Title: Maximum Entropy Reinforcement Learning via Energy-Based Normalizing Flow
Venue: NeurIPS 2024
Accepted: yes
Keywords: Reinforcement Learning

Maximum Entropy Reinforcement Learning

Normalizing Flows

Energy-Based Models

Energy-Based Normalizing Flow
Abstract: Existing Maximum-Entropy (MaxEnt) Reinforcement Learning (RL) methods for continuous action spaces are typically formulated based on actor-critic frameworks and optimized through alternating steps of policy evaluation and policy improvement. In the policy evaluation steps, the critic is updated to capture the soft Q-function. In the policy improvement steps, the actor is adjusted in accordance with the updated soft Q-function. In this paper, we introduce a new MaxEnt RL framework modeled using Energy-Based Normalizing Flows (EBFlow). This framework integrates the policy evaluation steps and the policy improvement steps, resulting in a single objective training process. Our method enables the calculation of the soft value function used in the policy evaluation target without Monte Carlo approximation. Moreover, this design supports the modeling of multi-modal action distributions while facilitating efficient action sampling. To evaluate the performance of our method, we conducted experiments on the MuJoCo benchmark suite and a number of high-dimensional robotic tasks simulated by Omniverse Isaac Gym. The evaluation results demonstrate that our method achieves superior performance compared to widely-adopted representative baselines.

Maximum Entropy Reinforcement Learning via Energy-Based Normalizing Flow

Abstract

Existing Maximum-Entropy (MaxEnt) Reinforcement Learning (RL) methods for continuous action spaces are typically formulated based on actor-critic frameworks and optimized through alternating steps of policy evaluation and policy improvement. In the policy evaluation steps, the critic is updated to capture the soft Q-function. In the policy improvement steps, the actor is adjusted in accordance with the updated soft Q-function. In this paper, we introduce a new MaxEnt RL framework modeled using Energy-Based Normalizing Flows (EBFlow). This framework integrates the policy evaluation steps and the policy improvement steps, resulting in a single objective training process. Our method enables the calculation of the soft value function used in the policy evaluation target without Monte Carlo approximation. Moreover, this design supports the modeling of multi-modal action distributions while facilitating efficient action sampling. To evaluate the performance of our method, we conducted experiments on the MuJoCo benchmark suite and a number of high-dimensional robotic tasks simulated by Omniverse Isaac Gym. The evaluation results demonstrate that our method achieves superior performance compared to widely-adopted representative baselines.

1 Introduction

Maximum-Entropy (MaxEnt) Reinforcement Learning (RL) [1-17] has emerged as a prominent method for modeling stochastic policies. Different from standard RL, MaxEnt RL integrates the entropy of policies as rewards, which leads to a balanced exploration-exploitation trade-off during training. This approach has demonstrated improved robustness both theoretically and empirically [17-19]. Building on this foundation, many studies leveraging MaxEnt RL have shown superior performance on continuous-control benchmark environments [8, 9] and real-world applications [20-22].

An active research domain in MaxEnt RL concentrates on the learning of the soft Q-function [8-15]. These methods follow the paradigm introduced in soft Q-learning (SQL) [8]. They parameterize the soft Q-function as an energy-based model [23] and optimize it based on the soft Bellman error [8] calculated from rewards and the soft value function. However, this approach presents two challenges. First, sampling from an energy-based model requires a costly Monte Carlo Markov Chain (MCMC) [24, 25] or variational inference [26] process, which can result in inefficient interactions with environments. Second, the calculation of the soft value function can involve computationally

infeasible integration, which requires an effective approximation method. To tackle these issues, various methods [8-15] were proposed, all grounded in a common design philosophy. To address the first challenge, these methods suggest operating on an actor-critic framework and optimizing it through alternating steps of policy evaluation and policy improvement. For the second challenge, they resort to Monte Carlo methods to approximate the soft value function using sets of random samples. Although these two issues can be circumvented, these methods still have their drawbacks. The actor-critic design introduces an additional optimization process for training the actor, which may lead to optimization errors in practice [27]. Moreover, the results of Monte Carlo approximation may be susceptible to estimation errors and variances when there are an insufficient number of samples [28-30].

Instead of using energy-based models to represent MaxEnt RL frameworks, this paper investigates an alternative method employing normalizing flows (i.e., flow-based models), which offer solutions to the aforementioned challenges. Our framework is inspired by the recently introduced Energy-Based Normalizing Flows (EBFlow) [31]. This design facilitates the derivation of an energy function from a flow-based model while supporting efficient sampling, which enables a unified representation of both the soft Q-function and its corresponding action sampling process. This feature allows the integration of the policy evaluation and policy improvement steps into a single objective training process. In addition, the probability density functions (pdf) of flow-based models can be calculated efficiently without approximation. This characteristic permits the derivation of an exact representation for the soft value function. Our experimental results demonstrate that the proposed framework exhibits superior performance on the commonly adopted MuJoCo benchmark [32, 33]. Furthermore, the evaluation results on the Omniverse Isaac Gym environments [34] indicate that our framework excels in performing challenging robotic tasks that simulate real-world scenarios.

2 Background and Related Works

In this section, we walk through the background material and the related works. We introduce the objective of MaxEnt RL in Section 2.1, describe existing actor-critic frameworks and soft value estimation methods in Section 2.2, and elaborate on the formulation of Energy-Based Normalizing Flow (EBFlow) in Section 2.3.

2.1 Maximum Entropy Reinforcement Learning

In this paper, we consider a Markov Decision Process (MDP) defined as a tuple ( S , A , p T , R , γ, p 0 ) , where S is a continuous state space, A is a continuous action space, p T : S × S × A → R ≥ 0 is the pdf of a next state s t +1 given a current state s t and a current action a t at timestep t , R : S × A → R is the reward function, 0 < γ < 1 is the discount factor, and p 0 is the pdf of the initial state s 0 . We adopt r t to denote R ( s t , a t ) , and use ρ π ( s t , a t ) to represent the state-action marginals of the trajectory distribution induced by a policy π ( a t | s t ) [8].

Standard RL defines the objective as π ∗ = argmax π ∑ t E ( s t , a t ) ∼ ρ π [ r t ] and has at least one deterministic optimal policy [35, 36]. In contrast, MaxEnt RL [4] augments the standard RL objective with the entropy of a policy at each visited state s t . The objective of MaxEnt RL is written as follows:

where H ( π ( · | s t )) ≜ E a ∼ π ( ·| s t ) [ -log π ( a | s t )] and α ∈ R > 0 is a temperature parameter for determining the relative importance of the entropy term against the reward. An extension of Eq. (1) defined with γ is discussed in [8]. To obtain π ∗ MaxEnt described in Eq. (1), the authors in [8] proposed to minimize the soft Bellman error for all states and actions. The solution can be expressed using the optimal soft Q-function Q ∗ soft : S × A → R and soft value function V ∗ soft : S → R as follows:

In practice, a policy can be modeled as π θ ( a t | s t ) = exp( 1 α ( Q θ ( s t , a t ) -V θ ( s t )) with parameter θ , where the soft Q-function and the soft value function are expressed as Q θ ( s t , a t ) and V θ ( s t ) =

α log ∫ exp ( 1 α Q θ ( s t , a t ) ) d a t , respectively. Given an experience reply buffer D that stores transition tuples ( s t , a t , r t , s t +1 ) , the training objective of Q θ (which can then be used to derive V θ and π θ ) can be written as the following equation according to the soft Bellman errors:

Nonetheless, directly using the objective in Eq. (4) presents challenges for two reasons. First, drawing samples from an energy-based model (i.e., π θ ( a t | s t ) ∝ exp( Q θ ( s t , a t ) /α ) ) requires a costly MCMC or variational inference process [26, 37], which makes the interaction with the environment inefficient. Second, the calculation of the soft value function involves integration, which require stochastic approximation methods [28-30] to accomplish. To address these issues, the previous MaxEnt RL methods [8-14] adopted actor-critic frameworks and introduced a number of techniques to estimate the soft value function. These methods are discussed in the next subsection.

2.2 Actor-Critic Frameworks and Soft Value Estimation in MaxEnt RL

Previous MaxEnt RL methods [8-15] employed actor-critic frameworks, in which the critic aims to capture the soft Q-function, while the actor learns to sample actions based on this soft Q-function. Available choices for modeling the actor include Gaussian models [9], Gaussian mixture models [38], variational autoencoders (VAE) [15, 13, 39], normalizing flows [10, 11], and amortized SVGD (A-SVGD) [8, 40], all of which support efficient sampling. The separation of the actor and the critic prevents the need for costly MCMC processes during sampling. However, this design induces additional training steps aimed to minimize the discrepancy between them. Let π θ ( a t | s t ) ∝ exp( 1 α Q θ ( s t , a t )) and π ϕ ( a t | s t ) denote the pdfs defined through the critic and the actor, respectively. The objective of this additional training process is formulated according to the reverse KL divergence D KL [ π ϕ ( · | s t ) || π θ ( · | s t )] between π ϕ and π θ , and is typically reduced as follows [9]:

The optimization processes defined by the objective functions L ( θ ) and L ( ϕ ) in Eqs. (4) and (5) are known as the policy evaluation steps and policy improvement steps [9], respectively. Through alternating updates according to ∇ θ L ( θ ) and ∇ ϕ L ( ϕ ) , the critic learns directly from the reward signals to estimate the soft Q-function, while the actor learns to draw samples based on the distribution defined by the critic.

Although the introduction of the actor enhances sampling efficiency, calculating the soft value function in Eq. (3) still requires Monte Carlo approximations for the computationally infeasible integration operation. Prior soft value estimation methods can be categorized into two groups: soft value estimation in Soft Q-Learning (SQL) and that in Soft Actor-Critic (SAC), with the former yielding a larger estimate than the latter, derived from Jensen's inequality (i.e., Proposition A.1 in Appendix A.1). These two soft value estimation methods are discussed in the following paragraphs.

Soft Value Estimation in SQL. Soft Q-Learning [8] leverages importance sampling to convert the integration in Eq. (3) into an expectation, which can be estimated using a set of independent and identically distributed (i.i.d.) samples. To ensure the estimation variance is small, the authors in [8] proposed to utilize samples drawn from π ϕ . Let { a ( i ) } i M =1 be a set of M samples drawn from π ϕ . The soft value function is approximated based on the following formula:

Eq. (6) has the least variance when π ϕ ( · | s t ) ∝ exp( Q θ ( s t , · ) /α ) [29]. In addition, as M →∞ , the law of large numbers ensures that this estimation converge to V θ ( s t ) [41].

Soft Value Estimation in SAC. Soft Actor-Critic [9] and its variants [10, 11, 13, 14, 12] reformulated the soft value function V θ ( s t ) = α log ∫ exp( Q θ ( s t , a ) /α ) d a as its equivalent form E a ∼ π θ [ Q θ ( s t , a ) -α log π θ ( a | s t )] based on the relationship that π θ ( a | s t ) = exp( 1 α ( Q θ ( s t , a )) -

V θ ( s t )) . By assuming that the policy improvement loss L ( ϕ ) is small (i.e., π θ ≈ π ϕ ), the soft value function V θ can be estimated as follows:

An inherent drawback of the estimation in Eq. (7) is its reliance on the assumption π ϕ ≈ π θ . In addition, the second approximation involves Monte Carlo estimation with M samples { a ( i ) } i M =1 , where the computational cost increases with the number of samples M .

2.3 Energy-Based Normalizing Flows

Normalizing flows (i.e., flow-based models) are universal representations for pdf [42]. Given input data x ∈ R D , a latent variable z ∈ R D with prior pdf p z , and an invertible function g θ = g L θ ◦· · · ◦ g 1 θ modeled as a neural network with L layers, where g i θ : R D → R D , ∀ i ∈ { 1 , · · · , L } . According to the change of variable theorem and the distributive property of the determinant operation, a parameterized pdf p θ can be described as follows:

where x 0 ≜ x is the input, x i = g i θ ◦ · · · ◦ g 1 θ ( x ) is the output of the i -th layer, and J g i θ ( x i -1 ) ≜ ∂ ∂ x i -1 g i θ ( x i -1 ) represents the Jacobian of the i -th layer of g θ with respect to x i -1 . To draw samples from p θ , one can first sample z from p z and then derive g -1 θ ( z ) . To facilitate efficient computation of the pdf and the inverse of g θ , one can adopt existing architectural designs [43-48] for g θ . Popular examples involve autoregressive layers [43-45] and coupling layers [46-48], which utilizes specially designed architectures to speed up the calculation.

Energy-Based Normalizing Flows (EBFlow) [31] were recently introduced to reinterpret flow-based models as energy-based models. In contrast to traditional normalizing flow research [46, 47, 49, 50] that focuses on the use of effective non-linearities, EBFlow emphasizes the use of both linear and non-linear transformations in the invertible transformation g θ . Such a concept was inspired by the development of normalizing flows with convolution layers [48, 51-54] or fully-connected layers [55, 56], linear independent component analysis (ICA) models [57, 58], as well as energy-based training techniques [58-60]. Let S l = { i | g i θ is linear } and S n = { i | g i θ is non-linear } represent the sets of indices of the linear and non-linear transformations in g θ , respectively. As shown in [31], the Jacobian determinant product in Eq. (8) can be decomposed according to S n and S l . This decomposition allows a flow-based model to be reinterpreted as an energy-based model, as illustrated in the following equation:

In EBFlow, the energy function E θ ( x ) is defined as -log( p z ( g θ ( x )) ∏ i ∈S n | det( J g i θ ( x i -1 )) | ) and the normalizing constant Z θ = ∫ exp( -E θ ( x )) d x = ∏ i ∈S l | det J g i θ | -1 is independent of x . The input-independence of Z θ holds since g i θ is either a first-degree or zero-degree polynomial for any i ∈ S l , and thus its Jacobian is a constant to x i -1 . This technique was originally proposed to reduce the training cost of maximum likelihood estimation for normalizing flows. However, we discovered that EBFlow is ideal for MaxEnt RL. Its unique capability to represent a parametric energy function with an associated sampler g -1 θ , and to calculate a normalizing constant Z θ without integration are able to address the challenges mentioned in Section 2.2. We discuss our insights in the next section.

3 Methodology

In this section, we introduce our proposed MaxEnt RL framework modeled using EBFlow. In Section 3.1, we describe the formulation and discuss its training and inference processes. In Section 3.2,

we present two techniques for improving the training of our framework. Ultimately, in Section 3.3, we offer an algorithm summary.

3.1 MaxEnt RL via EBFlow

We propose a new framework for modeling M ax E nt RL using EBFl ow , which we call MEow . This framework possesses several unique features. First, as EBFlow enables simultaneous modeling of an unnormalized density and its sampler, MEow can unify the actor and the critic previously separated in MaxEnt RL frameworks. This feature facilitates the integration of policy improexvement steps with policy evaluation steps, and results in a single objective training process. Second, the normalizing constant of EBFlow is expressed in closed form, which enables the calculation of the soft value function without resorting to the approximation methods mentioned in Eqs. (6) and (7). Third, given that normalizing flow is a universal approximator for probability density functions, our policy's expressiveness is not constrained, and can model multi-modal action distributions.

In MEow, the policy is described as a state-conditioned EBFlow, with its pdf presented as follows:

where the soft Q-function and the soft value function are selected as follows:

Such a selection satisfies V θ ( s t ) = α log ∫ exp( Q θ ( s t , a ) /α ) d a based on the property of EBFlow. In addition, both Q θ and V θ have a common output codomain R , which enables them to learn to output arbitrary real values. These properties are validated in Proposition 3.1, with the proof provided in Appendix A.2. The training and inference processes of MEow are summarized as follows.

Proposition 3.1. Eq. (11) satisfies the following statements: (1) Given that the Jacobian of g θ is non-singular, V θ ( s t ) ∈ R and Q θ ( s t , a t ) ∈ R , ∀ a t ∈ A , ∀ s t ∈ S . (2) V θ ( s t ) = α log ∫ exp( Q θ ( s t , a ) /α ) d a .

Training. With Q θ and V θ defined in Eq. (11), the loss L ( θ ) in Eq. (4) can be calculated without using Monte Carlo approximation of the soft value function target. Compared to the previous MaxEnt RL frameworks that rely on Monte Carlo estimation (i.e., Eqs. (6) and (7)), our framework offers the advantage of avoiding the errors induced by the approximation. In addition, MEow employs a unified policy rather than two separate roles (i.e., the actor and the critic), which eliminates the need for minimizing an additional policy improvement loss L ( ϕ ) to bridge the gap between π θ and π ϕ . This simplifies the training process of MaxEnt RL, and obviates the requirement of balancing between the two optimization loops.

Inference. The sampling process of π θ can be efficiently performed by deriving the inverse of g θ , as supported by several normalizing flow architectures [43-48]. In addition, unlike previous actor-critic frameworks susceptible to discrepancies between π θ and π ϕ , the distribution established via g -1 θ ( z | s t ) , where z ∼ p z , is consistently aligned with the pdf defined by Q θ . As a result, the actions taken by MEow can precisely reflect the learned soft Q-function.

3.2 Techniques for Improving the Training and Inference Processes of MEow

In this subsection, we introduce a number of training and inference techniques aimed at improving MEow while preserving its key features discussed in the previous subsection. For clarity, we refer to the MEow framework introduced in the last section as 'MEow (Vanilla)'.

Learnable Reward Shifting (LRS). Reward shifting [61-65] is a technique for shaping the reward function. This technique enhances the learning process by incorporating a shifting term in the reward function, which leads to a shifted optimal soft Q-function in MaxEnt RL. Inspired by this, this work proposes modeling a reward shifting function b θ : S → R with a neural network to enable the automatic learning of a reward shifting term. For notational simplicity, the parameters are denoted using θ , and the details of the architecture are presented in Appendix A.5.1. The soft Q-function Q b θ augmented by b θ is defined as follows:

The introduction of Q b θ results in a corresponding shifted soft value function V b θ ( s t ) ≜ α log ∫ exp( Q b θ ( s t , a ) /α ) d a = V θ ( s t ) + b θ ( s t ) (i.e., Proposition A.3 in Appendix A.2), which can be calculated without Monte Carlo estimation. Moreover, with the incorporation of b θ , the policy π θ remains invariant since exp( 1 α ( Q b θ ( s t , a t ) -V b θ ( s t ))) = exp( 1 α (( Q θ ( s t , a t ) + b θ ( s t )) -( V θ ( s t ) + b θ ( s t )))) = exp( 1 α ( Q θ ( s t , a t ) -V θ ( s t ))) , which allows the use of g -1 θ for efficiently sampling actions. As evidenced in Fig. 1, this method effectively addresses the issues of the significant growth and decay of Jacobian determinants of g θ (discussed in Appendix A.3). In Section 4.4, we further demonstrate that the performance of MEow can be significantly improved through this technique.

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Shifting-Based Clipped Double Q-Learning (SCDQ). As observed in [66], the overestimation of value functions often occurs in training. To address this issue, the authors in [66] propose clipped double Q-learning, which employs two separate Q-functions and uses the one with the smaller output to estimate the value function during training. This technique is also used in MaxEnt RL frameworks [9-13]. Inspired by this and our proposed learnable reward shifting, we further propose a shifting-based method that adopts two learnable reward shifting functions, b (1) θ and b (2) θ , without duplicating the soft Q-function Q θ and soft value function V θ defined by g θ . The soft Q-functions Q (1) θ and Q (2) θ with corresponding learnable reward shifting functions b (1) θ and b (2) θ can be obtained using Eq. (12), while the soft value function V clip θ is written as the following formula:

This design also prevents the production of two policies in MEow, as having two policies can complicate the inference procedure. In our ablation analysis presented in Section 4.4, we demonstrate that this technique can effectively improve the training process of MEow.

Deterministic Policy for Inference. As observed in [8], deterministic actors typically performed better as compared to its stochastic variant during the inference time. Such a problem can be formalized as finding an action a that maximizes Q ( s t , a ) for a given s t . Since A is a continuous space, finding such a value would require extensive calculation. In the MEow framework, this value can be derived by making assumptions on the model architecture construction. Our key observation is that if the Jacobian determinants of the non-linearities (i.e., g i θ ∈ S n ) are constants with respect to its inputs, and that argmax z p z ( z ) can be directly obtained, then the action a that maximizes Q ( s t , a ) can be efficiently derived according to the following proposition.

Proposition 3.2. Given that | det( J g i θ ( a i -1 | s t )) | is a constant with respect to a i -1 , then g -1 θ (argmax z p z ( z ) | s t ) = argmax a Q θ ( s t , a ) .

The proof is provided in Appendix A.2. It is important to note that g i θ can still be a non-linear transformation, given that | det( J g i θ ( a i -1 | s t )) | is a constant. To construct such a model, a Gaussian prior with the additive coupling transformations [46] can be used as non-linearities. Under such a design, an action can be derived by calculating g -1 θ ( µ | s t ) , where µ represents the mean of the

Algorithm 1 Pseudo Code of the Training Process of MEow

Gaussian distribution. We elaborate on our model architecture design in Appendix A.5.1, and provide a performance comparison between MEow evaluated using a stochastic policy (i.e., a t ∼ π θ ( · | s t ) ) and a deterministic policy (i.e., a t = argmax a Q θ ( s t , a ) ) in Section 4.4.

3.3 Algorithm Summary

We summarize the training process of MEow in Algorithm 1. The algorithm integrates the policy evaluation steps with the policy improvement steps, resulting in a single loss training process. This design differs from previous actor-critic frameworks, which typically perform two consecutive updates in each training step. In Algorithm 1, the learning rate is denoted as β . A set of shadow parameters θ ′ is maintained for calculating the delayed target values [67], and is updated according to the Polyak averaging [68] of θ , i.e., θ ′ ← (1 -τ ) θ ′ + τθ , where τ is the target smoothing factor.

4 Experiments

In the following sections, we first present an intuitive example of MEow trained in a two-dimensional multi-goal environment [8] in Section 4.1. We then compare MEow's performance against several continuous-action RL baselines in five MuJoCo environments [32, 33] in Section 4.2. Next, in Section 4.3, we evaluate MEow's performance on a number of Omniverse Isaac Gym environments [34] simulated based on real-world robotic application scenarios. Lastly, in Section 4.4, we provide an ablation analysis to inspect the effectiveness of each proposed technique. Among all experiments, we maintain the same model architecture, while adjusting inputs and outputs according to the state space and action space for each environment. We construct g θ using the additive coupling layers [46] with element-wise linear transformations, utilize a unit Gaussian as p z , and model the learnable adaptive reward shifting functions b θ as multi-layer perceptrons (MLPs). For detailed descriptions of the experimental setups, please refer to Appendix A.5.

4.1 Evaluation on a Multi-Goal Environment

In this subsection, we present an example of MEow trained in a two-dimensional multi-goal environment [8]. The environment involves four goals, indicated by the red dots in Fig. 2 (a). The reward function is defined by the negative Euclidean distance from each state to the nearest goal, and the corresponding reward landscape is depicted using contours in Fig. 2 (a). The gradient map in Fig. 2 (a) represents the soft value function predicted by our model. The blue lines extending from the center represent the trajectories produced using our policy.

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As illustrated in Fig. 2 (a), our model's soft value function predicts higher values around the goals, suggesting successful learning of the goal positions through rewards. In addition, the trajectories demonstrate our agent's correct transitions towards the goals, which validates the effectiveness of our learned policy. To illustrate the potential impact of approximation errors that might emerge when employing previous soft value estimation methods, we compare three calculation methods for the soft value function: (I) Our approach (i.e., Eq. (11)): V θ ( s t ) , (II) SQL-like (i.e., Eq. (6)): α log( 1 M ∑ M i =1 exp( Q θ ( s t , a ( i ) ) /α ) π ϕ ( a ( i ) | s t ) ) ,

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and (III) SAC-like (i.e., Eq. (7)): 1 M ∑ M i =1 ( Q θ ( s t , a ( i ) ) -α log π ϕ ( a ( i ) | s t )) , where { a ( i ) } i M =1 is sampled from π ϕ . The approximation errors of the soft value functions at the initial state are calculated using the Euclidean distances between (I) and (II), and between (I) and (III), for various values of M . As depicted in Fig. 2 (b), the blue line and the orange line decreases slowly with respect to M . These results suggest that Monte Carlo estimation converges slowly, making approximation methods such as Eqs. (6) and (7) challenging to achieve accurate predictions.

4.2 Performance Comparison on the MuJoCo Environments

In this experiment, we compare MEow with several commonly-used continuous control algorithms on five MuJoCo environments [32] from Gymnasium [33]. The baseline algorithms include SQL [8], SAC [9], deep deterministic policy gradient (DDPG) [69], twin delayed deep deterministic policy gradient (TD3) [66], and proximal policy optimization (PPO) [70]. The results for SAC, DDPG, TD3, and PPO were reproduced using Stable Baseline 3 (SB3) [71], utilizing SB3's refined hyperparameters. The results for SQL were reproduced using our own implementation, as SB3 does not support SQL and the official code is not reproducible. Our implementation adheres to SQL's original paper. Each method is trained independently under five different random seeds, and the evaluation curves for each environment are presented in the form of the means and the corresponding confidence intervals.

As depicted in Fig. 3, MEow performs comparably to SAC and outperforms the other baseline algorithms in most of the environments. Furthermore, in environments with larger action and state dimensionalities, such as 'Ant-v4' and 'Humanoid-v4', MEow offers performance improvements over SAC and exhibits fewer spikes in the evaluation curves. These results suggest that MEow is capable of performing high-dimensional tasks with stability. To further investigate the performance difference between MEow and SAC, we provide a thorough comparison between MEow, SAC [9], Flow-SAC [10, 11], and their variants in Appendix A.4.2. The results indicate that the training process involving policy evaluation and policy improvement steps may be inferior to our proposed training process with a single objective. In the next subsection, we provide a performance examination using the simulation environments from the Omniverse Isaac Gym [34].

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action dim. =   8

state dim. = 108

action dim. =   21

state dim. = 13

action dim. =   6

4.3 Performance Comparison on the Omniverse Issac Gym Environments

In this subsection, we examine the performance of MEow on a variety of robotic tasks simulated by Omniverse Isaac Gym [34], a GPU-based physics simulation platform. In addition to 'Ant' and 'Humanoid', we employ four additional tasks: 'Ingenuity', 'ANYmal', 'AllegroHand', and 'FrankaCabinet'. All of them are designed based on real-world robotic application scenarios. 'Ingenuity' and 'ANYmal' are locomotion environments inspired by NASA's Ingenuity helicopter and ANYbotics' industrial maintenance robots, respectively. On the other hand, 'AllegroHand' and 'FrankaCabinet' focus on executing specialized manipulative tasks with robotic hands and arms, respectively. A demonstration of these tasks is illustrated in Fig. 5.

In this experimental comparison, we adopt SAC as a baseline due to its excellent performance in the MuJoCo environments. The evaluation results are presented in Fig. 4. The results demonstrate that MEow exhibits superior performance on 'Ant (Isaac)' and 'Humanoid (Isaac)'. In addition, MEow consistently outperforms SAC across the four robotic environments (i.e., 'Ingenuity', 'ANYmal', 'AllegroHand', and 'FrankaCabinet'), indicating that our algorithm possesses the ability to perform challenging robotic tasks simulated based on real-world application scenarios.

4.4 Ablation Analysis

In this subsection, we provide an ablation analysis to examine the effectiveness of each technique introduced in Section 3.2.

Training Techniques. Fig. 6 compares the performance of three variants of MEow: 'MEow (Vanilla)', 'MEow (+LRS)', and 'MEow (+LRS & SCDQ)', across five MuJoCo environments. The results show that 'MEow (Vanilla)' consistently underperforms, with its total returns demonstrating negligible or no growth throughout the training period. In contrast, the variants incorporating translation functions demonstrate significant performance enhancements. This observation highlights the importance of including b θ in the model design. In addition, the comparison between 'MEow (+LRS)' and 'MEow (+LRS & SCDQ)' suggests that our reformulated approach to clipped double Q-learning [66] improves the final performance by a noticeable margin.

Inference Technique. Fig. 7 compares the performance of two variants of MEow: 'MEow (Stochastic)' and 'MEow (Deterministic)'. The former samples action based on a t ∼ π θ ( · | s t ) while the latter derive action according to a t = argmax a Q θ ( s t , a ) = g -1 θ ( µ | s t ) . As shown in the figure, MEow with a deterministic policy outperforms its stochastic variant, suggesting that a deterministic policy may be more effective for MEow's inference.

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5 Conclusion

In this paper, we introduce MEow, a unified MaxEnt RL framework that facilitates exact soft value calculations without the need for Monte Carlo estimation. We demonstrate that MEow can be optimized using a single objective function, which streamlines the training process. To further enhance MEow's performance, we incorporate two techniques, learnable reward shifting and shiftingbased clipped double Q-learning, into the design. We examine the effectiveness of MEow via experiments conducted in five MoJoCo environments and six robotic tasks simulated by Omniverse Isaac Gym. The results validate the superior performance of MEow compared to existing approaches.

Limitations and Discussions

As discussed in Section 3.2, deterministic policies typically offer better performance compared to their stochastic counterparts. Although our implementation of MEow supports deterministic inference, this capability is based on the assumptions that the Jacobian determinants of the non-linear transformations are constants with respect to their inputs, and that argmax z p z ( z ) can be efficiently derived. These assumptions may not hold for certain types of flow-based models. Therefore, exploring effective architectural choices for MEow represents a promising direction for further investigation.

On the other hand, the training speed of MEow is around 2 . 3 × slower than that of SAC, even though updates according to L ( ϕ ) are bypassed in MEow. According to our experimental observations, the computational bottleneck of MEow may lie in the inference speed of the flow-based model during interactions with environments. While this speed is significantly faster than many iterative methods, such as MCMC or variational inference, it is still slower compared to the inference speed of Gaussian models. As a result, enhancing the inference speed of flow-based models represents a potential avenue for further improving the training efficiency of MEow.

Finally, our hyperparameter sensitivity analysis, as presented in A.4.5, indicates that our current approach requires different values of τ to achieve optimal performance. Since hyperparameter tuning often demands significant computational resources, establishing a more generalized parameter setting or developing an automatic tuning mechanism for τ presents an important direction for future exploration.

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A Appendix

In this Appendix, we begin with a discussion of the soft value estimation methods used in SQL and SAC in Section A.1. We then derive a number of theoretical properties of MEow in Section A.2. Next, we discuss the issue of numerical instability in Section A.3. Then, we present additional experimental results in Section A.4, and summarize the experimental setups in Section A.5. Finally, we elaborate on the potential impacts of this work in Section A.6.

A.1 The Soft Value Estimation Methods in SAC and SQL

In this section, we elaborate on the soft value estimation methods mentioned in Section 2.2. We first show that V θ ( s t ) approximated using Eq. (6) is greater than that approximated using Eq. (7) for any given state s t in Proposition A.1 and Remark A.2. Then, we discuss their practical implementation.

Proposition A.1. For any s t ∈ S and α ∈ R > 0 , the following inequality holds:

Proof.

where ( i )

is due to Jensen's inequality.

Remark A.2 . The inequality in Eq. (A1) preserves after applying the Monte Carlo estimation. Namely,

where { a ( i ) } i M =1 represents a set of samples drawn from π ϕ .

Unlike the estimation in Eq. (7), the estimation in Eq. (6) is guaranteed to converge to V θ ( s t ) as M →∞ . However, empirically, the estimation method in Eq. (7) is preferred and widely used in the contemporary MaxEnt framework. One potential reason could be the required number of samples needed for effective approximation. According to [8], M = 32 is an effective choice for Eq. (6), whereas M = 1 works well for Eq. (7), as adopted by many previous works. [9-12, 14].

A.2 Theoretical Properties of MEow

In this section, we examine a number of key properties of the MEow framework. We begin by presenting a proposition to verify MEow's capability in modeling the soft Q-function and the soft value function. Then, we present a proposition to derive a deterministic policy in MEow. Finally, we offer a discussion of the impact of incorporating learnable reward shifting functions.

Proposition 3.1 Eq. (11) satisfies the following statements: (1) Given that the Jacobian of g θ is non-singular, V θ ( s t ) ∈ R and Q θ ( s t , a t ) ∈ R , ∀ a t ∈ A , ∀ s t ∈ S . (2) V θ ( s t ) = α log ∫ exp( Q θ ( s t , a ) /α ) d a .

Proof. (1) Given that the Jacobian of g θ is non-singular, V θ ( s t ) ∈ R and Q θ ( s t , a t ) ∈ R , ∀ a t ∈ A , ∀ s t ∈ S .

If the Jacobian of g θ is non-singular, then both ∏ i ∈S l | det( J g i θ ( s t )) | ∈ R > 0 and ∏ i ∈S n | det( J g i θ ( a i -1 t | s t )) | ∈ R > 0 . This suggests that α log ∏ i ∈S l | det( J g i θ ( s t )) | ∈ R and α log p z ( g θ ( a t | s t )) ∏ i ∈S n | det( J g i θ ( a i -1 t | s t )) | ∈ R . As a result, V θ ( s t ) ∈ R and Q θ ( s t , a t ) ∈ R .

(2) V θ ( s t ) = α log ∫ exp( Q θ ( s t , a ) /α ) d a .

In Section 3.2, we demonstrate that argmax a Q θ ( s t , a ) can be efficiently obtained through g -1 θ (argmax z p z ( z ) | s t ) . To provide theoretical support for this result, we include a proof for Proposition 3.2.

Proposition 3.2 Given that | det( J g i θ ( a i -1 | s t )) | is a constant with respect to a i -1 , then g -1 θ (argmax z p z ( z ) | s t ) = argmax a Q θ ( s t , a ) .

where ( i ) is because logarithm is strictly increasing, ( ii ) is due to z = g θ ( a | s t ) , and ( iii ) is due to a = g -1 θ ( z | s t ) .

In Section 3.2, we incorporate the learnable reward shifting function b θ in Q θ and V θ . This incorporation results in redefined soft Q-function Q b θ and soft value function V b θ . In Proposition A.3, we verify that V b θ ( s t ) = α log ∫ exp( Q b θ ( s t , a ) /α ) d a = V θ ( s t ) + b θ ( s t ) .

Proposition A.3. Given that Q and V satisfy V ( s t ) ≜ α log ∫ exp( Q ( s t , a ) /α ) d a . The augmented functions, Q b ( s t , a t ) = Q ( s t , a t ) + b ( s t ) and V b ( s t ) ≜ α log ∫ exp( Q b ( s t , a ) /α ) d a , where b ( s t ) is the the reward shifting function, satisfy V b ( s t ) = V ( s t ) + b ( s t ) .

Proof.

A.3 The Issue of Numerical Instability

In this section, we provide the motivation for employing the learnable reward shifting function described in Section 3.2. We show that while Q θ and V θ defined in Eq. (11) have the theoretical capability to learn arbitrary real values (i.e., Proposition 3.1), they may experience numerical instability in practice. This instability arises due to the exponential growth of ∏ i ∈S n | det( J g i θ ( a i -1 t | s t )) | and the exponential decay of ∏ i ∈S l | det( J g i θ ( s t )) | . We first examine the relationship between V θ ( s t ) and ∏ i ∈S l | det( J g i θ ( s t )) | according to the following equations:

The above equation suggests that the value of ∏ i ∈S l | det( J g i θ ( s t )) | decreases exponentially with respect to V θ ( s t ) , which may lead to numerical instability during training. On the other hand, the relationship between Q θ ( s t , a t ) and ∏ i ∈S n | det( J g i θ ( a i -1 t | s t )) | can be expressed according to the following equations:

The equations indicate that ∏ i ∈S n | det( J g i θ ( a i -1 t | s t )) | increases exponentially with respect to Q θ ( s t , a t ) -log p z ( g θ ( a t | s t )) . Therefore, increasing Q θ may also lead to an exponential growth of ∏ i ∈S n | det( J g i θ ( a i -1 t | s t )) | .

LRS makes our model less susceptible to numerical calculation errors since the learnable reward shifting function b θ , unlike Q θ and V θ , is not represented in logarithmic scale. Consider a case where FP32 precision is in use, 'MEow (Vanilla)' could fail to learn a target V θ ∗ ( s t ) > 38 , ∀ s t , since ∏ i ∈S l | det( J g i θ ( s t )) | = exp( -V θ ∗ ( s t )) < 2 -126 cannot be represented using FP32 precision. Therefore, without shifting the reward function, the loss sometimes becomes undefined values, and can lead to ineffective training (e.g., the green lines in Fig. 6). The reward shifting term can be designed as a (state-conditioned) function or a (non-state-conditioned) value. It can also be learnable or non-learnable. All of these designs (i.e., b θ ( s t ) , b ( s t ) , b θ , and b ) can be directly applied to MEow since none of them influences the action distribution. Based on our preliminary experiments, we identified that a learnable state-conditioned reward shifting delivers the best performance.

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A.4 Supplementary Experiments

In this section, we provide additional experimental results. In Section A.4.1, we offer a comparison between MEow with g θ modeled using additive coupling layers and that using affine coupling layers. In Section A.4.2, we compare the performance of MEow with four distinct types of actor-critic frameworks formulated based on prior works [9-11]. In Section A.4.3, we provide an example illustrating the ability of flow-based models to represent multi-modal distributions as policies. In Section A.4.4, we present a performance comparison between SAC and its variant with LRS. Finally, in Section A.4.5, we provide a sensitivity examination for the target smoothing parameter.

A.4.1 Comparison of Additive and Affine Transformations

In this section, we evaluate the performance of MEow with two commonly-adopted non-linear transformations, additive [46] and affine [47] coupling layers, for constructing g θ . The results are presented in Fig. A1. The results show that MEow with additive coupling layers achieves better performance than that with affine coupling layers. Based on this observation, we adopt additive coupling layers for constructing g θ throughout the experiments in Section 4 of the main manuscript.

A.4.2 Influences of Parameterization in MaxEnt RL Actor-Critic Frameworks

In this section, we compare the performance of MEow against four different actor-critic frameworks formulated based on prior works [9-11]. The first framework is the same as SAC [9], with the critic modeled as an energy-based model and the actor as a Gaussian. The second framework follows the approaches of [10, 11], where the critic is also an energy-based model, but the actor is a flow-based model. The third and fourth frameworks both utilize a flow-based model for the critic, with the actor modeled as a Gaussian and a flow-based model, respectively. These frameworks are denoted as: 'Energy Critic+Gaussian Actor' (ECGA), 'Energy Critic+Flow Actor' (ECFA), 'Flow Critic+Gaussian Actor' (FCGA), and 'Flow Critic+Flow Actor' (FCFA), respectively. Regarding the

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soft value calculation during training, the first and second frameworks adopt the value estimation method in SAC (i.e., Eq. (7)). For the third and the fourth frameworks, their training adopts the exact value calculation (i.e., Eq. (11)), which is the same as MEow. The results are presented in Fig. A2.

As depicted in Fig. A2, MEow exhibits superior performance and stability compared to other actor-critic frameworks in the 'Hopper-v4', 'Ant-v4', and 'Walker2d-v4' environments, and shows comparable performance with ECGA in most environments. In addition, the results that compare the frameworks with flow-based models as actors (i.e., ECFA and FCFA) to those with Gaussians as actors (i.e., ECGA and FCGA) suggest that Gaussians are more effective for modeling actors. This finding is similar to that in [10]. On the other hand, the comparisons between ECGA and FCGA, and between FCGA and FCFA, do not show a clear trend. These findings suggest that both flow-based and energy-based models can be suitable for modeling the soft Q-function. Furthermore, the comparison between FCFA and MEow reveals that the training process involving alternating policy evaluation and improvement steps may be inferior to our proposed training process with a single objective.

A.4.3 Modeling Multi-Modal Distributions using Flow-based Models

(b) (a) Action State Action State Figure A3: (a) The reward landscape of the one-step environment described in Section A.4.3. (b) The conditional pdf prediction using an NSF model. In this section, we use a one-dimensional example to demonstrate that flow-based models are capable of learning multi-modal action distributions. We employ a state-conditioned neural spline flow (NSF) [50] as the model, and train it in a single-step environment with one-dimensional state and action spaces. Fig. A3 (a) illustrates the reward landscape with the state and action denoted on the xaxis and y-axis, respectively. Fig. A3 (b) illustrates the probability density function (pdf) predicted by the model. The result demonstrates the capability of flow-based models to effectively learn multi-modal distributions.

A.4.4 Applying Learnable Reward Shifting to SAC

In this section, we examine the performance of SAC with the proposed LRS technique. Since the original implementation of SAC involves the clipped double Q-Learning technique, SAC with LRS is equivalent to SAC with the shifting-based clipped double Q-Learning (SCDQ) technique discussed in Section 3.2 of the main manuscript. The performance of 'SAC' and 'SAC (+LRS)' is presented in Fig. A4. The results indicate that applying LRS does not improve SAC's performance. Therefore, for a fair evaluation, the original implementation of SAC is adopted in the comparison in Section 4 of the main manuscript.

A.4.5 Sensitivity Examination for the Target Smoothing Parameter

In this section, we provide a performance comparison of SAC and MEow trained with different target smoothing parameter values (i.e., τ = 0 . 005 , 0 . 003 , 0 . 0005 , and 0 . 0001 ). The results shown in

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Fig. A5 indicate that SAC performs the best when τ = 0 . 005 , while MEow requires different τ values to achieve good performance across different tasks. Although both algorithms exhibit significant performance variations with different τ values, SAC demonstrates a more consistent trend in terms of the total returns among the tested values of τ .

A.5 Experimental Setups

In this section, we elaborate on the experimental configurations and provide the detailed hyperparameter setups for the experiments presented in Section 4 of the main manuscript. The code is implemented using PyTorch [72] and is available in the following repository: https: //github.com/ChienFeng-hub/meow .

A.5.1 Model Architecture

Among all experiments presented in Section 4 of the main manuscript, we maintain the same model architecture, while adjusting inputs and outputs according to the state space and action space for each environment. An illustration of this architecture is presented in Fig. A6. The model architecture comprises three main components: (I) normalizing flow, (II) hypernetwork, and (III) reward shifting function. For the first component, the transformation g θ includes four additive coupling layers [46] followed by an element-wise linear layer. The prior distribution p z is modeled as a unit Gaussian. For the second component, the hypernetwork involves two types of multi-layer perceptrons (MLPs), labeled as (a) and (b) in Fig. A6, which produce weights for the non-linear and linear transformations, respectively. Both MLPs employ swish activation functions [73] and have a hidden layer size of 64 . The MLPs labeled as (a) incorporate layer normalization [74] and a dropout layer [75] with a dropout rate of 0 . 1 . For the third component, the reward shifting functions (i.e., b (1) θ and b (2) θ ) are implemented using MLPs with swish activation and a hidden layer size of 256 . The parameters used in these components are collectively referred to as θ , and are optimized using the same objective function L ( θ ) defined in Eq. (4), with the soft Q-function and the soft value function replaced by Q b θ and V b θ , respectively. Please note that, for the sake of notational simplicity and conciseness, the parameters of each network are all represented using θ instead of distinct symbols (e.g., θ (II)-(a) , θ (II)-(b) , θ (III)-(1), and θ (III)-(2) ).

A.5.2 Experiments on the Multi-Goal Environment

The experiments in Section 4.1 are performed on a two-dimensional multi-goal environment [8]. This environment consists of four goals positioned at [0 , 5] , [0 , -5] , [5 , 0] , and [ -5 , 0] , denoted as g 1 , g 2 , g 3 , and g 4 , respectively. The reward is the sum of two components, r 1 ( s t ) and r 2 ( a t ) , which are

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formulated as follow:

According to Eq. (A3), r 1 ( s t ) encourages policies to reach states near the goals. On the other hand, r 2 ( a t ) encourages policies to produce actions with small magnitudes.

In this experiment, we adopt a temperature parameter α = 2 . 5 , a target smoothing factor τ = 0 . 0005 , a learning rate β = 0 . 001 , a discount factor γ = 0 . 9 , and a total of 4 , 000 training steps. The computation was carried out on NVIDIA TITAN V GPUs equipped with 12GB of memory. The training takes approximately four minutes.

A.5.3 Experiments on the MuJoCo Environments

Software and Hardware Setups. For the experiments on the MuJoCo environments, our implementation is built on CleanRL [76], with the normalizing flow component adapted from [77]. The computation was carried out on NVIDIA V100 GPUs equipped with 16GB of memory. The training takes approximately 13 hours per 1 million steps, with each GPU capable of executing four training sessions simultaneously.

Hyperparameter Setups. The shared and the environment-specific hyperparameters of MEow are summarized in Tables A1 and A3, respectively. The hyperparamers for the baseline methods are directly borrowed from Stable Baseline 3 (SB3) [71].

A.5.4 Experiments on the Omniverse Isaac Gym Environments

Software and Hardware Setups. For the experiments performed on Omniverse Isaac Gym, the implementation is built on SKRL [79] due to its compatibility with Omniverse Issac Gym [34]. The computation was carried out on NVIDIA L40 GPUs equipped with 48GB of memory. The training takes approximately 22 hours per 1 million training steps, with each GPU capable of executing three training sessions simultaneously. For 'Ant', 'Humanoid', 'Ingenuity', and 'ANYmal', each training step consists of 128 parallelizable interactions with the environments. For 'AllegroHand' and 'FrankaCabinet', each training step consists of 512 parallelizable interactions with the environments.

Hyperparameter Setups. The shared and the environment-specific hyperparameters of MEow are summarized in Tables A1 and A3, respectively. Those of SAC are summarized in Tables A2 and A4, respectively. Both SAC and MEow were tuned using the same search space for τ and α to ensure a fair comparison. Specifically, a grid search was conducted with τ values ranging from 0.1 to 0.00025 and α values from 0.8 to 0.0005 for both algorithms. The setups with the highest average return were selected for each environment.

A.6 Broader Impacts

This work represents a new research direction for MaxEnt RL. It discusses a unified method that can be trained using a single objective function and can avoid Monte Carlo estimation in the calculation of the soft value function, which addresses two issues in the existing MaxEnt RL methods. From a practical perspective, our experiments demonstrate that MEow can achieve superior performance compared to widely adopted representative baselines. In addition, the experimental results conducted in the Omniverse Isaac environments show that our framework can perform robotic tasks simulated based on real-world application scenarios. These results indicate the potential for deploying MEow in real robotic tasks. Given MEow's potential to be extended to perform challenging tasks, it is unlikely to have negative impacts on society.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a new framework for Maximum Entropy Reinforcement Learning using Energy-Based Normalizing Flows (EBFlow). The authors argue that this framework offers three main advantages: Firstly, the soft value function can be obtained without estimation. Second, the framework combines policy evaluation and policy improvement in one iteration. Finally, it does not use Monte Carlo methods and interacts with the environment effectively. To achieve this goal, Learnable Reward Shifting (LRS) technique which involves reward shifting for the value function using a reward shifting network and Shifting-Based Clipped Double Q-Learning (SCDQ) technique that employs two reward shifting networks for the value function and utilizes the minimum value function similar to double Q-learning have been proposed.\n\nSTRENGTHS:\n1. The paper is well-structured, and the overall content is easy to follow and intuitive. The detailed background explanations allow readers to progress through the main text with a solid understanding of the prior work.\n\n2. The experimental domains are diverse, clearly demonstrating the strengths of the proposed algorithm in an easy-to-understand manner. The experiments are well-structured, and the detailed explanations make it easier for readers to follow the setup.\n\n3. The proposed algorithm does not use Monte Carlo-based methods, allowing it to select the proper action for training during each step without waiting for the end of an episode. This enables the policy to interact with the environment more effectively\n\n4. Existing SAC-based algorithms often rely on value function estimation, which can introduce errors. The proposed algorithm can obtain the soft value function without estimation, thus avoiding such errors.\n\nWEAKNESSES:\n1. Despite the fact that the proposed method is the first to apply EBFlow to reinforcement learning (RL), the novelty of the algorithm may appear weak, as there are several existing RL methods based on flow-based models. Besides applying EBFlow to the RL structure, were there are any additional considerations or advantages that were taken into account to enhance its performance or applicability in the context of RL? If such points exist, highlighting them would help to emphasize the contributions of the authors' proposed technique.\n\n2. Introducing new frameworks generally necessitates providing sufficient mathematical proof to validate their efficacy, even for basic techniques or simple conjectures. It would be beneficial if the paper included mathematical proofs demonstrating that the proposed framework converges to the optimal policy. For example, the techniques presented in the paper, namely Learnable Reward Shifting (LRS) and Shifting-Based Clipped Double Q-Learning (SCDQ), have a significant impact on performance. However, the paper does not provide adequate mathematical justification for these techniques. Including such proofs would greatly enhance the confidence in the proposed method.\n\n3. It would be beneficial for the paper to compare its methods with more recent state-of-the-art algorithms. The comparison with SAC, published in 2018, feels a bit outdated. Given that reinforcement learning is a rapidly evolving field, including comparisons with more recent methods would provide a clearer understanding of where this framework stands in the current landscape. Such comparisons would greatly enhance the paper by placing the proposed method in the context of the latest advancements.\n\nQUESTIONS:\n1. Please answer the questions in the weakness section\n\n2. According to the appendix, the proposed method is said to be 2.3 times slower than SAC. Given this difference in speed, a time-wise comparison of the algorithm's performance would be very helpful to understand its efficiency relative to SAC. \n\n3. In Figure 3, SAC and MEow show similar performance in the Mujoco domain, but there is a noticeable difference in performance in the Isaac Gym domain in Figure 4. Could you provide a further analysis on the reasons for the occurrence of this difference? It would provide valuable insight into the strengths and limitations of the proposed method.\n\nMinor Issues\\\nTypo in line 149: \"Jocobian\" should be \"Jacobian.\"\\\nMisreference in section 4.1: It should refer to Fig. 2, not Fig. 1.\n\nLIMITATIONS:\nThey have handled it well.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-10"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes to use the energy-based normalizing flow (EBFlow) to represent the policy and value functions in maximum entropy (MaxEnt) RL. Specifically, the paper builds the connection between EBFlow and MaxEnt RL by linking the conditional unnormalized density that depends on the input to the action value function Q and the constant that is independent of the input to the value function V. As a consequence, the Boltzmann policy defined by Q and V can be expressed exactly using a mapping in EBFlow. By updating the current Q estimate to approximate the action value of the Boltzmann policy, the algorithm would spontaneously update the Boltzmann policy. Hypothetically, the algorithm should converge to the optimal policy under entropy regularization. Using an existing implementation of EBFlow in the relevant literature, the paper proposed MEow based on this formulation. Experiments on four MuJoCo environments and six Issac Gym environments demonstrate the learning performance of MEow.\n\nSTRENGTHS:\nThe strengths of this paper are as follows:\n1. The connection between EBFlow and MaxEnt RL revealed in this paper is very interesting and, to my knowledge, novel. Given that MaxEnt is a popular approach in the RL literature, the issue addressed in this paper, learning the action value function and the Boltzmann policy, as well as the proposed approach, will be interesting to a broad set of audiences in the RL community.\n2. The paper includes abundant comparative and ablation experiments. Notably, the ablation study covers the effect of the two components in the practical implementation, the difference between the deterministic and stochastic evaluation, etc.\n3. The paper is well-written and mostly clear.\n\nWEAKNESSES:\n1. There is a potential concern about the empirical investigation: The proposed method is tuned over the target smoothing parameter, while the baselines are not tuned. Why is the target smoothing parameter tuned per environment? What would happen if the proposed method also used the same, fixed smoothing parameter?\n2. This is a minor point. The organization of the paper can be improved. Specifically, the background section is too long, which takes up the space for more interesting discussions presented in the appendix. If space allows, the main text should include the network architecture in Figure A5, as it may be unfamiliar to a lot of readers. In addition, the limitation, especially the computational aspect of the proposed approach, should be discussed in the main paper.\n\nOther suggestions that do not affect the evaluation:\n* It would be great if there is at least some qualitative comparison between the proposed method and baselines using MaxEnt RL in the multi-goal environment.\n* Typos between Lines 254 and 276: The reference to Figure 1 should be to Figure 2.\n\nQUESTIONS:\nSome questions that may affect the evaluation:\n1. Are deterministic policies used for other algorithms in the evaluation in Figures 3 and 4?\n\nLIMITATIONS:\nThe limitations of the work are discussed in the appendix, which should be moved to the main text. Another unmentioned potential limitation is the sensitivity to the target network smoothing parameter.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-10"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a new framework for Maximum-Entropy (MaxEnt) Reinforcement Learning (RL) using Energy-Based Normalizing Flows (EBFlow). Traditional MaxEnt RL methods, particularly for continuous action spaces, typically utilize actor-critic frameworks and alternate between policy evaluation and policy improvement steps. The proposed EBFlow framework integrates these steps into a single objective, eliminating the need for Monte Carlo approximation while enabling efficient modeling of multi-modal action distributions. The framework was experimentally validated on the MuJoCo benchmark and high-dimensional robotic tasks in Omniverse Isaac Gym, demonstrating superior performance compared to existing baseline methods.\n\nSTRENGTHS:\n- By combining policy evaluation and policy improvement into a single training process, the proposed method simplifies the optimization process and avoids potential optimization errors inherent in alternating updates.\n- The use of normalizing flows facilitates efficient sampling and exact calculation of probability density functions, which addresses the inefficiencies associated with Monte Carlo methods in traditional MaxEnt RL approaches.\n- Experimental results indicate that the EBFlow framework outperforms widely-adopted baselines on both the MuJoCo benchmark and complex robotic tasks, showcasing its effectiveness and robustness.\n\nWEAKNESSES:\n- There are some unreasonable aspects in the organization of the content. For example, Section 3.2 appears somewhat abrupt; it is suggested to move the content of A.4 into the main text. The discussion about inference in Section A.3 is also very important and should be included in the main text.\n- The experimental section does not sufficiently highlight the advantages of the MEow method. The experiments in Section 4.1 on toy examples are not enough. Can comparisons be made from the perspectives of inference time, training time, representation capability, etc.?\n- While the integration of policy evaluation and improvement steps simplifies the training process conceptually, the implementation details of the EBFlow framework may introduce additional complexity that needs to be managed.\n\nQUESTIONS:\n- GFlowNet points out that the MaxEnt RL paradigm has deficiencies in terms of diversity. Specifically, MaxEnt RL tends to achieve target states that are \"further\" from the initial state. Does MEow also inherit this defect of the MaxEnt RL paradigm?\n- Energy-based normalizing flow is a generative model, and its original loss is to maximize log-likelihood. In Algorithm 1, this loss does not seem to be present. In my view, MEow uses the network structure of energy-based normalizing flow, but the loss still follows the MaxEnt RL paradigm. In this case, could the title of this paper be potentially misleading?\n- Figure A5 shows that the MEow network uses a hypernetwork module. Why was this module introduced? Is this a common architecture for energy-based normalizing flow?\n- What are the learning dynamics observed during the training process? Are there any notable differences in the convergence behavior compared to traditional actor-critic methods?\n- How does the proposed framework scale with increasing complexity of the environment or the state-action space? Are there any specific limitations or challenges observed during experimentation?\n- How robust is the EBFlow framework to different hyperparameters?\n\nLIMITATIONS:\nThe author has already discussed the limitations ofthe method and its potential impacts in the paper, and provided possible solutions.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces a new MaxEnt RL framework called Meow based on Energy-based Normalizing Flow, which integrates the policy evaluation steps and the policy improvement steps and results in a single objective training process. Besides Meow enables the calculation of the soft value function used in the policy evaluation target without Monte Carlo approximation and supports the modeling of multi-modal action distributions.\n\nSTRENGTHS:\n1.\tThe proposed method is innovative and interesting, which utilizes the special property of Energy-based Normalizing Flow, and thus enables the consistency between the actor and critic and a single objective training process.\n2.\tThe paper is overall well-written. The background and related works are introduced appropriately.\n3.\tThe experimental results are comprehensive, involving the mujoco and Issac gym benchmarks.\n\nWEAKNESSES:\n1.\tThe calculation of the derterminant is usually time-consuming. So the reviewer wonders how the training time of Meow is compared with that of the classical off-policy RL methods like SAC.\n2.\tIn the experiments, Meow is just compared with several classical RL methods for continuous control and lacks the comparison with advanced RL methods [1, 2] with other generative models like diffusion model and consistency model, which also supports the modeling of multi-modal action distribution.\n\n[1] Yang L, Huang Z, Lei F, et al. Policy representation via diffusion probability model for reinforcement learning[J]. arXiv preprint arXiv:2305.13122, 2023.\n\n[2] Yue Y, Kang B, Ma X, et al. Boosting offline reinforcement learning via data rebalancing[J]. arXiv preprint arXiv:2210.09241, 2022.\n\nQUESTIONS:\n1.\tHow is the training time of Meow compared with other methods?\n2.\tHow is Meow compared with other advanced RL methods with the utilization of generative-model-based policy?\n3.\tCould the authors explain why Meow can learn a good reward shifting rather than a trivial shifting, like for any $s_t$, $b(s_t)=0$?\n\nLIMITATIONS:\nThe authors have clearly presented the limitations in the paper.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a new method for MaxEnt RL based on the recently proposed energy-based normalizing flows (EBFlow). The adoption of EBFlow allows us to overcome two major issues with training MaxEnt RL algorithms: (I) sampling from an energy-based model (of the policy) and (ii) approximating the soft value function (with MC methods).\n\nSTRENGTHS:\n* **Contribution**: I think the contribution of this work is significant and novel. The issues discussed with MaxEnt RL are important. The idea proposed is novel, it naturally emerges from the EBFlow framework and it's seamlessly applied to the MaxEnt RL domain. The approach solves the problems mentioned and it demonstrates superior performance compared to other standard approaches.\n* **Presentation**: the presentation is clear and well-organised, and the method is described in detail.\n\nWEAKNESSES:\n* **New and old challenges**: while the approach resolves some issues with the current MaxEnt RL methods, it still suffers from typical deep RL issues (e.g. overestimation, for which the authors proposed SCDQ) and it introduces additional challenges (e.g. exploding Jacobian determinants, for which the authors introduce LRS). Normalizing flows also introduce additional constraints on the network's architecture, a problem which I believe is strongly mitigated by the introduced LRS (?)\n* **Required clarification on LRS**: see Questions\n\nQUESTIONS:\n* Typo at line 149, \"Jocobian\"\n* I am not sure if the role of the Learnable Reward Shifting (LRS) is clear to me. The authors say that they were inspired by the reward-shifting literature. However, I don't see the LRS term working the same way as classical reward shaping, as a state-dependent non-linear function ($b_\\theta$), without any particular constraints (I don't see any stated) wouldn't preserve the properties of the original functions. I think the role of the learnable reward shifting is to have some form of \"baseline\" in the Q and V functions that simplifies the estimation process (since it's using a more flexible learning architecture), and so it avoids the explosion of the Jacobian. Could the authors provide clarification on this?\n* I would be interested in seeing a comparison between the learned reward shifting term and the value function (V) in a plot (it would suffice to see this in the Appendix)\n\nLIMITATIONS:\nLimitations are addressed in Appendix. I think the authors should mention this in the main text.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-13"}