1 Introduction
Experience replay (ER) [Lin1992]
is currently the most common way to train value functions approximated as neural networks (NNs), in an online RL setting
[Adam et al.2012, Wawrzyński and Tanwani2013]. The buffer in ER is typically a recency buffer, storing the most recent transitions, composed of state, action, next state and reward. At each environment time step, the NN gets updated by using a minibatch of samples from the ER buffer, that is, the agent replays those transitions. ER enables the agent to be more sample efficient, and in fact can be seen as a simple form of modelbased RL [van Seijen and Sutton2015]. This connection is specific to the Dyna architecture [Sutton1990, Sutton1991], where the agent maintains a searchcontrol (SC) queue of pairs of states and actions and uses a model to generate next states and rewards. These simulated transitions are used to update values. ER, then, can be seen as a variant of Dyna with a nonparameteric model, where searchcontrol is determined by the observed states and actions.By moving beyond ER to Dyna with a learned model, we can potentially benefit from increased flexibility in obtaining simulated transitions. Having access to a model allows us to generate unobserved transitions, from a given stateaction pair. For example, a model allows the agent to obtain onpolicy or exploratory samples from a given state, which has been reported to have advantages [Gu et al.2016, Pan et al.2018, Santos et al.2012, Peng et al.2018]. More generally, models allow for a variety of choices for searchcontrol, which is critical as it emphasizes different states during the planning phase. Prioritized sweeping [Moore and Atkeson1993] uses the model to obtain predecessor states, with states sampled according to the absolute value of temporal difference error. This early work, and more recent work [Sutton et al.2008, Pan et al.2018, Corneil et al.2018], showed this addition significantly outperformed Dyna with states uniformly sampled from observed states. Most of the work on searchcontrol, however, has been limited to sampling visited or predecessor states. Predecessor states require a reverse model, which can be limiting. The range of possibilities has yet to be explored for searchcontrol and there is room for many more ideas.
In this work, we investigate using sampled trajectories by hill climbing on our learned value function to generate states for searchcontrol. Updating along such trajectories has the effect of propagating value from regions the agent currently believes to be highvalue. This strategy enables the agent to preemptively update regions where it is likely to visit next. Further, it focuses updates in areas where approximate values are high, and so important to the agent. To obtain such states for searchcontrol, we propose a noisy natural projected gradient algorithm. We show this has a connection to Langevin dynamics, whose distribution converges to the Gibbs distribution, where the density is proportional to the exponential of the state values. We empirically study different sampling distributions for populating the searchcontrol queue, and verify the effectiveness of hill climbing based on estimated values. We conduct experiments showing improved performance in four benchmark domains, as compared to DQN^{1}^{1}1We use DQN to refer to the algorithm by [Mnih et al.2015] that uses ER and target network, but not the exact original architecture., and illustrate the usage of our architecture for continuous control.
2 Background
We formalize the environment as a Markov Decision Process (MDP)
, where is the state space, is the action space,is the transition probabilities,
is the reward function, and is the discount factor. At each time step , the agent observes a state and takes an action , transitions to and receives a scalar reward according to the reward function .Typically, the goal is to learn a policy to maximize the expected return starting from some fixed initial state. One popular algorithm is Qlearning, by which we can obtain approximate actionvalues for parameters . The policy corresponds to acting greedily according to these actionvalues: for each state, select action . The Qlearning update for a sampled transition is
Though frequently used, such an update may not be sound with function approximation. Target networks [Mnih et al.2015] are typically used to improve stability when training NNs, where the bootstrap target on the next step is a fixed, older estimate of the actionvalues.
ER and Dyna can both be used to improve sample efficiency of DQN. Dyna is a modelbased method that simulates (or replays) transitions, to reduce the number of required interactions with the environment. A model is sometimes available a priori (e.g., from physical equations of the dynamics) or is learned using data collected through interacting with the environment. The generic Dyna architecture, with explicit pseudocode given by [Sutton and Barto2018, Chapter 8], can be summarized as follows. When the agent interacts with the real world, it updates both the actionvalue function and the learned model using the real transition. The agent then performs planning steps. In each planning step, the agent samples from the searchcontrol queue, generates next state and reward from using the model, and updates the actionvalues using Qlearning with the tuple .
3 A Motivating Example
In this section we provide an example of how the value function surface changes during learning on a simple continuousstate GridWorld domain. This provides intuition on why it is useful to populate the searchcontrol queue with states obtained by hill climbing on the estimated value function, as proposed in the next section.
Consider the GridWorld in Figure 1(a), which is a variant of the one introduced by [Peng and Williams1993]. In each episode, the agent starts from a uniformly sampled point from the area and terminates when it reaches the goal area . There are four actions ; each leads to a unit move towards the corresponding direction. As a costtogoal problem, the reward is per step.
In Figure 1, we plot the value function surface after , k and k minibatch updates to DQN. We visualize the gradient ascent trajectories with gradient steps starting from five states , , , , and . The gradient of the value function used in the gradient ascent is
(1) 
At the beginning, with a randomly initialized NN, the gradient with respect to state is almost zero, as seen in Figure 1(b). As the DQN agent updates its parameters, the gradient ascent generates trajectories directed towards the goal, though after only 14k steps, these are not yet contiguous, as seen Figure 1(c). After k steps, as in Figure 1(d), even though the value function is still inaccurate, the gradient ascent trajectories take all initial states to the goal area. This suggests that as long as the estimated value function roughly reflects the shape of the optimal value function, the trajectories provide a demonstration of how to reach the goal—or highvalue regions—and speed up learning by focusing updates on these relevant regions.
More generally, by focusing planning on regions the agent thinks are highvalue, it can quickly correct value function estimates before visiting those regions, and so avoid unnecessary interaction. We demonstrate this in Figure 1(e), where the agent obtains gains in performance by updating from highvalue states, even when its value estimates have the wrong shape. After 20k learning steps, the values are flipped by negating the sign of the parameters in the output layer of the NN. HCDyna, introduced in Section 5, quickly recovers compared to DQN and OnPolicy updates from the ER buffer. Planning steps help pushing down these erroneously highvalues, and the agent can recover much more quickly.
4 Effective Hill Climbing
To generate states for search control, we need an algorithm that can climb on the estimated value function surface. For general value function approximators, such as NNs, this can be difficult. The value function surface can be very flat or very rugged, causing the gradient ascent to get stuck in local optima and hence interrupt the gradient traveling process. Further, the state variables may have very different numerical scales. When using a regular gradient ascent method, it is likely for the state variables with a smaller numerical scale to immediately go out of the state space. Lastly, gradient ascent is unconstrained, potentially generating unrealizable states.
In this section, we propose solutions for all these issues. We provide a noisy invariant projected gradient ascent strategy to generate meaningful trajectories of states for searchcontrol. We then discuss connections to Langevin dynamics, a model for heat diffusion, which provides insight into the sampling distribution of our searchcontrol queue.
4.1 Noisy Natural Projected Gradient Ascent
To address the first issue, of flat or rugged function surfaces, we propose to add Gaussian noise on each gradient ascent step. Intuitively, this provides robustness to flat regions and avoids getting stuck in local maxima on the function surface, by diffusing across the surface to highvalue regions.
To address the second issue of vastly different numerical scales among state variables, we use a standard strategy to be invariant to scale: natural gradient ascent. A popular choice of natural gradient is derived by defining the metric tensor as the Fisher information matrix
[Amari and Douglas1998, Amari1998, Thomas et al.2016]. We introduce a simple and computationally efficient metric tensor: the inverse of covariance matrix of the states . This choice is simple, because the covariance matrix can easily be estimated online. We can define the following inner product:which induces a vector space—the Riemannian manifold—where we can compute the distance of two points and that are close to each other by . The steepest ascent updating rule based on this distance metric becomes , where is the gradient vector.
We demonstrate the utility of using the natural gradient scaling. Figure 2 shows the states from the searchcontrol queue filled by hill climbing in early stages of learning (after 8000 steps) on MountainCar. The domain has two state variables with very different numerical scale: position and velocity . Using a regular gradient update, the queue shows a state distribution with many states concentrated near the top since it is very easy for the velocity variable to go out of boundary. In contrast, the one with natural gradient, shows clear trajectories with an obvious tendency to the right top area (position ), which is the goal area.
We use projected gradient updates to address the third issue regarding unrealizable states. We explain the issue and solution using the Acrobot domain. The first two state variables are , where is the angle between the first robot arm’s link and the vector pointing downwards. This induces the restriction that . The hill climbing process could generate many states that do not satisfy this restriction. This could potentially degrade performance, since the NN needs to generalize to these states unnecessarily. We can use a projection operator to enforce such restrictions, whenever known, after each gradient ascent step. In Acrobot, is a simple normalization. In many settings, the constraints are simple box constraints, with projection just inside the boundary.
Now we are ready to introduce our final hill climbing rule:
(2) 
where is Gaussian noise and a stepsize. For simplicity, we set the stepsize to across all results in this work, though of course there could be better choices.
4.2 Connection to Langevin Dynamics
The proposed hill climbing procedure is similar to Langevin dynamics, which is frequently used as a tool to analyze optimization algorithms or to acquire an estimate of the expected parameter values w.r.t. some posterior distribution in Bayesian learning [Welling and Teh2011]. The overdamped Langevin dynamics can be described by a stochastic differential equation (SDE) , where is a dimensional Brownian motion and is a continuous differentiable function. Under some conditions, it turns out that the Langevin diffusion converges to a unique invariant distribution [Chiang et al.1987].
By apply the EulerMaruyama discretization scheme to the SDE, we acquire the discretized version where is an i.i.d. sequence of standard dimensional Gaussian random vectors and is a sequence of step sizes. This discretization scheme was used to acquire samples from the original invariant distribution
through the Markov chain
when it converges to the chain’s stationary distribution [Roberts1996]. The distance between the limiting distribution of and the invariant distribution of the underlying SDE has been characterized through various bounds [Durmus and Moulines2017].When we perform hill climbing, the parameter is constant at each time step . By choosing the function in the SDE above to be equal to , we see that the state distribution in our searchcontrol queue is approximately^{2}^{2}2Different assumptions on and properties of can give convergence claims with different strengths. Also refer to [Welling and Teh2011] for the discussion on the use of a preconditioner.
An important difference between the theoretical limiting distribution and the actual distribution acquired by our hill climbing method is that our trajectories would also include the states during the burnin or transient period, which refers to the period before the stationary behavior is reached. We would want to point out that those states play an essential role in improving learning efficiency as we will demonstrate in section 6.2.
5 Hill Climbing Dyna
In this section, we provide the full algorithm, called Hill Climbing Dyna, summarized in Algorithm 1. The key component is to use the Hill Climbing procedure developed in the previous section, to generate states for searchcontrol (SC). To ensure some separation between states in the searchcontrol queue, we use a threshold
to decide whether or not to add a state into the queue. We use a simple heuristic to set this threshold on each step, as the following sample average:
. The start state for the gradient ascent is randomly sampled from the ER buffer.In addition to using this new method for search control, we also found it beneficial to include updates on the experience generated in the real world. The minibatch sampled for training has proportion of transitions generated by states from the SC queue, and from the ER buffer. For example, for with a minibatch size of , the updates consists of transitions generated from states in the SC queue and 6 transitions from the ER buffer. Previous work using Dyna for learning NN value functions also used such mixed minibatches [Holland et al.2018].
One potential reason this addition is beneficial is that it alleviates issues with heavily skewing the sampling distribution to be offpolicy. Tabular Qlearning is an offpolicy learning algorithm, which has strong convergence guarantees under mild assumptions
[Tsitsiklis1994]. When moving to function approximation, however, convergence of Qlearning is much less well understood. The change in sampling distribution for the states could significantly impact convergence rates, and potentially even cause divergence. Empirically, previous prioritized ER work pointed out that skewing the sampling distribution from the ER buffer can lead to a biased solution [Schaul et al.2016]. Though the ER buffer is not onpolicy, because the policy is continually changing, the distribution of states is closer to the states that would be sampled by the current policy than those in SC. Using mixed states from the ER buffer, and those generated by Hill Climbing, could alleviate some of the issues with this skewness.Another possible reason that such mixed sampling could be necessary is due to model error. The use of real experience could mitigate issues with such error. We found, however, that this mixing has an effect even when using the true model. This suggests that this phenomenon indeed is related to the distribution over states.
We provide a small experiment in the GridWorld, depicted in Figure 1, using both a continuousstate and a discretestate version. We include a discrete state version, so we can demonstrate that the effect persists even in a tabular setting when Qlearning is known to be stable. The continuousstate setting uses NNs—as described more fully in Section 6—with a minibatch size of 32. For the tabular setting, the minibatch size is 1; updates are randomly selected to be from the SC queue or ER buffer proportional to . Figure 3 shows the performance of HCDyna as the mixing proportion increases from (ER only) to (SC only). In both cases, a mixing rate around provides the best results. Generally, using too few searchcontrol samples do not improve performance; focusing too many updates on searchcontrol samples seems to slightly speed up early learning, but then later learning suffers. In all further experiments in this paper, we set .
6 Experiments
In this section, we demonstrate the utility of (DQN)HCDyna in several benchmark domains, and then analyze the learning effect of different sampling distributions to generate states for the searchcontrol queue.
6.1 Results in Benchmark Domains
In this section, we present empirical results on four classic domains: the GridWorld (Figure 1(a)), MountainCar, CartPole and Acrobot. We present both discrete and continuous action results in the GridWorld, and compare to DQN for the discrete control and to Deep Deterministic Policy Gradient (DDPG) for the continuous control [Lillicrap et al.2016]
. The agents all use a twolayer NN, with ReLU activations and 32 nodes in each layer. We include results using both the true model and the learned model, on the same plots. We further include multiple planning steps
, where for each real environment step, the agent does updates with a minibatch of size 32.In addition to ER, we add an onpolicy baseline called OnPolicyDyna. This algorithm samples a minibatch of states (not the full transition) from the ER buffer, but then generates the next state and reward using an onpolicy action. This baseline distinguishes when the gain of HCDyna algorithm is due to onpolicy sampled actions, rather than because of the states in our searchcontrol queue.
6.1.1 Discrete Action
The results in Figure 4 show that (a) HCDyna never harms performance over ER and OnPolicyDyna, and in some cases significantly improves performance, (b) these gains persist even under learned models and (c) there are clear gains from HCDyna even with a small number of planning steps. Interestingly, using multiple minibatch updates per time step can significantly improve the performance of all the algorithms. DQN, however, has very limited gain when moving from to planning steps on all domains except GridWorld, whereas HCDyna seems to more noticeably improve from more planning steps. This implies a possible limit of the usefulness of only using samples in the ER buffer.
We observe that the onpolicy actions does not always help. The GridWorld domain is in fact the only one where onpolicy actions (OnPolicyDyna) shows an advantage as the number of planning steps increase. This result provides evidence that the gain of our algorithm is due to the states in our searchcontrol queue, rather than onpolicy sampled actions. We also see that even though both modelbased methods perform worse when the model has to be learned compared to when the true model is available, HCDyna is consistently better than OnPolicyDyna across all domains/settings.
To gain intuition for why our algorithm achieves superior performance, we visualize the states in the searchcontrol queue for HCDyna in the GridWorld domain (Figure 5). We also show the states in the ER buffer at the same time step, for both HCDyna and DQN to contrast. There are two interesting outcomes from this visualization. First, the modification to searchcontrol significantly changes where the agent explores, as evidenced by the ER buffer distribution. Second, HCDyna has many states in the SC queue that are near the goal region even when its ER buffer samples concentrate on the left part on the square. The agent can still update around the goal region even when it is physically in the left part of the domain.
6.1.2 Continuous Control
Our architecture can easily be used with continuous actions, as long as the algorithm estimates values. We use DDPG [Lillicrap et al.2016] as an example for use inside HCDyna. DDPG is an actorcritic algorithm that uses the deterministic policy gradient theorem [Silver et al.2014]. Let be the actor network parameterized by , and be the critic. Given an initial state value , the gradient ascent direction can be computed by . In fact, because the gradient step causes small changes, we can further approximate this gradient more efficiently using
, without backpropagating the gradient through the actor network. We modified the GridWorld in Figure
1(a) to have action space and an action is executed as . Figure 6 shows the learning curve of DDPG, and DDPG with OnPolicyDyna and with HCDyna. As before, HCDyna shows significant early learning benefits and also reaches a better solution. This highlights that improved searchcontrol could be particularly effective for algorithms that are known to be prone to local minima, like ActorCritic algorithms.6.2 Investigating Sampling Distributions for Searchcontrol
We next investigate the importance of two choices in HCDyna: (a) using trajectories to highvalue regions and (b) using the agent’s value estimates to identify these important regions. To test this, we include following sampling methods for comparison: (a) HCDyna: hill climbing by using (our algorithm); (b) Gibbs: sampling ; (c) HCDynaVstar: hill climbing by using and (d) GibbsVstar: sampling , where is a prelearned optimal value function. We also include the baselines OnPolicyDyna, ER and UniformDyna, which uniformly samples states from the whole state space. All strategies mix with ER, using , to better give insight into performance differences.
To facilitate sampling from the Gibbs distribution and computing the optimal value function, we test on a simplified TabularGridWorld domain of size , without any obstacles. Each state is represented by an integer , assigned from bottom to top, left to right on the square with grids. HCDyna and HCDynaVstar assume that the state space is continuous on the square and each grid can be represented by its center’s coordinates. We use the finite difference method for hill climbing.
6.2.1 Comparing to the Gibbs Distribution
As we pointed out the connection to the Langevin dynamics in Section 4.2, the limiting behavior of our hill climbing strategy is approximately a Gibbs distribution. Figure 7(a) shows that HCDyna performs the best among all sampling distributions, including Gibbs and other baselines. This result suggests that the states during the burnin period matter. Figure 7(b) shows the state count by randomly sampling the same number of states from the HCDyna’s searchcontrol queue and from that filled by Gibbs distribution. We can see that the Gibbs one concentrates its distribution only on very high value states.
6.2.2 Comparing to True Values
One hypothesis is that the value estimates guide the agent to the goal. A natural comparison, then, is to use the optimal values, which should point the agent directly to the goal. Figure 8(a) indicates that using the estimates, rather than true values, is more beneficial for planning. This result highlights that there does seem to be some additional value to focusing updates based on the agent’s current value estimates. Comparing state distribution of GibbsVstar and HCDynaVstar in Figure 8(b) to Gibbs and HCDyna in Figure 7(b), one can see that both distributions are even more concentrated, which seems to negatively impact performance.
7 Conclusion
We presented a new Dyna algorithm, called HCDyna, which generates states for searchcontrol by using hill climbing on value estimates. We proposed a noisy natural projected gradient ascent strategy for the hill climbing process. We demonstrate that using states from hill climbing can significantly improve sample efficiency in several benchmark domains. We empirically investigated, and validated, several choices in our algorithm, including the use of natural gradients, the utility of mixing with ER samples, the benefits of using estimated values for search control. A natural next step is to further investigate other criteria for assigning importance to states. Our HC strategy is generic for any smooth function; not only for value estimates. A possible alternative is to investigate importance based on error in a region, or based more explicitly on optimism or uncertainty, to encourage systematic exploration.
Acknowledgments
We would like to acknowledge funding from the Canada CIFAR AI Chairs Program, Amii and NSERC.
Appendix A Appendix
The appendix includes all algorithmic and experimental details.
a.1 Algorithmic details
We include the classic Dyna architecture [Sutton1991, Sutton and Barto2018] in Algorithm 2 and our algorithm with additional details in Algorithm 3.
a.2 Experimental details
Implementation details of common settings.
The GridWorld domain is written by ourselve, all other discrete action domains are from OpenAI Gym [Brockman et al.2016] with version
. The exact environment names we used are: MountainCarv0, CartPolev1, Acrobotv1. Deep learning implementation is based on tensorflow with version
[Abadi et al.2015]. On all domains, we use Adam optimizer, Xavier initializer, set minibatch size , buffer sizek. All activation functions are ReLU except the output layer of the
value is linear, and the output layer of the actor network is tanh. The output layer parameters were initialized from a uniform distribution
, all other parameters are initialized using Xavier initialization [Glorot and Bengio2010].As for model learning, we learn a difference model to alleviate the effect of outliers, that is, we learn a neural network model with input
and output . The neural network has two units hidden ReLUlayers. The model is learned in an online manner and by using samples from ER buffer with a fixed learning rate as and minibatch size across all experiments.Termination condition on OpenAI environments.
On OpenAI, each environment has a time limit and the termination flag will be true if either the time limit reached or the actual termination condition satisfied. However, theoretically we should truncate the return if and only if the actual termination condition satisfied. All of our experiments are conducted by setting discount rate if and only if the actual termination condition satisfied. For example, on mountain car, if and only if the position.
Experimental details of TabularGridWorld domain.
The purpose of using the tabular domain is to study the learning performances by using different sampling distribution to fill the searchcontrol queue. Our TabularGridWorld is similar to the continuous state domain introduced in 1(a) except that we do not have a wall and we introduce stochasticity to make it more representative. Four actions are available and can take the agent to the next grid respectively. An action can be executed successfully with probability otherwise a random action is taken.The TabularGridWorld size is and each episode start from leftbottom grid and would terminate if reached the righttop grid or k time steps. The return will not be truncated unless the righttop grid is reached. The discount rate is . For all algorithms, we fixed the exploration noise as and sweep over learning rate . We fix using exploration noise and mixing rate . We use planning steps for all algorithms. We evaluate each algorithm every environment time steps. Parameter is optimized by using the last evaluation episodes to ensure convergence.
For our algorithm HCDyna, we do not sweep any additional parameters. We fix doing gradient ascent steps per environment time step and the injected noise is gaussian . When adding the noise or using finite difference method for computing gradient, we logically regard the domain as and hence each grid is a square with length . Specifically, when in a grid, we find its corresponding center’s coordinates as its location to add noise. As for gradient ascent with finite difference approximation, given a state , we compute the value increasing rate from each of its neighbors and pick up the one with largest increasing rate as the next state. That is, . Both the searchcontrol queue size and ER buffer size are setted as .
The optimal value function used for HCDynaVstar and GibbsVstar on this domain is acquired by taking the value function at the end of training ER for steps and averaged over random seeds.
Experimental details of continuous state domains.
All continuous state domain, we set discount rate . We set the episode length limit as for both GridWorld and MountainCar, while keep other domains as the default setting. We use warmup steps for all algorithms.
For all Q networks, we consistently use a neural network with two units hidden ReLUlayers. We use target network moving frequency and sweep learning rate for vanilla DQN with ER with planning step , then we directly use the same best learning rate for all other experiments. For our particular parameters, we fixed the same setting across all domains: mixing rate and is sample average, number of gradient steps with gradient ascent step size and queue size . We incrementally update the empirical covariance matrix. When evaluating each algorithm, we keep a small noise when taking action and evaluate one episode every environment time steps for each run.
References

[Abadi et al.2015]
Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen,
and et al.
TensorFlow: Largescale machine learning on heterogeneous systems.
2015. Software available from tensorflow.org.  [Adam et al.2012] Sander Adam, Lucian Busoniu, and Robert Babuska. Experience replay for realtime reinforcement learning control. IEEE Transactions on Systems, Man, and Cybernetics, pages 201–212, 2012.
 [Amari and Douglas1998] ShunIchi Amari and Scott C. Douglas. Why natural gradient? IEEE International Conference on Acoustics, Speech and Signal Processing, pages 1213–1216, 1998.
 [Amari1998] ShunIchi Amari. Natural gradient works efficiently in learning. Neural Computation, 10(2):251–276, 1998.
 [Brockman et al.2016] Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. OpenAI Gym. arXiv:1606.01540, 2016.
 [Chiang et al.1987] TzuuShuh Chiang, ChiiRuey Hwang, and Shuenn Jyi Sheu. Diffusion for global optimization in . SIAM Journal on Control and Optimization, pages 737–753, 1987.
 [Corneil et al.2018] Dane S. Corneil, Wulfram Gerstner, and Johanni Brea. Efficient modelbased deep reinforcement learning with variational state tabulation. ICML, pages 1049–1058, 2018.
 [Durmus and Moulines2017] Alain Durmus and Eric Moulines. Nonasymptotic convergence analysis for the unadjusted Langevin algorithm. The Annals of Applied Probability, pages 1551–1587, 2017.

[Glorot and Bengio2010]
Xavier Glorot and Yoshua Bengio.
Understanding the difficulty of training deep feedforward neural
networks.
In
International Conference on Artificial Intelligence and Statistics
, 2010.  [Gu et al.2016] Shixiang Gu, Timothy P. Lillicrap, Ilya Sutskever, and Sergey Levine. Continuous Deep QLearning with Modelbased Acceleration. In ICML, pages 2829–2838, 2016.
 [Holland et al.2018] G. Zacharias Holland, Erik Talvitie, and Michael Bowling. The effect of planning shape on dynastyle planning in highdimensional state spaces. CoRR, abs/1806.01825, 2018.
 [Lillicrap et al.2016] Timothy P. Lillicrap, Jonathan J. Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. Continuous control with deep reinforcement learning. In ICLR, 2016.
 [Lin1992] LongJi Lin. SelfImproving Reactive Agents Based On Reinforcement Learning, Planning and Teaching. Machine Learning, 1992.
 [Mnih et al.2015] Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A. Rusu, and et al. Humanlevel control through deep reinforcement learning. Nature, 2015.
 [Moore and Atkeson1993] Andrew W. Moore and Christopher G. Atkeson. Prioritized sweeping: Reinforcement learning with less data and less time. Machine learning, pages 103–130, 1993.
 [Pan et al.2018] Yangchen Pan, Muhammad Zaheer, Adam White, Andrew Patterson, and Martha White. Organizing experience: a deeper look at replay mechanisms for samplebased planning in continuous state domains. In IJCAI, pages 4794–4800, 2018.
 [Peng and Williams1993] Jing Peng and Ronald J Williams. Efficient learning and planning within the Dyna framework. Adaptive behavior, 1993.
 [Peng et al.2018] Baolin Peng, Xiujun Li, Jianfeng Gao, Jingjing Liu, KamFai Wong, and ShangYu Su. Deep DynaQ: Integrating planning for taskcompletion dialogue policy learning. In Annual Meeting of the Association for Computational Linguistics, pages 2182–2192, 2018.
 [Roberts1996] Richard L. Roberts, Gareth O.and Tweedie. Exponential convergence of langevin distributions and their discrete approximations. Bernoulli, pages 341–363, 1996.
 [Santos et al.2012] Matilde Santos, José Antonio Martín H., Victoria López, and Guillermo Botella. DynaH: A heuristic planning reinforcement learning algorithm applied to roleplaying game strategy decision systems. KnowledgeBased Systems, 32:28–36, 2012.
 [Schaul et al.2016] Tom Schaul, John Quan, Ioannis Antonoglou, and David Silver. Prioritized Experience Replay. In ICLR, 2016.
 [Silver et al.2014] David Silver, Guy Lever, Nicolas Heess, Thomas Degris, Daan Wierstra, and Martin Riedmiller. Deterministic policy gradient algorithms. In ICML, pages I–387–I–395, 2014.
 [Sutton and Barto2018] Richard S. Sutton and Andrew G. Barto. Reinforcement Learning: An Introduction. The MIT Press, second edition, 2018.
 [Sutton et al.2008] Richard S. Sutton, Csaba Szepesvári, Alborz Geramifard, and Michael Bowling. Dynastyle planning with linear function approximation and prioritized sweeping. In UAI, pages 528–536, 2008.
 [Sutton1990] Richard S. Sutton. Integrated architectures for learning, planning, and reacting based on approximating dynamic programming. In ML, 1990.
 [Sutton1991] Richard S. Sutton. Dyna, an integrated architecture for learning, planning, and reacting. SIGART Bulletin, 2(4):160–163, 1991.
 [Thomas et al.2016] Philip Thomas, Bruno Castro Silva, Christoph Dann, and Emma Brunskill. Energetic natural gradient descent. In ICML, pages 2887–2895, 2016.
 [Tsitsiklis1994] John N. Tsitsiklis. Asynchronous stochastic approximation and Qlearning. Machine Learning, pages 185–202, 1994.
 [van Seijen and Sutton2015] Harm van Seijen and Richard S. Sutton. A deeper look at planning as learning from replay. In ICML, pages 2314–2322, 2015.
 [Wawrzyński and Tanwani2013] Paweł Wawrzyński and Ajay Kumar Tanwani. Autonomous reinforcement learning with experience replay. Neural Networks, pages 156–167, 2013.
 [Welling and Teh2011] Max Welling and Yee Whye Teh. Bayesian learning via stochastic gradient Langevin dynamics. In ICML, pages 681–688, 2011.