# a_simple_decentralized_crossentropy_method__64c8a14d.pdf

A Simple Decentralized Cross-Entropy Method

Zichen Zhang1, Jun Jin2 Martin Jagersand1 Jun Luo2, Dale Schuurmans1,

1University of Alberta 2Huawei Noah s Ark Lab equal advising {zichen2,mj7,daes}@ualberta.ca {jun.jin1,jun.luo1}@huawei.com

Cross-Entropy Method (CEM) is commonly used for planning in model-based reinforcement learning (MBRL) where a centralized approach is typically utilized to update the sampling distribution based on only the top-k operations results on samples. In this paper, we show that such a centralized approach makes CEM vulnerable to local optima, thus impairing its sample efﬁciency. To tackle this issue, we propose Decentralized CEM (Decent CEM), a simple but effective improvement over classical CEM, by using an ensemble of CEM instances running independently from one another, and each performing a local improvement of its own sampling distribution. We provide both theoretical and empirical analysis to demonstrate the effectiveness of this simple decentralized approach. We empirically show that, compared to the classical centralized approach using either a single or even a mixture of Gaussian distributions, our Decent CEM ﬁnds the global optimum much more consistently thus improves the sample efﬁciency. Furthermore, we plug in our Decent CEM in the planning problem of MBRL, and evaluate our approach in several continuous control environments, with comparison to the stateof-art CEM based MBRL approaches (PETS and POPLIN). Results show sample efﬁciency improvement by simply replacing the classical CEM module with our Decent CEM module, while only sacriﬁcing a reasonable amount of computational cost. Lastly, we conduct ablation studies for more in-depth analysis. Code is available at https://github.com/vincentzhang/decent CEM.

1 Introduction

Model-based reinforcement learning (MBRL) uses a model as a proxy of the environment for planning actions in multiple steps. This paper studies planning in MBRL with a speciﬁc focus on the Cross Entropy Method (CEM) [De Boer et al., 2005, Mannor et al., 2003], which is popular in MBRL due to its ease of use and strong empirical performance [Chua et al., 2018, Hafner et al., 2019, Wang and Ba, 2020, Zhang et al., 2021, Yang et al., 2020]. CEM is a stochastic, derivative-free optimization method. It uses a sampling distribution to generate imaginary trajectories of environment-agent interactions with the model. These trajectories are then ranked based on their returns computed from the rewards given by the model. The sampling distribution is updated to increase the likelihood of producing the top-k trajectories with higher returns. These steps are iterated and eventually yield an improved distribution over the action sequences to guide the action execution in the real environment.

Despite the strong empirical performance of CEM for planning, it is prone to two problems: (1) lower sample efﬁciency as the dimensionality of solution space increases, and (2) the Gaussian distribution that is commonly used for sampling may cause the optimization to get stuck in local optima of multi-modal solution spaces commonly seen in real-world problems. Previous works addressing these problems either add gradient-based updates of the samples to optimize the parameters of CEM,

Work partially done during Zichen s internship at Huawei Noah s Ark Lab.

36th Conference on Neural Information Processing Systems (Neur IPS 2022).

or adopt more expressive sampling distributions, such as using Gaussian Mixture Model [Okada and Taniguchi, 2020] or masked auto-regressive neural network [Hakhamaneshi et al., 2020]. Nevertheless, all CEM implementations to date are limited to a centralized formulation where the ranking step involves all samples. As analyzed below and in Section 3, such a centralized design makes CEM vulnerable to local optima and impairs its sample efﬁciency.

We propose Decentralized CEM (Decent CEM), a simple but effective improvement over classical CEM, to address the above problems. Rather than ranking all samples, as in the centralized design, our method distribute the sampling budget across an ensemble of CEM instances. These instances run independently from one another, and each performs a local improvement of its own sampling distribution based on the ranking of its generated samples. The best action is then aggregated by taking an arg max among the solution of the instances. It recovers the conventional CEM when the number of instance is one.

(a) Centralized CEM

(b) Decentralized CEM

Figure 1: Illustration of CEM approaches in optimization. Shades of red indicate relative value of the 2D optimization landscape: redder is better. and optimal solutions are near bottom left corner of the solution space. Blue dots are top-k samples, and black dots are other samples. Open dots represent the sampling distributions whose sizes indicate the number of generated samples.

We hypothesize that by shifting to this decentralized design, CEM can be less susceptible to premature convergence caused by the centralized ranking step. As illustrated in Fig. 1, the centralized sampling distribution exhibits a bias toward the sub-optimal solutions near top right, due to the global top-k ranking. This bias would occur regardless of the family of distributions used. In comparison, a decentralized approach could maintain enough diversity thanks to its local top-k ranking in each sampling instance.

Through a one-dimensional multi-modal optimization problem in Section 3, we show that Decent CEM empirically ﬁnds the global optimum more consistently than centralized CEM approaches that use either a single or a mixture of Gaussian distributions. Also we show that Decent CEM is theoretically sound that it converges almost surely to a local optimum. We further apply it to sequential decision making problems and use neural networks to parameterize the sampling distributions. Empirical results in several continuous control environments suggest that Decent CEM offers an effective mechanism to improve the sample efﬁciency over the baseline CEM under the same sample budget for planning.

2 Preliminaries

We consider a Markov Decision Process (MDP) speciﬁed by (S,A,R,P,γ,d0,T). S Rds is the state space, A Rda is the action space. ds, da are scalars denoting the dimensionality. R : S A ! R is the reward function that maps a state and action pair to a real-valued reward. P(s0|s, a) : S A S ! R+ is the transition probability from a state and action pair s, a to the next state s0. γ 2 [0, 1] is the discount factor. d0 denotes the distribution of the initial state s0. At time step t, the agent receives a state st 2 and takes an action at according to a policy ( |s) that maps the state to a probability distribution over the action space. The environment transitions to the next state st+1 P( |st, at) and gives a reward rt = R(st, at) to the agent. Following the settings from CEM-based MBRL papers (Sec. 6.1), we assume that the reward function is deterministic (a mild assumption [Agarwal et al., 2019]) and known. Note that they are not fundamental limitations of CEM and are adopted here so as to be consistent with the literature. The return Gt = PT

i=0 γirt+i, is the sum of discounted reward within an episode length of T. The agent aims to ﬁnd a policy that maximizes the expected return. We denote the learned model in MBRL as f!( |s, a), which is parameterized by ! and approximates P( |s, a).

Planning with the Cross Entropy Method Planning in MBRL is about leveraging the model to ﬁnd the best action in terms of its return. Model-Predictive-Control (MPC) performs decision-time

2We assume full observability, i.e. agent has access to the state.

planning at each time step up to a horizon to ﬁnd the optimal action sequence:

MPC(st) = arg max

i=0 γir(st+i, at+i) + γHV (s H)] (1)

where H is the planning horizon, at:t+H 1 denotes the action sequence from time step t to t+H 1, and V (s H) is the terminal value function at the end of the horizon. The ﬁrst action in this sequence is executed and the rest are discarded. The agent then re-plans at the next time step.

Figure 2: Cross Entropy Method (CEM) for Planning in MBRL

The Cross-Entropy Method (CEM) is a gradient-free optimization method that can be used for solving Eq. (1). The workﬂow is shown in Fig. 2. CEM planning starts by generating N samples { j}N

j=1 = {(ˆaj,0, ˆaj,1, , ˆaj,H 1)}N

j=1 from an initial sampling distribution gφ( ) parameterized by φ, where each sample j is an action sequence from the current time step up to the planning horizon H. The domain of gφ( ) has a dimension of d = da H.

Using a model f, CEM generates imaginary rollouts based on the action sequence { j} (in the case of a stochastic model) and estimate the associated value v( j) = E[ H 1

i=0 γir(sj,i, aj,i)] where sj,0 is the current state s and sj,i+1 f( |sj,i, aj,i). The terminal value γHV (sj,H) is omitted here following the convention in the CEM planning literature but the MPC performance can be further improved if paired with an accurate value predictor [Bertsekas, 2005, Lowrey et al., 2019]. The sampling distribution is then updated by ﬁtting to the current top-k samples in terms of their value estimates v( j), using the Maximum Likelihood Estimation (MLE) which solves:

φ0 = arg max

(v( j) vth) log gφ( j) (2)

where vth is the threshold equal to the value of the k-th best sample and

( ) is the indicator function. In practice, the update to the distribution parameters are smoothed by φl+1 = φ0 + (1 )φl where 2 [0, 1] is a smoothing parameter that balances between the solution to Eq. (2) and the parameter at the current internal iteration l. CEM repeats this process of sampling and distribution update in an inner-loop, until it reaches the stopping condition. In practice, it is stopped when either a maximum number of iterations has been reached or the parameters have not changed for a few iterations. The output of CEM is an action sequence, typically set as the expectation3 of the most recent sampling distribution for uni-modal distributions such as Gaussians ˆµ = E(gφ) = (ˆa0, ˆa1, , ˆa H 1).

Choices of Sampling Distributions in CEM: A common choice is a multivariate Gaussian distribution under which Eq.(2) has an analytical solution. But the uni-modal nature of Gaussian makes it inadequate in solving multi-modal optimization that often occurs in MBRL. To increase the capacity of the distribution, a Gaussian Mixture Model (GMM) can be used [Okada and Taniguchi, 2020]. We denote such an approach as CEM-GMM. Going forward, we use CEM to refer to the vanilla version that employs a Gaussian distribution. Computationally, the major difference between CEM and CEM-GMM is that the distribution update in CEM-GMM is more computation-intensive since it solves for more parameters. Detailed steps can be found in Okada and Taniguchi [2020].

3 Decentralized CEM

In this section, we ﬁrst introduce the formulation of the proposed decentralized approach called the Decentralized CEM (Decent CEM). Then we illustrate the intuition behind the proposed approach using a one-dimensional synthetic multi-modal optimization example where we show the issues of the existing CEM methods and how they can be addressed by Decent CEM.

3Other options are discussed in Appendix A.2

Formulation of Decent CEM Decent CEM is composed of an ensemble of M CEM instances indexed by i, each having its own sampling distributions gφi. They can be described by a set of distribution parameters Φ = {φi}M

i=1. Each instance i manages its own sampling and distribution update by the steps described in Section 2, independently from other instances.

Note that the N samples and k elites are evenly split among the M instances. The topk

M sample sets are decentralized and managed by each instance independently whereas the centralized approach only keeps one set of top-k samples regardless of the distribution family used.

After the stopping condition is reached for all instances, the ﬁnal sampling distribution is taken as the best distribution in the set Φ according to (the arg max uses a deterministic tie-breaking):

φDecent CEM = arg max

Eφi[v(x)] arg max

v( i,j) (3)

where Eφi[v(x)] denotes the expectation with respect to the distribution gφi, approximated by the sample mean of N

M samples. When M = 1, it recovers the conventional CEM.

Figure 3: Left: The objective function in a 1D optimization task. Right: Comparison of the proposed Decent CEM method to CEM and CEM-GMM, wherein the line and the shaded region denote the mean and the min/max cost from 10 independent runs. ˆx: solution of each method.

Motivational Example Consider a one dimensional multi-modal optimization problem shown in Fig. 3 (Left): arg minx sin(x) + sin(10x/3), 7.5 x 7.5. There are eight local optima, including one global optimum f(x ) = 1.9 where x = 5.146. This objective function mimics the RL value landscape that has many local optima, as shown by Wang and Ba [2020]. This optimization problem is easy in the sense that a grid search over the domain can get us a solution close to the global optimum. However, only our proposed Decent CEM method successfully converges to the global optimum consistently under varying population size (i.e., number of samples) and random runs, as shown in Fig. 3 (Right). For a fair comparison, hyperparameter search has been conducted on all methods for each population size (Appendix A).

Figure 4: How the sampling distributions evolve in the 1D optimization task, after the speciﬁed iteration. Symbols include samples , elites , local optima , global optimum . 2nd row in each ﬁgure shows the weighted p.d.f of individual distribution. Population size: 200.

Both CEM-GMM and the proposed Decent CEM are equipped with multiple sampling distributions. The fact that CEM-GMM is outperformed by Decent CEM may appear surprising. To gain some insights, we illustrate in Fig. 4 how the sampling distribution evolves during the iterative update (more details in Fig. 10 in Appendix). CEM updated the unimodal distribution toward a local optimum despite seeing the global optimum. CEM-GMM appears to have a similar issue. During MLE on the top-k samples, it moved most distribution components towards the same local optimum which quickly

led to mode collapse. On the contrary, Decent CEM successfully escaped the local optima thanks to its independent distribution update over decentralized top-k samples and was able to maintain a decent diversity among the distributions.

GMM suits density estimation problems like distribution-based clustering where the samples are drawn from a ﬁxed true distribution that can be represented by multi-modal Gaussians. However, in CEM for optimization, exploration is coupled with density estimation: the sampling distribution in CEM is not ﬁxed but rather gets updated iteratively toward the top-k samples. And the true distribution in optimization puts uniform non-zero densities to the global optima and zero densities everywhere else. When there is a unique global optimum, it degenerates into a Dirac measure that assigns the entire density to the optimum. Density estimation of such a distribution only needs one Gaussian but the exploration is challenging. In other words, the conditions for GMM to work well are not necessarily met when used as the sampling distribution in CEM. CEM-GMM is subject to mode collapse during the iterative top-k greediﬁcation, causing premature convergence, as observed in Fig 4. In comparison, our proposed decentralized approach takes care of the exploration aspect by running multiple CEM instances independently, each performing its own local improvement. This is shown to be effective from this optimization example and the benchmark results in Section 6. CEM-GMM only consistently converge to the global optimum when we increase the population size to the maximum 1,000 which causes expensive computations. Our proposed Decent CEM runs more than 100 times faster than CEM-GMM at this population size, shown in Table A.3 in Appendix.

Convergence of Decent CEM The convergence of Decent CEM requires the following assumption:

Assumption 1. Let M be the number of instances in Decent CEM and each instance has a sample size of Nk

M where Nk is the total number of samples that satisﬁes Nk = (kβ) where β > max{0, 1 2λ}. M and λ are some positive constant and 0 < M < Nk 8 k.

Here the assumption of the total sample size follows the same one as in the CEM literature. Other such standard assumptions are summarized in Appendix H.

Theorem 3.1 (Convergence of Decent CEM). If a CEM instance described in Algorithm 3 converges, and we decentralize it by evenly dividing its sample size Nk into M CEM instances which satisﬁes the Assumption 1, then the resulting Decent CEM converges almost surely to the best solution of the individual instances.

Proof. (sketch) We show that the previous convergence result of CEM [Hu et al., 2011] extends to Decent CEM under the same sample budget. The key observation is that the convergence property of each CEM instance still holds since the number of samples in each instance is only changed by a constant factor, i.e., number of instances. Each CEM instance converges to a local optimum. The convergence of Decent CEM to the best solution comes from the arg max operator and applying the strong law of large numbers. The detailed proof is left to Appendix H.

4 Decent CEM for Planning

In this section, we develop two instantiations of Decent CEM for planning in MBRL where the sampling distributions are parameterized by policy networks. For the dynamics model learning, we adopt the ensemble of probabilistic neural networks proposed in [Chua et al., 2018]. Each network predicts the mean and diagonal covariance of a Gaussian distribution and is trained by minimizing the negative log likelihood. Our work focuses on the planning aspect and refer to Chua et al. [2018] for further details on the model.

CEM Planning with a Policy Network In MBRL, CEM is applied to every state separately to solve the optimization problem stated in Eq. (1). The sampling distribution is typically initialized to a ﬁxed distribution at the beginning of every episode [Okada and Taniguchi, 2020, Pinneri et al., 2020], or more frequently at every time step [Hafner et al., 2019]. Such initialization schemes are sample inefﬁcient since there is no mechanism that allows the information of the high-value region in the value space of one state to generalize to nearby states. Also, the information is discarded after the initialization. It is hence difﬁcult to scale the approach to higher dimensional solution spaces, present in many continuous control environments. Wang and Ba [2020] proposed to use a policy network in CEM planning that helped to mitigate the issues above.

They developed two methods: POPLIN-A that plans in the action space, and POPLIN-P that plans in the parameter space of the policy network. In POPLIN-A, the policy network is used to learn to output the mean of a Gaussian sampling distribution of actions. In POPLIN-P, the policy network parameters serve as the initialization of the mean of the sampling distribution of parameters. The improved policy network can then be used to generate an action. They show that when compared to the vanilla method of using a ﬁxed sampling distribution in the action space, both modes of CEM planning with such a learned distribution perform better. The same principle of combining a policy network with CEM can be applied to the Decent CEM approach as well, which we describe next.

Policy Network 1

Policy Network M

Policy Network 2

Policy Network M-1

CEM Instance 1

CEM Instance 2

CEM Instance M-1

CEM Instance M

state action

Training Inference

Figure 5: The architecture of Decent CEM planning with M CEM instances. i = φi for planning in action space and i = i for planning in policy network parameter space with the instance index i 2 {1, , M}.

Decent CEM Planning with an Ensemble of Policy Networks For better sample efﬁciency in MBRL setting, we extend Decent CEM to use an ensemble of policy networks to learn the sampling distributions in the CEM instances. Similar to the POPLIN variations, we develop two instantiations of Decent CEM, namely Decent CEM-A and Decent CEM-P. The architecture of the proposed algorithm is illustrated in Fig. 5. Decent CEM-A plans in the action space. It consists of an ensemble of policy networks followed by CEM instances. Each policy network takes the current state st as input, outputs the parameters φi of the sampling distribution for CEM instance i in the action space. There is no fundamental difference from the Decent CEM formulation in Section 3 except that the initialization of sampling distributions is learned by the policy networks rather than a ﬁxed distribution.

The second instantiation Decent CEM-P plans in the parameter space of the policy network. The output of each policy network is the network parameter i. The initial sampling distribution of CEM instance i is a Gaussian distribution over the policy parameter space with the mean at i. In the arg max operation in Eq. (3), the sample i,j denotes the j-th parameter sample from the distribution for CEM instance i. Its value is approximated by the model-predicted value of the action sequence generated from the policy network with the parameters i,j.

The ensemble of policy networks in both instantiations Decent CEM-A and Decent CEM-P are initialized with random weights, which is empirically found to be adequate to ensure that the output of the networks do not collapse into the same distribution (Sec. 6.2 and Appendix F).

Training the Policy Networks in Decent CEM When planning in action space, the policy networks are trained by behavior cloning, similar to the scheme in POPLIN [Wang and Ba, 2020]. Denote the ﬁrst action in the planned action sequence at time step t by the i-th CEM instance as ˆat,i, the i-th policy network is trained to mimic ˆat,i and the training objective is min i Est,ˆat,i Dika i(st) ˆat,ik2

where Di denotes the replay buffer with the state and action pairs (st, ˆat,i). a i(st) is the action prediction at state st from the policy network parameterized by i.

While the above training scheme can be applied to both planning in action space and parameter space, we follow the setting parameter average (AVG) [Wang and Ba, 2020] training scheme when planning in parameter space. The parameter is updated as i = i + 1 |Di|

δi2Di δi where Di = {δi} is a dataset of policy network parameter updates planned from the i-th CEM instance previously. It is more effective than behavior cloning based on the experimental result reported by Wang and Ba [2020] and our own preliminary experiments.

Note that each policy network in the ensemble is trained independently from the data observed by its corresponding CEM instance rather than from the aggregated result after taking the arg max. This allows for enough diversity among the instances. More importantly, it increases the size of the training dataset for the policy networks compared to the approach taken in POPLIN. For example, with an ensemble of M instances, there would be M training data samples available from one real environment interaction, compared to the one data sample in POPLIN-A/P. As a result, Decent CEM is able to train larger policy networks than is otherwise possible, shown in Sec. 6.2 and Appendix F.

5 Related Work

We limit the scope of related works to CEM planning methods, which is one of the broad class of planning algorithms in MBRL. For a review of different families of planning algorithms, the readers are referred to Wang et al. [2019]. Vanilla CEM planning in action space with a single Gaussian distribution has been adopted as the planning method for both simulated and real-world robot control [Chua et al., 2018, Finn and Levine, 2017, Ebert et al., 2018, Hafner et al., 2019, Yang et al., 2020, Zhang et al., 2021]. Previous attempts to improving the performance of CEM-based planning can be grouped into two types of approaches. The ﬁrst type includes CEM in a hybrid of CEM+X where X is some other component or algorithm. POPLIN [Wang and Ba, 2020] is a prominent example where X is a policy network that learns a state conditioned distribution that initializes the subsequent CEM

process. Another common choice of X is gradient-based adjustment of the samples drawn in CEM. Grad CEM [Bharadhwaj et al., 2020] adjusts the samples in each iteration of CEM by taking gradient ascent of the return estimate w.r.t the actions. CEM-RL [Pourchot and Sigaud, 2019] also combines CEM with gradient based updates from RL algorithms but the samples are in the parameter space of the actor network. To improve computational efﬁciency, Lee et al. [2020] proposes an asynchronous version of CEM-RL where each CEM instance updates the sampling distribution asynchronously.

The second type of approach aims at improving CEM itself. Amos and Yarats [2020] proposes a fully-differentiable version of CEM called DCEM. The key is to make the top-k selection in CEM differentiable such that the entire CEM module can be trained in an end-to-end fashion. Despite cutting down the number of samples needed in CEM, this method does not beat the vanilla CEM in benchmark test. GACEM [Hakhamaneshi et al., 2020] increase the capacity of the sampling distribution by replacing the Gaussian distribution with an auto-regressive neural network. This change allows CEM to perform search in multi-modal solution space but it is only veriﬁed in toy examples and its computation seems too high to be scaled to MBRL tasks. Another method that increases the capacity of the sampling distribution is Pa ETS [Okada and Taniguchi, 2020] that uses a GMM with CEM. It is the approach that we followed for our CEM-GMM implementation. The running time results in the optimization task in Sec.3 shows that it is computationally heavier than the CEM and Decent CEM methods, limiting its use in complex environments. Macua et al. [2015] proposed a distributed CEM that is similar in spirit to our method in that they used multiple sampling distributions and applied the top-k selection locally to samples from each instance. However, their instances are cooperative as opposed to being independent as in our work. They applied collaborative smoothed projection steps to update each sampling distribution as an average of its neighboring instances including itself. The updating procedure is more complicated than our proposed method and proper network topology of the instances is needed: a naive approach of updating according to all instances will lead to mode collapse since the resulting sampling distributions will be identical. The method was tested in toy optimization examples only. Overall, this second type of approach did not outperform vanilla CEM, a phenomenon that motivated our move to a decentralized formulation.

6 Experiments

We evaluate the proposed Decent CEM methods in simulated environments with continuous action space. The experimental evaluation is mainly set up to understand if Decent CEM improves the performance and sample efﬁciency over conventional CEM approaches.

6.1 Benchmark Setup

Environments We run the benchmark in a set of Open AI Gym [Brockman et al., 2016] and Mu Jo Co [Todorov et al., 2012] environments commonly used in the MBRL literature: Pendulum,

Figure 6: Learning curves of the proposed Decent CEM methods and the baselines on continuous control environments. The line and the shaded region shows the mean and standard error of evaluation results from 5 training runs with different random seeds.

Inverted Pendulum, Cartpole, Acrobot, Fixed Swimmer4, Reacher, Hopper, Walker2D, Half Cheetah, PETS-Reacher3D, PETS-Half Cheetah, PETS-Pusher, Ant. The three environments preﬁxed by PETS are proposed by Chua et al. [2018]. Note that MBRL algorithms often make different assumptions about the dynamics model or the reward function. Their benchmark environments are often modiﬁed from the original Open AI gym environments such that the respective algorithm is runnable. Whenever possible, we inherit the same environment setup from that of the respective baseline methods. This is so that the comparison against the baselines is fair. More details on the environments and their reward functions are in Appendix B.

Algorithms The baseline algorithms are PETS [Chua et al., 2018] and POPLIN [Wang and Ba,

2020]. PETS uses CEM with a single Gaussian distribution for planning. The POPLIN algorithm combines a single policy network with CEM. As described in Sec. 4, POPLIN comes with two modes: POPLIN-A and POPLIN-P with the sufﬁx A denotes planning in action space and P for the network parameter space. We reuse default hyperparameters for these algorithms from the original papers if not mentioned speciﬁcally. For our proposed methods, we include two variations Decent CEM-A and Decent CEM-P as described in Sec. 4 where the sufﬁx carries the same meaning as in POPLIN-A/P. The ensemble size of Decent CEM-A/P as well as detailed hyperparameters for all algorithms are listed in the Appendix D.2. We also included Decentralized PETS, denoted by Decent PETS. All MBRL algorithms studied in this benchmark use the same ensemble networks proposed by Chua et al. [2018] for model learning. And a model-free RL baseline SAC [Haarnoja et al., 2018] was included.

Evaluation Protocol The learning curve shows the mean and standard error of the test performance out of 5 independent training runs. The test performance is an average return of 5 episodes of the evaluation environment, evaluated at every training episode. At the beginning of each training run, the evaluation environment is initialized with a ﬁxed random seed such that the evaluation environments are consistent across different methods and multiple runs to make it a fair comparison. All experiments were conducted using Tesla V100 Volta GPUs.

6.2 Results

Learning Curves The learning curves of the algorithms are shown in Fig. 6 for Inverted Pendulum, Acrobot, Reacher, Walker2D, PETS-Pusher and Ant, sorted by the state and action dimensionality

4a modiﬁed version of the original Gym Swimmer environment where the velocity sensor on the neck is moved to the head. This ﬁx was proposed by Wang and Ba [2020]

of task. The curves are smoothed with a sliding window of 10 data points. The full results for all environments are included in Appendix E. We can observe two main patterns from the results. One pattern was that in most environments, the decentralized methods Decent PETS, Decent CEM-A/P either matched or outperformed their counterpart that took a centralized approach. In fact, the former can be seen as a generalization of the later, by an additional hyperparameter that controls the ensemble size with size one recovering the centralized approach. The optimal ensemble size depends on the task. This additional hyperparameter offers ﬂexibility in ﬁne-tuning CEM for individual domains. For instance in Fig. 6, for the P-mode where the planning is performed in policy parameter space, an ensemble size of larger than one works better in most environments while an ensemble size of one works better in Ant. The other pattern was that using policy networks to learn the sampling distribution in general helped improving the performance of centralized CEM but not necessarily in decentralized formulation. This is perhaps due to that the added exploration from multiple instances makes it possible to identify good solutions in some environments. Using a policy net in such case may hinder the exploration due to overﬁtting to previous actions.

Ablation Study A natural question to ask about the Decent CEM-A/P methods is whether the increased performance is from the larger number of neural network parameters. We added two variations of the POPLIN baselines where a bigger policy network was used. The number of the network parameters was equivalent to that of the ensemble of policy networks in Decent CEM-A/P. We show the comparison in Reacher(2) and PETS-Pusher(7) (action dimension in parenthesis) in Fig. 7. In both action space and parameter space planning, a bigger policy network in POPLIN either did not help or signiﬁcantly impaired the performance (see the POPLIN-P results in reacher and PETS-Pusher). This is expected since unlike Decent CEM, the training data in POPLIN do not scale with the size of the policy network, as explained at the end of Sec. 4.

Figure 7: Ablation study on the policy network size where POPLIN-A&P have a bigger policy network equivalent in the total number of network weights to their Decent CEM counterparts.

Next, we look into the impact of ensemble size. Fig. 8 shows the learning curves of different ensemble sizes in the Reacher environments for action space planning (left), parameter space planning (middle) and planning without policy networks (right). Since we ﬁx the total number of samples the same across the methods, the larger the ensemble size is, the fewer samples that each instance has access to. As the ensemble size goes up, we are trading off the accuracy of the sample mean for better exploration by more instances. Varying this hyperparameter allows us to ﬁnd the best trade-off, as can be observed from the ﬁgure where increasing the ensemble size beyond a sweet spot yields diminishing or worse returns.

Figure 8: Ablation study on the ensemble size in Decent CEM-A/P and Decent PETS, e.g. E2 denotes an ensemble with 2 instances. The total number of samples during planning are the same across all variations.

(a) Selection Ratios (b) Action Statistics (c) Action Distance Statistics

Figure 9: Ablation of ensemble diversity in Pendulum during training of Decent CEM-A with 5 instances. (a) Cumulative selection ratio of each CEM instance. (b)(c) Statistics of the actions and pairwise action distances of the instances, respectively. The line and shaded region represent the mean and min/max.

Figure 9(a) shows the cumulative selection ratio of each CEM instance during training of Decent CEMA with an ensemble size of 5. It suggests that the random initialization of the policy network is

sufﬁcient to avoid mode collapse. We also plot the actions and pairwise action distances of the instances in Figure 9(b) and 9(c). For visual clarity, we show a time segment toward the end of the training rather than all the 10k steps. Decent CEM-A has maintained enough diversity in the instances even toward the end of the training. Decent CEM-P and Decent PETS share similar trends. The plots of their results along with other ablation results are included in Appendix F.

7 Conclusion and Future Work

In this paper, we study CEM planning in the context of continuous-action MBRL. We propose a novel decentralized formulation of CEM named Decent CEM, which generalizes CEM to run multiple independent instances and recovers the conventional CEM when the number of instances is one. We illustrate the strengths of the proposed Decent CEM approach in a motivational one-dimensional optimization task and show how it fundamentally differs from the CEM approach that uses a Gaussian or GMM. We also show that Decent CEM has almost sure convergence to a local optimum. We extend the proposed approach to MBRL by plugging in the decentralized CEM into three previous CEM-based methods: PETS, POPLIN-A, POPLIN-P. We show their efﬁcacy in benchmark control tasks and ablations studies.

There is a gap between the convergence result and practice where the theory assumes that the number of samples grow polynomially with the iterations whereas a constant sample size is commonly used in practice including our work. Investigating the convergence properties of CEM under a constant sample size makes an interesting direction for future work. Another interesting direction to pursue is ﬁnite-time analysis of CEM under both centralized and decentralized formulations. In addition, the implementation has room for optimization: the instances currently run serially but can be improved by a parallel implementation to take advantage of the parallelism of the ensemble.

Acknowledgments and Disclosure of Funding

We would like to thank the anonymous reviewers for their time and valuable suggestions. Zichen Zhang would like to thank Jincheng Mei for helpful discussions in the convergence analysis and Richard Sutton for raising a question about the model learning. This work was partially done during Zichen Zhang s internship at Huawei Noah s Ark Lab. Zichen Zhang gratefully acknowledges the ﬁnancial support by an NSERC CGSD scholarship and an Alberta Innovates Ph D scholarship. He is thankful for the compute resources generously provided by Digital Research Alliance of Canada (and formerly Compute Canada), which is sponsored through the accounts of Martin Jagersand and Dale Schuurmans. Dale Schuurmans gratefully acknowledges funding from the Canada CIFAR AI Chairs Program, Amii and NSERC.

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