# vime_variational_information_maximizing_exploration__18e2e120.pdf

VIME: Variational Information Maximizing Exploration

Rein Houthooft , Xi Chen , Yan Duan , John Schulman , Filip De Turck , Pieter Abbeel

UC Berkeley, Department of Electrical Engineering and Computer Sciences Ghent University - imec, Department of Information Technology Open AI

Scalable and effective exploration remains a key challenge in reinforcement learning (RL). While there are methods with optimality guarantees in the setting of discrete state and action spaces, these methods cannot be applied in high-dimensional deep RL scenarios. As such, most contemporary RL relies on simple heuristics such as ϵ-greedy exploration or adding Gaussian noise to the controls. This paper introduces Variational Information Maximizing Exploration (VIME), an exploration strategy based on maximization of information gain about the agent s belief of environment dynamics. We propose a practical implementation, using variational inference in Bayesian neural networks which efficiently handles continuous state and action spaces. VIME modifies the MDP reward function, and can be applied with several different underlying RL algorithms. We demonstrate that VIME achieves significantly better performance compared to heuristic exploration methods across a variety of continuous control tasks and algorithms, including tasks with very sparse rewards.

1 Introduction

Reinforcement learning (RL) studies how an agent can maximize its cumulative reward in a previously unknown environment, which it learns about through experience. A long-standing problem is how to manage the trade-off between exploration and exploitation. In exploration, the agent experiments with novel strategies that may improve returns in the long run; in exploitation, it maximizes rewards through behavior that is known to be successful. An effective exploration strategy allows the agent to generate trajectories that are maximally informative about the environment. For small tasks, this trade-off can be handled effectively through Bayesian RL [1] and PAC-MDP methods [2 6], which offer formal guarantees. However, these guarantees assume discrete state and action spaces. Hence, in settings where state-action discretization is infeasible, many RL algorithms use heuristic exploration strategies. Examples include acting randomly using ϵ-greedy or Boltzmann exploration [7], and utilizing Gaussian noise on the controls in policy gradient methods [8]. These heuristics often rely on random walk behavior which can be highly inefficient, for example Boltzmann exploration requires a training time exponential in the number of states in order to solve the well-known n-chain MDP [9]. In between formal methods and simple heuristics, several works have proposed to address the exploration problem using less formal, but more expressive methods [10 14]. However, none of them fully address exploration in continuous control, as discretization of the state-action space scales exponentially in its dimensionality. For example, the Walker2D task [15] has a 26-dim state-action space. If we assume a coarse discretization into 10 bins for each dimension, a table of state-action visitation counts would require 1026 entries.

This paper proposes a curiosity-driven exploration strategy, making use of information gain about the agent s internal belief of the dynamics model as a driving force. This principle can be traced back to the concepts of curiosity and surprise [16 18]. Within this framework, agents are encouraged to take actions that result in states they deem surprising i.e., states that cause large updates to the dynamics model distribution. We propose a practical implementation of measuring information gain using variational inference. Herein, the agent s current understanding of the environment dynamics is represented by a Bayesian neural networks (BNN) [19, 20]. We also show how this can be interpreted as measuring compression improvement, a proposed model of curiosity [21]. In contrast to previous curiosity-based approaches [10, 22], our model scales naturally to continuous state and action spaces. The presented approach is evaluated on a range of continuous control tasks, and multiple underlying RL algorithms. Experimental results show that VIME achieves significantly better performance than naïve exploration strategies.

2 Methodology

In Section 2.1, we establish notation for the subsequent equations. Next, in Section 2.2, we explain the theoretical foundation of curiosity-driven exploration. In Section 2.3 we describe how to adapt this idea to continuous control, and we show how to build on recent advances in variational inference for Bayesian neural networks (BNNs) to make this formulation practical. Thereafter, we make explicit the intuitive link between compression improvement and the variational lower bound in Section 2.4. Finally, Section 2.5 describes how our method is practically implemented.

2.1 Preliminaries

This paper assumes a finite-horizon discounted Markov decision process (MDP), defined by (S, A, P, r, ρ0, γ, T), in which S Rn is a state set, A Rm an action set, P : S A S R 0 a transition probability distribution, r : S A R a bounded reward function, ρ0 : S R 0 an initial state distribution, γ (0, 1] a discount factor, and T the horizon. States and actions viewed as random variables are abbreviated as S and A. The presented models are based on the optimization of a stochastic policy πα : S A R 0, parametrized by α. Let µ(πα) denote its expected discounted return: µ(πα) = Eτ[PT t=0 γtr(st, at)], where τ = (s0, a0, . . .) denotes the whole trajectory, s0 ρ0(s0), at πα(at|st), and st+1 P(st+1|st, at).

2.2 Curiosity

Our method builds on the theory of curiosity-driven exploration [16, 17, 21, 22], in which the agent engages in systematic exploration by seeking out state-action regions that are relatively unexplored. The agent models the environment dynamics via a model p(st+1|st, at; θ), parametrized by the random variable Θ with values θ Θ. Assuming a prior p(θ), it maintains a distribution over dynamic models through a distribution over θ, which is updated in a Bayesian manner (as opposed to a point estimate). The history of the agent up until time step t is denoted as ξt = {s1, a1, . . . , st}. According to curiosity-driven exploration [17], the agent should take actions that maximize the reduction in uncertainty about the dynamics. This can be formalized as maximizing the sum of reductions in entropy P

t (H(Θ|ξt, at) H(Θ|St+1, ξt, at)) , (1)

through a sequence of actions {at}. According to information theory, the individual terms equal the mutual information between the next state distribution St+1 and the model parameter Θ, namely I (St+1; Θ|ξt, at). Therefore, the agent is encouraged to take actions that lead to states that are maximally informative about the dynamics model. Furthermore, we note that

I (St+1; Θ|ξt, at) = Est+1 P( |ξt,at) DKL[p(θ|ξt, at, st+1) p(θ|ξt)] , (2)

the KL divergence from the agent s new belief over the dynamics model to the old one, taking expectation over all possible next states according to the true dynamics P. This KL divergence can be interpreted as information gain.

If calculating the posterior dynamics distribution is tractable, it is possible to optimize Eq. (2) directly by maintaining a belief over the dynamics model [17]. However, this is not generally the case. Therefore, a common practice [10, 23] is to use RL to approximate planning for maximal mutual information along a trajectory P

t I (St+1; Θ|ξt, at) by adding each term I (St+1; Θ|ξt, at) as an intrinsic reward, which captures the agent s surprise in the form of a reward function. This is practically realized by taking actions at πα(st) and sampling st+1 P( |st, at) in order to add DKL[p(θ|ξt, at, st+1) p(θ|ξt)] to the external reward. The trade-off between exploitation and exploration can now be realized explicitly as follows:

r (st, at, st+1) = r(st, at) + ηDKL[p(θ|ξt, at, st+1) p(θ|ξt)], (3)

with η R+ a hyperparameter controlling the urge to explore. In conclusion, the biggest practical issue with maximizing information gain for exploration is that the computation of Eq. (3) requires calculating the posterior p(θ|ξt, at, st+1), which is generally intractable.

2.3 Variational Bayes

We propose a tractable solution to maximize the information gain objective presented in the previous section. In a purely Bayesian setting, we can derive the posterior distribution given a new state-action pair through Bayes rule as

p(θ|ξt, at, st+1) = p(θ|ξt)p(st+1|ξt, at; θ)

p(st+1|ξt, at) , (4)

with p(θ|ξt, at) = p(θ|ξt) as actions do not influence beliefs about the environment [17]. Herein, the denominator is computed through the integral

p(st+1|ξt, at) = Z

Θ p(st+1|ξt, at; θ)p(θ|ξt)dθ. (5)

In general, this integral tends to be intractable when using highly expressive parametrized models (e.g., neural networks), which are often needed to accurately capture the environment model in high-dimensional continuous control.

We propose a practical solution through variational inference [24]. Herein, we embrace the fact that calculating the posterior p(θ|D) for a data set D is intractable. Instead we approximate it through an alternative distribution q(θ; φ), parameterized by φ, by minimizing DKL[q(θ; φ) p(θ|D)]. This is done through maximization of the variational lower bound L[q(θ; φ), D]:

L[q(θ; φ), D] = Eθ q( ;φ) [log p(D|θ)] DKL[q(θ; φ) p(θ)]. (6)

Rather than computing information gain in Eq. (3) explicitly, we compute an approximation to it, leading to the following total reward:

r (st, at, st+1) = r(st, at) + ηDKL[q(θ; φt+1) q(θ; φt)], (7)

with φt+1 the updated and φt the old parameters representing the agent s belief. Natural candidates for parametrizing the agent s dynamics model are Bayesian neural networks (BNNs) [19], as they maintain a distribution over their weights. This allows us to view the BNN as an infinite neural network ensemble by integrating out its parameters:

Θ p(y|x; θ)q(θ; φ)dθ. (8)

In particular, we utilize a BNN parametrized by a fully factorized Gaussian distribution [20]. Practical BNN implementation details are deferred to Section 2.5, while we give some intuition into the behavior of BNNs in the appendix.

2.4 Compression

It is possible to derive an interesting relationship between compression improvement an intrinsic reward objective defined in [25], and the information gain of Eq. (2). In [25], the agent s curiosity is

equated with compression improvement, measured through C(ξt; φt 1) C(ξt; φt), where C(ξ; φ) is the description length of ξ using φ as a model. Furthermore, it is known that the negative variational lower bound can be viewed as the description length [19]. Hence, we can write compression improvement as L[q(θ; φt), ξt] L[q(θ; φt 1), ξt]. In addition, an alternative formulation of the variational lower bound in Eq. (6) is given by

L[q(θ;φ),D] z }| { Z

Θ q(θ; φ) log p(θ, D)

q(θ; φ) dθ +DKL[q(θ; φ) p(θ|D)]. (9)

Thus, compression improvement can now be written as

(log p(ξt) DKL[q(θ; φt) p(θ|ξt)]) (log p(ξt) DKL[q(θ; φt 1) p(θ|ξt)]) . (10)

If we assume that φt perfectly optimizes the variational lower bound for the history ξt, then DKL[q(θ; φt) p(θ|ξt)] = 0, which occurs when the approximation equals the true posterior, i.e., q(θ; φt) = p(θ|ξt). Hence, compression improvement becomes DKL[p(θ|ξt 1) p(θ|ξt)]. Therefore, optimizing for compression improvement comes down to optimizing the KL divergence from the posterior given the past history ξt 1 to the posterior given the total history ξt. As such, we arrive at an alternative way to encode curiosity than information gain, namely DKL[p(θ|ξt) p(θ|ξt, at, st+1)], its reversed KL divergence. In experiments, we noticed no significant difference between the two KL divergence variants. This can be explained as both variants are locally equal when introducing small changes to the parameter distributions. Investigation of how to combine both information gain and compression improvement is deferred to future work.

2.5 Implementation

The complete method is summarized in Algorithm 1. We first set forth implementation and parametrization details of the dynamics BNN. The BNN weight distribution q(θ; φ) is given by the fully factorized Gaussian distribution [20]:

q(θ; φ) = Q|Θ| i=1 N(θi|µi; σ2 i ). (11)

Hence, φ = {µ, σ}, with µ the Gaussian s mean vector and σ the covariance matrix diagonal. This is particularly convenient as it allows for a simple analytical formulation of the KL divergence. This is described later in this section. Because of the restriction σ > 0, the standard deviation of the Gaussian BNN parameter is parametrized as σ = log(1 + eρ), with ρ R [20].

Now the training of the dynamics BNN through optimization of the variational lower bound is described. The second term in Eq. (6) is approximated through sampling Eθ q( ;φ) [log p(D|θ)] 1 N PN i=1 log p(D|θi) with N samples drawn according to θ q( ; φ) [20]. Optimizing the variational lower bound in Eq. (6) in combination with the reparametrization trick is called stochastic gradient variational Bayes (SGVB) [26] or Bayes by Backprop [20]. Furthermore, we make use of the local reparametrization trick proposed in [26], in which sampling at the weights is replaced by sampling the neuron pre-activations, which is more computationally efficient and reduces gradient variance. The optimization of the variational lower bound is done at regular intervals during the RL training process, by sampling D from a FIFO replay pool that stores recent samples (st, at, st+1). This is to break up the strong intratrajectory sample correlation which destabilizes learning in favor of obtaining i.i.d. data [7]. Moreover, it diminishes the effect of compounding posterior approximation errors.

The posterior distribution of the dynamics parameter, which is needed to compute the KL divergence in the total reward function r of Eq. (7), can be computed through the following minimization

φ = arg min φ

h ℓ(q(θ;φ),st) z }| { DKL[q(θ; φ) q(θ; φt 1)] | {z } ℓKL(q(θ;φ))

Eθ q( ;φ) [log p(st|ξt, at; θ)] i , (12)

where we replace the expectation over θ with samples θ q( ; φ). Because we only update the model periodically based on samples drawn from the replay pool, this optimization can be performed in parallel for each st, keeping φt 1 fixed. Once φ has been obtained, we can use it to compute the intrinsic reward.

Algorithm 1: Variational Information Maximizing Exploration (VIME) for each epoch n do

for each timestep t in each trajectory generated during n do

Generate action at πα(st) and sample state st+1 P( |ξt, at), get r(st, at). Add triplet (st, at, st+1) to FIFO replay pool R. Compute DKL[q(θ; φ n+1) q(θ; φn+1)] by approximation H 1 , following Eq. (16) for diagonal BNNs, or by optimizing Eq. (12) to obtain φ n+1 for general BNNs. Divide DKL[q(θ; φ n+1) q(θ; φn+1)] by median of previous KL divergences. Construct r (st, at, st+1) r(st, at) + ηDKL[q(θ; φ n+1) q(θ; φn+1)], following Eq. (7). Minimize DKL[q(θ; φn) p(θ)] Eθ q( ;φn) [log p(D|θ)] following Eq. (6), with D sampled randomly from R, leading to updated posterior q(θ; φn+1). Use rewards {r (st, at, st+1)} to update policy πα using any standard RL method.

To optimize Eq. (12) efficiently, we only take a single second-order step. This way, the gradient is rescaled according to the curvature of the KL divergence at the origin. As such, we compute DKL[q(θ; φ + λ φ) q(θ; φ)], with the update step φ defined as

φ = H 1(ℓ) φℓ(q(θ; φ), st), (13)

in which H(ℓ) is the Hessian of ℓ(q(θ; φ), st). Since we assume that the variational approximation is a fully factorized Gaussian, the KL divergence from posterior to prior has a particularly simple form:

DKL[q(θ; φ) q(θ; φ )] = 1

2 + 2 log σ i 2 log σi + (µ i µi)2

Because this KL divergence is approximately quadratic in its parameters and the log-likelihood term can be seen as locally linear compared to this highly curved KL term, we approximate H by only calculating it for the term KL term ℓKL(q(θ; φ)). This can be computed very efficiently in case of a fully factorized Gaussian distribution, as this approximation becomes a diagonal matrix. Looking at Eq. (14), we can calculate the following Hessian at the origin. The µ and ρ entries are defined as

µ2 i = 1 log2(1 + eρi) and 2ℓKL

ρ2 i = 2e2ρi

(1 + eρi)2 1 log2(1 + eρi), (15)

while all other entries are zero. Furthermore, it is also possible to approximate the KL divergence through a second-order Taylor expansion as 1

2 H 1 H H 1 , since both the value and gradient of the KL divergence are zero at the origin. This gives us

DKL[q(θ; φ + λ φ) q(θ; φ)] 1

2λ2 φℓ H 1(ℓKL) φℓ. (16)

Note that H 1(ℓKL) is diagonal, so this expression can be computed efficiently. Instead of using the KL divergence DKL[q(θ; φt+1) q(θ; φt)] directly as an intrinsic reward in Eq. (7), we normalize it by division through the average of the median KL divergences taken over a fixed number of previous trajectories. Rather than focusing on its absolute value, we emphasize relative difference in KL divergence between samples. This accomplishes the same effect since the variance of KL divergence converges to zero, once the model is fully learned.

3 Experiments

In this section, we investigate (i) whether VIME can succeed in domains that have extremely sparse rewards, (ii) whether VIME improves learning when the reward is shaped to guide the agent towards its goal, and (iii) how η, as used in in Eq. (3), trades off exploration and exploitation behavior. All experiments make use of the rllab [15] benchmark code base and the complementary continuous control tasks suite. The following tasks are part of the experimental setup: Cart Pole (S R4, A R1), Cart Pole Swingup (S R4, A R1), Double Pendulum (S R6, A R1), Mountain Car (S R3, A R1), locomotion tasks Half Cheetah (S R20, A R6), Walker2D (S R20, A R6), and the hierarchical task Swimmer Gather (S R33, A R2).

Performance is measured through the average return (not including the intrinsic rewards) over the trajectories generated (y-axis) at each iteration (x-axis). More specifically, the darker-colored lines in each plot represent the median performance over a fixed set of 10 random seeds while the shaded areas show the interquartile range at each iteration. Moreover, the number in each legend shows this performance measure, averaged over all iterations. The exact setup is described in the Appendix.

(a) Mountain Car

(b) Cart Pole Swingup

(c) Half Cheetah

(d) state space

Figure 1: (a,b,c) TRPO+VIME versus TRPO on tasks with sparse rewards; (d) comparison of TRPO+VIME (red) and TRPO (blue) on Mountain Car: visited states until convergence

Domains with sparse rewards are difficult to solve through naïve exploration behavior because, before the agent obtains any reward, it lacks a feedback signal on how to improve its policy. This allows us to test whether an exploration strategy is truly capable of systematic exploration, rather than improving existing RL algorithms by adding more hyperparameters. Therefore, VIME is compared with heuristic exploration strategies on the following tasks with sparse rewards. A reward of +1 is given when the car escapes the valley on the right side in Mountain Car; when the pole is pointed upwards in Cart Pole Swingup; and when the cheetah moves forward over five units in Half Cheetah. We compare VIME with the following baselines: only using Gaussian control noise [15] and using the ℓ2 BNN prediction error as an intrinsic reward, a continuous extension of [10]. TRPO [8] is used as the RL algorithm, as it performs very well compared to other methods [15]. Figure 1 shows the performance results. We notice that using a naïve exploration performs very poorly, as it is almost never able to reach the goal in any of the tasks. Similarly, using ℓ2 errors does not perform well. In contrast, VIME performs much better, achieving the goal in most cases. This experiment demonstrates that curiosity drives the agent to explore, even in the absence of any initial reward, where naïve exploration completely breaks down.

To further strengthen this point, we have evaluated VIME on the highly difficult hierarchical task Swimmer Gather in Figure 5 whose reward signal is naturally sparse. In this task, a two-link robot needs to reach apples while avoiding bombs that are perceived through a laser scanner. However, before it can make any forward progress, it has to learn complex locomotion primitives in the absence of any reward. None of the RL methods tested previously in [15] were able to make progress with naïve exploration. Remarkably, VIME leads the agent to acquire coherent motion primitives without any reward guidance, achieving promising results on this challenging task.

Next, we investigate whether VIME is widely applicable by (i) testing it on environments where the reward is well shaped, and (ii) pairing it with different RL methods. In addition to TRPO, we choose to equip REINFORCE [27] and ERWR [28] with VIME because these two algorithms usually suffer from premature convergence to suboptimal policies [15, 29], which can potentially be alleviated by better exploration. Their performance is shown in Figure 2 on several well-established continuous control tasks. Furthermore, Figure 3 shows the same comparison for the Walker2D locomotion task. In the majority of cases, VIME leads to a significant performance gain over heuristic exploration. Our exploration method allows the RL algorithms to converge faster, and notably helps REINFORCE and ERWR avoid converging to a locally optimal solution on Double Pendulum and Mountain Car. We note that in environments such as Cart Pole, a better exploration strategy is redundant as following the policy gradient direction leads to the globally optimal solution. Additionally, we tested adding Gaussian noise to the rewards as a baseline, which did not improve performance.

To give an intuitive understanding of VIME s exploration behavior, the distribution of visited states for both naïve exploration and VIME after convergence is investigated. Figure 1d shows that using Gaussian control noise exhibits random walk behavior: the state visitation plot is more condensed and ball-shaped around the center. In comparison, VIME leads to a more diffused visitation pattern, exploring the states more efficiently, and hence reaching the goal more quickly.

(a) Cart Pole

(b) Cart Pole Swingup

(c) Double Pendulum

(d) Mountain Car

Figure 2: Performance of TRPO (top row), ERWR (middle row), and REINFORCE (bottom row) with (+VIME) and without exploration for different continuous control tasks.

Figure 3: Performance of TRPO with and without VIME on the high-dimensional Walker2D locomotion task.

Figure 4: VIME: performance over the first few iterations for TRPO, REINFORCE, and ERWR i.f.o. η on Mountain Car.

Figure 5: Performance of TRPO with and without VIME on the challenging hierarchical task Swimmer Gather.

Finally, we investigate how η, as used in in Eq. (3), trades off exploration and exploitation behavior. On the one hand, higher η values should lead to a higher curiosity drive, causing more exploration. On the other hand, very low η values should reduce VIME to traditional Gaussian control noise. Figure 4 shows the performance on Mountain Car for different η values. Setting η too high clearly results in prioritizing exploration over getting additional external reward. Too low of an η value reduces the method to the baseline algorithm, as the intrinsic reward contribution to the total reward r becomes negligible. Most importantly, this figure highlights that there is a wide η range for which the task is best solved, across different algorithms.

4 Related Work

A body of theoretically oriented work demonstrates exploration strategies that are able to learn online in a previously unknown MDP and incur a polynomial amount of regret as a result, these algorithms find a near-optimal policy in a polynomial amount of time. Some of these algorithms are based on the principle of optimism under uncertainty: E3 [3], R-Max [4], UCRL [30]. An alternative approach is Bayesian reinforcement learning methods, which maintain a distribution over possible MDPs [1, 17, 23, 31]. The optimism-based exploration strategies have been extended to continuous state spaces, for example, [6, 9], however these methods do not accommodate nonlinear function approximators.

Practical RL algorithms often rely on simple exploration heuristics, such as ϵ-greedy and Boltzmann exploration [32]. However, these heuristics exhibit random walk exploratory behavior, which can lead

to exponential regret even in case of small MDPs [9]. Our proposed method of utilizing information gain can be traced back to [22], and has been further explored in [17, 33, 34]. Other metrics for curiosity have also been proposed, including prediction error [10, 35], prediction error improvement [36], leverage [14], neuro-correlates [37], and predictive information [38]. These methods have not been applied directly to high-dimensional continuous control tasks without discretization. We refer the reader to [21, 39] for an extensive review on curiosity and intrinsic rewards.

Recently, there have been various exploration strategies proposed in the context of deep RL. [10] proposes to use the ℓ2 prediction error as the intrinsic reward. [12] performs approximate visitation counting in a learned state embedding using Gaussian kernels. [11] proposes a form of Thompson sampling, training multiple value functions using bootstrapping. Although these approaches can scale up to high-dimensional state spaces, they generally assume discrete action spaces. [40] make use of mutual information for gait stabilization in continuous control, but rely on state discretization. Finally, [41] proposes a variational method for information maximization in the context of optimizing empowerment, which, as noted by [42], does not explicitly favor exploration.

5 Conclusions

We have proposed Variational Information Maximizing Exploration (VIME), a curiosity-driven exploration strategy for continuous control tasks. Variational inference is used to approximate the posterior distribution of a Bayesian neural network that represents the environment dynamics. Using information gain in this learned dynamics model as intrinsic rewards allows the agent to optimize for both external reward and intrinsic surprise simultaneously. Empirical results show that VIME performs significantly better than heuristic exploration methods across various continuous control tasks and algorithms. As future work, we would like to investigate measuring surprise in the value function and using the learned dynamics model for planning.

Acknowledgments

This work was supported in part by DARPA, the Berkeley Vision and Learning Center (BVLC), the Berkeley Artificial Intelligence Research (BAIR) laboratory, Berkeley Deep Drive (BDD), and ONR through a PECASE award. Rein Houthooft is supported by a Ph.D. Fellowship of the Research Foundation - Flanders (FWO). Xi Chen was also supported by a Berkeley AI Research lab Fellowship. Yan Duan was also supported by a Berkeley AI Research lab Fellowship and a Huawei Fellowship.

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