.47</td><td>72.70 ± 0.35</td></tr><tr><td>Almost No Inner Loop (ANIL)</td><td>50.23 ± 0.42</td><td>65.98 ± 0.38</td><td>59.43 ± 0.44</td><td>73.28 ± 0.33</td></tr><tr><td>Body Inner, Head Outer (BOHI)</td><td>50.61 ± 0.43</td><td>66.14 ± 0.37</td><td>59.69 ± 0.46</td><td>73.42 ± 0.37</td></tr><tr><td>MetaOptNet (Lee et al. (2019b))</td><td></td><td></td><td>62.64 ± 0.61</td><td>78.63 ± 0.46</td></tr><tr><td>Meta Contrastive Learning (MCL)</td><td>50.73 ± 0.43</td><td>66.25 ± 0.36</td><td>62.14 ± 0.43</td><td>78.34 ± 0.33</td></tr></table>
+
+# B MORE DETAILS ABOUT ALGORITHMS
+
+In this section, we provide further details about the training regimes and algorithms mentioned above. Note that we denote the meta parameters of the network as $\theta$ in previous sections. Considering the network is composed of the body (feature extractor) and head (classifier), we further rewrite $\theta$ as $\theta = [ \theta _ { f } , \theta _ { c } ]$ , where $\theta _ { f } , \theta _ { c }$ is the parameters of body and head respectively. For a given task $T _ { b } = \{ T _ { b } ^ { s } , \dot { T } _ { b } ^ { q } \}$ , the meta-initialization updating of MAML can be expressed as follows:
+
+$$
+\begin{array} { r l } & { \theta _ { f } ^ { t } = \theta _ { f } ^ { t - 1 } - \alpha \nabla _ { \theta _ { f } ^ { t - 1 } } \mathcal { L } _ { T _ { b } ^ { s } } ( \theta _ { f } ^ { t - 1 } , \theta _ { c } ^ { t - 1 } ) , \ \theta _ { c } ^ { t } = \theta _ { c } ^ { t - 1 } - \alpha \nabla _ { \theta _ { c } ^ { t - 1 } } \mathcal { L } _ { T _ { b } ^ { s } } ( \theta _ { f } ^ { t - 1 } , \theta _ { c } ^ { t - 1 } ) } \\ & { \theta _ { f } = \theta _ { f } - \beta \nabla _ { \theta _ { f } } \mathcal { L } _ { T _ { b } ^ { q } } ( \theta _ { f } ^ { t } , \theta _ { c } ^ { t } ) , \ \theta _ { c } = \theta _ { c } - \beta \nabla _ { \theta _ { c } } \mathcal { L } _ { T _ { b } ^ { q } } ( \theta _ { f } ^ { t } , \theta _ { c } ^ { t } ) } \end{array}
+$$
+
+where $\alpha$ is the step size of the inner loop, $\beta$ is the learning rate. In our work, we devise several methods using different training regimes including Multi-Task, Multi-Head, Almost No Inner Loop (ANIL) and Body Outer Loop, Head Inner Loop (BOHI) to study the role of network body and head during meta-training.
+
+The updating rules of Multi-Task can be expressed as follows
+
+$$
+\theta _ { f } = \theta _ { f } - \beta \nabla _ { \theta _ { f } } \mathcal { L } _ { T _ { b } ^ { q } } ( \theta _ { f } , \theta _ { c } ) , \theta _ { c } = \theta _ { c } - \beta \nabla _ { \theta _ { c } } \mathcal { L } _ { T _ { b } ^ { q } } ( \theta _ { f } , \theta _ { c } )
+$$
+
+For different tasks, the Multi-head has different heads for task specificity, given by
+
+$$
+\boldsymbol { \theta } _ { f } = \boldsymbol { \theta } _ { f } - \beta \boldsymbol { \nabla } _ { \boldsymbol { \theta } _ { f } } \mathcal { L } _ { T _ { b } ^ { q } } ( \boldsymbol { \theta } _ { f } , \boldsymbol { \theta } _ { c } ^ { T _ { b } } ) , \ \boldsymbol { \theta } _ { c } ^ { T _ { b } } = \boldsymbol { \theta } _ { c } ^ { T _ { b } } - \beta \boldsymbol { \nabla } _ { \boldsymbol { \theta } _ { c } ^ { T _ { b } } } \mathcal { L } _ { T _ { b } ^ { q } } ( \boldsymbol { \theta } _ { f } , \boldsymbol { \theta } _ { c } ^ { T _ { b } } )
+$$
+
+where $\theta _ { c } ^ { T _ { b } }$ is the specific parameters for task $T _ { b }$ . The network body is fixed during the inner loop for ANIL, given by
+
+$$
+\begin{array} { r l } & { { \theta } _ { c } ^ { t } = { \theta } _ { c } ^ { t - 1 } - \alpha \nabla _ { { \theta } _ { c } ^ { t - 1 } } \mathcal { L } _ { T _ { b } ^ { s } } ( { \theta } _ { f } ^ { t - 1 } , { \theta } _ { c } ^ { t - 1 } ) } \\ & { { \theta } _ { f } = { \theta } _ { f } - \beta \nabla _ { { \theta } _ { f } } \mathcal { L } _ { T _ { b } ^ { q } } ( { \theta } _ { f } , { \theta } _ { c } ^ { t } ) , \ { \theta } _ { c } = { \theta } _ { c } - \beta \nabla _ { { \theta } _ { c } } \mathcal { L } _ { T _ { b } ^ { q } } ( { \theta } _ { f } , { \theta } _ { c } ^ { t } ) } \end{array}
+$$
+
+For BOHI, the network body is updated only by the outer loop and the head is adapted only during the inner loop, making the head’s meta-initialized parameters unchanged, given by
+
+$$
+\begin{array} { r l } & { \theta _ { c } ^ { t } = \theta _ { c } ^ { t - 1 } - \alpha \nabla _ { \theta _ { c } ^ { t - 1 } } \mathcal { L } _ { T _ { b } ^ { s } } ( \theta _ { f } ^ { t - 1 } , \theta _ { c } ^ { t - 1 } ) } \\ & { \theta _ { f } = \theta _ { f } - \beta \nabla _ { \theta _ { f } } \mathcal { L } _ { T _ { b } ^ { q } } ( \theta _ { f } , \theta _ { c } ^ { t } ) } \end{array}
+$$
+
+# C MORE EXPERIMENTAL DETAILS
+
+# C.1 IMPLEMENTATION DETAILS
+
+For all training regimes, RDP and MCL, we use the Adam optimizer with weight decay of 5e-4 and the learning rate is set to 1e-3. For 4-layer convolution network with 64 filters, we flatten the output feature map of the network body, and obtain 1600-d features for miniImageNet and tieredImageNet, while 256-d features for CIFAR-FS and FC100. For ResNet12 network, we employ a global max pooling layer on the output feature map of the network body, and obtain 512-d features for four public datasets. During meta-training, we adopt horizontal flip, random crop and color (brightness, contrast, and saturation) jitter data augmentation as proposed in Gidaris & Komodakis (2018); Qiao et al. (2018). We train all models 100 epochs and take 500 batches per epoch. For MAML, BOHI and ANIL, both models are trained using 5 gradient steps of size $\alpha = 0 . 0 1$ for Conv4 and $\alpha = 0 . 1$ for ResNet12. For the Random Decision Planes algorithm, the number of decision planes $n _ { p }$ is set to 64. For the Meta Contrastive Learning (MCL) algorithm, we apply a two-layer nonlinear projection layer with hidden size of 512. Also, the query datapoints come from 10 different classes for each sampled task, which is helpful for accelerating model convergence.
+
+Table 5: The evaluation results of 5-way-K-shot learning on the TieredImageNet dataset.   
+
+<table><tr><td>Method</td><td>Conv4, K=1</td><td>Conv4, K=5</td><td>ResNet12, K=1</td><td>ResNet12, K=5</td></tr><tr><td>MAML (Finn et al. (2017))</td><td>52.45 ± 0.48</td><td>67.66 ± 0.42</td><td>63.05 ± 0.50</td><td>77.01 ± 0.40</td></tr><tr><td>Almost No Inner Loop (ANIL)</td><td>52.69 ± 0.47</td><td>67.44 ± 0.42</td><td>63.03 ± 0.49</td><td>76.98 ± 0.41</td></tr><tr><td>Body Inner, Head Outer (BOHI)</td><td>53.60 ± 0.48</td><td>68.39 ± 0.42</td><td>63.20 ± 0.51</td><td>77.11 ± 0.40</td></tr><tr><td>MetaOptNet (Lee et al. (2019b))</td><td></td><td></td><td>65.99 ± 0.72</td><td>81.56 ± 0.53</td></tr><tr><td>Meta Contrastive Learning (MCL)</td><td>53.12 ± 0.48</td><td>69.31 ± 0.41</td><td>65.98 ± 0.50</td><td>81.09 ± 0.37</td></tr></table>
+
+Table 6: The evaluation results of 5-way-K-shot learning on the CIFAR-FS dataset.   
+
+<table><tr><td>Method</td><td>Conv4, K=1</td><td>Conv4, K=5</td><td>ResNet12, K=1</td><td>ResNet12, K=5</td></tr><tr><td>MAML (Finn et al. (2017))</td><td>61.96 ± 0.51</td><td>76.16 ± 0.39</td><td>67.41 ± 0.51</td><td>80.94 ± 0.37</td></tr><tr><td>Almost No Inner Loop (ANIL)</td><td>63.27 ± 0.52</td><td>77.25 ± 0.38</td><td>69.23 ± 0.50</td><td>81.85 ± 0.36</td></tr><tr><td>Body Inner, Head Outer (BOHI)</td><td>63.58 ± 0.52</td><td>77.11 ± 0.38</td><td>68.41 ± 0.52</td><td>80.86 ± 0.37</td></tr><tr><td>MetaOptNet (Lee et al. (2019b))</td><td></td><td></td><td>72.00 ± 0.70</td><td>84.20 ± 0.50</td></tr><tr><td>Meta Contrastive Learning (MCL)</td><td>64.59 ± 0.52</td><td>77.24 ± 0.38</td><td>71.17 ± 0.49</td><td>84.18 ± 0.35</td></tr></table>
+
+Table 7: The evaluation results of 5-way-K-shot learning on the FC100 dataset.   
+
+<table><tr><td>Method</td><td>Conv4, K=1</td><td>Conv4, K=5</td><td>ResNet12,K=1</td><td>ResNet12,K=5</td></tr><tr><td>MAML (Finn et al. (2017))</td><td>34.75 ± 0.39</td><td>43.90 ± 0.38</td><td>38.43 ± 0.39</td><td>50.85 ± 0.38</td></tr><tr><td>Almost No Inner Loop (ANIL)</td><td>37.49 ± 0.38</td><td>49.58 ± 0.39</td><td>38.60 ± 0.40</td><td>50.70 ± 0.39</td></tr><tr><td>Body Inner, Head Outer (BOHI)</td><td>37.15 ± 0.41</td><td>48.68 ± 0.39</td><td>38.63 ± 0.40</td><td>50.65 ± 0.38</td></tr><tr><td>MetaOptNet (Lee et al. (2019b))</td><td></td><td></td><td>41.10 ± 0.60</td><td>55.50 ± 0.60</td></tr><tr><td>Meta Contrastive Learning (MCL)</td><td>37.73 ± 0.38</td><td>51.29 ± 0.39</td><td>41.38 ± 0.40</td><td>56.64 ± 0.39</td></tr></table>
+
+# C.2 MORE RESULTS FOR BOHI, ANIL, MAML, MCL
+
+In this section, we provide complete experimental results for BOHI, ANIL, MAML and MCL with different backbones on four datasets. The complete results on four datasets are presented in Table 4, Table 5, Table 6 and Table 7 respectively. The results can further verify our description mentioned above. Good features learning depends on the multi-step task-specific adaptation of head during the inner loop more than updating the meta-initialization of head in outer loop. With the lowperformance head, the update of body may even lead to a decline in the quality of features. In addition, the results on four datasets further demonstrate the effectiveness of our proposed MCL algorithm.
+
+# C.3 THE TIME-EFFICIENCY ANALYSIS FOR BOHI, ANIL, MAML, MCL
+
+It is obvious that ANIL, BOHI and MCL can speeds up training. The results about the comparison of computation time are presented in Table 8. We implement our methods based on PyTorch and the
+
+Table 8: The computation time of different methods.(tasks/sec)   
+
+<table><tr><td>Method</td><td>Conv4</td><td>ResNet12</td></tr><tr><td>MAML (Finn et al. (2017))</td><td>10.60</td><td>2.24</td></tr><tr><td>Random Decision Planes (RDP)</td><td>31.24</td><td>6.88</td></tr><tr><td>Almost No Inner Loop (ANIL)</td><td>30.98</td><td>6.86</td></tr><tr><td>Body Inner, Head Outer (BOHI)</td><td>31.12</td><td>6.88</td></tr><tr><td>Meta Contrastive Learning (MCL)</td><td>63.76</td><td>30.96</td></tr></table>
+
+training of models is run on two NVIDIA 1080Ti GPU. Notice that our MCL can run much faster than BOHI and ANIL while achieves better evaluation results. The training speedups also illustrate the significant computational benefit of MCL and prove its effectiveness.
+
+Table 9: The evaluation results about the quality of features extracted by the network body and projection layer. (5-way-5-shot on MiniImageNet, ResNet12)   
+
+<table><tr><td>Hidden size of gΦ</td><td>Network Body fe</td><td>Projection Layer gΦ</td></tr><tr><td>64</td><td>77.69 ± 0.33</td><td>61.83 ± 0.37</td></tr><tr><td>256</td><td>77.98 ± 0.34</td><td>63.46 ± 0.38</td></tr><tr><td>512</td><td>78.34 ± 0.33</td><td>66.27 ± 0.40</td></tr></table>
+
+# C.4 ABOUT THE PROJECTION LAYER OF MCL
+
+We have found that with a deeper backbone, the features learning can be facilitated a lot by the projection layer. We further evaluate the quality of features extracted by the network body and the projection layer. The evaluation results are given in Table 9. Even if the contrastive loss is applied to the projection layer, the network body learns better and general representations. We conjecture that during the meta-training, the projection layer may absorb more task-specific information while the backbone tends to learn task-independent representations.

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+        "text": "TO LEARN EFFECTIVE FEATURES: UNDERSTANDING THE TASK-SPECIFIC ADAPTATION OF MAML ",
+        "text_level": 1,
+        "bbox": [
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+        "text": "Anonymous authors Paper under double-blind review ",
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "text": "Meta learning, an effective way for learning unseen tasks with few samples, is an important research area in machine learning. Model Agnostic MetaLearning (MAML) (Finn et al. (2017)) is one of the most well-known gradientbased meta learning algorithms, that learns the meta-initialization through the inner and outer optimization loop. The inner loop is to perform fast adaptation in several gradient update steps with the support datapoints, while the outer loop to generalize the updated model to the query datapoints. Recently, it has been argued that instead of rapid learning and adaptation, the learned meta-initialization through MAML has already absorbed the high-quality features prior, where the task-specific head at training facilitates the feature learning. In this work, we investigate the impact of the task-specific adaptation of MAML and discuss the general formula for other gradient-based and metric-based meta-learning approaches. From our analysis, we further devise the Random Decision Planes (RDP) algorithm to find a suitable linear classifier without any gradient descent step and the Meta Contrastive Learning (MCL) algorithm to exploit the inter-samples relationship instead of the expensive inner-loop adaptation. We conduct sufficient experiments on various datasets to explore our proposed algorithms. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Few-shot learning, aiming to learn from few labelled examples, is a great challenge for modern machine learning systems. Meta learning, an effective way for tracking this challenge, enables the model to learn general knowledge across a distribution of tasks. Various ideas of meta learning have been proposed to address the few-shot problems. Gradient-based meta learning (Finn et al. (2017); Nichol et al. (2018)) learns the meta-parameters that can be quickly adapted to new tasks by few gradient descent steps. Metric-based meta learning (Koch et al. (2015); Vinyals et al. (2016); Snell et al. (2017)) proposes to learn a metric space by comparing different datapoints. Memorybased meta learning (Santoro et al. (2016)) can rapidly assimilate new data and leverage the stored information to make predictions. ",
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+    {
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+        "text": "Model Agnostic Meta-Learning (MAML) (Finn et al. (2017)) is one of the most well-known gradient-based meta learning algorithms, that learns the meta-initialization parameters through the inner optimization loop and the outer optimization loop. For a given task, the inner loop is to perform fast adaptation in several gradient descent steps with the support datapoints, while the outer loop to generalize the updated model to the query datapoints. With the learned meta-initialization, the model can be quickly adapted to the unseen tasks with few labelled samples. Following the MAML algorithm, many significant variants (Finn et al. (2018); Rusu et al. (2018); Oreshkin et al. (2018); Bertinetto et al. (2018); Lee et al. (2019b)) are studied under the few-shot setting. ",
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+    {
+        "type": "text",
+        "text": "To understand how the MAML works, Raghu et al. (2019) conduct a series of experiments and claim that rather than rapid learning and adaptation, the learned meta-initialization has already absorbed the high-quality features prior, thus the representations after fine-tuning are almost the same for the coming unseen tasks. Also, the task specific head of MAML at training facilitates the learning of better features. In this paper, we further design more representative experiments and present a formal argument to explain the importance of the task specific adaptation. Actually, the multi-step taskspecific adaptation, making the body and head have similar classification capabilities, can provide better gradient descent direction for the features learning of body. We also notice that for both the gradient-based methods (e.g. MAML (Finn et al. (2017)), MetaOptNet (Lee et al. (2019b))) and metric-based methods (e.g. Prototypical Networks (Snell et al. (2017))) that attempt to learn a taskspecific head using the support datapoints, the adaptation is a common mode for features learning of body but varied in different methods. ",
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+        "text": "Based on our analysis, we first propose a new training paradigm to find a decision plane (linear classifier) for guidance with no gradient descent step during the inner loop and get more supporting conclusions. Moreover, we devise another training paradigm that removes the inner loop and trains the model with only the query datapoints. Specifically, inspired by contrastive representation learning (Oord et al. (2018); Chen et al. (2020); He et al. (2020)), we exploit the inter-samples relationship of query set to find a guidance for the body across different tasks. This meta contrastive learning algorithm even achieves competitive results comparable to some state-of-the-art methods. In total, our contributions can be listed as follows: ",
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+        "text": "1. We present sufficient experiments and formal argument to explore the impact of the taskspecific adaptation for body features learning and discuss the general formula for other gradient-based and metric-based meta-learning approaches.   \n2. We devise a training algorithm to obtain a decision plane with no gradient descent step during the inner loop, named as Random Decision Planes (RDP), and get more supporting conclusions.   \n3. Unlike prior gradient-based methods, we propose the Meta Contrastive Learning (MCL) algorithm to exploit the inter-samples relations instead of training a task-specific head during the inner loop. Even without the task-specific adaptation for guidance, our algorithm still achieve better results with even less computation costs.   \n4. We empirically shows the effectiveness of the proposed algorithm with different backbones on four benchmark datasets: miniImageNet (Vinyals et al. (2016)), tieredImageNet (Ren et al. (2018)), CIFAR-FS (Bertinetto et al. (2018)) and FC100 (Oreshkin et al. (2018)). ",
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+        "type": "text",
+        "text": "2 RELATED WORKS ",
+        "text_level": 1,
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+        "text": "MAML (Finn et al. (2017)) is a highly influential gradient-based meta learning algorithm for fewshot learning. The amazing experiment results on several public few-shot datasets have proved its effectiveness. Following the core idea of MAML, there are numerous works to handle the data insufficiency problem in few-shot learning. Some works (Oreshkin et al. (2018); Vuorio et al. (2019)) introduce the task-dependent representations via conditioning the feature extractor on the specific task to improve the performance. Sun et al. (2019) also employ the meta-learned scaling and shifting parameters for transferring from another large-scale dataset. Others (Grant et al. (2018); Finn et al. (2018); Lee et al. (2019a)) study this problem from the perspective of Bayesian approach. Unlike prior methods, we provide two training paradigms, one with no gradient descent step during the inner loop and another removing the inner loop and exploiting the inter-sample relations for training. ",
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+    {
+        "type": "text",
+        "text": "Recent works also explore the key factors that makes the meta-learned model perform better than others at few-shot tasks. Chen et al. (2019) discovers that a deeper backbone has a large effect on the success of meta learning algorithm, while Goldblum et al. (2020) finds that the meta learning tends to cluster object classes more tightly in feature space for those methods that fix the backbone during the inner loop (Bertinetto et al. (2018); Rusu et al. (2018)). A very recent work (Raghu et al. (2019)) argues that the meta-trained model can be applied to new task due to the high-quality features prior learned by the meta-initialized parameters rather than rapid learning. In this paper, we further study the impact of the task-specific adaptation for feature learning. Based on the analysis, we devise two algorithms, Random Decision Planes (RDP) and Meta Contrastive Learning (MCL) requiring less computation cost but still with competitive performance. ",
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+    {
+        "type": "text",
+        "text": "3 MODEL-AGNOSTIC META LEARNING (MAML) ",
+        "text_level": 1,
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+        "text": "The MAML aims to learn the meta-initialized parameters $\\theta$ for the coming unseen tasks through the inner optimization loop and the outer optimization loop. Under the $N$ -way- $K$ -shot setting, for a task $T _ { b }$ sampled from the task distribution $P ( T )$ , we have a support set of $N \\times K$ examples $T _ { b } ^ { s }$ and a query set $T _ { b } ^ { q }$ , where $N$ is the number of sampled class and $K$ is the number of instances for each class. During the inner loop, with the support set $T _ { b } ^ { s }$ , we perform fast adaptation in several gradient descent steps and obtain the task-specific parameters $\\theta _ { T _ { b } } ^ { t }$ where $t$ is the number of gradient descent steps, given by: ",
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+    {
+        "type": "table",
+        "img_path": "images/251bedc5e0c41e6964d064bdbe68025d7abd8bdf2bc170f18c26075f4c51ed4e.jpg",
+        "table_caption": [
+            "Table 1: The evaluation results of 5-way-K-shot learning for methods with different training regimes on the MiniImageNet and TieredImageNet datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>MiniImageNet-5-way</td><td>TieredImageNet-5-way</td></tr><tr><td>Multi-Head(1)</td><td>38.66 ± 0.34</td><td>31.78 ± 0.37</td></tr><tr><td>Multi-Task(1)</td><td>40.14 ± 0.38</td><td>33.62 ± 0.38</td></tr><tr><td>MAML (2017)(1)</td><td>49.31 ± 0.40</td><td>52.45 ± 0.48</td></tr><tr><td>ANIL (Almost No Inner Loop)(1)</td><td>50.23 ± 0.42</td><td>52.69 ± 0.47</td></tr><tr><td>BOHI (Body Outer loop, Head Inner Loop)(1)</td><td>50.61 ± 0.43</td><td>53.60 ± 0.48</td></tr><tr><td>Multi-Head(5)</td><td>48.99 ± 0.33</td><td>41.48 ± 0.38</td></tr><tr><td>Multi-Task(5)</td><td>50.82 ± 0.35</td><td>44.94 ± 0.39</td></tr><tr><td>MAML (2017)(5)</td><td>64.77 ± 0.36</td><td>67.66 ± 0.42</td></tr><tr><td>ANIL (Almost No Inner Loop)(5)</td><td>65.98 ± 0.38</td><td>67.44 ± 0.43</td></tr><tr><td>BOHI (Body Outer loop,Head Inner Loop)(5)</td><td>66.14 ± 0.37</td><td>68.39 ± 0.42</td></tr></table>",
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+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/c76c14282647ce4b31836c287b8e8e9eb18d38b0dc86cf41db28f53ce95035f9.jpg",
+        "text": "$$\n\\theta _ { T _ { b } } ^ { t } = \\theta _ { T _ { b } } ^ { t - 1 } - \\alpha \\nabla _ { \\theta _ { T _ { b } } ^ { t - 1 } } \\mathcal { L } _ { T _ { b } ^ { s } } ( \\theta _ { T _ { b } } ^ { t - 1 } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\alpha$ is the step size for inner loop and $\\mathcal { L } _ { T _ { b } ^ { s } } ( \\theta _ { T _ { b } } ^ { t - 1 } )$ denoted as the loss on the support set $T _ { b } ^ { s }$ after $t - 1$ steps. With the query set $T _ { b } ^ { q }$ b , we compute the meta loss on the task-specific parameters $\\theta _ { T _ { b } } ^ { t }$ and backward to update the meta-initialized parameters $\\theta$ , given by ",
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+        "type": "equation",
+        "img_path": "images/540239d6aad4149f4571596e8e19e5db71dc6a984c249cf796c34dfd8cd09426.jpg",
+        "text": "$$\n\\theta = \\theta - \\beta \\nabla _ { \\theta } \\frac { 1 } { B } \\sum _ { b = 1 } ^ { B } \\mathcal { L } _ { T _ { b } ^ { q } } ( \\theta _ { T _ { b } } ^ { t } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\beta$ is the learning rate and $B$ is the number of sampled tasks in a batch. ",
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+    {
+        "type": "text",
+        "text": "4 IMPACT OF TASK-SPECIFIC ADAPTATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 THE MULTI-STEP TASK-SPECIFIC ADAPTATION IS IMPORTANT.",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To explore the effectiveness of MAML, Raghu et al. (2019) have conducted sufficient experiments, indicating that the network body (the representation layers) has already absorbed the high-quality features prior. During meta-testing, instead of fine tuning on the network head (the classifier), simply building the prototypes with the support set can achieve comparable performance to MAML. Raghu et al. (2019) also shows that the task specificity of head at training can facilitate feature learning and ensure good representation learning in the network body. In our work, we show that besides the task specificity of head, the multi-step adaptation is also essential, and further study the role of network body and head during meta-training. We devise several methods using different training regimes: (1) Multi-Task, where all the tasks simply share one common head and the model is trained in a traditional way without inner loop adaptation; (2) Multi-Head, where different tasks are equipped with different heads for task specificity and the model is trained in a traditional way without inner loop adaptation; (3) Almost No Inner Loop (ANIL), where the network body is fixed during the inner loop; (4) Body Outer Loop, Head Inner Loop (BOHI), where the network body is updated only by the outer loop and the head is adapted only during the inner loop, making the head’s meta-initialized parameters unchanged. More algorithms’ details can be found in Appendix B, and implementation details can be found in Appendix C.1. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Following Raghu et al. (2019), we employ the cosine similarities between prototypes and the query datapoints to evaluate the quality of features learned. As Table 1 shows, even equipped with taskspecific head, the Multi-Head training still performs worse than the standard MAML algorithm by a large margin, indicating the multi-step adaptation of MAML is helpful for features learning. The results of Multi-Head and Multi-Task show the importance of multi-step task-specific adaptation. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/323b27e20fba56523cde9743ed75492f6ca7f5fff7291b139b0b1cd66eb3f61e.jpg",
+        "image_caption": [
+            "Figure 1: The adaptation of the random initialized model for the sampled tasks in different steps. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
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+        "image_caption": [
+            "Figure 2: The adaptation after 5,000 iterations for the sampled tasks in different steps . "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As the results shown in Table 1, the ANIL training remains effective comparable to the standard MAML algorithm, indicating that the task-specific adaptation of network body is unnecessary to learn good features. More interestingly, the BOHI training that keeps the meta-initialization of head unchanged even performs better than MAML, further demonstrating that good features learning depends on the multi-step task-specific adaptation of head during inner loop more than updating the meta-initialization of head in outer loop. Also, the ANIL and BOHI have similar performance, indicating that compared with learned prior knowledge in head, the inner loop adaptation, as a guidance, contributes more to the features learning. More experimental results can be found in Appendix C.2. ",
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+    {
+        "type": "text",
+        "text": ".2 WHY IS MULTI-STEP TASK-SPECIFIC ADAPTATION IMPORTANT?",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Having observed that the MAML algorithm outperforms the Multi-Task training by a large margin and the multi-step task-specific adaptation is important for features learning, we extend our analysis to explore the reason why the inner loop adaptation is essential for MAML at different stages of meta training. Specifically, we freeze the initialized MAML model and model at 5,000 iterations, sample validation tasks from the task distribution, and record the test accuracy of model in different inner loop steps. Both the body accuracy based on prototypes construction and head accuracy based on fine-tuning are given in Figure 1 and Figure 2, where “Task ID” stands for different tasks. As the results shows, at different stages of meta training, the head accuracy increases significantly in the first few adaptation steps since the model has learnt the correspondence between sample and label. However, at the beginning of training, there is only a small improvement on the body accuracy after first adaptation step. In Figure 2, as the model converges, the body accuracy even decreases in the first few adaptation steps. In the following steps, with the task-specific adaptation of head, the network body then learns better representations, further demonstrating that the multistep task-specific adaptation, making the body and head have similar classification capabilities, can be regarded as a guidance to provide better gradient descent direction for the feature learning of body. ",
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+    {
+        "type": "table",
+        "img_path": "images/3c642052c626bd4036baeac97062dd5a66f4c007ed106c97cb943964f425b156.jpg",
+        "table_caption": [
+            "Algorithm 1 The Random Decision Planes (RDP) Algorithm for N-way-K-shot learning "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm1 The Random Decision Planes (RDP) Algorithm for N-way-K-shot learning Input: Network Body fe,Learning Rate β, Task Distribution P(T) while not done do Sample a batch of tasks {Tb}b=1, where Tb ~ P(T) forb∈{1,..,B} do</td></tr><tr><td>for each sample x in {TTdo</td></tr><tr><td>z = |lfe(x)ll end for define CrossEntropyLoss(H,D) as the cross entropy loss</td></tr><tr><td>on the features representations set D with head H. W*= argmin CrossEntropyLoss(W,{(z,)))</td></tr><tr><td>WEP Lb = CrossEntropyLoss(W*,{(z&#x27;,y)}K)</td></tr><tr><td>end for 0=θ-βVθB∑b=1Lb B end while</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "To understand this intuitive argument better, we consider a sample $( { \\pmb x } , y )$ for few-shot classification where the cross entropy loss is employed, formulated as: ",
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+        "img_path": "images/e7db673f5b635bab0012de6532358278c06e3219e519e8bf908a2dbb2af3c06b.jpg",
+        "text": "$$\n\\mathcal { L } _ { c } = - \\mathrm { l o g } ( \\frac { \\mathrm { e x p } ( w _ { y } ^ { \\top } h ) } { \\sum _ { k } \\mathrm { e x p } ( w _ { k } ^ { \\top } h ) } ) = - w _ { y } ^ { \\top } h + \\mathrm { l o g } ( \\sum _ { k } \\mathrm { e x p } ( w _ { k } ^ { \\top } h ) )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\{ \\pmb { w } _ { 1 } , \\pmb { w } _ { 2 } , . . . , \\pmb { w } _ { k } \\}$ is the weights of the classifier head, $^ { h }$ is the body representation of $_ { \\textbf { \\em x } }$ . The gradients of loss $\\mathcal { L } _ { c }$ with respect to the body representation $^ { h }$ are denoted by, ",
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+    {
+        "type": "equation",
+        "img_path": "images/a4a8fcaf6909f2085242ef2c0be21059da081a727032519ec476ffdbf1a9db44.jpg",
+        "text": "$$\n\\frac { \\partial \\mathcal { L } _ { c } } { \\partial \\pmb { h } } = - \\pmb { w } _ { y } + \\frac { \\sum _ { k } \\pmb { w } _ { k } \\mathrm { e x p } ( \\pmb { w } _ { k } ^ { \\top } \\pmb { h } ) } { \\sum _ { k } \\mathrm { e x p } ( \\pmb { w } _ { k } ^ { \\top } \\pmb { h } ) } = - \\pmb { w } _ { y } + \\bar { \\pmb { w } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\bar { \\pmb w }$ is exactly the weighted average of the weights $\\{ \\pmb { w } _ { 1 } , \\pmb { w } _ { 2 } , . . . , \\pmb { w } _ { k } \\}$ . As shown in Equation 4, a reasonable direction for the network body to minimize the target loss $\\mathcal { L } _ { c }$ is to make the representation $^ { h }$ closer to the corresponding class weight ${ \\pmb w } _ { y }$ , given by $\\pmb { h } = \\pmb { h } + \\lambda ( \\pmb { w } _ { y } - \\pmb { \\bar { w } } )$ . As the model converges, in the first few adaptation steps, there is a significant margin between the performance of head and body, and the classifier weights contain little knowledge about correspondence between samples and labels and differences between different classes. With the low-performance head, this updating rule for body may lead to a decline in the quality of features, which also explains why the simpler BOHI, ANIL even performs better than MAML in Table 1. After several adaptation steps during the inner loop, the body then receives the useful guidance for features learning from the taskspecific head since ${ \\pmb w } _ { y }$ can better express its corresponding class. The formulation above shows that the multi-step task-specific adaptation, making the body and head have similar classification capabilities, can provide better gradient descent direction for the features learning of body. ",
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+    {
+        "type": "text",
+        "text": "4.3 TASK-SPECIFIC ADAPTATION IN OTHER META-LEARNING ALGORITHMS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Having noticed that the multi-step task-specific adaptation of MAML, which promotes the performance of head, can facilitate the features learning of body. It works similarly for other gradientbased methods that use end-to-end fine-tuning, such as Reptile (Nichol et al. (2018)). In the case of meta-learning methods that fix the network body and only update the head during the inner loop, such as MetaOptNet (Lee et al. (2019b)) and R2-D2 (Bertinetto et al. (2018)), the convex optimization of head also aims to provide a classifier with better classification capabilities. For metric-based methods, such as Prototypical Networks (Snell et al. (2017)), the adaptation of head is actually conducted through the nearest neighbor algorithm. In conclusion, the adaptation is a common mode but varied in different methods. These meta-learning algorithms reveal a general formula that the inner loop is for building a task-specific head that matches the classification capabilities of body and the outer loop for task-independent features learning. ",
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+    {
+        "type": "table",
+        "img_path": "images/684ed0d107828e0ce810ccc8b8f91f621d58c60ae51de77418191ec991110f52.jpg",
+        "table_caption": [
+            "Table 2: The evaluation results of 5-way-K-shot learning for the standard MAML and Random Decision Planes (RDP) with different backbones. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Backbone</td><td>MiniImageNet-5-way</td><td> TieredImageNet-5-way</td><td>FC100-5-way</td></tr><tr><td>MAML (2017)(1)</td><td>Conv4</td><td>49.31 ± 0.40</td><td>52.45 ± 0.48</td><td>34.75 ± 0.39</td></tr><tr><td>RDP(1)</td><td>Conv4</td><td>46.12 ± 0.38</td><td>47.63 ± 0.44</td><td>36.63 ± 0.38</td></tr><tr><td>RDP(1)</td><td>ResNet12</td><td>51.16 ± 0.43</td><td>51.37 ± 0.46</td><td>37.54 ± 0.40</td></tr><tr><td>MAML (2017)(5)</td><td>Conv4</td><td>64.77 ± 0.36</td><td>67.66 ± 0.42</td><td>43.90 ± 0.38</td></tr><tr><td>RDP(5)</td><td>Conv4</td><td>63.34 ± 0.36</td><td>65.19 ± 0.42</td><td>49.46 ± 0.39</td></tr><tr><td>RDP(5)</td><td>ResNet12</td><td>65.72 ± 0.36</td><td>66.31 ± 0.41</td><td>50.29 ± 0.38</td></tr></table>",
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+    {
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+        "img_path": "images/97034f91d46151f9006f5649e7a8354abc351edc92e686aa641234f661b23922.jpg",
+        "image_caption": [
+            "Figure 3: The effect of the number of decision planes on the miniImageNet and FC100 datasets. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 THE RANDOM DECISION PLANES ALGORITHM ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "As discussed above, the multi-step adaptation based on gradient descent during the inner loop aims to provide guidance for features learning of body. From this consideration, we suppose that if a suitable linear classifier is given, the feature learning can be facilitated even without gradient descent during the inner loop. From this consideration, we devise such an algorithm named Random Decision Planes (RDP), where a classifier is chosen from a predefined set $\\mathcal { P }$ according to the target loss on the support set. The predefined set of classifier $\\mathcal { P }$ consists of $n _ { p }$ different orthonormal matrices that are generated through the Gram-Schmidt method from random matrices. During the inner loop, without gradient descent, we directly choose a most suitable classifier as the network head which minimizes the cross entropy loss on the support set. In the outer loop, we compute the loss based on the chosen head and run backward to update the network body. A formal description of RDP is presented in Algorithm 1. The implementation details can be found in Appendix C.1. ",
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+    {
+        "type": "text",
+        "text": "The overall evaluation results on three datasets are presented in Table 2. Note that we also remove the head and construct the prototypes from the body network $f _ { \\theta }$ for predictions during meta-testing. The proposed RDP algorithm performs comparably to the standard MAML method on three datasets, especially on the FC100 dataset. Without any task-specific adaptation for the network body, a best performing classifier chosen from a set of randomly generated subspaces can also be a guidance to facilitate the features learning, further suggesting that a head with better classification capabilities, is key factor to learn good representations even if the chosen approximate head performs worse than a gradient-based head, and the main purpose of task-specific adaptation is to adjust the lowperformance head for features learning of body. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 2 The Meta Contrastive Learning (MCL) Algorithm for N-way learning ",
+        "text_level": 1,
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+        "table_caption": [],
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+        "table_body": "<table><tr><td>Input: Network Body fo, Projection Layer gφ,Learning Rate β, Constant T,Task Distribution P(T) while not done do Sample a batch of tasks {Tb}B=1, where Tb ~ P(T)</td></tr><tr><td>for b ∈ {1,...,B} do</td></tr><tr><td>and y2k-1= y2k where k ∈ {1,.,N}). for i ∈ {1,...,2N} do</td></tr><tr><td>zi=gΦ(fe(x)） end for</td></tr><tr><td>for i ∈{1.,..., 2N} and j ∈{1,...,2N} do Si,j= zzj/(zil|lzjl)</td></tr><tr><td>end for define l(i,j)=-log( exp(si,j/T) (∑11xp(s/</td></tr><tr><td>Lb=2∑_1[l（2k -1,2k)+ (2k,2k -1)] N end for θ=0-βθB∑B=1Lb JB</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Also, we conduct experiments to explore the impact of the number of decision planes. Results are shown in Figure 3 on two datasets. With a small set of decision planes, it can be more difficult to find a suitable head to guide the features learning, while with enough decision planes, the performance then reaches the upper limit. ",
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+    {
+        "type": "text",
+        "text": "6 THE META CONTRASTIVE LEARNING ALGORITHM ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We have already seen that the multi-step task-specific adaptation to improve the classifier head can essentially facilitate the features learning of body. In total, prior gradient-based methods based on the cross-entropy loss proposes to learn the correspondence between samples and assigned labels for different tasks, thus requiring the task-specific adaptation for the classifier head during inner loop. Since the task-specific head also serves for features learning of body, we wonder if we can remove the inner loop or adaptation, and make full use of the labels information in other way to be a guidance for features learning. From this consideration and inspired by recent works (Chen et al. (2020); He et al. (2020)) about self-supervised contrastive learning, we further devise the Meta Contrastive Learning (MCL) algorithm that directly removes the inner loop and exploits the inter-sample relationship with only the query set. ",
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+        "text": "Specifically, rather than using cross entropy loss for task-specific adaptation, we simply impose that normalized representations from the same class are closer together than representations from different classes. For $N$ -way few-shot learning, we sample two examples per class to build the query set. Next, for a given anchor example, the meta contrastive loss pulls it closer to the point of same class while pushes the anchor farther away from the negative examples of other classes. Following Chen et al. (2020), we also employ a small neural network projection layer that maps the body features to the space where contrastive loss is applied. A formal description of MCL is presented in Algorithm 2. The implementation details can be found in Appendix C.1. ",
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+        "type": "text",
+        "text": "During meta-testing, we discard the projection layer $g _ { \\phi }$ and construct the prototypes from the body network $f _ { \\theta }$ for predictions. The overall evaluation results on the MiniImageNet, TieredImageNet and FC100 datasets are presented in Table 3. Note that TADAM (Oreshkin et al. (2018)) employs a extra task embedding network (TEN) block to predict element-wise scale and shift vectors, and MetaOptNet (Lee et al. (2019b)) proposes to learn a linear support vector machine (SVM) as classifier head during the inner loop. Unlike those methods, our MCL method is arguably simpler. By exploiting the relationship between different samples, we are able to remove the inner loop which contains a complex adaptation process, and devise a contrastive loss to train the network body directly. As the results shows, our method outperforms almost previous well-designed methods and also achieves results comparable to MetaOptNet. More experimental results and time-efficiency analysis can be found in Appendix C.2 and C.3. ",
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+    {
+        "type": "table",
+        "img_path": "images/04f60df9d99f2bcb8e5b1b5c4e9951d23fc033c09e02421e579ad464d0f29a75.jpg",
+        "table_caption": [
+            "Table 3: The evaluation results of 5-way-K-shot learning for the Meta Contrastive Learning (MCL) and other baselines with different backbones. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Backbone</td><td>MiniImageNet-5-way</td><td>TieredImageNet-5-way</td><td>FC100-5-way</td></tr><tr><td>MAML (2017)(1)</td><td>Conv4</td><td>49.31 ± 0.40</td><td>52.45 ± 0.48</td><td>34.75 ± 0.39</td></tr><tr><td>MCL(1)</td><td>Conv4</td><td>50.73 ± 0.43</td><td>53.12 ± 0.48</td><td>37.73 ± 0.38</td></tr><tr><td>TADAM (2018)(1)</td><td>ResNet12</td><td>58.50 ± 0.30</td><td></td><td>40.10 ± 0.40</td></tr><tr><td>MetaOptNet (2019b)(1)</td><td>ResNet12</td><td>62.64 ± 0.61</td><td>65.99 ± 0.72</td><td>41.10 ± 0.60</td></tr><tr><td>MCL(1)</td><td>ResNet12</td><td>62.14 ± 0.43</td><td>65.98 ± 0.50</td><td>41.38 ± 0.40</td></tr><tr><td>MAML (2017)(5)</td><td>Conv4</td><td>64.77 ± 0.36</td><td>67.66 ± 0.42</td><td>43.90 ± 0.38</td></tr><tr><td>MCL(5)</td><td>Conv4</td><td>66.25 ± 0.36</td><td>69.31 ± 0.41</td><td>51.29 ± 0.39</td></tr><tr><td>TADAM (2018)(5)</td><td>ResNet12</td><td>76.70 ± 0.30</td><td></td><td>56.10 ± 0.40</td></tr><tr><td>MetaOptNet (2019b)(5)</td><td>ResNet12</td><td>78.63 ± 0.46</td><td>81.56 ± 0.53</td><td>55.50 ± 0.60</td></tr><tr><td>MCL(5)</td><td>ResNet12</td><td>78.34 ± 0.33</td><td>81.09 ± 0.37</td><td>56.64 ± 0.39</td></tr></table>",
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+    {
+        "type": "image",
+        "img_path": "images/ba3d9472b7185a0c80973c9148357d7d8caf73f0516c3efb9d5c9089fce9606d.jpg",
+        "image_caption": [
+            "Figure 4: The effect of output dimension of $g _ { \\phi }$ on the MiniImageNet dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "We also study the impact of the projection layer $g _ { \\phi }$ . Figure 4 shows the evaluation results with different output dimensions. Note that “None” means that there is no projection layer for loss computation. As the results show, for a deeper ResNet12 backbone, the projection layer facilitates the features learning a lot ( $56 \\%$ for 5-shot, ${ > } 5 \\%$ for 1-shot). We conjecture that the projection layer is trained to extract task-specific information useful for the contrastive loss, while the body representations $^ { h }$ learns more general information. More analysis can be found in Appendix C.4. ",
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+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In this paper, based on the hypothesis that feature reuse is the dominant factor for the success of MAML algorithm, we further study the impact of task-specific adaptation and devise several training regimes including BOHI, Multi-Head and so on. Also, we provide a more formal argument from the perspective of gradient descent optimization. Based on analysis above, we find that the multistep task-specific adaptation, making the body and head have similar classification capabilities, can provide better gradient descent direction for the features learning of body. We further connect our results to other meta-learning algorithm, showing the adaptation is a common mode but varied in different methods. From our consideration, we devise the RDP algorithm where a suitable linear classifier is chosen without gradient descent and get more supporting conclusions. We also build the ",
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+    {
+        "type": "text",
+        "text": "MCL algorithm that removes the inner loop and exploit the inter-sample relationship, and achieve results comparable to some state-of-the-art methods. ",
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+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A FEW-SHOT IMAGE CLASSIFICATION DATASETS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we introduce four benchmark datasets often used for few-shot image classification: the miniImageNet (Vinyals et al. (2016)), tieredImageNet (Ren et al. (2018)), CIFAR-FS (Bertinetto et al. (2018)) and FC100 (Oreshkin et al. (2018)). ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The miniImageNet (Vinyals et al. (2016)) dataset is standard benchmark for few-shot image classification, comprises 100 classes randomly chosen from the original ImageNet (Russakovsky et al. (2015)) dataset, where 64 classes is used for meta-training, 16 classes for meta-validation and 20 classes for meta-testing. Each class contains 600 images of size $8 4 \\times 8 4$ . Since the original class splits are unavailable, we use the commonly-used split proposed in Ravi & Larochelle (2016). ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The tieredImageNet (Ren et al. (2018)) dataset is another larger subset of ImageNet (Russakovsky et al. (2015)). This dataset contains 608 classes that are grouped into 34 high-level categories, where 20 categories (351 classes) are used for meta-training, 6 categories (97 classes) for meta-validation and 8 categories(160 classes) for meta-testing. All images are also size of $8 4 \\times 8 4$ . ",
+        "bbox": [
+            174,
+            728,
+            823,
+            784
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The CIFAR-FS (Bertinetto et al. (2018)) dataset is a few-shot image classification benchmark, consisting of all 100 classes from CIFAR-100 (Krizhevsky et al. (2010)). These classes are randomly split into 64, 16, and 20 separately for meta-training, meta-validation and meta-testing. Each class contains 600 images of size $3 2 \\times 3 2$ . ",
+        "bbox": [
+            174,
+            790,
+            823,
+            847
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The FC100 (Oreshkin et al. (2018)) dataset is another benchmark derived from CIFAR100 (Krizhevsky et al. (2010)). This dataset comprises 100 classes that are grouped into 20 highlevel categories, where 12 categories (60 classes) are used for meta-training, 4 categories (20 classes) for meta-validation and 4 categories (20 classes) for meta-testing. Each class contains 600 images of size $3 2 \\times 3 2$ . ",
+        "bbox": [
+            176,
+            854,
+            823,
+            922
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/962a209e32a4abf6b66bcea03718e02757f3905d96cbfc07e4a15038bfeb42a4.jpg",
+        "table_caption": [
+            "Table 4: The evaluation results of 5-way-K-shot learning on the MiniImageNet dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Conv4, K=1</td><td>Conv4, K=5</td><td>ResNet12, K=1</td><td>ResNet12, K=5</td></tr><tr><td>MAML (Finn et al. (2017))</td><td>49.31 ± 0.40</td><td>64.77 ± 0.36</td><td>57.45 ± 0.47</td><td>72.70 ± 0.35</td></tr><tr><td>Almost No Inner Loop (ANIL)</td><td>50.23 ± 0.42</td><td>65.98 ± 0.38</td><td>59.43 ± 0.44</td><td>73.28 ± 0.33</td></tr><tr><td>Body Inner, Head Outer (BOHI)</td><td>50.61 ± 0.43</td><td>66.14 ± 0.37</td><td>59.69 ± 0.46</td><td>73.42 ± 0.37</td></tr><tr><td>MetaOptNet (Lee et al. (2019b))</td><td></td><td></td><td>62.64 ± 0.61</td><td>78.63 ± 0.46</td></tr><tr><td>Meta Contrastive Learning (MCL)</td><td>50.73 ± 0.43</td><td>66.25 ± 0.36</td><td>62.14 ± 0.43</td><td>78.34 ± 0.33</td></tr></table>",
+        "bbox": [
+            174,
+            142,
+            872,
+            243
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B MORE DETAILS ABOUT ALGORITHMS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            272,
+            524,
+            289
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide further details about the training regimes and algorithms mentioned above. Note that we denote the meta parameters of the network as $\\theta$ in previous sections. Considering the network is composed of the body (feature extractor) and head (classifier), we further rewrite $\\theta$ as $\\theta = [ \\theta _ { f } , \\theta _ { c } ]$ , where $\\theta _ { f } , \\theta _ { c }$ is the parameters of body and head respectively. For a given task $T _ { b } = \\{ T _ { b } ^ { s } , \\dot { T } _ { b } ^ { q } \\}$ , the meta-initialization updating of MAML can be expressed as follows: ",
+        "bbox": [
+            173,
+            304,
+            825,
+            375
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/25cc76bbc337a4a4bbfa927653822ec84e0e1e4285e69fe4aa4a4565ab9dc588.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\theta _ { f } ^ { t } = \\theta _ { f } ^ { t - 1 } - \\alpha \\nabla _ { \\theta _ { f } ^ { t - 1 } } \\mathcal { L } _ { T _ { b } ^ { s } } ( \\theta _ { f } ^ { t - 1 } , \\theta _ { c } ^ { t - 1 } ) , \\ \\theta _ { c } ^ { t } = \\theta _ { c } ^ { t - 1 } - \\alpha \\nabla _ { \\theta _ { c } ^ { t - 1 } } \\mathcal { L } _ { T _ { b } ^ { s } } ( \\theta _ { f } ^ { t - 1 } , \\theta _ { c } ^ { t - 1 } ) } \\\\ & { \\theta _ { f } = \\theta _ { f } - \\beta \\nabla _ { \\theta _ { f } } \\mathcal { L } _ { T _ { b } ^ { q } } ( \\theta _ { f } ^ { t } , \\theta _ { c } ^ { t } ) , \\ \\theta _ { c } = \\theta _ { c } - \\beta \\nabla _ { \\theta _ { c } } \\mathcal { L } _ { T _ { b } ^ { q } } ( \\theta _ { f } ^ { t } , \\theta _ { c } ^ { t } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            227,
+            381,
+            738,
+            428
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha$ is the step size of the inner loop, $\\beta$ is the learning rate. In our work, we devise several methods using different training regimes including Multi-Task, Multi-Head, Almost No Inner Loop (ANIL) and Body Outer Loop, Head Inner Loop (BOHI) to study the role of network body and head during meta-training. ",
+        "bbox": [
+            176,
+            431,
+            821,
+            489
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The updating rules of Multi-Task can be expressed as follows ",
+        "bbox": [
+            174,
+            494,
+            578,
+            510
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a2ca97749a398a9adac36cb259d5169a37a17f2daf1b2f51246f42d6dd43dfeb.jpg",
+        "text": "$$\n\\theta _ { f } = \\theta _ { f } - \\beta \\nabla _ { \\theta _ { f } } \\mathcal { L } _ { T _ { b } ^ { q } } ( \\theta _ { f } , \\theta _ { c } ) , \\theta _ { c } = \\theta _ { c } - \\beta \\nabla _ { \\theta _ { c } } \\mathcal { L } _ { T _ { b } ^ { q } } ( \\theta _ { f } , \\theta _ { c } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            277,
+            515,
+            669,
+            535
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For different tasks, the Multi-head has different heads for task specificity, given by ",
+        "bbox": [
+            173,
+            547,
+            714,
+            563
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/751dd133bc596538f0f106dd36308626e065fdcb66b1c8f1f32be43482f41b19.jpg",
+        "text": "$$\n\\boldsymbol { \\theta } _ { f } = \\boldsymbol { \\theta } _ { f } - \\beta \\boldsymbol { \\nabla } _ { \\boldsymbol { \\theta } _ { f } } \\mathcal { L } _ { T _ { b } ^ { q } } ( \\boldsymbol { \\theta } _ { f } , \\boldsymbol { \\theta } _ { c } ^ { T _ { b } } ) , \\ \\boldsymbol { \\theta } _ { c } ^ { T _ { b } } = \\boldsymbol { \\theta } _ { c } ^ { T _ { b } } - \\beta \\boldsymbol { \\nabla } _ { \\boldsymbol { \\theta } _ { c } ^ { T _ { b } } } \\mathcal { L } _ { T _ { b } ^ { q } } ( \\boldsymbol { \\theta } _ { f } , \\boldsymbol { \\theta } _ { c } ^ { T _ { b } } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            258,
+            568,
+            691,
+            590
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "where $\\theta _ { c } ^ { T _ { b } }$ is the specific parameters for task $T _ { b }$ . The network body is fixed during the inner loop for ANIL, given by ",
+        "bbox": [
+            174,
+            598,
+            826,
+            627
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fe4bf9065bc501011cf5d7f8a7b6748e99956d1492bf24a437f0e79a984f61af.jpg",
+        "text": "$$\n\\begin{array} { r l } & { { \\theta } _ { c } ^ { t } = { \\theta } _ { c } ^ { t - 1 } - \\alpha \\nabla _ { { \\theta } _ { c } ^ { t - 1 } } \\mathcal { L } _ { T _ { b } ^ { s } } ( { \\theta } _ { f } ^ { t - 1 } , { \\theta } _ { c } ^ { t - 1 } ) } \\\\ & { { \\theta } _ { f } = { \\theta } _ { f } - \\beta \\nabla _ { { \\theta } _ { f } } \\mathcal { L } _ { T _ { b } ^ { q } } ( { \\theta } _ { f } , { \\theta } _ { c } ^ { t } ) , \\ { \\theta } _ { c } = { \\theta } _ { c } - \\beta \\nabla _ { { \\theta } _ { c } } \\mathcal { L } _ { T _ { b } ^ { q } } ( { \\theta } _ { f } , { \\theta } _ { c } ^ { t } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            285,
+            632,
+            678,
+            678
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For BOHI, the network body is updated only by the outer loop and the head is adapted only during the inner loop, making the head’s meta-initialized parameters unchanged, given by ",
+        "bbox": [
+            171,
+            681,
+            823,
+            710
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a4eeed8db669447dc1caee0de9cba473b193b09139d54bd4d5cb101a9a5c7c43.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\theta _ { c } ^ { t } = \\theta _ { c } ^ { t - 1 } - \\alpha \\nabla _ { \\theta _ { c } ^ { t - 1 } } \\mathcal { L } _ { T _ { b } ^ { s } } ( \\theta _ { f } ^ { t - 1 } , \\theta _ { c } ^ { t - 1 } ) } \\\\ & { \\theta _ { f } = \\theta _ { f } - \\beta \\nabla _ { \\theta _ { f } } \\mathcal { L } _ { T _ { b } ^ { q } } ( \\theta _ { f } , \\theta _ { c } ^ { t } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            714,
+            607,
+            760
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "C MORE EXPERIMENTAL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            782,
+            477,
+            799
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "C.1 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            814,
+            408,
+            829
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For all training regimes, RDP and MCL, we use the Adam optimizer with weight decay of 5e-4 and the learning rate is set to 1e-3. For 4-layer convolution network with 64 filters, we flatten the output feature map of the network body, and obtain 1600-d features for miniImageNet and tieredImageNet, while 256-d features for CIFAR-FS and FC100. For ResNet12 network, we employ a global max pooling layer on the output feature map of the network body, and obtain 512-d features for four public datasets. During meta-training, we adopt horizontal flip, random crop and color (brightness, contrast, and saturation) jitter data augmentation as proposed in Gidaris & Komodakis (2018); Qiao et al. (2018). We train all models 100 epochs and take 500 batches per epoch. For MAML, BOHI and ANIL, both models are trained using 5 gradient steps of size $\\alpha = 0 . 0 1$ for Conv4 and $\\alpha = 0 . 1$ for ResNet12. For the Random Decision Planes algorithm, the number of decision planes $n _ { p }$ is set to 64. For the Meta Contrastive Learning (MCL) algorithm, we apply a two-layer nonlinear projection layer with hidden size of 512. Also, the query datapoints come from 10 different classes for each sampled task, which is helpful for accelerating model convergence. ",
+        "bbox": [
+            173,
+            839,
+            825,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/08ed173b54befeca2b11ec4f6436a11f67d563119ef42c41f6f6910f085030b1.jpg",
+        "table_caption": [
+            "Table 5: The evaluation results of 5-way-K-shot learning on the TieredImageNet dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Conv4, K=1</td><td>Conv4, K=5</td><td>ResNet12, K=1</td><td>ResNet12, K=5</td></tr><tr><td>MAML (Finn et al. (2017))</td><td>52.45 ± 0.48</td><td>67.66 ± 0.42</td><td>63.05 ± 0.50</td><td>77.01 ± 0.40</td></tr><tr><td>Almost No Inner Loop (ANIL)</td><td>52.69 ± 0.47</td><td>67.44 ± 0.42</td><td>63.03 ± 0.49</td><td>76.98 ± 0.41</td></tr><tr><td>Body Inner, Head Outer (BOHI)</td><td>53.60 ± 0.48</td><td>68.39 ± 0.42</td><td>63.20 ± 0.51</td><td>77.11 ± 0.40</td></tr><tr><td>MetaOptNet (Lee et al. (2019b))</td><td></td><td></td><td>65.99 ± 0.72</td><td>81.56 ± 0.53</td></tr><tr><td>Meta Contrastive Learning (MCL)</td><td>53.12 ± 0.48</td><td>69.31 ± 0.41</td><td>65.98 ± 0.50</td><td>81.09 ± 0.37</td></tr></table>",
+        "bbox": [
+            174,
+            142,
+            872,
+            243
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/bae7cf943859ca8d034b448f31e3a29f175fcc6507ccce95b372213cc9caa3ea.jpg",
+        "table_caption": [
+            "Table 6: The evaluation results of 5-way-K-shot learning on the CIFAR-FS dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Conv4, K=1</td><td>Conv4, K=5</td><td>ResNet12, K=1</td><td>ResNet12, K=5</td></tr><tr><td>MAML (Finn et al. (2017))</td><td>61.96 ± 0.51</td><td>76.16 ± 0.39</td><td>67.41 ± 0.51</td><td>80.94 ± 0.37</td></tr><tr><td>Almost No Inner Loop (ANIL)</td><td>63.27 ± 0.52</td><td>77.25 ± 0.38</td><td>69.23 ± 0.50</td><td>81.85 ± 0.36</td></tr><tr><td>Body Inner, Head Outer (BOHI)</td><td>63.58 ± 0.52</td><td>77.11 ± 0.38</td><td>68.41 ± 0.52</td><td>80.86 ± 0.37</td></tr><tr><td>MetaOptNet (Lee et al. (2019b))</td><td></td><td></td><td>72.00 ± 0.70</td><td>84.20 ± 0.50</td></tr><tr><td>Meta Contrastive Learning (MCL)</td><td>64.59 ± 0.52</td><td>77.24 ± 0.38</td><td>71.17 ± 0.49</td><td>84.18 ± 0.35</td></tr></table>",
+        "bbox": [
+            174,
+            304,
+            872,
+            405
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/771d2f4e8ba122cdff3b749264ca72eb3699d50f4bbc471207b30015c4e42ac3.jpg",
+        "table_caption": [
+            "Table 7: The evaluation results of 5-way-K-shot learning on the FC100 dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Conv4, K=1</td><td>Conv4, K=5</td><td>ResNet12,K=1</td><td>ResNet12,K=5</td></tr><tr><td>MAML (Finn et al. (2017))</td><td>34.75 ± 0.39</td><td>43.90 ± 0.38</td><td>38.43 ± 0.39</td><td>50.85 ± 0.38</td></tr><tr><td>Almost No Inner Loop (ANIL)</td><td>37.49 ± 0.38</td><td>49.58 ± 0.39</td><td>38.60 ± 0.40</td><td>50.70 ± 0.39</td></tr><tr><td>Body Inner, Head Outer (BOHI)</td><td>37.15 ± 0.41</td><td>48.68 ± 0.39</td><td>38.63 ± 0.40</td><td>50.65 ± 0.38</td></tr><tr><td>MetaOptNet (Lee et al. (2019b))</td><td></td><td></td><td>41.10 ± 0.60</td><td>55.50 ± 0.60</td></tr><tr><td>Meta Contrastive Learning (MCL)</td><td>37.73 ± 0.38</td><td>51.29 ± 0.39</td><td>41.38 ± 0.40</td><td>56.64 ± 0.39</td></tr></table>",
+        "bbox": [
+            174,
+            467,
+            872,
+            566
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            598,
+            825,
+            695
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C.2 MORE RESULTS FOR BOHI, ANIL, MAML, MCL ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            713,
+            563,
+            728
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide complete experimental results for BOHI, ANIL, MAML and MCL with different backbones on four datasets. The complete results on four datasets are presented in Table 4, Table 5, Table 6 and Table 7 respectively. The results can further verify our description mentioned above. Good features learning depends on the multi-step task-specific adaptation of head during the inner loop more than updating the meta-initialization of head in outer loop. With the lowperformance head, the update of body may even lead to a decline in the quality of features. In addition, the results on four datasets further demonstrate the effectiveness of our proposed MCL algorithm. ",
+        "bbox": [
+            174,
+            739,
+            825,
+            852
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C.3 THE TIME-EFFICIENCY ANALYSIS FOR BOHI, ANIL, MAML, MCL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            868,
+            686,
+            883
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "It is obvious that ANIL, BOHI and MCL can speeds up training. The results about the comparison of computation time are presented in Table 8. We implement our methods based on PyTorch and the ",
+        "bbox": [
+            173,
+            895,
+            820,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/821ce0159d5c94edde7e49cbd25fdbe397739e4d49a7f9861fa246938dc1d8d5.jpg",
+        "table_caption": [
+            "Table 8: The computation time of different methods.(tasks/sec) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Conv4</td><td>ResNet12</td></tr><tr><td>MAML (Finn et al. (2017))</td><td>10.60</td><td>2.24</td></tr><tr><td>Random Decision Planes (RDP)</td><td>31.24</td><td>6.88</td></tr><tr><td>Almost No Inner Loop (ANIL)</td><td>30.98</td><td>6.86</td></tr><tr><td>Body Inner, Head Outer (BOHI)</td><td>31.12</td><td>6.88</td></tr><tr><td>Meta Contrastive Learning (MCL)</td><td>63.76</td><td>30.96</td></tr></table>",
+        "bbox": [
+            300,
+            131,
+            697,
+            232
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "training of models is run on two NVIDIA 1080Ti GPU. Notice that our MCL can run much faster than BOHI and ANIL while achieves better evaluation results. The training speedups also illustrate the significant computational benefit of MCL and prove its effectiveness. ",
+        "bbox": [
+            176,
+            263,
+            825,
+            306
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/698072539d770a8e4208c2cb2c7d0413975ec4e79b031804c60034732997ebce.jpg",
+        "table_caption": [
+            "Table 9: The evaluation results about the quality of features extracted by the network body and projection layer. (5-way-5-shot on MiniImageNet, ResNet12) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hidden size of gΦ</td><td>Network Body fe</td><td>Projection Layer gΦ</td></tr><tr><td>64</td><td>77.69 ± 0.33</td><td>61.83 ± 0.37</td></tr><tr><td>256</td><td>77.98 ± 0.34</td><td>63.46 ± 0.38</td></tr><tr><td>512</td><td>78.34 ± 0.33</td><td>66.27 ± 0.40</td></tr></table>",
+        "bbox": [
+            287,
+            363,
+            710,
+            435
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C.4 ABOUT THE PROJECTION LAYER OF MCL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            465,
+            503,
+            481
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We have found that with a deeper backbone, the features learning can be facilitated a lot by the projection layer. We further evaluate the quality of features extracted by the network body and the projection layer. The evaluation results are given in Table 9. Even if the contrastive loss is applied to the projection layer, the network body learns better and general representations. We conjecture that during the meta-training, the projection layer may absorb more task-specific information while the backbone tends to learn task-independent representations. ",
+        "bbox": [
+            173,
+            492,
+            825,
+            577
+        ],
+        "page_idx": 12
+    }
+]

parse/train/SkeK3s0qKQ/SkeK3s0qKQ.md

@@ -0,0 +1,389 @@
+# EPISODIC CURIOSITY THROUGH REACHABILITY
+
+Nikolay Savinov∗1 Anton Raichuk∗1 Raphael Marinier¨ ∗1 Damien Vincent ∗1   
+Marc Pollefeys 3 Timothy Lillicrap 2 Sylvain Gelly 1
+
+1Google Brain, 2DeepMind, 3ETH Zurich ¨
+
+# ABSTRACT
+
+Rewards are sparse in the real world and most of today’s reinforcement learning algorithms struggle with such sparsity. One solution to this problem is to allow the agent to create rewards for itself — thus making rewards dense and more suitable for learning. In particular, inspired by curious behaviour in animals, observing something novel could be rewarded with a bonus. Such bonus is summed up with the real task reward — making it possible for RL algorithms to learn from the combined reward. We propose a new curiosity method which uses episodic memory to form the novelty bonus. To determine the bonus, the current observation is compared with the observations in memory. Crucially, the comparison is done based on how many environment steps it takes to reach the current observation from those in memory — which incorporates rich information about environment dynamics. This allows us to overcome the known “couch-potato” issues of prior work — when the agent finds a way to instantly gratify itself by exploiting actions which lead to hardly predictable consequences. We test our approach in visually rich 3D environments in VizDoom, DMLab and MuJoCo. In navigational tasks from VizDoom and DMLab, our agent outperforms the state-of-the-art curiosity method ICM. In MuJoCo, an ant equipped with our curiosity module learns locomotion out of the first-person-view curiosity only. The code is available at https://github.com/google-research/episodic-curiosity.
+
+# 1 INTRODUCTION
+
+Many real-world tasks have sparse rewards. For example, animals searching for food may need to go many miles without any reward from the environment. Standard reinforcement learning algorithms struggle with such tasks because of reliance on simple action entropy maximization as a source of exploration behaviour.
+
+Multiple approaches were proposed to achieve better explorative policies. One way is to give a reward bonus which facilitates exploration by rewarding novel observations. The reward bonus is summed up with the original task reward and optimized by standard RL algorithms. Such an approach is motivated by neuroscience studies of animals: an animal has an ability to reward itself for something novel – the mechanism biologically built into its dopamine release system. How exactly this bonus is formed remains an open question.
+
+Many modern curiosity formulations aim at maximizing “surprise” — inability to predict the future. This approach makes perfect sense but, in fact, is far from perfect. To show why, let us consider a thought experiment. Imagine an agent is put into a 3D maze. There is a precious goal somewhere in the maze which would give a large reward. Now, the agent is also given a remote control to a TV and can switch the channels. Every switch shows a random image (say, from a fixed set of images). The curiosity formulations which optimize surprise would rejoice because the result of the channel switching action is unpredictable. The agent would be drawn to the TV instead of looking for a goal in the environment (this was indeed observed in (Burda et al., 2018a)). So, should we call the channel switching behaviour curious? Maybe, but it is unproductive for the original sparsereward goal-reaching task. What would be a definition of curiosity which does not suffer from such “couch-potato” behaviour?
+
+We propose a new curiosity definition based on the following intuition. If the agent knew the observation after changing a TV channel is only one step away from the observation before doing that — it probably would not be so interesting to change the channel in the first place (too easy). This intuition can be formalized as giving a reward only for those observations which take some effort to reach (outside the already explored part of the environment). The effort is measured in the number of environment steps. To estimate it we train a neural network approximator: given two observations, it would predict how many steps separate them. The concept of novelty via reachability is illustrated in Figure 1. To make the description above practically implementable, there is still one piece missing though. For determining the novelty of the current observation, we need to keep track of what was already explored in the environment. A natural candidate for that purpose would be episodic memory: it stores instances of the past which makes it easy to apply the reachability approximator on pairs of current and past observations.
+
+![](images/8bb78acec2571c80946e144fd8c4a2928dad7dead21262892a2878c6a0cba13a.jpg)  
+Figure 1: We define novelty through reachability. The nodes in the graph are observations, the edges — possible transitions. The blue nodes are already in memory, the green nodes are reachable from the memory within $k = 2$ steps (not novel), the orange nodes are further away — take more than $k$ steps to reach (novel). In practice, the full possible transition graph is not available, so we train a neural network approximator to predict if the distance in steps between observations is larger or smaller than $k$ .
+
+Our method works as follows. The agent starts with an empty memory at the beginning of the episode and at every step compares the current observation with the observations in memory to determine novelty. If the current observation is indeed novel — takes more steps to reach from observations in memory than a threshold — the agent rewards itself with a bonus and adds the current observation to the episodic memory. The process continues until the end of the episode, when the memory is wiped clean.
+
+We benchmark our method on a range of tasks from visually rich 3D environments VizDoom, DMLab and MuJoCo. We conduct the comparison with other methods — including the state-of-the-art curiosity method ICM (Pathak et al., 2017) — under the same budget of environment interactions. First, we use the VizDoom environments from prior work to establish that our re-implementation of the ICM baseline is correct — and also demonstrate at least 2 times faster convergence of our method with respect to the baseline. Second, in the randomized procedurally generated environments from DMLab our method turns out to be more robust to spurious behaviours than the method ICM: while the baseline learns a persistent firing behaviour in navigational tasks (thus creating interesting pictures for itself), our method learns a reasonable explorative behaviour. In terms of quantitative evaluation, our method reaches the goal at least 2 times more often in the procedurally generated test levels in DMLab with a very sparse reward. Third, when comparing the behaviour of the agent in the complete absence of rewards, our method covers at least 4 times more area (measured in discrete $( x , y )$ coordinate cells) than the baseline ICM. Fourth, we demonstrate that our curiosity bonus does not significantly deteriorate performance of the plain PPO algorithm (Schulman et al., 2017) in two tasks with dense reward in DMLab. Finally, we demonstrate that an ant in a MuJoCo environment can learn locomotion purely from our curiosity reward computed based on the first-person view.
+
+# 2 EPISODIC CURIOSITY
+
+We consider an agent which interacts with an environment. The interactions happen at discrete time steps over the episodes of limited duration $T$ . At each time step $t$ , the environment provides the agent with an observation $\mathbf { o } _ { t }$ from the observational space $\mathcal { O }$ (we consider images), samples an action $a _ { t }$ from a set of actions $\mathcal { A }$ using a probabilistic policy $\pi ( \mathbf { o } _ { t } )$ and receives a scalar reward $r _ { t } ~ \in \mathbb { R }$ together with the new observation $\mathbf { o } _ { t + 1 }$ and an end-of-episode indicator. The goal of the agent is to optimize the expectation of the discounted sum of rewards during the episode $\begin{array} { r } { \mathbf { \bar { \boldsymbol { S } } } = \sum _ { t } \gamma ^ { t } \bar { \boldsymbol { r } _ { t } } } \end{array}$ .
+
+In this work we primarily focus on the tasks where rewards $r _ { t }$ are sparse — that is, zero for most of the time steps $t$ . Under such conditions commonly used RL algorithms (e.g., PPO Schulman et al. (2017)) do not work well. We further introduce an episodic curiosity (EC) module which alleviates this problem. The purpose of this module is to produce a reward bonus $b _ { t }$ which is further summed up with the task reward $r _ { t }$ to give an augmented reward $\widehat { r } _ { t } = r _ { t } + b _ { t }$ . The augmented reward has a bnice property from the RL point of view — it is a dense reward. Learning with such reward is faster, more stable and often leads to better final performance in terms of the cumulative task reward $S$ .
+
+![](images/f5b26daba6d4f270ed00487ec3e61377a5ec67c3a3888291d596d0792ad828d2.jpg)  
+Figure 2: Left: siamese architecture of reachability (R) network. Right: R-network is trained based on a sequence of observations that the agent encounters while acting. The temporally close (within threshold) pairs of observations are positive examples, while temporally far ones — negatives.
+
+In the following section we describe the key components of our episodic curiosity module.
+
+# 2.1 EPISODIC CURIOSITY MODULE
+
+The episodic curiosity (EC) module takes the current observation $\mathbf { o }$ as input and produces a reward bonus $b$ . The module consists of both parametric and non-parametric components. There are two parametric components: an embedding network $E : \mathcal { O }  \mathbb { R } ^ { n }$ and a comparator network $C$ : $\mathbb { R } ^ { n } \times \mathbb { R } ^ { n } \to [ 0 , 1 ]$ . Those parametric components are trained together to predict reachability as parts of the reachability network — shown in Figure 2. There are also two non-parametric components: an episodic memory buffer $\mathbf { M }$ and a reward bonus estimation function $B$ . The high-level overview of the system is shown in Figure 3. Next, we give a detailed explanation of all the components.
+
+Embedding and comparator networks. Both networks are designed to function jointly for estimating within- $k$ -step-reachability of one observation $\mathbf { o } _ { i }$ from another observation $\mathbf { o } _ { j }$ as parts of a reachability network $\mathbf { \bar { \mathit { R } } } ( \mathbf { o } _ { i } , \mathbf { o } _ { j } ) = \mathbf { \bar { \mathit { C } } } ( E ( \mathbf { o } _ { i } ) , E ( \mathbf { o } _ { j } ) )$ . This is a siamese architecture similar to (Zagoruyko & Komodakis, 2015). The architecture is shown in Figure 2. R-network is a classifier trained with a logistic regression loss: it predicts values close to 0 if probability of two observations being reachable from one another within $k$ steps is low, and values close to 1 when this probability is high. Inside the episodic curiosity the two networks are used separately to save up computation and memory.
+
+Episodic memory. The episodic memory buffer $\mathbf { M }$ stores embeddings of past observations from the current episode, computed with the embedding network $E$ . The memory buffer has a limited capacity $K$ to avoid memory and performance issues. At every step, the embedding of the current observation might be added to the memory. What to do when the capacity is exceeded? One solution we found working well in practice is to substitute a random element in memory with the current element. This way there are still more fresh elements in memory than older ones, but the older elements are not totally neglected.
+
+Reward bonus estimation module. The purpose of this module is to check for reachable observations in memory and if none is found — assign larger reward bonus to the current time step. The check is done by comparing embeddings in memory to the current embedding via comparator network. Essentially, this check insures that no observation in memory can be reached by taking only a few actions from the current state — our characterization of novelty.
+
+# 2.2 BONUS COMPUTATION ALGORITHM.
+
+At every time step, the current observation o goes through the embedding network producing the embedding vector ${ \bf e } = E ( { \bf o } )$ . This embedding vector is compared with the stored embeddings in the memory buffer $\mathbf { M } = \left. \mathbf { e } _ { 1 } , \ldots , \mathbf { e } _ { | \mathbf { M } | } \right.$ via the comparator network $C$ where $| \mathbf { M } |$ is the current number of elements in memory. This comparator network fills the reachability buffer with values
+
+$$
+c _ { i } = C ( \mathbf { e } _ { i } , \mathbf { e } ) , \quad i = 1 , | \mathbf { M } | .
+$$
+
+![](images/fe553ae84666c820b2b22f915708261f314a64fb08a4e1896ecd8eee94538393.jpg)  
+Figure 3: The use of episodic curiosity (EC) module for reward bonus computation. The module take a current observation as input and computes a reward bonus which is higher for novel observations. This bonus is later summed up with the task reward and used for training an RL agent.
+
+Then the similarity score between the memory buffer and the current embedding is computed from the reachability buffer as (with a slight abuse of notation)
+
+$$
+C ( \mathbf { M } , \mathbf { e } ) = F \left( c _ { 1 } , \ldots , c _ { | \mathbf { M } | } \right) \in [ 0 , 1 ] .
+$$
+
+where the aggregation function $F$ is a hyperparameter of our method. Theoretically, $F = { \mathrm { m a x } }$ would be a good choice, however, in practice it is prone to outliers coming from the parametric embedding and comparator networks. Empirically, we found that 90-th percentile works well as a robust substitute to maximum.
+
+As a curiosity bonus, we take
+
+$$
+b = B ( \mathbf { M } , \mathbf { e } ) = \alpha ( \beta - C ( \mathbf { M } , \mathbf { e } ) ) ,
+$$
+
+where $\alpha \in \mathbb { R } ^ { + }$ and $\beta \in \mathbb { R }$ are hyperparameters of our method. The value of $\alpha$ depends on the scale of task rewards — we will discuss how to select it in the experimental section. The value of $\beta$ determines the sign of the reward — and thus could bias the episodes to be shorter or longer. Empirically, $\beta = 0 . 5$ works well for fixed-duration episodes, and $\beta = 1$ is preferred if an episode could have variable length.
+
+After the bonus computation, the observation embedding is added to memory if the bonus $b$ is larger than a novelty threshold $b _ { n o v e l t y }$ . This check is necessary for the following reason. If every observation embedding is added to the memory buffer, the observation from the current step will always be reachable from the previous step. Thus, the reward would never be granted. The threshold $b _ { n o v e l t y }$ induces a discretization in the embedding space. Intuitively, this makes sense: only “distinct enough” memories are stored. As a side benefit, the memory buffer stores information with much less redundancy. We refer the reader to the video1 which visualizes the curiosity reward bonus and the memory state during the operation of the algorithm.
+
+# 2.3 REACHABILITY NETWORK TRAINING
+
+If the full transition graph in Figure 1 was available, there would be no need of a reachability network and the novelty could be computed analytically through the shortest-path algorithm. However, normally we have access only to the sequence of observations which the agent receives while acting. Fortunately, as suggested by (Savinov et al., 2018), even a simple observation sequence graph could still be used for training a reasonable approximator to the real step-distance. This procedure is illustrated in Figure 2. This procedure takes as input a sequence of observations $\mathbf { o } _ { 1 } , \ldots , \mathbf { o } _ { N }$ and forms pairs from those observations. The pairs $( \mathbf { o } _ { i } , \mathbf { o } _ { j } )$ where $| i - j | \leq k$ are taken as positive (reachable) examples while the pairs with $| i - j | > \gamma k$ become negative examples. The hyperparameter $\gamma$ is necessary to create a gap between positive and negative examples. In the end, the network is trained with logistic regression loss to output the probability of the positive (reachable) class.
+
+In our work, we have explored two settings for training a reachability network: using a random policy and together with the task-solving policy (online training). The first version generally follows the training protocol proposed by (Savinov et al., 2018). We put the agent into exactly the same conditions where it will be eventually tested: same episode duration and same action set. The agent takes random actions from the action set. Given the environment interaction budget (2.5M 4-repeated steps in DMLab, 300K 4-repeated steps in VizDoom), the agent fills in the replay buffer with observations coming from its interactions with the environment, and forms training pairs by sampling from this replay buffer randomly. The second version collects the data on-policy, and re-trains the reachability network every time after a fixed number of environment interactions is performed. We provide the details of R-network training in the supplementary material.
+
+![](images/7bc1149c05ae19ecbfb228b2f6acd49ac54440234b7b7a2ce2986bf49364efec.jpg)  
+Figure 4: Examples of tasks considered in our experiments: (a) VizDoom static maze goal reaching, (b) DMLab randomized maze goal reaching, (c) DMLab key-door puzzle, (d) MuJoCo ant locomotion out of first-person-view curiosity.
+
+# 3 EXPERIMENTAL SETUP
+
+We test our method in multiple environments from VizDoom (Kempka et al., 2016), DMLab (Beattie et al., 2016) and MuJoCo (Todorov et al., 2012; Schulman et al., 2015). The experiments in $V _ { l Z } .$ - Doom allow us to verify that our re-implementation of the previous state-of-the-art curiosity method ICM (Pathak et al., 2017) is correct. The experiments in DMLab allow us to extensively test the generalization of our method as well as baselines — DMLab provides convenient procedural level generation capabilities which allows us to train and test RL methods on hundreds of levels. The experiments in MuJoCo allow us to show the generality of our method. Due to space limits, the MuJoCo experiments are described in the supplementary material. The examples of tasks are shown in Figure 4.
+
+Environments. Both VizDoom and DMLab environments provide rich maze-like 3D environments. The observations are given to the agent in the form of images. For VizDoom, we use $8 4 \times 8 4$ grayscale images as input. For DMLab, we use $8 4 \times 8 4$ RGB images as input. The agent operates with a discrete action set which comprises different navigational actions. For VizDoom, the standard action set consists of 3 actions: move forward, turn left/right. For DMLab, it consists of 9 actions: move forward/backward, turn left/right, strafe left/right, turn left/righ $^ +$ move forward, fire. For both VizDoom and DMLab we use all actions with a repeat of 4, as typical in the prior work. We only use RGB input of the provided RGBD observations and remove all head-on display information from the screen, leaving only the plain first-person view images of the maze. The rewards and episode durations differ between particular environments and will be further specified in the corresponding experimental sections.
+
+Basic RL algorithm. We choose the commonly used PPO algorithm from the open-source implementation2 as our basic RL algorithm. The policy and value functions are represented as CNNs to reduce number of hyperparameters — LSTMs are harder to tune and such tuning is orthogonal to the contribution of the paper. We apply PPO to the sum of the task reward and the bonus reward coming from specific curiosity algorithms. The hyperparameters of the PPO algorithm are given in the supplementary material. We use only two sets of hyperparameters: one for all VizDoom environments and the other one for all DMLab environments.
+
+Baseline methods. The simplest baseline for our approach is just the basic RL algorithm applied to the task reward. As suggested by the prior work and our experiments, this is a relatively weak baseline in the tasks where reward is sparse.
+
+As the second baseline, we take the state-of-the-art curiosity method ICM (Pathak et al., 2017). As follows from the results in (Pathak et al., 2017; Fu et al., 2017), ICM is superior to methods VIME (Houthooft et al., 2016), #Exploration (Tang et al., 2017) and $E X ^ { 2 }$ (Fu et al., 2017) on the curiosity tasks in visually rich 3D environments.
+
+![](images/4b4ec1c71c7a6aeb17662e33b371a1b670e3ba65f63ed9b7b43b2c4515e35f00.jpg)  
+Figure 5: Examples of maze types used in our experiments: (a) VizDoom static maze goal reaching, (b) DMLab randomized maze goal reaching, (c) DMLab randomized maze goal reaching with doors.
+
+Finally, as a sanity check, we introduce a novel baseline method which we call Grid Oracle. Since we can access current $( x , y )$ coordinates of the agent in all environments, we are able to directly discretize the world in 2D cells and reward the agent for visiting as many cells as possible during the episode (the reward bonus is proportional to the number of cells visited). At the end of the episode, cell visit counts are zeroed. The reader should keep in mind that this baseline uses privileged information not available to other methods (including our own method EC). While this privileged information is not guaranteed to lead to success in any particular RL task, we do observe this baseline to perform strongly in many tasks, especially in complicated DMLab environments. The Grid Oracle baseline has two hyperparameters: the weight for combining Grid Oracle reward with the task reward and the cell size.
+
+Hyperparameter tuning. As DMLab environments are procedurally generated, we perform tuning on the validation set, disjoint with the training and test sets. The tuning is done on one of the environments and then the same hyperparameters are re-used for all other environments. VizDoom environments are not procedurally generated, so there is no trivial way to have proper training/validation/test splits — so we tune on the same environment (as typical in the prior RL work for the environments without splits). When tuning, we consider the mean final reward of 10 training runs with the same set of hyperparameters as the objective — thus we do not perform any seed tuning. All hyperparameter values are listed in the supplementary material. Note that although bonus scalar $\alpha$ depends on the range of task rewards, the environments in VizDoom and DMLab have similar ranges within each platform — so our approach with re-using $\alpha$ for multiple environments works.
+
+# 4 EXPERIMENTS
+
+In this section, we describe the specific tasks we are solving and experimental results for all considered methods on those tasks. There are 4 methods to report: PPO, $\mathrm { P P O } + \mathrm { I C M }$ , PPO $^ +$ Grid Oracle and $\mathrm { P P O } + \mathrm { E C }$ (our method). First, we test static-maze goal reaching in VizDoom environments from prior work to verify that our baseline re-implementation is correct. Second, we test the goal-reaching behaviour in procedurally generated mazes in DMLab. Third, we train no-reward (pure curiosity) maze exploration on the levels from DMLab and report Grid Oracle reward as an approximate measure of the maze coverage. Finally, we demonstrate that our curiosity bonus does not significantly deteriorate performance in two dense reward tasks in DMLab. All the experiments were conducted under the same environment interaction budget for all methods (R-network pre-training is included in this budget). The videos of all trained agents in all environments are available online3.
+
+For additional experiments we refer the reader to the supplementary material: there we show that R-network can successfully generalize between environments, demonstrate stability of our method to hyperparameters and present an ablation study.
+
+# 4.1 STATIC MAZE GOAL REACHING.
+
+The goal of this experiment is to verify our re-implementation of the baseline method is correct. We use the MyWayHome task from VizDoom. The agent has to reach the goal in a static 3D maze in the time limit of 525 4-repeated steps (equivalent to 1 minute). It only gets a reward of $+ 1$ when it reaches the goal (episode ends at that moment), the rest of the time the reward is zero.
+
+The task has three sub-tasks (following the setup in (Pathak et al., 2017)): “Dense”, “Sparse” and “Very Sparse”. The layout of the maze is demonstrated in Figure 5(c). The goal is always at the same room but the starting points are different in those sub-tasks. For the “Dense” subtask, the agent starts in one of the random locations in the maze, some of which are close to the goal. In this sub-task, the reward is relatively dense (hence the name): the agent is likely to bump into the goal by a short random walk. Thus, this is an easy task even for standard RL methods. The other two sub-tasks are harder: the agent starts in a medium-distant room from the goal (“Sparse”) or in a very distant room (“Very Sparse”). Those tasks are hard for standard RL algorithms because the probability of bumping into a rewarding state by a random walk is very low.
+
+![](images/7a7f0a2a8c5d62a3afa36a1d440e1c159733fc1f9262137063054b93a4cf2f54.jpg)  
+Figure 6: Task reward as a function of training step for VizDoom tasks. Higher is better. We use the offline version of our algorithm and shift the curves for our method by the number of environment steps used to train R-network — so the comparison is fair. We run every method with a repeat of 3 (same as in prior work (Pathak et al., 2017)) and show all runs. No seed tuning is performed.
+
+The training curves are shown in Figure 6. By analysing them, we draw a few conclusions. First, our re-implementation of the ICM baseline is correct and the results are in line with those published in (Pathak et al., 2017). Second, our method works on-par with the ICM baseline in terms of final performance, quickly reaching $1 0 0 \%$ success rate in all three sub-tasks. Finally, in terms of convergence speed, our algorithm is significantly faster than the state-of-the-art method ICM — our method reaches $1 0 0 \%$ success rate at least 2 times faster. Note that to make the comparison of the training speed fair, we shift our training curves by the environment interaction budget used for training R-network.
+
+# 4.2 PROCEDURALLY GENERATED RANDOM MAZE GOAL REACHING.
+
+In this experiment we aim to evaluate maze goal reaching task generalization on a large scale. We train on hundreds of levels and then test also on hundreds of hold-out levels. We use “Explore Goal Locations Large” (we will denote it “Sparse”) and “Explore Obstructed Goals Large” (we will denote it “Sparse $^ +$ Doors”) levels in the DMLab simulator. In those levels, the agent starts in a random location in a randomly generated maze (both layout and textures are randomized at the beginning of the episode). Within the time limit of 1800 4-repeated steps (equivalent to 2 minutes), the agent has to reach the goal as many times as possible. Every time it reaches a goal, it is respawned into another random location in the maze and has to go to the goal again. Every time the goal is reached, the agent gets a reward $+ 1 0$ , the rest of the time the reward is zero. The second level is a variation of the first one with doors which make the paths in the maze longer. The layouts of the levels are demonstrated in Figure 5(b,c).
+
+We found out that the standard task “Sparse” is actually relatively easy even for the plain PPO algorithm. The reason is that the agent starting point and the goal are sampled on the map independently of each other — and sometimes both happen to be in the same room which simplifies the task. To test the limits of the algorithms, we create a gap between the starting point and the goal which eliminates same-room initialization. We report the results for both the original task “Sparse” and its harder version “Very Sparse”. Thus, there are overall three tasks considered in this section: “Sparse”, “Very Sparse” and “Sparse $^ +$ Doors”.
+
+The results demonstrate that our method can reasonably adapt to ever-changing layouts and textures — see Table 1 and training curves in Figure 7. We outperform the baseline method ICM in all three environments using the same environment interaction budget of 20M 4-repeated steps. The environment “Sparse” is relatively easy and all methods work reasonably. In the “Very Sparse” and “Sparse $^ +$ Doors” settings our advantage with respect to PPO and ICM is more clear. On those levels, the visual inspection of the ICM learnt behaviour reveals an important property of this method: it is confused by the firing action and learns to entertain itself by firing until it runs out of ammunition. A similar finding was reported in a concurrent work (Burda et al., 2018a): the agent was given an action which switched the content on a TV screen in a maze, along with the movement actions. Instead of moving, the agent learns to switch channels forever. While one might intuitively accept such “couch-potato” behaviour in intelligent creatures, it does not need to be a consequence of curious behaviour. In particular, we are not observing such dramatic firing behaviour for our curiosity formulation: according to Figure 1, an observation after firing is still one step away from the one before firing, so it is not novel (note that firing still could happen in practice because of the entropy term in PPO). Thus, our formulation turns out to be more robust than ICM’s prediction error in this scenario. Note that we do not specifically look for an action set which breaks the baseline — just use the standard one for DMLab, in line with the prior work (e.g., (Espeholt et al., 2018)).
+
+![](images/36cc35956581da1d343973680cdeb87e5dffe96d9999679f04ee98cacf28fa1b.jpg)  
+Figure 7: Reward as a function of training step for DMLab tasks. Higher is better. “ECO” stands for the online version of our method, which trains R-network and the policy at the same time. We run every method 30 times and show 5 randomly selected runs. No seed tuning is performed.
+
+The result of this experiment suggests to look more into how methods behave in extremely-sparse reward scenarios. The limiting case would be no reward at all — we consider it in the next section.
+
+# 4.3 NO REWARD/AREA COVERAGE.
+
+This experiment aims to quantitatively establish how good our method is in the scenario when no task reward is given. One might question why this scenario is interesting — however, before the task reward is found for the first time, the agent lives in the no-reward world. How it behaves in this case will also determine how likely it is to stumble into the task reward in the first place.
+
+We use one of the DMLab levels — “Sparse” from the previous experiment. We modify the task to eliminate the reward and name the new task “No Reward”. To quantify success in this task, we report the reward coming from Grid Oracle for all compared methods. This reward provides a discrete approximation to the area covered by the agent while exploring.
+
+The training curves are shown in Figure 7 and the final test results in Table 1. The result of this experiment is that our method and Grid Oracle both work, while the ICM baseline is not working — and the qualitative difference in behaviour is bigger than in the previous experiments. As can be seen from the training curves, after a temporary increase, ICM quality actually decreases over time, rendering a sharp disagreement between the prediction-error-based bonus and the area coverage metric. By looking at the video3, we observe that the firing behaviour of ICM becomes even more prominent, while our method still shows reasonable exploration.
+
+Finally, we try to find out if the ICM baseline behaviour above is due to the firing action only. Could it learn exploration of randomized mazes if the Fire action is excluded from the action set? For that purpose, we create a new version of the task — we call it “No Reward - Fire”. This task demonstrates qualitatively similar results to the one with the full action set — see Table 1. By looking at the videos3, we hypothesise that the agent can most significantly change its current view when it is close to the wall — thus increasing one-step prediction error — so it tends to get stuck near “interesting” diverse textures on the walls.
+
+The results suggest that in an environment completely without reward, the ICM method will exhaust its curiosity very quickly — passing through a sharp peak and then degrading into undesired behaviour. This observation raises concerns: what if ICM passes the peak before it reaches the first task reward in the cases of real tasks? Supposedly, it would require careful tuning per-game. Furthermore, in some cases, it would take a lot of time with a good exploration behaviour to reach the first reward, which would require to stay at the top performance for longer — which is problematic for the ICM method but still possible for our method.
+
+Table 1: Reward in DMLab tasks (mean $\pm$ std) for all compared methods. Higher is better. “ECO” stands for the online version of our method, which trains R-network and the policy at the same time. We report Grid Oracle reward in tasks with no reward. The Grid Oracle method is given for reference — it uses privileged information unavailable to other methods. Results are averaged over 30 random seeds. No seed tuning is performed.   
+
+<table><tr><td>Method</td><td>Sparse</td><td>Very Sparse</td><td>Sparse+Doors</td><td>No Reward</td><td>No Reward - Fire</td><td>Dense 1</td><td>Dense 2</td></tr><tr><td>PPO</td><td>27.0 ± 5.1</td><td>8.6±4.3</td><td>1.5 ± 0.1</td><td>191 ± 12</td><td>217 ±19</td><td>22.8±0.5</td><td>9.41 ± 0.02</td></tr><tr><td>PPO + ICM</td><td>23.8±2.8</td><td>11.2 ± 3.9</td><td>2.7±0.2</td><td>72±2</td><td>87±3</td><td>20.9±0.6</td><td>9.39 ± 0.02</td></tr><tr><td>PPO + EC (ours)</td><td>26.2 ± 1.9</td><td>24.7 ± 2.2</td><td>8.5±0.6</td><td>475±8</td><td>492 ± 10</td><td>19.9 ± 0.7</td><td>9.53 ± 0.03</td></tr><tr><td>PPO + ECO (ours)</td><td>41.6 ± 1.7</td><td>40.5 ± 1.1</td><td>19.8 ± 0.5</td><td>472±18</td><td>457±32</td><td>22.9 ± 0.4</td><td>9.60 ± 0.02</td></tr><tr><td>PPO + Grid Oracle</td><td>56.7 ± 1.3</td><td>54.3 ±1.2</td><td>29.4± 0.5</td><td>796±2</td><td>795±3</td><td>20.9±0.6</td><td>8.97 ±0.04</td></tr></table>
+
+# 4.4 DENSE REWARD TASKS.
+
+A desirable property of a good curiosity bonus is to avoid hurting performance in dense-reward tasks (in addition to improving performance for sparse-reward tasks). We test this scenario in two levels in the DMLab simulator: “Rooms Keys Doors Puzzle” (which we denote “Dense 1”) and “Rooms Collect Good Objects Train” (which we denote “Dense $2 ^ { \circ }$ ). In the first task, the agent has to collect keys and reach the goal object behind a few doors openable by those keys. The rewards in this task are rather dense (key collection/door opening is rewarded). In the second task the agent has to collect good objects (give positive reward) and avoid bad objects (give negative reward). The episode lasts for 900 4-repeated steps (equivalent to 1 minute) in both tasks.
+
+The results show that our method indeed does not significantly deteriorate performance of plain PPO in those dense-reward tasks — see Table 1. The training curves for “Dense 1” are shown in Figure 7 and for “Dense 2” — in the supplementary material. Note that we use the same bonus weight in this task as in other DMLab tasks before. All methods work similarly besides the Grid Oracle in the “Dense 2” task — which performs slightly worse. Video inspection3 reveals that Grid Oracle — the only method which has ground-truth knowledge about area it covers during training — sometimes runs around excessively and occasionally fails to collect all good objects.
+
+# 5 DISCUSSION
+
+Our method is at the intersection of multiple topics: curiosity, episodic memory and temporal distance prediction. In the following, we discuss the relation to the prior work on those topics.
+
+Curiosity in visually rich 3D environments. Recently, a few works demonstrated the possibility to learn exploration behaviour in visually rich 3D environments like DMLab (Beattie et al., 2016) and VizDoom (Kempka et al., 2016). (Pathak et al., 2017) trains a predictor for the embedding of the next observation and if the reality is significantly different from the prediction — rewards the agent. In that work, the embedding is trained with the purpose to be a good embedding for predicting action taken between observations — unlike an earlier work (Stadie et al., 2015) which obtains an embedding from an autoencoder. It was later shown by (Burda et al., 2018a) that the perceptive prediction approach has a downside — the agent could become a “couch-potato” if given an action to switch TV channels. This observation is confirmed in our experiments by observing a persistent firing behaviour of the ICM baseline in the navigational tasks with very sparse or no reward. By contrast, our method does not show this behaviour. Another work (Fu et al., 2017) trains a temporal distance predictor and then uses this predictor to establish novelty: if the observation is easy to classify versus previous observations, it is novel. This method does not use episodic memory, however, and the predictor is used in way which is different from our work.
+
+General curiosity. Curiosity-based exploration for RL has been extensively studied in the literature. For an overview, we refer the reader to the works (Oudeyer & Kaplan, 2009; Oudeyer et al., 2007). The most common practical approaches could be divided into three branches: predictionerror-based, count-based and goal-generation-based. Since the prediction-based approaches were discussed before, in the following we focus on the latter two branches.
+
+The count-based approach suggests to keep visit counts for observations and concentrate on visiting states which has been rarely visited before — which bears distant similarity to how we use episodic memory. This idea is natural for discrete observation spaces and has solid theoretical foundations. Its extension to continuous observation spaces is non-trivial, however. The notable step in this direction was taken by works (Bellemare et al., 2016; Ostrovski et al., 2017) which introduce a trained observation density model which is later converted to a function behaving similarly to counts. The way conversion is done has some similarity to prediction-error-based approaches: it is the difference of the density in the example before and after training of this example which is converted to count. The experiments in the original works operate on Atari games (Bellemare et al., 2013) and were not benchmarked on visually rich 3D environments. Another approach (Tang et al., 2017) discretises the continuous observation space by hashing and then uses the count-based approach in this discretised space. This method is appealing in its simplicity, however, the experiments in (Pathak et al., 2017; Fu et al., 2017) show that it does not perform well in visually rich 3D environments. Another line of work, Novelty Search (Lehman & Stanley, 2011) and its recent follow-up (Conti et al., 2018), proposed maintaining an archive of behaviours and comparing current behaviour to those — however, the comparison is done by euclidean distance and behaviours are encoded using coordinates, while we learn the comparison function and only use pixels.
+
+Finally, our concept of novelty through reachability is reminiscent of generating the goals which are reachable but not too easy — a well-studied topic in the prior work. The work (Held et al., 2017) uses a GAN to differentiate what is easy to reach from what is not and then generate goals at the boundary. Another work (Baranes & Oudeyer, 2013) defines new goals according to the expected progress the agent will make if it learns to solve the associated task. The recent work (Per´ e et al. ´ , 2018) learns an embedding for the goal space and then samples increasingly difficult goals from that space. In a spirit similar to those works, our method implicitly defines goals that are at least some fixed number of steps away by using the reachability network. However, our method is easier to implement than other goal-generation methods and quite general.
+
+Episodic memory. Two recent works (Blundell et al., 2016; Pritzel et al., 2017) were inspired by the ideas of episodic memory in animals and proposed an approach to learn the functioning of episodic memory along with the task for which this memory is applied. Those works are more focused on repeating successful strategies than on exploring environments — and are not designed to work in the absence of task rewards.
+
+Temporal distance prediction. The idea to predict the distance between video frames has been studied extensively. Usually this prediction is an auxiliary task for solving another problem. (Sermanet et al., 2017) trains an embedding such that closer in time frames are also closer in the embedding space. Multiple works (Fu et al., 2017; Savinov et al., 2018; Aytar et al., 2018) train a binary classifier for predicting if the distance in time between frames is within a certain threshold or not. While (Sermanet et al., 2017; Aytar et al., 2018) use only the embedding for their algorithms, (Fu et al., 2017; Savinov et al., 2018) also use the classifier trained together with the embedding. As mentioned earlier, (Fu et al., 2017) uses this classifier for density estimation instead of comparison to episodic memory. (Savinov et al., 2018) does compare to the episodic memory buffer but solves a different task — given an already provided exploration video, navigate to a goal — which is complementary to the task in our work.
+
+# 6 CONCLUSION
+
+In this work we propose a new model of curiosity based on episodic memory and the ideas of reachability. This allows us to overcome the known “couch-potato” issues of prior work and outperform the previous curiosity state-of-the-art method ICM in visually rich 3D environments from VizDoom and DMLab. Our method also allows a MuJoCo ant to learn locomotion purely out of first-personview curiosity. In the future, we want to make policy aware of memory not only in terms of receiving reward, but also in terms of acting. Can we use memory content retrieved based on reachability to guide exploration behaviour in the test time? This could open opportunities to learn exploration in new tasks in a few-shot style — which is currently a big scientific challenge.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank Olivier Pietquin, Alexey Dosovitskiy, Vladlen Koltun, Carlos Riquelme, Charles Blundell, Sergey Levine and Matthieu Geist for the valuable discussions about our work.
+
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+
+# SUPPLEMENTARY MATERIAL
+
+The supplementary material is organized as follows. First, we describe the MuJoCo locomotion experiments. Then we provide training details for R-network. After that, we list hyperparameter values and the details of hyperparameter search for all methods. Then we show experimental results which suggest that R-network can generalize between environments: we transfer one general Rnetwork from all available DMLab30 levels to our tasks of interest and also transfer R-networks between single environments. After that, we present the results from a stability/ablation study which suggests our method is stable with respect to its most important hyperparameters and the components we used in the method are actually necessary for its performance (and measure their influence). Then we demonstrate the robustness of our method to the environments where every state has a stochastic next state. After that, we discuss computational considerations for our method. Finally, we provide the training curves for the “Dense $2 ^ { \circ }$ task in the main text.
+
+# S1 MuJoCo ANT LOCOMOTION OUT OF FIRST-PERSON-VIEW CURIOSITY
+
+Equipped with our curiosity module, a MuJoCo ant has learned4 to move out of curiosity based on the first-person view5.
+
+First, let us describe the setup:
+
+• Environment: the standard MuJoCo environment is a plane with a uniform or repetitive texture on it — nothing to be visually curious about. To fix that, we tiled the $4 0 0 \times 4 0 0$ floor into squares of size $4 \times 4$ . Each tile is assigned a random texture from a set of 190 textures at the beginning of every episode. The ant is initialized at a random location in the $2 0 0 \times 2 0 0$ central square of the floor. The episode lasts for 1000 steps (no action repeat is used). If the $z$ -coordinate of the center of mass of the ant is above 1.0 or below 0.2 — the episode ends prematurely (standard termination condition). Observation space: for computing the curiosity reward, we only use a first-person view camera mounted on the ant (that way we can use the same architecture of our curiosity module as in VizDoom and DMLab). For policy, we use the standard body features from Ant-v2 in gym-mujoco6 (joint angles, velocities, etc.).   
+• Action space: standard continuous space from Ant-v2 in gym-mujoco.   
+• Basic RL solver: PPO (same as in the main text of the paper).   
+• Baselines: PPO on task reward, PPO on task reward plus constant reward 1 at every step as a trivial curiosity bonus (which we denote $\mathrm { P P O } { + } 1$ , it optimizes for longer survival).
+
+Second, we present quantitative results for the setting with no task reward after 10M training steps in Table S1 (the first row). Our method outperforms the baselines. As seen in the videos7, PPO (random policy) dies quickly, $\mathrm { P P O } { + } 1$ survives for longer but does not move much and our method moves around the environment.
+
+Additionally, we performed an experiment with an extremely sparse task reward — which we call “Escape Circle”. The reward is given as follows: 0 reward inside the circle of radius 10, and starting from 10, we give a one-time reward of 1 every time an agent goes through a concentric circle of radius $1 0 + 0 . 5 k$ (for integer $k \geq 0$ ). The results at 10M training steps are shown in Table S1 (the second row). Our method significantly outperforms the baselines (better than the best baseline by a factor of 10).
+
+Finally, let us discuss the relation to some other works in the field of learning locomotion from intrinsic reward. The closest work in terms of task setup is the concurrent work (Burda et al., 2018a). The authors demonstrate slow motion8 of the ant learned from pixel-based curiosity only.
+
+Other works use state features (joint angles, velocities, etc.) for formulating intrinsic reward, not pixels — which is a different setup. One work in this direction is the concurrent work (Eysenbach et al., 2018) — which also contains a good overview of the literature on intrinsic reward from state features.
+
+Table S1: Learning locomotion for MuJoCo Ant. For “No reward”, the task reward is 0 (so plain PPO is a random policy), and Grid Oracle rewards are reported (with cell size 5). Results are averaged over 30 random seeds for “No reward” and over 10 random seeds for “Escape Circle”. No seed tuning is performed.   
+
+<table><tr><td>Task</td><td>PPO</td><td>PPO+1</td><td>PPO + EC (ours)</td></tr><tr><td>No Reward</td><td>1.4 ± 0.02</td><td>1.7 ± 0.06</td><td>5.0 ± 0.27</td></tr><tr><td>Escape Circle</td><td>0.59 ± 0.54</td><td>0.45 ± 0.39</td><td>6.53 ± 3.57</td></tr></table>
+
+# S2 REACHABILITY NETWORK TRAINING DETAILS
+
+For training R-network, we use mini-batches of 64 observation pairs (matched within episodes). The training is run for 50K mini-batch iterations for VizDoom and 200K mini-batch iterations for DMLab. At the beginning of every pass through the buffer, we re-shuffle it. We use Adam optimizer with learning rate $\bar { 1 } 0 ^ { - 4 }$ . The R-network uses a siamese architecture with two branches (see Figure 2 in the main text), each branch is Resnet-18 with 512 outputs, with a fully-connected network applied to the concatenated output of the branches. The fully-connected network has four hidden layers with 512 units, batch-normalization and ReLU is applied after each layer besides the last one, which is a softmax layer. Observations are RGB-images with resolution $1 6 0 \times 1 2 0$ pixels.
+
+For online training of the R-network, we collect the experience and perform training every 720K 4- repeated environment steps. Every time the experience is collected, we make 10 epochs of training on this experience. Before every epoch, the data is shuffled.
+
+# S3 HYPERPARAMETERS
+
+The hyperparameters of different methods are given in Table S2 for VizDoom environment, in Table S3 for DMLab environment, and in Tables S4, S5 for MuJoCo Ant environment. The hyperparameters for DMLab are tuned on the “Sparse” environment for all methods — because all methods work reasonably on this environment (it is unfair to tune a method on an environment where it fails and also unfair to tune different methods on different environments). We use the PPO algorithm from the open-source implementation9. For implementation convenience, we scale both the bonus and the task reward (with a single balancing coefficient it would not be possible to turn off one of those rewards).
+
+Table S2: Hyper-parameters used for VizDoom environment.   
+
+<table><tr><td></td><td>PPO</td><td>PPO +ICM</td><td>PPO + EC</td></tr><tr><td>Learning rate</td><td>0.00025</td><td>0.00025</td><td>0.00025</td></tr><tr><td>PPO entropy coefficient</td><td>0.01</td><td>0.01</td><td>0.01</td></tr><tr><td>Task reward scale</td><td>5</td><td>5</td><td>5</td></tr><tr><td>Curiosity bonus scale α</td><td>0</td><td>0.01</td><td>1</td></tr><tr><td>ICM forward inverse ratio</td><td>-</td><td>0.2</td><td>-</td></tr><tr><td>ICM curiosity loss strength</td><td>-</td><td>10</td><td>-</td></tr><tr><td>EC memory size</td><td></td><td>1</td><td>200</td></tr><tr><td>EC reward shift β</td><td></td><td></td><td>0.5</td></tr><tr><td>EC novelty threshold bnovelty</td><td></td><td></td><td>0</td></tr><tr><td>EC aggregation function F</td><td></td><td></td><td>percentile-90</td></tr></table>
+
+Table S3: Hyper-parameters used for DMLab environment.   
+
+<table><tr><td></td><td>PPO</td><td>PPO+ICM</td><td>PPO + Grid Oracle</td><td>PPO + EC</td></tr><tr><td>Learning rate</td><td>0.00019</td><td>0.00025</td><td>0.00025</td><td>0.00025</td></tr><tr><td>PPO entropy coefficient</td><td>0.0011</td><td>0.0042</td><td>0.0066</td><td>0.0021</td></tr><tr><td>Task reward scale</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Curiosity bonus scale α</td><td>0</td><td>0.55</td><td>0.052</td><td>0.030</td></tr><tr><td>Grid Oracle cell size</td><td>-</td><td>=</td><td>30</td><td>-</td></tr><tr><td>ICM forward inverse ratio</td><td>=</td><td>0.96</td><td>-</td><td>-</td></tr><tr><td>ICM curiosity loss strength</td><td></td><td>64</td><td>-</td><td>-</td></tr><tr><td>EC memory size</td><td></td><td>-</td><td></td><td>200</td></tr><tr><td>EC reward shift β</td><td></td><td>-</td><td>1</td><td>0.5</td></tr><tr><td>EC novelty threshold bnovelty</td><td></td><td></td><td></td><td>0</td></tr><tr><td>EC aggregation function F</td><td></td><td></td><td></td><td>percentile-90</td></tr></table>
+
+Table S4: Hyper-parameters used for MuJoCo Ant “No Reward” environment. For the $\mathrm { P P O } { + } 1$ baseline, the curiosity reward is substituted by $+ 1$ (optimizes for survival). The curiosity bonus scale is applied to this reward.   
+
+<table><tr><td></td><td>PPO</td><td>PPO+1</td><td>PPO +EC</td></tr><tr><td>Learning rate</td><td>0.0003</td><td>0.00007</td><td>0.00007</td></tr><tr><td>PPO entropy coefficient</td><td>8e-6</td><td>0.0001</td><td>0.00002</td></tr><tr><td>Task reward scale</td><td>0</td><td>0</td><td>0</td></tr><tr><td>Curiosity bonus scale α</td><td>0</td><td>1</td><td>1</td></tr><tr><td>EC memory size</td><td>-</td><td>1</td><td>1000</td></tr><tr><td>EC reward shift β</td><td></td><td></td><td>1</td></tr><tr><td>EC novelty threshold bnovelty</td><td></td><td></td><td>0</td></tr><tr><td>EC aggregation function F</td><td></td><td></td><td>10th largest</td></tr></table>
+
+Table S5: Hyper-parameters used for MuJoCo Ant “Escape Circle” environment. For the $\mathrm { P P O } { + } 1$ baseline, the curiosity reward is substituted by $+ 1$ (optimizes for survival). The curiosity bonus scale is applied to this reward.   
+
+<table><tr><td></td><td>PPO</td><td>PPO+1</td><td>PPO + EC</td></tr><tr><td>Learning rate</td><td>0.0001</td><td>0.0001</td><td>4.64e-05</td></tr><tr><td>PPO entropy coefficient</td><td>1.21e-06</td><td>1.43e-06</td><td>1.78e-06</td></tr><tr><td>Task reward scale</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Curiosity bonus scale α</td><td>0</td><td>0.85</td><td>0.25</td></tr><tr><td>EC memory size</td><td>1</td><td>-</td><td>1000</td></tr><tr><td>EC reward shift β</td><td></td><td>=</td><td>1</td></tr><tr><td>EC novelty threshold bnovelty</td><td></td><td></td><td>0</td></tr><tr><td>EC aggregation function F</td><td></td><td></td><td>10th largest</td></tr></table>
+
+# S4 R-NETWORK GENERALIZATION STUDY
+
+One of the promises of our approach is its potential ability to generalize between tasks. In this section we verify if this promise holds.
+
+# S4.1 TRAINING R-NETWORK ON ALL DMLab-30 TASKS
+
+Could we train a universal R-network for all available levels — and then use this network for all our tasks of interest? Since different games have different dynamics models, the notion of closely reachable or far observations also changes from game to game. Can R-network successfully handle this variability? Table S6 suggests that using a universal R-network slightly hurts the performance compared to using a specialized R-network trained specifically for the task. However, it still definitely helps to get higher reward compared to using the plain PPO. The R-network is trained using 10M environment interactions equally split across all 30 DMLab-30 tasks.
+
+Table S6: Reward on the tasks “No Reward” and “Very Sparse” using a universal R-network. Two baselines (PPO and $\mathrm { P P O } + \mathrm { E C }$ with a specialized R-network) are also provided.   
+
+<table><tr><td>Method</td><td>No Reward</td><td>Very Sparse</td></tr><tr><td>PPO</td><td>191 ±12</td><td>8.6±4.3</td></tr><tr><td>PPO + EC with specialized R-network</td><td>475±8</td><td>24.7 ± 2.2</td></tr><tr><td>PPO + EC with universal R-network</td><td>348±8</td><td>19.3 ± 1.0</td></tr></table>
+
+# S4.2 TRAINING R-NETWORK ON ONE LEVEL AND TESTING ON ANOTHER
+
+This experiment is similar to the previous one but in a sense is more extreme. Instead of training on all levels (including the levels of interest and other unrelated levels), can we train R-network on just one task and use if for a different task? Table S7 suggests we can obtain reasonable performance by transferring the R-network between similar enough environments. The performance is unsatisfactory only in one case (using the R-network trained on “Dense 2”). Our hypothesis is that the characteristics of the environments are sufficiently different in that case: single room versus maze, static textures on the walls versus changing textures.
+
+Table S7: Reward on the environments “No Reward” and “Very Sparse” (columns) when the Rnetwork is trained on different environments (rows). We provide a result with a matching R-network for reference (bottom).   
+
+<table><tr><td>R-network training environment</td><td>No Reward</td><td>Very Sparse</td></tr><tr><td>Dense 1</td><td>320±5</td><td>18.5 ± 1.4</td></tr><tr><td>Dense 2</td><td>43±2</td><td>0.8 ± 0.5</td></tr><tr><td>Sparse + Doors</td><td>376±7</td><td>16.2 ± 0.7</td></tr><tr><td>Matching environment</td><td>475 ±8</td><td>24.7 ± 2.2</td></tr></table>
+
+# S5 STABILITY/ABLATION STUDY
+
+The experiments are done both in “No Reward” and “Very Sparse” environments. The “No Reward” environment is useful to avoid the situations where task reward would hide important behavioural differences between different flavors of our method (this “hiding” effect can be easily observed for different methods comparison in the dense reward tasks — but the influence of task reward still remains even in sparser cases). As in the main text, for the “No Reward” task we report the Grid Oracle reward as a discrete approximation to the area covered by the agent trajectories.
+
+# S5.1 POSITIVE EXAMPLE THRESHOLD IN R-NETWORK TRAINING
+
+Training the R-network requires a threshold $k$ to separate negative from positive pairs. The trained policy implicitly depends on this threshold. Ideally, the policy performance should not be too sensitive to this hyper-parameter. We conduct a study where the threshold is varied from 2 to 10 actions (as in all experiments before, each action is repeated 4 times). Table S8 shows that the EC performance is reasonably robust to the choice of this threshold.
+
+Table S8: Reward in the “No Reward” and “Very Sparse“ tasks using different positive example thresholds $k$ when training the R-network.   
+
+<table><tr><td>Threshold k</td><td>No Reward</td><td>Very Sparse</td></tr><tr><td>2</td><td>378±18</td><td>28.3 ± 1.6</td></tr><tr><td>3</td><td>395 ±10</td><td>20.9 ± 1.6</td></tr><tr><td>4</td><td>412±8</td><td>31.1 ± 1.2</td></tr><tr><td>5</td><td>475±8</td><td>24.7 ± 2.2</td></tr><tr><td>7</td><td>451±4</td><td>23.6 ± 1.0</td></tr><tr><td>10</td><td>455±7</td><td>20.8 ± 0.8</td></tr></table>
+
+# S5.2 MEMORY SIZE IN EC MODULE
+
+The EC-module relies on an explicit memory buffer to store the embeddings of past observations and define novelty. One legitimate question is to study the impact of the size of this memory buffer on the performance of the EC-module. As observed in table S9, the memory size has little impact on the performance.
+
+Table S9: Reward for different values of the memory size for the tasks “No Reward” and “Very Sparse”.   
+S5.3 ENVIRONMENT INTERACTION BUDGET FOR TRAINING R-NETWORK   
+
+<table><tr><td>Memory size</td><td>No Reward</td><td> Very Sparse</td></tr><tr><td>100</td><td>447±6</td><td>19.4 ± 1.9</td></tr><tr><td>200</td><td>475±8</td><td>24.7 ± 2.2</td></tr><tr><td>350</td><td>459±6</td><td>23.5 ± 1.4</td></tr><tr><td>500</td><td>452±6</td><td>23.8± 2.0</td></tr></table>
+
+The sample complexity of our EC method includes two parts: the sample complexity to train the Rnetwork and the sample complexity of the policy training. In the worst case – when the R-network does not generalize across environments – the R-network has to be trained for each environment and the total sample complexity is then the sum of the previous two sample complexities. It is then crucial to see how many steps are needed to train R-network such that it can capture the notion of reachability. R-network trained using a number of environment steps as low as 1M already gives good performance, see Table S10.
+
+Table S10: Reward of the policy trained on the “No Reward” and “Very Sparse“ tasks with an R-network trained using a varying number of environment interactions (from 100K to 5M).   
+
+<table><tr><td>Interactions</td><td>No Reward</td><td>Very Sparse</td></tr><tr><td>100K</td><td>357±18</td><td>12.2 ± 1.3</td></tr><tr><td>300K</td><td>335±9</td><td>16.2 ± 0.7</td></tr><tr><td>1M</td><td>383±13</td><td>18.6 ± 0.9</td></tr><tr><td>2.5M</td><td>475±8</td><td>24.7 ± 2.2</td></tr><tr><td>5M</td><td>416±5</td><td>20.7 ± 1.4</td></tr></table>
+
+# S5.4 IMPORTANCE OF TRAINING DIFFERENT PARTS OF R-NETWORK
+
+The R-network is composed of an Embedding network and a Comparator network. How important is each for the final performance of our method? To establish that, we conduct two experiments. First, we fix the Embedding network at the random initialization and train only the Comparator. Second, we substitute the Comparator network applied to embeddings $\mathbf { e } _ { 1 } , \mathbf { e } _ { 2 }$ with the sigmoid function $\sigma ( \mathbf { e } _ { 1 } ^ { T } \mathbf { e } _ { 2 } )$ and train only the Embedding. According to the results in Table S11, we get a reasonable performance with a random embedding: the results are still better than the plain PPO (but worse than with the complete R-network). However, without the Comparator the quality drops below the plain PPO.
+
+![](images/a248302ba7eb1ac222582a5dd6cdee1baa4afe01d6f60c99d36971f27882c303.jpg)  
+Figure S1: Examples of randomized environments: (a) Image Action, (b) Noise.
+
+This experiment leads us to two conclusions. First, training the Embedding network is desired but not necessary for our method to work. Second, using the Comparator is essential and cannot be omitted in the current setup. Apparently, predicting reachability requires fine-grained access to both embeddings at the same time — and a simple comparison function does not work.
+
+Table S11: Reward on the “No Reward” and “Very Sparse“ tasks using ablated versions of the R-network.   
+
+<table><tr><td>Method</td><td>No Reward</td><td> Very Sparse</td></tr><tr><td>PPO</td><td>191 ± 12</td><td>8.6 ± 4.3</td></tr><tr><td>PPO + EC with complete R-network</td><td>475±8</td><td>24.7 ± 2.2</td></tr><tr><td>PPO + EC with random Embedding</td><td>392 ±12</td><td>16.2 ± 1.4</td></tr><tr><td>PPO + EC without Comparator network</td><td>48±3</td><td>5.8± 2.4</td></tr></table>
+
+# S6 RANDOMIZED ENVIRONMENTS
+
+In the main text of the paper we observed how the firing action confused the surprise-based curiosity method ICM. This was a manifestation of the hardness of future prediction performed by ICM. Importantly, there could be more than one reason why future prediction is hard (as observed in the concurrent work (Burda et al., 2018b)): partial observability of the environment, insufficiently rich future prediction model or randomized transitions in the environment. Since our own method EC relies on comparisons to the past instead of predictions of the future, one could expect it to be more robust to those factors (intuitively, comparison to the past is an easier problem). The goal of this section is to provide additional evidence for that.
+
+We are going to experiment with one source of future prediction errors which we have used in the thought experiment from the introduction: environment stochasticity. In particular, we analyze how different methods behave when all the states in the environment provide stochastic next state. For that, we create versions of the DMLab environments “Sparse” and “Very Sparse” with added strong source of stochasticity: randomized TV on the head-on display of the agent. It is implemented as follows: the lower right quadrant of the agent’s first person view is occupied with random images. We try a few settings:
+
+• “Image Action $k ^ { \prime \prime }$ : there are $k$ images of animals retrieved from the internet, an agent has a special action which changes an image on the TV screen to a random one from this set. An example is shown in Figure S1(a).   
+“Noise”: at every step a different noise pattern is shown on the TV screen, independently from agent’s actions. The noise is sampled uniformly from $[ 0 , 2 5 5 ]$ independently for each pixel. An example is shown in Figure S1(b).   
+• “Noise Action”: same as “Noise”, but the noise pattern only changes if the agent uses a special action.
+
+Table S12: Reward in the randomized-TV versions of DMLab task “Sparse” (mean $\pm$ std) for all compared methods. Higher is better. “Original” stands for the non-randomized standard version of the task which we used in the main text. “ECO” stands for the online version of our method, which trains R-network and the policy at the same time. The Grid Oracle method is given for reference — it uses privileged information unavailable to other methods. Results are averaged over 30 random seeds. No seed tuning is performed.   
+
+<table><tr><td>Method</td><td colspan="3">Image Action</td><td>Noise</td><td>Noise Action</td><td>Original</td></tr><tr><td></td><td>3</td><td>10</td><td>30</td><td></td><td></td><td></td></tr><tr><td>PPO</td><td>11.5 ± 2.1</td><td>10.9 ± 1.8</td><td>8.5± 1.5</td><td>11.6 ± 1.9</td><td>9.8 ± 1.5</td><td>27.0 ± 5.1</td></tr><tr><td>PPO + ICM</td><td>10.0 ± 1.2</td><td>10.5 ± 1.2</td><td>6.9 ± 1.0</td><td>7.7 ± 1.1</td><td>7.6 ± 1.1</td><td>23.8± 2.8</td></tr><tr><td>PPO + EC (ours)</td><td>19.8 ± 0.7</td><td>15.3 ± 0.4</td><td>13.1 ± 0.3</td><td>18.7±0.8</td><td>14.8± 0.4</td><td>26.2 ± 1.9</td></tr><tr><td>PPO + ECO (ours)</td><td>24.3 ± 2.1</td><td>26.6 ± 2.8</td><td>18.5± 0.6</td><td>28.2 ± 2.4</td><td>18.9 ± 1.9</td><td>41.6 ± 1.7</td></tr><tr><td>PPO + Grid Oracle</td><td>37.7± 0.7</td><td>37.1± 0.7</td><td>37.4± 0.7</td><td>38.8± 0.8</td><td>39.3 ± 0.8</td><td>56.7 ± 1.3</td></tr></table>
+
+Table S13: Reward in the randomized-TV versions of DMLab task “Very Sparse” (mean $\pm$ std) for all compared methods. Higher is better. “Original” stands for the non-randomized standard version of the task which we used in the main text. “ECO” stands for the online version of our method, which trains R-network and the policy at the same time. The Grid Oracle method is given for reference — it uses privileged information unavailable to other methods. Results are averaged over 30 random seeds. No seed tuning is performed.   
+
+<table><tr><td>Method</td><td colspan="3">Image Action</td><td>Noise</td><td>Noise Action</td><td>Original</td></tr><tr><td></td><td>3</td><td>10</td><td>30</td><td></td><td></td><td></td></tr><tr><td>PPO</td><td>6.5 ± 1.6</td><td>8.3±1.8</td><td>6.3 ± 1.8</td><td>8.7 ±1.9</td><td>6.1 ± 1.8</td><td>8.6±4.3</td></tr><tr><td>PPO + ICM</td><td>3.8±0.8</td><td>4.7±0.9</td><td>4.9 ± 0.7</td><td>6.0 ±1.3</td><td>5.7 ± 1.4</td><td>11.2 ± 3.9</td></tr><tr><td>PPO +EC (ours)</td><td>13.8± 0.5</td><td>10.2 ± 0.8</td><td>7.4 ± 0.5</td><td>13.4 ± 0.6</td><td>11.3 ± 0.4</td><td>24.7± 2.2</td></tr><tr><td>PPO + ECO (ours)</td><td>20.5± 1.3</td><td>17.8 ± 0.8</td><td>16.8 ± 1.4</td><td>26.0 ± 1.6</td><td>12.5 ± 1.3</td><td>40.5 ± 1.1</td></tr><tr><td>PPO + Grid Oracle</td><td>35.4± 0.6</td><td>35.9 ± 0.6</td><td>36.3± 0.7</td><td>35.5± 0.6</td><td>35.4 ± 0.8</td><td>54.3 ± 1.2</td></tr></table>
+
+The results at 20M 4-repeated environment steps are shown in Tables S12, S13. In almost all cases, the performance of all methods deteriorates because of any source of stochasticity. However, our method turns out to be reasonably robust to all sources of stochasticity and still outperforms the baselines in all settings. The videos10,11 demonstrate that our method still explores the maze reasonably well.
+
+# S7 COMPUTATIONAL CONSIDERATIONS
+
+The most computationally intensive parts of our algorithm are the memory reachability queries. Reachabilities to past memories are computed in parallel via mini-batching. We have shown the algorithm to work reasonably fast with a memory size of 200. For orders of magnitude larger memory sizes, one would need to better parallelize reachability computations — which should in principle be possible. Memory consumption for the stored memories is very modest $\mathbf { \zeta } _ { 4 0 0 \ K B ) }$ , as we only store 200 of 512-float-embeddings, not the observations.
+
+As for the speed comparison between different methods, $\mathrm { P P O } + \mathrm { I C M }$ is $1 . 0 9 \mathbf { x }$ slower than PPO and $\mathrm { P P O } + \mathrm { E C }$ (our method) is $1 . 8 4 \mathbf { x }$ slower than PPO. In terms of the number of parameters, R-network brings 13M trainable variables, while PPO alone was 1.7M and $\mathrm { P P O } + \mathrm { I C M }$ was 2M. That said, there was almost no effort spent on optimizing the pipeline in terms of speed/parameters, so it is likely easy to make improvements in this respect. It is quite likely that a resource-consuming Resnet-18 is not needed for the R-network — a much simpler model may work as well. In this paper, we followed the setup for the R-network from prior work (Savinov et al., 2018) because it was shown to perform well, but there is no evidence that this setup is necessary.
+
+# S8 ADDITIONAL DMLab TRAINING CURVES
+
+![](images/66472d7f181a4749164d8c0020144436017f2978d9550d694759779604442b52.jpg)  
+Figure S2: Reward as a function of training step for the DMLab task “Dense $2 ^ { \circ }$ . Higher is better. We shift the curves for our method by the number of environment steps used to train R-network — so the comparison between different methods is fair. We run every method 30 times and show 5 randomly selected runs. No seed tuning is performed.
+
+We show additional training curves from the main text experimental section in Figure S2.

parse/train/SnONpXZ_uQ_/SnONpXZ_uQ_.md

@@ -0,0 +1,215 @@
+# Learning Graph Models for Retrosynthesis Prediction
+
+Vignesh Ram Somnath1
+
+Charlotte Bunne1
+
+Connor W. Coley2
+
+Andreas Krause1 Regina Barzilay3
+
+1Department of Computer Science, ETH 2Department of Chemical Engineering, MIT 3Computer Science and Artificial Intelligence Lab, MIT 1{vsomnath, bunnec, krausea}@ethz.ch, 2ccoley@mit.edu, 3regina@csail.mit.edu
+
+# Abstract
+
+Retrosynthesis prediction is a fundamental problem in organic synthesis, where the task is to identify precursor molecules that can be used to synthesize a target molecule. A key consideration in building neural models for this task is aligning model design with strategies adopted by chemists. Building on this viewpoint, this paper introduces a graph-based approach that capitalizes on the idea that the graph topology of precursor molecules is largely unaltered during a chemical reaction. The model first predicts the set of graph edits transforming the target into incomplete molecules called synthons. Next, the model learns to expand synthons into complete molecules by attaching relevant leaving groups. This decomposition simplifies the architecture, making its predictions more interpretable, and also amenable to manual correction. Our model achieves a top-1 accuracy of $5 3 . 7 \%$ , outperforming previous template-free and semi-template-based methods.
+
+# 1 Introduction
+
+Retrosynthesis prediction, first formalized by E. J. Corey [Corey, 1991] is a fundamental problem in organic synthesis that attempts to identify a series of chemical transformations for synthesizing a target molecule. In the single-step formulation, the task is to identify a set of reactant molecules given a target. Beyond simple reactions, many practical tasks involving complex organic molecules are difficult even for expert chemists. As a result, substantial experimental exploration is needed to cover for deficiencies of analytical approaches. This has motivated interest in computer-assisted retrosynthesis [Corey and Wipke, 1969], with a recent surge in machine learning methods [Chen et al., 2019, Coley et al., 2017b, Dai et al., 2019, Zheng et al., 2019, Genheden et al., 2020].
+
+Computationally, the main challenge is how to explore the combinatorial space of reactions that can yield the target molecule. Largely, previous methods for retrosynthesis prediction can be divided into template-based [Coley et al., 2017b, Dai et al., 2019, Segler and Waller, 2017] and template-free [Chen et al., 2019, Zheng et al., 2019] approaches. Template-based methods match a target molecule against a large set of templates, which are molecular subgraph patterns that highlight changes during a chemical reaction. Despite their interpretability, these methods fail to generalize to new reactions. Template-free methods bypass templates by learning a direct mapping from the SMILES [Weininger, 1988] representations of the product to reactants. Despite their greater generalization potential, these methods generate reactant SMILES character by character, increasing generation complexity.
+
+![](images/e988f2e4a9f13ae6fa41934806006c24c87183942b7fd7f824274aee3a6237f9.jpg)  
+Figure 1: Overview of Our Approach. a. Edit Prediction. We train a model to learn a distribution over possible graph edits. In this case, the correct edit corresponds to breaking the bond marked in red. Applying this edit produces two synthons. b. Synthon Completion. Another model is trained to pick candidate leaving groups (blue) for each synthon from a discrete vocabulary, which are then attached to produce the final reactants.
+
+Another important consideration in building retrosynthesis models is aligning model design with strategies adopted by expert chemists. These strategies are influenced by fundamental properties of chemical reactions, independent of complexity level: (i.) the product atoms are always a subset of the reactant atoms1, and (ii.) the molecular graph topology is largely unaltered from products to reactants. For example, in the standard retrosynthesis dataset, only $6 . 3 \%$ of the atoms in the product undergo any change in connectivity.
+
+This consideration has received more attention in recent semi-template-based methods [Shi et al., 2020, Yan et al., 2020], that generate reactants from a product in two stages: (i.) first identify intermediate molecules called synthons, (ii.) and then complete synthons into reactants by sequential generation of atoms or SMILES characters.. Our model GRAPHRETRO also uses a similar workflow. However, we avoid sequential generation for completing synthons by instead selecting subgraphs called leaving groups from a precomputed vocabulary. This vocabulary is constructed during preprocessing by extracting subgraphs that differ between a synthon and the corresponding reactant. The vocabulary has a small size (170 for USPTO-50k) indicating remarkable redundancy, while covering $9 9 . 7 \%$ of the test set. Operating at the level of these subgraphs greatly reduces the complexity of reactant generation, with improved empirical performance. This formulation also simplifies our architecture, and makes our predictions more transparent, interpretable and amenable to manual correction.
+
+The benchmark dataset for evaluating retrosynthesis models is USPTO-50k [Schneider et al., 2016], which consists of 50000 reactions across 10 reaction classes. The dataset contains an unexpected shortcut towards predicting the edit, in that the product atom with atom-mapping 1 is part of the edit in $7 5 \%$ of the cases, allowing predictions that depend on the position of the atom to overestimate performance. We canonicalize the product SMILES and remap the existing dataset, thereby removing the shortcut. On this remapped dataset, GRAPHRETRO achieves a top-1 accuracy of $5 3 . 7 \%$ when the reaction class is not known, outperforming both template-free and semi-template-based methods.
+
+# 2 Related Work
+
+Retrosynthesis Prediction Existing machine learning methods for retrosynthesis prediction can be divided into template-based, template-free and recent semi-template-based approaches.
+
+Template-Based: Templates are either hand-crafted by experts [Hartenfeller et al., 2011, Szymkuc´ et al., 2016], or extracted algorithmically from large databases Coley et al. [2017a], Law et al. [2009]. Exhaustively applying large template sets is expensive due to the involved subgraph matching procedure. Template-based methods therefore utilize different ways of prioritizing templates, by either learning a conditional distribution over the template set [Segler and Waller, 2017], ranking templates based on molecular similarities to precedent reactions [Coley et al., 2017b] or directly modelling the joint distribution of templates and reactants using logic variables [Dai et al., 2019]. Despite their interpretability, these methods fail to generalize outside their rule set.
+
+Template-Free: Template-free methods [Liu et al., 2017, Zheng et al., 2019, Chen et al., 2019] learn a direct transformation from products to reactants using architectures from neural machine translation and a string based representation of molecules called SMILES [Weininger, 1988]. Linearizing molecules as strings does not utilize the inherently rich chemical structure. In addition, the reactant SMILES are generated from scratch, character by character. Attempts have been made to improve validity by adding a syntax correcter [Zheng et al., 2019] and a mixture model to improve diversity of suggestions [Chen et al., 2019], but the performance remains worse than [Dai et al., 2019] on the standard retrosynthesis dataset. Sun et al. [2021] formulate retrosynthesis using energy-based models, with additional parameterizations and loss terms to enforce the duality between forward (reaction prediction) and backward (retrosynthesis) prediction.
+
+Semi-Template-Based: Our work is closely related to recently proposed semi-template-based methods [Shi et al., 2020, Yan et al., 2020], which first identify synthons and then expand synthons into reactants through sequential generation using either a graph generative model [Shi et al., 2020] or a Transformer [Yan et al., 2020]. To reduce the complexity of reactant generation, we instead complete synthons using subgraphs called leaving groups selected from a precomputed vocabulary. This allows us to view synthon completion as a classification problem instead of a generative one. We also utilize the dependency graph between possible edits, and update edit predictions using a message passing network (MPN) [Gilmer et al., 2017] on this graph. Both innovations together yield a $4 . 8 \%$ and $3 . 3 \%$ performance improvement respectively over previous semi-template-based methods.
+
+Reaction Center Identification The reaction center covers a small number of participating atoms involved in the reaction. Our work is also related to models that predict reaction outcomes by learning to rank atom pairs based on their likelihood to be in the reaction center [Coley et al., 2019, Jin et al., 2017]. The task of identifying the reaction center is related to the step of deriving the synthons in our formulation. Our work departs from [Coley et al., 2019, Jin et al., 2017] as we utilize the property that new bond formations occur rarely $( \sim 0 . 1 \% )$ from products to synthons, allowing us to predict a score only for existing bonds and atoms and reduce prediction complexity from $O ( N ^ { \bar { 2 } } )$ to $O ( N )$ . We also utilize the dependency graph between possible edits, and update edit predictions using a MPN on this graph.
+
+Utilizing Substructures Substructures have been utilized in various tasks from sentence generation by fusing phrases to molecule generation and optimization [Jin et al., 2018, 2020]. Our work is closely related to [Jin et al., 2020] which uses precomputed substructures as building blocks for property-conditioned molecule generation. However, instead of precomputing, synthons —analogous building blocks for reactants— are indirectly learnt during training.
+
+# 3 Model Design
+
+Our approach leverages the property that graph topology is largely unaltered from products to reactants. To achieve this, we first derive suitable building blocks from the product called synthons, and then complete them into valid reactants by adding specific functionalities called leaving groups. These derivations, called edits, are characterized by modifications to bonds or hydrogen counts on atoms. We first train a neural network to predict a score for possible edits (Section 3.1). The edit with the highest score is then applied to the product to obtain synthons. Since the number of unique leaving groups are small, we model leaving group selection as a classification problem over a precomputed vocabulary (Section 3.2). To produce candidate reactants, we attach the predicted leaving group to the corresponding synthon through chemically constrained rules. The overall process is outlined in Figure 1. Before describing the two modules, we introduce relevant preliminaries that set the background for the remainder of the paper.
+
+Retrosynthesis Prediction A retrosynthesis pair $R$ is described by a pair of molecular graphs $( \mathcal { G } _ { p } , \mathcal { G } _ { r } )$ , where $\mathcal { G } _ { p }$ are the products and $\mathcal { G } _ { r }$ the reactants. A molecular graph is described as $\mathcal { G } = \mathbf { \bar { \rho } } ( \mathcal { V } , \mathcal { E } )$ with atoms $\nu$ as nodes and bonds $\mathcal { E }$ as edges. Prior work has focused on the single product case, while reactants can have multiple connected components, i.e. $\mathcal { G } _ { r } = \{ \mathcal { G } _ { r _ { c } } \} _ { c = 1 } ^ { C }$ . Retrosynthesis pairs are atom-mapped so that each product atom has a unique corresponding reactant atom. The retrosynthesis task then, is to infer $\{ \mathcal { G } _ { r _ { c } } \} _ { c = 1 } ^ { C }$ given $\mathcal { G } _ { p }$ .
+
+Edits Edits consist of (i.) atom pairs $\left\{ \left( a _ { i } , a _ { j } \right) \right\}$ where the bond type changes from products to reactants, and (ii.) atoms $\left\{ { a } _ { i } \right\}$ where the number of hydrogens attached to the atom change from products to reactants. We denote the set of edits by $E$ . Since retrosynthesis pairs in the training set are atom-mapped, edits can be automatically identified by comparing the atoms and atom pairs in the product to their corresponding reactant counterparts.
+
+Synthons and Leaving Groups Applying edits $E$ to the product $\mathcal { G } _ { p }$ results in incomplete molecules called synthons. Synthons are analogous to rationales or building blocks, which are expanded into valid reactants by adding specific functionalities called leaving groups that are responsible for its reactivity. We denote synthons by $\mathcal { G } _ { s }$ and leaving groups by $\mathcal { G } _ { l }$ . We further assume that synthons and leaving groups have the same number of connected components as the reactants, i.e $\mathcal { G } _ { s } \doteq \{ \mathcal { G } _ { s _ { c } } \} _ { c = 1 } ^ { C }$ and $\mathcal { G } _ { l } = \{ \mathcal { G } _ { l _ { c } } ^ { \star } \} _ { c = 1 } ^ { C }$ . This assumption holds for $9 9 . 9 7 \%$ reactions in the training set.
+
+Formally, our model generates reactants by first predicting the set of edits $E$ that transform $\mathcal { G } _ { p }$ into $\mathcal { G } _ { s }$ , followed by predicting a leaving group $\mathcal { G } _ { l _ { c } }$ to attach to each synthon $\mathcal { G } _ { s _ { c } }$ . The model is defined as
+
+$$
+P ( \mathcal G _ { r } | \mathcal G _ { p } ) = \sum _ { E , \mathcal G _ { l } } P ( E | \mathcal G _ { p } ) P ( \mathcal G _ { l } | \mathcal G _ { p } , \mathcal G _ { s } ) ,
+$$
+
+where $\mathcal { G } _ { s } , \mathcal { G } _ { r }$ are deterministic given $E , { \mathcal { G } } _ { l }$ , and $\mathcal { G } _ { p }$ .
+
+# 3.1 Edit Prediction
+
+For a given retrosynthesis pair $R = ( \mathcal G _ { p } , \mathcal G _ { r } )$ , we predict an edit score only for existing bonds and atoms, instead of every atom pair as in [Coley et al., 2019, Jin et al., 2017]. This choice is motivated by the low frequency $( \sim 0 . 1 \% )$ of new bond formations in the training set examples. Coupled with the sparsity of molecular graphs, this reduces the prediction complexity from $O ( N ^ { 2 } )$ to $O ( N )$ for a product with $N$ atoms. Our edit prediction model has variants tailored to single and multiple edit prediction. Since $9 5 \%$ of the training set consists of single edit examples, the remainder of this section describes the setup for single edit prediction. A detailed description of our multiple edit prediction model can be found in Appendix ??.
+
+Each bond $( u , v )$ in $\mathcal { G } _ { p }$ is associated with a label $y _ { u v k } \in \{ 0 , 1 \}$ indicating whether its bond type $k$ has changed from the products to reactants. Each atom $u$ is associated with a label $y _ { u } \in \{ 0 , \bar { 1 } \}$ indicating a change in hydrogen count. We predict edit scores using representations that are learnt using a graph encoder.
+
+Graph Encoder To obtain atom representations, we use a variant of the message passing network (MPN) described in [Gilmer et al., 2017]. Each atom $u$ has a feature vector $\mathbf { x } _ { u }$ indicating its atom type, degree and other properties. Each bond $( u , v )$ has a feature vector $\mathbf { x } _ { u v }$ indicating its aromaticity, bond type and ring membership. For simplicity, we denote the encoding process by $\mathrm { M P N } ( \cdot )$ and describe architectural details in Appendix ??. The MPN computes atom representations $\{ \mathbf { c } _ { u } | u \in \mathcal { G } \}$ via
+
+$$
+\{ \mathbf { c } _ { u } \} = \mathrm { M P N } ( \mathcal { G } , \{ \mathbf { x } _ { u } \} , \{ \mathbf { x } _ { u v } \} _ { v \in \mathcal { N } ( u ) } ) ,
+$$
+
+where $\mathcal { N } ( u )$ denotes the neighbors of atom $u$ . The graph representation $\mathbf { c } _ { \mathcal { G } }$ is an aggregation of atom representations, i.e. $\mathbf { c } _ { \mathcal { G } } = \dot { \sum _ { { u } \in \mathcal { V } } } \mathbf { c } _ { u }$ . When $\mathcal { G }$ has connected components $\left\{ { \mathcal { G } } _ { i } \right\}$ , we get a set of graph representations $\left\{ \mathbf { c } _ { \mathcal { G } _ { i } } \right\}$ . For a bond $( u , v )$ , we define its representation $\mathbf { c } _ { u v } = ( \operatorname { A B S } ( \mathbf { c } _ { u } , \mathbf { c } _ { v } ) | | \mathbf { c } _ { u } + \mathbf { c } _ { v } )$ , where ABS denotes absolute difference and $| |$ refers to concatenation. This ensures our representations are permutation invariant. These representations are then used to predict atom and bond edit scores using corresponding neural networks,
+
+$$
+\begin{array} { r } { \boldsymbol { s } _ { u } = \mathbf { u _ { a } } ^ { T } \boldsymbol { \tau } ( \mathbf { W _ { a } } \mathbf { c } _ { u } + b ) \quad } \\ { \boldsymbol { s } _ { u v k } = \mathbf { u _ { k } } ^ { T } \boldsymbol { \tau } ( \mathbf { W _ { k } } \mathbf { c } _ { u v } + b _ { k } ) , } \end{array}
+$$
+
+where $\tau ( \cdot )$ is the ReLU activation function.
+
+Updating Bond Edit Scores Unlike a typical classification problem where the labels are independent, edits can have possible dependencies between each other. For example, bonds part of a stable system such as an aromatic ring have a greater tendency to remain unchanged (label 0). We attempt to leverage such dependencies to update initial edit scores. To this end, we build a graph with bonds $( u , v )$ as nodes, and introduce an edge between bonds sharing an atom. We use another $\mathrm { M P N } ( \cdot )$ on this graph to learn aggregated neighborhood messages $\mathbf { m } _ { u v }$ , and update the edit scores $s _ { u v k }$ in a manner similar to how LSTMs update representations,
+
+$$
+\begin{array} { r l } & { f _ { u v k } = \sigma ( \mathbf { W _ { k x } ^ { f } } \mathbf { x } _ { u v } + \mathbf { W _ { k m } ^ { f } } \mathbf { m } _ { u v } ) } \\ & { i _ { u v k } = \sigma ( \mathbf { W _ { k x } ^ { i } } \mathbf { x } _ { u v } + \mathbf { W _ { k m } ^ { i } } \mathbf { m } _ { u v } ) } \\ & { \tilde { m } _ { u v k } = \mathbf { u _ { m } } \tau ( \mathbf { W _ { k x } ^ { m } } \mathbf { x } _ { u v } + \mathbf { W _ { k m } ^ { m } } \mathbf { m } _ { u v } ) } \\ & { \tilde { s } _ { u v k } = f _ { u v k } \cdot s _ { u v k } + i _ { u v k } \cdot \tilde { m } _ { u v k } . } \end{array}
+$$
+
+Training We train by minimizing the cross-entropy loss over possible bond and atom edits
+
+$$
+\mathcal { L } _ { e } = - \sum _ { ( \mathcal { G } _ { p } , E ) } \left( \sum _ { ( ( u , v ) , k ) \in E } y _ { u v k } \mathrm { l o g } ( \widetilde s _ { u v k } ) + \sum _ { u \in E } y _ { u } \mathrm { l o g } ( s _ { u } ) \right) .
+$$
+
+The cross-entropy loss enforces the model to learn a distribution over possible edits instead of reasoning about each edit independently, as with the binary cross entropy loss used in [Jin et al., 2017, Coley et al., 2019].
+
+# 3.2 Synthon Completion
+
+Synthons are completed into valid reactants by adding specific functionalities called leaving groups. This involves two complementary tasks: (i.) selecting the appropriate leaving group, and (ii.) attaching the leaving group to the synthon. As ground truth leaving groups are not directly provided, we extract the leaving groups and construct a vocabulary $\mathcal { X }$ of unique leaving groups during preprocessing.
+
+The vocabulary has a limited size ( $| \mathcal { X } | = 1 7 0$ for a standard dataset with 50, 000 examples, and 72000 synthons) indicating the redundancy of leaving groups used in accomplishing retrosynthetic transformations. This redundancy also allows us to formulate leaving group selection as a classification problem over $\mathcal { X }$ , while retaining the ability to generate diverse reactants using different combinations of leaving groups.
+
+Vocabulary Construction Before constructing the vocabulary, we align connected components of synthon and reactant graphs by comparing atom mapping overlaps. Using aligned pairs $\mathcal { G } _ { s _ { c } } =$ $( \gamma _ { s _ { c } } , \mathcal { E } _ { s _ { c } } )$ and $\mathcal { G } _ { r _ { c } } = ( \nu _ { r _ { c } } , \mathcal { E } _ { r _ { c } } )$ as input, the leaving group vocabulary $\mathcal { X }$ is constructed by extracting subgraphs $\mathcal { G } _ { l _ { c } } = ( \nu _ { l _ { c } } , \mathcal { E } _ { l _ { c } } )$ such that $\smash { \gamma _ { l _ { c } } = \gamma _ { r _ { c } } \setminus \gamma _ { s _ { c } } }$ . Atoms $\left\{ { a } _ { i } \right\}$ in the leaving groups that attach to synthons are marked with a special symbol. We also add three tokens to $\mathcal { X }$ namely START, which indicates the start of synthon completion, END, which indicates that there is no leaving group to add and PAD, which is used to handle variable numbers of synthon components in a minibatch.
+
+Leaving Group Selection For synthon component $c \leq C$ , where $C$ is the number of connected components in the synthon graph, we use three inputs for leaving group selection – the product representation $\mathbf { c } _ { \mathcal { G } _ { p } }$ , the synthon component representation $\mathbf { c } _ { \mathcal { G } _ { s _ { c } } }$ , and the leaving group representation for the previous synthon component, ${ \bf e } _ { l _ { c - 1 } }$ . The product and synthon representations are learnt using the $\mathrm { M P N } ( \cdot )$ . For each $x _ { i } \in { \mathcal { X } }$ , representations can be learnt by either training independent embedding vectors (ind) or by treating each $x _ { i }$ as a subgraph and using the $\mathrm { M P N } ( \cdot )$ (shared). In the shared setting, we use the same $\mathrm { M P N } ( \cdot )$ as the product and synthons.
+
+The leaving group probabilities are then computed by combining $\mathbf { c } _ { \mathcal { G } _ { p } } , \mathbf { c } _ { \mathcal { G } _ { s _ { c } } }$ and $\mathbf { e } _ { l _ { c - 1 } }$ via a single layer neural network and softmax function
+
+$$
+\hat { q } _ { l _ { c } } = \mathrm { s o f t m a x } \left( \mathbf { U } \tau \left( \mathbf { W } _ { 1 } \mathbf { c } _ { \mathcal { G } _ { p } } + \mathbf { W } _ { 2 } \mathbf { c } _ { \mathcal { G } _ { s _ { c } } } + \mathbf { W } _ { 3 } \mathbf { e } _ { l _ { \left( c - 1 \right) } } \right) \right) ,
+$$
+
+where $\hat { q } _ { l _ { c } }$ is distribution learnt over $\mathcal { X }$ . Using the representation of the previous leaving group ${ \bf e } _ { l _ { c - 1 } }$ allows the model to understand combinations of leaving groups that generate the desired product from the reactants. We also include the product representation $\mathbf { c } _ { \mathcal { G } _ { p } }$ as the synthon graphs are derived from the product graph.
+
+Training For step $c$ , given the one hot encoding of the true leaving group $q _ { l _ { c } }$ , we minimize the cross-entropy loss
+
+$$
+\mathcal { L } _ { s } = \sum _ { c = 1 } ^ { C } \mathcal { L } ( \hat { q } _ { l _ { c } } , q _ { l _ { c } } ) .
+$$
+
+Training utilizes teacher-forcing [Williams and Zipser, 1989] so that the model makes predictions given correct histories. During inference, at every step, we use the representation of leaving group from the previous step with the highest predicted probability.
+
+Leaving Group Attachment Attaching leaving groups to synthons is a deterministic process and not learnt during training. The task involves identification of the type of bonds to add between attaching atoms in the leaving group (marked during vocabulary construction), and the atom(s) participating in the edit. These bonds can be inferred by applying the valency constraint, which determines the maximum number of neighbors for each atom. The attachment process does not modify any stereochemistry. Given synthons and leaving groups, the attachment process has a $100 \%$ accuracy. The detailed procedure is described in Appendix ??.
+
+# 3.3 Inference
+
+Inference is performed using beam search with a log-likelihood scoring function. For a beam width $n$ , we select $n$ edits with highest scores and apply them to the product to obtain $n$ synthons, where each synthon can consist of multiple connected components. The synthons form the nodes for beam search. Each node maintains a cumulative score by aggregating the log-likelihoods of the edit and predicted leaving groups. Leaving group inference starts with a connected component for each synthon, and selects $n$ leaving groups with highest log-likelihoods. From the $n ^ { 2 }$ possibilities, we select $n$ nodes with the highest cumulative scores. This process is repeated until all nodes have a leaving group predicted for each synthon component.
+
+# 4 Evaluation
+
+Evaluating retrosynthesis models is challenging as multiple sets of reactants can be generated from the same product. To deal with this, previous works [Coley et al., 2017b, Dai et al., 2019] evaluate the ability of the model to recover retrosynthetic strategies recorded in the dataset.
+
+Data We use the benchmark dataset USPTO-50k [Schneider et al., 2016] for all our experiments. The dataset contains 50, 000 atom-mapped reactions across 10 reaction classes. We use the same dataset version and splits as provided by [Dai et al., 2019]. The USPTO-50k dataset contains a shortcut in that the product atom with atom-mapping 1 is part of the edit in $\sim 7 5 \%$ of the cases. If the product SMILES is not canonicalized, predictions utilizing operations that depend on the position of the atom or bond will be able to use the shortcut, and overestimate performance. We canonicalize the product SMILES, and reassign atom-mappings to the reactant atoms based on the canonical ordering, which removes the shortcut. Details on the remapping procedure can be found in Appendix ??.
+
+Evaluation We use the top- $\mathbf { \nabla } \cdot n$ accuracy $( n = 1 , 3 , 5 , 1 0 )$ ) as our evaluation metric, defined as the fraction of examples where the recorded reactants are suggested by the model with rank $\leq n$ . Following prior work [Coley et al., 2017b, Zheng et al., 2019, Dai et al., 2019], we compute the accuracy by comparing the canonical SMILES of predicted reactants to the ground truth. Atom-mapping is excluded from this comparison, but stereochemistry, which describes the relative orientation of atoms in the molecule, is retained. The evaluation is carried out for two settings, with the reaction class being known or unknown.
+
+Table 1: Top- $n$ exact match accuracy. Best values within each section are highlighted in bold.   
+
+<table><tr><td rowspan="3">Model</td><td colspan="8">Top-n Accuracy (%)</td></tr><tr><td colspan="4">Reaction class known</td><td colspan="4">Reaction class unknown</td></tr><tr><td>1</td><td>3</td><td>5</td><td>10</td><td>1</td><td>3</td><td>5</td><td>10</td></tr><tr><td colspan="9">Template-Based</td></tr><tr><td>RETROSIM [Coley et al.,2017b]</td><td>52.9</td><td>73.8</td><td>81.2</td><td>88.1</td><td>37.3</td><td>54.7</td><td>63.3</td><td>74.1</td></tr><tr><td>NEURALSYM [Segler and Waller,2017]</td><td>55.3</td><td>76.0</td><td>81.4</td><td>85.1</td><td>44.4</td><td>65.3</td><td>72.4</td><td>78.9</td></tr><tr><td>GLN [Dai et ai., 2019]</td><td>64.2</td><td>79.1</td><td>85.2</td><td>90.0</td><td>52.5</td><td>69.0</td><td>75.6</td><td>83.7</td></tr><tr><td>DUALTB [Sun et al.,2021]</td><td>67.7</td><td>84.8</td><td>88.9</td><td>92.0</td><td>55.2</td><td>74.6</td><td>80.5</td><td>86.9</td></tr><tr><td colspan="9">Template-Free</td></tr><tr><td>SCROP [Zheng et al.,2019]</td><td>59.0</td><td>74.8</td><td>78.1</td><td>81.1</td><td>43.7</td><td>60.0</td><td>65.2</td><td>68.7</td></tr><tr><td>LV-TRANSFORMER [Chen et al.,2019]</td><td>-</td><td>-</td><td>-</td><td>-</td><td>40.5</td><td>65.1</td><td>72.8</td><td>79.4</td></tr><tr><td>DUALTF [Sun et al., 2021]</td><td>65.7</td><td>81.9</td><td>84.7</td><td>85.9</td><td>53.6</td><td>70.7</td><td>74.6</td><td>77.0</td></tr><tr><td colspan="9">Semi-Template-Based</td></tr><tr><td>G2Gs [Shi et al.,2020]</td><td>61.0</td><td>81.3</td><td>86.0</td><td>88.7</td><td>48.9</td><td>67.6</td><td>72.5</td><td>75.5</td></tr><tr><td>RETROXPERT [Yan et al.,2020]</td><td>62.1</td><td>75.8</td><td>78.5</td><td>80.9</td><td>50.4</td><td>61.1</td><td>62.3</td><td>63.4</td></tr><tr><td>GRAPHRETRO (ours)</td><td>63.9</td><td>81.5</td><td>85.2</td><td>88.1</td><td>53.7</td><td>68.3</td><td>72.2</td><td>75.5</td></tr></table>
+
+Baselines For evaluating overall performance, we compare GRAPHRETRO to nine baselines — four template-based, three template-free, and two semi-template-based methods. These include:
+
+Template-Based: RETROSIM Coley et al. [2017b] ranks templates for a given target molecule by computing molecular similarities to precedent reactions. NEURALSYM [Segler and Waller, 2017] trains a model to rank templates given a target molecule. GLN [Dai et al., 2019] models the joint distribution of templates and reactants in a hierarchical fashion using logic variables. DUALTB [Sun et al., 2021] uses an energy-based model formulation for retrosynthesis, with additional parameterizations and loss terms to enforce the duality between forward (reaction prediction) and backward (retrosynthesis prediction). Inference is carried out using reactant candidates obtained by applying an extracted template set to the products.
+
+Template-Free: SCROP [Zheng et al., 2019], LV-TRANSFORMER [Chen et al., 2019] and DUALTF [Sun et al., 2021] use the Transformer architecture [Vaswani et al., 2017] to output reactant SMILES given a product SMILES. To improve the validity of their suggestions, SCROP include a second Transformer that functions as a syntax correcter. LV-TRANSFORMER uses a latent variable mixture model to improve diversity of suggestions. DUALTF utilizes additional parameterizations and loss terms to enforce the duality between forward (reaction prediction) and backward (retrosynthesis prediction).
+
+Semi-Template-Based: G2GS [Shi et al., 2020] and RETROXPERT [Yan et al., 2020] first identify synthons, and then expand the synthons into reactants by either sequential generation of atoms and bonds (G2Gs), or using the Transformer architecture (RETROXPERT). The training dataset for the Transformer in [Yan et al., 2020] is augmented with incorrectly predicted synthons with the goal of learning a correction mechanism.
+
+Results for NEURALSYM are taken from [Dai et al., 2019]. The authors in [Yan et al., 2020] report their performance being affected by the dataset leakage2. Thus, we use the most recent results from their website on the canonicalized dataset. For remaining baselines, we directly use the values reported in their paper. For the synthon completion module, we use the ind configuration given its better empirical performance.
+
+# 4.1 Overall Performance
+
+Reaction class unknown As shown in Table 1, when the reaction class is unknown, GRAPHRETRO outperforms G2GS by $4 . 8 \%$ and and RETROXPERT by $3 . 3 \%$ in top-1 accuracy. Performance improvements are also seen for larger $n$ , except for $n = 5$ . Barring DUALTB, the top-1 accuracy is also better than other template-free and template-based methods. For larger $n$ , one reason for lower top- $n$ accuracies than most template-based methods is that templates already contain combinations of leaving group patterns. In contrast, our model learns to discover these during training. A second hypothesis to this end is that simply adding log-likelihood scores from edit prediction and synthon completion models may be suboptimal and bias the beam search in the direction of the more dominating term. We leave it to future work to investigate scoring functions that rank the attachment.
+
+Reaction class known When the reaction class is known, GRAPHRETRO outperforms G2GS and RETROXPERT by a margin of $3 \%$ and $2 \%$ respectively in top-1 accuracy. GRAPHRETRO also outperforms all the template-free methods in top- $^ n$ accuracy. for GRAPHRETRO are also better than most template-based and template-free methods. When the reaction class is known, RETROSIM and GLN restrict template sets corresponding to the reaction class, thus improving performance. The increased edit prediction performance (Section 4.2) for GRAPHRETRO helps outweigh this factor, achieving comparable or better performance till $n = 5$ .
+
+# 4.2 Individual Module Performance
+
+To gain more insight into the working of GRAPHRETRO, we evaluate the top- $\boldsymbol { n }$ accuracy $\mathbf { \nabla } _ { n } =$ $1 , 2 , 3 , 5 )$ of edit prediction and synthon completion modules, along with corresponding ablation studies, with results shown in Table 2.
+
+Edit Prediction For the edit prediction module, we compare the true edit(s) to top- $\boldsymbol { n }$ edits predicted by the model. We also consider two ablation studies, one where we directly use the initial edit scores without updating them, and the other where we predict edits using atom-pairs instead of existing bonds and atoms. Both design choices lead to improvements in performance, as shown in Table 2. We hypothesize that the larger improvement compared to edit prediction using atom-pairs is due to the easier optimization procedure, with lesser imbalance between labels 1 and 0.
+
+Synthon Completion For evaluating the synthon completion module, we first apply the true edits to obtain synthons, and compare the true leaving groups to top- $\mathbf { \nabla } \cdot n$ leaving groups predicted by the model. We test the performance of both the ind and shared configurations. Both configurations perform similarly, and are able to identify $\sim 9 7 \%$ (close to its upper bound of $9 9 . 7 \%$ ) of the true leaving groups in its top-5 choices, when the reaction class is known. The top-1, 3 and 5 accuracies of the synthon completion for unknown reaction classes for G2Gs are $6 1 . 1 \%$ , $8 1 . 5 \%$ and $8 6 . 7 \%$ respectively, while ours are $7 5 . 6 \%$ , $9 2 . 5 \%$ and $9 6 . 1 \%$ , indicating a $10 \%$ performance improvement using a classification formulation over the generative one adopted by G2Gs.
+
+Table 2: Performance Study of edit prediction and synthon completion modules   
+
+<table><tr><td rowspan="3"> Setting</td><td colspan="8">Top-n Accuracy (%)</td></tr><tr><td colspan="4">Reaction class known</td><td colspan="4">Reaction class unknown</td></tr><tr><td>1</td><td>2</td><td>3</td><td>5</td><td>1</td><td>2</td><td>3</td><td>5</td></tr><tr><td>Edit Prediction</td><td>84.6</td><td>92.2</td><td>93.7</td><td>94.5</td><td>70.8</td><td>85.1</td><td>89.5</td><td>92.7</td></tr><tr><td>- without edit score updates</td><td>84.3</td><td>92.1</td><td>93.7</td><td>94.5</td><td>70.1</td><td>84.8</td><td>89.4</td><td>92.6</td></tr><tr><td>- predicting on atom pairs</td><td>81.9</td><td>89.5</td><td>90.9</td><td>92.1</td><td>68.6</td><td>83.2</td><td>88.3</td><td>91.8</td></tr><tr><td>Synthon Completion (ind)</td><td>77.4</td><td>89.5</td><td>94.2</td><td>97.6</td><td>75.6</td><td>87.4</td><td>92.5</td><td>96.1</td></tr><tr><td>Synthon Completion (shared)</td><td>76.9</td><td>89.6</td><td>93.9</td><td>97.4</td><td>74.9</td><td>87.7</td><td>92.9</td><td>96.3</td></tr></table>
+
+# 4.3 Example Predictions
+
+In Figure 2, we visualize the model predictions and the ground truth for three cases. Figure 2a shows an example where the model identifies both the edits and leaving groups correctly. In Figure 2b, the correct edit is identified but the predicted leaving groups are incorrect. We hypothesize this is due to the fact that in the training set, leaving groups attaching to the carbonyl carbon $\scriptstyle ( \mathbf { C } = \mathbf { O } )$ are small (e.g. -OH, - $\mathrm { - N H _ { 2 } }$ , halides). The true leaving group in this example, however, is large. The model is unable to reason about this and predicts the small leaving group -I. In Figure 2c, the model identifies the edit and consequently the leaving group incorrectly. This highlights a limitation of our model. If the edit is predicted incorrectly, the model cannot suggest the true precursors.
+
+# 4.4 Limitations
+
+The simplified and interpretable construction of GRAPHRETRO comes with certain limitations. First, the overall performance of the model is limited by the performance of the edit prediction step. If the predicted edit is incorrect, the true reactants cannot be salvaged. This limitation is partly remedied by our model design, that allows for user intervention to correct the edit. Second, our method is reliant on atom-mapping for extracting edits and leaving groups. Extracting edits directly based on substructure matching currently suffer from false positives, and heuristics to correct for these result in correct edits in only ${ \sim } 9 0 \%$ of the cases. Third, our formulation assumes that we have as many synthons as reactants, which is violated in some reactions. We leave it to future work to extend the model to realize a single reactant from multiple synthons, and introduce more chemically meaningful edit correction mechanisms.
+
+![](images/9df65c90e84b75ac55b8dc08692726ee93cf492df4b05d29c79c8c792f98fccd.jpg)  
+Figure 2: Example Predictions. The true edit and incorrect edit (if any) are highlighted in green and red respectively. The true and predicted leaving groups are highlighted in blue. a. Correctly predicted example by the model. b. Correctly predicted edit but incorrectly predicted leaving groups. c. Incorrectly predicted edit and leaving group.
+
+# 5 Conclusion
+
+Previous methods for single-step retrosynthesis either restrict prediction to a template set, are insensitive to molecular graph structure or generate molecules from scratch. We address these shortcomings by introducing a graph-based semi-template-based model inspired by a chemist’s workflow, enhancing the interpretability of retrosynthesis models. Given a target molecule, we first identify synthetic building blocks (synthons) which are then realized into valid reactants, thus avoiding molecule generation from scratch. Our model outperforms previous semi-template-methods by significant margins on the benchmark dataset. Future work aims to extend the model to realize a single reactant from multiple synthons, and introduce more chemically meaningful components to improve the synergy between such tools for retrosynthesis prediction and a practitioner’s expertise.
+
+# Acknowledgements
+
+This research was supported by the Machine Learning for Pharmaceutical Discovery and Synthesis Consortium at MIT. V.R.S. was also supported by the Zeno Karl Schindler Foundation. C.B. was supported by the Swiss National Science Foundation under the National Center of Competence in Research (NCCR) Catalysis under grant agreement 51NF40 180544. We thank the Leonhard scientific computing cluster at ETH Zürich for providing computational resources.
+
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+C. W. Coley, L. Rogers, W. H. Green, and K. F. Jensen. Computer-Assisted Retrosynthesis Based on Molecular Similarity. ACS Central Science, 3, 2017b.   
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+E. Corey and W. T. Wipke. Computer-assisted design of complex organic syntheses. Science, 166 (3902):178–192, 1969.   
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+        "text": "Charlotte Bunne1 ",
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+        "text": "Connor W. Coley2 ",
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+        "text": "Andreas Krause1 Regina Barzilay3 ",
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+        "text": "1Department of Computer Science, ETH 2Department of Chemical Engineering, MIT 3Computer Science and Artificial Intelligence Lab, MIT 1{vsomnath, bunnec, krausea}@ethz.ch, 2ccoley@mit.edu, 3regina@csail.mit.edu ",
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+        "type": "text",
+        "text": "Abstract ",
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+        "text": "Retrosynthesis prediction is a fundamental problem in organic synthesis, where the task is to identify precursor molecules that can be used to synthesize a target molecule. A key consideration in building neural models for this task is aligning model design with strategies adopted by chemists. Building on this viewpoint, this paper introduces a graph-based approach that capitalizes on the idea that the graph topology of precursor molecules is largely unaltered during a chemical reaction. The model first predicts the set of graph edits transforming the target into incomplete molecules called synthons. Next, the model learns to expand synthons into complete molecules by attaching relevant leaving groups. This decomposition simplifies the architecture, making its predictions more interpretable, and also amenable to manual correction. Our model achieves a top-1 accuracy of $5 3 . 7 \\%$ , outperforming previous template-free and semi-template-based methods. ",
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+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "Retrosynthesis prediction, first formalized by E. J. Corey [Corey, 1991] is a fundamental problem in organic synthesis that attempts to identify a series of chemical transformations for synthesizing a target molecule. In the single-step formulation, the task is to identify a set of reactant molecules given a target. Beyond simple reactions, many practical tasks involving complex organic molecules are difficult even for expert chemists. As a result, substantial experimental exploration is needed to cover for deficiencies of analytical approaches. This has motivated interest in computer-assisted retrosynthesis [Corey and Wipke, 1969], with a recent surge in machine learning methods [Chen et al., 2019, Coley et al., 2017b, Dai et al., 2019, Zheng et al., 2019, Genheden et al., 2020]. ",
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+        "text": "Computationally, the main challenge is how to explore the combinatorial space of reactions that can yield the target molecule. Largely, previous methods for retrosynthesis prediction can be divided into template-based [Coley et al., 2017b, Dai et al., 2019, Segler and Waller, 2017] and template-free [Chen et al., 2019, Zheng et al., 2019] approaches. Template-based methods match a target molecule against a large set of templates, which are molecular subgraph patterns that highlight changes during a chemical reaction. Despite their interpretability, these methods fail to generalize to new reactions. Template-free methods bypass templates by learning a direct mapping from the SMILES [Weininger, 1988] representations of the product to reactants. Despite their greater generalization potential, these methods generate reactant SMILES character by character, increasing generation complexity. ",
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+        "img_path": "images/e988f2e4a9f13ae6fa41934806006c24c87183942b7fd7f824274aee3a6237f9.jpg",
+        "image_caption": [
+            "Figure 1: Overview of Our Approach. a. Edit Prediction. We train a model to learn a distribution over possible graph edits. In this case, the correct edit corresponds to breaking the bond marked in red. Applying this edit produces two synthons. b. Synthon Completion. Another model is trained to pick candidate leaving groups (blue) for each synthon from a discrete vocabulary, which are then attached to produce the final reactants. "
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+        "text": "Another important consideration in building retrosynthesis models is aligning model design with strategies adopted by expert chemists. These strategies are influenced by fundamental properties of chemical reactions, independent of complexity level: (i.) the product atoms are always a subset of the reactant atoms1, and (ii.) the molecular graph topology is largely unaltered from products to reactants. For example, in the standard retrosynthesis dataset, only $6 . 3 \\%$ of the atoms in the product undergo any change in connectivity. ",
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+        "text": "This consideration has received more attention in recent semi-template-based methods [Shi et al., 2020, Yan et al., 2020], that generate reactants from a product in two stages: (i.) first identify intermediate molecules called synthons, (ii.) and then complete synthons into reactants by sequential generation of atoms or SMILES characters.. Our model GRAPHRETRO also uses a similar workflow. However, we avoid sequential generation for completing synthons by instead selecting subgraphs called leaving groups from a precomputed vocabulary. This vocabulary is constructed during preprocessing by extracting subgraphs that differ between a synthon and the corresponding reactant. The vocabulary has a small size (170 for USPTO-50k) indicating remarkable redundancy, while covering $9 9 . 7 \\%$ of the test set. Operating at the level of these subgraphs greatly reduces the complexity of reactant generation, with improved empirical performance. This formulation also simplifies our architecture, and makes our predictions more transparent, interpretable and amenable to manual correction. ",
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+        "text": "The benchmark dataset for evaluating retrosynthesis models is USPTO-50k [Schneider et al., 2016], which consists of 50000 reactions across 10 reaction classes. The dataset contains an unexpected shortcut towards predicting the edit, in that the product atom with atom-mapping 1 is part of the edit in $7 5 \\%$ of the cases, allowing predictions that depend on the position of the atom to overestimate performance. We canonicalize the product SMILES and remap the existing dataset, thereby removing the shortcut. On this remapped dataset, GRAPHRETRO achieves a top-1 accuracy of $5 3 . 7 \\%$ when the reaction class is not known, outperforming both template-free and semi-template-based methods. ",
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+        "text": "2 Related Work ",
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+        "text": "Retrosynthesis Prediction Existing machine learning methods for retrosynthesis prediction can be divided into template-based, template-free and recent semi-template-based approaches. ",
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+        "text": "Template-Based: Templates are either hand-crafted by experts [Hartenfeller et al., 2011, Szymkuc´ et al., 2016], or extracted algorithmically from large databases Coley et al. [2017a], Law et al. [2009]. Exhaustively applying large template sets is expensive due to the involved subgraph matching procedure. Template-based methods therefore utilize different ways of prioritizing templates, by either learning a conditional distribution over the template set [Segler and Waller, 2017], ranking templates based on molecular similarities to precedent reactions [Coley et al., 2017b] or directly modelling the joint distribution of templates and reactants using logic variables [Dai et al., 2019]. Despite their interpretability, these methods fail to generalize outside their rule set. ",
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+        "text": "Template-Free: Template-free methods [Liu et al., 2017, Zheng et al., 2019, Chen et al., 2019] learn a direct transformation from products to reactants using architectures from neural machine translation and a string based representation of molecules called SMILES [Weininger, 1988]. Linearizing molecules as strings does not utilize the inherently rich chemical structure. In addition, the reactant SMILES are generated from scratch, character by character. Attempts have been made to improve validity by adding a syntax correcter [Zheng et al., 2019] and a mixture model to improve diversity of suggestions [Chen et al., 2019], but the performance remains worse than [Dai et al., 2019] on the standard retrosynthesis dataset. Sun et al. [2021] formulate retrosynthesis using energy-based models, with additional parameterizations and loss terms to enforce the duality between forward (reaction prediction) and backward (retrosynthesis) prediction. ",
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+        "text": "Semi-Template-Based: Our work is closely related to recently proposed semi-template-based methods [Shi et al., 2020, Yan et al., 2020], which first identify synthons and then expand synthons into reactants through sequential generation using either a graph generative model [Shi et al., 2020] or a Transformer [Yan et al., 2020]. To reduce the complexity of reactant generation, we instead complete synthons using subgraphs called leaving groups selected from a precomputed vocabulary. This allows us to view synthon completion as a classification problem instead of a generative one. We also utilize the dependency graph between possible edits, and update edit predictions using a message passing network (MPN) [Gilmer et al., 2017] on this graph. Both innovations together yield a $4 . 8 \\%$ and $3 . 3 \\%$ performance improvement respectively over previous semi-template-based methods. ",
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+        "type": "text",
+        "text": "Reaction Center Identification The reaction center covers a small number of participating atoms involved in the reaction. Our work is also related to models that predict reaction outcomes by learning to rank atom pairs based on their likelihood to be in the reaction center [Coley et al., 2019, Jin et al., 2017]. The task of identifying the reaction center is related to the step of deriving the synthons in our formulation. Our work departs from [Coley et al., 2019, Jin et al., 2017] as we utilize the property that new bond formations occur rarely $( \\sim 0 . 1 \\% )$ from products to synthons, allowing us to predict a score only for existing bonds and atoms and reduce prediction complexity from $O ( N ^ { \\bar { 2 } } )$ to $O ( N )$ . We also utilize the dependency graph between possible edits, and update edit predictions using a MPN on this graph. ",
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+    {
+        "type": "text",
+        "text": "Utilizing Substructures Substructures have been utilized in various tasks from sentence generation by fusing phrases to molecule generation and optimization [Jin et al., 2018, 2020]. Our work is closely related to [Jin et al., 2020] which uses precomputed substructures as building blocks for property-conditioned molecule generation. However, instead of precomputing, synthons —analogous building blocks for reactants— are indirectly learnt during training. ",
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+    {
+        "type": "text",
+        "text": "3 Model Design ",
+        "text_level": 1,
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+        "text": "Our approach leverages the property that graph topology is largely unaltered from products to reactants. To achieve this, we first derive suitable building blocks from the product called synthons, and then complete them into valid reactants by adding specific functionalities called leaving groups. These derivations, called edits, are characterized by modifications to bonds or hydrogen counts on atoms. We first train a neural network to predict a score for possible edits (Section 3.1). The edit with the highest score is then applied to the product to obtain synthons. Since the number of unique leaving groups are small, we model leaving group selection as a classification problem over a precomputed vocabulary (Section 3.2). To produce candidate reactants, we attach the predicted leaving group to the corresponding synthon through chemically constrained rules. The overall process is outlined in Figure 1. Before describing the two modules, we introduce relevant preliminaries that set the background for the remainder of the paper. ",
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+        "text": "Retrosynthesis Prediction A retrosynthesis pair $R$ is described by a pair of molecular graphs $( \\mathcal { G } _ { p } , \\mathcal { G } _ { r } )$ , where $\\mathcal { G } _ { p }$ are the products and $\\mathcal { G } _ { r }$ the reactants. A molecular graph is described as $\\mathcal { G } = \\mathbf { \\bar { \\rho } } ( \\mathcal { V } , \\mathcal { E } )$ with atoms $\\nu$ as nodes and bonds $\\mathcal { E }$ as edges. Prior work has focused on the single product case, while reactants can have multiple connected components, i.e. $\\mathcal { G } _ { r } = \\{ \\mathcal { G } _ { r _ { c } } \\} _ { c = 1 } ^ { C }$ . Retrosynthesis pairs are atom-mapped so that each product atom has a unique corresponding reactant atom. The retrosynthesis task then, is to infer $\\{ \\mathcal { G } _ { r _ { c } } \\} _ { c = 1 } ^ { C }$ given $\\mathcal { G } _ { p }$ . ",
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+        "text": "Edits Edits consist of (i.) atom pairs $\\left\\{ \\left( a _ { i } , a _ { j } \\right) \\right\\}$ where the bond type changes from products to reactants, and (ii.) atoms $\\left\\{ { a } _ { i } \\right\\}$ where the number of hydrogens attached to the atom change from products to reactants. We denote the set of edits by $E$ . Since retrosynthesis pairs in the training set are atom-mapped, edits can be automatically identified by comparing the atoms and atom pairs in the product to their corresponding reactant counterparts. ",
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+        "text": "Synthons and Leaving Groups Applying edits $E$ to the product $\\mathcal { G } _ { p }$ results in incomplete molecules called synthons. Synthons are analogous to rationales or building blocks, which are expanded into valid reactants by adding specific functionalities called leaving groups that are responsible for its reactivity. We denote synthons by $\\mathcal { G } _ { s }$ and leaving groups by $\\mathcal { G } _ { l }$ . We further assume that synthons and leaving groups have the same number of connected components as the reactants, i.e $\\mathcal { G } _ { s } \\doteq \\{ \\mathcal { G } _ { s _ { c } } \\} _ { c = 1 } ^ { C }$ and $\\mathcal { G } _ { l } = \\{ \\mathcal { G } _ { l _ { c } } ^ { \\star } \\} _ { c = 1 } ^ { C }$ . This assumption holds for $9 9 . 9 7 \\%$ reactions in the training set. ",
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+        "type": "text",
+        "text": "Formally, our model generates reactants by first predicting the set of edits $E$ that transform $\\mathcal { G } _ { p }$ into $\\mathcal { G } _ { s }$ , followed by predicting a leaving group $\\mathcal { G } _ { l _ { c } }$ to attach to each synthon $\\mathcal { G } _ { s _ { c } }$ . The model is defined as ",
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+    {
+        "type": "equation",
+        "img_path": "images/e5989ffe48f40113cab8f00a06662537b577c1fdbf3a8f9d8153b9ad86f85418.jpg",
+        "text": "$$\nP ( \\mathcal G _ { r } | \\mathcal G _ { p } ) = \\sum _ { E , \\mathcal G _ { l } } P ( E | \\mathcal G _ { p } ) P ( \\mathcal G _ { l } | \\mathcal G _ { p } , \\mathcal G _ { s } ) ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $\\mathcal { G } _ { s } , \\mathcal { G } _ { r }$ are deterministic given $E , { \\mathcal { G } } _ { l }$ , and $\\mathcal { G } _ { p }$ . ",
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+        "type": "text",
+        "text": "3.1 Edit Prediction ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "For a given retrosynthesis pair $R = ( \\mathcal G _ { p } , \\mathcal G _ { r } )$ , we predict an edit score only for existing bonds and atoms, instead of every atom pair as in [Coley et al., 2019, Jin et al., 2017]. This choice is motivated by the low frequency $( \\sim 0 . 1 \\% )$ of new bond formations in the training set examples. Coupled with the sparsity of molecular graphs, this reduces the prediction complexity from $O ( N ^ { 2 } )$ to $O ( N )$ for a product with $N$ atoms. Our edit prediction model has variants tailored to single and multiple edit prediction. Since $9 5 \\%$ of the training set consists of single edit examples, the remainder of this section describes the setup for single edit prediction. A detailed description of our multiple edit prediction model can be found in Appendix ??. ",
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+    {
+        "type": "text",
+        "text": "Each bond $( u , v )$ in $\\mathcal { G } _ { p }$ is associated with a label $y _ { u v k } \\in \\{ 0 , 1 \\}$ indicating whether its bond type $k$ has changed from the products to reactants. Each atom $u$ is associated with a label $y _ { u } \\in \\{ 0 , \\bar { 1 } \\}$ indicating a change in hydrogen count. We predict edit scores using representations that are learnt using a graph encoder. ",
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+        "type": "text",
+        "text": "Graph Encoder To obtain atom representations, we use a variant of the message passing network (MPN) described in [Gilmer et al., 2017]. Each atom $u$ has a feature vector $\\mathbf { x } _ { u }$ indicating its atom type, degree and other properties. Each bond $( u , v )$ has a feature vector $\\mathbf { x } _ { u v }$ indicating its aromaticity, bond type and ring membership. For simplicity, we denote the encoding process by $\\mathrm { M P N } ( \\cdot )$ and describe architectural details in Appendix ??. The MPN computes atom representations $\\{ \\mathbf { c } _ { u } | u \\in \\mathcal { G } \\}$ via ",
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+        "type": "equation",
+        "img_path": "images/b517ef315352815d369d32f3af958c21a2ae42d976e66bdc837220e4bb39c388.jpg",
+        "text": "$$\n\\{ \\mathbf { c } _ { u } \\} = \\mathrm { M P N } ( \\mathcal { G } , \\{ \\mathbf { x } _ { u } \\} , \\{ \\mathbf { x } _ { u v } \\} _ { v \\in \\mathcal { N } ( u ) } ) ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $\\mathcal { N } ( u )$ denotes the neighbors of atom $u$ . The graph representation $\\mathbf { c } _ { \\mathcal { G } }$ is an aggregation of atom representations, i.e. $\\mathbf { c } _ { \\mathcal { G } } = \\dot { \\sum _ { { u } \\in \\mathcal { V } } } \\mathbf { c } _ { u }$ . When $\\mathcal { G }$ has connected components $\\left\\{ { \\mathcal { G } } _ { i } \\right\\}$ , we get a set of graph representations $\\left\\{ \\mathbf { c } _ { \\mathcal { G } _ { i } } \\right\\}$ . For a bond $( u , v )$ , we define its representation $\\mathbf { c } _ { u v } = ( \\operatorname { A B S } ( \\mathbf { c } _ { u } , \\mathbf { c } _ { v } ) | | \\mathbf { c } _ { u } + \\mathbf { c } _ { v } )$ , where ABS denotes absolute difference and $| |$ refers to concatenation. This ensures our representations are permutation invariant. These representations are then used to predict atom and bond edit scores using corresponding neural networks, ",
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+        "img_path": "images/43a6f38c88896496f6b8c1e00b965fa739609189983ee4e111bdef371168642c.jpg",
+        "text": "$$\n\\begin{array} { r } { \\boldsymbol { s } _ { u } = \\mathbf { u _ { a } } ^ { T } \\boldsymbol { \\tau } ( \\mathbf { W _ { a } } \\mathbf { c } _ { u } + b ) \\quad } \\\\ { \\boldsymbol { s } _ { u v k } = \\mathbf { u _ { k } } ^ { T } \\boldsymbol { \\tau } ( \\mathbf { W _ { k } } \\mathbf { c } _ { u v } + b _ { k } ) , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\tau ( \\cdot )$ is the ReLU activation function. ",
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+        "text": "Updating Bond Edit Scores Unlike a typical classification problem where the labels are independent, edits can have possible dependencies between each other. For example, bonds part of a stable system such as an aromatic ring have a greater tendency to remain unchanged (label 0). We attempt to leverage such dependencies to update initial edit scores. To this end, we build a graph with bonds $( u , v )$ as nodes, and introduce an edge between bonds sharing an atom. We use another $\\mathrm { M P N } ( \\cdot )$ on this graph to learn aggregated neighborhood messages $\\mathbf { m } _ { u v }$ , and update the edit scores $s _ { u v k }$ in a manner similar to how LSTMs update representations, ",
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+        "img_path": "images/190ca6c12e9d0551c6c507ca3bb0a358e64c6db98378f96a20d19207a9bfc801.jpg",
+        "text": "$$\n\\begin{array} { r l } & { f _ { u v k } = \\sigma ( \\mathbf { W _ { k x } ^ { f } } \\mathbf { x } _ { u v } + \\mathbf { W _ { k m } ^ { f } } \\mathbf { m } _ { u v } ) } \\\\ & { i _ { u v k } = \\sigma ( \\mathbf { W _ { k x } ^ { i } } \\mathbf { x } _ { u v } + \\mathbf { W _ { k m } ^ { i } } \\mathbf { m } _ { u v } ) } \\\\ & { \\tilde { m } _ { u v k } = \\mathbf { u _ { m } } \\tau ( \\mathbf { W _ { k x } ^ { m } } \\mathbf { x } _ { u v } + \\mathbf { W _ { k m } ^ { m } } \\mathbf { m } _ { u v } ) } \\\\ & { \\tilde { s } _ { u v k } = f _ { u v k } \\cdot s _ { u v k } + i _ { u v k } \\cdot \\tilde { m } _ { u v k } . } \\end{array}\n$$",
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+        "type": "text",
+        "text": "Training We train by minimizing the cross-entropy loss over possible bond and atom edits ",
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+        "img_path": "images/d39b464af053c1ddc554cc80cf10fc8a8cf6434c8681b4503820a67f7447deed.jpg",
+        "text": "$$\n\\mathcal { L } _ { e } = - \\sum _ { ( \\mathcal { G } _ { p } , E ) } \\left( \\sum _ { ( ( u , v ) , k ) \\in E } y _ { u v k } \\mathrm { l o g } ( \\widetilde s _ { u v k } ) + \\sum _ { u \\in E } y _ { u } \\mathrm { l o g } ( s _ { u } ) \\right) .\n$$",
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+    {
+        "type": "text",
+        "text": "The cross-entropy loss enforces the model to learn a distribution over possible edits instead of reasoning about each edit independently, as with the binary cross entropy loss used in [Jin et al., 2017, Coley et al., 2019]. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.2 Synthon Completion ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Synthons are completed into valid reactants by adding specific functionalities called leaving groups. This involves two complementary tasks: (i.) selecting the appropriate leaving group, and (ii.) attaching the leaving group to the synthon. As ground truth leaving groups are not directly provided, we extract the leaving groups and construct a vocabulary $\\mathcal { X }$ of unique leaving groups during preprocessing. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The vocabulary has a limited size ( $| \\mathcal { X } | = 1 7 0$ for a standard dataset with 50, 000 examples, and 72000 synthons) indicating the redundancy of leaving groups used in accomplishing retrosynthetic transformations. This redundancy also allows us to formulate leaving group selection as a classification problem over $\\mathcal { X }$ , while retaining the ability to generate diverse reactants using different combinations of leaving groups. ",
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+    },
+    {
+        "type": "text",
+        "text": "Vocabulary Construction Before constructing the vocabulary, we align connected components of synthon and reactant graphs by comparing atom mapping overlaps. Using aligned pairs $\\mathcal { G } _ { s _ { c } } =$ $( \\gamma _ { s _ { c } } , \\mathcal { E } _ { s _ { c } } )$ and $\\mathcal { G } _ { r _ { c } } = ( \\nu _ { r _ { c } } , \\mathcal { E } _ { r _ { c } } )$ as input, the leaving group vocabulary $\\mathcal { X }$ is constructed by extracting subgraphs $\\mathcal { G } _ { l _ { c } } = ( \\nu _ { l _ { c } } , \\mathcal { E } _ { l _ { c } } )$ such that $\\smash { \\gamma _ { l _ { c } } = \\gamma _ { r _ { c } } \\setminus \\gamma _ { s _ { c } } }$ . Atoms $\\left\\{ { a } _ { i } \\right\\}$ in the leaving groups that attach to synthons are marked with a special symbol. We also add three tokens to $\\mathcal { X }$ namely START, which indicates the start of synthon completion, END, which indicates that there is no leaving group to add and PAD, which is used to handle variable numbers of synthon components in a minibatch. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Leaving Group Selection For synthon component $c \\leq C$ , where $C$ is the number of connected components in the synthon graph, we use three inputs for leaving group selection – the product representation $\\mathbf { c } _ { \\mathcal { G } _ { p } }$ , the synthon component representation $\\mathbf { c } _ { \\mathcal { G } _ { s _ { c } } }$ , and the leaving group representation for the previous synthon component, ${ \\bf e } _ { l _ { c - 1 } }$ . The product and synthon representations are learnt using the $\\mathrm { M P N } ( \\cdot )$ . For each $x _ { i } \\in { \\mathcal { X } }$ , representations can be learnt by either training independent embedding vectors (ind) or by treating each $x _ { i }$ as a subgraph and using the $\\mathrm { M P N } ( \\cdot )$ (shared). In the shared setting, we use the same $\\mathrm { M P N } ( \\cdot )$ as the product and synthons. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The leaving group probabilities are then computed by combining $\\mathbf { c } _ { \\mathcal { G } _ { p } } , \\mathbf { c } _ { \\mathcal { G } _ { s _ { c } } }$ and $\\mathbf { e } _ { l _ { c - 1 } }$ via a single layer neural network and softmax function ",
+        "bbox": [
+            173,
+            90,
+            823,
+            119
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d2fa89696df25b480905fd11b89872899b1206510d6da72f11724cb4e151b106.jpg",
+        "text": "$$\n\\hat { q } _ { l _ { c } } = \\mathrm { s o f t m a x } \\left( \\mathbf { U } \\tau \\left( \\mathbf { W } _ { 1 } \\mathbf { c } _ { \\mathcal { G } _ { p } } + \\mathbf { W } _ { 2 } \\mathbf { c } _ { \\mathcal { G } _ { s _ { c } } } + \\mathbf { W } _ { 3 } \\mathbf { e } _ { l _ { \\left( c - 1 \\right) } } \\right) \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            131,
+            691,
+            151
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { q } _ { l _ { c } }$ is distribution learnt over $\\mathcal { X }$ . Using the representation of the previous leaving group ${ \\bf e } _ { l _ { c - 1 } }$ allows the model to understand combinations of leaving groups that generate the desired product from the reactants. We also include the product representation $\\mathbf { c } _ { \\mathcal { G } _ { p } }$ as the synthon graphs are derived from the product graph. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Training For step $c$ , given the one hot encoding of the true leaving group $q _ { l _ { c } }$ , we minimize the cross-entropy loss ",
+        "bbox": [
+            173,
+            233,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c5878d96a04183a5e264e043e021073accc4e3106f900490a5939023894394b7.jpg",
+        "text": "$$\n\\mathcal { L } _ { s } = \\sum _ { c = 1 } ^ { C } \\mathcal { L } ( \\hat { q } _ { l _ { c } } , q _ { l _ { c } } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            275,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Training utilizes teacher-forcing [Williams and Zipser, 1989] so that the model makes predictions given correct histories. During inference, at every step, we use the representation of leaving group from the previous step with the highest predicted probability. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Leaving Group Attachment Attaching leaving groups to synthons is a deterministic process and not learnt during training. The task involves identification of the type of bonds to add between attaching atoms in the leaving group (marked during vocabulary construction), and the atom(s) participating in the edit. These bonds can be inferred by applying the valency constraint, which determines the maximum number of neighbors for each atom. The attachment process does not modify any stereochemistry. Given synthons and leaving groups, the attachment process has a $100 \\%$ accuracy. The detailed procedure is described in Appendix ??. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "3.3 Inference ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Inference is performed using beam search with a log-likelihood scoring function. For a beam width $n$ , we select $n$ edits with highest scores and apply them to the product to obtain $n$ synthons, where each synthon can consist of multiple connected components. The synthons form the nodes for beam search. Each node maintains a cumulative score by aggregating the log-likelihoods of the edit and predicted leaving groups. Leaving group inference starts with a connected component for each synthon, and selects $n$ leaving groups with highest log-likelihoods. From the $n ^ { 2 }$ possibilities, we select $n$ nodes with the highest cumulative scores. This process is repeated until all nodes have a leaving group predicted for each synthon component. ",
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+    },
+    {
+        "type": "text",
+        "text": "4 Evaluation ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Evaluating retrosynthesis models is challenging as multiple sets of reactants can be generated from the same product. To deal with this, previous works [Coley et al., 2017b, Dai et al., 2019] evaluate the ability of the model to recover retrosynthetic strategies recorded in the dataset. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Data We use the benchmark dataset USPTO-50k [Schneider et al., 2016] for all our experiments. The dataset contains 50, 000 atom-mapped reactions across 10 reaction classes. We use the same dataset version and splits as provided by [Dai et al., 2019]. The USPTO-50k dataset contains a shortcut in that the product atom with atom-mapping 1 is part of the edit in $\\sim 7 5 \\%$ of the cases. If the product SMILES is not canonicalized, predictions utilizing operations that depend on the position of the atom or bond will be able to use the shortcut, and overestimate performance. We canonicalize the product SMILES, and reassign atom-mappings to the reactant atoms based on the canonical ordering, which removes the shortcut. Details on the remapping procedure can be found in Appendix ??. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Evaluation We use the top- $\\mathbf { \\nabla } \\cdot n$ accuracy $( n = 1 , 3 , 5 , 1 0 )$ ) as our evaluation metric, defined as the fraction of examples where the recorded reactants are suggested by the model with rank $\\leq n$ . Following prior work [Coley et al., 2017b, Zheng et al., 2019, Dai et al., 2019], we compute the accuracy by comparing the canonical SMILES of predicted reactants to the ground truth. Atom-mapping is excluded from this comparison, but stereochemistry, which describes the relative orientation of atoms in the molecule, is retained. The evaluation is carried out for two settings, with the reaction class being known or unknown. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/b50491b8ada14574c6239d28c43d14386b5d359906cdb5f2c186422249a03113.jpg",
+        "table_caption": [
+            "Table 1: Top- $n$ exact match accuracy. Best values within each section are highlighted in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\">Model</td><td colspan=\"8\">Top-n Accuracy (%)</td></tr><tr><td colspan=\"4\">Reaction class known</td><td colspan=\"4\">Reaction class unknown</td></tr><tr><td>1</td><td>3</td><td>5</td><td>10</td><td>1</td><td>3</td><td>5</td><td>10</td></tr><tr><td colspan=\"9\">Template-Based</td></tr><tr><td>RETROSIM [Coley et al.,2017b]</td><td>52.9</td><td>73.8</td><td>81.2</td><td>88.1</td><td>37.3</td><td>54.7</td><td>63.3</td><td>74.1</td></tr><tr><td>NEURALSYM [Segler and Waller,2017]</td><td>55.3</td><td>76.0</td><td>81.4</td><td>85.1</td><td>44.4</td><td>65.3</td><td>72.4</td><td>78.9</td></tr><tr><td>GLN [Dai et ai., 2019]</td><td>64.2</td><td>79.1</td><td>85.2</td><td>90.0</td><td>52.5</td><td>69.0</td><td>75.6</td><td>83.7</td></tr><tr><td>DUALTB [Sun et al.,2021]</td><td>67.7</td><td>84.8</td><td>88.9</td><td>92.0</td><td>55.2</td><td>74.6</td><td>80.5</td><td>86.9</td></tr><tr><td colspan=\"9\">Template-Free</td></tr><tr><td>SCROP [Zheng et al.,2019]</td><td>59.0</td><td>74.8</td><td>78.1</td><td>81.1</td><td>43.7</td><td>60.0</td><td>65.2</td><td>68.7</td></tr><tr><td>LV-TRANSFORMER [Chen et al.,2019]</td><td>-</td><td>-</td><td>-</td><td>-</td><td>40.5</td><td>65.1</td><td>72.8</td><td>79.4</td></tr><tr><td>DUALTF [Sun et al., 2021]</td><td>65.7</td><td>81.9</td><td>84.7</td><td>85.9</td><td>53.6</td><td>70.7</td><td>74.6</td><td>77.0</td></tr><tr><td colspan=\"9\">Semi-Template-Based</td></tr><tr><td>G2Gs [Shi et al.,2020]</td><td>61.0</td><td>81.3</td><td>86.0</td><td>88.7</td><td>48.9</td><td>67.6</td><td>72.5</td><td>75.5</td></tr><tr><td>RETROXPERT [Yan et al.,2020]</td><td>62.1</td><td>75.8</td><td>78.5</td><td>80.9</td><td>50.4</td><td>61.1</td><td>62.3</td><td>63.4</td></tr><tr><td>GRAPHRETRO (ours)</td><td>63.9</td><td>81.5</td><td>85.2</td><td>88.1</td><td>53.7</td><td>68.3</td><td>72.2</td><td>75.5</td></tr></table>",
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Baselines For evaluating overall performance, we compare GRAPHRETRO to nine baselines — four template-based, three template-free, and two semi-template-based methods. These include: ",
+        "bbox": [
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+            465,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Template-Based: RETROSIM Coley et al. [2017b] ranks templates for a given target molecule by computing molecular similarities to precedent reactions. NEURALSYM [Segler and Waller, 2017] trains a model to rank templates given a target molecule. GLN [Dai et al., 2019] models the joint distribution of templates and reactants in a hierarchical fashion using logic variables. DUALTB [Sun et al., 2021] uses an energy-based model formulation for retrosynthesis, with additional parameterizations and loss terms to enforce the duality between forward (reaction prediction) and backward (retrosynthesis prediction). Inference is carried out using reactant candidates obtained by applying an extracted template set to the products. ",
+        "bbox": [
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+            507,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Template-Free: SCROP [Zheng et al., 2019], LV-TRANSFORMER [Chen et al., 2019] and DUALTF [Sun et al., 2021] use the Transformer architecture [Vaswani et al., 2017] to output reactant SMILES given a product SMILES. To improve the validity of their suggestions, SCROP include a second Transformer that functions as a syntax correcter. LV-TRANSFORMER uses a latent variable mixture model to improve diversity of suggestions. DUALTF utilizes additional parameterizations and loss terms to enforce the duality between forward (reaction prediction) and backward (retrosynthesis prediction). ",
+        "bbox": [
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+            722
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Semi-Template-Based: G2GS [Shi et al., 2020] and RETROXPERT [Yan et al., 2020] first identify synthons, and then expand the synthons into reactants by either sequential generation of atoms and bonds (G2Gs), or using the Transformer architecture (RETROXPERT). The training dataset for the Transformer in [Yan et al., 2020] is augmented with incorrectly predicted synthons with the goal of learning a correction mechanism. ",
+        "bbox": [
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+            729,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Results for NEURALSYM are taken from [Dai et al., 2019]. The authors in [Yan et al., 2020] report their performance being affected by the dataset leakage2. Thus, we use the most recent results from their website on the canonicalized dataset. For remaining baselines, we directly use the values reported in their paper. For the synthon completion module, we use the ind configuration given its better empirical performance. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 Overall Performance ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            90,
+            357,
+            106
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Reaction class unknown As shown in Table 1, when the reaction class is unknown, GRAPHRETRO outperforms G2GS by $4 . 8 \\%$ and and RETROXPERT by $3 . 3 \\%$ in top-1 accuracy. Performance improvements are also seen for larger $n$ , except for $n = 5$ . Barring DUALTB, the top-1 accuracy is also better than other template-free and template-based methods. For larger $n$ , one reason for lower top- $n$ accuracies than most template-based methods is that templates already contain combinations of leaving group patterns. In contrast, our model learns to discover these during training. A second hypothesis to this end is that simply adding log-likelihood scores from edit prediction and synthon completion models may be suboptimal and bias the beam search in the direction of the more dominating term. We leave it to future work to investigate scoring functions that rank the attachment. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Reaction class known When the reaction class is known, GRAPHRETRO outperforms G2GS and RETROXPERT by a margin of $3 \\%$ and $2 \\%$ respectively in top-1 accuracy. GRAPHRETRO also outperforms all the template-free methods in top- $^ n$ accuracy. for GRAPHRETRO are also better than most template-based and template-free methods. When the reaction class is known, RETROSIM and GLN restrict template sets corresponding to the reaction class, thus improving performance. The increased edit prediction performance (Section 4.2) for GRAPHRETRO helps outweigh this factor, achieving comparable or better performance till $n = 5$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2 Individual Module Performance ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            396,
+            436,
+            411
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To gain more insight into the working of GRAPHRETRO, we evaluate the top- $\\boldsymbol { n }$ accuracy $\\mathbf { \\nabla } _ { n } =$ $1 , 2 , 3 , 5 )$ of edit prediction and synthon completion modules, along with corresponding ablation studies, with results shown in Table 2. ",
+        "bbox": [
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+            426,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Edit Prediction For the edit prediction module, we compare the true edit(s) to top- $\\boldsymbol { n }$ edits predicted by the model. We also consider two ablation studies, one where we directly use the initial edit scores without updating them, and the other where we predict edits using atom-pairs instead of existing bonds and atoms. Both design choices lead to improvements in performance, as shown in Table 2. We hypothesize that the larger improvement compared to edit prediction using atom-pairs is due to the easier optimization procedure, with lesser imbalance between labels 1 and 0. ",
+        "bbox": [
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+            493,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Synthon Completion For evaluating the synthon completion module, we first apply the true edits to obtain synthons, and compare the true leaving groups to top- $\\mathbf { \\nabla } \\cdot n$ leaving groups predicted by the model. We test the performance of both the ind and shared configurations. Both configurations perform similarly, and are able to identify $\\sim 9 7 \\%$ (close to its upper bound of $9 9 . 7 \\%$ ) of the true leaving groups in its top-5 choices, when the reaction class is known. The top-1, 3 and 5 accuracies of the synthon completion for unknown reaction classes for G2Gs are $6 1 . 1 \\%$ , $8 1 . 5 \\%$ and $8 6 . 7 \\%$ respectively, while ours are $7 5 . 6 \\%$ , $9 2 . 5 \\%$ and $9 6 . 1 \\%$ , indicating a $10 \\%$ performance improvement using a classification formulation over the generative one adopted by G2Gs. ",
+        "bbox": [
+            173,
+            603,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/a7464e4972dde4bef7b44f291b1851eda3f6ff349da317bc359b2a73e891edfc.jpg",
+        "table_caption": [
+            "Table 2: Performance Study of edit prediction and synthon completion modules "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\"> Setting</td><td colspan=\"8\">Top-n Accuracy (%)</td></tr><tr><td colspan=\"4\">Reaction class known</td><td colspan=\"4\">Reaction class unknown</td></tr><tr><td>1</td><td>2</td><td>3</td><td>5</td><td>1</td><td>2</td><td>3</td><td>5</td></tr><tr><td>Edit Prediction</td><td>84.6</td><td>92.2</td><td>93.7</td><td>94.5</td><td>70.8</td><td>85.1</td><td>89.5</td><td>92.7</td></tr><tr><td>- without edit score updates</td><td>84.3</td><td>92.1</td><td>93.7</td><td>94.5</td><td>70.1</td><td>84.8</td><td>89.4</td><td>92.6</td></tr><tr><td>- predicting on atom pairs</td><td>81.9</td><td>89.5</td><td>90.9</td><td>92.1</td><td>68.6</td><td>83.2</td><td>88.3</td><td>91.8</td></tr><tr><td>Synthon Completion (ind)</td><td>77.4</td><td>89.5</td><td>94.2</td><td>97.6</td><td>75.6</td><td>87.4</td><td>92.5</td><td>96.1</td></tr><tr><td>Synthon Completion (shared)</td><td>76.9</td><td>89.6</td><td>93.9</td><td>97.4</td><td>74.9</td><td>87.7</td><td>92.9</td><td>96.3</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.3 Example Predictions ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            92,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In Figure 2, we visualize the model predictions and the ground truth for three cases. Figure 2a shows an example where the model identifies both the edits and leaving groups correctly. In Figure 2b, the correct edit is identified but the predicted leaving groups are incorrect. We hypothesize this is due to the fact that in the training set, leaving groups attaching to the carbonyl carbon $\\scriptstyle ( \\mathbf { C } = \\mathbf { O } )$ are small (e.g. -OH, - $\\mathrm { - N H _ { 2 } }$ , halides). The true leaving group in this example, however, is large. The model is unable to reason about this and predicts the small leaving group -I. In Figure 2c, the model identifies the edit and consequently the leaving group incorrectly. This highlights a limitation of our model. If the edit is predicted incorrectly, the model cannot suggest the true precursors. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.4 Limitations ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The simplified and interpretable construction of GRAPHRETRO comes with certain limitations. First, the overall performance of the model is limited by the performance of the edit prediction step. If the predicted edit is incorrect, the true reactants cannot be salvaged. This limitation is partly remedied by our model design, that allows for user intervention to correct the edit. Second, our method is reliant on atom-mapping for extracting edits and leaving groups. Extracting edits directly based on substructure matching currently suffer from false positives, and heuristics to correct for these result in correct edits in only ${ \\sim } 9 0 \\%$ of the cases. Third, our formulation assumes that we have as many synthons as reactants, which is violated in some reactions. We leave it to future work to extend the model to realize a single reactant from multiple synthons, and introduce more chemically meaningful edit correction mechanisms. ",
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+    {
+        "type": "image",
+        "img_path": "images/9df65c90e84b75ac55b8dc08692726ee93cf492df4b05d29c79c8c792f98fccd.jpg",
+        "image_caption": [
+            "Figure 2: Example Predictions. The true edit and incorrect edit (if any) are highlighted in green and red respectively. The true and predicted leaving groups are highlighted in blue. a. Correctly predicted example by the model. b. Correctly predicted edit but incorrectly predicted leaving groups. c. Incorrectly predicted edit and leaving group. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Previous methods for single-step retrosynthesis either restrict prediction to a template set, are insensitive to molecular graph structure or generate molecules from scratch. We address these shortcomings by introducing a graph-based semi-template-based model inspired by a chemist’s workflow, enhancing the interpretability of retrosynthesis models. Given a target molecule, we first identify synthetic building blocks (synthons) which are then realized into valid reactants, thus avoiding molecule generation from scratch. Our model outperforms previous semi-template-methods by significant margins on the benchmark dataset. Future work aims to extend the model to realize a single reactant from multiple synthons, and introduce more chemically meaningful components to improve the synergy between such tools for retrosynthesis prediction and a practitioner’s expertise. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            194,
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This research was supported by the Machine Learning for Pharmaceutical Discovery and Synthesis Consortium at MIT. V.R.S. was also supported by the Zeno Karl Schindler Foundation. C.B. was supported by the Swiss National Science Foundation under the National Center of Competence in Research (NCCR) Catalysis under grant agreement 51NF40 180544. We thank the Leonhard scientific computing cluster at ETH Zürich for providing computational resources. ",
+        "bbox": [
+            174,
+            224,
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+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
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+        "bbox": [
+            171,
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+    },
+    {
+        "type": "text",
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+    }
+]

parse/train/ryHlUtqge/ryHlUtqge.md

@@ -0,0 +1,260 @@
+# GENERALIZING SKILLS WITH SEMI-SUPERVISED REINFORCEMENT LEARNING
+
+Chelsea $\mathbf { F i n n } \dagger$ , Tianhe $\mathbf { V } \mathbf { u } \dagger$ , Justin $\mathbf { F u } \dagger$ , Pieter Abbeel $^ { \dagger \ddagger }$ , Sergey Levine† † Berkeley AI Research (BAIR), University of California, Berkeley ‡ OpenAI {cbfinn,tianhe.yu,justinfu,pabbeel,svlevine}@berkeley.edu
+
+# ABSTRACT
+
+Deep reinforcement learning (RL) can acquire complex behaviors from low-level inputs, such as images. However, real-world applications of such methods require generalizing to the vast variability of the real world. Deep networks are known to achieve remarkable generalization when provided with massive amounts of labeled data, but can we provide this breadth of experience to an RL agent, such as a robot? The robot might continuously learn as it explores the world around it, even while it is deployed and performing useful tasks. However, this learning requires access to a reward function, to tell the agent whether it is succeeding or failing at its task. Such reward functions are often hard to measure in the real world, especially in domains such as robotics and dialog systems, where the reward could depend on the unknown positions of objects or the emotional state of the user. On the other hand, it is often quite practical to provide the agent with reward functions in a limited set of situations, such as when a human supervisor is present, or in a controlled laboratory setting. Can we make use of this limited supervision, and still benefit from the breadth of experience an agent might collect in the unstructured real world? In this paper, we formalize this problem setting as semisupervised reinforcement learning (SSRL), where the reward function can only be evaluated in a set of “labeled” MDPs, and the agent must generalize its behavior to the wide range of states it might encounter in a set of “unlabeled” MDPs, by using experience from both settings. Our proposed method infers the task objective in the unlabeled MDPs through an algorithm that resembles inverse RL, using the agent’s own prior experience in the labeled MDPs as a kind of demonstration of optimal behavior. We evaluate our method on challenging, continuous control tasks that require control directly from images, and show that our approach can improve the generalization of a learned deep neural network policy by using experience for which no reward function is available. We also show that our method outperforms direct supervised learning of the reward.
+
+# 1 INTRODUCTION
+
+Reinforcement learning (RL) provides a powerful framework for learning behavior from highlevel goals. RL has been combined with deep networks to learn policies for problems such as Atari games (Mnih et al., 2015), simple Minecraft tasks (Oh et al., 2016), and simulated locomotion (Schulman et al., 2015). To apply reinforcement learning (RL) to real-world scenarios, however, the learned policy must be able to handle the variability of the real-world and generalize to scenarios that it has not seen previously. In many such domains, such as robotics and dialog systems, the variability of the real-world poses a significant challenge. Methods for training deep, flexible models combined with massive amounts of labeled data are known to enable wide generalization for supervised learning tasks (Russakovsky et al., 2015). Lifelong learning aims to address this data challenge in the context of RL by enabling the agent to continuously learn as it collects new experiences “on the job,” directly in the real world (Thrun & Mitchell, 1995). However, this learning requires access to a reward function, to tell the agent whether it is succeeding or failing at its task. Although the reward is a high-level supervision signal that is in principle easier to provide than detailed labels, in practice it often depends on information that is extrinsic to the agent and is therefore difficult to measure in the real world. For example, in robotics, the reward may depend on the poses of all of the objects in the environment, and in dialog systems, the reward may depend on the happiness of the user. This reward supervision is practical to measure in a small set of instrumented training scenarios, in laboratory settings, or under the guidance of a human teacher, but quickly becomes impractical to provide continuously to a lifelong learning system, when the agent is deployed in varied and diverse real-world settings.
+
+![](images/f56d4c48e37ccac7e48d7456f17a8bfccbe7bb25b978c67ac9e39a5d12d6973c.jpg)  
+Figure 1: We consider the problem of semi-supervised reinforcement learning, where a reward function can be evaluated in some small set of labeled MDPs $\mathcal { M } \in L$ , but the resulting policy must be successful on a larger set of unlabeled MDPs $\mathcal { M } \in L$ for which the reward function is not known. In standard RL, the policy is trained only on the labeled MDPs, while in transfer learning, the policy is finetuned using a known reward function in the unlabeled MDP set. Semi-supervised RL is distinct in that it involves using experience from the unlabeled set without access to the reward function.
+
+Conceptually, we might imagine that this challenge should not exist, since reinforcement learning should, at least in principle, be able to handle high-level delayed rewards that can always be measured. For example, a human or animal might have their reward encode some higher-level intrinsic goals such as survival, reproduction, or the absence of pain and hunger. However, most RL methods do not operate at the level of such extremely sparse and high-level rewards, and most of the successes of RL have been in domains with natural sources of detailed external feedback, such as the score in a video game. In most real-world scenarios, such a natural and convenient score typically does not exist. It therefore seems that intelligent agents in the real world should be able to cope with only partial reward supervision, and that algorithms that enable this are of both of practical and conceptual value, since they bring us closer to real-world lifelong reinforcement learning, and can help us understand adaptive intelligent systems that can learn even under limited supervisory feedback. So how can an agent continue to learn in the real world without access to a reward function?
+
+In this work, we formalize this as the problem of semi-supervised reinforcement learning, where the agent must perform RL when the reward function is known in some settings, but cannot be evaluated in others. As illustrated in Figure 1, we assume that the agent can first learn in a small range of “labeled” scenarios, where the reward is available, and then experiences a wider range of “unlabeled” scenarios where it must learn to act successfully, akin to lifelong learning in the real world. This problem statement can be viewed as being analogous to the problem of semi-supervised learning, but with the additional complexity of sequential decision making. Standard approaches to RL simply learn a policy in the scenarios where a reward function is available, and hope that it generalizes to new unseen conditions. However, it should be possible to leverage unlabeled experiences to find a more general policy, and to achieve continuous improvement from lifelong real-world experience.
+
+Our main contribution is to propose and evaluate the first algorithm for performing semi-supervised reinforcement learning, which we call semi-supervised skill generalization (S3G). Our approach can leverage unlabeled experience to learn a policy that can succeed in a wider variety of scenarios than a policy trained only with labeled experiences. In our method, we train an RL policy in settings where a reward function is available, and then run an algorithm that resembles inverse reinforcement learning, to simultaneously learn a reward and a more general policy in the wider range of unlabeled settings. Unlike traditional applications of inverse RL algorithms, we use roll-outs from the RL policy in the labeled conditions as demonstrations, rather than a human expert, making our method completely autonomous. Although our approach is compatible with any choice of reinforcement learning and inverse reinforcement learning algorithm, we use the guided cost learning method in our experimental evaluation, which allows us to evaluate on high-dimensional, continuous robotic manipulation tasks with unknown dynamics while using a relatively modest number of samples (Finn et al., 2016). We compare our method to two baselines: (a) a policy trained with RL in settings where reward labels are available (as is standard), and (b) a policy trained in the unlabeled settings using a reward function trained to regress to available reward labels. We find that S3G recovers a policy that is substantially more effective than the prior, standard approach in a wide variety of settings, without using any additional labeled information. We also find that, by using an inverse RL objective, our method achieves superior generalization to the reward regression approach.
+
+# 2 RELATED WORK
+
+Utilizing both labeled and unlabeled data is a well-known technique that can improve learning performance when data is limited (Zhu & Goldberg, 2009). These techniques are especially important in domains where large, supervised datasets are difficult to acquire, but unlabeled data is plentiful. This problem is generally known as semi-supervised learning. Methods for solving this problem often include propagating known labels to the unlabeled examples (Zhu & Ghahramani, 2002) and using regularizing side information (Szummer & Jaakkola, 2002) such as the structure of the data. Semi-supervised learning has been performed with deep models, either by blending unsupervised and supervised objectives (Rasmus et al., 2016; Zhang et al., 2016) or by using generative models, with the labels treated as missing data (Kingma et al., 2014). Semi-supervised learning is particularly relevant in robotics and control, where collecting labeled experience on real hardware is expensive. However, while semi-supervised learning has been successful in domains such as object tracking and detection (Teichman & Thrun, 2007), applications to action and control have not been applied to the objective of the task itself.
+
+The generalization capabilities of policies learned through RL (and deep RL) has been limited, as pointed out by Oh et al. Oh et al. (2016). That is, typically the settings under which the agent is tested do not vary from those under which it was trained. We develop a method for generalizing skills to a wider range of settings using unlabeled experience. A related but orthogonal problem is transfer learning (Taylor & Stone, 2009; Barrett et al., 2010), which attempts to use prior experience in one domain to improve training performance in another. Transfer learning has been applied to RL domains for transferring information across environments (Mordatch et al., 2016; Tzeng et al., 2016), robots (Devin et al., 2016), and tasks (Konidaris & Barto, 2006; Stolle & Atkeson, 2007; Dragan et al., 2011; Parisotto et al., 2016; Rusu et al., 2016). The goal of these approaches is typically to utilize experience in a source domain to learn faster or better in the target domain. Unlike most transfer learning scenarios, we assume that supervision cannot be obtained in many scenarios. We are also not concerned with large, systematic domain shift: we assume that the labeled and unlabeled settings come from the same underlying distribution. Note, however, that the method that we develop could be used for transfer learning problems where the state and reward are consistent across domains.
+
+To the best of our knowledge, this paper is the first to provide a practical and tractable algorithm for semi-supervised RL with large, expressive function approximators, and illustrate that such learning actually improves the generalization of the learned policy. However, the idea of semi-supervised reinforcement learning procedures has been previously discussed as a compelling research direction by Christiano (2016) and Amodei et al. (2016).
+
+To accomplish semi-supervised reinforcement learning, we propose a method that resembles an inverse reinforcement learning (IRL) algorithm, in that it imputes the reward function in the unlabeled settings by learning from the successful trials in the labeled settings. IRL was first introduced by $\mathrm { N g }$ et al. (2000) as the problem of learning reward functions from expert, human demonstrations, typically with the end goal of learning a policy that can succeed from states that are not in the set of demonstrations (Abbeel & Ng, 2004). We use IRL to infer the reward function underlying a policy previously learned in a small set of labeled scenarios, rather than using expert demonstrations. We build upon prior methods, including guided cost learning, which propose to learn a cost and a policy simultaneously (Finn et al., 2016; Ho et al., 2016). Note that the problem that we are considering is distinct from semi-supervised inverse reinforcement learning Audiffren et al. (2015), which makes use of expert and non-expert trajectories for learning. We require a reward function in some instances, rather than expert demonstrations.
+
+# 3 SEMI-SUPERVISED REINFORCEMENT LEARNING
+
+We first define semi-supervised reinforcement learning. We would like the problem definition to be able to capture situations where supervision, via the reward function, is only available in a small set of labeled Markov decision processes (MDPs), but where we want our agent to be able to continue to learn to perform successfully in a much larger set of unlabeled MDPs, where reward labels are unavailable. For example, if the task corresponds to an autonomous car learning to drive, the labeled MDPs might correspond to a range of closed courses, while the unlabeled MDPs might involve driving on real-world highways and city streets. We use the terms labeled and unlabeled in analogy to semi-supervised learning, but note a reward observation is not as directly informative as a label.
+
+Formally, we consider a distribution $p ( \mathcal { M } )$ over undiscounted finite-horizon MDPs, each defined as a 4-tuple $\mathcal { M } _ { i } = ( S , A , T , R )$ over states, actions, transition dynamics (which are generally unknown), and reward. The states and actions may be continuous or discrete, and the reward function $R$ is assumed to the same across MDPs in the distribution $p ( \mathcal { M } )$ . Let $L$ and $U$ denote two sets of MDPs sampled from the distribution $p ( \mathcal { M } )$ . Experience may be collected in both sets of MDPs, but the reward can only be evaluated in the set of labeled MDPs $L$ . The objective is to find a policy $\pi ^ { * }$ that maximizes expected reward in the distribution over MDPs:
+
+$$
+\pi ^ { * } = \underset { \pi } { \arg \operatorname* { m a x } } ~ \mathbb { E } _ { \pi , p ( \mathcal { M } ) } \left[ \sum _ { t = 0 } ^ { H } R ( s _ { t } , a _ { t } ) \right] ,
+$$
+
+where $H$ denotes the horizon. Note that the notion of finding a policy that succeeds on a distribution of MDPs is very natural in many real-world reinforcement learning problems. For example, in the earlier autonomous driving example, our goal is not to find a policy that succeeds on one particular road or in one particular city, but on all roads that the car might encounter. Note that the problem can also be formalized in terms of a single large MDP with a large diversity of initial states, but viewing the expectation as being over a distribution of MDPs provides a more natural analogue with semi-supervised learning, as we discuss below.
+
+In standard semi-supervised learning, it is assumed that the data distribution is the same across both labeled and unlabeled examples, and the amount of labeled data is limited. Similarly, semisupervised reinforcement learning assumes that the labeled and unlabeled MDPs are sampled from the same distribution. In SSRL, however, it is the set of labeled MDPs that is limited, whereas acquiring large amounts of experience within the set of labeled MDPs is permissible, though unlimited experience in the labeled MDPs is not sufficient on its own for good performance on the entire MDP distribution. This is motivated by real-world lifelong learning, where an agent (e.g. a robot) may be initially trained with detailed reward information in a small set of scenarios (e.g. with a human teacher), and is then deployed into a much larger set of scenarios, without reward labels. One natural question is how much variation can exist in the distribution over MDPs. We empirically answer this question in our experimental evaluation in Section 5.
+
+The standard paradigm in reinforcement learning is to learn a policy in the labeled MDPs and apply it directly to new MDPs from the same distribution, hoping that the original policy will generalize (Oh et al., 2016). An alternative approach is to train a reward function with supervised learning to regress from the agent’s observations to the reward labels, and then use this reward function for learning in the unlabeled settings. In our experiments, we find that this approach is often more effective because, unlike the policy, the reward function is decoupled from the rest of the MDP, and can thus generalize more readily. The agent can then continue to learn from unlabeled experiences using the learned reward function. However, because the state distributions in the two sets of MDPs may be different, a function approximator trained on the reward function in the labeled MDPs may not necessarily generalize well to the unlabeled one, due to the domain shift. A more effective solution would be to incorporate the unlabeled experience sampled from $U$ when learning the reward. Unlike typical semi-supervised learning, the goal is not to learn the reward labels per se, but to learn a policy that optimizes the reward. By incorporating both labeled and unlabeled experience, we can develop an algorithm that alternates between inferring the reward function and updating the policy, which effectively provides a shaping, or curriculum, for learning to perform well in the unlabeled settings. In the following section, we discuss our proposed algorithm in detail.
+
+# 4 SEMI-SUPERVISED SKILL GENERALIZATION
+
+We now present our approach for performing semi-supervised reinforcement learning for generalizing previously learned skills. As discussed previously, our goal is to learn a policy that maximizes expected reward in ${ \mathcal { M } } \in U$ , using both unlabeled experience in $U$ and labeled experience in $L$ . We will use the formalism adopted in the previous section; however, note that performing RL in a set of MDPs can be equivalently be viewed as a single MDP with a large diversity of initial conditions.
+
+In order to perform semi-supervised reinforcement learning, we use the framework of maximum entropy control (Ziebart, 2010; Kappen et al., 2012), also called linear-solvable MDPs (Dvijotham & Todorov, 2010). This framework is a generalization of the standard reinforcement learning formulation, where instead of optimizing the expected reward, we optimize an entropy-regularized objective of the form
+
+$$
+\pi _ { \mathrm { R L } } = \underset { \pi } { \arg \operatorname* { m a x } } \mathbb { E } _ { \pi , \mathcal { M } \in L } \left[ \sum _ { t = 0 } ^ { H } R ( s _ { t } , a _ { t } ) \right] - \mathcal { H } ( \pi ) .
+$$
+
+To see that this is a generalization of the standard RL setting, observe that, as the magnitude of the reward increases, the relative weight on the entropy regularizer decreases, so the classic RL objective can be recovered by putting a temperature $\beta$ on the reward, and taking the limit as $\beta \to \infty$ . For finite rewards, this objective encourages policies to take random actions when all options have roughly equal value. Under the optimal policy $\pi _ { \mathrm { R L } }$ , samples with the highest reward $R$ have the highest likelihood, and the likelihood decreases exponentially with decrease in reward. In our work, this framework helps to produce policies in the labeled MDP that are diverse, and therefore better suited for inferring reward functions that transfer effectively to the unlabeled MDP.
+
+After training $\pi _ { \mathrm { R L } }$ , we generate a set of samples from $\pi _ { \mathrm { R L } }$ in $L$ , which we denote as $\mathcal { D } _ { \pi _ { \mathrm { R L } } }$ . The objective of S3G is to use $\mathcal { D } _ { \pi _ { \mathrm { R L } } }$ to find a policy that maximizes expected reward in $U$ ,
+
+$$
+\operatorname* { m a x } _ { \theta } \ \mathbb { E } _ { \pi _ { \theta } , \mathcal { M } \in U } \left[ \sum _ { t = 0 } ^ { T } R ( s _ { t } , a _ { t } ) \right] - \mathcal { H } ( \pi _ { \theta } ) ,
+$$
+
+where the reward $R$ is not available. By using the agent’s prior experience $\mathcal { D } _ { \pi _ { \mathrm { R L } } }$ , as well as unlabeled experience in $U$ , we aim to learn a well-shaped reward function to facilitate learning in $U$ . To do so, S3G simultaneously learns a reward function ${ \tilde { R } } _ { \phi }$ with parameters $\phi$ and optimizes a policy $\pi _ { \theta }$ with parameters $\theta$ in the unlabeled MDP $U$ . This consists of iteratively taking samples ${ \mathcal { D } } _ { \pi _ { \theta } }$ from the current policy $\pi _ { \theta }$ in $U$ , updating the reward ${ \tilde { R } } _ { \phi }$ , and updating the policy $\pi$ using reward values imputed using ${ \tilde { R } } _ { \phi }$ . At the end of the procedure, we end up with a policy $\pi _ { \theta }$ optimized in $U$ . As shown in prior work, this procedure corresponds to an inverse reinforcement learning algorithm that converges to a policy that matches the performance observed in $\mathcal { D } _ { \pi _ { \mathrm { R L } } }$ (Finn et al., 2016). We next go over the objectives used for updating the reward and the policy.
+
+Reward update: Because of the entropy regularized objective in Equation 1, it follows that the samples $\mathcal { D } _ { \pi _ { \mathrm { R L } } }$ are generated from the following maximum entropy distribution (Ziebart, 2010):
+
+$$
+p ( \tau ) = \frac { 1 } { Z } \exp ( R ( \tau ) ) ,
+$$
+
+where $\tau$ denotes a single trajectory sample $\left\{ s _ { 0 } , a _ { 0 } , s _ { 1 } , a _ { 1 } , . . . , s _ { T } \right\}$ and $\begin{array} { r } { R ( \tau ) = \sum _ { t } R \big ( s _ { t } , a _ { t } \big ) } \end{array}$ . Thus, the objective of the reward optimization phase is to maximize the log likelihood of the agent’s prior experience $\mathcal { D } _ { \pi _ { \mathrm { R L } } }$ under this exponential model. The computational challenge here is to estimate the partition function $Z$ which is intractable to compute in high-dimensional spaces. We thus use importance sampling, using samples to estimate the partition function $Z$ as follows:
+
+$$
+\mathcal { L } ( \phi ) = \sum _ { \tau \sim \mathcal { D } _ { \pi _ { \mathrm { L } } } } \tilde { R } _ { \phi } ( \tau ) - \log Z ~ \approx \sum _ { \tau \sim \mathcal { D } _ { \pi _ { \mathrm { R } } } } \tilde { R } _ { \phi } ( \tau ) - \log \sum _ { \tau \sim \mathcal { D } _ { \mathrm { s a m p } } } \frac { \exp ( \tilde { R } _ { \phi } ( \tau ) ) } { q ( \tau ) } ,
+$$
+
+where $\mathcal { D } _ { \mathrm { s a m p } }$ is the set of samples used for estimating the partition function $Z$ and $q ( \tau )$ is the probability of sampling $\tau$ under the policy it was generated from. Note that the distribution of this set of samples is crucial for effectively estimating $Z$ . The optimal distribution for importance sampling is the one that is proportional to $\dot { q ( \tau ) } \propto | \exp ( \tilde { R } _ { \phi } ( \tau ) ) | = \exp ( \tilde { R } _ { \phi } ( \tau ) )$ . Conveniently, this is also the optimal behavior when the reward function is fully optimized such that $\tilde { R } _ { \phi } \approx R$ . Thus, we adaptively update the policy to minimize the KL-divergence between its own distribution and the distribution induced by the current reward, $\tilde { R } _ { \phi } ( \tau )$ , and use samples from the policy to estimate the partition function. Since the importance sampling estimate of $Z$ will be high variance at the beginning of training when fewer policy samples have been collected, we also use the samples from the RL policy $\pi _ { \mathrm { R L } }$ . Thus we set $\mathcal { D } _ { \mathrm { s a m p } }$ to be $\{ \mathcal { D } _ { \pi _ { \theta } } \cup \mathcal { D } _ { \pi _ { \mathrm { R L } } } \}$ .
+
+# Algorithm 1 Semi-Supervised Skill Generalization
+
+0: inputs: Set of unlabeled MDPs $U$ ; reward $R$ for labeled MDPs $\mathcal { M } \in L$   
+1: Optimize $\pi _ { \mathrm { R L } }$ to maximize $R$ in $\mathcal { M } \in L$   
+2: Generate samples $\mathcal { D } _ { \pi _ { \mathrm { R L } } }$ from $\pi _ { \mathrm { R L } }$ in $\mathcal { M } \in L$   
+3: Initialize $\mathcal { D } _ { \mathrm { s a m p } }  \mathcal { D } _ { \pi _ { \mathrm { R L } } }$   
+4: for iteration $i = 1$ to $I$ do   
+5: Run $\pi _ { \theta }$ in ${ \mathcal { M } } \in U$ to generate samples ${ \mathcal { D } } _ { \pi _ { \theta } }$   
+6: Append samples ${ \mathcal { D } } _ { \mathrm { s a m p } } \gets { \mathcal { D } } _ { \mathrm { s a m p } } \cup { \mathcal { D } } _ { \pi _ { \theta } }$   
+7: Update reward ${ \tilde { R } } _ { \phi }$ according to Equation 3 using $\mathcal { D } _ { \pi _ { \mathrm { R L } } }$ and $\mathcal { D } _ { \mathrm { s a m p } }$   
+8: Update policy $\pi _ { \theta }$ according to Equation 4, using ${ \tilde { R } } _ { \phi }$ and ${ \mathcal { D } } _ { \pi _ { \theta } }$   
+9: end for   
+10: return generalized policy $\pi _ { \theta }$
+
+We parameterize the reward using a neural network, and update it using mini-batch stochastic gradient descent, by backpropagating the gradient of the Equation 3 to the parameters of the reward.
+
+Policy update: Our goal with the policy is two-fold. First, we of course need a policy that succeeds in MDPs ${ \mathcal { M } } \in U$ . But since the reward in these MDPs is unavailable, the policy must also serve to generate samples for more accurately estimating the partition function in Equation 2, so that the reward update step can improve the accuracy of the estimated reward function. The policy optimization objective to achieve both of these is to maximize the expected reward ${ \tilde { R } } _ { \phi }$ , augmented with an entropy term as before:
+
+$$
+\mathcal { L } ( \theta ) = \ \mathbb { E } _ { \pi _ { \theta } , \mathcal { M } \in U } \left[ \sum _ { t = 0 } ^ { T } \tilde { R } _ { \phi } ( s _ { t } , a _ { t } ) \right] - \mathcal { H } ( \pi _ { \theta } )
+$$
+
+While we could in principle use any policy optimization method in this step, our prototype uses mirror descent guided policy search (MDGPS), a sample-efficient policy optimization method suitable for training complex neural network policies that has been validated on real-world physical robots (Montgomery & Levine, 2016; Montgomery et al., 2016). We interleave reward function updates using the objective in Equation 3 within the policy optimization method. We describe the policy optimization procedure in detail in Appendix A.
+
+The full algorithm is presented in Algorithm 1. Note that this iterative procedure of comparing the current policy to the optimal behavior provides a form of shaping or curriculum to learning. Our method is structured similarly to the recently proposed guided cost learning method (Finn et al., 2016), and inherits its convergence properties and theoretical foundations. Guided cost learning is an inverse RL algorithm that interleaves policy learning and reward learning directly in the target domain, which in our case is the unlabeled MDPs. Unlike guided cost learning, however, the cost (or reward) is not inferred from expert human-provided demonstrations, but from the agent’s own prior experience in the labeled MDPs.
+
+# 5 EXPERIMENTAL EVALUATION
+
+Since the aim of S3G is to improve the generalization performance of a learned policy by leveraging data from the unlabeled MDPs, our experiments focus on domains where generalization is critical for success. Despite the focus on generalization in many machine learning problems, the generalization capabilities of policies trained with RL have frequently been overlooked. For example, in recent RL benchmarks such as the Arcade Learning Environment (Bellemare et al., 2012) and OpenAI Gym (Brockman et al., 2016), the training conditions perfectly match the testing conditions. Thus, we define our own set of simulated control tasks for this paper, explicitly considering the types of variation that a robot might encounter in the real world. Through our evaluation, we seek to measure how well semi-supervised methods can leverage unlabeled experiences to improve the generalization of a deep neural network policy learned only in only labeled scenarios.
+
+Code for reproducing the simulated experiments is available online1. Videos of the learned policies can be viewed at sites.google.com/site/semisupervisedrl.
+
+![](images/8556a0eaf2dd40289c25f21974b75288d55b93d65120bef567e208b9d4732c86.jpg)  
+Figure 2: Illustrations of the tasks. For the reacher with vision, the range of the target for the labeled MDPs is shown with a red dotted line, and for the unlabeled MDPs with a green dashed line. For the obstacle and cheetah tasks, we show the highest obstacle height.
+
+# 5.1 TASKS
+
+Each of the tasks are modeled using the MuJoCo simulator, and involve continuous state and action spaces with unknown dynamics. The task difficulty ranges from simple, low-dimensional problems to tasks with complex dynamics and high-dimensional observations. In each experiment, the reward function is available in some settings but not others, and the unlabeled MDPs generally involve a wider variety of conditions. We visualize the tasks in Figure 2 and describe them in detail below:
+
+obstacle navigation / obstacle height: The goal of this task is to navigate a point robot around an obstacle to a goal position in 2D. The observation is the robot’s position and velocity, and does not include the height of the obstacle. The height of the obstacle is 0.2 in the labeled MDP, and 0.5 in the unlabeled MDP.
+
+2-link reacher / mass: This task involves moving the end-effector of a two-link reacher to a specified goal position. The observation is the robot’s joint angles, end-effector pose, and their timederivatives. In the labeled MDPs, the mass of the arm varies between $7 \times 1 0 ^ { - 9 }$ and $7 \times 1 0 ^ { 1 }$ , whereas the unlabeled MDPs involve a range of $7 \times 1 0 ^ { - 9 }$ to $7 \times 1 0 ^ { 3 }$ .
+
+2-link reacher with vision / target position: The task objective is the same as the 2-link reacher, except, in this task, the MDPs involve a wide 2D range of target positions, shown in Figure 2. Instead of passing in the coordinate of the target position, the policy and the reward function receive a raw $6 4 \times 8 0$ RGB image of the environment at the first time step.
+
+half-cheetah jump / wall height: In this task, the goal is for a simulated 6-DOF cheetah-like robot with to jump over a wall, with $10 \%$ gravity. The observation is the robot’s joint angles, global pose, and their velocities, for a total dimension of 20. The unlabeled MDP involves jumping over a 0.5 meter wall, compared to the labeled MDP with a 0.2 meter wall. Success is measured based on whether or not the cheetah fully clears the wall. Policies for reward regression, S3G, and oracle were initialized from the RL policy.
+
+In all tasks, the continuous action vector corresponds to the torques or forces applied to each of the robot’s joints. For the first three tasks, reaching the goal position within $5 \mathrm { { c m } }$ is considered a success. For the non-visual tasks, the policy was represented using a neural network with 2 hidden layers of 40 units each. The vision task used 3 convolutional layers with 15 filters of size $5 \times 5$ each, followed by the spatial feature point transformation proposed by Levine et al. (2016), and lastly 3 fully-connected layers of 20 units each. The reward function architecture mirrored the architecture as the policy, but using a quadratic norm on the output, as done by Finn et al. (2016).
+
+# 5.2 EVALUATION
+
+In our evaluation, we compare the performance of S3G to that of (i) the RL policy $\pi _ { \mathrm { R L } }$ , trained only in the labeled MDPs, (ii) a policy learned using a reward function fitted with supervised learning, and (iii) an oracle policy which can access the true reward function in all scenarios. The architecture of the reward function fitted with supervised learning is the same as that used in S3G.
+
+To extensively test the generalization capabilities of the policies learned with each method, we measure performance on a wide range of settings that is a superset of the unlabeled and labeled MDPs, as indicated in Figure 3. We report the success rate of policies learned with each method in Table 1, and visualize the generalization performance in the 2-link reacher, cheetah, and obstacle tasks in Figure 3. The sample complexity of each method is reported in Appendix B.
+
+![](images/ceecfb1ee13f08db3dd5e34eb794e859dddda6497b1c7d5cbe2c6fcf516f9335.jpg)  
+Figure 3: Generalization capability of the obstacle, 2-link reacher, and half-cheetah tasks as a function of the task variation. Performance for these tasks is averaged over 3 random seeds.
+
+In all four tasks, the RL policy $\pi _ { \mathrm { R L } }$ generalizes worse than S3G, which demonstrates that, by using unlabeled experience, we can indeed improve generalization to different masses, target positions, and obstacle sizes. In the obstacle and both reacher tasks, S3G also outperforms reward regression, suggesting that it is also useful to use unlabeled experience to learn the reward.
+
+In the obstacle task, the results demonstrate that the reward functions learned using S3G actually produce better generalization in some cases than learning on both the labeled and unlabeled MDPs with full knowledge of the true reward function. While this may at first seem counterintuitive, this agrees with the observation in prior work Guo et al. (2013) that the true reward function is not always the best one when learning with limited samples, computational power, or representational capacity (i.e. because it is not sufficiently shaped). S3G also outperforms the oracle and reward regression in the 2-link reacher task, indicating that the learned reward shaping is also beneficial in that task.
+
+For the vision task, the visual features learned via RL in the labeled MDPs were used to initialize the vision layers of the reward and policy. We trained the vision-based reacher with S3G with both end-to-end finetuning of the visual features and with the visual features frozen and only the fully-connected layers trained on the unlabeled MDPs. We found performance to be similar in both cases, suggesting that the visual features learned with RL were good enough, though fine-tuning the features end-to-end with the inverse RL objective did not hurt the performance.
+
+# 6 CONCLUSION & FUTURE WORK
+
+We presented the first method for semi-supervised reinforcement learning, motivated by real-world lifelong learning. By inferring the reward in settings where one is not available, S3G can improve the generalization of a learned neural network policy trained only in the “labeled” settings. Additionally, we find that, compared to using supervised regression to reward labels, we can achieve higher performance using an inverse RL objective for inferring the reward underlying the agent’s prior experience. Interestingly, this does not directly make use of the reward labels when inferring the reward of states in the unlabeled MDPs, and our results on the obstacle navigation task in fact suggest that the rewards learned with S3G exhibit better shaping.
+
+As we discuss previously, the reward and policy optimization methods that we build on in this work are efficient enough to learn complex tasks with hundreds of trials, making them well suited for learning on physical systems such as robots. Indeed, previous work has evaluated similar methods on real physical systems, in the context of inverse RL (Finn et al., 2016) and vision-based policy learning (Levine et al., 2016). Thus, it is likely feasible to apply this method for semi-supervised reinforcement learning on a real robotic system. Applying S3G on physical systems has the potential to enable real-world lifelong learning, where an agent is initialized using a moderate amount of labeled experience in a constrained setting, such as a robot learning a skill for the first time in the lab, and then allowed to explore the real world while continuous improving its capabilities without additional supervision. This type of continuous semi-supervised reinforcement learning has the potential to remove the traditional distinction between a training and test phase for reinforcement learning agents, providing us with autonomous systems that continue to get better with use.
+
+# ACKNOWLEDGMENTS
+
+The authors would like to thank Anca Dragan for insightful discussions, and Aviv Tamar and Roberto Calandra for helpful feedback on the paper. Funding was provided by the NSF GRFP, the DARPA Simplex program, and Berkeley DeepDrive.
+
+# REFERENCES
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+
+# A MIRROR DESCENT GUIDED POLICY SEARCH
+
+To optimize policies with S3G, we chose to use mirror-descent guided policy search (MDGPS), for its superior sample efficiency over other policy optimization methods. MDGPS belongs to a class of guided policy search methods, which simplify policy search by decomposing the problem into two phases: a) a trajectory-centric RL phase (C-phase) and b) a supervised learning phase (S-phase). During the C-phase, a trajectory-centric RL method is used to train ”local” controllers for each of M initial positions. In the S-phase, a global policy $\pi _ { \theta } ( a | s )$ is trained using supervised learning to match the output of each of the local policies.
+
+MDGPS can be interpreted as an approximate variant of mirror-descent on the expected cost $\begin{array} { r } { J ( \theta ) = \sum _ { t = 1 } ^ { T } \mathbb { E } _ { \pi _ { \theta } ( s _ { t } , a _ { t } ) } [ - R ( s _ { t } , a _ { t } ) ] } \end{array}$ under policy’s trajectory distribution, where $\pi _ { \theta } ( s _ { t } , a _ { t } )$ denotes the marginal of $\begin{array} { r } { \pi _ { \boldsymbol { \theta } } ( \tau ) = p ( s _ { 1 } ) \prod _ { t = 1 } ^ { T } p ( s _ { t + 1 } | s _ { t } , a _ { t } ) \pi ( a _ { t } | s _ { t } ) } \end{array}$ and $\tau = \{ s _ { 1 } , a _ { 1 } , \ldots , s _ { T } , a _ { T } \}$ denotes the trajectory. In the C-phase, we learn new local policies for each initial position, and in the S-phase we project the local policies down to a single global policy $\pi _ { \theta }$ , using KL divergence as the distance metric.
+
+To produce local policies, we make use of the iterative linear quadratic regulator (iLQR) algorithm to train time-varying linear-Gaussian controllers. iLQR makes up for its weak representational power by being sample efficient under regimes where it is capable of learning. Usage of iLQR requires a twice-differentiable cost function and linearized dynamics.
+
+In order to fit a dynamics model, we use the recent samples to fit a gaussian mixture model (GMM) on $\left( s _ { t } , a _ { t } , s _ { t + 1 } \right)$ tuples. We then use linear regression to fit time-varying linear dynamics of the form $s _ { t + 1 } = F _ { t } s _ { t } + f _ { t }$ on local policy samples from the most recent iteration, using the clusters from the GMM as a normal-inverse Wishart prior.
+
+During the C-step, for each initial condition $m$ , we optimize the entropy-augmented of the form, objective constrained against the global policy:
+
+$$
+q _ { m } = \underset { q } { \arg \operatorname* { m a x } } \mathbb { E } _ { q , p _ { m } ( s _ { 0 } ) } \left[ \sum _ { t = 0 } ^ { T } R ( s _ { t } , a _ { t } ) \right] - \mathcal { H } ( q ) \mathrm { s . t . } \mathcal { D } _ { K L } ( q | | \pi _ { \theta } ) \leq \varepsilon
+$$
+
+Where $R ( s _ { t } , a _ { t } )$ is a twice-differentiable objective such as $L 2$ -distance from a target state.
+
+This optimization results in a local time-varying linear-Gaussian controller $q _ { m } ( \mathbf { s } _ { t } | \mathbf { a } _ { t } ) \sim \mathcal { N } ( K _ { m , t } s _ { t } +$ $k _ { m , t } , C _ { m , t } )$ which is executed to obtain supervised learning examples for the S-step.
+
+# B SAMPLE COMPLEXITY OF EXPERIMENTS
+
+Because we use guided policy search to optimize the policy, we inherit its sample efficiency. In Table 2, we report the number of samples used in both labeled and unlabeled scenarios for all tasks and all methods. Note that the labeled samples used by the oracle are in from the “unlabeled” MDPs $U$ , where we generally assume that reward labels are not available.
+
+Table 2: Sample complexity of each experiment. This table records the total number of samples used to train policies in the labeled setting (RL and oracle), and the unlabeled setting (reward regression, S3G). The sample complexity of unlabeled experiments is denoted as (unlabeled samples $^ +$ labeled samples)   
+
+<table><tr><td></td><td>Labeled</td><td>Unlabeled+Labeled</td><td></td></tr><tr><td></td><td>RL oracle</td><td>reward regression</td><td>S3G</td></tr><tr><td>obstacle 2-link reacher</td><td>250 250</td><td>300+250</td><td>300+250</td></tr><tr><td rowspan="3">2-link reacherwith vision half-cheetah</td><td>200 300</td><td>900+200</td><td>900+200°</td></tr><tr><td>250 650</td><td>1170+250</td><td>1300+250</td></tr><tr><td>600 600</td><td>1400+600</td><td>1400+600</td></tr></table>

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+    {
+        "type": "text",
+        "text": "GENERALIZING SKILLS WITH SEMI-SUPERVISED REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Chelsea $\\mathbf { F i n n } \\dagger$ , Tianhe $\\mathbf { V } \\mathbf { u } \\dagger$ , Justin $\\mathbf { F u } \\dagger$ , Pieter Abbeel $^ { \\dagger \\ddagger }$ , Sergey Levine† † Berkeley AI Research (BAIR), University of California, Berkeley ‡ OpenAI {cbfinn,tianhe.yu,justinfu,pabbeel,svlevine}@berkeley.edu ",
+        "bbox": [
+            184,
+            170,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            262,
+            544,
+            277
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep reinforcement learning (RL) can acquire complex behaviors from low-level inputs, such as images. However, real-world applications of such methods require generalizing to the vast variability of the real world. Deep networks are known to achieve remarkable generalization when provided with massive amounts of labeled data, but can we provide this breadth of experience to an RL agent, such as a robot? The robot might continuously learn as it explores the world around it, even while it is deployed and performing useful tasks. However, this learning requires access to a reward function, to tell the agent whether it is succeeding or failing at its task. Such reward functions are often hard to measure in the real world, especially in domains such as robotics and dialog systems, where the reward could depend on the unknown positions of objects or the emotional state of the user. On the other hand, it is often quite practical to provide the agent with reward functions in a limited set of situations, such as when a human supervisor is present, or in a controlled laboratory setting. Can we make use of this limited supervision, and still benefit from the breadth of experience an agent might collect in the unstructured real world? In this paper, we formalize this problem setting as semisupervised reinforcement learning (SSRL), where the reward function can only be evaluated in a set of “labeled” MDPs, and the agent must generalize its behavior to the wide range of states it might encounter in a set of “unlabeled” MDPs, by using experience from both settings. Our proposed method infers the task objective in the unlabeled MDPs through an algorithm that resembles inverse RL, using the agent’s own prior experience in the labeled MDPs as a kind of demonstration of optimal behavior. We evaluate our method on challenging, continuous control tasks that require control directly from images, and show that our approach can improve the generalization of a learned deep neural network policy by using experience for which no reward function is available. We also show that our method outperforms direct supervised learning of the reward. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+            336,
+            700
+        ],
+        "page_idx": 0
+    },
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+        "text": "Reinforcement learning (RL) provides a powerful framework for learning behavior from highlevel goals. RL has been combined with deep networks to learn policies for problems such as Atari games (Mnih et al., 2015), simple Minecraft tasks (Oh et al., 2016), and simulated locomotion (Schulman et al., 2015). To apply reinforcement learning (RL) to real-world scenarios, however, the learned policy must be able to handle the variability of the real-world and generalize to scenarios that it has not seen previously. In many such domains, such as robotics and dialog systems, the variability of the real-world poses a significant challenge. Methods for training deep, flexible models combined with massive amounts of labeled data are known to enable wide generalization for supervised learning tasks (Russakovsky et al., 2015). Lifelong learning aims to address this data challenge in the context of RL by enabling the agent to continuously learn as it collects new experiences “on the job,” directly in the real world (Thrun & Mitchell, 1995). However, this learning requires access to a reward function, to tell the agent whether it is succeeding or failing at its task. Although the reward is a high-level supervision signal that is in principle easier to provide than detailed labels, in practice it often depends on information that is extrinsic to the agent and is therefore difficult to measure in the real world. For example, in robotics, the reward may depend on the poses of all of the objects in the environment, and in dialog systems, the reward may depend on the happiness of the user. This reward supervision is practical to measure in a small set of instrumented training scenarios, in laboratory settings, or under the guidance of a human teacher, but quickly becomes impractical to provide continuously to a lifelong learning system, when the agent is deployed in varied and diverse real-world settings. ",
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+            "Figure 1: We consider the problem of semi-supervised reinforcement learning, where a reward function can be evaluated in some small set of labeled MDPs $\\mathcal { M } \\in L$ , but the resulting policy must be successful on a larger set of unlabeled MDPs $\\mathcal { M } \\in L$ for which the reward function is not known. In standard RL, the policy is trained only on the labeled MDPs, while in transfer learning, the policy is finetuned using a known reward function in the unlabeled MDP set. Semi-supervised RL is distinct in that it involves using experience from the unlabeled set without access to the reward function. "
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+        "text": "Conceptually, we might imagine that this challenge should not exist, since reinforcement learning should, at least in principle, be able to handle high-level delayed rewards that can always be measured. For example, a human or animal might have their reward encode some higher-level intrinsic goals such as survival, reproduction, or the absence of pain and hunger. However, most RL methods do not operate at the level of such extremely sparse and high-level rewards, and most of the successes of RL have been in domains with natural sources of detailed external feedback, such as the score in a video game. In most real-world scenarios, such a natural and convenient score typically does not exist. It therefore seems that intelligent agents in the real world should be able to cope with only partial reward supervision, and that algorithms that enable this are of both of practical and conceptual value, since they bring us closer to real-world lifelong reinforcement learning, and can help us understand adaptive intelligent systems that can learn even under limited supervisory feedback. So how can an agent continue to learn in the real world without access to a reward function? ",
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+        "text": "In this work, we formalize this as the problem of semi-supervised reinforcement learning, where the agent must perform RL when the reward function is known in some settings, but cannot be evaluated in others. As illustrated in Figure 1, we assume that the agent can first learn in a small range of “labeled” scenarios, where the reward is available, and then experiences a wider range of “unlabeled” scenarios where it must learn to act successfully, akin to lifelong learning in the real world. This problem statement can be viewed as being analogous to the problem of semi-supervised learning, but with the additional complexity of sequential decision making. Standard approaches to RL simply learn a policy in the scenarios where a reward function is available, and hope that it generalizes to new unseen conditions. However, it should be possible to leverage unlabeled experiences to find a more general policy, and to achieve continuous improvement from lifelong real-world experience. ",
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+        "text": "Our main contribution is to propose and evaluate the first algorithm for performing semi-supervised reinforcement learning, which we call semi-supervised skill generalization (S3G). Our approach can leverage unlabeled experience to learn a policy that can succeed in a wider variety of scenarios than a policy trained only with labeled experiences. In our method, we train an RL policy in settings where a reward function is available, and then run an algorithm that resembles inverse reinforcement learning, to simultaneously learn a reward and a more general policy in the wider range of unlabeled settings. Unlike traditional applications of inverse RL algorithms, we use roll-outs from the RL policy in the labeled conditions as demonstrations, rather than a human expert, making our method completely autonomous. Although our approach is compatible with any choice of reinforcement learning and inverse reinforcement learning algorithm, we use the guided cost learning method in our experimental evaluation, which allows us to evaluate on high-dimensional, continuous robotic manipulation tasks with unknown dynamics while using a relatively modest number of samples (Finn et al., 2016). We compare our method to two baselines: (a) a policy trained with RL in settings where reward labels are available (as is standard), and (b) a policy trained in the unlabeled settings using a reward function trained to regress to available reward labels. We find that S3G recovers a policy that is substantially more effective than the prior, standard approach in a wide variety of settings, without using any additional labeled information. We also find that, by using an inverse RL objective, our method achieves superior generalization to the reward regression approach. ",
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+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "text": "Utilizing both labeled and unlabeled data is a well-known technique that can improve learning performance when data is limited (Zhu & Goldberg, 2009). These techniques are especially important in domains where large, supervised datasets are difficult to acquire, but unlabeled data is plentiful. This problem is generally known as semi-supervised learning. Methods for solving this problem often include propagating known labels to the unlabeled examples (Zhu & Ghahramani, 2002) and using regularizing side information (Szummer & Jaakkola, 2002) such as the structure of the data. Semi-supervised learning has been performed with deep models, either by blending unsupervised and supervised objectives (Rasmus et al., 2016; Zhang et al., 2016) or by using generative models, with the labels treated as missing data (Kingma et al., 2014). Semi-supervised learning is particularly relevant in robotics and control, where collecting labeled experience on real hardware is expensive. However, while semi-supervised learning has been successful in domains such as object tracking and detection (Teichman & Thrun, 2007), applications to action and control have not been applied to the objective of the task itself. ",
+        "bbox": [
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+        "type": "text",
+        "text": "The generalization capabilities of policies learned through RL (and deep RL) has been limited, as pointed out by Oh et al. Oh et al. (2016). That is, typically the settings under which the agent is tested do not vary from those under which it was trained. We develop a method for generalizing skills to a wider range of settings using unlabeled experience. A related but orthogonal problem is transfer learning (Taylor & Stone, 2009; Barrett et al., 2010), which attempts to use prior experience in one domain to improve training performance in another. Transfer learning has been applied to RL domains for transferring information across environments (Mordatch et al., 2016; Tzeng et al., 2016), robots (Devin et al., 2016), and tasks (Konidaris & Barto, 2006; Stolle & Atkeson, 2007; Dragan et al., 2011; Parisotto et al., 2016; Rusu et al., 2016). The goal of these approaches is typically to utilize experience in a source domain to learn faster or better in the target domain. Unlike most transfer learning scenarios, we assume that supervision cannot be obtained in many scenarios. We are also not concerned with large, systematic domain shift: we assume that the labeled and unlabeled settings come from the same underlying distribution. Note, however, that the method that we develop could be used for transfer learning problems where the state and reward are consistent across domains. ",
+        "bbox": [
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+        "type": "text",
+        "text": "To the best of our knowledge, this paper is the first to provide a practical and tractable algorithm for semi-supervised RL with large, expressive function approximators, and illustrate that such learning actually improves the generalization of the learned policy. However, the idea of semi-supervised reinforcement learning procedures has been previously discussed as a compelling research direction by Christiano (2016) and Amodei et al. (2016). ",
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+    {
+        "type": "text",
+        "text": "To accomplish semi-supervised reinforcement learning, we propose a method that resembles an inverse reinforcement learning (IRL) algorithm, in that it imputes the reward function in the unlabeled settings by learning from the successful trials in the labeled settings. IRL was first introduced by $\\mathrm { N g }$ et al. (2000) as the problem of learning reward functions from expert, human demonstrations, typically with the end goal of learning a policy that can succeed from states that are not in the set of demonstrations (Abbeel & Ng, 2004). We use IRL to infer the reward function underlying a policy previously learned in a small set of labeled scenarios, rather than using expert demonstrations. We build upon prior methods, including guided cost learning, which propose to learn a cost and a policy simultaneously (Finn et al., 2016; Ho et al., 2016). Note that the problem that we are considering is distinct from semi-supervised inverse reinforcement learning Audiffren et al. (2015), which makes use of expert and non-expert trajectories for learning. We require a reward function in some instances, rather than expert demonstrations. ",
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+        "type": "text",
+        "text": "3 SEMI-SUPERVISED REINFORCEMENT LEARNING",
+        "text_level": 1,
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+        "text": "We first define semi-supervised reinforcement learning. We would like the problem definition to be able to capture situations where supervision, via the reward function, is only available in a small set of labeled Markov decision processes (MDPs), but where we want our agent to be able to continue to learn to perform successfully in a much larger set of unlabeled MDPs, where reward labels are unavailable. For example, if the task corresponds to an autonomous car learning to drive, the labeled MDPs might correspond to a range of closed courses, while the unlabeled MDPs might involve driving on real-world highways and city streets. We use the terms labeled and unlabeled in analogy to semi-supervised learning, but note a reward observation is not as directly informative as a label. ",
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+        "text": "Formally, we consider a distribution $p ( \\mathcal { M } )$ over undiscounted finite-horizon MDPs, each defined as a 4-tuple $\\mathcal { M } _ { i } = ( S , A , T , R )$ over states, actions, transition dynamics (which are generally unknown), and reward. The states and actions may be continuous or discrete, and the reward function $R$ is assumed to the same across MDPs in the distribution $p ( \\mathcal { M } )$ . Let $L$ and $U$ denote two sets of MDPs sampled from the distribution $p ( \\mathcal { M } )$ . Experience may be collected in both sets of MDPs, but the reward can only be evaluated in the set of labeled MDPs $L$ . The objective is to find a policy $\\pi ^ { * }$ that maximizes expected reward in the distribution over MDPs: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/6f600440c7c7a884a671d8cad9dadc578be173009c8e994aa05a88a94d4296f9.jpg",
+        "text": "$$\n\\pi ^ { * } = \\underset { \\pi } { \\arg \\operatorname* { m a x } } ~ \\mathbb { E } _ { \\pi , p ( \\mathcal { M } ) } \\left[ \\sum _ { t = 0 } ^ { H } R ( s _ { t } , a _ { t } ) \\right] ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $H$ denotes the horizon. Note that the notion of finding a policy that succeeds on a distribution of MDPs is very natural in many real-world reinforcement learning problems. For example, in the earlier autonomous driving example, our goal is not to find a policy that succeeds on one particular road or in one particular city, but on all roads that the car might encounter. Note that the problem can also be formalized in terms of a single large MDP with a large diversity of initial states, but viewing the expectation as being over a distribution of MDPs provides a more natural analogue with semi-supervised learning, as we discuss below. ",
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+        "type": "text",
+        "text": "In standard semi-supervised learning, it is assumed that the data distribution is the same across both labeled and unlabeled examples, and the amount of labeled data is limited. Similarly, semisupervised reinforcement learning assumes that the labeled and unlabeled MDPs are sampled from the same distribution. In SSRL, however, it is the set of labeled MDPs that is limited, whereas acquiring large amounts of experience within the set of labeled MDPs is permissible, though unlimited experience in the labeled MDPs is not sufficient on its own for good performance on the entire MDP distribution. This is motivated by real-world lifelong learning, where an agent (e.g. a robot) may be initially trained with detailed reward information in a small set of scenarios (e.g. with a human teacher), and is then deployed into a much larger set of scenarios, without reward labels. One natural question is how much variation can exist in the distribution over MDPs. We empirically answer this question in our experimental evaluation in Section 5. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The standard paradigm in reinforcement learning is to learn a policy in the labeled MDPs and apply it directly to new MDPs from the same distribution, hoping that the original policy will generalize (Oh et al., 2016). An alternative approach is to train a reward function with supervised learning to regress from the agent’s observations to the reward labels, and then use this reward function for learning in the unlabeled settings. In our experiments, we find that this approach is often more effective because, unlike the policy, the reward function is decoupled from the rest of the MDP, and can thus generalize more readily. The agent can then continue to learn from unlabeled experiences using the learned reward function. However, because the state distributions in the two sets of MDPs may be different, a function approximator trained on the reward function in the labeled MDPs may not necessarily generalize well to the unlabeled one, due to the domain shift. A more effective solution would be to incorporate the unlabeled experience sampled from $U$ when learning the reward. Unlike typical semi-supervised learning, the goal is not to learn the reward labels per se, but to learn a policy that optimizes the reward. By incorporating both labeled and unlabeled experience, we can develop an algorithm that alternates between inferring the reward function and updating the policy, which effectively provides a shaping, or curriculum, for learning to perform well in the unlabeled settings. In the following section, we discuss our proposed algorithm in detail. ",
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+        "type": "text",
+        "text": "4 SEMI-SUPERVISED SKILL GENERALIZATION ",
+        "text_level": 1,
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+        "text": "We now present our approach for performing semi-supervised reinforcement learning for generalizing previously learned skills. As discussed previously, our goal is to learn a policy that maximizes expected reward in ${ \\mathcal { M } } \\in U$ , using both unlabeled experience in $U$ and labeled experience in $L$ . We will use the formalism adopted in the previous section; however, note that performing RL in a set of MDPs can be equivalently be viewed as a single MDP with a large diversity of initial conditions. ",
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+        "text": "In order to perform semi-supervised reinforcement learning, we use the framework of maximum entropy control (Ziebart, 2010; Kappen et al., 2012), also called linear-solvable MDPs (Dvijotham & Todorov, 2010). This framework is a generalization of the standard reinforcement learning formulation, where instead of optimizing the expected reward, we optimize an entropy-regularized objective of the form ",
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+        "text": "$$\n\\pi _ { \\mathrm { R L } } = \\underset { \\pi } { \\arg \\operatorname* { m a x } } \\mathbb { E } _ { \\pi , \\mathcal { M } \\in L } \\left[ \\sum _ { t = 0 } ^ { H } R ( s _ { t } , a _ { t } ) \\right] - \\mathcal { H } ( \\pi ) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "To see that this is a generalization of the standard RL setting, observe that, as the magnitude of the reward increases, the relative weight on the entropy regularizer decreases, so the classic RL objective can be recovered by putting a temperature $\\beta$ on the reward, and taking the limit as $\\beta \\to \\infty$ . For finite rewards, this objective encourages policies to take random actions when all options have roughly equal value. Under the optimal policy $\\pi _ { \\mathrm { R L } }$ , samples with the highest reward $R$ have the highest likelihood, and the likelihood decreases exponentially with decrease in reward. In our work, this framework helps to produce policies in the labeled MDP that are diverse, and therefore better suited for inferring reward functions that transfer effectively to the unlabeled MDP. ",
+        "bbox": [
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+        "type": "text",
+        "text": "After training $\\pi _ { \\mathrm { R L } }$ , we generate a set of samples from $\\pi _ { \\mathrm { R L } }$ in $L$ , which we denote as $\\mathcal { D } _ { \\pi _ { \\mathrm { R L } } }$ . The objective of S3G is to use $\\mathcal { D } _ { \\pi _ { \\mathrm { R L } } }$ to find a policy that maximizes expected reward in $U$ , ",
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+        "type": "equation",
+        "img_path": "images/c8c1e3f9186fa079c6555348840d72ef2ea21fff0ff5ee3bbafce6b497db4cdd.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\theta } \\ \\mathbb { E } _ { \\pi _ { \\theta } , \\mathcal { M } \\in U } \\left[ \\sum _ { t = 0 } ^ { T } R ( s _ { t } , a _ { t } ) \\right] - \\mathcal { H } ( \\pi _ { \\theta } ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where the reward $R$ is not available. By using the agent’s prior experience $\\mathcal { D } _ { \\pi _ { \\mathrm { R L } } }$ , as well as unlabeled experience in $U$ , we aim to learn a well-shaped reward function to facilitate learning in $U$ . To do so, S3G simultaneously learns a reward function ${ \\tilde { R } } _ { \\phi }$ with parameters $\\phi$ and optimizes a policy $\\pi _ { \\theta }$ with parameters $\\theta$ in the unlabeled MDP $U$ . This consists of iteratively taking samples ${ \\mathcal { D } } _ { \\pi _ { \\theta } }$ from the current policy $\\pi _ { \\theta }$ in $U$ , updating the reward ${ \\tilde { R } } _ { \\phi }$ , and updating the policy $\\pi$ using reward values imputed using ${ \\tilde { R } } _ { \\phi }$ . At the end of the procedure, we end up with a policy $\\pi _ { \\theta }$ optimized in $U$ . As shown in prior work, this procedure corresponds to an inverse reinforcement learning algorithm that converges to a policy that matches the performance observed in $\\mathcal { D } _ { \\pi _ { \\mathrm { R L } } }$ (Finn et al., 2016). We next go over the objectives used for updating the reward and the policy. ",
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+        "type": "text",
+        "text": "Reward update: Because of the entropy regularized objective in Equation 1, it follows that the samples $\\mathcal { D } _ { \\pi _ { \\mathrm { R L } } }$ are generated from the following maximum entropy distribution (Ziebart, 2010): ",
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+    {
+        "type": "equation",
+        "img_path": "images/043a9bf77598c3201095c9a7a26939bb3d45fcad1f24c1610a5d4042351991b8.jpg",
+        "text": "$$\np ( \\tau ) = \\frac { 1 } { Z } \\exp ( R ( \\tau ) ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\tau$ denotes a single trajectory sample $\\left\\{ s _ { 0 } , a _ { 0 } , s _ { 1 } , a _ { 1 } , . . . , s _ { T } \\right\\}$ and $\\begin{array} { r } { R ( \\tau ) = \\sum _ { t } R \\big ( s _ { t } , a _ { t } \\big ) } \\end{array}$ . Thus, the objective of the reward optimization phase is to maximize the log likelihood of the agent’s prior experience $\\mathcal { D } _ { \\pi _ { \\mathrm { R L } } }$ under this exponential model. The computational challenge here is to estimate the partition function $Z$ which is intractable to compute in high-dimensional spaces. We thus use importance sampling, using samples to estimate the partition function $Z$ as follows: ",
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+        "text": "$$\n\\mathcal { L } ( \\phi ) = \\sum _ { \\tau \\sim \\mathcal { D } _ { \\pi _ { \\mathrm { L } } } } \\tilde { R } _ { \\phi } ( \\tau ) - \\log Z ~ \\approx \\sum _ { \\tau \\sim \\mathcal { D } _ { \\pi _ { \\mathrm { R } } } } \\tilde { R } _ { \\phi } ( \\tau ) - \\log \\sum _ { \\tau \\sim \\mathcal { D } _ { \\mathrm { s a m p } } } \\frac { \\exp ( \\tilde { R } _ { \\phi } ( \\tau ) ) } { q ( \\tau ) } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathcal { D } _ { \\mathrm { s a m p } }$ is the set of samples used for estimating the partition function $Z$ and $q ( \\tau )$ is the probability of sampling $\\tau$ under the policy it was generated from. Note that the distribution of this set of samples is crucial for effectively estimating $Z$ . The optimal distribution for importance sampling is the one that is proportional to $\\dot { q ( \\tau ) } \\propto | \\exp ( \\tilde { R } _ { \\phi } ( \\tau ) ) | = \\exp ( \\tilde { R } _ { \\phi } ( \\tau ) )$ . Conveniently, this is also the optimal behavior when the reward function is fully optimized such that $\\tilde { R } _ { \\phi } \\approx R$ . Thus, we adaptively update the policy to minimize the KL-divergence between its own distribution and the distribution induced by the current reward, $\\tilde { R } _ { \\phi } ( \\tau )$ , and use samples from the policy to estimate the partition function. Since the importance sampling estimate of $Z$ will be high variance at the beginning of training when fewer policy samples have been collected, we also use the samples from the RL policy $\\pi _ { \\mathrm { R L } }$ . Thus we set $\\mathcal { D } _ { \\mathrm { s a m p } }$ to be $\\{ \\mathcal { D } _ { \\pi _ { \\theta } } \\cup \\mathcal { D } _ { \\pi _ { \\mathrm { R L } } } \\}$ . ",
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+        "type": "text",
+        "text": "Algorithm 1 Semi-Supervised Skill Generalization ",
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+        "text": "0: inputs: Set of unlabeled MDPs $U$ ; reward $R$ for labeled MDPs $\\mathcal { M } \\in L$   \n1: Optimize $\\pi _ { \\mathrm { R L } }$ to maximize $R$ in $\\mathcal { M } \\in L$   \n2: Generate samples $\\mathcal { D } _ { \\pi _ { \\mathrm { R L } } }$ from $\\pi _ { \\mathrm { R L } }$ in $\\mathcal { M } \\in L$   \n3: Initialize $\\mathcal { D } _ { \\mathrm { s a m p } }  \\mathcal { D } _ { \\pi _ { \\mathrm { R L } } }$   \n4: for iteration $i = 1$ to $I$ do   \n5: Run $\\pi _ { \\theta }$ in ${ \\mathcal { M } } \\in U$ to generate samples ${ \\mathcal { D } } _ { \\pi _ { \\theta } }$   \n6: Append samples ${ \\mathcal { D } } _ { \\mathrm { s a m p } } \\gets { \\mathcal { D } } _ { \\mathrm { s a m p } } \\cup { \\mathcal { D } } _ { \\pi _ { \\theta } }$   \n7: Update reward ${ \\tilde { R } } _ { \\phi }$ according to Equation 3 using $\\mathcal { D } _ { \\pi _ { \\mathrm { R L } } }$ and $\\mathcal { D } _ { \\mathrm { s a m p } }$   \n8: Update policy $\\pi _ { \\theta }$ according to Equation 4, using ${ \\tilde { R } } _ { \\phi }$ and ${ \\mathcal { D } } _ { \\pi _ { \\theta } }$   \n9: end for   \n10: return generalized policy $\\pi _ { \\theta }$ ",
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+        "type": "text",
+        "text": "We parameterize the reward using a neural network, and update it using mini-batch stochastic gradient descent, by backpropagating the gradient of the Equation 3 to the parameters of the reward. ",
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+        "type": "text",
+        "text": "Policy update: Our goal with the policy is two-fold. First, we of course need a policy that succeeds in MDPs ${ \\mathcal { M } } \\in U$ . But since the reward in these MDPs is unavailable, the policy must also serve to generate samples for more accurately estimating the partition function in Equation 2, so that the reward update step can improve the accuracy of the estimated reward function. The policy optimization objective to achieve both of these is to maximize the expected reward ${ \\tilde { R } } _ { \\phi }$ , augmented with an entropy term as before: ",
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+        "type": "equation",
+        "img_path": "images/0edf74b800c584d208e64e19a53921de6d2129a8638941c6cfea1e7ab9fcfd89.jpg",
+        "text": "$$\n\\mathcal { L } ( \\theta ) = \\ \\mathbb { E } _ { \\pi _ { \\theta } , \\mathcal { M } \\in U } \\left[ \\sum _ { t = 0 } ^ { T } \\tilde { R } _ { \\phi } ( s _ { t } , a _ { t } ) \\right] - \\mathcal { H } ( \\pi _ { \\theta } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "While we could in principle use any policy optimization method in this step, our prototype uses mirror descent guided policy search (MDGPS), a sample-efficient policy optimization method suitable for training complex neural network policies that has been validated on real-world physical robots (Montgomery & Levine, 2016; Montgomery et al., 2016). We interleave reward function updates using the objective in Equation 3 within the policy optimization method. We describe the policy optimization procedure in detail in Appendix A. ",
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+    {
+        "type": "text",
+        "text": "The full algorithm is presented in Algorithm 1. Note that this iterative procedure of comparing the current policy to the optimal behavior provides a form of shaping or curriculum to learning. Our method is structured similarly to the recently proposed guided cost learning method (Finn et al., 2016), and inherits its convergence properties and theoretical foundations. Guided cost learning is an inverse RL algorithm that interleaves policy learning and reward learning directly in the target domain, which in our case is the unlabeled MDPs. Unlike guided cost learning, however, the cost (or reward) is not inferred from expert human-provided demonstrations, but from the agent’s own prior experience in the labeled MDPs. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTAL EVALUATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Since the aim of S3G is to improve the generalization performance of a learned policy by leveraging data from the unlabeled MDPs, our experiments focus on domains where generalization is critical for success. Despite the focus on generalization in many machine learning problems, the generalization capabilities of policies trained with RL have frequently been overlooked. For example, in recent RL benchmarks such as the Arcade Learning Environment (Bellemare et al., 2012) and OpenAI Gym (Brockman et al., 2016), the training conditions perfectly match the testing conditions. Thus, we define our own set of simulated control tasks for this paper, explicitly considering the types of variation that a robot might encounter in the real world. Through our evaluation, we seek to measure how well semi-supervised methods can leverage unlabeled experiences to improve the generalization of a deep neural network policy learned only in only labeled scenarios. ",
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+        "type": "text",
+        "text": "Code for reproducing the simulated experiments is available online1. Videos of the learned policies can be viewed at sites.google.com/site/semisupervisedrl. ",
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+        "img_path": "images/8556a0eaf2dd40289c25f21974b75288d55b93d65120bef567e208b9d4732c86.jpg",
+        "image_caption": [
+            "Figure 2: Illustrations of the tasks. For the reacher with vision, the range of the target for the labeled MDPs is shown with a red dotted line, and for the unlabeled MDPs with a green dashed line. For the obstacle and cheetah tasks, we show the highest obstacle height. "
+        ],
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+    {
+        "type": "text",
+        "text": "5.1 TASKS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Each of the tasks are modeled using the MuJoCo simulator, and involve continuous state and action spaces with unknown dynamics. The task difficulty ranges from simple, low-dimensional problems to tasks with complex dynamics and high-dimensional observations. In each experiment, the reward function is available in some settings but not others, and the unlabeled MDPs generally involve a wider variety of conditions. We visualize the tasks in Figure 2 and describe them in detail below: ",
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+    {
+        "type": "text",
+        "text": "obstacle navigation / obstacle height: The goal of this task is to navigate a point robot around an obstacle to a goal position in 2D. The observation is the robot’s position and velocity, and does not include the height of the obstacle. The height of the obstacle is 0.2 in the labeled MDP, and 0.5 in the unlabeled MDP. ",
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+    {
+        "type": "text",
+        "text": "2-link reacher / mass: This task involves moving the end-effector of a two-link reacher to a specified goal position. The observation is the robot’s joint angles, end-effector pose, and their timederivatives. In the labeled MDPs, the mass of the arm varies between $7 \\times 1 0 ^ { - 9 }$ and $7 \\times 1 0 ^ { 1 }$ , whereas the unlabeled MDPs involve a range of $7 \\times 1 0 ^ { - 9 }$ to $7 \\times 1 0 ^ { 3 }$ . ",
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+    {
+        "type": "text",
+        "text": "2-link reacher with vision / target position: The task objective is the same as the 2-link reacher, except, in this task, the MDPs involve a wide 2D range of target positions, shown in Figure 2. Instead of passing in the coordinate of the target position, the policy and the reward function receive a raw $6 4 \\times 8 0$ RGB image of the environment at the first time step. ",
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+    {
+        "type": "text",
+        "text": "half-cheetah jump / wall height: In this task, the goal is for a simulated 6-DOF cheetah-like robot with to jump over a wall, with $10 \\%$ gravity. The observation is the robot’s joint angles, global pose, and their velocities, for a total dimension of 20. The unlabeled MDP involves jumping over a 0.5 meter wall, compared to the labeled MDP with a 0.2 meter wall. Success is measured based on whether or not the cheetah fully clears the wall. Policies for reward regression, S3G, and oracle were initialized from the RL policy. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In all tasks, the continuous action vector corresponds to the torques or forces applied to each of the robot’s joints. For the first three tasks, reaching the goal position within $5 \\mathrm { { c m } }$ is considered a success. For the non-visual tasks, the policy was represented using a neural network with 2 hidden layers of 40 units each. The vision task used 3 convolutional layers with 15 filters of size $5 \\times 5$ each, followed by the spatial feature point transformation proposed by Levine et al. (2016), and lastly 3 fully-connected layers of 20 units each. The reward function architecture mirrored the architecture as the policy, but using a quadratic norm on the output, as done by Finn et al. (2016). ",
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+    {
+        "type": "text",
+        "text": "5.2 EVALUATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In our evaluation, we compare the performance of S3G to that of (i) the RL policy $\\pi _ { \\mathrm { R L } }$ , trained only in the labeled MDPs, (ii) a policy learned using a reward function fitted with supervised learning, and (iii) an oracle policy which can access the true reward function in all scenarios. The architecture of the reward function fitted with supervised learning is the same as that used in S3G. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "To extensively test the generalization capabilities of the policies learned with each method, we measure performance on a wide range of settings that is a superset of the unlabeled and labeled MDPs, as indicated in Figure 3. We report the success rate of policies learned with each method in Table 1, and visualize the generalization performance in the 2-link reacher, cheetah, and obstacle tasks in Figure 3. The sample complexity of each method is reported in Appendix B. ",
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+    {
+        "type": "image",
+        "img_path": "images/ceecfb1ee13f08db3dd5e34eb794e859dddda6497b1c7d5cbe2c6fcf516f9335.jpg",
+        "image_caption": [
+            "Figure 3: Generalization capability of the obstacle, 2-link reacher, and half-cheetah tasks as a function of the task variation. Performance for these tasks is averaged over 3 random seeds. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "In all four tasks, the RL policy $\\pi _ { \\mathrm { R L } }$ generalizes worse than S3G, which demonstrates that, by using unlabeled experience, we can indeed improve generalization to different masses, target positions, and obstacle sizes. In the obstacle and both reacher tasks, S3G also outperforms reward regression, suggesting that it is also useful to use unlabeled experience to learn the reward. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the obstacle task, the results demonstrate that the reward functions learned using S3G actually produce better generalization in some cases than learning on both the labeled and unlabeled MDPs with full knowledge of the true reward function. While this may at first seem counterintuitive, this agrees with the observation in prior work Guo et al. (2013) that the true reward function is not always the best one when learning with limited samples, computational power, or representational capacity (i.e. because it is not sufficiently shaped). S3G also outperforms the oracle and reward regression in the 2-link reacher task, indicating that the learned reward shaping is also beneficial in that task. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For the vision task, the visual features learned via RL in the labeled MDPs were used to initialize the vision layers of the reward and policy. We trained the vision-based reacher with S3G with both end-to-end finetuning of the visual features and with the visual features frozen and only the fully-connected layers trained on the unlabeled MDPs. We found performance to be similar in both cases, suggesting that the visual features learned with RL were good enough, though fine-tuning the features end-to-end with the inverse RL objective did not hurt the performance. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION & FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We presented the first method for semi-supervised reinforcement learning, motivated by real-world lifelong learning. By inferring the reward in settings where one is not available, S3G can improve the generalization of a learned neural network policy trained only in the “labeled” settings. Additionally, we find that, compared to using supervised regression to reward labels, we can achieve higher performance using an inverse RL objective for inferring the reward underlying the agent’s prior experience. Interestingly, this does not directly make use of the reward labels when inferring the reward of states in the unlabeled MDPs, and our results on the obstacle navigation task in fact suggest that the rewards learned with S3G exhibit better shaping. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "As we discuss previously, the reward and policy optimization methods that we build on in this work are efficient enough to learn complex tasks with hundreds of trials, making them well suited for learning on physical systems such as robots. Indeed, previous work has evaluated similar methods on real physical systems, in the context of inverse RL (Finn et al., 2016) and vision-based policy learning (Levine et al., 2016). Thus, it is likely feasible to apply this method for semi-supervised reinforcement learning on a real robotic system. Applying S3G on physical systems has the potential to enable real-world lifelong learning, where an agent is initialized using a moderate amount of labeled experience in a constrained setting, such as a robot learning a skill for the first time in the lab, and then allowed to explore the real world while continuous improving its capabilities without additional supervision. This type of continuous semi-supervised reinforcement learning has the potential to remove the traditional distinction between a training and test phase for reinforcement learning agents, providing us with autonomous systems that continue to get better with use. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The authors would like to thank Anca Dragan for insightful discussions, and Aviv Tamar and Roberto Calandra for helpful feedback on the paper. Funding was provided by the NSF GRFP, the DARPA Simplex program, and Berkeley DeepDrive. ",
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+        "page_idx": 8
+    },
+    {
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+    {
+        "type": "text",
+        "text": "A MIRROR DESCENT GUIDED POLICY SEARCH ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "To optimize policies with S3G, we chose to use mirror-descent guided policy search (MDGPS), for its superior sample efficiency over other policy optimization methods. MDGPS belongs to a class of guided policy search methods, which simplify policy search by decomposing the problem into two phases: a) a trajectory-centric RL phase (C-phase) and b) a supervised learning phase (S-phase). During the C-phase, a trajectory-centric RL method is used to train ”local” controllers for each of M initial positions. In the S-phase, a global policy $\\pi _ { \\theta } ( a | s )$ is trained using supervised learning to match the output of each of the local policies. ",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "MDGPS can be interpreted as an approximate variant of mirror-descent on the expected cost $\\begin{array} { r } { J ( \\theta ) = \\sum _ { t = 1 } ^ { T } \\mathbb { E } _ { \\pi _ { \\theta } ( s _ { t } , a _ { t } ) } [ - R ( s _ { t } , a _ { t } ) ] } \\end{array}$ under policy’s trajectory distribution, where $\\pi _ { \\theta } ( s _ { t } , a _ { t } )$ denotes the marginal of $\\begin{array} { r } { \\pi _ { \\boldsymbol { \\theta } } ( \\tau ) = p ( s _ { 1 } ) \\prod _ { t = 1 } ^ { T } p ( s _ { t + 1 } | s _ { t } , a _ { t } ) \\pi ( a _ { t } | s _ { t } ) } \\end{array}$ and $\\tau = \\{ s _ { 1 } , a _ { 1 } , \\ldots , s _ { T } , a _ { T } \\}$ denotes the trajectory. In the C-phase, we learn new local policies for each initial position, and in the S-phase we project the local policies down to a single global policy $\\pi _ { \\theta }$ , using KL divergence as the distance metric. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "To produce local policies, we make use of the iterative linear quadratic regulator (iLQR) algorithm to train time-varying linear-Gaussian controllers. iLQR makes up for its weak representational power by being sample efficient under regimes where it is capable of learning. Usage of iLQR requires a twice-differentiable cost function and linearized dynamics. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In order to fit a dynamics model, we use the recent samples to fit a gaussian mixture model (GMM) on $\\left( s _ { t } , a _ { t } , s _ { t + 1 } \\right)$ tuples. We then use linear regression to fit time-varying linear dynamics of the form $s _ { t + 1 } = F _ { t } s _ { t } + f _ { t }$ on local policy samples from the most recent iteration, using the clusters from the GMM as a normal-inverse Wishart prior. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "During the C-step, for each initial condition $m$ , we optimize the entropy-augmented of the form, objective constrained against the global policy: ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/802a77049a0860c106ab7b0db8671caabe49a63fcfa3c40527598ec59a01e9b2.jpg",
+        "text": "$$\nq _ { m } = \\underset { q } { \\arg \\operatorname* { m a x } } \\mathbb { E } _ { q , p _ { m } ( s _ { 0 } ) } \\left[ \\sum _ { t = 0 } ^ { T } R ( s _ { t } , a _ { t } ) \\right] - \\mathcal { H } ( q ) \\mathrm { s . t . } \\mathcal { D } _ { K L } ( q | | \\pi _ { \\theta } ) \\leq \\varepsilon\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Where $R ( s _ { t } , a _ { t } )$ is a twice-differentiable objective such as $L 2$ -distance from a target state. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This optimization results in a local time-varying linear-Gaussian controller $q _ { m } ( \\mathbf { s } _ { t } | \\mathbf { a } _ { t } ) \\sim \\mathcal { N } ( K _ { m , t } s _ { t } +$ $k _ { m , t } , C _ { m , t } )$ which is executed to obtain supervised learning examples for the S-step. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B SAMPLE COMPLEXITY OF EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Because we use guided policy search to optimize the policy, we inherit its sample efficiency. In Table 2, we report the number of samples used in both labeled and unlabeled scenarios for all tasks and all methods. Note that the labeled samples used by the oracle are in from the “unlabeled” MDPs $U$ , where we generally assume that reward labels are not available. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/4ade004504e58b7bbfb97c47a9203970ebc88dc45a749a5e17c88df1efa1395a.jpg",
+        "table_caption": [
+            "Table 2: Sample complexity of each experiment. This table records the total number of samples used to train policies in the labeled setting (RL and oracle), and the unlabeled setting (reward regression, S3G). The sample complexity of unlabeled experiments is denoted as (unlabeled samples $^ +$ labeled samples) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Labeled</td><td>Unlabeled+Labeled</td><td></td></tr><tr><td></td><td>RL oracle</td><td>reward regression</td><td>S3G</td></tr><tr><td>obstacle 2-link reacher</td><td>250 250</td><td>300+250</td><td>300+250</td></tr><tr><td rowspan=\"3\">2-link reacherwith vision half-cheetah</td><td>200 300</td><td>900+200</td><td>900+200°</td></tr><tr><td>250 650</td><td>1170+250</td><td>1300+250</td></tr><tr><td>600 600</td><td>1400+600</td><td>1400+600</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    }
+]