1 Introduction
Modelfree deep reinforcement learning (RL) algorithms have been applied in a range of challenging domains, from games (Mnih et al., 2013; Silver et al., 2016) to robotic control (Schulman et al., 2015)
. The combination of RL and highcapacity function approximators such as neural networks holds the promise of automating a wide range of decision making and control tasks, but widespread adoption of these methods in realworld domains has been hampered by two major challenges. First, modelfree deep RL methods are notoriously expensive in terms of their sample complexity. Even relatively simple tasks can require millions of steps of data collection, and complex behaviors with highdimensional observations might need substantially more. Second, these methods are often brittle with respect to their hyperparameters: learning rates, exploration constants, and other settings must be set carefully for different problem settings to achieve good results. Both of these challenges severely limit the applicability of modelfree deep RL to realworld tasks.
One cause for the poor sample efficiency of deep RL methods is onpolicy learning: some of the most commonly used deep RL algorithms, such as TRPO (Schulman et al., 2015), PPO (Schulman et al., 2017b) or A3C (Mnih et al., 2016), require new samples to be collected for each gradient step. This quickly becomes extravagantly expensive, as the number of gradient steps and samples per step needed to learn an effective policy increases with task complexity. Offpolicy algorithms aim to reuse past experience. This is not directly feasible with conventional policy gradient formulations, but is relatively straightforward for Qlearning based methods (Mnih et al., 2015). Unfortunately, the combination of offpolicy learning and highdimensional, nonlinear function approximation with neural networks presents a major challenge for stability and convergence (Bhatnagar et al., 2009). This challenge is further exacerbated in continuous state and action spaces, where a separate actor network is often used to perform the maximization in Qlearning. A commonly used algorithm in such settings, deep deterministic policy gradient (DDPG) (Lillicrap et al., 2015), provides for sampleefficient learning but is notoriously challenging to use due to its extreme brittleness and hyperparameter sensitivity (Duan et al., 2016; Henderson et al., 2017).
We explore how to design an efficient and stable modelfree deep RL algorithm for continuous state and action spaces. To that end, we draw on the maximum entropy framework, which augments the standard maximum reward reinforcement learning objective with an entropy maximization term (Ziebart et al., 2008; Toussaint, 2009; Rawlik et al., 2012; Fox et al., 2016; Haarnoja et al., 2017). Maximum entropy reinforcement learning alters the RL objective, though the original objective can be recovered using a temperature parameter (Haarnoja et al., 2017). More importantly, the maximum entropy formulation provides a substantial improvement in exploration and robustness: as discussed by Ziebart (2010)
, maximum entropy policies are robust in the face of model and estimation errors, and as demonstrated by
(Haarnoja et al., 2017), they improve exploration by acquiring diverse behaviors. Prior work has proposed modelfree deep RL algorithms that perform onpolicy learning with entropy maximization (O’Donoghue et al., 2016), as well as offpolicy methods based on soft Qlearning and its variants (Schulman et al., 2017a; Nachum et al., 2017a; Haarnoja et al., 2017). However, the onpolicy variants suffer from poor sample complexity for the reasons discussed above, while the offpolicy variants require complex approximate inference procedures in continuous action spaces.In this paper, we demonstrate that we can devise an offpolicy maximum entropy actorcritic algorithm, which we call soft actorcritic (SAC), which provides for both sampleefficient learning and stability. This algorithm extends readily to very complex, highdimensional tasks, such as the Humanoid benchmark (Duan et al., 2016) with 21 action dimensions, where offpolicy methods such as DDPG typically struggle to obtain good results (Gu et al., 2016). SAC also avoids the complexity and potential instability associated with approximate inference in prior offpolicy maximum entropy algorithms based on soft Qlearning (Haarnoja et al., 2017). We present a convergence proof for policy iteration in the maximum entropy framework, and then introduce a new algorithm based on an approximation to this procedure that can be practically implemented with deep neural networks, which we call soft actorcritic. We present empirical results that show that soft actorcritic attains a substantial improvement in both performance and sample efficiency over both offpolicy and onpolicy prior methods. We also compare to twin delayed deep deterministic (TD3) policy gradient algorithm (Fujimoto et al., 2018), which is a concurrent work that proposes a deterministic algorithm that substantially improves on DDPG.
2 Related Work
Our soft actorcritic algorithm incorporates three key ingredients: an actorcritic architecture with separate policy and value function networks, an offpolicy formulation that enables reuse of previously collected data for efficiency, and entropy maximization to enable stability and exploration. We review prior works that draw on some of these ideas in this section. Actorcritic algorithms are typically derived starting from policy iteration, which alternates between policy evaluation—computing the value function for a policy—and policy improvement—using the value function to obtain a better policy (Barto et al., 1983; Sutton & Barto, 1998). In largescale reinforcement learning problems, it is typically impractical to run either of these steps to convergence, and instead the value function and policy are optimized jointly. In this case, the policy is referred to as the actor, and the value function as the critic. Many actorcritic algorithms build on the standard, onpolicy policy gradient formulation to update the actor (Peters & Schaal, 2008), and many of them also consider the entropy of the policy, but instead of maximizing the entropy, they use it as an regularizer (Schulman et al., 2017b, 2015; Mnih et al., 2016; Gruslys et al., 2017). Onpolicy training tends to improve stability but results in poor sample complexity.
There have been efforts to increase the sample efficiency while retaining robustness by incorporating offpolicy samples and by using higher order variance reduction techniques
(O’Donoghue et al., 2016; Gu et al., 2016). However, fully offpolicy algorithms still attain better efficiency. A particularly popular offpolicy actorcritic method, DDPG (Lillicrap et al., 2015), which is a deep variant of the deterministic policy gradient (Silver et al., 2014) algorithm, uses a Qfunction estimator to enable offpolicy learning, and a deterministic actor that maximizes this Qfunction. As such, this method can be viewed both as a deterministic actorcritic algorithm and an approximate Qlearning algorithm. Unfortunately, the interplay between the deterministic actor network and the Qfunction typically makes DDPG extremely difficult to stabilize and brittle to hyperparameter settings (Duan et al., 2016; Henderson et al., 2017). As a consequence, it is difficult to extend DDPG to complex, highdimensional tasks, and onpolicy policy gradient methods still tend to produce the best results in such settings (Gu et al., 2016). Our method instead combines offpolicy actorcritic training with a stochastic actor, and further aims to maximize the entropy of this actor with an entropy maximization objective. We find that this actually results in a considerably more stable and scalable algorithm that, in practice, exceeds both the efficiency and final performance of DDPG. A similar method can be derived as a zerostep special case of stochastic value gradients (SVG(0)) (Heess et al., 2015). However, SVG(0) differs from our method in that it optimizes the standard maximum expected return objective, and it does not make use of a separate value network, which we found to make training more stable.Maximum entropy reinforcement learning optimizes policies to maximize both the expected return and the expected entropy of the policy. This framework has been used in many contexts, from inverse reinforcement learning (Ziebart et al., 2008) to optimal control (Todorov, 2008; Toussaint, 2009; Rawlik et al., 2012). In guided policy search (Levine & Koltun, 2013; Levine et al., 2016), the maximum entropy distribution is used to guide policy learning towards highreward regions. More recently, several papers have noted the connection between Qlearning and policy gradient methods in the framework of maximum entropy learning (O’Donoghue et al., 2016; Haarnoja et al., 2017; Nachum et al., 2017a; Schulman et al., 2017a). While most of the prior modelfree works assume a discrete action space, Nachum et al. (2017b) approximate the maximum entropy distribution with a Gaussian and Haarnoja et al. (2017) with a sampling network trained to draw samples from the optimal policy. Although the soft Qlearning algorithm proposed by Haarnoja et al. (2017) has a value function and actor network, it is not a true actorcritic algorithm: the Qfunction is estimating the optimal Qfunction, and the actor does not directly affect the Qfunction except through the data distribution. Hence, Haarnoja et al. (2017) motivates the actor network as an approximate sampler, rather than the actor in an actorcritic algorithm. Crucially, the convergence of this method hinges on how well this sampler approximates the true posterior. In contrast, we prove that our method converges to the optimal policy from a given policy class, regardless of the policy parameterization. Furthermore, these prior maximum entropy methods generally do not exceed the performance of stateoftheart offpolicy algorithms, such as DDPG, when learning from scratch, though they may have other benefits, such as improved exploration and ease of finetuning. In our experiments, we demonstrate that our soft actorcritic algorithm does in fact exceed the performance of prior stateoftheart offpolicy deep RL methods by a wide margin.
3 Preliminaries
We first introduce notation and summarize the standard and maximum entropy reinforcement learning frameworks.
3.1 Notation
We address policy learning in continuous action spaces. We consider an infinitehorizon Markov decision process (MDP), defined by the tuple
, where the state space and the action spaceare continuous, and the unknown state transition probability
represents the probability density of the next state given the current state and action . The environment emits a bounded reward on each transition. We will use and to denote the state and stateaction marginals of the trajectory distribution induced by a policy .3.2 Maximum Entropy Reinforcement Learning
Standard RL maximizes the expected sum of rewards . We will consider a more general maximum entropy objective (see e.g. Ziebart (2010)), which favors stochastic policies by augmenting the objective with the expected entropy of the policy over :
(1) 
The temperature parameter determines the relative importance of the entropy term against the reward, and thus controls the stochasticity of the optimal policy. The maximum entropy objective differs from the standard maximum expected reward objective used in conventional reinforcement learning, though the conventional objective can be recovered in the limit as . For the rest of this paper, we will omit writing the temperature explicitly, as it can always be subsumed into the reward by scaling it by .
This objective has a number of conceptual and practical advantages. First, the policy is incentivized to explore more widely, while giving up on clearly unpromising avenues. Second, the policy can capture multiple modes of nearoptimal behavior. In problem settings where multiple actions seem equally attractive, the policy will commit equal probability mass to those actions. Lastly, prior work has observed improved exploration with this objective (Haarnoja et al., 2017; Schulman et al., 2017a), and in our experiments, we observe that it considerably improves learning speed over stateofart methods that optimize the conventional RL objective function. We can extend the objective to infinite horizon problems by introducing a discount factor to ensure that the sum of expected rewards and entropies is finite. Writing down the maximum entropy objective for the infinite horizon discounted case is more involved (Thomas, 2014) and is deferred to Appendix A.
Prior methods have proposed directly solving for the optimal Qfunction, from which the optimal policy can be recovered (Ziebart et al., 2008; Fox et al., 2016; Haarnoja et al., 2017). We will discuss how we can devise a soft actorcritic algorithm through a policy iteration formulation, where we instead evaluate the Qfunction of the current policy and update the policy through an offpolicy gradient update. Though such algorithms have previously been proposed for conventional reinforcement learning, our method is, to our knowledge, the first offpolicy actorcritic method in the maximum entropy reinforcement learning framework.
4 From Soft Policy Iteration to Soft ActorCritic
Our offpolicy soft actorcritic algorithm can be derived starting from a maximum entropy variant of the policy iteration method. We will first present this derivation, verify that the corresponding algorithm converges to the optimal policy from its density class, and then present a practical deep reinforcement learning algorithm based on this theory.
4.1 Derivation of Soft Policy Iteration
We will begin by deriving soft policy iteration, a general algorithm for learning optimal maximum entropy policies that alternates between policy evaluation and policy improvement in the maximum entropy framework. Our derivation is based on a tabular setting, to enable theoretical analysis and convergence guarantees, and we extend this method into the general continuous setting in the next section. We will show that soft policy iteration converges to the optimal policy within a set of policies which might correspond, for instance, to a set of parameterized densities.
In the policy evaluation step of soft policy iteration, we wish to compute the value of a policy according to the maximum entropy objective in Equation 1. For a fixed policy, the soft Qvalue can be computed iteratively, starting from any function and repeatedly applying a modified Bellman backup operator given by
(2) 
where
(3) 
is the soft state value function. We can obtain the soft value function for any policy by repeatedly applying as formalized below.
Lemma 1 (Soft Policy Evaluation).
Consider the soft Bellman backup operator in Equation 2 and a mapping with , and define . Then the sequence will converge to the soft Qvalue of as .
Proof.
See Appendix B.1. ∎
In the policy improvement step, we update the policy towards the exponential of the new Qfunction. This particular choice of update can be guaranteed to result in an improved policy in terms of its soft value. Since in practice we prefer policies that are tractable, we will additionally restrict the policy to some set of policies , which can correspond, for example, to a parameterized family of distributions such as Gaussians. To account for the constraint that
, we project the improved policy into the desired set of policies. While in principle we could choose any projection, it will turn out to be convenient to use the information projection defined in terms of the KullbackLeibler divergence. In the other words, in the policy improvement step, for each state, we update the policy according to
(4) 
The partition function normalizes the distribution, and while it is intractable in general, it does not contribute to the gradient with respect to the new policy and can thus be ignored, as noted in the next section. For this projection, we can show that the new, projected policy has a higher value than the old policy with respect to the objective in Equation 1. We formalize this result in Lemma 2.
Lemma 2 (Soft Policy Improvement).
Let and let be the optimizer of the minimization problem defined in Equation 4. Then for all with .
Proof.
See Appendix B.2. ∎
The full soft policy iteration algorithm alternates between the soft policy evaluation and the soft policy improvement steps, and it will provably converge to the optimal maximum entropy policy among the policies in (Theorem 1). Although this algorithm will provably find the optimal solution, we can perform it in its exact form only in the tabular case. Therefore, we will next approximate the algorithm for continuous domains, where we need to rely on a function approximator to represent the Qvalues, and running the two steps until convergence would be computationally too expensive. The approximation gives rise to a new practical algorithm, called soft actorcritic.
Theorem 1 (Soft Policy Iteration).
Repeated application of soft policy evaluation and soft policy improvement from any converges to a policy such that for all and , assuming .
Proof.
See Appendix B.3. ∎
4.2 Soft ActorCritic
As discussed above, large continuous domains require us to derive a practical approximation to soft policy iteration. To that end, we will use function approximators for both the Qfunction and the policy, and instead of running evaluation and improvement to convergence, alternate between optimizing both networks with stochastic gradient descent. We will consider a parameterized state value function
, soft Qfunction , and a tractable policy . The parameters of these networks are , and. For example, the value functions can be modeled as expressive neural networks, and the policy as a Gaussian with mean and covariance given by neural networks. We will next derive update rules for these parameter vectors.
The state value function approximates the soft value. There is no need in principle to include a separate function approximator for the state value, since it is related to the Qfunction and policy according to Equation 3. This quantity can be estimated from a single action sample from the current policy without introducing a bias, but in practice, including a separate function approximator for the soft value can stabilize training and is convenient to train simultaneously with the other networks. The soft value function is trained to minimize the squared residual error
(5) 
where is the distribution of previously sampled states and actions, or a replay buffer. The gradient of Equation 5
can be estimated with an unbiased estimator
(6) 
where the actions are sampled according to the current policy, instead of the replay buffer. The soft Qfunction parameters can be trained to minimize the soft Bellman residual
(7) 
with
(8) 
which again can be optimized with stochastic gradients
(9) 
The update makes use of a target value network , where can be an exponentially moving average of the value network weights, which has been shown to stabilize training (Mnih et al., 2015). Alternatively, we can update the target weights to match the current value function weights periodically (see Appendix E). Finally, the policy parameters can be learned by directly minimizing the expected KLdivergence in Equation 4:
(10) 
There are several options for minimizing . A typical solution for policy gradient methods is to use the likelihood ratio gradient estimator (Williams, 1992)
, which does not require backpropagating the gradient through the policy and the target density networks. However, in our case, the target density is the Qfunction, which is represented by a neural network an can be differentiated, and it is thus convenient to apply the reparameterization trick instead, resulting in a lower variance estimator. To that end, we reparameterize the policy using a neural network transformation
(11) 
where is an input noise vector, sampled from some fixed distribution, such as a spherical Gaussian. We can now rewrite the objective in Equation 10 as
(12) 
where is defined implicitly in terms of , and we have noted that the partition function is independent of and can thus be omitted. We can approximate the gradient of Equation 12 with
(13) 
where is evaluated at . This unbiased gradient estimator extends the DDPG style policy gradients (Lillicrap et al., 2015) to any tractable stochastic policy.
Our algorithm also makes use of two Qfunctions to mitigate positive bias in the policy improvement step that is known to degrade performance of value based methods (Hasselt, 2010; Fujimoto et al., 2018). In particular, we parameterize two Qfunctions, with parameters , and train them independently to optimize . We then use the minimum of the Qfunctions for the value gradient in Equation 6 and policy gradient in Equation 13, as proposed by Fujimoto et al. (2018). Although our algorithm can learn challenging tasks, including a 21dimensional Humanoid, using just a single Qfunction, we found two Qfunctions significantly speed up training, especially on harder tasks. The complete algorithm is described in Algorithm 1. The method alternates between collecting experience from the environment with the current policy and updating the function approximators using the stochastic gradients from batches sampled from a replay buffer. In practice, we take a single environment step followed by one or several gradient steps (see Appendix D for all hyperparameter). Using offpolicy data from a replay buffer is feasible because both value estimators and the policy can be trained entirely on offpolicy data. The algorithm is agnostic to the parameterization of the policy, as long as it can be evaluated for any arbitrary stateaction tuple.
5 Experiments
The goal of our experimental evaluation is to understand how the sample complexity and stability of our method compares with prior offpolicy and onpolicy deep reinforcement learning algorithms. We compare our method to prior techniques on a range of challenging continuous control tasks from the OpenAI gym benchmark suite (Brockman et al., 2016) and also on the rllab implementation of the Humanoid task (Duan et al., 2016). Although the easier tasks can be solved by a wide range of different algorithms, the more complex benchmarks, such as the 21dimensional Humanoid (rllab), are exceptionally difficult to solve with offpolicy algorithms (Duan et al., 2016). The stability of the algorithm also plays a large role in performance: easier tasks make it more practical to tune hyperparameters to achieve good results, while the already narrow basins of effective hyperparameters become prohibitively small for the more sensitive algorithms on the hardest benchmarks, leading to poor performance (Gu et al., 2016).
We compare our method to deep deterministic policy gradient (DDPG) (Lillicrap et al., 2015), an algorithm that is regarded as one of the more efficient offpolicy deep RL methods (Duan et al., 2016); proximal policy optimization (PPO) (Schulman et al., 2017b), a stable and effective onpolicy policy gradient algorithm; and soft Qlearning (SQL) (Haarnoja et al., 2017), a recent offpolicy algorithm for learning maximum entropy policies. Our SQL implementation also includes two Qfunctions, which we found to improve its performance in most environments. We additionally compare to twin delayed deep deterministic policy gradient algorithm (TD3) (Fujimoto et al., 2018), using the authorprovided implementation. This is an extension to DDPG, proposed concurrently to our method, that first applied the double Qlearning trick to continuous control along with other improvements. We have included trust region path consistency learning (TrustPCL) (Nachum et al., 2017b) and two other variants of SAC in Appendix E. We turned off the exploration noise for evaluation for DDPG and PPO. For maximum entropy algorithms, which do not explicitly inject exploration noise, we either evaluated with the exploration noise (SQL) or use the mean action (SAC). The source code of our SAC implementation^{1}^{1}1github.com/haarnoja/sac and videos^{2}^{2}2sites.google.com/view/softactorcritic are available online.
5.1 Comparative Evaluation
Figure 1 shows the total average return of evaluation rollouts during training for DDPG, PPO, and TD3. We train five different instances of each algorithm with different random seeds, with each performing one evaluation rollout every 1000 environment steps. The solid curves corresponds to the mean and the shaded region to the minimum and maximum returns over the five trials.
The results show that, overall, SAC performs comparably to the baseline methods on the easier tasks and outperforms them on the harder tasks with a large margin, both in terms of learning speed and the final performance. For example, DDPG fails to make any progress on Antv1, Humanoidv1, and Humanoid (rllab), a result that is corroborated by prior work (Gu et al., 2016; Duan et al., 2016). SAC also learns considerably faster than PPO as a consequence of the large batch sizes PPO needs to learn stably on more highdimensional and complex tasks. Another maximum entropy RL algorithm, SQL, can also learn all tasks, but it is slower than SAC and has worse asymptotic performance. The quantitative results attained by SAC in our experiments also compare very favorably to results reported by other methods in prior work (Duan et al., 2016; Gu et al., 2016; Henderson et al., 2017), indicating that both the sample efficiency and final performance of SAC on these benchmark tasks exceeds the state of the art. All hyperparameters used in this experiment for SAC are listed in Appendix D.
5.2 Ablation Study
The results in the previous section suggest that algorithms based on the maximum entropy principle can outperform conventional RL methods on challenging tasks such as the humanoid tasks. In this section, we further examine which particular components of SAC are important for good performance. We also examine how sensitive SAC is to some of the most important hyperparameters, namely reward scaling and target value update smoothing constant.
Stochastic vs. deterministic policy.
Soft actorcritic learns stochastic policies via a maximum entropy objective. The entropy appears in both the policy and value function. In the policy, it prevents premature convergence of the policy variance (Equation 10). In the value function, it encourages exploration by increasing the value of regions of state space that lead to highentropy behavior (Equation 5). To compare how the stochasticity of the policy and entropy maximization affects the performance, we compare to a deterministic variant of SAC that does not maximize the entropy and that closely resembles DDPG, with the exception of having two Qfunctions, using hard target updates, not having a separate target actor, and using fixed rather than learned exploration noise. Figure 2 compares five individual runs with both variants, initialized with different random seeds. Soft actorcritic performs much more consistently, while the deterministic variant exhibits very high variability across seeds, indicating substantially worse stability. As evident from the figure, learning a stochastic policy with entropy maximization can drastically stabilize training. This becomes especially important with harder tasks, where tuning hyperparameters is challenging. In this comparison, we updated the target value network weights with hard updates, by periodically overwriting the target network parameters to match the current value network (see Appendix E for a comparison of average performance on all benchmark tasks).
Policy evaluation.
Since SAC converges to stochastic policies, it is often beneficial to make the final policy deterministic at the end for best performance. For evaluation, we approximate the maximum a posteriori action by choosing the mean of the policy distribution. Figure (a) compares training returns to evaluation returns obtained with this strategy indicating that deterministic evaluation can yield better performance. It should be noted that all of the training curves depict the sum of rewards, which is different from the objective optimized by SAC and other maximum entropy RL algorithms, including SQL and TrustPCL, which maximize also the entropy of the policy.
Reward scale.
Soft actorcritic is particularly sensitive to the scaling of the reward signal, because it serves the role of the temperature of the energybased optimal policy and thus controls its stochasticity. Larger reward magnitudes correspond to lower entries. Figure (b) shows how learning performance changes when the reward scale is varied: For small reward magnitudes, the policy becomes nearly uniform, and consequently fails to exploit the reward signal, resulting in substantial degradation of performance. For large reward magnitudes, the model learns quickly at first, but the policy then becomes nearly deterministic, leading to poor local minima due to lack of adequate exploration. With the right reward scaling, the model balances exploration and exploitation, leading to faster learning and better asymptotic performance. In practice, we found reward scale to be the only hyperparameter that requires tuning, and its natural interpretation as the inverse of the temperature in the maximum entropy framework provides good intuition for how to adjust this parameter.
Target network update.
It is common to use a separate target value network that slowly tracks the actual value function to improve stability. We use an exponentially moving average, with a smoothing constant , to update the target value network weights as common in the prior work (Lillicrap et al., 2015; Mnih et al., 2015). A value of one corresponds to a hard update where the weights are copied directly at every iteration and zero to not updating the target at all. In Figure (c), we compare the performance of SAC when varies. Large can lead to instabilities while small can make training slower. However, we found the range of suitable values of to be relatively wide and we used the same value (0.005) across all of the tasks. In Figure 4 (Appendix E) we also compare to another variant of SAC, where instead of using exponentially moving average, we copy over the current network weights directly into the target network every 1000 gradient steps. We found this variant to benefit from taking more than one gradient step between the environment steps, which can improve performance but also increases the computational cost.
6 Conclusion
We present soft actorcritic (SAC), an offpolicy maximum entropy deep reinforcement learning algorithm that provides sampleefficient learning while retaining the benefits of entropy maximization and stability. Our theoretical results derive soft policy iteration, which we show to converge to the optimal policy. From this result, we can formulate a soft actorcritic algorithm, and we empirically show that it outperforms stateoftheart modelfree deep RL methods, including the offpolicy DDPG algorithm and the onpolicy PPO algorithm. In fact, the sample efficiency of this approach actually exceeds that of DDPG by a substantial margin. Our results suggest that stochastic, entropy maximizing reinforcement learning algorithms can provide a promising avenue for improved robustness and stability, and further exploration of maximum entropy methods, including methods that incorporate second order information (e.g., trust regions (Schulman et al., 2015)) or more expressive policy classes is an exciting avenue for future work.
Acknowledgments
We would like to thank Vitchyr Pong for insightful discussions and help in implementing our algorithm as well as providing the DDPG baseline code; Ofir Nachum for offering support in running TrustPCL experiments; and George Tucker for his valuable feedback on an early version of this paper. This work was supported by Siemens and Berkeley DeepDrive.
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Appendix A Maximum Entropy Objective
The exact definition of the discounted maximum entropy objective is complicated by the fact that, when using a discount factor for policy gradient methods, we typically do not discount the state distribution, only the rewards. In that sense, discounted policy gradients typically do not optimize the true discounted objective. Instead, they optimize average reward, with the discount serving to reduce variance, as discussed by Thomas (2014). However, we can define the objective that is optimized under a discount factor as
(14) 
This objective corresponds to maximizing the discounted expected reward and entropy for future states originating from every stateaction tuple weighted by its probability under the current policy.
Appendix B Proofs
b.1 Lemma 1
Lemma 1 (Soft Policy Evaluation). Consider the soft Bellman backup operator in Equation 2 and a mapping with , and define . Then the sequence will converge to the soft Qvalue of as .
Proof.
Define the entropy augmented reward as and rewrite the update rule as
(15) 
and apply the standard convergence results for policy evaluation (Sutton & Barto, 1998). The assumption is required to guarantee that the entropy augmented reward is bounded. ∎
b.2 Lemma 2
Lemma 2 (Soft Policy Improvement). Let and let be the optimizer of the minimization problem defined in Equation 4. Then for all with .
Proof.
Let and let and be the corresponding soft stateaction value and soft state value, and let be defined as
(16) 
It must be the case that , since we can always choose . Hence
(17) 
and since partition function depends only on the state, the inequality reduces to
(18) 
Next, consider the soft Bellman equation:
(19) 
where we have repeatedly expanded on the RHS by applying the soft Bellman equation and the bound in Equation 18. Convergence to follows from Lemma 1. ∎
b.3 Theorem 1
Theorem 1 (Soft Policy Iteration). Repeated application of soft policy evaluation and soft policy improvement to any converges to a policy such that for all and , assuming .
Proof.
Let be the policy at iteration . By Lemma 2, the sequence is monotonically increasing. Since is bounded above for (both the reward and entropy are bounded), the sequence converges to some . We will still need to show that is indeed optimal. At convergence, it must be case that for all , . Using the same iterative argument as in the proof of Lemma 2, we get for all , that is, the soft value of any other policy in is lower than that of the converged policy. Hence is optimal in . ∎
Appendix C Enforcing Action Bounds
We use an unbounded Gaussian as the action distribution. However, in practice, the actions needs to be bounded to a finite interval. To that end, we apply an invertible squashing function () to the Gaussian samples, and employ the change of variables formula to compute the likelihoods of the bounded actions. In the other words, let
be a random variable and
the corresponding density with infinite support. Then , where is applied elementwise, is a random variable with support in with a density given by(20) 
Since the Jacobian is diagonal, the loglikelihood has a simple form
(21) 
where is the element of .
Appendix D Hyperparameters
Table 1 lists the common SAC parameters used in the comparative evaluation in Figure 1 and Figure 4. Table 2 lists the reward scale parameter that was tuned for each environment.
Parameter  Value  
Shared  
optimizer  Adam (Kingma & Ba, 2015)  
learning rate  
discount ()  0.99  
replay buffer size  
number of hidden layers (all networks)  2  
number of hidden units per layer  256  
number of samples per minibatch  256  
nonlinearity  ReLU  
SAC  
target smoothing coefficient ()  0.005  
target update interval  1  
gradient steps  1  
SAC (hard target update)  
target smoothing coefficient ()  1  
target update interval  1000  
gradient steps (except humanoids)  4  
gradient steps (humanoids)  1 
Environment  Action Dimensions  Reward Scale 

Hopperv1  3  5 
Walker2dv1  6  5 
HalfCheetahv1  6  5 
Antv1  8  5 
Humanoidv1  17  20 
Humanoid (rllab)  21  10 
Appendix E Additional Baseline Results
Figure 4 compares SAC to TrustPCL (Figure 4. TrustPC fails to solve most of the task within the given number of environment steps, although it can eventually solve the easier tasks (Nachum et al., 2017b) if ran longer. The figure also includes two variants of SAC: a variant that periodically copies the target value network weights directly instead of using exponentially moving average, and a deterministic ablation which assumes a deterministic policy in the value update (Equation 6) and the policy update (Equation 13), and thus strongly resembles DDPG with the exception of having two Qfunctions, using hard target updates, not having a separate target actor, and using fixed exploration noise rather than learned. Both of these methods can learn all of the tasks and they perform comparably to SAC on all but Humanoid (rllab) task, on which SAC is the fastest.
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