1 Introduction
Reinforcement learning has long been considered as a general framework applicable to a broad range of problems. Reinforcement learning algorithms have been categorized in various ways. There is an important distinction, however, arises between discrete and continuous action spaces. In discrete domains, there are several algorithms, such as Qlearning, that leverage backups through Bellman equations and dynamic programming to solve problems effectively. These strategies have led to the use of deep neural networks to learn policies and value functions that can achieve superhuman accuracy in several games mnih2013playing ; silver2016mastering where actions lie in discrete domains. This success spurred the development of RL techniques that use deep neural networks for continuous control problems lillicrap2015continuous ; gu2016continuous ; levine2016end . The gains in these domains, however, have not been as outsized as they have been for discrete action domains, and superhuman performance seems unachievable with current techniques.
This disparity is in part a result of the inherent difficulty in maximizing an arbitrary function on a continuous domain, even in low dimensional settings; further, techniques such as beam searches do not exist or do not apply directly. This makes it difficult to perform inference and learning in such settings. Additionally, it becomes harder to apply dynamic programming methods to back up value function estimates from successor states to parent states. Several of the recent continuous control reinforcement learning approaches attempt to borrow characteristics from discrete problems by proposing models that allow maximization and backups more easily
gu2016continuous .One way in which continuous control can avail itself of the above advantages is to discretize each of the dimensions of continuous control action spaces. As noted in lillicrap2015continuous , doing this naively, however, would create an exponentially large discrete space of actions. For example with dimensions being discretized into bins, the problem would balloon to a discrete space with possible actions.
We leverage the recent success of sequencetosequence type models seq2seq to train such discretized models, without falling into the trap of requiring an exponentially large number of actions. Our method relies on a technique that was first introduced in bengio1999modeling
, which allows us to escape the curse of dimensionality in high dimensional spaces by modeling complicated probability distributions using the chain rule decomposition. In this paper, we similarly parameterize functions of interest – Qvalues – using a decomposition of the joint function into a sequence of conditional approximations. Families of models that exhibit these sequential prediction properties have also been referred to as autoregressive. With this formulation, we are able to achieve finegrained discretization of individual domains, without an explosion in the number of parameters; at the same time we can model arbitrarily complex distributions while maintaining the ability to perform (approximate) maximization.
While this strategy can be applied to most function approximation settings in RL, we focus on offpolicy settings with a DQN inspired algorithm for Qlearning. Complementary results for an actorcritic model sutton1998reinforcement ; sutton1999policy based on the same autoregressive concept are presented in Appendix D. Empirical results on an illustrative multimodal problem demonstrates how our model is able to perform global maximization, avoiding the exploration problems faced by algorithms like NAF naf and DDPG lillicrap2015continuous . We also show the effectiveness of our method in solving a range of benchmark continuous control problems including hopper to humanoid.
2 Method
In this paper, we introduce the idea of building continuous control algorithms utilizing sequential, or autoregressive, models that predict over action spaces one dimension at a time. Here, we use discrete distributions over each dimension (achieved by discretizing each continuous dimension into bins) and apply it on offpolicy learning. We explore one instantiation of such a model in the body of this work and discuss three additional variants in Appendix
C, Appendix D and Appendix E.2.1 Preliminaries
We briefly describe the notation we use in this paper. Let be the observed state of the agent, be the dimensional action space, and be the stochastic environment in which the agent operates. Finally, let
be the vector obtained by taking the subrange of a vector
.At each step , the agent takes an action , receives a reward from the environment and transitions stochastically to a new state according to (possibly unknown) dynamics . An episode consists of a sequence of such steps , with where is the last time step. An episode terminates when a stopping criterion is true (for example when a game is lost, or when the number of steps is greater than some threshold length ).
Let be the discounted reward received by the agent starting at step of an episode. As with standard reinforcement learning, the goal of our agent is to learn a policy that maximizes the expected total reward it would receive from the environment by following this policy.
Because this paper is focused on offpolicy learning with QLearning watkins1992q , we will provide a brief description of the algorithm.
2.1.1 QLearning
Qlearning is an offpolicy algorithm that learns an actionvalue function and a corresponding greedypolicy, . The model is trained by finding the fixed point of the transition operator, i.e.
(1) 
This is done by minimizing the Bellman Error, over the exploration distribution,
(2) 
Traditionally, is represented as a table of state action pairs or with linear function approximators or shallow neural networks watkins1992q ; tesauro1995temporal . Recently, there has been an effort to apply these techniques to more complex domains using nonlinear function approximators that are deep neural networks mnih2013playing ; mnih2015human . In these models, a Deep QNetwork (DQN) parameterized by parameters, , is used to predict Qvalues, i.e. . The DQN parameters, , are trained by performing gradient descent on the error in equation 2, without taking a gradient through the Qvalues of the successor states (although, see Baird95residualalgorithms for an approach that takes this into account).
Since the greedy policy, , uses the action value with the maximum Qvalue, it is essential that any parametric form of be able to find a maxima easily with respect to actions. For a DQN where the output layer predicts the Qvalues for each of the discrete outputs, it is easy to find this max – it is simply the action corresponding to the index of the output with the highest estimated Qvalue. In continuous action problems, it can be tricky to formulate a parametric form of the Qvalue where it is easy to find such a maxima. Techniques like Normalized Advantage Function (NAF)naf constrain the function approximator such that it is easy to compute a max analytically at the cost of reduced flexibility. Another approach has been followed by the Deep Deterministic Policy Gradients algorithm, which uses an actorcritic algorithm in an offpolicy setting to train a deterministic policy lillicrap2015continuous . The model trains a critic to estimate the function and a separate policy to try to approximate a max over the critic.
In this work, we develop a similar technique where we modify the form of our Qvalue function while still retaining the ability to find local maxima over actions for use in a greedy policy.
2.2 Sequential DQN
We propose a Sequential DQN (SDQN) model that structures a function in the form of a sequential model over action dimensions. To introduce this idea, we first show a transformation done on the environment which splits a D action into a sequence of 1D actions. We then shift this modification from an environment transformation to the construction of a function resulting in our final technique.
2.2.1 Transformed Environment
Consider an environment with an dimensional action space. We can perform a transformation to this environment into a similar environment inserting fictitious states. Each new fictitious state has a 1D action space. Each new action choice is conditioned on the previous state as well as on all previously selected actions. When the th action is selected, the environment dynamics take into effect using the previous actions to perform one step. The process is depicted for in Figure 1.
This transformation reduces the D actions to a series of 1D actions. We can now discretize the 1D output space and directly apply learning. Note that we could apply this strategy to continuous values, without discretization, by choosing a conditional distribution, such as a mixture of 1D Gaussians, over which a maxima can easily be found. As such, this approach is equally applicable to pure continuous domains as compared to discrete approximations.
The downside to this transformation is that it increases the amount of steps needed to solve a MDP. In practice, this transformation makes learning a function considerably harder. The extra steps of dynamic programming coupled with learned function approximators causes large overestimation and stability issues.
2.2.2 Model
In this section, we shift the previous section’s ideas into a model that acts on the original, untransformed environment. We learn a deterministic policy defined as:
(3) 
Here is the number of action dimensions, and is the deterministic policy for the dimension action choice, i.e. . A greedy policy from learning is used to compute these policies:
(4) 
where and is the set of possible actions for values for . Here is a "partial Qvalue function" that we estimate using learning and is defined recursively because is an input to . These "partial Qvalue functions" correspond to the Qvalues occurring on the fictitious states introduced in the previous section.
We parameterize as a neural network that receives as input. The neural network outputs a dimensional vector, where is the number of bins that we discretized action space into. This parameterization allows for arbitrarily complex resolution at the cost of increasing the number of bins, . We found to be a good tradeoff between performance and tractability.
This model is similar to doing learning on the transformed environment described in the previous section. As before, the entire chain of ’s can then be trained with learning. In practice, however, we found that the values were significantly overestimated, due to the increased time dependencies and the max operator in learning. When training, there is an approximation error in value predictions. This approximation error and a max operator in learning causes overestimation. To overcome this issue we introduce a second function, , inspired by double DQN hasselt2016deep , which we use to produce a final estimate of the value of all action dimensions. Note, in the traditional double DQN work, a past version of the function is used. In our setting, we train an entirely separate network, , on the original, untransformed environment. By doing this, learning becomes easier as there are less long term dependencies.
There are several components we must train for our model. First, minimizes the Bellman Error:
(5) 
Next, is trained to match the predicted value from :
(6) 
Finally, learns to satisfy the Bellman equation ensuring that . We do this by creating an internal consistency loss for each dimension :
(7) 
We sample transitions from a replay buffer, , to minimize all three losses using SGD mnih2013playing . As in mnih2013playing , at each training step, is filled with one transition sampled using the current policy, .
The policy action selection, , is illustrated in Figure 2, while training is depicted in Figure A.1.
Another motivation for structuring our losses in this way is to decouple the model that learns values from the model that chooses max actions – or the policy. In this way our approach is similar to DDPG. A value is learned in one network, , and a policy to approximate the max is learned in another, .
2.3 Exploration
Numerous exploration strategies have been explored in reinforcement learning osband2016deep ; blundell2015weight ; houthooft2016vime . When computing actions from our sequential policies, we can additionally inject noise at each choice and then select the remaining actions based on this choice. We found this yields better performance as compared to epsilon greedy.
2.4 Neural Network Parameterization
We implemented each learned function in the above model as a deep neural network. We explored two parameterizations of the sequential components, the s. First, we looked at a recurrent LSTM model hochreiter1997long . This model has shared weights and passes information via hidden activations from one action to another. The input at each time step is a function of the current state, , and action,
. Second, we looked at a version with no shared weights, passing all information into each sequential prediction model. These models are feed forward neural networks that take as input a concatenation of all previous action selections as well as the state. In more complex domains, such as vision based control tasks for example, it would make sense to untie only a subset of the weights. In practice, we found that the untied version performed better. Optimizing these model families is still an ongoing work. For full detail of model architectures and training procedures selection see Appendix
F.3 Related Work
Our work was motivated by two distinct desires – to learn policies over exponentially large discrete action spaces, and to approximate value functions over high dimensional continuous action spaces effectively. In our paper we used a sequential parameterization of policies that help us to achieve this without making an assumption about the actual functional form of the model. Other prior work attempts to handle high dimensional action spaces by assuming specific decompositions. For example, sallans2004reinforcement were able to scale up learning to extremely large action sizes by factoring the action value function and use product of experts to learn policies. An alternative strategy was proposed in dulac2015deep using action embeddings and applying knearest neighbors to reduce scaling of action sizes. By laying out actions on a hypercube, pazis2011generalized are able to perform a binary search over actions resulting in a logarithmic search for the optimal action. Their method is similar to SDQN, as both construct a value from sub
values. Their approach presupposes these constraints, however, and optimizes the Bellman equation by optimizing hyperplanes independently thus enabling optimizing via linear programming. Our approach is iterative and refines the action selection, which contrasts to their independent subplane maximization.
Along with the development of discrete space algorithms, researchers have innovated specialized solutions to learn over continuous state and action environments including lever2014deterministic ; lillicrap2015continuous ; naf . More recently, novel deep RL approaches have been developed for continuous state and action problems. TRPO schulman2015trust and A3C mnih2016asynchronous
uses a stocastic policy parameterized by diagonal covariance Gaussian distributions. NAF
naf relies on quadratic advantage function enabling closed form optimization of the optimal action. Other methods structure the network in a way such that they are convex in the actions while being nonconvex with respect to states amos2016input or use a linear policy rajeswaran2017towards .In the context of reinforcement learning, sequential or autoregressive policies have previously been used to describe exponentially large action spaces such as the space of neural architectures, zoph2016neural and over sequences of words norouzi2016reward ; shen2015minimum . These approaches rely on policy gradient methods whereas we explore offpolicy methods. Hierarchical/options based methods, including dayan1993feudal which perform spatial abstraction or sutton1999option that perform temporal abstraction pose another way to factor action spaces. These methods refine their action selection from time where our approaches operates on the same timescale and factors the action space.
A vast literature on constructing sequential models to solve tasks exists outside of RL. These models are a natural fit when the data is generated in a sequential process such as in language modeling bengio2003neural
. One of the first and most effective deep learned sequencetosequence models for language modeling was proposed in
sutskever2014sequence , which used an encoderdecoder architecture. In other domains, techniques such as NADE larochelle2011neural have been developed to compute tractable likelihood. Techniques like Pixel RNN oord2016pixel have been used to great success in the image domain where there is no clear generation sequence. Hierarchical softmax morin2005hierarchical performs a hierarchical decomposition based on WordNet semantic information.The second motivation of our work was to enable learning over more flexible, possibly multimodal policy landscape. Existing methods use stochastic neural networks florensa2017 or construct energy models haarnoja2017reinforcement sampled with Stein variational gradient descent liu2016stein ; wang2016learning . In our work instead of sampling, we construct a secondary network to evaluate a max.
4 Experiments
4.1 Multimodal Example Environment
To consider the effectiveness of our algorithm, we consider a deterministic environment with a single time step, and a 2D action space. This can be thought of as being a twoarmed bandit problem with deterministic rewards, or as a search problem in 2D action space. We chose our reward function to be a multimodal distribution as shown in the first column in Figure 3. A large suboptimal mode and a smaller optimal mode exist.
As with bandit problems, this formulation helps us isolate the ability of our method to find an optimal policy, without the confounding effect that arises from backing up rewards via the Bellman operator for sequential problems. We look at the behavior of SDQN as well as that of DDPG and NAF on this task. As in traditional RL, we do exploration while learning. We consider uniformly sampling (greedy with
) as well as sampling data from a normal distribution centered at the current policy – we refer to this as "local." A visualization of the final
surfaces as well as training curves can be found in Figure 3.First, we considered the performance of DDPG. DDPG uses local optimization to learn a policy on a constantly changing estimate of values predicted by a critic. The form of the distribution is flexible and as such there is no closed form properties we can make use of for learning a policy. As such, we resort to gradient descent, a local optimization algorithm. Due to its local nature, it is possible for this algorithm to get stuck in suboptimal policies that are local maximum of the current critic. We hypothesize that these local maximum in policy space exist in more realistic simulated environments as well. Traditionally, deep learning methods use local optimizers and avoid local minima or maxima by working in a high dimensional parameter space choromanska2015loss . In RL, however, the action space of a policy is relatively small dimensionally thus it is much more likely that they exist. For example, in the hopper environment, a common failure mode we experienced when training algorithms like DDPG is to learn to balance instead of moving forward and hopping.
Next, we consider how NAF behaves on this environment. NAF makes an assumption that the function is quadratic in action space. During training, NAF fits this quadratic surface to minimize the expected
loss evaluated with transitions from a replay buffer. Given the restricted functional form of the model, it is no longer possible to model the entire space without error. As such, the distribution of sampled points used for learning (i.e. the behavior policy) matters greatly. When the behavior policy is a uniform distribution over the action space, the quadratic approximation yields a surface where the maximum is in a low reward region with no path to improve. Interestingly though, this is no longer the case when the behavior policy is a stochastic Gaussian policy whose mean is the greedy policy w.r.t. the previous estimate of the
values. In this setting, NAF only models the quadratic surface around where samples are taken, i.e. locally around the maxima of the estimated values. This nonstationary optimization results in behavior quite similar to DDPG in that it performs a local optimization. This experiment suggests that in NAF there is a balance between a global quadratic model, and exploiting local structure when fitting a function.Finally, we consider a SDQN. As expected, this model is capable of completely representing the surface (under the limits of discretization) and does not suffer from inflexibility of the previous methods. The optimization of the policy is not done locally – both the uniform behavior policy and the stochastic Gaussian behavior policy converge to the optimal solution. Much like DDPG, the loss surface learned can be stationary – it does not need to shift over time to learn the optimal solution. Unlike DDPG, however, the policy will not get stuck in a local maximum. With the uniform behavior policy setting, the model slowly reaches the right solution, as there are many wasted samples. With a behavior policy that is closer to being onpolicy (such as the stochastic Gaussian greedy policy referred to above), this slowness is reduced. Much of the error occurs from selecting over estimated actions. When sampling more on policy, the over estimated data points get sampled more frequently converging to the optimal solution. Unlike the other global algorithms like NAF, performance will not decrease if we increase sampling noise to cover more of the action space. This allows us to balance exploration and learning of the Q function more easily. ^{1}^{1}1This assumes that the models have enough capacity. In a limited capacity setting, one would still want to explore locally. Much like NAF, SDQN models will shift capacity to modeling these spaces, which are sampled, thus making better use of the capacity.
4.2 Mujoco environments
To evaluate the relative performance of these models we perform a series of experiments on common continuous control tasks. We test the hopper, swimmer, half cheetah, walker2d and humanoid environments from the OpenAI gym suite gym . ^{2}^{2}2 For technical reasons, our simulations use a different numerical simulation strategy provided by Mujoco mujoco . In practice though, we found the differences in final reward to be within the expected variability of rerunning an algorithm with a different random seed.
We performed a wide hyper parameter search over various parameters in our models (described in Appendix F), and selected the best performing runs. We then ran 10 random seeds of the same hyper parameters to evaluate consistency and to get a more realistic estimate of performance. We believe this replication is necessary as many of these algorithms are not only sensitive to both hyper parameters but random seeds. This is not a quality we would like in our RL algorithms and by doing the 10x replications, we are able to detect this phenomena.
First, we look at learning curves of some of the environments tested in Figure 4. Our method quickly achieves good policies much faster than DDPG.
Next, for a more qualitative analysis, we use the best reward achieved while training averaged across over 25,000 steps and with evaluations sampled every 5,000 steps. Again we perform an average over 10 different random seeds. This metric gives a much better sense of stability than the traditionally reported instantaneous max reward achieved during training.
We compare our algorithm to the current stateoftheart in offpolicy continuous control: DDPG. Through careful model selection and extensive hyper parameter tuning, we train models with performance better than previously published for DDPG on some of these tasks. Despite this search, however, we believe that there is still space for significant performance gain for all the models given different neural network architectures and hyper parameters. Results can be seen in Figure 5. Our algorithm achieves better performance on four of the five environments we tested.
agent  hopper  swimmer  half cheetah  humanoid  walker2d 

SDQN  3342.62  179.23  7774.77  3096.71  3227.73 
DDPG  3296.49  133.52  6614.26  3055.98  3640.93 
5 Discussion
Conceptually, our approach centers on the idea that action selection at each stage can be factored and sequentially selected using an autoregressive formulation. In this work we use 1D action spaces that are discretized. Existing work in the image modeling domain suggests that using a mixture of logistic units salimans2017pixelcnn++ greatly speeds up training and would also satisfy our need for a closed form max. Additionally, this work imposes a prespecified ordering of actions which may negatively impact training for certain classes of problems. To address this, we could learn to factor the action space into the sequential order for continuous action spaces or learn to group action sets for discrete action spaces. Another promising direction is to combine this approximate max action with gradient based optimization procedure. This would relieve some of the complexity of the modeling task of the maxing network, at the cost of increased compute when sampling from the policy. Finally, the work presented here is exclusively on offpolicy methods. Use of an autoregressive policy with discretized actions could also be used as the policy for any stochastic policy optimization algorithm such as TRPO schulman2015trust or A3C mnih2016asynchronous .
6 Conclusion
In this work we present a continuous control algorithm that utilize discretized action spaces and sequential models. The technique we propose is an offpolicy RL algorithm that utilizes sequential prediction and discretization. We decompose our model into a function and an auxiliary network that acts as a policy and is responsible for computing an approximate max over actions. The effectiveness of our method is demonstrated on illustrative and benchmark tasks, as well as on more complex continuous control tasks. Two additional formulations of discretized sequential prediction models are presented in Appendix C and Appendix D
Acknowledgements
We would like to thank Nicolas Heess for his insight on exploration and assistance in scaling up task complexity, and Oscar Ramirez for his assistance running some experiments. We would like to thank Eric Jang, Sergey Levine, Mohammad Norouzi, Leslie Phillips, Chase Roberts, and Vincent Vanhoucke for their comments and feedback. Finally we would like to thank the entire Brain Team for their support.
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Appendix A Model Diagrams
Appendix B Model Visualization
To gain insight into the characteristics of that our SDQN algorithm learns, we visualized results from the hopper environment, because these are easier to comprehend.
First we compute each action dimension’s distribution, , and compare those distributions to that of the double DQN network for the full action taken, . A figure containing these distributions and corresponding state visualization can be found in Figure B.2.
For most states in the hopper walk cycle, the distribution is very flat. This implies that small changes in the action taken in a state will have little impact on future reward. This makes sense as the system can recover from any action taken in this frame. However, this is not true for all states – certain critical states exist, such as when the hopper is pushing off, where not selecting the correct action value greatly degrades performance. This can be seen in frame 466.
Our algorithm is trained with a number of soft constraints. First, if fully converged, we would expect >= as every new subaction taken should maintain or improve the expected future discounted reward. Additionally, we would expect (from eq. 2.2.2). In the majority of frames these properties seem correct, but there is certainly room for improvement.
Next, we attempt to look at surfaces in a more global manner. We plot 2D cross sections for each pair of actions and assume the third dimension is zero. Figure B.3 shows the results.
As seen in the previous visualization, the surface of both the autoregressive surface and the is not smooth, which is expected as the environment action space for Hopper is expected to be highly nonlinear. Some regions of the surface seem quite noisy which is not expected. Interestingly though, these regions of noise do not seem to lower the performance of the final policy. In learning, only the maximum value regions have any impact on the taken policy. Future work is needed to better characterize this effect. We would like to explore techniques that use "soft" Qlearning [29, schulman2017equivalencee, 14]. These techniques will use more of the surface thus smooth the representations.
Additionally, we notice that the dimensions of the autoregressive model are modeled differently. The last action, has considerably more noise than the previous two action dimensions. This large difference in the smoothness and shape of the surfaces demonstrates that the order of the actions dimensions matters. This figure suggests that the model has a harder time learning sharp features in the dimension. In future work, we would like to explore learned orderings, or bidirectional models, to combat this.
Finally, the form of is extremely noisy and has many cube artifacts. The input of this function is both a one hot quantized action, as well as the floating point representation. It appears the model uses the quantization as its main feature and learns a sharp surface.
Appendix C Add SDQN
In this section, we discuss a different model that also makes use of sequential prediction and quantization. The SDQN model uses partialDQN’s, , to define sequential greedy policies over all dimensions, . In this setting, one can think of it as acting similarly to an environment transformed to predict one dimension at a time. Thus reward signals from the final reward must be propagated back through a chain of partialDQN. This results in a more difficult credit assignment problem that needs to be solved. This model attempts to solve this by changing the structure of networks. This formulation, called Add SDQN replaces the series of maxes from the Bellman backup with a summation over learned functions.
Results for this method and the others presented in the appendices can be found in Appendix E.2.
c.1 Method
As before, we aim to learn a deterministic policy of the DQN, where
(8) 
Here the value is defined as the sum of :
(9) 
Unlike before, the sequential components no longer represent functions, so we will swap the notation of our compositional function from to .
The parameters of all the models are trained by matching , in equation 9 to as follows:
(10) 
We train with learning, as shown in equation 5.
Unlike the sequential max policy of previous section, here we find the optimal action by beam search to maximize equation 9. In practice, the learning dynamics of the neural network parameterizations we use yield solutions that are amenable to this shallow search and do not require a search of the full exponential space.
A figure showing this network’s training procedure can be found in Figure C.4.
c.2 Network Parameterization Note
At this point, we have only tested the LSTM variant and not the untied parameterization. Optimizing these model families is an ongoing work and we could assume that Add SDQN could potentially perform better if it were using the untied version as we did for the originally presented SDQN algorithm.
Appendix D Prob SDQN
In the previous sections we showed the use of our technique within the Qlearning framework. Here we describe its use within the offpolicy, actorcritic framework to learn a sequential policy [42, 43, 10]. We name this model Prob SDQN.
Results for this method and the others presented in the appendices can be found in Appendix E.2.
d.1 Method
We define a policy, , recursively as follows where is defined as:
(11) 
Unlike in the previous two models, is not some form of of another function but a learned function.
As in previous work, we optimize the expected reward of policy (as predicted by a function) under data collected from some other policy, [10, 19].
(12) 
where is an estimate of values and is trained to minimize equation 5.
We use policy gradients / REINFORCE to compute gradients of equation 12 [49]. Because is trained offpolicy, we include an importance sampling term to correct for the mismatch of the state distribution, , from the learned policy and the state distribution, from the behavior policy. To reduce variance, we employ a Monte Carlo baseline, , which is the mean reward from samples from [23].
(13) 
(14) 
In practice, using importance sampling ratios like this have been known to introduce high variance in gradients [28]. In this work, we make the assumption that and are very similar – the smaller the replay buffer is, the better this assumption. This term can be removed and we are no longer required to compute . This assumption introduces bias, but drastically lowers the variance of this gradient in practice.
During training, we approximate the highest probability path with a beam search and treat our policy as deterministic.
As is the case with offpolicy algorithms, training the policy does not require samples from the environment under the target policy – only evaluations of . This makes it much more attractive for tasks where sampling the environment is challenging – such as robotics.
A figure showing the training procedure can be found in figure D.5.
d.2 Network Parameterization Note
At this time, we have only tested the LSTM variant and not the untied parameterization. Optimizing these model families is ongoing and we could assume that Prob SDQN could potentially perform better if it were using the untied version as we did for the originally presented SDQN algorithm.
Appendix E Independent DQN
In the previous work, all previous methods contain both discretization and sequential prediction to enable arbitrarily complex distributions. We wished to separate these two factors, so we constructed a model that just performed discretization and keeps the independence assumption that is commonly used in RL.
Results for this method and the others presented in the appendices can be found in Appendix E.2.
e.1 Method
We define a function as the mean of many independent functions, :
(15) 
Because each is independent of all other actions, a tractable max exists and as such we define our policy as an over each independent dimension:
(16) 
As in previous models, is then trained with learning as in equation 5.
e.2 Results
Results for the additional techniques can be seen in table 1. SDQN and DDPG are copied from the previous section for ease of reference.
The Add SDQN method performs about 800 reward better on our hardest task, humanoid, but performs worse on the simpler environments. The IDQN method, somewhat surprisingly, is able to learn reasonable policies in spite of its limited functional form. In the case of swimmer, the independent model performs slightly better than SDQN, but worse than the other versions of our models. Prob SDQN performs the best on the swimmer task by large margin, but underperforms dramatically on half cheetah and humanoid. Looking into trade offs of model design with respect to environments is of interest to us for future work.
agent  hopper  swimmer  half cheetah  humanoid  walker2d 

SDQN  3342.62  179.23  7774.77  3096.71  3227.73 
Prob SDQN  3056.35  268.88  650.33  691.11  2342.97 
Add SDQN  1624.33  202.04  4051.47  3811.44  1517.17 
IDQN  2135.47  189.52  2563.25  1032.60  668.28 
DDPG  3296.49  133.52  6614.26  3055.98  3640.93 

Appendix F Training and Model details
f.1 Hyper Parameter Sensitivity
The most sensitive hyper parameters were the learning rate of the two networks, reward scale, and finally, discount factor. Parameters such as batch size, quantization amounts, and network sizes mattered to a lesser extent. We found it best to have exploration be done on less than 10% of the transitions. We didn’t see any particular value below this that gave better performance. In our experiments, the structure of model used also played a large impact in performance, such as, using tied versus untied weights for each sequential prediction model.
In future work, we would like to lower the amount of hyper parameters needed in these algorithms and study the effects of various hyper parameters more thoroughly.
f.2 Sdqn
In this model, we looked at a number of configurations. Of the greatest importance is the form of the model itself. We looked at an LSTM parameterization as well as an untied weight variant. The untied weight variant’s parameter search is listed below.
To compute we first take the state and previous actions and do one fully connected layer of size "embedding size". We then concatenate these representations and feed them into a 2 hidden layer MLP with "hidden size" units. The output of this network is "quantization bins" wide with no activation.
uses the same embedding of state and action and then passes it though a 1 hidden layer fully connected network finally outputting a single scalar.
Hyper Parameter  Range  Notes 
use batchnorm  on, off  use batchnorm on the networks 
replay capacity:  2e4, 2e5, inf  
batch size  256, 512  
quantization bins  32  We found higher values generally converged to better final solutions. 
hidden size  256, 512  
embedding size  128  
reward scaling  0.05, 0.1  
target network moving average  0.99, 0.99, 0.98  
adam learning rate for TD updates  1e3, 1e4, 1e5  
adam learning rate for maxing network  1e3, 1e4, 1e5  
gradient clipping  off, 10  
l2 reguralization  off, 1e1, 1e2, 1e3, 1e4  
learning rate decay  log linear, none  
learning rate decay delta  2  Decay 2 orders of magnitude down from 0 to 1m steps. 
td term multiplier  0.2, 0.5,  
using target network on double q network  on, off  
tree consistency multiplier  5  Scalar on the tree loss 
energy use penalty  0, 0.05, 0.1, 0.2  Factor multiplied by velocity and subtracted from reward 
gamma (discount factor)  0.9, 0.99, 0.999  
drag down reguralizer  0.0, 0.1  Constant factor to penalize high q values. This is used to control over exploration. It has a very small effect in final performance. 
tree target greedy penalty  1.0  A penalty on MSE or Q predictions from greedy net to target. This acts to prevent taking too large steps in function space 
exploration type  boltzmann or epsilon greedy  
boltzman temperature  1.0, 0.5, 0.1, 0.01, 0.001  
prob to sample from boltzman (vs take max)  1.0, 0.5, 0.2, 0.1, 0.0  
boltzman decay  decay both prob to sample and boltzman temperature to 0.001  
epsilon noise  0.5, 0.2, 0.1, 0.05, 0.01  
epsilon decay  linearly to 0.001 over the first 1M steps 
Best hyper parameters for a subset of environments.
Hyper Parameter  Hopper  Cheetah 
use batchnorm  off  off 
replay capacity:  inf  inf 
batch size  512  512 
quantization bins  32  32 
hidden size  256  512 
embedding size  128  128 
reward scaling  0.1  0.1 
target network moving average  0.99  0.9 
adam learning rate for TD updates  1e3  1e3 
adam learning rate for maxing network  5e5  1e4 
gradient clipping  off  off 
l2 regularization  1e4  1e4 
learning rate decay for q  log linear  log linear 
learning rate decay delta for q  2 orders of magnitude down from interpolated over 1M steps. 
2 orders down interpolated over 1M steps 
learning rate decay for tree  none  none 
learning rate decay delta for tree  NA  NA 
td term multiplier  0.5  0.5 
useing target network on double q network  off  on 
tree consistency multiplier  5  5 
energy use penalty  0  0.0 
gamma (discount factor)  0.995  0.99 
drag down reguralizer  0.1  0.1 
tree target greedy penalty  1.0  1.0 
exploration type  boltzmann  boltzmann 
boltzman temperature  1.0  0.1 
prob to sample from boltzman (vs take max)  0.2  1.0 
boltzman decay  decay both prob to sample and boltzman temperature to 0.001 over 1M steps  decay both prob to sample and boltzman temperature to 0.001 over 1M steps 
epsilon noise  NA  NA 
epsilon decay  NA  NA 
f.3 Add SDQN
is parameterized the same as in F.2. The policy is parameterized by a multi layer LSTM. Each step of the LSTM takes in the previous action value, and outputs some number of "quantization bins." An action is selected, converted back to a one hot representation, and fed into an embedding module. The result of the embedding is then fed into the LSTM.
When predicting the first action, a learned initial state variable is fed into the LSTM as the embedding.
Hyper Parameter  Range  Notes 
replay capacity:  2e4, 2e5, inf  
batch size  256, 512  
quantization bins  8, 16, 32  We found higher values generally converged to better final solutions. 
lstm hidden size  128, 256, 512  
number of lstm layers  1, 2  
embedding size  128  
Adam learning rate for TD updates  1e3, 1e4, 1e5  
Adam learning rate for maxing network  1e3, 1e4, 1e5  
td term multiplier  1.0, 0.2, 0.5,  
target network moving average  0.99, 0.99, 0.999  
using target network on double q network  on, off  
reward scaling  0.01, 0.05, 0.1, 0.12, 0.08  
train number beams  1,2,3  number of beams used when evaluating argmax during training. 
eval number beams  1,2,3  number of beams used when evaluating the argmax during evaluation. 
exploration type  boltzmann or epsilon greedy  Epsilon noise injected after each action choice 
boltzmann temperature  1.0, 0.5, 0.1, 0.01, 0.001  
prob to sample from boltzmann (vs take max)  1.0, 0.5, 0.2, 0.1, 0.05  
epsilon noise  0.5, 0.2, 0.1, 0.05, 0.01 
Best hyper parameters for a subset of environments.
Hyper Parameter  Hopper  Cheetah 
replay capacity:  inf  inf 
batch size  256  256 
quantization bins  16  32 
lstm hidden size  128  256 
number of lstm layers  1  1 
embedding size  128  
Adam learning rate for TD updates  1e4  5e3 
Adam learning rate for maxing network  1e5  5e5 
td term multiplier  0.2  1.0 
target network moving average  0.999  0.99 
using target network on double q network  on  on 
reward scaling  0.05  0.12 
train number beams  2  1 
eval number beams  2  2 
exploration type  boltzmann  
boltzmann temperature  0.1  0.1 
prob to sample from boltzmann (vs take max)  0.5  0.5 
epsilon noise  NA  NA 
f.4 Prob SDQN
is parameterized the same as in F.2. The policy is parameterized by a multi layer LSTM. Each step of the LSTM takes in the previous action value, and outputs some number of "quantization bins". A softmax activation is applied thus normalizing this distribution. An action is selected, converted back to a one hot representation and fed into an embedding module. The result is then fed into the next time step.
When predicting the first action, a learned initial state variable is fed into the LSTM as the embedding.
Hyper Parameter  Range  Notes 
replay capacity:  2e4, 2e5, inf  
batch size  256, 512  
quantization bins  8, 16, 32  We found higher values generally converged to better final solutions. 
hidden size  256  
embedding size  128  
adam learning rate for TD updates  1e3, 1e4  
adam learning rate for maxing network  1e4, 1e5, 1e6  
td term multiplier  10, 1.0, 0.5, 0.1,  
target network moving average  0.995, 0.99, 0.999, 0.98  
number of baseline samples  2, 3, 4  
train number beams  1,2,3  number of beams used when evaluating argmax during training. 
eval number beams  1,2,3  number of beams used when evaluating the argmax during evaluation. 
epsilon noise  0.5, 0.2, 0.1, 0.05, 0.01  
epsilon noise decay  linearly move to 0.001 over 1m steps  
reward scaling  0.0005, 0.001, 0.01, 0.015, 0.1, 1  
energy use penalty  0, 0.05, 0.1, 0.2  Factor multiplied by velocity and subtracted from reward 
entropy regularizer penalty  1.0 
Best parameters from for a subset of environments.
Hyper Parameter  Hopper  Cheetah 
replay capacity:  2e4  2e4 
batch size  512  256 
quantization bins  32  32 
hidden size  256  256 
embedding size  128  128 
adam learning rate for TD updates  1e4  1e3 
adam learning rate for maxing network  1e5  1e4 
td term multiplier  10  10 
target network moving average  0.98  0.99 
number of baseline samples  2  4 
train number beams  1  1 
eval number beams  1  1 
epsilon noise  0.1  0.0 
epsilon noise decay  linearly move to 0.001 over 1m steps  NA 
reward scaling  0.1  0.5 
energy use penalty  0.05  0.0 
entropy regularizer penalty  1.0  1.0 
f.5 Idqn
We construct 1 hidden layer MLP, one for each action dimension. Each mlp has "hidden size" units. We perform Bellman updates using the same strategy done in DQN [25].
We perform our initial hyper parameter search with points sampled from the following grid.
Hyper Parameter  Range  Notes 
replay capacity:  inf  
batch size  256, 512  
quantization bins  8, 16, 32  
hidden size  128, 256, 512  
gamma (discount factor)  0.95, 0.99, 0.995, 0.999  
reward scaling  0.05, 0.1  
target network moving average  0.99, 0.99, 0.98  
l2 reguralization  off, 1e1, 1e2, 1e3, 1e4  
noise type  epsilon greedy  
epsilon amount(percent of time noise)  0.01, 0.05, 0.1, 0.2  
adam learning rate  1e3, 1e4, 1e5 
Best hyper parameters on a subset of environments.
Hyper Parameter  Hopper  Cheetah 
replay capacity:  inf  inf 
batch size  512  256 
quantization bins  16  8 
hidden size  512  128 
gamma (discount factor)  0.99  0.995 
reward scaling  0.1  0.1 
target network moving average  0.99  0.99 
l2 reguralization  off  1e4 
noise type  epsilon greedy  epsilon greedy 
epsilon amount(percent of time noise)  0.05  0.01 
adam learning rate  1e4  1e4 
f.6 Ddpg
Due to our DDPG implementation, we chose to do a mix of random search and parameter selection on a grid.
Hyper Parameter  Range  Notes 
learning rate :  [1e5, 1e3]  Done on log scale 
gamma (discount factor)  0.95, 0.99, 0.995, 0.999  
batch size  [10, 500]  
actor hidden 1 layer units  [10, 300]  
actor hidden 2 layer units  [5, 200]  
critic hidden 1 layer units  [10, 400]  
critic hidden 2 layer units  [4, 300]  
reward scaling  0.0005, 0.001, 0.01, 0.015, 0.1, 1  
target network update rate  [10, 500]  
target network update fraction  [1e3, 1e1]  Done on log scale 
gradient clipping from critic to target  [0, 10]  
OU noise damping  [0, 1]  
OU noise std  [0, 1] 
Best hyper parameters on a subset of environments.
Hyper Parameter  Hopper  Cheetah 
learning rate :  0.00026  0.000086 
gamma (discount factor)  0.995  .995 
batch size  451  117 
actor hidden 1 layer units  48  11 
actor hidden 2 layer units  107  199 
critic hidden 1 layer units  349  164 
critic hidden 2 layer units  299  256 
reward scaling  0.01  0.01 
target network update rate  10  445 
target network update fraction  0.0103  0.0677 
gradient clipping from critic to target  8.49  0.600 
OU noise damping  0.0367  0.6045 
OU noise std  0.074  0.255 
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