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+# Discovering and Achieving Goals via World Models
+
+Russell Mendonca\* Carnegie Mellon University
+
+Oleh Rybkin\* University of Pennsylvania
+
+Kostas Daniilidis University of Pennsylvania
+
+Danijar Hafner University of Toronto
+
+Deepak Pathak Carnegie Mellon University
+
+# Abstract
+
+How can artificial agents learn to solve many diverse tasks in complex visual environments without any supervision? We decompose this question into two challenges: discovering new goals and learning to reliably achieve them. Our proposed agent, Latent Explorer Achiever (LEXA), addresses both challenges by learning a world model from image inputs and using it to train an explorer and an achiever policy via imagined rollouts. Unlike prior methods that explore by reaching previously visited states, the explorer plans to discover unseen surprising states through foresight, which are then used as diverse targets for the achiever to practice. After the unsupervised phase, LEXA solves tasks specified as goal images zero-shot without any additional learning. LEXA substantially outperforms previous approaches to unsupervised goal reaching, both on prior benchmarks and on a new challenging benchmark with 40 test tasks spanning across four robotic manipulation and locomotion domains. LEXA further achieves goals that require interacting with multiple objects in sequence.
+
+# 1 Introduction
+
+How can we build an agent that learns to solve hundreds of tasks in complex visual environments, such as rearranging objects with a robot arm or completing chores in a kitchen? While traditional reinforcement learning (RL) has been successful for individual tasks, it requires a substantial amount of human effort for every new task. Specifying task rewards requires domain knowledge, access to object positions, is timeconsuming, and prone to human errors. Moreover, traditional RL would require environment interaction to explore and practice in the environment for every new task. Instead, we approach learning hundreds of tasks through the paradigm of unsupervised goal-conditioned RL, where the agent learns many diverse skills in the environment in the complete absence of supervision, to later solve tasks via user-specified goal images immediately without further training [2, 26, 40].
+
+Challenges Exploring the environment and learning to solve many different tasks is substantially more challenging than traditional RL with a dense reward function or learning from
+
+![](images/e9bbfaebb889bb3a54b1b8ad20dd535aa53cac17ad26150666466df10a82aed2.jpg)  
+Figure 1: LEXA learns a world model without any supervision, and leverages it to train two policies in imagination. The explorer finds new images and the achiever learns to reliably reach them. Once trained, the achiever reaches user-specified goals zero-shot without further training at test time.
+
+![](images/ab01d515c5043880effb1e5122d6556dd9c7a184f2bc430aef795af4ecd91478.jpg)  
+Figure 2: We benchmark LEXA across four visual control environments. A representative sample of the test-time goals is shown here. RoboYoga features complex locomotion and precise control of high-dimensional agents, RoboBins manipulation with multiple objects, and RoboKitchen a variety of diverse tasks that require complex control strategies such as opening a cabinet.
+
+expert demonstrations. Existing methods are limited to simple tasks, such as picking or pushing a puck [13, 32, 37] or controlling simple 2D robots [50]. The key challenge in improving the performance of unsupervised RL is exploration. In particular, previous approaches explore by either revisiting previously seen rare goals [14, 18, 55] or sampling goals from a generative model [32, 37]. However, in both these approaches, the policy as well as the generative model are trained on previously visited states from the replay buffer, and hence the sampled goals are either within or near the frontier of agent’s experience. Ideally, we would like the agent to discover goals much beyond its frontier for efficient exploration, but how does an agent generate goals that it is yet to encounter? This is an open question not just for AI but for cognitive science too [42].
+
+Approach To rectify this issue, we leverage a learned world model to train a separate explorer and achiever policy in imagination. Instead of randomly sampling or generating goals, our explorer policy discovers distant goals by first planning a sequence of actions optimized in imagination of the world model to find novel states with high expected information gain [30, 43, 44]. It then executes those imagined actions in the environment to discover interesting states without the need to generate them. Note these actions are likely to lead the agent to states which are several steps outside the frontier because otherwise the model wouldn’t have had high uncertainty or information gain. Finally, these discovered states are used as diverse targets for the achiever to practice. We train the achiever from on-policy imagination rollouts within the world model and without relying on experience relabeling, therefore leveraging foresight over hindsight. After this unsupervised training phase, the achiever solves tasks specified as goal images zero-shot without any additional learning at deployment. Unlike in the conventional RL paradigm [31, 47], our method is trained once and then used to achieve several tasks at test time without any supervision during training or testing.
+
+Contributions We introduce Latent Explorer Achiever (LEXA), an unsupervised goal reaching agent that trains an explorer and an achiever within a shared world model. At training, LEXA unlocks diverse data for goal reaching in environments where exploration is nontrivial. At test time, the achiever solves challenging locomotion and manipulation tasks provided as user-specified goal images. Our contributions are summarized as follows:
+
+• We propose to learn separate explorer and achiever policies as an approach to overcome the exploration problem of unsupervised goal-conditioned RL.   
+• We show that forward-looking exploration by planning with a learned world model substantially outperforms previous strategies for goal exploration.   
+• To evaluate on challenging tasks, we introduce a new goal reaching benchmark with a total of 40   
+diverse goal images across 4 different robot locomotion and manipulation environments.   
+• LEXA outperforms prior methods, being the first to show success in the Kitchen robotic manipulation environment, and achieves goal images where multiple objects need to be moved.
+
+![](images/261b4f4a6881305424cba99bcd987917803949b8856632917f699dbdedfd6295.jpg)  
+Figure 3: Latent Explorer Achiever (LEXA) learns a general world model that is used to train an explorer and a goal achiever policy. The explorer (left) is trained on imagined latent state rollouts of the world model $s _ { t : T }$ to maximize the disagreement objective $r _ { t } ^ { e } = \mathrm { V a r } ( \bar { s ^ { \prime } } )$ . The goal achiever (right) is conditioned on a goal $g$ and is also trained on imagined rollouts to minimize a distance function $d ( s _ { t } , e _ { g } )$ . Goals are sampled randomly from replay buffer images. For training a temporal distance, we use the imagined rollouts of the achiever and predict the number of time steps between each two states. By combining forward-looking exploration and data-efficient training of the achiever, LEXA provides a simple and powerful solution for unsupervised reinforcement learning.
+
+# 2 Latent Explorer Achiever (LEXA)
+
+Our aim is to build an agent that can achieve arbitrary user-specified goals after learning in the environment without any supervision. This presents two challenges - collecting trajectories that contain diverse goals and learning to achieve these goals when specified as a goal image. We introduce a simple solution based on a world model and imagination training that addresses both challenges. The world model represents the agent’s current knowledge about the environment and is used for training two policies, the explorer and the achiever. To explore novel situations, we construct an estimate of which states the world model is still uncertain about. To achieve goals, we train the goal-conditioned achiever in imagination, using the images found so far as unsupervised goals. At test time, the achiever is deployed to reach user-specified goals. The training procedure is in Algorithm 1.
+
+# 2.1 World Model
+
+To efficiently predict potential outcomes of future actions in environments with high-dimensional image inputs, we leverage a Recurrent State Space Model (RSSM) [23] that learns to predict forward using compact model states that facilitate planning [7, 51]. In contrast to predicting forward in image space, the model states enable efficient parallel planning with a large batch size and can reduce accumulating errors [39]. The world model consists of the following components:
+
+$$
+{ \begin{array} { l l l } { e _ { t } = \operatorname { e n c } _ { \phi } ( x _ { t } ) } & { { \mathrm { P o s t e r i o r : } } } & { \ q _ { \phi } ( s _ { t } \mid s _ { t - 1 } , a _ { t - 1 } , e _ { t } ) } \\ { p _ { \phi } ( s _ { t } \mid s _ { t - 1 } , a _ { t - 1 } ) } & { { \mathrm { I m a g e ~ d e c o d e r : } } } & { \ p _ { \phi } ( x _ { t } \mid s _ { t } ) } \end{array} }
+$$
+
+The model states $s _ { t }$ contain a deterministic component $h _ { t }$ and a stochastic component $z _ { t }$ with diagonalcovariance Gaussian distribution. $h _ { t }$ is the recurrent state of a Gated Recurrent Unit (GRU) [11]. The encoder and decoder are convolutional neural networks (CNNs) and the remaining components are multi-layer perceptrons (MLPs). The world model is trained end-to-end by optimizing the evidence lower bound (ELBO) via stochastic backpropagation [28, 38] with the Adam optimizer [27].
+
+# 2.2 Explorer
+
+To efficiently explore, we seek out surprising states imagined by the world model [6, 41, 43, 44, 46], as opposed to retrospectively exploring by revisiting previously novel states [4, 5, 8, 34]. As the world model can predict model states that correspond to unseen situations in the environment, the imagined trajectories contain more novel goals, compared to model-free exploration that is limited to the replay buffer. To collect informative novel trajectories in the environment, we train an exploration
+
+1: initialize: World model $\mathcal { M }$ , Replay buffer $\mathcal { D }$ , Explorer $\pi ^ { \mathrm { e } } ( a _ { t } \mid z _ { t } )$ , Achiever $\pi ^ { \mathbf { g } } ( a _ { t } \mid z _ { t } , g )$   
+2: while exploring do   
+3: Train $\mathcal { M }$ on $\mathcal { D }$   
+4: Train $\pi ^ { \mathrm { e } }$ in imagination of $\mathcal { M }$ to maximize exploration rewards $\textstyle \sum _ { t } r _ { t } ^ { \mathrm { e } }$ .   
+5: Train $\pi ^ { \mathrm { g } }$ in imagination of $\mathcal { M }$ to maximize $\textstyle \sum _ { t } r _ { t } ^ { \mathrm { g } } ( z _ { t } , g )$ for images $g \sim \mathcal { D }$ .   
+6: (Optional) Train $d ( z _ { i } , z _ { j } )$ to predict distances $j - i$ on the imagination data from last step.   
+7: Deploy $\pi ^ { \mathrm { e } }$ in the environment to explore and grow $\mathcal { D }$ .   
+8: Deploy $\pi ^ { \mathrm { g } }$ in the environment to achieve a goal image $g \sim \mathcal { D }$ to grow $\mathcal { D }$ .   
+9: end while   
+10: while evaluating do   
+11: given: Evaluation goal $g$   
+12: Deploy $\pi ^ { \mathrm { g } }$ in the world to reach $g$ .   
+13: end while
+
+policy $\pi ^ { e }$ from the model states $s _ { t }$ in imagination of the world model to maximize an exploration reward:
+
+$$
+{ \mathrm { E x p l o r e r : } } \qquad \pi ^ { e } ( a _ { t } \mid s _ { t } ) \qquad { \mathrm { E x p l o r e r ~ V a l u e : } } \qquad v ^ { e } ( s _ { t } )
+$$
+
+To explore the most informative model states, we estimate the epistemic uncertainty as a disagreement of an ensemble of transition functions. We train an ensemble of 1-step models to predict the next model state from the current model state. The ensemble model is trained alongside the world model on model states produced by the encoder $q _ { \phi }$ . Because the ensemble models are initialized at random, they will differ, especially for inputs that they have not been trained on [29, 36]:
+
+$$
+\mathrm { E n s e m b l e : } \quad f ( s _ { t } , \theta ^ { k } ) = \hat { z } _ { t + 1 } ^ { k } \quad \mathrm { f o r } \quad k = 1 . . K
+$$
+
+Leveraging the ensemble, we estimate the epistemic uncertainty as the ensemble disagreement. The exploration reward is the variance of the ensemble predictions averaged across dimension of the model state, which approximates the expected information gain [3, 43]:
+
+$$
+r _ { t } ^ { \mathrm { e } } ( s _ { t } ) \doteq \frac { 1 } { N } \sum _ { n } \operatorname { V a r } _ { \{ \mathrm { k } \} } \left[ f ( s _ { t } , \theta _ { k } ) \right] _ { n }
+$$
+
+The explorer $\pi ^ { e }$ maximizes the sum of future exploration rewards $\boldsymbol { r } _ { t } ^ { e }$ using the Dreamer algorithm [24], which considers long-term rewards into the future by maximizing $\lambda$ -returns under a learned value function. As a result, the explorer is trained to seek out situations are as informative as possible from imagined latent trajectories of the world model, and is periodically deployed in the environment to add novel trajectories to the replay buffer, so the world model and goal achiever policy can improve.
+
+# 2.3 Achiever
+
+To leverage the knowledge obtained by exploration for learning to reach goals, we train a goal achiever policy $\pi ^ { g }$ that receives a model state and a goal as input. Our aim is to train a general policy that is capable of reaching many diverse goals. To achieve this in a data-efficient way, it is crucial that environment trajectories that were collected with one goal in mind are reused to also learn how to reach other goals. While prior work addressed this by goal relabeling which makes off-policy policy optimization a necessity [2], we instead leverage past trajectories via the world model trained on them that lets us generate an unlimited amount of new imagined trajectories for training the goal achiever on-policy in imagination. This simplifies policy optimization and can improve stability, while still sharing all collected experience across many goals.
+
+$$
+\pi ^ { g } ( a _ { t } \mid s _ { t } , e _ { g } ) \qquad \mathrm { A c h i e v e r ~ V a l u e } ; \qquad v ^ { g } ( s _ { t } , e _ { g } )
+$$
+
+To train the goal achiever, we sample a goal image $x _ { g }$ from the replay buffer and compute its embedding $e _ { g } = \mathrm { e n c } _ { \phi } ( x _ { g } )$ . The achiever aims to maximize an unsupervised goal-reaching reward $r ^ { g } ( s _ { t } , e _ { g } )$ . We discuss different choices for this reward in Section 2.4. We again use the Dreamer algorithm [24] for training, where now the value function also receives the goal embedding as input. In addition to imagination training, it can also be important to perform practice trials with the goal achiever in the true environment, so that any model inaccuracies along the goal reaching trajectories may be corrected. To perform practice trials, we sample a goal from the replay buffer and execute the goal achiever policy for that goal in the environment. These trials are interleaved with exploration episodes collected by the exploration policy in equal proportion. We note that the goal achiever learning is entirely unsupervised because the practice goals are simply images the agent encountered through exploration or during previous practice trails.
+
+![](images/8833211f2fc46aac053a0c19271c2751e3ee461c189b8c3a201cf26bcb97a95e.jpg)  
+Figure 4: Successful LEXA trajectories. When given a goal image from the test set, LEXA’s achiever is used in the environment to reach that image. On RoboKitchen, LEXA manipulates up to three different objects together from a single goal image (kettle, light switch, and cabinet). On RoboBins, LEXA performs temporally extended tasks such as picking and placing two objects in a row.
+
+# 2.4 Latent Distances
+
+Training the achiever policy requires us to define a goal achievement reward $r ^ { g } ( s _ { t } , e _ { g } )$ that measures how close the latent state $s _ { t }$ should be considered to the goal $e _ { g }$ . One simple measure is the cosine distance in the latent space obtained by inputting image observations into the world-model. However, such a distance function brings visually similar states together even if they could be farther apart in temporal manner as measured by actions needed to reach from one to other. This bias makes this suitable only to scenarios where most of pixels in the observations are directly controllable, e.g., trying to arrange robot’s body in certain shape, such as RoboYoga poses in Figure 2. However, many environments contain agent as well as the world, such as manipulation involves interacting with objects that are not directly controllable. The cosine distance would try matching the entire goal image, and thus places a large weight on both matching the robot and object positions with the desired goal. Since the robot position is directly controllable it is much easier to match, but this metric overly focuses on it, yielding poor policies that ignore objects. We address this is by using the number of timesteps it takes to move from one image to another as a distance measure [25, 26]. This ignores large changes in robot position, since these can be completed in very few steps, and will instead focus more on the objects. This temporal cost function can be learned purely in imagination rollouts from our world model allowing as much data as needed without taking any steps in the real world.
+
+Cosine Distance To use cosine distance with LEXA, for a latent state $s _ { t }$ , and a goal embedding $e ^ { g }$ , we use the latent inference network $q$ to infer $s ^ { g }$ , and define the reward as the cosine similarity [54]:
+
+$$
+r _ { t } ^ { g } ( s _ { t } , e _ { g } ) \doteq \sum _ { i } \overline { { s } } _ { t i } \overline { { s } } _ { g i } , \quad \mathrm { w h e r e } \quad \overline { { s } } _ { t } = s _ { t } / \| s _ { t } \| _ { 2 } , \quad \overline { { s } } _ { g } = s _ { g } / \| s _ { g } \| _ { 2 } ,
+$$
+
+i.e. the cosine of the angle between the two vectors $s _ { t } , s _ { g }$ in the $N -$ dimensional latent space.
+
+Temporal Distance To use temporal distances with LEXA, we train a neural network $d$ to predict the number of time steps between two embeddings. We train it by sampling pairs of states $s _ { t }$ , $s _ { t + k }$ from an imagined rollout of the achiever and predicting the distance $k$ . We implement the temporal distance in terms of predicted image embeddings $\boldsymbol { \hat { e } } _ { t + k }$ in order to remove extra recurrent information:
+
+Predicted embedding: $\mathrm { e m b } ( s _ { t } ) = \hat { e } _ { t } \approx e _ { t } { \quad } \mathrm { T e m p o r a l ~ d i s t a n c e } ; \quad d _ { \omega } ( \hat { e } _ { t } , \hat { e } _ { t + k } ) \approx k / H ,$ where $H$ is the maximum distance equal to the imagination horizon. Training distance function only on imagination data from the same trajectory would cause it to predict poor distance to far away states coming from other trajectories, such as images that are impossible to reach during one episode. In order to incorporate learning signal from such far-away goals, we include them by sampling images from a different trajectory. We annotate these negative samples with the maximum possible distance, so that the agent always prefers images that were seen in the same trajectory.
+
+![](images/b02095cdb02ada544f65d40a4cf2610dc2db333769a1afaf5a6454034a90d495.jpg)  
+Figure 5: Coincidental goal success achieved during the unsupervised exploration phase. The forwardlooking explorer policy of LEXA results in substantially better coverage compared to SkewFit, a popular method for goal based exploration.
+
+$$
+r _ { t } ^ { g } ( s _ { t } , e _ { g } ) = - d _ { \omega } ( \hat { e } _ { t } , e _ { g } ) , \quad \mathrm { w h e r e } \quad \hat { e } _ { t } = \mathrm { e m b } ( s _ { t } ) , \quad e _ { g } = \mathrm { e n c } _ { \phi } ( x _ { g } )
+$$
+
+The learned distance function depends on the training data policy. However, as the policy becomes more competent, the distance estimates will be closer to the optimal number of time steps to reach a particular goal, and the policy converges to the optimal solution [25]. LEXA always uses the latest data to train the distance function using imagination, ensuring that the convergence is fast.
+
+# 3 Experiments
+
+Our evaluation focuses on the following scientific questions:
+
+1. Does LEXA outperform prior work on previous benchmarks and a new challenging benchmark?   
+2. How does forward-looking exploration of goals compare to previous goal exploration strategies?   
+3. How does the distance function affect the ability to reach goals in different types of environments?   
+4. Can we train one general LEXA to control different robots across visually distinct environments?   
+5. What components of LEXA are important for performance?
+
+We evaluate LEXA on prior benchmarks used by SkewFit [37], DISCERN [50], and Plan2Explore [43] in Section 3.3. Since these benchmarks are largely saturated, we also introduce a new challenging benchmark shown in Figure 2. We evaluate LEXA on this benchmark is Section 3.2.
+
+# 3.1 Experimental setup
+
+As not many prior methods have shown success on reaching diverse goals from image inputs, we perform an apples-to-apples comparison by implementing the baselines using the same world model and policy optimization as our method:
+
+• SkewFit SkewFit [37] uses model-free hindsight experience replay and explores by sampling goals from the latent space of a variational autoencoder [28, 38]. Being one of the state-of-the-art agents, we use the original implementation that does not use a world model or explorer policy.   
+• DDL Dynamic Distance Learning [25] trains a temporal distance function similar to our method. Following the original algorithm, DDL uses greedy exploration and trains the distance function on the replay buffer instead of in imagination.   
+• DIAYN Diversity is All You Need [15] learns a latent skill space and uses mutual information between skills and reached states as the objective. We augment DIAYN with our explorer policy and train a learned skill predictor to obtain a skill for a given test image [12].   
+• GCSL Goal-Conditioned Supervised Learning [20] trains the goal policy on replay buffer goals and mimics the actions that previously led to the goal. We also augment GCSL with our explorer policy, as we found no learning success without it.
+
+Our new benchmark defines goal images for a diverse set of four existing environments as follows:
+
+![](images/1310ecf4c91afe8a97fdd0a2ff16844020c1a947729ce1b05b3d97a0c5844f2a.jpg)  
+Figure 6: Evaluation of goal reaching agents on our four benchmarks. A single agent is trained from images without rewards and then evaluated on reaching goal images from the test set (see Figure 1). Both LEXA agents solve many of the tasks and significantly outperform prior work. SkewFit and DLL struggle with exploration, while DIAYN and GCSL use our explorer but still are not able to learn a good downstream policy. Refer table 1 for final success percentage (averaged across tasks) for each method and benchmark domain.
+
+• RoboYoga We use the walker and quadruped domains of the DeepMind Control Suite [48] to define the RoboYoga benchmark, consisting of 12 goal images that correspond to different body poses for each of the two environments, such as lying down, standing up, and balancing. • RoboBins Based on MetaWorld [53], we create a scene with a Sawyer robotic arm, two bins, and two blocks of different colors. The goal images specify tasks that include reaching, manipulating only one block, and manipulating both blocks. • RoboKitchen The last benchmark involves the challenging kitchen environment from [22], where a franka robot can interact with various objects including a burner, light switch, sliding cabinet, hinge cabinet, microwave, or kettle. The goal images we include describe tasks that require interacting with only one object, as well as interacting with two objects.
+
+# 3.2 Performance on New Benchmark
+
+We show the results on our main benchmark in Figure 6 and include heatmaps that show per-task success on each of the evaluation tasks from the benchmarks in the Appendix. Further, we report success averaged across tasks for each domain at the end of training in Table 1. We visualize example successful trajectory executions for tasks that require manipulating multiple objects in Fig. 4.
+
+RoboYoga The environments in this benchmark are directly controllable since they contain no other objects except the robot. We recall that for such settings we expect the cosine distance to be effective, as perceptual distance is quite accurate. Training is thus faster compared to using learned temporal distances, where the metric is learned from scratch. From Table 1 and Figure 6 we see that this is indeed the case for these environments (Walker and Quadruped), as LEXA with the cosine metric outperforms all prior approaches. Furthermore with temporal distances LEXA makes better progress compared to prior work on a much larger number of goals as can be seen from the per-task performance (Figures ??, ??), even though average success over goals looks similar to that of DDL.
+
+RoboBins This environment involves interaction with block objects, and thus is not directly controllable, and so we expect LEXA to perform better with the temporal distance metric. From Table 1 and
+
+<table><tr><td>Method</td><td>Kitchen</td><td>RoboBins</td><td>Quadruped</td><td>Walker</td></tr><tr><td>DDL</td><td>0.00</td><td>35.42</td><td>22.50</td><td>40.00</td></tr><tr><td>DIAYN</td><td>0.00</td><td>13.69</td><td>13.81</td><td>0.28</td></tr><tr><td>GCSL</td><td>0.00</td><td>7.94</td><td>15.83</td><td>1.11</td></tr><tr><td>SkewFit</td><td>0.23</td><td>15.77</td><td>5.52</td><td>0.01</td></tr><tr><td>LEXA + Temporal (Ours)</td><td>37.50</td><td>69.44</td><td>31.39</td><td>36.72</td></tr><tr><td>LEXA + Cosine (Ours)</td><td>6.02</td><td>45.83</td><td>56.11</td><td>73.06</td></tr></table>
+
+Table 1: Performance on our new challenging benchmark, spanning across the four domains shown in Figure 2. The number are goal success rates, averaged over test goals within each environment.
+
+![](images/1ef7c10b47d9d98a8e1a64e538f880461c2ee0273a63c335f48dbd65f5cbaaed.jpg)  
+Figure 7: Success rates on RoboBin. In line with the prior literature, previous methods are successful at reaching and sometimes pushing. LEXA pushes the state-of-the-art by picking and placing multiple objects to reach challenging goal images. Analogous heat maps for the other domains are included in the appendix.
+
+Figure 6, we see that LEXA gets higher average success than all prior approaches. Further from the per-task performance in 7, LEXA with the temporal distance metric is the only approach that makes progress on all goals in the benchmark. The main difference in performance between using temporal and cosine distance can be seen in the tasks involving two blocks, which are the most complex tasks in this environment (the last 3 columns of the per-task plot). The best performing prior method is DDL which solves reaching, and can perform simple pushing tasks. This method performs poorly due to poor exploration, as shown in Figure 5. We see that while other prior methods make some progress on reaching, they fail on harder tasks.
+
+RoboKitchen This benchmark involves diverse objects that require different manipulation behavior. From Table 1 and Figure 6 and ?? we find that LEXA with temporal distance is able to learn multiple RoboKitchen tasks, some of which require sequentially completing 2 tasks in the environment. All prior methods barely make progress due to the challenging nature of this benchmark, and furthermore using the cosine distance function makes very limited progress. The gap in performance between using the two distance functions is much larger in this environment compared to RoboBins since there are many more objects and they are not as clearly visible as the blocks.
+
+# Single Agent Across All Environments
+
+In the previous sections we have shown that our approach can achieve diverse goals in different environments. However, we trained a new agent for every new environment, which doesn’t scale well to large numbers of environments. Thus we investigate if we can train a train a single agent across four environments in the benchmark. From Figure ?? we see that our approach with learned temporal distance is able to make progress on tasks from RoboKitchen, RoboBins Reaching, RoboBins Pick & Place and Walker, while the best prior method on the single-environment tasks (DDL) mainly solves walker tasks and reaching from RoboBin.
+
+# 3.3 Performance on Prior benchmarks
+
+To further verify the results obtained on our benchmark, we evaluate LEXA on previously used benchmarks. We observe that LEXA significantly outperforms prior work on these benchmarks, and is often close to the optimal policy. Additional details are provided in ??????.
+
+SkewFit Benchmark SkewFit [37] introduces a robotic manipulation benchmark for unsupervised methods with simple tasks like planar pushing or picking. We evaluate on this benchmark in Table 2.
+
+Table 2: Goal distance for SkewFit goals [37].   
+
+<table><tr><td>Method</td><td>Pusher</td><td>Pickup</td></tr><tr><td>RIG [32]</td><td>7.7cm</td><td>3.7cm</td></tr><tr><td>RIG + HER [2]</td><td>7.5cm</td><td>3.5cm</td></tr><tr><td>Skew-Fit [37]</td><td>4.9cm</td><td>1.8cm</td></tr><tr><td>LEXA + Temporal</td><td>2.3cm</td><td>1.4cm</td></tr></table>
+
+Baseline results are taken from [37]. LEXA significantly outperforms prior work on these tasks. Pushing and picking up blocks from images is largely solved and future work can focus on harder benchmarks such as those introduced in our paper.
+
+DISCERN Benchmark We attempted to replicate the tasks described in [50] that are based on simple two-dimensional robots [48]. While the original tasks are not released, we followed the procedure for generating the goals described in the paper. Despite following the exact procedure, we were not able to obtain similar goals to the ones used in the original paper. Nevertheless, we show the goal completion percentage results obtained with our reproduced evaluation compared to DISCERN results from the original paper. LEXA results were obtained with early stopping. In Table 3 we see that our agent solves many of the tasks in this benchmark.
+
+Table 3: Success for DISCERN goals [50].   
+
+<table><tr><td>Task</td><td>LEXA</td><td>DISCERN</td></tr><tr><td>Cup</td><td>84.0%</td><td>76.5%</td></tr><tr><td>Cartpole</td><td>35.9%</td><td>21.3%</td></tr><tr><td>Finger</td><td>40.9%</td><td>21.8%</td></tr><tr><td>Pendulum</td><td>79.1%</td><td>75.7%</td></tr><tr><td>Pointmass</td><td>83.2%</td><td>49.6%</td></tr><tr><td>Reacher</td><td>100.0%</td><td>87.1%</td></tr></table>
+
+Plan2Explore Benchmark We provide a comparison on the standard reward-based DM control tasks [48] in Table 4. To compare on this benchmark, we create goal images that correspond to the reward functions. This setup is arguably harder for our agent, but is much more practical. Note our agent never observes the reward function and only observes the goal at test time. Plan2Explore adapts to new tasks but it needs the reward function to be known at test time, while DrQV2 is an oracle agent that observes the reward at training time. Baseline results are taken from [43, 52]. LEXA results were obtained with early stopping. LEXA outperforms Plan2Explore on most tasks and even performs comparably to state of the art oracle agents (DrQ, DrQv2, Dreamer) that use true task rewards during training.
+
+Table 4: Zero-shot return on P2E tasks [43].   
+
+<table><tr><td>Task Zero-Shot</td><td>LEXA [</td><td>P2E</td><td>DrQv2 X</td></tr><tr><td>Walker Stand</td><td>957</td><td>331</td><td>968</td></tr><tr><td>Hopper Stand</td><td>840</td><td>841</td><td>957</td></tr><tr><td>Cartpole Balance</td><td>886</td><td>950</td><td>989</td></tr><tr><td>Cartpole Bal. Sparse</td><td>996</td><td>860</td><td>983</td></tr><tr><td>Pendulum Swing Up</td><td>788</td><td>792</td><td>837</td></tr><tr><td>Cup Catch</td><td>969</td><td>962</td><td>909</td></tr><tr><td>Reacher Hard</td><td>937</td><td>66</td><td>970</td></tr></table>
+
+# 3.4 Analysis
+
+Prior work Most work we compared against struggles with exploration, such as SkewFit and DLL methods. DIAYN is augmented with our explorer, but still fails to leverage the exploration data to learn a diverse set of skills. GCSL struggles to fit the exploration data and produces behavior that does not solve the task, perhaps because the exploration data is too diverse. We observed that all baselines make progress on the simple reaching, but struggle with other tasks. We have experimented with several versions and improvements to the baselines and report the best obtained performance.
+
+Ablation of different components We ablated components of LEXA on the RoboBins environment in Figure 8. Using a separate explorer policy crucial as without it the agent does not discover the more interesting tasks. Without negative sampling the agent learns slower, perhaps because the distance function doesn’t produce reasonable outputs when queried on images that are more than horizon length apart. Training the distance function with real data converges to slightly lower success than using imagination data, since real data is sampled in an off-policy manner due to its limited quantity.
+
+![](images/e74e622cfc1a92b2076d60433d82e8c9de54acaa1e72a8a8ad12dea1ef790bc6.jpg)  
+Figure 8: Ablations on RoboBins. A separate explorer is crucial for most tasks. Training temporal distance on negative samples speeds up learning, and both negative sampling and training in imagination as opposed to real data are important for the hardest tasks.
+
+Exploration performance Due to importance of exploration, we further examine the diversity of the data collected during training. We log the instances where the agent coincidentally solves an evaluation task during exploration, for the RoboKitchen and RoboBins environments. In Figure 5, we see that our method encounters harder tasks involving multiple objects much more often.
+
+# 4 Related Work
+
+Learning to Achieve Goals The problem of learning to reach many different goals has been commonly addressed with model-free methods that learn a single goal-conditioned policy [2, 26, 40]. Recent work has combined these approaches with various ways to generate training goals, such as asymmetric self-play [33, 45] or by sampling goals of intermediate difficulty [14, 18]. These approaches can achieve remarkable performance in simulated robotic domains, however, they focus on the settings where the agent can directly perceive the low-dimensional environment state.
+
+A few works have attempted to scale these model-free methods to visual goals by using contrastive [50] or reconstructive [32, 37] representation learning. However, these approaches struggle to perform meaningful exploration as no clear reward signal is available to guide the agent toward solving interesting tasks. Some works [10, 49] avoid this challenge by using a large dataset of interesting behaviors. Other works [37, 55] attempt to explore by generating goals similar to those that have already been seen, but do not try to explore truly novel states.
+
+A particularly relevant set of approaches used model-based methods to achieve goals via planning [13, 17] or learning model-regularized policies [35]. However, these approaches are limited by short planning horizons. In contrast, we learn long-horizon goal-conditioned value functions which allows us to solve more challenging tasks. More generally, most of the above approaches are limited by simplistic exploration, while our method leverages model imagination to search for novel states, which significantly improves exploration and in turn the downstream capabilities of the agent.
+
+Learning Distance Functions A crucial challenge for visual goal reaching is the choice of the reward or the cost function for the goal achieving policy. Several approaches use representation learning to create a distance in the feature space [9, 32, 50, 51]. However, this naive distance may not be most reflective of how hard a particular goal is to reach. One line of research has proposed using the mutual information between the current state and the goal as the distance metric [1, 12, 15, 21], however, it remains to be seen whether this approach can scale to more complex tasks.
+
+Other works proposed temporal distances that measure the amount of time it takes to reach the goal. One approach is to learn the distance with approximate dynamic programming using Q-learning methods [16, 19, 26]. Our distance function is most similar to Hartikainen et al. [25], who learn a temporal distance with supervised learning on recent policy experience. In contrast to [25], we always train the distance on-policy in imagination, and we further integrate this achiever policy into our latent explorer achiever framework to discover novel goals for the achiever to practice on.
+
+# 5 Conclusion
+
+We presented Latent Explorer Achiever (LEXA), an agent for unsupervised RL that explores its environment, learns to achieve the discovered goals, and solves image-based tasks in a zero-shot way. By planning for novelty in imagination, LEXA prospectively explores to discover meaningful behaviors in substantially more diverse environments than considered by prior work. Further, LEXA is able to solve challenging downstream tasks specified as images without any supervision such as rewards or demonstrations. By proposing a challenging benchmark and the first agent to achieve meaningful performance on these tasks, we hope to stimulate future research on unsupervised agents, which we believe are fundamentally more scalable than traditional agents that require a human to design the tasks and rewards for learning.
+
+Many challenges remain for building unsupervised agents. Many tasks in our benchmark are still unsolved and there remains room for progress on the algorithmic side both for the world model and policy optimization. Further, it is important to demonstrate the benefits of unsupervised agents on real-world systems to verify their scalability. Finally, for widespread adoption, it is crucial to consider the problem of goal specification and design methods that act on goals that are easy to specify, such as via natural language. We believe LEXA will enable future work to tackle these goals effectively.
+
+Acknowledgements We thank Ben Eysenbach, Stephen Tian, Sergey Levine, Dinesh Jayaraman, Karl Pertsch, Ed Hu and the members of GRASP lab and Pathak lab for insightful discussions. We also thank Murtaza Dalal and Chuning Zhu for help with MuJoCo environments. Finally, DP would like to thank Laura Schulz, Josh Tenenbaum and Alison Gopnik for seeding the idea of goal-setting in children, and Pulkit Agrawal for several crucial discussions (over gelato) since then. OR and KD were supported by ARL DCIST CRA W911NF-17-2-0181, ONR N00014-17-1-2093, and by Honda Research Institute. This work was partially supported by GoodAI Research Award and DARPA Machine Common Sense grant.
+
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+        "text": "Discovering and Achieving Goals via World Models ",
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+        "type": "text",
+        "text": "Russell Mendonca\\* Carnegie Mellon University ",
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+    {
+        "type": "text",
+        "text": "Oleh Rybkin\\* University of Pennsylvania ",
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+    },
+    {
+        "type": "text",
+        "text": "Kostas Daniilidis University of Pennsylvania ",
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+    },
+    {
+        "type": "text",
+        "text": "Danijar Hafner University of Toronto ",
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deepak Pathak Carnegie Mellon University ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "How can artificial agents learn to solve many diverse tasks in complex visual environments without any supervision? We decompose this question into two challenges: discovering new goals and learning to reliably achieve them. Our proposed agent, Latent Explorer Achiever (LEXA), addresses both challenges by learning a world model from image inputs and using it to train an explorer and an achiever policy via imagined rollouts. Unlike prior methods that explore by reaching previously visited states, the explorer plans to discover unseen surprising states through foresight, which are then used as diverse targets for the achiever to practice. After the unsupervised phase, LEXA solves tasks specified as goal images zero-shot without any additional learning. LEXA substantially outperforms previous approaches to unsupervised goal reaching, both on prior benchmarks and on a new challenging benchmark with 40 test tasks spanning across four robotic manipulation and locomotion domains. LEXA further achieves goals that require interacting with multiple objects in sequence. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "How can we build an agent that learns to solve hundreds of tasks in complex visual environments, such as rearranging objects with a robot arm or completing chores in a kitchen? While traditional reinforcement learning (RL) has been successful for individual tasks, it requires a substantial amount of human effort for every new task. Specifying task rewards requires domain knowledge, access to object positions, is timeconsuming, and prone to human errors. Moreover, traditional RL would require environment interaction to explore and practice in the environment for every new task. Instead, we approach learning hundreds of tasks through the paradigm of unsupervised goal-conditioned RL, where the agent learns many diverse skills in the environment in the complete absence of supervision, to later solve tasks via user-specified goal images immediately without further training [2, 26, 40]. ",
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+    {
+        "type": "text",
+        "text": "Challenges Exploring the environment and learning to solve many different tasks is substantially more challenging than traditional RL with a dense reward function or learning from ",
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+    {
+        "type": "image",
+        "img_path": "images/e9bbfaebb889bb3a54b1b8ad20dd535aa53cac17ad26150666466df10a82aed2.jpg",
+        "image_caption": [
+            "Figure 1: LEXA learns a world model without any supervision, and leverages it to train two policies in imagination. The explorer finds new images and the achiever learns to reliably reach them. Once trained, the achiever reaches user-specified goals zero-shot without further training at test time. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "image",
+        "img_path": "images/ab01d515c5043880effb1e5122d6556dd9c7a184f2bc430aef795af4ecd91478.jpg",
+        "image_caption": [
+            "Figure 2: We benchmark LEXA across four visual control environments. A representative sample of the test-time goals is shown here. RoboYoga features complex locomotion and precise control of high-dimensional agents, RoboBins manipulation with multiple objects, and RoboKitchen a variety of diverse tasks that require complex control strategies such as opening a cabinet. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "expert demonstrations. Existing methods are limited to simple tasks, such as picking or pushing a puck [13, 32, 37] or controlling simple 2D robots [50]. The key challenge in improving the performance of unsupervised RL is exploration. In particular, previous approaches explore by either revisiting previously seen rare goals [14, 18, 55] or sampling goals from a generative model [32, 37]. However, in both these approaches, the policy as well as the generative model are trained on previously visited states from the replay buffer, and hence the sampled goals are either within or near the frontier of agent’s experience. Ideally, we would like the agent to discover goals much beyond its frontier for efficient exploration, but how does an agent generate goals that it is yet to encounter? This is an open question not just for AI but for cognitive science too [42]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Approach To rectify this issue, we leverage a learned world model to train a separate explorer and achiever policy in imagination. Instead of randomly sampling or generating goals, our explorer policy discovers distant goals by first planning a sequence of actions optimized in imagination of the world model to find novel states with high expected information gain [30, 43, 44]. It then executes those imagined actions in the environment to discover interesting states without the need to generate them. Note these actions are likely to lead the agent to states which are several steps outside the frontier because otherwise the model wouldn’t have had high uncertainty or information gain. Finally, these discovered states are used as diverse targets for the achiever to practice. We train the achiever from on-policy imagination rollouts within the world model and without relying on experience relabeling, therefore leveraging foresight over hindsight. After this unsupervised training phase, the achiever solves tasks specified as goal images zero-shot without any additional learning at deployment. Unlike in the conventional RL paradigm [31, 47], our method is trained once and then used to achieve several tasks at test time without any supervision during training or testing. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Contributions We introduce Latent Explorer Achiever (LEXA), an unsupervised goal reaching agent that trains an explorer and an achiever within a shared world model. At training, LEXA unlocks diverse data for goal reaching in environments where exploration is nontrivial. At test time, the achiever solves challenging locomotion and manipulation tasks provided as user-specified goal images. Our contributions are summarized as follows: ",
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+        "text": "• We propose to learn separate explorer and achiever policies as an approach to overcome the exploration problem of unsupervised goal-conditioned RL.   \n• We show that forward-looking exploration by planning with a learned world model substantially outperforms previous strategies for goal exploration.   \n• To evaluate on challenging tasks, we introduce a new goal reaching benchmark with a total of 40   \ndiverse goal images across 4 different robot locomotion and manipulation environments.   \n• LEXA outperforms prior methods, being the first to show success in the Kitchen robotic manipulation environment, and achieves goal images where multiple objects need to be moved. ",
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+        "img_path": "images/261b4f4a6881305424cba99bcd987917803949b8856632917f699dbdedfd6295.jpg",
+        "image_caption": [
+            "Figure 3: Latent Explorer Achiever (LEXA) learns a general world model that is used to train an explorer and a goal achiever policy. The explorer (left) is trained on imagined latent state rollouts of the world model $s _ { t : T }$ to maximize the disagreement objective $r _ { t } ^ { e } = \\mathrm { V a r } ( \\bar { s ^ { \\prime } } )$ . The goal achiever (right) is conditioned on a goal $g$ and is also trained on imagined rollouts to minimize a distance function $d ( s _ { t } , e _ { g } )$ . Goals are sampled randomly from replay buffer images. For training a temporal distance, we use the imagined rollouts of the achiever and predict the number of time steps between each two states. By combining forward-looking exploration and data-efficient training of the achiever, LEXA provides a simple and powerful solution for unsupervised reinforcement learning. "
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+        "type": "text",
+        "text": "2 Latent Explorer Achiever (LEXA) ",
+        "text_level": 1,
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+        "text": "Our aim is to build an agent that can achieve arbitrary user-specified goals after learning in the environment without any supervision. This presents two challenges - collecting trajectories that contain diverse goals and learning to achieve these goals when specified as a goal image. We introduce a simple solution based on a world model and imagination training that addresses both challenges. The world model represents the agent’s current knowledge about the environment and is used for training two policies, the explorer and the achiever. To explore novel situations, we construct an estimate of which states the world model is still uncertain about. To achieve goals, we train the goal-conditioned achiever in imagination, using the images found so far as unsupervised goals. At test time, the achiever is deployed to reach user-specified goals. The training procedure is in Algorithm 1. ",
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+        "type": "text",
+        "text": "2.1 World Model ",
+        "text_level": 1,
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+        "text": "To efficiently predict potential outcomes of future actions in environments with high-dimensional image inputs, we leverage a Recurrent State Space Model (RSSM) [23] that learns to predict forward using compact model states that facilitate planning [7, 51]. In contrast to predicting forward in image space, the model states enable efficient parallel planning with a large batch size and can reduce accumulating errors [39]. The world model consists of the following components: ",
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+        "img_path": "images/58e48f6a2376d8a71828658c0a6a6f47419cb58a5a06de85dde3e53c525e8e41.jpg",
+        "text": "$$\n{ \\begin{array} { l l l } { e _ { t } = \\operatorname { e n c } _ { \\phi } ( x _ { t } ) } & { { \\mathrm { P o s t e r i o r : } } } & { \\ q _ { \\phi } ( s _ { t } \\mid s _ { t - 1 } , a _ { t - 1 } , e _ { t } ) } \\\\ { p _ { \\phi } ( s _ { t } \\mid s _ { t - 1 } , a _ { t - 1 } ) } & { { \\mathrm { I m a g e ~ d e c o d e r : } } } & { \\ p _ { \\phi } ( x _ { t } \\mid s _ { t } ) } \\end{array} }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "The model states $s _ { t }$ contain a deterministic component $h _ { t }$ and a stochastic component $z _ { t }$ with diagonalcovariance Gaussian distribution. $h _ { t }$ is the recurrent state of a Gated Recurrent Unit (GRU) [11]. The encoder and decoder are convolutional neural networks (CNNs) and the remaining components are multi-layer perceptrons (MLPs). The world model is trained end-to-end by optimizing the evidence lower bound (ELBO) via stochastic backpropagation [28, 38] with the Adam optimizer [27]. ",
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+    {
+        "type": "text",
+        "text": "2.2 Explorer ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To efficiently explore, we seek out surprising states imagined by the world model [6, 41, 43, 44, 46], as opposed to retrospectively exploring by revisiting previously novel states [4, 5, 8, 34]. As the world model can predict model states that correspond to unseen situations in the environment, the imagined trajectories contain more novel goals, compared to model-free exploration that is limited to the replay buffer. To collect informative novel trajectories in the environment, we train an exploration ",
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+        "type": "text",
+        "text": "1: initialize: World model $\\mathcal { M }$ , Replay buffer $\\mathcal { D }$ , Explorer $\\pi ^ { \\mathrm { e } } ( a _ { t } \\mid z _ { t } )$ , Achiever $\\pi ^ { \\mathbf { g } } ( a _ { t } \\mid z _ { t } , g )$   \n2: while exploring do   \n3: Train $\\mathcal { M }$ on $\\mathcal { D }$   \n4: Train $\\pi ^ { \\mathrm { e } }$ in imagination of $\\mathcal { M }$ to maximize exploration rewards $\\textstyle \\sum _ { t } r _ { t } ^ { \\mathrm { e } }$ .   \n5: Train $\\pi ^ { \\mathrm { g } }$ in imagination of $\\mathcal { M }$ to maximize $\\textstyle \\sum _ { t } r _ { t } ^ { \\mathrm { g } } ( z _ { t } , g )$ for images $g \\sim \\mathcal { D }$ .   \n6: (Optional) Train $d ( z _ { i } , z _ { j } )$ to predict distances $j - i$ on the imagination data from last step.   \n7: Deploy $\\pi ^ { \\mathrm { e } }$ in the environment to explore and grow $\\mathcal { D }$ .   \n8: Deploy $\\pi ^ { \\mathrm { g } }$ in the environment to achieve a goal image $g \\sim \\mathcal { D }$ to grow $\\mathcal { D }$ .   \n9: end while   \n10: while evaluating do   \n11: given: Evaluation goal $g$   \n12: Deploy $\\pi ^ { \\mathrm { g } }$ in the world to reach $g$ .   \n13: end while ",
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+    {
+        "type": "text",
+        "text": "policy $\\pi ^ { e }$ from the model states $s _ { t }$ in imagination of the world model to maximize an exploration reward: ",
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+        "type": "equation",
+        "img_path": "images/7ba64821b38f462d0dca465160657471479c64f31661612c406c54c58e2ee0f1.jpg",
+        "text": "$$\n{ \\mathrm { E x p l o r e r : } } \\qquad \\pi ^ { e } ( a _ { t } \\mid s _ { t } ) \\qquad { \\mathrm { E x p l o r e r ~ V a l u e : } } \\qquad v ^ { e } ( s _ { t } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To explore the most informative model states, we estimate the epistemic uncertainty as a disagreement of an ensemble of transition functions. We train an ensemble of 1-step models to predict the next model state from the current model state. The ensemble model is trained alongside the world model on model states produced by the encoder $q _ { \\phi }$ . Because the ensemble models are initialized at random, they will differ, especially for inputs that they have not been trained on [29, 36]: ",
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+        "type": "equation",
+        "img_path": "images/9eaf94fbc2a8cb25b1753cea7aae4403ee0ce1fa7e767fda661a545d5f5b9b04.jpg",
+        "text": "$$\n\\mathrm { E n s e m b l e : } \\quad f ( s _ { t } , \\theta ^ { k } ) = \\hat { z } _ { t + 1 } ^ { k } \\quad \\mathrm { f o r } \\quad k = 1 . . K\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Leveraging the ensemble, we estimate the epistemic uncertainty as the ensemble disagreement. The exploration reward is the variance of the ensemble predictions averaged across dimension of the model state, which approximates the expected information gain [3, 43]: ",
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+    {
+        "type": "equation",
+        "img_path": "images/e563d21593e05baec600751267cd9bf05f0136899c02a8bd8b7c13d11096854c.jpg",
+        "text": "$$\nr _ { t } ^ { \\mathrm { e } } ( s _ { t } ) \\doteq \\frac { 1 } { N } \\sum _ { n } \\operatorname { V a r } _ { \\{ \\mathrm { k } \\} } \\left[ f ( s _ { t } , \\theta _ { k } ) \\right] _ { n }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The explorer $\\pi ^ { e }$ maximizes the sum of future exploration rewards $\\boldsymbol { r } _ { t } ^ { e }$ using the Dreamer algorithm [24], which considers long-term rewards into the future by maximizing $\\lambda$ -returns under a learned value function. As a result, the explorer is trained to seek out situations are as informative as possible from imagined latent trajectories of the world model, and is periodically deployed in the environment to add novel trajectories to the replay buffer, so the world model and goal achiever policy can improve. ",
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+    {
+        "type": "text",
+        "text": "2.3 Achiever ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To leverage the knowledge obtained by exploration for learning to reach goals, we train a goal achiever policy $\\pi ^ { g }$ that receives a model state and a goal as input. Our aim is to train a general policy that is capable of reaching many diverse goals. To achieve this in a data-efficient way, it is crucial that environment trajectories that were collected with one goal in mind are reused to also learn how to reach other goals. While prior work addressed this by goal relabeling which makes off-policy policy optimization a necessity [2], we instead leverage past trajectories via the world model trained on them that lets us generate an unlimited amount of new imagined trajectories for training the goal achiever on-policy in imagination. This simplifies policy optimization and can improve stability, while still sharing all collected experience across many goals. ",
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+    {
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+        "img_path": "images/09edeeeee38f4f3ef02c2b840b130771ce97529492066b7d71e0d48de6e85f00.jpg",
+        "text": "$$\n\\pi ^ { g } ( a _ { t } \\mid s _ { t } , e _ { g } ) \\qquad \\mathrm { A c h i e v e r ~ V a l u e } ; \\qquad v ^ { g } ( s _ { t } , e _ { g } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To train the goal achiever, we sample a goal image $x _ { g }$ from the replay buffer and compute its embedding $e _ { g } = \\mathrm { e n c } _ { \\phi } ( x _ { g } )$ . The achiever aims to maximize an unsupervised goal-reaching reward $r ^ { g } ( s _ { t } , e _ { g } )$ . We discuss different choices for this reward in Section 2.4. We again use the Dreamer algorithm [24] for training, where now the value function also receives the goal embedding as input. In addition to imagination training, it can also be important to perform practice trials with the goal achiever in the true environment, so that any model inaccuracies along the goal reaching trajectories may be corrected. To perform practice trials, we sample a goal from the replay buffer and execute the goal achiever policy for that goal in the environment. These trials are interleaved with exploration episodes collected by the exploration policy in equal proportion. We note that the goal achiever learning is entirely unsupervised because the practice goals are simply images the agent encountered through exploration or during previous practice trails. ",
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+    {
+        "type": "image",
+        "img_path": "images/8833211f2fc46aac053a0c19271c2751e3ee461c189b8c3a201cf26bcb97a95e.jpg",
+        "image_caption": [
+            "Figure 4: Successful LEXA trajectories. When given a goal image from the test set, LEXA’s achiever is used in the environment to reach that image. On RoboKitchen, LEXA manipulates up to three different objects together from a single goal image (kettle, light switch, and cabinet). On RoboBins, LEXA performs temporally extended tasks such as picking and placing two objects in a row. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "2.4 Latent Distances ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Training the achiever policy requires us to define a goal achievement reward $r ^ { g } ( s _ { t } , e _ { g } )$ that measures how close the latent state $s _ { t }$ should be considered to the goal $e _ { g }$ . One simple measure is the cosine distance in the latent space obtained by inputting image observations into the world-model. However, such a distance function brings visually similar states together even if they could be farther apart in temporal manner as measured by actions needed to reach from one to other. This bias makes this suitable only to scenarios where most of pixels in the observations are directly controllable, e.g., trying to arrange robot’s body in certain shape, such as RoboYoga poses in Figure 2. However, many environments contain agent as well as the world, such as manipulation involves interacting with objects that are not directly controllable. The cosine distance would try matching the entire goal image, and thus places a large weight on both matching the robot and object positions with the desired goal. Since the robot position is directly controllable it is much easier to match, but this metric overly focuses on it, yielding poor policies that ignore objects. We address this is by using the number of timesteps it takes to move from one image to another as a distance measure [25, 26]. This ignores large changes in robot position, since these can be completed in very few steps, and will instead focus more on the objects. This temporal cost function can be learned purely in imagination rollouts from our world model allowing as much data as needed without taking any steps in the real world. ",
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+    {
+        "type": "text",
+        "text": "Cosine Distance To use cosine distance with LEXA, for a latent state $s _ { t }$ , and a goal embedding $e ^ { g }$ , we use the latent inference network $q$ to infer $s ^ { g }$ , and define the reward as the cosine similarity [54]: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/e88f6979b53255a72be4873ad55353e204741ed389c04e6460ef8ffeee08d936.jpg",
+        "text": "$$\nr _ { t } ^ { g } ( s _ { t } , e _ { g } ) \\doteq \\sum _ { i } \\overline { { s } } _ { t i } \\overline { { s } } _ { g i } , \\quad \\mathrm { w h e r e } \\quad \\overline { { s } } _ { t } = s _ { t } / \\| s _ { t } \\| _ { 2 } , \\quad \\overline { { s } } _ { g } = s _ { g } / \\| s _ { g } \\| _ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "i.e. the cosine of the angle between the two vectors $s _ { t } , s _ { g }$ in the $N -$ dimensional latent space. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Temporal Distance To use temporal distances with LEXA, we train a neural network $d$ to predict the number of time steps between two embeddings. We train it by sampling pairs of states $s _ { t }$ , $s _ { t + k }$ from an imagined rollout of the achiever and predicting the distance $k$ . We implement the temporal distance in terms of predicted image embeddings $\\boldsymbol { \\hat { e } } _ { t + k }$ in order to remove extra recurrent information: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Predicted embedding: $\\mathrm { e m b } ( s _ { t } ) = \\hat { e } _ { t } \\approx e _ { t } { \\quad } \\mathrm { T e m p o r a l ~ d i s t a n c e } ; \\quad d _ { \\omega } ( \\hat { e } _ { t } , \\hat { e } _ { t + k } ) \\approx k / H ,$ where $H$ is the maximum distance equal to the imagination horizon. Training distance function only on imagination data from the same trajectory would cause it to predict poor distance to far away states coming from other trajectories, such as images that are impossible to reach during one episode. In order to incorporate learning signal from such far-away goals, we include them by sampling images from a different trajectory. We annotate these negative samples with the maximum possible distance, so that the agent always prefers images that were seen in the same trajectory. ",
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+        "text": "",
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+    {
+        "type": "image",
+        "img_path": "images/b02095cdb02ada544f65d40a4cf2610dc2db333769a1afaf5a6454034a90d495.jpg",
+        "image_caption": [
+            "Figure 5: Coincidental goal success achieved during the unsupervised exploration phase. The forwardlooking explorer policy of LEXA results in substantially better coverage compared to SkewFit, a popular method for goal based exploration. "
+        ],
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+        "bbox": [
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+        "text": "",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/3852fdf70971e5fa1441eba39e645683b4ca9de4a9fbc1050f24b0f3ed4a5bdf.jpg",
+        "text": "$$\nr _ { t } ^ { g } ( s _ { t } , e _ { g } ) = - d _ { \\omega } ( \\hat { e } _ { t } , e _ { g } ) , \\quad \\mathrm { w h e r e } \\quad \\hat { e } _ { t } = \\mathrm { e m b } ( s _ { t } ) , \\quad e _ { g } = \\mathrm { e n c } _ { \\phi } ( x _ { g } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The learned distance function depends on the training data policy. However, as the policy becomes more competent, the distance estimates will be closer to the optimal number of time steps to reach a particular goal, and the policy converges to the optimal solution [25]. LEXA always uses the latest data to train the distance function using imagination, ensuring that the convergence is fast. ",
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+    {
+        "type": "text",
+        "text": "3 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our evaluation focuses on the following scientific questions: ",
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+    {
+        "type": "text",
+        "text": "1. Does LEXA outperform prior work on previous benchmarks and a new challenging benchmark?   \n2. How does forward-looking exploration of goals compare to previous goal exploration strategies?   \n3. How does the distance function affect the ability to reach goals in different types of environments?   \n4. Can we train one general LEXA to control different robots across visually distinct environments?   \n5. What components of LEXA are important for performance? ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluate LEXA on prior benchmarks used by SkewFit [37], DISCERN [50], and Plan2Explore [43] in Section 3.3. Since these benchmarks are largely saturated, we also introduce a new challenging benchmark shown in Figure 2. We evaluate LEXA on this benchmark is Section 3.2. ",
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+    {
+        "type": "text",
+        "text": "3.1 Experimental setup ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "As not many prior methods have shown success on reaching diverse goals from image inputs, we perform an apples-to-apples comparison by implementing the baselines using the same world model and policy optimization as our method: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• SkewFit SkewFit [37] uses model-free hindsight experience replay and explores by sampling goals from the latent space of a variational autoencoder [28, 38]. Being one of the state-of-the-art agents, we use the original implementation that does not use a world model or explorer policy.   \n• DDL Dynamic Distance Learning [25] trains a temporal distance function similar to our method. Following the original algorithm, DDL uses greedy exploration and trains the distance function on the replay buffer instead of in imagination.   \n• DIAYN Diversity is All You Need [15] learns a latent skill space and uses mutual information between skills and reached states as the objective. We augment DIAYN with our explorer policy and train a learned skill predictor to obtain a skill for a given test image [12].   \n• GCSL Goal-Conditioned Supervised Learning [20] trains the goal policy on replay buffer goals and mimics the actions that previously led to the goal. We also augment GCSL with our explorer policy, as we found no learning success without it. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our new benchmark defines goal images for a diverse set of four existing environments as follows: ",
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/1310ecf4c91afe8a97fdd0a2ff16844020c1a947729ce1b05b3d97a0c5844f2a.jpg",
+        "image_caption": [
+            "Figure 6: Evaluation of goal reaching agents on our four benchmarks. A single agent is trained from images without rewards and then evaluated on reaching goal images from the test set (see Figure 1). Both LEXA agents solve many of the tasks and significantly outperform prior work. SkewFit and DLL struggle with exploration, while DIAYN and GCSL use our explorer but still are not able to learn a good downstream policy. Refer table 1 for final success percentage (averaged across tasks) for each method and benchmark domain. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "• RoboYoga We use the walker and quadruped domains of the DeepMind Control Suite [48] to define the RoboYoga benchmark, consisting of 12 goal images that correspond to different body poses for each of the two environments, such as lying down, standing up, and balancing. • RoboBins Based on MetaWorld [53], we create a scene with a Sawyer robotic arm, two bins, and two blocks of different colors. The goal images specify tasks that include reaching, manipulating only one block, and manipulating both blocks. • RoboKitchen The last benchmark involves the challenging kitchen environment from [22], where a franka robot can interact with various objects including a burner, light switch, sliding cabinet, hinge cabinet, microwave, or kettle. The goal images we include describe tasks that require interacting with only one object, as well as interacting with two objects. ",
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+    {
+        "type": "text",
+        "text": "3.2 Performance on New Benchmark ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We show the results on our main benchmark in Figure 6 and include heatmaps that show per-task success on each of the evaluation tasks from the benchmarks in the Appendix. Further, we report success averaged across tasks for each domain at the end of training in Table 1. We visualize example successful trajectory executions for tasks that require manipulating multiple objects in Fig. 4. ",
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+    {
+        "type": "text",
+        "text": "RoboYoga The environments in this benchmark are directly controllable since they contain no other objects except the robot. We recall that for such settings we expect the cosine distance to be effective, as perceptual distance is quite accurate. Training is thus faster compared to using learned temporal distances, where the metric is learned from scratch. From Table 1 and Figure 6 we see that this is indeed the case for these environments (Walker and Quadruped), as LEXA with the cosine metric outperforms all prior approaches. Furthermore with temporal distances LEXA makes better progress compared to prior work on a much larger number of goals as can be seen from the per-task performance (Figures ??, ??), even though average success over goals looks similar to that of DDL. ",
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+    {
+        "type": "text",
+        "text": "RoboBins This environment involves interaction with block objects, and thus is not directly controllable, and so we expect LEXA to perform better with the temporal distance metric. From Table 1 and ",
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+    {
+        "type": "table",
+        "img_path": "images/a408384474e00ab844d1211531dfdf5c3c91a48468085be2d8527eb2438c6fb6.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Performance on our new challenging benchmark, spanning across the four domains shown in Figure 2. The number are goal success rates, averaged over test goals within each environment. "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>Kitchen</td><td>RoboBins</td><td>Quadruped</td><td>Walker</td></tr><tr><td>DDL</td><td>0.00</td><td>35.42</td><td>22.50</td><td>40.00</td></tr><tr><td>DIAYN</td><td>0.00</td><td>13.69</td><td>13.81</td><td>0.28</td></tr><tr><td>GCSL</td><td>0.00</td><td>7.94</td><td>15.83</td><td>1.11</td></tr><tr><td>SkewFit</td><td>0.23</td><td>15.77</td><td>5.52</td><td>0.01</td></tr><tr><td>LEXA + Temporal (Ours)</td><td>37.50</td><td>69.44</td><td>31.39</td><td>36.72</td></tr><tr><td>LEXA + Cosine (Ours)</td><td>6.02</td><td>45.83</td><td>56.11</td><td>73.06</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/1ef7c10b47d9d98a8e1a64e538f880461c2ee0273a63c335f48dbd65f5cbaaed.jpg",
+        "image_caption": [
+            "Figure 7: Success rates on RoboBin. In line with the prior literature, previous methods are successful at reaching and sometimes pushing. LEXA pushes the state-of-the-art by picking and placing multiple objects to reach challenging goal images. Analogous heat maps for the other domains are included in the appendix. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Figure 6, we see that LEXA gets higher average success than all prior approaches. Further from the per-task performance in 7, LEXA with the temporal distance metric is the only approach that makes progress on all goals in the benchmark. The main difference in performance between using temporal and cosine distance can be seen in the tasks involving two blocks, which are the most complex tasks in this environment (the last 3 columns of the per-task plot). The best performing prior method is DDL which solves reaching, and can perform simple pushing tasks. This method performs poorly due to poor exploration, as shown in Figure 5. We see that while other prior methods make some progress on reaching, they fail on harder tasks. ",
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+    {
+        "type": "text",
+        "text": "RoboKitchen This benchmark involves diverse objects that require different manipulation behavior. From Table 1 and Figure 6 and ?? we find that LEXA with temporal distance is able to learn multiple RoboKitchen tasks, some of which require sequentially completing 2 tasks in the environment. All prior methods barely make progress due to the challenging nature of this benchmark, and furthermore using the cosine distance function makes very limited progress. The gap in performance between using the two distance functions is much larger in this environment compared to RoboBins since there are many more objects and they are not as clearly visible as the blocks. ",
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+    {
+        "type": "text",
+        "text": "Single Agent Across All Environments ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In the previous sections we have shown that our approach can achieve diverse goals in different environments. However, we trained a new agent for every new environment, which doesn’t scale well to large numbers of environments. Thus we investigate if we can train a train a single agent across four environments in the benchmark. From Figure ?? we see that our approach with learned temporal distance is able to make progress on tasks from RoboKitchen, RoboBins Reaching, RoboBins Pick & Place and Walker, while the best prior method on the single-environment tasks (DDL) mainly solves walker tasks and reaching from RoboBin. ",
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+    {
+        "type": "text",
+        "text": "3.3 Performance on Prior benchmarks ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "To further verify the results obtained on our benchmark, we evaluate LEXA on previously used benchmarks. We observe that LEXA significantly outperforms prior work on these benchmarks, and is often close to the optimal policy. Additional details are provided in ??????. ",
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+    {
+        "type": "text",
+        "text": "SkewFit Benchmark SkewFit [37] introduces a robotic manipulation benchmark for unsupervised methods with simple tasks like planar pushing or picking. We evaluate on this benchmark in Table 2. ",
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+    {
+        "type": "table",
+        "img_path": "images/618a99a8eebbc909bcc8480480e11f5c2c2e3cdb46ac0a839edabb1385d30fc5.jpg",
+        "table_caption": [
+            "Table 2: Goal distance for SkewFit goals [37]. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Pusher</td><td>Pickup</td></tr><tr><td>RIG [32]</td><td>7.7cm</td><td>3.7cm</td></tr><tr><td>RIG + HER [2]</td><td>7.5cm</td><td>3.5cm</td></tr><tr><td>Skew-Fit [37]</td><td>4.9cm</td><td>1.8cm</td></tr><tr><td>LEXA + Temporal</td><td>2.3cm</td><td>1.4cm</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Baseline results are taken from [37]. LEXA significantly outperforms prior work on these tasks. Pushing and picking up blocks from images is largely solved and future work can focus on harder benchmarks such as those introduced in our paper. ",
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+    {
+        "type": "text",
+        "text": "DISCERN Benchmark We attempted to replicate the tasks described in [50] that are based on simple two-dimensional robots [48]. While the original tasks are not released, we followed the procedure for generating the goals described in the paper. Despite following the exact procedure, we were not able to obtain similar goals to the ones used in the original paper. Nevertheless, we show the goal completion percentage results obtained with our reproduced evaluation compared to DISCERN results from the original paper. LEXA results were obtained with early stopping. In Table 3 we see that our agent solves many of the tasks in this benchmark. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/9dbe6942689ae58e63d21aca585954939140886d90393a2f8da727367c842a83.jpg",
+        "table_caption": [
+            "Table 3: Success for DISCERN goals [50]. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Task</td><td>LEXA</td><td>DISCERN</td></tr><tr><td>Cup</td><td>84.0%</td><td>76.5%</td></tr><tr><td>Cartpole</td><td>35.9%</td><td>21.3%</td></tr><tr><td>Finger</td><td>40.9%</td><td>21.8%</td></tr><tr><td>Pendulum</td><td>79.1%</td><td>75.7%</td></tr><tr><td>Pointmass</td><td>83.2%</td><td>49.6%</td></tr><tr><td>Reacher</td><td>100.0%</td><td>87.1%</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Plan2Explore Benchmark We provide a comparison on the standard reward-based DM control tasks [48] in Table 4. To compare on this benchmark, we create goal images that correspond to the reward functions. This setup is arguably harder for our agent, but is much more practical. Note our agent never observes the reward function and only observes the goal at test time. Plan2Explore adapts to new tasks but it needs the reward function to be known at test time, while DrQV2 is an oracle agent that observes the reward at training time. Baseline results are taken from [43, 52]. LEXA results were obtained with early stopping. LEXA outperforms Plan2Explore on most tasks and even performs comparably to state of the art oracle agents (DrQ, DrQv2, Dreamer) that use true task rewards during training. ",
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+    {
+        "type": "table",
+        "img_path": "images/51d5fc875ba9f8aa14d97f8359720c5d4e5ca2112bacb09ed20a084ec9bc5fc4.jpg",
+        "table_caption": [
+            "Table 4: Zero-shot return on P2E tasks [43]. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Task Zero-Shot</td><td>LEXA [</td><td>P2E</td><td>DrQv2 X</td></tr><tr><td>Walker Stand</td><td>957</td><td>331</td><td>968</td></tr><tr><td>Hopper Stand</td><td>840</td><td>841</td><td>957</td></tr><tr><td>Cartpole Balance</td><td>886</td><td>950</td><td>989</td></tr><tr><td>Cartpole Bal. Sparse</td><td>996</td><td>860</td><td>983</td></tr><tr><td>Pendulum Swing Up</td><td>788</td><td>792</td><td>837</td></tr><tr><td>Cup Catch</td><td>969</td><td>962</td><td>909</td></tr><tr><td>Reacher Hard</td><td>937</td><td>66</td><td>970</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "3.4 Analysis ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Prior work Most work we compared against struggles with exploration, such as SkewFit and DLL methods. DIAYN is augmented with our explorer, but still fails to leverage the exploration data to learn a diverse set of skills. GCSL struggles to fit the exploration data and produces behavior that does not solve the task, perhaps because the exploration data is too diverse. We observed that all baselines make progress on the simple reaching, but struggle with other tasks. We have experimented with several versions and improvements to the baselines and report the best obtained performance. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Ablation of different components We ablated components of LEXA on the RoboBins environment in Figure 8. Using a separate explorer policy crucial as without it the agent does not discover the more interesting tasks. Without negative sampling the agent learns slower, perhaps because the distance function doesn’t produce reasonable outputs when queried on images that are more than horizon length apart. Training the distance function with real data converges to slightly lower success than using imagination data, since real data is sampled in an off-policy manner due to its limited quantity. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/e74e622cfc1a92b2076d60433d82e8c9de54acaa1e72a8a8ad12dea1ef790bc6.jpg",
+        "image_caption": [
+            "Figure 8: Ablations on RoboBins. A separate explorer is crucial for most tasks. Training temporal distance on negative samples speeds up learning, and both negative sampling and training in imagination as opposed to real data are important for the hardest tasks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Exploration performance Due to importance of exploration, we further examine the diversity of the data collected during training. We log the instances where the agent coincidentally solves an evaluation task during exploration, for the RoboKitchen and RoboBins environments. In Figure 5, we see that our method encounters harder tasks involving multiple objects much more often. ",
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+    {
+        "type": "text",
+        "text": "4 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Learning to Achieve Goals The problem of learning to reach many different goals has been commonly addressed with model-free methods that learn a single goal-conditioned policy [2, 26, 40]. Recent work has combined these approaches with various ways to generate training goals, such as asymmetric self-play [33, 45] or by sampling goals of intermediate difficulty [14, 18]. These approaches can achieve remarkable performance in simulated robotic domains, however, they focus on the settings where the agent can directly perceive the low-dimensional environment state. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A few works have attempted to scale these model-free methods to visual goals by using contrastive [50] or reconstructive [32, 37] representation learning. However, these approaches struggle to perform meaningful exploration as no clear reward signal is available to guide the agent toward solving interesting tasks. Some works [10, 49] avoid this challenge by using a large dataset of interesting behaviors. Other works [37, 55] attempt to explore by generating goals similar to those that have already been seen, but do not try to explore truly novel states. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "A particularly relevant set of approaches used model-based methods to achieve goals via planning [13, 17] or learning model-regularized policies [35]. However, these approaches are limited by short planning horizons. In contrast, we learn long-horizon goal-conditioned value functions which allows us to solve more challenging tasks. More generally, most of the above approaches are limited by simplistic exploration, while our method leverages model imagination to search for novel states, which significantly improves exploration and in turn the downstream capabilities of the agent. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Learning Distance Functions A crucial challenge for visual goal reaching is the choice of the reward or the cost function for the goal achieving policy. Several approaches use representation learning to create a distance in the feature space [9, 32, 50, 51]. However, this naive distance may not be most reflective of how hard a particular goal is to reach. One line of research has proposed using the mutual information between the current state and the goal as the distance metric [1, 12, 15, 21], however, it remains to be seen whether this approach can scale to more complex tasks. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Other works proposed temporal distances that measure the amount of time it takes to reach the goal. One approach is to learn the distance with approximate dynamic programming using Q-learning methods [16, 19, 26]. Our distance function is most similar to Hartikainen et al. [25], who learn a temporal distance with supervised learning on recent policy experience. In contrast to [25], we always train the distance on-policy in imagination, and we further integrate this achiever policy into our latent explorer achiever framework to discover novel goals for the achiever to practice on. ",
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We presented Latent Explorer Achiever (LEXA), an agent for unsupervised RL that explores its environment, learns to achieve the discovered goals, and solves image-based tasks in a zero-shot way. By planning for novelty in imagination, LEXA prospectively explores to discover meaningful behaviors in substantially more diverse environments than considered by prior work. Further, LEXA is able to solve challenging downstream tasks specified as images without any supervision such as rewards or demonstrations. By proposing a challenging benchmark and the first agent to achieve meaningful performance on these tasks, we hope to stimulate future research on unsupervised agents, which we believe are fundamentally more scalable than traditional agents that require a human to design the tasks and rewards for learning. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Many challenges remain for building unsupervised agents. Many tasks in our benchmark are still unsolved and there remains room for progress on the algorithmic side both for the world model and policy optimization. Further, it is important to demonstrate the benefits of unsupervised agents on real-world systems to verify their scalability. Finally, for widespread adoption, it is crucial to consider the problem of goal specification and design methods that act on goals that are easy to specify, such as via natural language. We believe LEXA will enable future work to tackle these goals effectively. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Acknowledgements We thank Ben Eysenbach, Stephen Tian, Sergey Levine, Dinesh Jayaraman, Karl Pertsch, Ed Hu and the members of GRASP lab and Pathak lab for insightful discussions. We also thank Murtaza Dalal and Chuning Zhu for help with MuJoCo environments. Finally, DP would like to thank Laura Schulz, Josh Tenenbaum and Alison Gopnik for seeding the idea of goal-setting in children, and Pulkit Agrawal for several crucial discussions (over gelato) since then. OR and KD were supported by ARL DCIST CRA W911NF-17-2-0181, ONR N00014-17-1-2093, and by Honda Research Institute. This work was partially supported by GoodAI Research Award and DARPA Machine Common Sense grant. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "References [1] J. Achiam, H. Edwards, D. Amodei, and P. Abbeel. Variational option discovery algorithms. arXiv preprint arXiv:1807.10299, 2018. 10 [2] M. Andrychowicz, F. Wolski, A. Ray, J. Schneider, R. Fong, P. Welinder, B. McGrew, J. Tobin, P. Abbeel, and W. Zaremba. Hindsight experience replay. arXiv preprint arXiv:1707.01495,   \n2017. 1, 4, 8, 10 [3] P. Ball, J. Parker-Holder, A. Pacchiano, K. Choromanski, and S. Roberts. Ready policy one: World building through active learning. In International Conference on Machine Learning, pages 591–601. PMLR, 2020. 4 [4] M. Bellemare, S. Srinivasan, G. Ostrovski, T. Schaul, D. Saxton, and R. Munos. Unifying count-based exploration and intrinsic motivation. In Advances in Neural Information Processing Systems, pages 1471–1479, 2016. 3 [5] L. Beyer, D. Vincent, O. Teboul, S. Gelly, M. Geist, and O. Pietquin. Mulex: Disentangling exploitation from exploration in deep rl. arXiv preprint arXiv:1907.00868, 2019. 3 [6] B. 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Advances in child development and behavior, 43:269–294, 2012. 2   \n[43] R. Sekar, O. Rybkin, K. Daniilidis, P. Abbeel, D. Hafner, and D. Pathak. Planning to explore via self-supervised world models. ICML, 2020. 2, 3, 4, 6, 9   \n[44] P. Shyam, W. Jaskowski, and F. Gomez. Model-based active exploration. ´ arXiv preprint arXiv:1810.12162, 2018. 2, 3   \n[45] S. Sukhbaatar, Z. Lin, I. Kostrikov, G. Synnaeve, A. Szlam, and R. Fergus. Intrinsic motivation and automatic curricula via asymmetric self-play. ICLR, 2018. 10   \n[46] Y. Sun, F. Gomez, and J. Schmidhuber. Planning to be surprised: Optimal bayesian exploration in dynamic environments. In International Conference on Artificial General Intelligence, pages 41–51. Springer, 2011. 3   \n[47] R. S. Sutton and A. G. Barto. Reinforcement learning: An introduction. MIT press, 2018. 2   \n[48] Y. Tassa, Y. Doron, A. Muldal, T. Erez, Y. Li, D. d. L. Casas, D. Budden, A. Abdolmaleki, J. Merel, A. Lefrancq, et al. 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parse/train/B1nLkl-0Z/B1nLkl-0Z.md

@@ -0,0 +1,368 @@
+# LEARNING GAUSSIAN POLICIES FROM SMOOTHED ACTION VALUE FUNCTIONS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+State-action value functions (i.e., Q-values) are ubiquitous in reinforcement learning (RL), giving rise to popular algorithms such as SARSA and Q-learning. We propose a new notion of action value defined by a Gaussian smoothed version of the expected Q-value. We show that such smoothed Q-values still satisfy a Bellman equation, making them learnable from experience sampled from an environment. Moreover, the gradients of expected reward with respect to the mean and covariance of a parameterized Gaussian policy can be recovered from the gradient and Hessian of the smoothed Q-value function. Based on these relationships we develop new algorithms for training a Gaussian policy directly from a learned smoothed Q-value approximator. Our approach is amenable to proximal optimization techniques by augmenting the objective with a penalty on KLdivergence from a previous policy. We find that the ability to learn both a mean and covariance during training allows this approach to achieve much better results on standard continuous control benchmarks.
+
+# 1 INTRODUCTION
+
+Model-free reinforcement learning algorithms often alternate between two concurrent but interacting processes: (1) policy evaluation, where an action value function (i.e., a Q-value) is updated to obtain a better estimate of the return associated with taking a specific action, and (2) policy improvement, where the policy is updated aiming to maximize the current value function. In the past, different notions of Q-value have led to distinct but important families of RL methods. For example, SARSA (Rummery & Niranjan, 1994; Sutton & Barto, 1998; Van Seijen et al., 2009) uses the expected Q-value, defined as the expected return of following the current policy. Q-learning (Watkins, 1989) exploits a hard-max notion of Q-value, defined as the expected return of following an optimal policy. Soft Q-learning (Haarnoja et al., 2017) and PCL (Nachum et al., 2017a) both use a soft-max form of Q-value, defined as the future return of following an optimal entropy regularized policy. Clearly, the choice of Q-value function has a considerable effect on the resulting algorithm; for example, restricting the types of policies that can be expressed, and determining the type of exploration that can be naturally applied.
+
+In this work we introduce a new notion of action value: the smoothed action value function ${ \tilde { Q } } ^ { \pi }$ . Unlike previous notions, which associate a value with a specific action at each state, the smoothed Qvalue associates a value with a specific distribution over actions. In particular, the smoothed Q-value of a state-action pair $( s , a )$ is defined as the expected return of first taking an action sampled from a normal distribution $N ( \dot { a } , \Sigma ( s ) )$ , centered at $a$ , then following actions sampled from the current policy thereafter. In this way, the smoothed Q-value can also be interpreted as a Gaussian-smoothed or noisy version of the expected Q-value.
+
+We show that smoothed Q-values possess a number of interesting properties that make them attractive for use in RL algorithms. For one, the smoothed Q-values satisfy a single-step Bellman consistency, which allows bootstrapping to be used to train a function approximator. Secondly, for Gaussian policies, the standard optimization objective (expected return) can be expressed in terms of smoothed Q-values. Moreover, the gradient of this objective with respect to the mean and covariance of the Gaussian policy is equivalent to the gradient and the Hessian of the smoothed Q-value function, which allows one to derive updates to the policy parameters by having access to the derivatives of a sufficiently accurate smoothed Q-value function.
+
+This observation leads us to propose an algorithm called Smoothie, which in the spirit of (Deep) Deterministic Policy Gradient (DDPG) (Silver et al., 2014; Lillicrap et al., 2016), trains a policy using the derivatives of a trained (smoothed) Q-value function, thus avoiding the high-variance of stochastic updates used in standard policy gradient algorithms (Williams & Peng, 1991; Konda & Tsitsiklis, 2000). Unlike DDPG, which is well-known to have poor exploratory behavior (Haarnoja et al., 2017), the approach we develop is able to utilize a non-deterministic Gaussian policy parameterized by both a mean and a covariance, thus allowing the policy to be exploratory by default and alleviating the need for excessive hyperparameter tuning.
+
+Furthermore, we show that Smoothie can be easily adapted to incorporate proximal policy optimization techniques by augmenting the objective with a penalty on KL-divergence from a previous version of the policy. The inclusion of a KL-penalty is not feasible in the standard DDPG algorithm, but we show that it is possible with our formulation, and it significantly improves stability and overall performance. On standard continuous control benchmarks, our results are competitive with or exceed state-of-the-art, especially for more difficult tasks in the low-data regime.
+
+# 2 NOTATION & BACKGROUND
+
+We consider the standard model-free RL framework, where an agent interacts with a stochastic black-box environment by sequentially observing the state of the environment, emitting an action, and receiving a reward feedback; the goal is to find an agent that achieves maximal cumulative discounted reward. This problem can be expressed in terms of a Markov decision process (MDP) that consists of a state space $s$ and an action space $\mathcal { A }$ , where at iteration $t$ the agent encounters a state $s _ { t } ~ \in ~ S$ and emits an action $a _ { t } \in \mathcal A$ , after which the environment returns a scalar reward $\boldsymbol { r } _ { t } \sim R ( s _ { t } , \boldsymbol { a } _ { t } )$ and places the agent in a new state $s _ { t + 1 } \sim P ( s _ { t } , a _ { t } )$ .
+
+We model the behavior of the agent using a stochastic policy $\pi$ that produces a distribution over feasible actions at each state $s$ as $\pi ( a \mid s )$ . The optimization objective (expected discounted return), as a function of the policy, can then be expressed in terms of the expected action value function $Q ^ { \pi } ( s , a )$ by,
+
+$$
+O _ { \tt E R } ( \pi ) = \int _ { \cal S } \int _ { \cal A } \pi ( a \mid s ) Q ^ { \pi } ( s , a ) \mathrm { d } a \mathrm { d } \rho ^ { \pi } ( s ) ,
+$$
+
+where $\rho ^ { \pi } ( s )$ is the stationary distribution of the states under $\pi$ , and $Q ^ { \pi } ( s , a )$ is recursively defined using the Bellman equation,
+
+$$
+Q ^ { \pi } ( s , a ) = \mathbb { E } _ { r , s ^ { \prime } } \left[ r + \gamma \int _ { \mathcal { A } } Q ^ { \pi } ( s ^ { \prime } , a ^ { \prime } ) \pi ( a ^ { \prime } \mid s ^ { \prime } ) \mathrm { d } a \right] \mathrm { , }
+$$
+
+where $\gamma \in [ 0 , 1 ]$ is the discount factor. For brevity, we will often suppress explicit denotation of the sampling distribution $R$ over immediate rewards and the distribution $P$ over state transitions.
+
+The policy gradient theorem (Sutton et al., 2000) expresses the gradient of $O _ { \mathrm { E R } } ( \pi _ { \theta } )$ w.r.t. $\theta$ , the tunable parameters of a policy $\pi _ { \theta }$ , as,
+
+$$
+\begin{array} { r c l } { { \nabla _ { \theta } { \cal O } _ { \mathrm { E R } } ( \pi _ { \theta } ) } } & { { = } } & { { \displaystyle \int _ { S } \int _ { { \cal A } } \nabla _ { \theta } \pi _ { \theta } ( a \mid s ) Q ^ { \pi } ( s , a ) \mathrm { d } a \mathrm { d } \rho ^ { \pi } ( s ) } } \\ { { } } & { { = } } & { { \displaystyle \int _ { S } \mathbb { E } _ { a \sim \pi _ { \theta } ( a \mid s ) } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a \mid s ) Q ^ { \pi } ( s , a ) \right] \mathrm { d } \rho ^ { \pi } ( s ) . } } \end{array}
+$$
+
+Many reinforcement learning algorithms, including policy gradient and actor-critic variants, trade off variance and bias when estimating the random variable inside the expectation in (4); for example, by attempting to estimate $Q ^ { \pi } ( s , a )$ accurately using function approximation. In the simplest scenario, an unbiased estimate of $Q ^ { \pi } ( s , a )$ is formed by accumulating discounted rewards from each state forward using a single Monte Carlo sample.
+
+In this paper, we focus on multivariate Gaussian policies over continuous action spaces, $\mathcal { A } \equiv \mathbb { R } ^ { d _ { a } }$ . We represent the observed state of the MDP as a $d _ { s }$ -dimensional feature vector $\bar { \Phi } ( s ) \in \mathbb { R } ^ { d _ { s } }$ , and parametrize the Gaussian policy by a mean and covariance function, respectively $\mu ( \boldsymbol { s } ) : \mathbb { R } ^ { d _ { \boldsymbol { s } } }  \mathbb { R } ^ { d _ { a } }$ and $\Sigma ( s ) : \mathbb { R } ^ { d _ { s } }  \mathbb { R } ^ { d _ { a } } \times \mathbb { R } ^ { \tilde { d } _ { a } }$ . These map the observed state of the environment to a Gaussian distribution,
+
+$$
+\pi ( a | s ) = N ( a | \mu ( s ) , \Sigma ( s ) ) = | 2 \pi \Sigma ( s ) | ^ { - 1 / 2 } \exp \left\{ - \frac { 1 } { 2 } \| a - \mu ( s ) \| _ { \Sigma ( s ) ^ { - 1 } } ^ { 2 } \right\} ,
+$$
+
+where $\| v \| _ { A } ^ { 2 } = v ^ { \mathsf { T } } A v$ . Below we develop new RL training methods for this family of parametric policies, but some of the ideas presented may generalize to other families of policies as well. We begin the formulation by reviewing some prior work on learning Gaussian policies.
+
+# 2.1 DETERMINISTIC POLICY GRADIENT
+
+Silver et al. (2014) present a new formulation of the policy gradient, called the deterministic policy gradient, for the family of Gaussian policies in the limit where the policy covariance approaches zero. In such a scenario, the policy becomes deterministic because sampling from the policy always returns the Gaussian mean. The key observation of (Silver et al., 2014) is that under a deterministic policy $\pi \equiv ( \mu , \Sigma \to 0 )$ , one can estimate the expected future return from a state $s$ as,
+
+$$
+\operatorname * { l i m } _ { \Sigma \to 0 } \int _ { A } \pi ( a \mid s ) Q ^ { \pi } ( s , a ) \mathrm { d } a = Q ^ { \pi } ( s , \mu ( s ) ) .
+$$
+
+Then, one can express the gradient of the optimization objective (expected discounted return) for a parameterized $\pi _ { \boldsymbol { \theta } } \equiv \mu _ { \boldsymbol { \theta } }$ as,
+
+$$
+\nabla _ { \theta } { \cal O } _ { \mathrm { E R } } ( \pi _ { \theta } ) = \int _ { S } \nabla _ { \theta } Q ^ { \pi } ( s , \mu _ { \theta } ( s ) ) \mathrm { d } \rho ^ { \pi } ( s ) = \int _ { S } \frac { \partial Q ^ { \pi } ( s , a ) } { \partial a } | _ { a = \mu _ { \theta } ( s ) } \nabla _ { \theta } \mu _ { \theta } ( s ) \mathrm { d } \rho ^ { \pi } ( s ) .
+$$
+
+This can be thought of as a characterization of the policy gradient theorem for deterministic policies.
+
+In the limit of $\Sigma  0$ , one can also re-express the Bellman equation (2) as,
+
+$$
+Q ^ { \pi } ( s , a ) = \mathbb { E } _ { r , s ^ { \prime } } \left[ r + Q ^ { \pi } ( s ^ { \prime } , \mu ( s ^ { \prime } ) ) \right] .
+$$
+
+Therefore, a value function approximator $Q _ { w } ^ { \pi }$ can be optimized by minimizing the Bellman error,
+
+$$
+E ( w ) = \sum _ { ( s , a , r , s ^ { \prime } ) \in \mathcal { D } } ( Q _ { w } ^ { \pi } ( s , a ) - r - \gamma Q _ { w } ^ { \pi } ( s ^ { \prime } , \mu ( s ^ { \prime } ) ) ^ { 2 } ,
+$$
+
+for transitions $( s , a , r , s ^ { \prime } )$ sampled from a dataset $\mathcal { D }$ of interactions of the agent with the environment. Algorithms like DDPG (Lillicrap et al., 2016) alternate between improving the value function by gradient descent on (9), and improving the policy based on (7).
+
+In practice, to gain better sample efficiency, Degris et al. (2012) and Silver et al. (2014) replace the on-policy state distribution $\rho ^ { \pi } ( s )$ in (7) with an off-policy distribution $\rho ^ { \beta } ( s )$ based on a replay buffer. After this substitution, the policy gradient identity in (7) does not hold exactly, however, prior work finds that this works well in practice and improves sample efficiency. We also adopt a similar approximation in our method to make use of off-policy data.
+
+# 3 SMOOTHED ACTION VALUE FUNCTIONS
+
+In this paper, we introduce smoothed action value functions, the gradients of which provide an effective signal for optimizing the parameters of a Gaussian policy. Our notion of smoothed Qvalues, denoted $\tilde { Q } ^ { \pi } ( s , a )$ , differs from ordinary Q-values $Q ^ { \pi } ( s , a )$ in that smoothed Q-values do not assume the first action of the agent is fully specified, but rather they assume that only the mean of the distribution of the first action is known. Hence, to compute $\tilde { Q } ^ { \pi } ( s , a )$ , one has to perform an expectation of $Q ^ { \pi } ( s , { \tilde { a } } )$ for actions $\tilde { a }$ drawn in the vicinity of $a$ . More formally, smoothed action values are defined as,
+
+$$
+\tilde { Q } ^ { \pi } ( s , a ) = \int _ { A } N ( \tilde { a } | a , \Sigma ( s ) ) Q ^ { \pi } ( s , \tilde { a } ) \mathrm { d } \tilde { a } .
+$$
+
+With this definition of ${ \tilde { Q } } ^ { \pi }$ , one can re-express the expected reward objective for a Gaussian policy $\pi \equiv ( \mu , \Sigma )$ as,
+
+$$
+O _ { \mathrm { E R } } ( \pi ) = \int _ { S } \tilde { Q } ^ { \pi } ( s , \mu ( s ) ) \mathrm { d } \rho ^ { \pi } ( s ) .
+$$
+
+The insight that differentiates this approach from prior work including Heess et al. (2015); Ciosek & Whiteson (2017) is that instead of learning a function approximator for $Q ^ { \pi } ( s , a )$ and then drawing samples to approximate the expectation in (10) and its derivative, we directly learn a function approximator for $\bar { Q } ^ { \pi } ( s , a )$ .
+
+The key observation that enables direct bootstrapping of smoothed $\mathrm { Q }$ -values, $\tilde { Q } ^ { \pi } ( s , a )$ , is that their form allows a notion of Bellman consistency. First, note that for Gaussian policies $\pi \equiv ( \mu , \Sigma )$ we have
+
+$$
+Q ^ { \pi } ( s , a ) = \mathbb { E } _ { r , s ^ { \prime } } [ r + \gamma \tilde { Q } ^ { \pi } ( s ^ { \prime } , \mu ( s ^ { \prime } ) ) ] .
+$$
+
+Then, combining (10) and (12), one can derive the following one-step Bellman equation for smoothed Q-values,
+
+$$
+\tilde { Q } ^ { \pi } ( s , a ) = \int _ { \cal A } N ( \tilde { a } \mid a , \Sigma ( s ) ) \mathbb { E } _ { \tilde { r } , \tilde { s } ^ { \prime } } \left[ \tilde { r } + \gamma \tilde { Q } ^ { \pi } ( \tilde { s } ^ { \prime } , \mu ( \tilde { s } ^ { \prime } ) ) \right] \mathrm { d } \tilde { a } ,
+$$
+
+where $\tilde { r }$ and ${ \tilde { s } } ^ { \prime }$ are sampled from $R ( s , { \tilde { a } } )$ and $P ( s , { \tilde { a } } )$ . Below, we elaborate on how one can make use of the derivatives of ${ \tilde { Q } } ^ { \pi }$ to learn $\mu$ and $\Sigma$ , and how the Bellman equation in (13) enables direct optimization of ${ \tilde { Q } } ^ { \pi }$ .
+
+# 3.1 POLICY IMPROVEMENT - OPTIMIZING $\left( \mu _ { \theta } , \Sigma _ { \phi } \right)$ GIVEN ${ \tilde { Q } } ^ { \pi }$
+
+We parameterize a Gaussian policy $\pi _ { \boldsymbol { \theta } , \boldsymbol { \phi } } \equiv ( \mu _ { \boldsymbol { \theta } } , \Sigma _ { \boldsymbol { \phi } } )$ in terms of two sets of parameters $\theta$ and $\phi$ for the mean and the covariance. The gradient of the objective $w . r . t .$ . mean parameters follows from the policy gradient theorem and is almost identical to (7),
+
+$$
+\nabla _ { \theta } { \cal O } _ { \mathrm { E R } } ( \pi _ { \theta , \phi } ) = \int _ { S } \frac { \partial \tilde { Q } ^ { \pi } ( s , a ) } { \partial a } \big | _ { a = \mu _ { \theta } ( s ) } \nabla _ { \theta } \mu _ { \theta } ( s ) \mathrm { d } \rho ^ { \pi } ( s ) .
+$$
+
+Estimating the derivative of the objective w.r.t. covariance parameters is not as straightforward, since ${ \tilde { Q } } ^ { \pi }$ is not a direct function of $\Sigma$ . However, a key observation of this work is that the second derivative of ${ \tilde { Q } } ^ { \pi } w . r . t .$ . actions is sufficient to exactly compute the derivative of $\tilde { Q } ^ { \pi } w . r . t . \Sigma$ ,
+
+$$
+\frac { \partial \tilde { Q } ^ { \pi } ( s , a ) } { \partial \Sigma ( s ) } = \frac { 1 } { 2 } \cdot \frac { \partial ^ { 2 } \tilde { Q } ^ { \pi } ( s , a ) } { \partial a ^ { 2 } } .
+$$
+
+A proof of this identity is provided in the Appendix. The proof may be easily derived by expressing both sides of the equation using standard matrix calculus like $\begin{array} { r } { \frac { \partial } { \partial A } | A | ^ { - 1 / 2 } = - \frac { 1 } { 2 } | A | ^ { - 1 / 2 } \bar { A } ^ { - 1 } } \end{array}$ and $\begin{array} { r } { \frac { \partial } { \partial A } | | v | | _ { A ^ { - 1 } } ^ { 2 } = - A ^ { - 1 } v v ^ { T } A ^ { - 1 } } \end{array}$ .
+
+Then, the full derivative w.r.t. $\phi$ takes the form,
+
+$$
+\nabla _ { \phi } { \cal O } _ { \mathrm { E R } } ( \pi _ { \theta , \phi } ) = \frac { 1 } { 2 } \int _ { S } \frac { \partial ^ { 2 } { \tilde { Q } } ^ { \pi } ( s , a ) } { \partial a ^ { 2 } } \big | _ { a = \mu _ { \theta } ( s ) } \nabla _ { \phi } \Sigma _ { \phi } ( s ) \mathrm { d } \rho ^ { \pi } ( s ) .
+$$
+
+# 3.2 POLICY EVALUATION - OPTIMIZING $\tilde { Q } _ { w } ^ { \pi }$ GIVEN $( \mu , \Sigma )$
+
+We can think of two ways to optimize $\tilde { Q } _ { w } ^ { \pi }$ . The first approach leverages (10) to update ${ \tilde { Q } } ^ { \pi }$ based on expected Q-value function $Q ^ { \pi }$ . In such an approach, one trains a parameterized $Q _ { w } ^ { \pi }$ to approximate the standard expected Q-value function $Q ^ { \pi }$ using standard methods (see e.g., Rummery $\&$ Niranjan (1994); Sutton & Barto (1998); Van Seijen et al. (2009)). Then, one fits $\tilde { Q } _ { w } ^ { \pi }$ based on $Q _ { w } ^ { \pi }$ . In particular, given transitions $( s , a , r , s ^ { \prime } )$ sampled from interactions with the environment, one can train $Q _ { w } ^ { \pi }$ to minimize the Bellman error $( Q _ { w } ^ { \pi } ( s , a ) - r - \gamma Q _ { w } ^ { \pi } ( s ^ { \prime } , a ^ { \prime } ) ) ^ { 2 }$ where $a ^ { \prime } \sim N ( \mu ( s ^ { \prime } ) , \Sigma ( s ^ { \prime } ) )$ . Then, $\tilde { Q } _ { w } ^ { \pi }$ can be optimized to minimize the squared error $( \tilde { Q } _ { w } ^ { \pi } ( s , a ) - \mathbb { E } _ { \tilde { a } } Q _ { w } ^ { \pi } ( s , \tilde { a } ) ) ^ { 2 }$ where $\tilde { a } \sim$ $N ( a , \Sigma ( s ) )$ , using several samples. When the target values in these residuals are treated as fixed (i.e., using a target network), such a training procedure will achieve a fixed point when $\tilde { Q } _ { w } ^ { \pi } ( s , a )$ satisfies the recursion in the Bellman equation (10).
+
+The second approach requires a single function approximator for $\tilde { Q } _ { w } ^ { \pi } ( s , a )$ , resulting in a simpler implementation, and thus we use this approach in our experimental evaluation. Suppose one has access to a tuple $( s , \tilde { a } , \tilde { r } , \tilde { s } ^ { \prime } )$ sampled from a replay buffer with knowledge of the sampling probability $q ( \tilde { a } \mid s )$ (possibly unnormalized). Then assuming that this sampling distribution has a full support, we draw a phantom action $a \sim N ( \tilde { a } , \Sigma ( s ) )$ and optimize $\tilde { Q } _ { w } ^ { \pi } ( s , a )$ by minimizing a weighted Bellman error $\begin{array} { r } { \frac { 1 } { q ( \tilde { a } | s ) } ( \tilde { Q } _ { w } ^ { \pi } ( s , a ) - \tilde { r } - \gamma \tilde { Q } _ { w } ^ { \pi } ( \tilde { s } ^ { \prime } , \mu ( \tilde { s } ^ { \prime } ) ) ^ { 2 } } \end{array}$ . For a specific pair of state and action $( s , a )$ the
+
+expected value of the objective is,
+
+$$
+E ( w \mid ( s , a ) ) ~ = ~ \mathbb { E } _ { q ( \bar { a } \mid s ) , \bar { r } , \bar { s } ^ { \prime } } \left[ \frac { N ( a \mid \tilde { a } , \Sigma ( s ) ) } { q ( \tilde { a } \mid s ) } ( \tilde { Q } _ { w } ^ { \pi } ( s , a ) - \tilde { r } - \gamma \tilde { Q } _ { w } ^ { \pi } ( \tilde { s } ^ { \prime } , \mu ( \tilde { s } ^ { \prime } ) ) ) ^ { 2 } \right] ~ .
+$$
+
+Note that $N ( a | \tilde { a } , \Sigma ( s ) ) = N ( \tilde { a } | a , \Sigma ( s ) )$ . Therefore, when the target value $\tilde { r } + \gamma \tilde { Q } _ { w } ^ { \pi } ( \tilde { s } ^ { \prime } , \mu ( \tilde { s } ^ { \prime } ) )$ is treated as fixed (e.g., when using target networks) this training procedure reaches an optimum when $\tilde { Q } _ { w } ^ { \pi } ( s , a )$ satisfies the recursion in the Bellman equation (13).
+
+In practice, we find that it is unnecessary to keep track of the probabilities $q ( \tilde { a } \mid s )$ , and assume the replay buffer provides a near-uniform distribution of actions conditioned on states. Other recent work has also benefited from ignoring or heavily damping importance weights (Munos et al., 2016; Wang et al., 2017; Schulman et al., 2017). However, it is possible when interacting with the environment to save the probability of sampled actions along with their transitions, and thus have access to $q ( \tilde { \boldsymbol { a } } \mid \boldsymbol { s } ) \approx N ( \tilde { \boldsymbol { a } } \mid \bar { \mu } _ { \mathrm { o l d } } ( \boldsymbol { s } ) , \dot { \Sigma _ { \mathrm { o l d } } } ( \boldsymbol { s } ) )$ .
+
+# 3.3 PROXIMAL POLICY OPTIMIZATION
+
+Policy gradient algorithms are notoriously unstable, particularly in continuous control problems. Such instability has motivated the development of trust region methods that attempt to mitigate the issue by constraining each gradient step to lie within a trust region (Schulman et al., 2015), or augmenting the expected reward objective with a penalty on KL-divergence from a previous policy (Nachum et al., 2017b; Schulman et al., 2017; Azar et al., 2012). These stabilizing techniques have thus far not been applicable to algorithms like DDPG, since the policy is deterministic. The formulation we propose in this paper, however, is easily amenable to trust region optimization. Specifically, we may augment the objective (11) with a penalty
+
+$$
+{ \cal O } _ { \mathrm { T R } } ( \pi ) = { \cal O } _ { \mathrm { E R } } ( \pi ) - \lambda \int _ { \cal S } \mathrm { K L } ( \pi \parallel \pi _ { \mathrm { o l d } } ) \mathrm { d } \rho ^ { \pi } ( s ) ,
+$$
+
+where $\pi _ { \mathrm { o l d } } \equiv \left( \mu _ { \mathrm { o l d } } , \Sigma _ { \mathrm { o l d } } \right)$ is a previous parameterization of the policy. The optimization is straightforward, since the KL-divergence of two Gaussians can be expressed analytically.
+
+# 4 RELATED WORK
+
+This paper follows a long line of work that uses Q-value functions to stably learn a policy, which in the past has been used to either approximate expected (Rummery & Niranjan, 1994; Van Seijen et al., 2009; Gu et al., 2017) or optimal (Watkins, 1989; Silver et al., 2014; Nachum et al., 2017a; Haarnoja et al., 2017; Metz et al., 2017) future value.
+
+Work that is most similar to what we present are methods that exploit gradient information from the Q-value function to train a policy. Deterministic policy gradient (Silver et al., 2014) is perhaps the best known of these. The method we propose can be interpreted as a generalization of the deterministic policy gradient. Indeed, if one takes the limit of the policy covariance $\Sigma ( s )$ as it goes to 0, the proposed Q-value function becomes the deterministic value function of DDPG, and the updates for training the Q-value approximator and the policy mean are identical.
+
+Stochastic Value Gradient (SVG) (Heess et al., 2015) also trains stochastic policies using an update that is similar to DDPG (i.e., SVG(0) with replay). The key differences with our approach are that SVG does not provide an update for the covariance, and the mean update in SVG estimates the gradient with a noisy Monte Carlo sample, which we avoid by estimating the smoothed $\mathrm { Q }$ -value function. Although a covariance update could be derived using the same reparameterization trick as in the mean update, that would also require a noisy Monte Carlo estimate. Methods for updating the covariance along the gradient of expected reward are essential for applying the subsequent trust region and proximal policy techniques.
+
+More recently, Ciosek & Whiteson (2017) introduced expected policy gradients (EPG), a generalization of DDPG that provides updates for the mean and covariance of a stochastic Gaussian policy using gradients of an estimated Q-value function. In that work, the expected Q-value used in standard policy gradient algorithms such as SARSA (Sutton & Barto, 1998; Rummery & Niranjan, 1994; Van Seijen et al., 2009) is estimated. The updates in EPG therefore require approximating an integral of the expected Q-value function. Our analogous process directly estimates an integral (via the smoothed Q-value function) and avoids approximate integrals, thereby making the updates simpler. Moreover, while Ciosek & Whiteson (2017) rely on a quadratic Taylor expansion of the estimated Q-value function, we instead rely on the strength of neural network function approximators to directly estimate the smoothed Q-value function.
+
+The novel training scheme we propose for learning the covariance of a Gaussian policy relies on properties of Gaussian integrals (Bonnet, 1964; Price, 1958). Similar identities have been used in the past to derive updates for variational auto-encoders (Kingma & Welling, 2014) and Gaussian back-propagation (Rezende et al., 2014).
+
+Finally, the perspective presented in this paper, where Q-values represent the averaged return of a distribution of actions rather than a single action, is distinct from recent advances in distributional RL (Bellemare et al., 2017). Those approaches focus on the distribution of returns of a single action, whereas we consider the single average return of a distribution of actions. Although we restrict our attention in this paper to Gaussian policies, an interesting topic for further investigation is to study the applicability of this new perspective to a wider class of policy distributions.
+
+# 5 EXPERIMENTS
+
+We utilize the insights from Section 3 to introduce a new RL algorithm, Smoothie. Smoothie maintains a parameterized $\tilde { Q } _ { w } ^ { \pi }$ trained via the procedure described in Section 3.2. It then uses the gradient and Hessian of this approximation to train a Gaussian policy $\mu _ { \theta } , \Sigma _ { \phi }$ using the updates stated in (14) and (16). See Algorithm 1 for a simplified pseudocode of our algorithm.
+
+# Algorithm 1 Smoothie
+
+<table><tr><td>Algorithm1Smoothie Input: Environment ENV,learning rates 7π, nQ,discount factor y,KL-penalty 入,l</td></tr><tr><td>number of training steps N, target network lag T. Initialize0,,w,set0&#x27;=0,Φ&#x27;=,w&#x27;=w.</td></tr><tr><td>fori=OtoN-1do</td></tr><tr><td>/ Collect experience</td></tr><tr><td>Sample action a ~ N(μe(s),∑(s)) and apply to ENV to yield r and s&#x27;. Insert transition (s,a,r,s&#x27;) to replay buffer.</td></tr><tr><td></td></tr><tr><td>// Train μ,∑</td></tr><tr><td>Sample batch{(k,@k,Tk,S)}1 fromreplay buffer aQ(ska) Compute gradients gk =</td></tr><tr><td>da la=μe(sk)*</td></tr><tr><td>²(s@） Compute Hessians Hk = a² la=μe(sk)*</td></tr><tr><td>Compute KL-penalties KLk =KL(μθ,Σ𝜙llμe,Σ).</td></tr><tr><td>Compute updates 1B</td></tr><tr><td>1B</td></tr><tr><td>△=B 1 ∑k=1 Updateθ←θ+ηπ△0,←+ηπ△.</td></tr><tr><td></td></tr><tr><td>// Train Qπ</td></tr><tr><td>Sample batch {(sk,k,Tk,S)}1 from replay bufer. Sample phantom actions ak ~ N(ák,Σ(sk)). Compute lossL(u)=∑1((s)-r-qQ(s,ue()2.</td></tr></table>
+
+We perform a number of evaluations of Smoothie compared to DDPG. We choose DDPG as a baseline because it (1) utilizes gradient information of a Q-value approximator, much like our algorithm; and (2) is a standard algorithm well-known to have achieve good, sample-efficient performance on continuous control benchmarks.
+
+# 5.1 A SYNTHETIC TASK
+
+To evaluate Smoothie we begin with a simple synthetic task which allows us to study its behavior in a restricted setting. We devised a simple single-action one-shot environment in which the reward function is a mixture of two Gaussians, one better than the other (see Figure 1 (Right)). We initialize the policy mean to be centered on the worse of the two Gaussians. We plot the learnable policy mean and standard deviation during training for Smoothie and DDPG in Figure 1 (Left). Smoothie learns both the mean and variance, while DDPG learns only the mean and the variance plotted is the exploratory noise, whose scale is kept fixed during training.
+
+As expected we observe that DDPG cannot escape the local optimum. At the beginning of training it exhibits some movement away from the local optimum (likely due to the initial noisy approximation given by $Q _ { w } ^ { \pi } \mathrm { . }$ ), it is unable to progress very far from the initial mean. Note that this is not an issue of exploration. The exploration scale is high enough that $Q _ { w } ^ { \pi }$ is aware of the better Gaussian. The issue is in the update for $\mu _ { \theta }$ , which is only with regard to the derivative of $Q _ { w } ^ { \pi }$ at the current mean.
+
+On the other hand, we find Smoothie is successfully able to solve the task. This is because the smoothed reward function approximated by $\tilde { Q } _ { w } ^ { \pi }$ has a derivative which clearly points $\mu _ { \theta }$ towards the better Gaussian. We also observe that Smoothie is able to suitably adjust the covariance $\Sigma _ { \phi }$ during training. Initially, $\Sigma _ { \phi }$ decreases due to the concavity of the smoothed reward function. As a region of convexity is entered, it begins to increase, before again decreasing to near-zero as $\mu _ { \theta }$ approaches the global optimum.
+
+![](images/3e62a205722715b4eae018cb617c9164cd435c130be75b6bf1c3addc835c777e.jpg)  
+Figure 1: Left: The learnable policy mean and standard deviation during training for Smoothie and DDPG on a simple one-shot synthetic task. The standard deviation for DDPG is the exploratory noise kept constant during training. Right: The reward function for the synthetic task along with its Gaussian-smoothed version. We find that Smoothie can successfully escape the lower-reward local optimum. We also notice Smoothie increases and decreases its policy variance as the convexity/concavity of the smoothed reward function changes.
+
+# 5.2 CONTINUOUS CONTROL
+
+We now turn our attention to standard continuous control benchmarks available on OpenAI Gym (Brockman et al., 2016) utilizing the MuJoCo environment (Todorov et al., 2012).
+
+Our implementations utilize feed forward neural networks for policy and Q-values. We parameterize the covariance $\Sigma _ { \phi }$ as a diagonal given by $e ^ { \phi }$ . The exploration for DDPG is determined by an Ornstein-Uhlenbeck process (Uhlenbeck & Ornstein, 1930; Lillicrap et al., 2016). Additional implementation details are provided in the Appendix.
+
+![](images/f21887a3716c42d41ffbbba35b3c52cf13877d7b56bb86c24b7339f4a95daf67.jpg)  
+Figure 2: Results of Smoothie, DDPG, and TRPO on continuous control benchmarks. The $\mathbf { X }$ -axis is in millions of environment steps. Each plot shows the average reward and standard deviation clipped at the min and max of six randomly seeded runs after choosing best hyperparameters. We see that Smoothie is competitive with DDPG even when DDPG uses a hyperparameter-tuned noise scale, and Smoothie learns the optimal noise scale (the covariance) during training. Moreoever, we observe significant advantages in terms of final reward performance, especially in the more difficult tasks like Hopper, Walker2d, and Humanoid. Across all tasks, TRPO is not sufficiently sampleefficient to provide a competitive baseline.
+
+We compare the results of Smoothie and DDPG in Figure 2. For each task we performed a hyperparameter search over actor learning rate, critic learning rate and reward scale, and plot the average of six runs for the best hyperparameters. For DDPG we extended the hyperparameter search to also consider the scale and damping of exploratory noise provided by the Ornstein-Uhlenbeck process. Smoothie, on the other hand, contains an additional hyperparameter to determine the weight on KL-penalty.
+
+Despite DDPG having the advantage of its exploration decided by a hyperparameter search while Smoothie must learn its exploration without supervision, we find that Smoothie performs competitively or better across all tasks, exhibiting a slight advantage in Swimmer and Ant, while showing more dramatic improvements in Hopper, Walker2d, and Humanoid. The improvement is especially dramatic for Hopper, where the average reward is doubled. We also highlight the results for Humanoid, which as far as we know, are the best published results for a method that only trains on the order of millions of environment steps. In contrast, TRPO, which to the best of our knowledge is the only other algorithm which can achieve better performance, requires on the order of tens of millions of environment steps to achieve comparable reward. This gives added evidence to the benefits of using a learnable covariance and not restricting a policy to be deterministic.
+
+Empirically, we found the introduction of a KL-penalty to improve performance of Smoothie, especially on harder tasks. We present a comparison of results of Smoothie with and without the KL-penalty on the four harder tasks in Figure 3. A KL-penalty to encourage stability is not possible in DDPG. Thus, our algorithm provides a much needed solution to the inherent instability in DDPG training.
+
+![](images/31e9c2cf700c5e3ed94cc8e3ab513890a7ad5439462138652b2f32dd4b21ec5f.jpg)  
+Figure 3: Results of Smoothie with and without a KL-penalty. The $\mathbf { X } ^ { } -$ -axis is in millions of environment steps. We observe benefits of using a proximal policy optimization method, especially in Hopper and Humanoid, where the performance improvement is significant without sacrificing sample efficiency.
+
+# 6 CONCLUSION
+
+We have presented a new Q-value function, ${ \tilde { Q } } ^ { \pi }$ , that is a Gaussian-smoothed version of the standard expected Q-value, $Q ^ { \pi }$ . The advantage of using ${ \tilde { Q } } ^ { \pi }$ over $Q ^ { \pi }$ is that its gradient and Hessian possess an intimate relationship with the gradient of expected reward with respect to mean and covariance of a Gaussian policy. The resulting algorithm, Smoothie, is able to successfully learn both mean and covariance during training, leading to performance that can match or surpass that of DDPG, especially when incorporating a penalty on divergence from a previous policy.
+
+The success of ${ \tilde { Q } } ^ { \pi }$ is encouraging. Intuitively it may be argued that learning ${ \tilde { Q } } ^ { \pi }$ is more sensible than learning $Q ^ { \pi }$ . The smoothed Q-values by definition make the true reward surface smoother, thus possibly easier to learn; moreover the smoothed Q-values have a more direct relationship with the expected discounted return objective. We encourage future work to further investigate these claims as well as techniques to apply the underlying motivations for ${ \tilde { Q } } ^ { \pi }$ to other types of policies.
+
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+
+# A PROOF OF EQUATION (15)
+
+We note that similar identities for Gaussian integrals exist in the literature (Price, 1958; Rezende et al., 2014) and point the reader to these works for further information.
+
+The specific identity we state may be derived using standard matrix calculus. We make use of the fact that
+
+$$
+\frac { \partial } { \partial A } | A | ^ { - 1 / 2 } = - \frac { 1 } { 2 } | A | ^ { - 3 / 2 } \frac { \partial } { \partial A } | A | = - \frac { 1 } { 2 } | A | ^ { - 1 / 2 } A ^ { - 1 } ,
+$$
+
+and for symmetric $A$ ,
+
+$$
+\frac \partial { \partial A } | | \boldsymbol { v } | | _ { A ^ { - 1 } } ^ { 2 } = - A ^ { - 1 } { \boldsymbol { v } } { \boldsymbol { v } } ^ { T } A ^ { - 1 } .
+$$
+
+We omit $s$ from $\Sigma ( s )$ in the following equations for succinctness. The LHS of (15) is
+
+$$
+\begin{array} { l } { \displaystyle \int _ { A } Q ^ { \pi } ( s , \tilde { a } ) \frac { \partial } { \partial \Sigma } N ( \tilde { a } | a , \Sigma ) \mathrm { d } \tilde { a } } \\ { \displaystyle = \int _ { A } Q ^ { \pi } ( s , \tilde { a } ) \exp \left\{ - \frac { 1 } { 2 } | | \tilde { a } - a | | _ { \Sigma ^ { - 1 } } ^ { 2 } \right\} \left( \frac { \partial } { \partial \Sigma } | 2 \pi \Sigma | ^ { - 1 / 2 } - \frac { 1 } { 2 } | 2 \pi \Sigma | ^ { - 1 / 2 } \frac { \partial } { \partial \Sigma } | | \tilde { a } - a | | _ { \Sigma ^ { - 1 } } ^ { 2 } \right) \mathrm { d } \tilde { a } } \\ { \displaystyle \qquad = \frac { 1 } { 2 } \int _ { A } Q ^ { \pi } ( s , \tilde { a } ) N ( \tilde { a } | a , \Sigma ) \left( - \Sigma ^ { - 1 } + \Sigma ^ { - 1 } ( \tilde { a } - a ) ( \tilde { a } - a ) ^ { T } \Sigma ^ { - 1 } \right) \mathrm { d } \tilde { a } . \quad ( 2 \pi \Sigma ) \mathrm { d } \tilde { a } , } \end{array}
+$$
+
+Meanwhile, towards tackling the RHS of (15) we note that
+
+$$
+\frac { \partial \tilde { Q } ^ { \pi } ( s , a ) } { \partial a } = \int _ { \cal A } Q ^ { \pi } ( s , \tilde { a } ) N ( \tilde { a } | a , \Sigma ) \Sigma ^ { - 1 } ( \tilde { a } - a ) \mathrm { d } \tilde { a } .
+$$
+
+Thus we have
+
+$$
+\frac { \partial ^ { 2 } \tilde { Q } ^ { \pi } ( s , a ) } { \partial a ^ { 2 } } = \int _ { \mathcal { A } } Q ^ { \pi } ( s , \tilde { a } ) \left( \Sigma ^ { - 1 } ( \tilde { a } - a ) \frac { \partial } { \partial a } N ( \tilde { a } | a , \Sigma ) + N ( \tilde { a } | a , \Sigma ) \frac { \partial } { \partial a } \Sigma ^ { - 1 } ( \tilde { a } - a ) \right) \mathrm { d } \tilde { a } .
+$$
+
+$$
+= \int _ { \mathcal { A } } Q ^ { \pi } ( s , \tilde { a } ) N ( \tilde { a } | a , \Sigma ) ( \Sigma ^ { - 1 } ( \tilde { a } - a ) ( \tilde { a } - a ) ^ { T } \Sigma ^ { - 1 } - \Sigma ^ { - 1 } ) \mathrm { d } \tilde { a } .
+$$
+
+# B COMPATIBLE FUNCTION APPROXIMATION
+
+A function approximator $\tilde { Q } _ { w } ^ { \pi }$ of ${ \tilde { Q } } ^ { \pi }$ should be sufficiently accurate so that updates for $\mu _ { \theta } , \Sigma _ { \phi }$ are not affected by substituting $\frac { \partial \tilde { Q } _ { w } ^ { \pi } ( s , a ) } { \partial a }$ and $\frac { \partial ^ { 2 } \tilde { Q } _ { w } ^ { \pi } ( s , a ) } { \partial a ^ { 2 } }$ for $\frac { \partial \tilde { Q } ^ { \pi } ( s , a ) } { \partial a }$ and $\frac { \partial ^ { 2 } \tilde { Q } ^ { \pi } ( s , a ) } { \partial a ^ { 2 } }$ , respectively.
+
+We claim that a $\tilde { Q } _ { w } ^ { \pi }$ is compatible with respect to $\mu _ { \theta }$ if
+
+1 $\left. \nabla _ { a } \tilde { Q } _ { w } ^ { \pi } ( s , a ) \right| _ { a = \mu _ { \theta } ( s ) } = \nabla _ { \theta } \mu _ { \theta } ( s ) ^ { T } w ,$   
+2. $\begin{array} { r } { \nabla _ { w } \int _ { \mathcal { S } } \Big ( \nabla _ { a } \tilde { Q } _ { w } ^ { \pi } ( s , a ) \big | _ { a = \mu _ { \theta } ( s ) } - \nabla _ { a } \tilde { Q } ^ { \pi } ( s , a ) \big | _ { a = \mu _ { \theta } ( s ) } \Big ) ^ { 2 } \mathrm { d } \rho ^ { \pi } ( s ) = 0 } \end{array}$ (i.e., $w$ minimizes the expected squared error of the gradients).
+
+Additionally, $\tilde { Q } _ { w } ^ { \pi }$ is compatible with respect to $\Sigma _ { \phi }$ if
+
+1. $\nabla _ { a } ^ { 2 } \tilde { Q } _ { w } ^ { \pi } ( s , a ) \big | _ { a = \mu _ { \theta } ( s ) } = \nabla _ { \phi } \Sigma _ { \phi } ( s ) ^ { T } w ,$   
+2. $\begin{array} { r } { \nabla _ { w } \int _ { \mathcal { S } } \left( \nabla _ { a } ^ { 2 } \tilde { Q } _ { w } ^ { \pi } ( s , a ) \big \vert _ { a = \mu _ { \theta } ( s ) } - \nabla _ { a } ^ { 2 } \tilde { Q } ^ { \pi } ( s , a ) \big \vert _ { a = \mu _ { \theta } ( s ) } \right) ^ { 2 } \mathrm { d } \rho ^ { \pi } ( s ) = 0 } \end{array}$ (i.e., $w$ minimizes the expected squared error of the Hessians).
+
+One possible parameterization of $\tilde { Q } _ { w } ^ { \pi }$ may be achieved by taking $w = [ w _ { 0 } , w _ { 1 } , w _ { 2 } ]$ and parameterizing
+
+$$
+\begin{array} { r } { \tilde { Q } _ { w } ^ { \pi } ( s , a ) = V _ { w _ { 0 } } ( s ) + ( a - \mu _ { \theta } ( s ) ) ^ { T } \nabla _ { \theta } \mu _ { \theta } ( s ) ^ { T } w _ { 1 } + ( a - \mu _ { \theta } ( s ) ) ^ { T } \nabla _ { \phi } \Sigma _ { \phi } ( s ) ^ { T } w _ { 2 } ( a - \mu _ { \theta } ( s ) ) . } \end{array}
+$$
+
+<table><tr><td>Hyperparameter</td><td>Range</td><td>Sampling</td></tr><tr><td>actor learningrate critic learning rate</td><td>[1e-6,1e-3] [1e-6,1e-3]</td><td>log</td></tr><tr><td>reward scale</td><td>[0.01,0.3]</td><td>l0g</td></tr><tr><td>OU damping</td><td>[1e-4,1e-3]</td><td>l0g</td></tr><tr><td>OU stddev</td><td></td><td>10g</td></tr><tr><td>入</td><td>[1e-3,1.0]</td><td>log</td></tr><tr><td>discount factor</td><td>[1e-6, 4e-2] 0.995</td><td>log</td></tr><tr><td>target network lag</td><td>0.01</td><td>fixed</td></tr><tr><td>batch size</td><td>128</td><td>fixed</td></tr><tr><td>clipping on gradients of Q</td><td></td><td>fixed</td></tr><tr><td></td><td>4.0</td><td>fixed</td></tr><tr><td>num gradient updates per observation</td><td>1</td><td>fixed</td></tr><tr><td>Huber loss clipping</td><td>1.0</td><td>fixed</td></tr></table>
+
+Proof. We shall show how the conditions stated for compatibility with respect to $\Sigma _ { \phi }$ are sufficient. The reasoning for $\mu _ { \theta }$ follows via a similar argument. We also refer the reader to Silver et al. (2014) which includes a similar procedure for showing compatibility.
+
+From the second condition for compatibility with respect to $\Sigma _ { \phi }$ we have
+
+$$
+\int _ { S } \left( \nabla _ { a } ^ { 2 } \tilde { Q } _ { w } ^ { \pi } ( s , a ) \big | _ { a = \mu _ { \theta } ( s ) } - \nabla _ { a } ^ { 2 } \tilde { Q } ^ { \pi } ( s , a ) \big | _ { a = \mu _ { \theta } ( s ) } \right) \nabla _ { w } \left( \nabla _ { a } ^ { 2 } \tilde { Q } _ { w } ^ { \pi } ( s , a ) \big | _ { a = \mu _ { \theta } ( s ) } \right) \mathrm { d } \rho ^ { \pi } ( s ) = 0 .
+$$
+
+We may combine this with the first condition to find
+
+$$
+\int _ { S } \nabla _ { a } ^ { 2 } \tilde { Q } _ { w } ^ { \pi } ( s , a ) \big | _ { a = \mu _ { \theta } ( s ) } \nabla _ { \phi } \Sigma _ { \phi } ( s ) \mathrm { d } \rho ^ { \pi } ( s ) = \int _ { S } \nabla _ { a } ^ { 2 } \tilde { Q } ^ { \pi } ( s , a ) \big | _ { a = \mu _ { \theta } ( s ) } \nabla _ { \phi } \Sigma _ { \phi } ( s ) \mathrm { d } \rho ^ { \pi } ( s ) ,
+$$
+
+which is the desired property for compatibility.
+
+While it is reassuring to know that there exists a class of function approximators which are compatible, this fact is largely ignored in practice. Not only is the class of compatible functions heavily restricted in terms of expressiveness, due to the first set of conditions for $\mu _ { \theta } , \Sigma _ { \phi }$ , it is also impossible to satisfy the second set of conditions without access to derivative and Hessian information of the true ${ \tilde { Q } } ^ { \pi }$ . This problem is also present in DDPG, and we feel this issue merits additional investigation in future work.
+
+# C IMPLEMENTATION DETAILS
+
+We utilize feed forward networks for both policy and Q-value approximator. For $\mu _ { \boldsymbol { \theta } } ( s )$ we use two hidden layers of dimensions (400, 300) and relu activation functions. For $\tilde { Q } _ { w } ^ { \pi } ( s , a )$ and $Q _ { w } ^ { \pi } ( s , a )$ we first embed the state into a 400 dimensional vector using a fully-connected layer and tanh nonlinearity. We then concatenate the embedded state with $a$ and pass the result through a 1-hidden layer neural network of dimension 300 with tanh activations. We use a diagonal $\Sigma _ { \phi } ^ { \bar { } } ( s ) = e ^ { \phi }$ for Smoothie, with $\phi$ initialized to $- 1$ .
+
+To find optimal hyperparameters we perform a 100-trial random search over the hyperparameters specified in Table 1. The OU exploration parameters only apply to DDPG. The $\lambda$ coefficient on KL-penalty only applies to Smoothie with a KL-penalty.

parse/train/B1nLkl-0Z/B1nLkl-0Z_content_list.json

@@ -0,0 +1,1818 @@
+[
+    {
+        "type": "text",
+        "text": "LEARNING GAUSSIAN POLICIES FROM SMOOTHED ACTION VALUE FUNCTIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            99,
+            656,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            171,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "State-action value functions (i.e., Q-values) are ubiquitous in reinforcement learning (RL), giving rise to popular algorithms such as SARSA and Q-learning. We propose a new notion of action value defined by a Gaussian smoothed version of the expected Q-value. We show that such smoothed Q-values still satisfy a Bellman equation, making them learnable from experience sampled from an environment. Moreover, the gradients of expected reward with respect to the mean and covariance of a parameterized Gaussian policy can be recovered from the gradient and Hessian of the smoothed Q-value function. Based on these relationships we develop new algorithms for training a Gaussian policy directly from a learned smoothed Q-value approximator. Our approach is amenable to proximal optimization techniques by augmenting the objective with a penalty on KLdivergence from a previous policy. We find that the ability to learn both a mean and covariance during training allows this approach to achieve much better results on standard continuous control benchmarks. ",
+        "bbox": [
+            233,
+            266,
+            764,
+            460
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            487,
+            336,
+            503
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Model-free reinforcement learning algorithms often alternate between two concurrent but interacting processes: (1) policy evaluation, where an action value function (i.e., a Q-value) is updated to obtain a better estimate of the return associated with taking a specific action, and (2) policy improvement, where the policy is updated aiming to maximize the current value function. In the past, different notions of Q-value have led to distinct but important families of RL methods. For example, SARSA (Rummery & Niranjan, 1994; Sutton & Barto, 1998; Van Seijen et al., 2009) uses the expected Q-value, defined as the expected return of following the current policy. Q-learning (Watkins, 1989) exploits a hard-max notion of Q-value, defined as the expected return of following an optimal policy. Soft Q-learning (Haarnoja et al., 2017) and PCL (Nachum et al., 2017a) both use a soft-max form of Q-value, defined as the future return of following an optimal entropy regularized policy. Clearly, the choice of Q-value function has a considerable effect on the resulting algorithm; for example, restricting the types of policies that can be expressed, and determining the type of exploration that can be naturally applied. ",
+        "bbox": [
+            174,
+            518,
+            825,
+            699
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work we introduce a new notion of action value: the smoothed action value function ${ \\tilde { Q } } ^ { \\pi }$ . Unlike previous notions, which associate a value with a specific action at each state, the smoothed Qvalue associates a value with a specific distribution over actions. In particular, the smoothed Q-value of a state-action pair $( s , a )$ is defined as the expected return of first taking an action sampled from a normal distribution $N ( \\dot { a } , \\Sigma ( s ) )$ , centered at $a$ , then following actions sampled from the current policy thereafter. In this way, the smoothed Q-value can also be interpreted as a Gaussian-smoothed or noisy version of the expected Q-value. ",
+        "bbox": [
+            174,
+            708,
+            825,
+            805
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We show that smoothed Q-values possess a number of interesting properties that make them attractive for use in RL algorithms. For one, the smoothed Q-values satisfy a single-step Bellman consistency, which allows bootstrapping to be used to train a function approximator. Secondly, for Gaussian policies, the standard optimization objective (expected return) can be expressed in terms of smoothed Q-values. Moreover, the gradient of this objective with respect to the mean and covariance of the Gaussian policy is equivalent to the gradient and the Hessian of the smoothed Q-value function, which allows one to derive updates to the policy parameters by having access to the derivatives of a sufficiently accurate smoothed Q-value function. ",
+        "bbox": [
+            174,
+            813,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This observation leads us to propose an algorithm called Smoothie, which in the spirit of (Deep) Deterministic Policy Gradient (DDPG) (Silver et al., 2014; Lillicrap et al., 2016), trains a policy using the derivatives of a trained (smoothed) Q-value function, thus avoiding the high-variance of stochastic updates used in standard policy gradient algorithms (Williams & Peng, 1991; Konda & Tsitsiklis, 2000). Unlike DDPG, which is well-known to have poor exploratory behavior (Haarnoja et al., 2017), the approach we develop is able to utilize a non-deterministic Gaussian policy parameterized by both a mean and a covariance, thus allowing the policy to be exploratory by default and alleviating the need for excessive hyperparameter tuning. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            215
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, we show that Smoothie can be easily adapted to incorporate proximal policy optimization techniques by augmenting the objective with a penalty on KL-divergence from a previous version of the policy. The inclusion of a KL-penalty is not feasible in the standard DDPG algorithm, but we show that it is possible with our formulation, and it significantly improves stability and overall performance. On standard continuous control benchmarks, our results are competitive with or exceed state-of-the-art, especially for more difficult tasks in the low-data regime. ",
+        "bbox": [
+            174,
+            222,
+            825,
+            305
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 NOTATION & BACKGROUND",
+        "text_level": 1,
+        "bbox": [
+            176,
+            324,
+            439,
+            342
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We consider the standard model-free RL framework, where an agent interacts with a stochastic black-box environment by sequentially observing the state of the environment, emitting an action, and receiving a reward feedback; the goal is to find an agent that achieves maximal cumulative discounted reward. This problem can be expressed in terms of a Markov decision process (MDP) that consists of a state space $s$ and an action space $\\mathcal { A }$ , where at iteration $t$ the agent encounters a state $s _ { t } ~ \\in ~ S$ and emits an action $a _ { t } \\in \\mathcal A$ , after which the environment returns a scalar reward $\\boldsymbol { r } _ { t } \\sim R ( s _ { t } , \\boldsymbol { a } _ { t } )$ and places the agent in a new state $s _ { t + 1 } \\sim P ( s _ { t } , a _ { t } )$ . ",
+        "bbox": [
+            173,
+            354,
+            825,
+            454
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We model the behavior of the agent using a stochastic policy $\\pi$ that produces a distribution over feasible actions at each state $s$ as $\\pi ( a \\mid s )$ . The optimization objective (expected discounted return), as a function of the policy, can then be expressed in terms of the expected action value function $Q ^ { \\pi } ( s , a )$ by, ",
+        "bbox": [
+            174,
+            459,
+            825,
+            515
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c8cd7a89e53b1d3793cf1570c985f6d22970baa4281b62cb5b9aea8f11e54055.jpg",
+        "text": "$$\nO _ { \\tt E R } ( \\pi ) = \\int _ { \\cal S } \\int _ { \\cal A } \\pi ( a \\mid s ) Q ^ { \\pi } ( s , a ) \\mathrm { d } a \\mathrm { d } \\rho ^ { \\pi } ( s ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            339,
+            513,
+            656,
+            547
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\rho ^ { \\pi } ( s )$ is the stationary distribution of the states under $\\pi$ , and $Q ^ { \\pi } ( s , a )$ is recursively defined using the Bellman equation, ",
+        "bbox": [
+            176,
+            547,
+            825,
+            575
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4935471ef55f310e3e24877e897a38f4f10940cefaaf19c3e367eff3cb69679d.jpg",
+        "text": "$$\nQ ^ { \\pi } ( s , a ) = \\mathbb { E } _ { r , s ^ { \\prime } } \\left[ r + \\gamma \\int _ { \\mathcal { A } } Q ^ { \\pi } ( s ^ { \\prime } , a ^ { \\prime } ) \\pi ( a ^ { \\prime } \\mid s ^ { \\prime } ) \\mathrm { d } a \\right] \\mathrm { , }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            320,
+            577,
+            676,
+            612
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\gamma \\in [ 0 , 1 ]$ is the discount factor. For brevity, we will often suppress explicit denotation of the sampling distribution $R$ over immediate rewards and the distribution $P$ over state transitions. ",
+        "bbox": [
+            176,
+            614,
+            821,
+            642
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The policy gradient theorem (Sutton et al., 2000) expresses the gradient of $O _ { \\mathrm { E R } } ( \\pi _ { \\theta } )$ w.r.t. $\\theta$ , the tunable parameters of a policy $\\pi _ { \\theta }$ , as, ",
+        "bbox": [
+            174,
+            648,
+            821,
+            678
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0a901f690d4f358615e33840232298f5dac446623f98273090e286de6d11634e.jpg",
+        "text": "$$\n\\begin{array} { r c l } { { \\nabla _ { \\theta } { \\cal O } _ { \\mathrm { E R } } ( \\pi _ { \\theta } ) } } & { { = } } & { { \\displaystyle \\int _ { S } \\int _ { { \\cal A } } \\nabla _ { \\theta } \\pi _ { \\theta } ( a \\mid s ) Q ^ { \\pi } ( s , a ) \\mathrm { d } a \\mathrm { d } \\rho ^ { \\pi } ( s ) } } \\\\ { { } } & { { = } } & { { \\displaystyle \\int _ { S } \\mathbb { E } _ { a \\sim \\pi _ { \\theta } ( a \\mid s ) } \\left[ \\nabla _ { \\theta } \\log \\pi _ { \\theta } ( a \\mid s ) Q ^ { \\pi } ( s , a ) \\right] \\mathrm { d } \\rho ^ { \\pi } ( s ) . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            679,
+            727,
+            747
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Many reinforcement learning algorithms, including policy gradient and actor-critic variants, trade off variance and bias when estimating the random variable inside the expectation in (4); for example, by attempting to estimate $Q ^ { \\pi } ( s , a )$ accurately using function approximation. In the simplest scenario, an unbiased estimate of $Q ^ { \\pi } ( s , a )$ is formed by accumulating discounted rewards from each state forward using a single Monte Carlo sample. ",
+        "bbox": [
+            176,
+            746,
+            825,
+            816
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we focus on multivariate Gaussian policies over continuous action spaces, $\\mathcal { A } \\equiv \\mathbb { R } ^ { d _ { a } }$ . We represent the observed state of the MDP as a $d _ { s }$ -dimensional feature vector $\\bar { \\Phi } ( s ) \\in \\mathbb { R } ^ { d _ { s } }$ , and parametrize the Gaussian policy by a mean and covariance function, respectively $\\mu ( \\boldsymbol { s } ) : \\mathbb { R } ^ { d _ { \\boldsymbol { s } } }  \\mathbb { R } ^ { d _ { a } }$ and $\\Sigma ( s ) : \\mathbb { R } ^ { d _ { s } }  \\mathbb { R } ^ { d _ { a } } \\times \\mathbb { R } ^ { \\tilde { d } _ { a } }$ . These map the observed state of the environment to a Gaussian distribution, ",
+        "bbox": [
+            173,
+            821,
+            825,
+            893
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/30e174d88444c4e6a80ccbede2e10a2cc7673ac23a5d544af7e72083cd2e3091.jpg",
+        "text": "$$\n\\pi ( a | s ) = N ( a | \\mu ( s ) , \\Sigma ( s ) ) = | 2 \\pi \\Sigma ( s ) | ^ { - 1 / 2 } \\exp \\left\\{ - \\frac { 1 } { 2 } \\| a - \\mu ( s ) \\| _ { \\Sigma ( s ) ^ { - 1 } } ^ { 2 } \\right\\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            236,
+            893,
+            759,
+            929
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\| v \\| _ { A } ^ { 2 } = v ^ { \\mathsf { T } } A v$ . Below we develop new RL training methods for this family of parametric policies, but some of the ideas presented may generalize to other families of policies as well. We begin the formulation by reviewing some prior work on learning Gaussian policies. ",
+        "bbox": [
+            174,
+            102,
+            825,
+            146
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 DETERMINISTIC POLICY GRADIENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            162,
+            464,
+            176
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Silver et al. (2014) present a new formulation of the policy gradient, called the deterministic policy gradient, for the family of Gaussian policies in the limit where the policy covariance approaches zero. In such a scenario, the policy becomes deterministic because sampling from the policy always returns the Gaussian mean. The key observation of (Silver et al., 2014) is that under a deterministic policy $\\pi \\equiv ( \\mu , \\Sigma \\to 0 )$ , one can estimate the expected future return from a state $s$ as, ",
+        "bbox": [
+            173,
+            188,
+            825,
+            258
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5bbd75b4612abde13afa02d3c4fd27ad64c7f82ad3480efe30ec063b6a7aeef2.jpg",
+        "text": "$$\n\\operatorname * { l i m } _ { \\Sigma \\to 0 } \\int _ { A } \\pi ( a \\mid s ) Q ^ { \\pi } ( s , a ) \\mathrm { d } a = Q ^ { \\pi } ( s , \\mu ( s ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            263,
+            651,
+            297
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Then, one can express the gradient of the optimization objective (expected discounted return) for a parameterized $\\pi _ { \\boldsymbol { \\theta } } \\equiv \\mu _ { \\boldsymbol { \\theta } }$ as, ",
+        "bbox": [
+            174,
+            301,
+            823,
+            330
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/edb5b4ed43a4f1a0e669d90a2efa86e348e4bfe334dbf8cd55ff8c9d7fefcff8.jpg",
+        "text": "$$\n\\nabla _ { \\theta } { \\cal O } _ { \\mathrm { E R } } ( \\pi _ { \\theta } ) = \\int _ { S } \\nabla _ { \\theta } Q ^ { \\pi } ( s , \\mu _ { \\theta } ( s ) ) \\mathrm { d } \\rho ^ { \\pi } ( s ) = \\int _ { S } \\frac { \\partial Q ^ { \\pi } ( s , a ) } { \\partial a } | _ { a = \\mu _ { \\theta } ( s ) } \\nabla _ { \\theta } \\mu _ { \\theta } ( s ) \\mathrm { d } \\rho ^ { \\pi } ( s ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            194,
+            335,
+            784,
+            369
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This can be thought of as a characterization of the policy gradient theorem for deterministic policies. ",
+        "bbox": [
+            173,
+            373,
+            823,
+            388
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the limit of $\\Sigma  0$ , one can also re-express the Bellman equation (2) as, ",
+        "bbox": [
+            176,
+            395,
+            666,
+            410
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2f00868dda5fac9664e78a8235cb650a820dc280c48e8bb4598ed26563080e3e.jpg",
+        "text": "$$\nQ ^ { \\pi } ( s , a ) = \\mathbb { E } _ { r , s ^ { \\prime } } \\left[ r + Q ^ { \\pi } ( s ^ { \\prime } , \\mu ( s ^ { \\prime } ) ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            414,
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+            433
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Therefore, a value function approximator $Q _ { w } ^ { \\pi }$ can be optimized by minimizing the Bellman error, ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/33f14e5df15606e8aea149929ac6bcadca43b1040e98716946a23e09f91a9fe2.jpg",
+        "text": "$$\nE ( w ) = \\sum _ { ( s , a , r , s ^ { \\prime } ) \\in \\mathcal { D } } ( Q _ { w } ^ { \\pi } ( s , a ) - r - \\gamma Q _ { w } ^ { \\pi } ( s ^ { \\prime } , \\mu ( s ^ { \\prime } ) ) ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "for transitions $( s , a , r , s ^ { \\prime } )$ sampled from a dataset $\\mathcal { D }$ of interactions of the agent with the environment. Algorithms like DDPG (Lillicrap et al., 2016) alternate between improving the value function by gradient descent on (9), and improving the policy based on (7). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In practice, to gain better sample efficiency, Degris et al. (2012) and Silver et al. (2014) replace the on-policy state distribution $\\rho ^ { \\pi } ( s )$ in (7) with an off-policy distribution $\\rho ^ { \\beta } ( s )$ based on a replay buffer. After this substitution, the policy gradient identity in (7) does not hold exactly, however, prior work finds that this works well in practice and improves sample efficiency. We also adopt a similar approximation in our method to make use of off-policy data. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 SMOOTHED ACTION VALUE FUNCTIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we introduce smoothed action value functions, the gradients of which provide an effective signal for optimizing the parameters of a Gaussian policy. Our notion of smoothed Qvalues, denoted $\\tilde { Q } ^ { \\pi } ( s , a )$ , differs from ordinary Q-values $Q ^ { \\pi } ( s , a )$ in that smoothed Q-values do not assume the first action of the agent is fully specified, but rather they assume that only the mean of the distribution of the first action is known. Hence, to compute $\\tilde { Q } ^ { \\pi } ( s , a )$ , one has to perform an expectation of $Q ^ { \\pi } ( s , { \\tilde { a } } )$ for actions $\\tilde { a }$ drawn in the vicinity of $a$ . More formally, smoothed action values are defined as, ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/8713600fb539a4a0fd028c7442c7fd55bcda9b1e7c291fef1adceb76fdfeb0f3.jpg",
+        "text": "$$\n\\tilde { Q } ^ { \\pi } ( s , a ) = \\int _ { A } N ( \\tilde { a } | a , \\Sigma ( s ) ) Q ^ { \\pi } ( s , \\tilde { a } ) \\mathrm { d } \\tilde { a } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "With this definition of ${ \\tilde { Q } } ^ { \\pi }$ , one can re-express the expected reward objective for a Gaussian policy $\\pi \\equiv ( \\mu , \\Sigma )$ as, ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/746ea57183920be259738989cf19f6bf70a85484858cb13172e34626faa37835.jpg",
+        "text": "$$\nO _ { \\mathrm { E R } } ( \\pi ) = \\int _ { S } \\tilde { Q } ^ { \\pi } ( s , \\mu ( s ) ) \\mathrm { d } \\rho ^ { \\pi } ( s ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The insight that differentiates this approach from prior work including Heess et al. (2015); Ciosek & Whiteson (2017) is that instead of learning a function approximator for $Q ^ { \\pi } ( s , a )$ and then drawing samples to approximate the expectation in (10) and its derivative, we directly learn a function approximator for $\\bar { Q } ^ { \\pi } ( s , a )$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The key observation that enables direct bootstrapping of smoothed $\\mathrm { Q }$ -values, $\\tilde { Q } ^ { \\pi } ( s , a )$ , is that their form allows a notion of Bellman consistency. First, note that for Gaussian policies $\\pi \\equiv ( \\mu , \\Sigma )$ we have ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/21a3aedaa734f3c25543a623fa9c74af9b5602d1607eca5b4b1dda92b52eb8f7.jpg",
+        "text": "$$\nQ ^ { \\pi } ( s , a ) = \\mathbb { E } _ { r , s ^ { \\prime } } [ r + \\gamma \\tilde { Q } ^ { \\pi } ( s ^ { \\prime } , \\mu ( s ^ { \\prime } ) ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Then, combining (10) and (12), one can derive the following one-step Bellman equation for smoothed Q-values, ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fe18d88621a98303e4f2b541b01df7598dfd6b3292ac4b2c5c26bfe54af28ffb.jpg",
+        "text": "$$\n\\tilde { Q } ^ { \\pi } ( s , a ) = \\int _ { \\cal A } N ( \\tilde { a } \\mid a , \\Sigma ( s ) ) \\mathbb { E } _ { \\tilde { r } , \\tilde { s } ^ { \\prime } } \\left[ \\tilde { r } + \\gamma \\tilde { Q } ^ { \\pi } ( \\tilde { s } ^ { \\prime } , \\mu ( \\tilde { s } ^ { \\prime } ) ) \\right] \\mathrm { d } \\tilde { a } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\tilde { r }$ and ${ \\tilde { s } } ^ { \\prime }$ are sampled from $R ( s , { \\tilde { a } } )$ and $P ( s , { \\tilde { a } } )$ . Below, we elaborate on how one can make use of the derivatives of ${ \\tilde { Q } } ^ { \\pi }$ to learn $\\mu$ and $\\Sigma$ , and how the Bellman equation in (13) enables direct optimization of ${ \\tilde { Q } } ^ { \\pi }$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 POLICY IMPROVEMENT - OPTIMIZING $\\left( \\mu _ { \\theta } , \\Sigma _ { \\phi } \\right)$ GIVEN ${ \\tilde { Q } } ^ { \\pi }$ ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We parameterize a Gaussian policy $\\pi _ { \\boldsymbol { \\theta } , \\boldsymbol { \\phi } } \\equiv ( \\mu _ { \\boldsymbol { \\theta } } , \\Sigma _ { \\boldsymbol { \\phi } } )$ in terms of two sets of parameters $\\theta$ and $\\phi$ for the mean and the covariance. The gradient of the objective $w . r . t .$ . mean parameters follows from the policy gradient theorem and is almost identical to (7), ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6d9aef1b0a5509659a282090238279c9da22df639f2282e7c6aa94387ffb0566.jpg",
+        "text": "$$\n\\nabla _ { \\theta } { \\cal O } _ { \\mathrm { E R } } ( \\pi _ { \\theta , \\phi } ) = \\int _ { S } \\frac { \\partial \\tilde { Q } ^ { \\pi } ( s , a ) } { \\partial a } \\big | _ { a = \\mu _ { \\theta } ( s ) } \\nabla _ { \\theta } \\mu _ { \\theta } ( s ) \\mathrm { d } \\rho ^ { \\pi } ( s ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            307,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Estimating the derivative of the objective w.r.t. covariance parameters is not as straightforward, since ${ \\tilde { Q } } ^ { \\pi }$ is not a direct function of $\\Sigma$ . However, a key observation of this work is that the second derivative of ${ \\tilde { Q } } ^ { \\pi } w . r . t .$ . actions is sufficient to exactly compute the derivative of $\\tilde { Q } ^ { \\pi } w . r . t . \\Sigma$ , ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5b5740281c05060410f4bd717038d516178564d8e3793b5af4abb32a9f57cbf7.jpg",
+        "text": "$$\n\\frac { \\partial \\tilde { Q } ^ { \\pi } ( s , a ) } { \\partial \\Sigma ( s ) } = \\frac { 1 } { 2 } \\cdot \\frac { \\partial ^ { 2 } \\tilde { Q } ^ { \\pi } ( s , a ) } { \\partial a ^ { 2 } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A proof of this identity is provided in the Appendix. The proof may be easily derived by expressing both sides of the equation using standard matrix calculus like $\\begin{array} { r } { \\frac { \\partial } { \\partial A } | A | ^ { - 1 / 2 } = - \\frac { 1 } { 2 } | A | ^ { - 1 / 2 } \\bar { A } ^ { - 1 } } \\end{array}$ and $\\begin{array} { r } { \\frac { \\partial } { \\partial A } | | v | | _ { A ^ { - 1 } } ^ { 2 } = - A ^ { - 1 } v v ^ { T } A ^ { - 1 } } \\end{array}$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Then, the full derivative w.r.t. $\\phi$ takes the form, ",
+        "bbox": [
+            174,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f41c8adcf7c19ddab43a7d258f46d084293a60a0cb7aa0628a9bb7df0f7b6d52.jpg",
+        "text": "$$\n\\nabla _ { \\phi } { \\cal O } _ { \\mathrm { E R } } ( \\pi _ { \\theta , \\phi } ) = \\frac { 1 } { 2 } \\int _ { S } \\frac { \\partial ^ { 2 } { \\tilde { Q } } ^ { \\pi } ( s , a ) } { \\partial a ^ { 2 } } \\big | _ { a = \\mu _ { \\theta } ( s ) } \\nabla _ { \\phi } \\Sigma _ { \\phi } ( s ) \\mathrm { d } \\rho ^ { \\pi } ( s ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 POLICY EVALUATION - OPTIMIZING $\\tilde { Q } _ { w } ^ { \\pi }$ GIVEN $( \\mu , \\Sigma )$ ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We can think of two ways to optimize $\\tilde { Q } _ { w } ^ { \\pi }$ . The first approach leverages (10) to update ${ \\tilde { Q } } ^ { \\pi }$ based on expected Q-value function $Q ^ { \\pi }$ . In such an approach, one trains a parameterized $Q _ { w } ^ { \\pi }$ to approximate the standard expected Q-value function $Q ^ { \\pi }$ using standard methods (see e.g., Rummery $\\&$ Niranjan (1994); Sutton & Barto (1998); Van Seijen et al. (2009)). Then, one fits $\\tilde { Q } _ { w } ^ { \\pi }$ based on $Q _ { w } ^ { \\pi }$ . In particular, given transitions $( s , a , r , s ^ { \\prime } )$ sampled from interactions with the environment, one can train $Q _ { w } ^ { \\pi }$ to minimize the Bellman error $( Q _ { w } ^ { \\pi } ( s , a ) - r - \\gamma Q _ { w } ^ { \\pi } ( s ^ { \\prime } , a ^ { \\prime } ) ) ^ { 2 }$ where $a ^ { \\prime } \\sim N ( \\mu ( s ^ { \\prime } ) , \\Sigma ( s ^ { \\prime } ) )$ . Then, $\\tilde { Q } _ { w } ^ { \\pi }$ can be optimized to minimize the squared error $( \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) - \\mathbb { E } _ { \\tilde { a } } Q _ { w } ^ { \\pi } ( s , \\tilde { a } ) ) ^ { 2 }$ where $\\tilde { a } \\sim$ $N ( a , \\Sigma ( s ) )$ , using several samples. When the target values in these residuals are treated as fixed (i.e., using a target network), such a training procedure will achieve a fixed point when $\\tilde { Q } _ { w } ^ { \\pi } ( s , a )$ satisfies the recursion in the Bellman equation (10). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The second approach requires a single function approximator for $\\tilde { Q } _ { w } ^ { \\pi } ( s , a )$ , resulting in a simpler implementation, and thus we use this approach in our experimental evaluation. Suppose one has access to a tuple $( s , \\tilde { a } , \\tilde { r } , \\tilde { s } ^ { \\prime } )$ sampled from a replay buffer with knowledge of the sampling probability $q ( \\tilde { a } \\mid s )$ (possibly unnormalized). Then assuming that this sampling distribution has a full support, we draw a phantom action $a \\sim N ( \\tilde { a } , \\Sigma ( s ) )$ and optimize $\\tilde { Q } _ { w } ^ { \\pi } ( s , a )$ by minimizing a weighted Bellman error $\\begin{array} { r } { \\frac { 1 } { q ( \\tilde { a } | s ) } ( \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) - \\tilde { r } - \\gamma \\tilde { Q } _ { w } ^ { \\pi } ( \\tilde { s } ^ { \\prime } , \\mu ( \\tilde { s } ^ { \\prime } ) ) ^ { 2 } } \\end{array}$ . For a specific pair of state and action $( s , a )$ the ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "expected value of the objective is, ",
+        "bbox": [
+            173,
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c76fa0ee66173ea378e379d523736869626d6fb2a9f21c00ecef6ee7a64e1c84.jpg",
+        "text": "$$\nE ( w \\mid ( s , a ) ) ~ = ~ \\mathbb { E } _ { q ( \\bar { a } \\mid s ) , \\bar { r } , \\bar { s } ^ { \\prime } } \\left[ \\frac { N ( a \\mid \\tilde { a } , \\Sigma ( s ) ) } { q ( \\tilde { a } \\mid s ) } ( \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) - \\tilde { r } - \\gamma \\tilde { Q } _ { w } ^ { \\pi } ( \\tilde { s } ^ { \\prime } , \\mu ( \\tilde { s } ^ { \\prime } ) ) ) ^ { 2 } \\right] ~ .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            207,
+            125,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that $N ( a | \\tilde { a } , \\Sigma ( s ) ) = N ( \\tilde { a } | a , \\Sigma ( s ) )$ . Therefore, when the target value $\\tilde { r } + \\gamma \\tilde { Q } _ { w } ^ { \\pi } ( \\tilde { s } ^ { \\prime } , \\mu ( \\tilde { s } ^ { \\prime } ) )$ is treated as fixed (e.g., when using target networks) this training procedure reaches an optimum when $\\tilde { Q } _ { w } ^ { \\pi } ( s , a )$ satisfies the recursion in the Bellman equation (13). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In practice, we find that it is unnecessary to keep track of the probabilities $q ( \\tilde { a } \\mid s )$ , and assume the replay buffer provides a near-uniform distribution of actions conditioned on states. Other recent work has also benefited from ignoring or heavily damping importance weights (Munos et al., 2016; Wang et al., 2017; Schulman et al., 2017). However, it is possible when interacting with the environment to save the probability of sampled actions along with their transitions, and thus have access to $q ( \\tilde { \\boldsymbol { a } } \\mid \\boldsymbol { s } ) \\approx N ( \\tilde { \\boldsymbol { a } } \\mid \\bar { \\mu } _ { \\mathrm { o l d } } ( \\boldsymbol { s } ) , \\dot { \\Sigma _ { \\mathrm { o l d } } } ( \\boldsymbol { s } ) )$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 PROXIMAL POLICY OPTIMIZATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            319,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Policy gradient algorithms are notoriously unstable, particularly in continuous control problems. Such instability has motivated the development of trust region methods that attempt to mitigate the issue by constraining each gradient step to lie within a trust region (Schulman et al., 2015), or augmenting the expected reward objective with a penalty on KL-divergence from a previous policy (Nachum et al., 2017b; Schulman et al., 2017; Azar et al., 2012). These stabilizing techniques have thus far not been applicable to algorithms like DDPG, since the policy is deterministic. The formulation we propose in this paper, however, is easily amenable to trust region optimization. Specifically, we may augment the objective (11) with a penalty ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ce671f74d9ac9a165e85048a8cd34aa0136ad30376cb6d7e13762ea2132990ef.jpg",
+        "text": "$$\n{ \\cal O } _ { \\mathrm { T R } } ( \\pi ) = { \\cal O } _ { \\mathrm { E R } } ( \\pi ) - \\lambda \\int _ { \\cal S } \\mathrm { K L } ( \\pi \\parallel \\pi _ { \\mathrm { o l d } } ) \\mathrm { d } \\rho ^ { \\pi } ( s ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\pi _ { \\mathrm { o l d } } \\equiv \\left( \\mu _ { \\mathrm { o l d } } , \\Sigma _ { \\mathrm { o l d } } \\right)$ is a previous parameterization of the policy. The optimization is straightforward, since the KL-divergence of two Gaussians can be expressed analytically. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            550,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This paper follows a long line of work that uses Q-value functions to stably learn a policy, which in the past has been used to either approximate expected (Rummery & Niranjan, 1994; Van Seijen et al., 2009; Gu et al., 2017) or optimal (Watkins, 1989; Silver et al., 2014; Nachum et al., 2017a; Haarnoja et al., 2017; Metz et al., 2017) future value. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Work that is most similar to what we present are methods that exploit gradient information from the Q-value function to train a policy. Deterministic policy gradient (Silver et al., 2014) is perhaps the best known of these. The method we propose can be interpreted as a generalization of the deterministic policy gradient. Indeed, if one takes the limit of the policy covariance $\\Sigma ( s )$ as it goes to 0, the proposed Q-value function becomes the deterministic value function of DDPG, and the updates for training the Q-value approximator and the policy mean are identical. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Stochastic Value Gradient (SVG) (Heess et al., 2015) also trains stochastic policies using an update that is similar to DDPG (i.e., SVG(0) with replay). The key differences with our approach are that SVG does not provide an update for the covariance, and the mean update in SVG estimates the gradient with a noisy Monte Carlo sample, which we avoid by estimating the smoothed $\\mathrm { Q }$ -value function. Although a covariance update could be derived using the same reparameterization trick as in the mean update, that would also require a noisy Monte Carlo estimate. Methods for updating the covariance along the gradient of expected reward are essential for applying the subsequent trust region and proximal policy techniques. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "More recently, Ciosek & Whiteson (2017) introduced expected policy gradients (EPG), a generalization of DDPG that provides updates for the mean and covariance of a stochastic Gaussian policy using gradients of an estimated Q-value function. In that work, the expected Q-value used in standard policy gradient algorithms such as SARSA (Sutton & Barto, 1998; Rummery & Niranjan, 1994; Van Seijen et al., 2009) is estimated. The updates in EPG therefore require approximating an integral of the expected Q-value function. Our analogous process directly estimates an integral (via the smoothed Q-value function) and avoids approximate integrals, thereby making the updates simpler. Moreover, while Ciosek & Whiteson (2017) rely on a quadratic Taylor expansion of the estimated Q-value function, we instead rely on the strength of neural network function approximators to directly estimate the smoothed Q-value function. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The novel training scheme we propose for learning the covariance of a Gaussian policy relies on properties of Gaussian integrals (Bonnet, 1964; Price, 1958). Similar identities have been used in the past to derive updates for variational auto-encoders (Kingma & Welling, 2014) and Gaussian back-propagation (Rezende et al., 2014). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Finally, the perspective presented in this paper, where Q-values represent the averaged return of a distribution of actions rather than a single action, is distinct from recent advances in distributional RL (Bellemare et al., 2017). Those approaches focus on the distribution of returns of a single action, whereas we consider the single average return of a distribution of actions. Although we restrict our attention in this paper to Gaussian policies, an interesting topic for further investigation is to study the applicability of this new perspective to a wider class of policy distributions. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We utilize the insights from Section 3 to introduce a new RL algorithm, Smoothie. Smoothie maintains a parameterized $\\tilde { Q } _ { w } ^ { \\pi }$ trained via the procedure described in Section 3.2. It then uses the gradient and Hessian of this approximation to train a Gaussian policy $\\mu _ { \\theta } , \\Sigma _ { \\phi }$ using the updates stated in (14) and (16). See Algorithm 1 for a simplified pseudocode of our algorithm. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Smoothie ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/eb3fa1f9a495240797e692645eace753d0aebfc6f323140585dd7b69aaea1d55.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm1Smoothie Input: Environment ENV,learning rates 7π, nQ,discount factor y,KL-penalty 入,l</td></tr><tr><td>number of training steps N, target network lag T. Initialize0,,w,set0&#x27;=0,Φ&#x27;=,w&#x27;=w.</td></tr><tr><td>fori=OtoN-1do</td></tr><tr><td>/ Collect experience</td></tr><tr><td>Sample action a ~ N(μe(s),∑(s)) and apply to ENV to yield r and s&#x27;. Insert transition (s,a,r,s&#x27;) to replay buffer.</td></tr><tr><td></td></tr><tr><td>// Train μ,∑</td></tr><tr><td>Sample batch{(k,@k,Tk,S)}1 fromreplay buffer aQ(ska) Compute gradients gk =</td></tr><tr><td>da la=μe(sk)*</td></tr><tr><td>²(s@） Compute Hessians Hk = a² la=μe(sk)*</td></tr><tr><td>Compute KL-penalties KLk =KL(μθ,Σ𝜙llμe,Σ).</td></tr><tr><td>Compute updates 1B</td></tr><tr><td>1B</td></tr><tr><td>△=B 1 ∑k=1 Updateθ←θ+ηπ△0,←+ηπ△.</td></tr><tr><td></td></tr><tr><td>// Train Qπ</td></tr><tr><td>Sample batch {(sk,k,Tk,S)}1 from replay bufer. Sample phantom actions ak ~ N(ák,Σ(sk)). Compute lossL(u)=∑1((s)-r-qQ(s,ue()2.</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We perform a number of evaluations of Smoothie compared to DDPG. We choose DDPG as a baseline because it (1) utilizes gradient information of a Q-value approximator, much like our algorithm; and (2) is a standard algorithm well-known to have achieve good, sample-efficient performance on continuous control benchmarks. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 A SYNTHETIC TASK ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To evaluate Smoothie we begin with a simple synthetic task which allows us to study its behavior in a restricted setting. We devised a simple single-action one-shot environment in which the reward function is a mixture of two Gaussians, one better than the other (see Figure 1 (Right)). We initialize the policy mean to be centered on the worse of the two Gaussians. We plot the learnable policy mean and standard deviation during training for Smoothie and DDPG in Figure 1 (Left). Smoothie learns both the mean and variance, while DDPG learns only the mean and the variance plotted is the exploratory noise, whose scale is kept fixed during training. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As expected we observe that DDPG cannot escape the local optimum. At the beginning of training it exhibits some movement away from the local optimum (likely due to the initial noisy approximation given by $Q _ { w } ^ { \\pi } \\mathrm { . }$ ), it is unable to progress very far from the initial mean. Note that this is not an issue of exploration. The exploration scale is high enough that $Q _ { w } ^ { \\pi }$ is aware of the better Gaussian. The issue is in the update for $\\mu _ { \\theta }$ , which is only with regard to the derivative of $Q _ { w } ^ { \\pi }$ at the current mean. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "On the other hand, we find Smoothie is successfully able to solve the task. This is because the smoothed reward function approximated by $\\tilde { Q } _ { w } ^ { \\pi }$ has a derivative which clearly points $\\mu _ { \\theta }$ towards the better Gaussian. We also observe that Smoothie is able to suitably adjust the covariance $\\Sigma _ { \\phi }$ during training. Initially, $\\Sigma _ { \\phi }$ decreases due to the concavity of the smoothed reward function. As a region of convexity is entered, it begins to increase, before again decreasing to near-zero as $\\mu _ { \\theta }$ approaches the global optimum. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/3e62a205722715b4eae018cb617c9164cd435c130be75b6bf1c3addc835c777e.jpg",
+        "image_caption": [
+            "Figure 1: Left: The learnable policy mean and standard deviation during training for Smoothie and DDPG on a simple one-shot synthetic task. The standard deviation for DDPG is the exploratory noise kept constant during training. Right: The reward function for the synthetic task along with its Gaussian-smoothed version. We find that Smoothie can successfully escape the lower-reward local optimum. We also notice Smoothie increases and decreases its policy variance as the convexity/concavity of the smoothed reward function changes. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 CONTINUOUS CONTROL ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We now turn our attention to standard continuous control benchmarks available on OpenAI Gym (Brockman et al., 2016) utilizing the MuJoCo environment (Todorov et al., 2012). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Our implementations utilize feed forward neural networks for policy and Q-values. We parameterize the covariance $\\Sigma _ { \\phi }$ as a diagonal given by $e ^ { \\phi }$ . The exploration for DDPG is determined by an Ornstein-Uhlenbeck process (Uhlenbeck & Ornstein, 1930; Lillicrap et al., 2016). Additional implementation details are provided in the Appendix. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/f21887a3716c42d41ffbbba35b3c52cf13877d7b56bb86c24b7339f4a95daf67.jpg",
+        "image_caption": [
+            "Figure 2: Results of Smoothie, DDPG, and TRPO on continuous control benchmarks. The $\\mathbf { X }$ -axis is in millions of environment steps. Each plot shows the average reward and standard deviation clipped at the min and max of six randomly seeded runs after choosing best hyperparameters. We see that Smoothie is competitive with DDPG even when DDPG uses a hyperparameter-tuned noise scale, and Smoothie learns the optimal noise scale (the covariance) during training. Moreoever, we observe significant advantages in terms of final reward performance, especially in the more difficult tasks like Hopper, Walker2d, and Humanoid. Across all tasks, TRPO is not sufficiently sampleefficient to provide a competitive baseline. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We compare the results of Smoothie and DDPG in Figure 2. For each task we performed a hyperparameter search over actor learning rate, critic learning rate and reward scale, and plot the average of six runs for the best hyperparameters. For DDPG we extended the hyperparameter search to also consider the scale and damping of exploratory noise provided by the Ornstein-Uhlenbeck process. Smoothie, on the other hand, contains an additional hyperparameter to determine the weight on KL-penalty. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Despite DDPG having the advantage of its exploration decided by a hyperparameter search while Smoothie must learn its exploration without supervision, we find that Smoothie performs competitively or better across all tasks, exhibiting a slight advantage in Swimmer and Ant, while showing more dramatic improvements in Hopper, Walker2d, and Humanoid. The improvement is especially dramatic for Hopper, where the average reward is doubled. We also highlight the results for Humanoid, which as far as we know, are the best published results for a method that only trains on the order of millions of environment steps. In contrast, TRPO, which to the best of our knowledge is the only other algorithm which can achieve better performance, requires on the order of tens of millions of environment steps to achieve comparable reward. This gives added evidence to the benefits of using a learnable covariance and not restricting a policy to be deterministic. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Empirically, we found the introduction of a KL-penalty to improve performance of Smoothie, especially on harder tasks. We present a comparison of results of Smoothie with and without the KL-penalty on the four harder tasks in Figure 3. A KL-penalty to encourage stability is not possible in DDPG. Thus, our algorithm provides a much needed solution to the inherent instability in DDPG training. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/31e9c2cf700c5e3ed94cc8e3ab513890a7ad5439462138652b2f32dd4b21ec5f.jpg",
+        "image_caption": [
+            "Figure 3: Results of Smoothie with and without a KL-penalty. The $\\mathbf { X } ^ { } -$ -axis is in millions of environment steps. We observe benefits of using a proximal policy optimization method, especially in Hopper and Humanoid, where the performance improvement is significant without sacrificing sample efficiency. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have presented a new Q-value function, ${ \\tilde { Q } } ^ { \\pi }$ , that is a Gaussian-smoothed version of the standard expected Q-value, $Q ^ { \\pi }$ . The advantage of using ${ \\tilde { Q } } ^ { \\pi }$ over $Q ^ { \\pi }$ is that its gradient and Hessian possess an intimate relationship with the gradient of expected reward with respect to mean and covariance of a Gaussian policy. The resulting algorithm, Smoothie, is able to successfully learn both mean and covariance during training, leading to performance that can match or surpass that of DDPG, especially when incorporating a penalty on divergence from a previous policy. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The success of ${ \\tilde { Q } } ^ { \\pi }$ is encouraging. Intuitively it may be argued that learning ${ \\tilde { Q } } ^ { \\pi }$ is more sensible than learning $Q ^ { \\pi }$ . The smoothed Q-values by definition make the true reward surface smoother, thus possibly easier to learn; moreover the smoothed Q-values have a more direct relationship with the expected discounted return objective. We encourage future work to further investigate these claims as well as techniques to apply the underlying motivations for ${ \\tilde { Q } } ^ { \\pi }$ to other types of policies. ",
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+        "page_idx": 8
+    },
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+            162
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The specific identity we state may be derived using standard matrix calculus. We make use of the fact that ",
+        "bbox": [
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+            167,
+            823,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/abe39c2a97779def484e4eba142c60209fd9e9f36be5544140f1ba14e1bd2e4a.jpg",
+        "text": "$$\n\\frac { \\partial } { \\partial A } | A | ^ { - 1 / 2 } = - \\frac { 1 } { 2 } | A | ^ { - 3 / 2 } \\frac { \\partial } { \\partial A } | A | = - \\frac { 1 } { 2 } | A | ^ { - 1 / 2 } A ^ { - 1 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            191,
+            678,
+            223
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "and for symmetric $A$ , ",
+        "bbox": [
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+            223,
+            316,
+            238
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/18034b9d4af3f0775f1fdad3b9f4dd9d28f0abf8f059744b1cc914ba9d87feaf.jpg",
+        "text": "$$\n\\frac \\partial { \\partial A } | | \\boldsymbol { v } | | _ { A ^ { - 1 } } ^ { 2 } = - A ^ { - 1 } { \\boldsymbol { v } } { \\boldsymbol { v } } ^ { T } A ^ { - 1 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            236,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We omit $s$ from $\\Sigma ( s )$ in the following equations for succinctness. The LHS of (15) is ",
+        "bbox": [
+            173,
+            267,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6b52af45a4ad5887c919cf42e53a40ce44cfb2c6eb63716e3df1ab2d181c4ad4.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\int _ { A } Q ^ { \\pi } ( s , \\tilde { a } ) \\frac { \\partial } { \\partial \\Sigma } N ( \\tilde { a } | a , \\Sigma ) \\mathrm { d } \\tilde { a } } \\\\ { \\displaystyle = \\int _ { A } Q ^ { \\pi } ( s , \\tilde { a } ) \\exp \\left\\{ - \\frac { 1 } { 2 } | | \\tilde { a } - a | | _ { \\Sigma ^ { - 1 } } ^ { 2 } \\right\\} \\left( \\frac { \\partial } { \\partial \\Sigma } | 2 \\pi \\Sigma | ^ { - 1 / 2 } - \\frac { 1 } { 2 } | 2 \\pi \\Sigma | ^ { - 1 / 2 } \\frac { \\partial } { \\partial \\Sigma } | | \\tilde { a } - a | | _ { \\Sigma ^ { - 1 } } ^ { 2 } \\right) \\mathrm { d } \\tilde { a } } \\\\ { \\displaystyle \\qquad = \\frac { 1 } { 2 } \\int _ { A } Q ^ { \\pi } ( s , \\tilde { a } ) N ( \\tilde { a } | a , \\Sigma ) \\left( - \\Sigma ^ { - 1 } + \\Sigma ^ { - 1 } ( \\tilde { a } - a ) ( \\tilde { a } - a ) ^ { T } \\Sigma ^ { - 1 } \\right) \\mathrm { d } \\tilde { a } . \\quad ( 2 \\pi \\Sigma ) \\mathrm { d } \\tilde { a } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            191,
+            290,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Meanwhile, towards tackling the RHS of (15) we note that ",
+        "bbox": [
+            173,
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+            560,
+            420
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/97ad7296a6d1cc5351d9efa7676e3b98d2abdfd6dc40f2d5825b5cc8fba010c2.jpg",
+        "text": "$$\n\\frac { \\partial \\tilde { Q } ^ { \\pi } ( s , a ) } { \\partial a } = \\int _ { \\cal A } Q ^ { \\pi } ( s , \\tilde { a } ) N ( \\tilde { a } | a , \\Sigma ) \\Sigma ^ { - 1 } ( \\tilde { a } - a ) \\mathrm { d } \\tilde { a } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            422,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Thus we have ",
+        "bbox": [
+            174,
+            463,
+            267,
+            478
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f2b6236837facdae824c64e99a49ae86f2e0eb661f82effd25f97f310e1950d1.jpg",
+        "text": "$$\n\\frac { \\partial ^ { 2 } \\tilde { Q } ^ { \\pi } ( s , a ) } { \\partial a ^ { 2 } } = \\int _ { \\mathcal { A } } Q ^ { \\pi } ( s , \\tilde { a } ) \\left( \\Sigma ^ { - 1 } ( \\tilde { a } - a ) \\frac { \\partial } { \\partial a } N ( \\tilde { a } | a , \\Sigma ) + N ( \\tilde { a } | a , \\Sigma ) \\frac { \\partial } { \\partial a } \\Sigma ^ { - 1 } ( \\tilde { a } - a ) \\right) \\mathrm { d } \\tilde { a } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            191,
+            482,
+            779,
+            517
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/930d5d06c21a4bd3186c1ee7d89ae0dba775aea5ad5fb4f3ec4efe3838e6785c.jpg",
+        "text": "$$\n= \\int _ { \\mathcal { A } } Q ^ { \\pi } ( s , \\tilde { a } ) N ( \\tilde { a } | a , \\Sigma ) ( \\Sigma ^ { - 1 } ( \\tilde { a } - a ) ( \\tilde { a } - a ) ^ { T } \\Sigma ^ { - 1 } - \\Sigma ^ { - 1 } ) \\mathrm { d } \\tilde { a } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            522,
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+            556
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B COMPATIBLE FUNCTION APPROXIMATION ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            590,
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+            608
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A function approximator $\\tilde { Q } _ { w } ^ { \\pi }$ of ${ \\tilde { Q } } ^ { \\pi }$ should be sufficiently accurate so that updates for $\\mu _ { \\theta } , \\Sigma _ { \\phi }$ are not affected by substituting $\\frac { \\partial \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) } { \\partial a }$ and $\\frac { \\partial ^ { 2 } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) } { \\partial a ^ { 2 } }$ for $\\frac { \\partial \\tilde { Q } ^ { \\pi } ( s , a ) } { \\partial a }$ and $\\frac { \\partial ^ { 2 } \\tilde { Q } ^ { \\pi } ( s , a ) } { \\partial a ^ { 2 } }$ , respectively. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We claim that a $\\tilde { Q } _ { w } ^ { \\pi }$ is compatible with respect to $\\mu _ { \\theta }$ if ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "1 $\\left. \\nabla _ { a } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) \\right| _ { a = \\mu _ { \\theta } ( s ) } = \\nabla _ { \\theta } \\mu _ { \\theta } ( s ) ^ { T } w ,$   \n2. $\\begin{array} { r } { \\nabla _ { w } \\int _ { \\mathcal { S } } \\Big ( \\nabla _ { a } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } - \\nabla _ { a } \\tilde { Q } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } \\Big ) ^ { 2 } \\mathrm { d } \\rho ^ { \\pi } ( s ) = 0 } \\end{array}$ (i.e., $w$ minimizes the expected squared error of the gradients). ",
+        "bbox": [
+            210,
+            690,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Additionally, $\\tilde { Q } _ { w } ^ { \\pi }$ is compatible with respect to $\\Sigma _ { \\phi }$ if ",
+        "bbox": [
+            174,
+            767,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "1. $\\nabla _ { a } ^ { 2 } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } = \\nabla _ { \\phi } \\Sigma _ { \\phi } ( s ) ^ { T } w ,$   \n2. $\\begin{array} { r } { \\nabla _ { w } \\int _ { \\mathcal { S } } \\left( \\nabla _ { a } ^ { 2 } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) \\big \\vert _ { a = \\mu _ { \\theta } ( s ) } - \\nabla _ { a } ^ { 2 } \\tilde { Q } ^ { \\pi } ( s , a ) \\big \\vert _ { a = \\mu _ { \\theta } ( s ) } \\right) ^ { 2 } \\mathrm { d } \\rho ^ { \\pi } ( s ) = 0 } \\end{array}$ (i.e., $w$ minimizes the expected squared error of the Hessians). ",
+        "bbox": [
+            210,
+            795,
+            825,
+            861
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "One possible parameterization of $\\tilde { Q } _ { w } ^ { \\pi }$ may be achieved by taking $w = [ w _ { 0 } , w _ { 1 } , w _ { 2 } ]$ and parameterizing ",
+        "bbox": [
+            174,
+            871,
+            823,
+            902
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b8e0b8ed230238272a76a4d3fc82242545e3918890d0fbb8ad9f69a4b959e7a1.jpg",
+        "text": "$$\n\\begin{array} { r } { \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) = V _ { w _ { 0 } } ( s ) + ( a - \\mu _ { \\theta } ( s ) ) ^ { T } \\nabla _ { \\theta } \\mu _ { \\theta } ( s ) ^ { T } w _ { 1 } + ( a - \\mu _ { \\theta } ( s ) ) ^ { T } \\nabla _ { \\phi } \\Sigma _ { \\phi } ( s ) ^ { T } w _ { 2 } ( a - \\mu _ { \\theta } ( s ) ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            179,
+            906,
+            789,
+            926
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/c90628bc28b19c55096f492408d94d5ecf169eef242d75e9f07e24db6cf27564.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter</td><td>Range</td><td>Sampling</td></tr><tr><td>actor learningrate critic learning rate</td><td>[1e-6,1e-3] [1e-6,1e-3]</td><td>log</td></tr><tr><td>reward scale</td><td>[0.01,0.3]</td><td>l0g</td></tr><tr><td>OU damping</td><td>[1e-4,1e-3]</td><td>l0g</td></tr><tr><td>OU stddev</td><td></td><td>10g</td></tr><tr><td>入</td><td>[1e-3,1.0]</td><td>log</td></tr><tr><td>discount factor</td><td>[1e-6, 4e-2] 0.995</td><td>log</td></tr><tr><td>target network lag</td><td>0.01</td><td>fixed</td></tr><tr><td>batch size</td><td>128</td><td>fixed</td></tr><tr><td>clipping on gradients of Q</td><td></td><td>fixed</td></tr><tr><td></td><td>4.0</td><td>fixed</td></tr><tr><td>num gradient updates per observation</td><td>1</td><td>fixed</td></tr><tr><td>Huber loss clipping</td><td>1.0</td><td>fixed</td></tr></table>",
+        "bbox": [
+            271,
+            101,
+            722,
+            285
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proof. We shall show how the conditions stated for compatibility with respect to $\\Sigma _ { \\phi }$ are sufficient. The reasoning for $\\mu _ { \\theta }$ follows via a similar argument. We also refer the reader to Silver et al. (2014) which includes a similar procedure for showing compatibility. ",
+        "bbox": [
+            173,
+            348,
+            825,
+            391
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "From the second condition for compatibility with respect to $\\Sigma _ { \\phi }$ we have ",
+        "bbox": [
+            174,
+            397,
+            648,
+            412
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/482a65447c1c827eb4abc41f71898558ca37abec243b140aaac7a7ec6b1504e3.jpg",
+        "text": "$$\n\\int _ { S } \\left( \\nabla _ { a } ^ { 2 } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } - \\nabla _ { a } ^ { 2 } \\tilde { Q } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } \\right) \\nabla _ { w } \\left( \\nabla _ { a } ^ { 2 } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } \\right) \\mathrm { d } \\rho ^ { \\pi } ( s ) = 0 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            183,
+            420,
+            784,
+            452
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We may combine this with the first condition to find ",
+        "bbox": [
+            174,
+            458,
+            517,
+            473
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fead4ffc0f1712d604674150fb03c9293a85fb979a783002ea72c59f0f461aa1.jpg",
+        "text": "$$\n\\int _ { S } \\nabla _ { a } ^ { 2 } \\tilde { Q } _ { w } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } \\nabla _ { \\phi } \\Sigma _ { \\phi } ( s ) \\mathrm { d } \\rho ^ { \\pi } ( s ) = \\int _ { S } \\nabla _ { a } ^ { 2 } \\tilde { Q } ^ { \\pi } ( s , a ) \\big | _ { a = \\mu _ { \\theta } ( s ) } \\nabla _ { \\phi } \\Sigma _ { \\phi } ( s ) \\mathrm { d } \\rho ^ { \\pi } ( s ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            202,
+            479,
+            764,
+            513
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "which is the desired property for compatibility. ",
+        "bbox": [
+            176,
+            518,
+            482,
+            534
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "While it is reassuring to know that there exists a class of function approximators which are compatible, this fact is largely ignored in practice. Not only is the class of compatible functions heavily restricted in terms of expressiveness, due to the first set of conditions for $\\mu _ { \\theta } , \\Sigma _ { \\phi }$ , it is also impossible to satisfy the second set of conditions without access to derivative and Hessian information of the true ${ \\tilde { Q } } ^ { \\pi }$ . This problem is also present in DDPG, and we feel this issue merits additional investigation in future work. ",
+        "bbox": [
+            173,
+            540,
+            825,
+            626
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            646,
+            439,
+            662
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We utilize feed forward networks for both policy and Q-value approximator. For $\\mu _ { \\boldsymbol { \\theta } } ( s )$ we use two hidden layers of dimensions (400, 300) and relu activation functions. For $\\tilde { Q } _ { w } ^ { \\pi } ( s , a )$ and $Q _ { w } ^ { \\pi } ( s , a )$ we first embed the state into a 400 dimensional vector using a fully-connected layer and tanh nonlinearity. We then concatenate the embedded state with $a$ and pass the result through a 1-hidden layer neural network of dimension 300 with tanh activations. We use a diagonal $\\Sigma _ { \\phi } ^ { \\bar { } } ( s ) = e ^ { \\phi }$ for Smoothie, with $\\phi$ initialized to $- 1$ . ",
+        "bbox": [
+            174,
+            676,
+            825,
+            763
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "To find optimal hyperparameters we perform a 100-trial random search over the hyperparameters specified in Table 1. The OU exploration parameters only apply to DDPG. The $\\lambda$ coefficient on KL-penalty only applies to Smoothie with a KL-penalty. ",
+        "bbox": [
+            174,
+            770,
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+            814
+        ],
+        "page_idx": 11
+    }
+]