Efficient Off-Policy Learning for High-Dimensional Action Spaces

## Abstract

Abstract Existing off-policy reinforcement learning algorithms often rely on an explicit state-action-value function representation, which can be problematic in high-dimensional action spaces due to the curse of dimensionality.
This reliance results in data inefficiency as maintaining a state-action-value function in such spaces is challenging.
We present an efficient approach that utilizes only a state-value function as the critic for off-policy deep reinforcement learning.
This approach, which we refer to as Vlearn, effectively circumvents the limitations of existing methods by eliminating the necessity for an explicit state-action-value function.
To this end, we introduce a novel importance sampling loss for learning deep value functions from off-policy data.
While this is common for linear methods, it has not been combined with deep value function networks.
This transfer to deep methods is not straightforward and requires novel design choices such as robust policy updates, twin value function networks to avoid an optimization bias, and importance weight clipping.
We also present a novel analysis of the variance of our estimate compared to commonly used importance sampling estimators such as V-trace.
Our approach improves sample complexity as well as final performance and ensures consistent and robust performance across various benchmark tasks.
Eliminating the state-action-value function in Vlearn facilitates a streamlined learning process, enabling more effective exploration and exploitation in complex environments.

## 1 Introduction

Reinforcement learning (RL) has emerged as a powerful paradigm for training intelligent agents through interaction with their environment [, ].
On-policy and off-policy are the two primary approaches within model-free RL On-policy methods rely on newly generated (quasi-)online samples in each iteration [, ], whereas off-policy methods leverage a replay buffer populated by a behavior policy [, , ].
Although on-policy methods can compensate for off-policy data to some extent via importance sampling [], they cannot exploit it fully.
To harness the full potential of off-policy data, off-policy methods traditionally focus on learning state-action-value functions (Q-functions) as critics [].
However, the Q-function’s dependency on states and actions prevents it from updating actions that were not used when acting in the environment with the behavior policy.
Furthermore, the associated complexity with learning Q-functions is often undesirable, especially in high-dimensional action spaces.
Thus, alternatives similar to the on-policy setting based on state-value functions (V-functions), would be preferable.

In this work, we present Vlearn, a novel off-policy policy gradient method that exclusively utilizes V-functions.
While existing methods, such as V-trace [], attempt to increase the amount of stale data in on-policy methods, we found them to be less effective in a fully off-policy context.
These methods typically focus on reweighting Bellman targets, but prior research [] indicates that applying importance sampling to the entire Bellman error is more advantageous.
Inspired by this finding for linear models, we show that optimizing this objective in deep RL effectively optimizes an upper bound of the original naive Bellman error, which can be derived using Jensen’s inequality and yields the same location of the minimum.
This loss shifts the importance weights from the Bellman targets to the objective, simplifying V-function updates and enhancing the stability of learning a V-function from off-policy data, which posed significant challenges in earlier methods.
Our analysis in the bandit setting shows that this Bellman error estimator has lower variance compared to the estimators from common deep RL importance sampling schemes like V-trace [].
Additionally, we improve the stability of policy learning by using the trust region update from [] and other commonly used techniques from off-policy RL, forming an efficient off-policy trust region method.
Our experiments demonstrate the importance of these design choices in achieving high performance.
Notably, our method excels in environments with complex action spaces, such as the challenging dog locomotion tasks in DeepMind control (DMC) [], which most standard off-policy actor-critic methods fail to solve.

Figure: Figure 1: To provide intuition on the differences between Vlearn and V-trace, we consider the following example for different importance ratios $\rho$. Assume that for a state $s_{t}$, the Bellman target is $r(s,a)+\gamma V_{\bar{\theta}}(s_{t+1})=4$, the target critic predicts $V_{\bar{\theta}}(s_{t})=-6$ and we plot the loss for values of $V_{\theta}(s)$ for Vlearn and V-trace. For on-policy samples ($\rho=1.0$) both losses are the same. However, for samples that are progressively more off-policy ($\rho\to 0)$, we see how V-trace increasingly relies on the target critic, shifting the optimal value towards it. Vlearn on the other hand simply reduces the scale of the loss and thus the importance of the sample. This makes Vlearn more robust to errors in the target critic.
Refer to caption: x1.png

## 2 Related Work

To enhance sample efficiency in RL, off-policy algorithms aim to leverage historical data stored in a replay buffer [].
While deep Q-Learning-based methods [, , , ] excel in efficiently learning from discrete action spaces, they are often not directly applicable to continuous problems.
For such problems, off-policy actor-critic methods optimize a policy network that generates continuous actions based on the state-action-value function estimator [, ].
This approach accommodates both deterministic [, ] and probabilistic policies [, ].

In the on-policy setting, trust-region methods [, , ] have proven effective in stabilizing the policy gradient.
However, these methods have not received the same level of attention in the off-policy setting.
Past efforts have focused on training the value function with off-policy data [] or extended trust region policy optimization [] to the off-policy setting [, ].
While [], for instance, employs a more standard approach by combining off-policy trust regions with additional advancements, such as Retrace [] and truncated importance sampling, it is still not able to compete with modern actor-critic approaches.
[], on the other hand, provide a different perspective and split policy and value-function learning into separate supervised learning steps.
Generally, standard off-policy trust region methods often fall behind modern actor-critic approaches [, ] and cannot achieve competitive performance.
However, maximum aposteriori policy optimisation (MPO) [] can be seen as a non-classical trust region method.
Its EM-based formulation offers more flexibility, with trust regions akin to optimizing a parametric E-step without an explicit M-step.

While on-policy trust region methods estimate the less complex state-value function, most off-policy methods rely on estimating state-action-value functions to learn from replay buffer data [, , ].
Methods exclusively utilizing state-value functions must employ importance sampling to address the distributional discrepancy between target and behavior policies.
However, this has been primarily studied in the context of asynchronous on-policy settings, where the policy distributions differ only slightly [, ].
In particular, V-trace [] aims to compensate for a limited distributional shift due to asynchronous computations and is not designed for a full off-policy setting, such as in this work.
Further, such an off-policy correction is computationally expensive as it relies on storing and processing trajectories.
Moreover, it is widely acknowledged that the truncation technique used in importance weight calculations suffers from a substantial bias.
Consequently, the implementation of trajectory-based target estimators would exacerbate the already pronounced issue of bias propagation.

In the linear case, the impact of varying the placement of the importance weight in the off-policy case has already been studied.
Applying importance weighting to the entire Bellman error, rather than just the Bellman target, yields better performance and may even lead to reduced variance.
This approach is naturally employed in methods such as off-policy TD(0) [, Chapter 11.1], and further supported by [, , ] on its effectiveness in the linear setting.
However, translating these findings to deep RL presents a challenge, as insights from linear models often do not directly apply [, ].

## 3 Efficient State-Value Function Learning from Off-Policy Data

We seek to determine an optimal policy $\pi(a|s)$ within the framework of a Markov decision process (MDP), which is defined by a tuple $(\mathcal{S},\mathcal{A},\mathcal{T},r,\rho_{0},\gamma)$.
Here, both the state space $\mathcal{S}$ and action space $\mathcal{A}$ are continuous.
The transition density function $\mathcal{T}:\mathcal{S}\times\mathcal{A}\times\mathcal{S}\rightarrow\mathbb{R}
^{+}$ is defined as a function mapping from the current state ${s}_{t}\in\mathcal{S}$ and action ${a}_{t}\in\mathcal{A}$ to the probability density of transitioning to the subsequent state ${s}_{t+1}\in\mathcal{S}$.
The initial state distribution’s density is denoted as $\mu:\mathcal{S}\rightarrow\mathbb{R}^{+}$. The reward attained from interactions with the environment is determined by the function $r:\mathcal{S}\times\mathcal{A}\rightarrow\mathbb{R}$, and the parameter $\gamma\in[0,1)$ represents the discount factor applied to future rewards.
Our primary objective is to maximize the expected cumulative discounted reward $G_{t}=\mathbb{E}_{{\cal T},\mu,\pi}\left[\sum_{k=t}^{\infty}\gamma^{k-t}r({s}_
{k},{a}_{k})\right]$.
Most popular off-policy actor-critic methods [, , ] now aim to find a policy that maximizes the cumulative discounted reward by making use of a learnable state-action value estimate $Q_{\theta}^{\pi}(s,a)=\mathbb{E}_{\pi}\left[G_{t}|{s}_{t}={s},{a}_{t}={a}\right]$.
To train this estimator, they rely on a dataset $\mathcal{D}=\{(s_{t},a_{t},r_{t},s_{t+1})_{t=1\dots N}\}$ and a behavioral policy $\pi_{b}(\cdot|s)$ responsible for generating this dataset.
Typically, $\mathcal{D}$ takes the form of a replay buffer [], with the corresponding behavior policy $\pi_{b}$ being a mixture of the historical policies used to populate the buffer.
Training the state-action value function then usually involves temporal difference learning [, ], with updates grounded in the Bellman equation [].
The objective commonly optimized is

$$ $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{E}_{(s_{t},a_{t} )\sim\mathcal{D}}\Big{[}\big{(}Q_{\theta}^{\pi}(s_{t},a_{t})-\big{(}r(s_{t},a_ {t})+\gamma\mathbb{E}_{s_{t+1}\sim\mathcal{T}(\cdot|s_{t},a_{t}),a_{t+1}\sim \pi(\cdot|s_{t+1})}Q_{\bar{\theta}}(s_{t+1},a_{t+1})\big{)}\big{)}^{2}\Big{]},$ (1) $$

where $Q_{\bar{\theta}}(s,a)$ is a frozen target network.
Furthermore, to mitigate overestimation bias, most methods employ two state-action value functions [].
To maintain the stability of this approach, the target network’s weights are updated via polyak averaging at each time step as $\bar{\theta}\leftarrow\tau\theta+(1-\tau)\bar{\theta}$.

### 3.1 Importance Sampling Estimators for the Bellman Error

Instead of using the state-action function $Q_{\theta}^{\pi}(s_{t},a_{t})$, an alternative approach is to work solely with the state-value function $V_{\theta}^{\pi}(s_{t})$.
This base estimator can be trained by minimizing the following loss function using importance sampling

$$ $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L_{\text{base}}(\theta)= \mathbb{E}_{s_{t}\sim\mathcal{D}}\Bigg{[}\bigg{(}V_{\theta}^{\pi}(s_{t})- \mathbb{E}_{a_{t}\sim\mathcal{D},s_{t+1}\sim\mathcal{T}(\cdot|s_{t},a_{t})} \left[\frac{\pi(a_{t}|s_{t})}{\pi_{b}(a_{t}|s_{t})}\bar{V}\right]\bigg{)}^{2} \Bigg{]}.$ (2) $$

where $\bar{V}=r(s_{t},a_{t})+\gamma V_{\bar{\theta}}(s_{t+1})$.
Directly optimizing this loss can be difficult, as approximating the inner expectation with one Monte Carlo sample yields a high variance.
Getting a more reliable estimate requires multiple samples, which implies either multiple action executions per state or occurrences of the same state in the replay buffer, an unrealistic assumption for most RL problems.
A key distinction from the Q-function based objective (Equation 1) is the introduction of the importance weight, which compensates for the discrepancy between the behavior distribution $\pi_{b}(\cdot|s)$ and the current policy $\pi(\cdot|s)$.
Unlike the Q-function that updates estimates solely for the selected action, the V-function does not depend on specific actions and implicitly updates estimates for all actions.
The difference between the current and the data-generating policy must therefore be taken into account.

Equation 2 is closely related to the 1-step V-trace estimate [].
It can be seen as a naive version of V-trace because a large difference between the target and behavior policies may result in importance weights approaching either zero or infinity, consequently impacting the Bellman optimization target for the state-value function $V_{\theta}^{\pi}(s_{t})$.
While truncated importance weights can prevent excessively large values for the Bellman target, the importance ratios and target may still approach values close to zero.
V-trace reformulates this objective and effectively interpolates between the Bellman target and the current (target) value function in the one-step case $n=1$

$$ $L_{\text{V-trace}}(\theta)=\mathbb{E}_{s_{t}\sim\mathcal{D}}\Bigg{[}\left(V_{ \theta}^{\pi}(s_{t})-\Big{(}\mathbb{E}_{a_{t}\sim\mathcal{D},s_{t+1}\sim \mathcal{T}(\cdot|s_{t},a_{t})}\left[(1-\rho_{t})V_{\bar{\theta}}(s_{t})+\rho_ {t}\bar{V}\Big{)}\right]\right)^{2}\Bigg{]},$ (3) $$

where $\rho_{t}=\min({\pi(a_{t}|s_{t})}/{\pi_{b}(a_{t}|s_{t})},\epsilon_{\rho})$ are the truncated importance weights with a user-specified upper bound (typically $\epsilon_{\rho}=1$).
This objective can then also be extended for off-policy corrections of n-step returns [].
Yet, this formulation has two main drawbacks.
Since executing multiple actions for the same state is impractical, or even impossible, V-trace approximates the inner expectation with just one action sample, resulting in potentially undesirable high variance estimates.
Moreover, while the V-trace reformulation avoids optimizing the value estimate toward zero for small importance ratios, the interpolation now optimizes it increasingly toward the current (target) value function (see Figure 1).
This interpolation also leads to a shift of the optimum, which means for samples with small importance ratios the value function is barely making any learning progress and maintains its current estimate.
It is important to note that only the optimum changes; the scale of the loss function, and thus the scale of the gradient, remains the same for all importance ratios.
In the original work, this issue is less pronounced, as V-trace provides off-policy corrections for asynchronous distributed workers of an on-policy method.
In this setup, the policy distributions are expected to stay relatively close, mitigating the impact of any potential divergence.
However, in a complete off-policy setting, such as in our case, the samples in the replay buffer and the current policy deviate significantly faster and stronger from each other.
The size of the replay buffer, learning speed, or entropy of different policies can all influence the importance ratio.

Finally, [, ] introduced a normalized importance weighting estimator for the Bellmann error which has been used for linear value function estimation methods such as LSTD [].
This loss is given by

$$ $\displaystyle L_{\text{Vlearn}}(\theta)=\mathbb{E}_{(s_{t},a_{t})\sim\mathcal{ D},s_{t+1}\sim\mathcal{T}(\cdot|s_{t},a_{t})}\left[\frac{\pi(a_{t}|s_{t})}{\pi _{b}(a_{t}|s_{t})}\Big{(}V_{\theta}^{\pi}(s_{t})-\bar{V}\Big{)}^{2}\right].$ (4) $$

Previous research in the linear domain has already demonstrated that weighting only the Bellman targets is less effective than applying importance weights to the entire loss function [, ].
Despite this evidence, deep learning approaches have predominantly continued to use the former method [, ]
Thus, this work aims to adapt these insights from [, ] for deep RL applications.

### 3.2 VLearn: Deep Off-Policy State-Value Function Learning

To address the above-mentioned issues of V-trace, we propose a more effective approach to training the value function by following [, ] and shifting the importance ratio to the full loss function by extending their results to the general function approximation case.
We first show that both loss functions, i. e. Equation 2 and Equation 5 share the same optimum.

###### Theorem 1

Consider the following loss, minimized with respect to the V-function’s parameters $\theta$

$$ $\displaystyle\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L_{\text{Vlearn}}(\theta )=\mathbb{E}_{(s_{t},a_{t})\sim\mathcal{D},s_{t+1}\sim\mathcal{T}(\cdot|s_{t}, a_{t})}\left[\frac{\pi(a_{t}|s_{t})}{\pi_{b}(a_{t}|s_{t})}\Big{(}V_{\theta}^{ \pi}(s_{t})-\bar{V}\Big{)}^{2}\right].$ (5) $$

This objective serves as an upper bound to the importance-weighted Bellman loss (Equation 2). Furthermore, this upper bound is consistent; that is, a value function minimizing Equation 5 also minimizes Equation 2.

The first statement’s proof relies on Jensen’s Inequality, while the proof of the second statement is an extension of [].
The complete proofs are provided in Appendix A.1 and A.2, respectively.

This upper bound closely resembles the standard state-action-value loss functions used in other off-policy methods but is based on the action-independent state-value function.
It introduces importance weights to account for the mismatch between behavior and policy distribution.
Importantly, evaluating Equation 5 becomes straightforward using the provided replay buffer $\mathcal{D}$.

###### Theorem 1

#### Variance Analysis of Importance Sampling Estimators of the Bellman Error.

When we draw samples from the joint state-action distribution, it simplifies handling a single action per state.
This helps to reduce variance compared to the original and V-trace objectives.
Moreover, one of the challenges with the use of importance sampling is that its estimator can exhibit high variance when the target and behavior policies significantly diverge.
However, evaluating the variance of the importance sampling estimator is generally intractable [, ].
Therefore, we consider the stateless MDP, the multi-armed bandit problem, as a simplified scenario.
The critic’s empirical losses for the base, Vlearn, and V-trace objective (Equation 2, 3 and 5) can then be simplified for $N$ samples as

$$ $\displaystyle L_{\text{base}}=\left(V-\frac{1}{N}\sum_{a\sim\pi_{b}}\rho_{a}r( a)\right)^{2}$ $\displaystyle\qquad L_{\text{V-trace}}=\frac{1}{N}\sum_{a\sim\pi_{b}}\Big{(}V- \big{(}\left(1-\rho_{a}\right)V+\rho_{a}r(a)\big{)}\Big{)}^{2}$ $\displaystyle L_{\text{Vlearn}}=$ $\displaystyle\frac{1}{N}\sum_{a\sim\pi_{b}}\rho_{a}\Big{(}V-r(a)\Big{)}^{2},$ $$

where we denote weights $\rho_{a}={\pi(a)}/{\pi_{b}(a)}$.
Taking the derivatives of the above losses with respect to the estimator $V$ and solving the derivative of zero for $V$ yields the following three estimators

$$ $\displaystyle\hat{V}_{\text{base}}=\frac{\sum_{a\sim\pi_{b}}\rho_{a}r(a)}{N} \qquad\hat{V}_{\text{V-trace}}=\frac{\sum_{a\sim\pi_{b}}\rho_{a}^{2}r(a)}{\sum _{a\sim\pi_{b}}\rho_{a}^{2}}\qquad\hat{V}_{\text{Vlearn}}=$ $\displaystyle\frac{\sum_{a\sim\pi_{b}}\rho_{a}r(a)}{\sum_{a\sim\pi_{b}}\rho_{a }}.$ $$

This simplified scenario illustrates the intuition that our proposed weighted loss yields the self-normalized importance weighting estimator, while V-trace results in a squared self-normalized estimator and the base objective yields a standard importance weighting estimator.
It is well established that the self-normalized estimator is more robust in the presence of extreme importance weights and often yields a lower value than the standard importance weighting estimator in general machine learning [] and in RL settings [, ].
However, the squared estimator from V-trace yields higher variances, as discussed in Appendix A.3.

Furthermore, unlike V-trace, each sample optimizes towards the Bellman target but has an impact on the total loss per step depending on its importance weight, as shown in Figure 1.
The smaller importance weights mainly scale the gradient without causing a shift in optimizing to a different optimum, such as the current (target) value function.
This approach makes learning the state-value function in an off-policy setting more stable and efficient.

#### Choice of the Behavior Policy.

An essential consideration is defining the behavior policy $\pi_{b}$ in an off-policy setting, where $\pi_{b}$ is a mixture of all past policies that contributed to the replay buffer.
Computationally, storing and evaluating this mixture policy would be expensive.
However, since we rely on importance sampling, which only requires access to the (log-)probabilities of each action, we can easily extend the replay buffer with this single entry at minimal additional cost.
In addition, this (log-)probability can then be used directly for importance sampling in the policy update step.

### 3.3 Off-Policy Policy Learning with VLearn

Figure: Algorithm 1

To find the optimal policy, conventional policy gradient techniques frequently use the gradient of the likelihood ratio and an importance sampling estimator to optimize the estimated Q-function.
In particular, a more effective approach involves optimizing the advantage function, denoted as $A^{\pi}({s},{a})=Q^{\pi}({s},{a})-V^{\pi}({s})$.
Adding the V-function as a baseline yields an unbiased gradient estimator with reduced variance.
The optimization can then be formulated as follows

$$ $\max_{\phi}\hat{J}(\pi_{\phi},\pi_{b})=\max_{\phi}\mathbb{E}_{(s,a)\sim\cal D} \left[\frac{\pi_{\phi}({a}|{s})}{\pi_{b}({a}|{s})}A^{\pi}({s},{a})\right].$ (6) $$

In the on-policy case, the advantage values are commonly estimated via Monte Carlo approaches, such as general advantage estimation [].
Yet, in our setting, we cannot reliably compute the Monte Carlo estimate and do not have a Q-function estimate [].
Further, the direct use of the V-function to improve the policy is not feasible.
Nevertheless, using a replay buffer enables a strategy akin to the standard policy gradient.
This approach enables us to utilize an off-policy estimate of the V-function, resulting in a significant improvement in sample efficiency.
The advantage estimate is computed using the one-step return of our off-policy evaluated value function $A^{\pi}(s_{t},a_{t})=r_{t}+\gamma V_{\theta}^{\pi}(s_{t+1})-V_{\theta}^{\pi}(s
_{t})$.
Equation 6 can then be optimized by any policy gradient algorithm, e. g. , proximal policy optimization (PPO) [] or trust region projection layer (TRPL) [].

In this work, we use TRPL [] as it has been shown to stabilize learning even for complex and high-dimensional action spaces [, ].
Unlike PPO, it provides a mathematically sound and scalable approach to enforce trust regions exactly per state.
Moreover, TRPL allows us to use the constraint policy also during the value function update, which now requires importance sampling.
PPO has only the clipped objective for the policy update, and using a clipped value function has been shown to potentially degrade performance [].

TRPL efficiently enforces a trust region for each input state of the policy using differentiable convex optimization layers [], providing more stability and control during training and at the same time reducing the dependency on implementation choices [].
Intuitively, the layer ensures the predicted Gaussian distribution from the policy network always satisfies the trust region constraint.
This way, the objective from Equation 6 can directly be optimized as the trust region always holds.
The layer receives the network’s initial prediction for the mean $\bm{\mu}$ and covariance $\bm{\Sigma}$ of a Gaussian distribution and projects them into the trust region when either exceeds their respective bounds.
This is done individually for each state provided to the network.
The resulting Gaussian policy distribution, characterized by the projected parameters $\tilde{\bm{\mu}}$ and $\tilde{\bm{\Sigma}}$, is then used for subsequent computations, e. g. sampling and loss computation.
Formally, as part of the forward pass, the layer solves the following two optimization problems for each state $\bm{s}$

$$ $\displaystyle\operatorname*{arg\,min}_{\tilde{\bm{\mu}}_{s}}d_{\textrm{mean}} \left(\tilde{\bm{\mu}}_{s},\bm{\mu}(s)\right),\,$ $\displaystyle\textrm{s.t.}\,d_{\textrm{mean}}\left(\tilde{\bm{\mu}}_{s},{\bm{ \mu}}_{\textrm{old}}(s)\right)\leq\epsilon_{\bm{\mu}}$ $\displaystyle\operatorname*{arg\,min}_{\tilde{\bm{\Sigma}}_{s}}d_{\textrm{cov} }\left(\tilde{\bm{\Sigma}}_{s},\bm{\Sigma}(\bm{s})\right),\,$ $\displaystyle\textrm{s.t.}\,d_{\textrm{cov}}\left(\tilde{\bm{\Sigma}}_{s},{\bm {\Sigma}}_{\textrm{old}}(\bm{s})\right)\leq\epsilon_{\Sigma},$ $$

where $\tilde{\bm{\mu}}_{s}$ and $\tilde{\bm{\Sigma}}_{s}$ are the optimization variables for input state $\bm{s}$.
The trust region bounds $\epsilon_{\mu}$, $\epsilon_{\Sigma}$ are for the mean and covariance of the Gaussian distribution, respectively.
For the dissimilarities between means $d_{\textrm{mean}}$ and covariances $d_{\textrm{cov}}$, we use the decomposed KL-divergence.
How to receive the gradients for the backward pass via implicit differentiation is described in [].

As the original V-trace algorithm is primarily used for off-policy corrections in distributed on-policy learning methods, it typically does not incorporate a target network.
In contrast, it is common practice for off-policy methods [, , ] to incorporate a target network.
This practice is also followed by V-trace’s predecessor, Retrace [], and we include it for VLearn as well.

### 3.4 Behavior Policy vs Old Policy

For Vlearn, we keep track of three different policies: The current policy $\pi_{\phi}$, which is optimized, the old policy ${\pi}_{\textrm{old}}$, which is used as a reference for the trust region, and the behavioral policy $\pi_{b}$, which is needed for off-policy correction using importance sampling.
Similar to most on-policy trust region methods, ${\pi}_{\textrm{old}}$ is a copy of the main policy network from the previous iteration.
Choosing the behavioral policy for the trust region would be detrimental to performance because it would slow down the policy network update or, in the worst case, force the policy back to a much older and worse policy distribution.
Conversely, we cannot use the copied old policy as a behavior policy because the actual behavior policy $\pi_{b}$ can be arbitrarily far away, especially for older samples.
Generally, the behavior policy is a mixture of all past policies that have been used to populate the replay buffer.
However, storing and/or evaluating this would be expensive, so we assume that each sample belongs to only one mixture component, specifically the policy that originally created the action.
Since we rely on importance sampling for the off-policy correction, we can simply store the (log-)probability for each action as part of the replay buffer to represent $\pi_{b}$ during training.

To stabilize our method further, we make use of improvements of other actor-critic and policy gradient methods, including twin critics and delayed policy updates [], $\tanh$ squashing [], as well as advantage normalization [, ].
While the overestimation bias is not a direct problem when using just state-value functions, we found them beneficial in practice.
We assume the twin network acts as a small ensemble that provides a form of regularization.
Given there is no significant drawback and the widespread adoption of the twin network approach in other baseline methods, we chose to maintain it in our case as well.
Additionally, similar to prior works [, ] we aim to reduce the variance by replacing the standard importance sampling ratio with truncated importance sampling $\min({\pi(a_{t}|s_{t})}/{\pi_{b}(a_{t}|s_{t})},\epsilon_{\rho})$.
An overview of our full approach is shown in Algorithm 1.

## 4 Experiments

Figure: Figure 2: Mean over $10$ seeds and 95% bootstrapped confidence intervals for the Gymnasium tasks. While SAC performs well on the lower dimensional tasks, Vlearn achieves better asymptotic performance for the higher dimensional problems - Ant-v4 and Humanoid-v4. Compared to V-trace our method is significantly more stable and achieves a better final performance.
Refer to caption: x3.png

For our experiments, we evaluate Vlearn on different continuous control tasks from Gymnasium [] and DMC [].
As baselines, we trained the standard off-policy methods SAC [] and MPO [] as well as PPO [] and the on-policy version of TRPL [].
In addition, we compare our method to 1-step V-trace [] by replacing our lower bound objective from Equation 5 with the objective in Equation 3.
This comparison aims to highlight the difference in the placement of the importance ratios.
To ensure a fair assessment, we want to eliminate any external factors that could influence the results and thus do not use n-step returns.
While the n-step V-trace can be computed recursively, our approach can leverage the product of the n-step importance weights, preserving consistency with V-trace.
All other components of the method remain the same as in Vlearn.

The evaluation protocol follows the approach described in [].
All methods are evaluated for $10$ different seeds each, and their performance is aggregated using mean values and 95% bootstrapped confidence intervals for individual tasks.
We maintain uniformity in the architecture of the policy and critic networks for all methods, incorporating layer normalization [] before the first hidden layer.
Hyperparameters are kept constant and only adjusted appropriately for the higher dimensional dog tasks.
Detailed hyperparameter information for all methods can be found in Appendix B.

### 4.1 Gymnasium

As illustrated in Figure 2, Vlearn performs well on lower dimensional tasks, achieving asymptotic performance comparable to SAC, except for HalfCheetah-v4.
While Vlearn outperforms on-policy methods on the HalfCheetah-v4, it exhibits a slower convergence rate compared to SAC and reaches a lower local optimum, similar to the on-policy methods.
Generally, HalfCheetah-v4 seems to be an outlier that challenges the learning capabilities of all trust-region methods.
Even MPO, employing a related KL regularization concept, does not match the performance of SAC.
Moreover, the environment itself has shown extreme behavior in the past [].
However, for the other lower dimensional tasks, Vlearn and MPO perform comparably to the remaining baseline methods.
In the two higher-dimensional tasks, Ant-v4 and Humanoid-v4, Vlearn outperforms all other baselines.
Across both environments, Vlearn demonstrates superior convergence speed and asymptotic performance.
While the improvement is more subtle for Ant-v4 with its $8$-dimensional action space, Vlearn achieves a 25% increase over SAC in the larger $17$-dimensional action space of Humanoid-v4.
Here, focusing solely on learning a state-value function proves to be a less complex and more effective approach than attempting to learn the full state-action value function.
MPO is unable to achieve competitive performance for either Ant-v4 nor Humanoid-v4.

In direct comparisons between Vlearn and V-trace, Vlearn consistently demonstrates superior performance across all tasks.
In particular, V-trace demonstrates minimal learning within a fixed number of environment interactions.
Our investigations revealed that while V-trace is eventually able to learn, it experiences a significant drop in performance after an extended training period.
This discrepancy can be attributed to the sole difference between the two experiments, namely the V-trace objective.
The key distinction between our objective and V-trace lies in how they handle situations where the importance ratio approaches zero and the position of the expectations.
In our objective, samples get assigned a weight close to zero, minimizing their influence on the gradient.
Conversely, V-trace attempts to bring these samples closer to the target network, potentially leading to performance degradation.
Additionally, estimating the joint state-action expectation is typically more stable.
This observation aligns with findings in [, , ], which shows that importance sampling exclusively for the Bellman targets can lead to inferior performance.
While they evaluated this only for the linear case, we found similar results for general nonlinear function approximation as shown in Figure 2.

### 4.2 DeepMind Control Suite

Figure: Figure 3: Performance on the $38$-dimensional DMC dog tasks. Shown are the mean over $10$ seeds and 95% bootstrapped confidence intervals. While SAC and MPO struggle to learn a consistent policy, Vlearn excels across all three tasks. On-policy methods show modest improvements and V-trace even struggles to make any meaningful progress.
Refer to caption: x9.png

The DMC [] dog tasks pose the most challenging problems in our evaluation, with a $38$-dimensional action space modeling a realistic pharaoh dog.
Consistent with the Gymnasium setting, Vlearn exhibits superior performance on these high-dimensional dog locomotion tasks.
Although we found that SAC benefits from layer normalization for these tasks, it struggles to learn well-performing policies for all three movement types.
While SAC improves for “easier" motions, its convergence remains highly unstable, and its final performance often falls below that of on-policy methods.
MPO has similar problems as with the higher dimensional Gymnasium tasks and is unable to solve the tasks.
In contrast, Vlearn learns reliably for all three distinct dog movement types (see Figure 3).

Comparing our V-function learning approach to the V-trace estimator, the disparities are even more pronounced for the DMC dog tasks.
For these high-dimensional problems, the V-trace estimator fails to learn and is among the worst baselines, even falling behind on-policy methods.
Generally, we observe a performance decline across the board, even in the case of lower-dimensional problems.

### 4.3 Ablation Studies

Figure: Figure 4: Top. Ablation study of the replay buffer size has on the policy’s performance. For smaller replay buffers learning becomes unstable or does not converge, while larger sizes tend to lead to similar final performances. Bottom. Ablation study for various variations of our proposed method.
Refer to caption: x13.png

In our ablation study, we investigate the effect of replay buffer size as well as the individual components of our method on the learning process.
We trained multiple agents on the Gymnasium Humanoid-v4 with varying replay buffer sizes of
$\{2e3,1e4,5e4,1e5,2.5e5,5e5,1e6\}$
, for $10$ seeds each.
Note that while the original V-trace [] relies on a relatively large replay buffer, the massively parallel computation used there implies that policy distribution differences are not as pronounced as in standard off-policy methods like SAC.
As shown in Figure 4 (top), significant improvement occurs when transitioning from updating the policy and critical setting near an on-policy setting, using the smallest replay buffer size, to using a medium buffer.
In particular, moving from a small buffer to a medium buffer yields significant benefits, while there is little difference between buffer sizes at the upper end of the range.
In our experiments, we found that a replay buffer size of $5e5$ consistently produced optimal performance across all tasks.

For the second ablation study, we trained several variations of Vlearn for Humanoid-v4 to investigate the effects of each component (Figure 4, bottom).
As a naive baseline, we removed importance sampling (No IS) and assumed that all samples in the replay buffer were from the current policy.
Without the importance sampling correction, the agent was unable to learn.
Replacing the TRPL policy loss with the clipped PPO loss (PPO loss), our results suggest that the heuristic trust region provided by PPO is insufficient for the off-policy case, where stabilizing learning is even more important.
While PPO can achieve reasonable performance on the task, its asymptotic performance lags behind that of the agent using TRPL.
For importance weight truncation, we followed previous work [, ] by choosing $\epsilon_{\rho}=1$ to reduce variance and potentially avoid exploding gradients.
Similar to the PPO loss, we saw that learning becomes unstable for a larger truncation level ($\epsilon_{\rho}=20$).
Finally, we trained an agent without the twin critic networks (No Twin), which performs significantly worse.
We assume that the twin network can be seen as a small ensemble that provides a form of regularization.

## 5 Conclusion and Limitations

In this work, we have demonstrated an efficient approach for learning V-functions from off-policy data using an upper bound objective to subsequently update the policy network.
Our approach not only ensures computational efficiency, but also demonstrates improved stability and performance compared to existing methods.
Integrating this idea with TRPL yields an effective off-policy method that is particularly well-suited for scenarios involving high-dimensional action spaces.

Although our method excels in handling high-dimensional action spaces, it may still require a substantial amount of data to achieve optimal performance.
Hence, sample efficiency still remains a challenge that needs to be addressed further.
Additionally, in specific environments, such as HalfCheetah-v4, Vlearn did not achieve a competitive performance compared to existing approaches.
Improving the method’s reliability, particularly in lower-dimensional scenarios, represents an ongoing challenge.
For future work, we are looking to combine this work with other advances in RL, such as distributional critics [] or ensembles [].
Furthermore, we are looking to extend our approach to the realm of offline RL, which offers the opportunity to leverage pre-collected data efficiently, opening doors to real-world applications and minimizing the need for extensive data collection.

## Appendix A Derivation

We now show the proof of our new objective function.
We start with a standard loss function for learning a V-value function for policy $\pi$ as defined in Equation 2.
Then we derive its upper bound shown in Equation 5.
For simplicity, we denote $\bar{V}=r(s,a)+\gamma V_{\bar{\theta}}(s^{\prime})$.

### A.1 Upper Bound Objective

$$ $\displaystyle\mathbb{E}_{s\sim d^{\pi}(s)}\Bigg{[}\left(V_{\theta}^{\pi}(s)- \mathbb{E}_{a\sim\pi(\cdot|s)}\bar{V}\right)^{2}\Bigg{]}$ $\displaystyle=$ $\displaystyle\mathbb{E}_{s\sim d^{\pi}(s)}\Bigg{[}\left(V_{\theta}^{\pi}(s)- \mathbb{E}_{a\sim\pi_{b}(\cdot|s)}\left[\frac{\pi(a|s)}{\pi_{b}(a|s)}\bar{V} \right]\right)^{2}\Bigg{]}$ $\displaystyle=$ $\displaystyle\mathbb{E}_{s\sim d^{\pi}(s)}\Bigg{[}\Bigg{.}\Bigg{(}\mathbb{E}_{ a\sim\pi_{b}(\cdot|s)}\left[\frac{\pi(a|s)}{\pi_{b}(a|s)}V_{\theta}^{\pi}(s) \right]-\mathbb{E}_{a\sim\pi_{b}(\cdot|s)}\left[\frac{\pi(a|s)}{\pi_{b}(a|s)} \bar{V}\right]\Bigg{)}^{2}\Bigg{.}\Bigg{]}$ $\displaystyle(V_{\theta}^{\pi}\,\text{ does not depend on a})$ $\displaystyle=$ $\displaystyle\mathbb{E}_{s\sim d^{\pi}(s)}\Bigg{[}\left(\mathbb{E}_{a\sim\pi_{ b}(\cdot|s)}\left[\frac{\pi(a|s)}{\pi_{b}(a|s)}\left(V_{\theta}^{\pi}(s)-\bar{ V}\right)\right]\right)^{2}\Bigg{]}$ $\displaystyle=$ $\displaystyle\mathbb{E}_{s\sim d^{\pi}(s)}\left[\left(\int{\underbrace{\pi_{b} (a|s)\frac{\pi(a|s)}{\pi_{b}(a|s)}}_{\text{{\color[rgb]{0,0,0}\definecolor[ named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0} \pgfsys@color@gray@fill{0} weight terms t }}}\underbrace{\left(V_{\theta}^{\pi }(s)-\bar{V}\right)}_{x}}da\right)^{2}\right]$ $\displaystyle=$ $\displaystyle\mathbb{E}_{s\sim d^{\pi}(s)}\Bigg{[}f\left(\int txda\right)\Bigg {]},$ $\displaystyle\leq$ $\displaystyle\mathbb{E}_{s\sim d^{\pi}(s)}\Bigg{[}\int tf\left(x\right)da\Bigg {]},$ $\displaystyle=$ $\displaystyle\mathbb{E}_{s\sim d^{\pi}(s)}\Bigg{[}\mathbb{E}_{a\sim\pi_{b}( \cdot|s)}\left[\frac{\pi(a|s)}{\pi_{b}(a|s)}\left(V_{\theta}^{\pi}(s)-\bar{V} \right)^{2}\right]\Bigg{]}$ $\displaystyle(\text{Jensen's inequality})$ $\displaystyle=$ $\displaystyle\mathbb{E}_{s\sim d^{\pi}(s),a\sim\pi_{b}(\cdot|s)}\Bigg{[}\frac{ \pi(a|s)}{\pi_{b}(a|s)}\left(V_{\theta}^{\pi}(s)-\bar{V}\right)^{2}\Bigg{]}.$ $$

Here, we denote the convex function $f(x)=x^{2}$, the weight terms are in $[0,1]$ and normalized, $\int tda=\int\pi_{b}(a|s)\frac{\pi(a|s)}{\pi_{b}(a|s)}da=\int\pi(a|s)da=1$
and $d^{\pi}(s)$ is the stationary distribution induced by policy $\pi$.
Therefore, one can estimate the loss using Monte Carlo samples from the joint state-action distribution

$$ $\sum_{t}\sum_{j}\frac{\pi(a_{t,j}|s_{t})}{\pi_{b}(a_{t,j}|s_{t})}\Big{(}V_{ \theta}^{\pi}(s_{t})-\big{(}r_{t,j}+\gamma V_{\bar{\theta}}(s_{t+1,j})\big{)} \Big{)}^{2}.$ $$

### A.2 Consistency of Upper Bound Objective

We follow a similar derivation as in [] (for the case of state-action value function $Q(s,a)$) to prove the consistency between $L^{*}(\theta)$ and $L(\theta)$, i. e. the solution for minimizing $L(\theta)$ is the same for the original objective $L^{*}(\theta)$.

$$ $\displaystyle L^{*}(\theta)$ $\displaystyle=\mathbb{E}_{s\sim d^{\pi}(s)}\left[\left(V_{\theta}^{\pi}(s)- \mathbb{E}_{a\sim\pi(\cdot|s)}[\bar{V}]\right)^{2}\right]$ $\displaystyle=\mathbb{E}_{s\sim d^{\pi}(s)}[V_{\theta}^{\pi}(s)^{2}-2V_{\theta }^{\pi}(s)\mathbb{E}_{a\sim\pi(\cdot|s)}[\bar{V}]+\mathbb{E}_{a\sim\pi(\cdot|s )}[\bar{V}]^{2}]$ $\displaystyle L(\theta)$ $\displaystyle=\mathbb{E}_{s\sim d^{\pi}(s)}\left[\mathbb{E}_{a\sim\pi_{b}( \cdot|s)}\left[\rho\left(V_{\theta}^{\pi}(s)-\bar{V}\right)^{2}\right]\right]$ $\displaystyle=\mathbb{E}_{s\sim d^{\pi}(s)}[\mathbb{E}_{a\sim\pi_{b}(\cdot|s)} [\rho(V_{\theta}^{\pi}(s)^{2}-2V_{\theta}^{\pi}(s)\bar{V}+\bar{V}^{2})]]$ $\displaystyle=\mathbb{E}_{s\sim d^{\pi}(s)}[V_{\theta}^{\pi}(s)^{2}-2V_{\theta }^{\pi}(s)\mathbb{E}_{a\sim\pi_{b}(\cdot|s)}\left[\rho\bar{V}\right]+\mathbb{E }_{a\sim\pi_{b}(\cdot|s)}\left[\rho\bar{V}^{2}\right]]$ $\displaystyle(V_{\theta}^{\pi}\,\text{ does not depend on a})$ $\displaystyle=\mathbb{E}_{s\sim d^{\pi}(s)}[V_{\theta}^{\pi}(s)^{2}-2V_{\theta }^{\pi}(s)\mathbb{E}_{a\sim\pi(\cdot|s)}\left[\bar{V}\right]+\mathbb{E}_{a\sim \pi(\cdot|s)}\left[\bar{V}^{2}\right]]$ $$

where $\rho={\pi(a|s)}/{\pi_{b}(a|s)}$.
The above results show that $L^{*}(\theta)$ and $L(\theta)$ are identical except for a constant term added, which remains independent of $V_{\theta}^{\pi}$.

### A.3 Analyzing the Variance of the different Importance Sampling Estimators

As shown in Section 3.2, for the bandit case the V-trace and Vlearn estimators are given by

$$ $\hat{V}_{\text{V-trace}}=\dfrac{\sum_{a\sim\pi_{b}}\rho_{a}^{2}r(a)}{\sum_{a \sim\pi_{b}}\rho_{a}^{2}}\quad\text{and}\quad\hat{V}_{\text{Vlearn}}=\dfrac{ \sum_{a\sim\pi_{b}}\rho_{a}r(a)}{\sum_{a\sim\pi_{b}}\rho_{a}}$ $$

with the crucial difference that the importance weights are squared for V-trace.
We can now consider the effective sample sizes for both, the standard self-normalized importance sampling estimator, ${\rho_{i}x_{i}}/{\sum_{j}\rho_{j}}$ and the squared version ${\rho_{i}^{2}x_{i}}/{\sum_{j}\rho_{j}^{2}}$.
They are given by

$$ $n_{\text{eff}}^{\rho}=\dfrac{\left(\sum_{i=1}^{n}\rho_{i}\right)^{2}}{\sum_{i= 1}^{n}\rho_{i}^{2}}\quad\text{and}\quad n_{\text{eff}}^{\rho^{2}}=\dfrac{\left (\sum_{i=1}^{n}\rho_{i}^{2}\right)^{2}}{\sum_{i=1}^{n}\rho_{i}^{4}},$ $$

respectively.
Given that $\rho_{i}\geq 0$ with $\sum_{i=1}^{n}\rho_{i}$ being strictly positive and $n>1$, it holds that $n_{\text{eff}}^{\rho}\geq n_{\text{eff}}^{\rho^{2}}$ (cf. [] Chapter 9.3).
Consequently, the variance in the squared case is always larger or equal to that of the standard estimator.

## Appendix B Hyperparameters

We conducted a random grid search to tune all hyperparameters for both Gymnasium(^1^11Published under MIT license at [https://github.com/Farama-Foundation/Gymnasium](https://github.com/Farama-Foundation/Gymnasium)) [] and DMC (^2^22Published under Apache-2.0 license at [https://github.com/google-deepmind/dm_control](https://github.com/google-deepmind/dm_control)) [], which are based on Mujoco(^3^33Published under Apache-2.0 license at [https://github.com/google-deepmind/mujoco](https://github.com/google-deepmind/mujoco)) [].
We included learning rates within the range $[10^{-5},5\times 10^{-5},10^{-4},3\times 10^{-4},5\times 10^{-4},10^{-3}]$, mean and covariance bounds within $[10^{-5},10^{-4},5\times 10^{-4},10^{-3},5\times 10^{-3},10^{-2},5\times 10^{-
2},10^{-1},5\times 10^{-1}]$, batch sizes of $[16,32,64,128,256,512]$, policy update intervals of $[1,2]$, and network sizes consisting of two hidden layers with $[64,128,256,512]$ neurons each.
All models were trained on an internal cluster on one Nvidia $V100$ for approximately 1-3 days, depending on the task.
The best hyperparameter sets, as shown in Tables 1 and 2, were then trained for multiple seeds.

We found it is important of achieving a harmonious balance between the selected learning rate and the mean/covariance bound of the trust region.
We observed that an aggressive learning rate does not pair well with a tight bound, and vice versa.
Interestingly, when compared to other off-policy methods, we found that Vlearn and off-policy V-trace performed better with smaller batch sizes, and we did not observe any benefits from using warm start samples.
In general, we noticed that the hyperparameters were a blend of configurations established from existing off-policy and on-policy methods.
We attribute this to the structural similarities between our approach and both on-policy methods, such as PPO [] and TRPL [], as well as off-policy methods, such as SAC [] and MPO (^4^44Published under Apache-2.0 license at [https://github.com/google-deepmind/acme/](https://github.com/google-deepmind/acme/)) [].

Table: Table 1: Hyperparameters for the Gymnasium [] experiments in Figure 2. The larger sample size for the the two on-policy methods is for the Humanoid-v4 experiments.

Table: Table 2: Hyperparameters for the dog tasks from DMC [] experiments in Figure 3.

## Appendix C Broader Impacts

The research presented in this paper is mainly centered around fundamental research for general RL.
However, it might still carry potential implications for society, encompassing both positive and negative impacts on a more general level.

Positive impacts include enhancing systems and processes in sectors like transportation, healthcare, robotics, and finance through optimized decision-making, leading to benefits like safer transport and more efficient healthcare.
Negative impacts involve possible job displacement due to automation and environmental concerns from high computational demands. Additionally, RL algorithms could be misused for cyber-attacks or autonomous weapons.
To mitigate these issues, we advocate for responsible usage of RL algorithms, recommend research on improving their efficiency and accessibility, and encourage minimizing environmental impacts to ensure widespread benefits.

## Peer Reviews

confidence: 3
contribution: 2
initial_rating: 8
presentation: 4
questions: - Can the authors comment on the first weakness?
- A lot of work seems to be necessary to ensure stable training of Vlearn. How is it expected to perform in noisy settings, or under non-stationary rewards? Would its performance be impact similarly to that of baselines, or more?
- In Appendix C, \citet commands should be replaced with \citep.
soundness: 3
strengths: - The paper is clearly written, and the contribution is well outlined.
- Figure 1 is very helpful in describing the different behavior of Vlearn and Vtrace.
- The experimental evaluation is rigorous, involving multiple seeds and controlling for implementation differences and hyperparameter selection.
- Authors are transparent regarding limitations of the method on standard benchmarks (see Figure 4).
summary: This work proposes a fully off-policy reinforcement learning algorithm based on policy gradient and importance sampling. Crucially, the algorithm does not attempt to estimate a state-action value function, but only a value function. As a consequence, the algorithm is well-suited for large action spaces. The authors build upon existing analysis in the linear case suggesting that the entire Bellman error should be reweighted, instead of the Bellman target alone. A simple analysis in bandit settings is provided, confirming that the chosen objective has less variance. The authors follow this idea in the deep reinforcement learning settings. The algorithm is evaluated in high-dimensional continuous control environments, showing that it is essential to combine the objective with a stable algorithm such as TRPL. The algorithm outperforms all baselines in large action spaces, and remains competitive in standard environments with smaller action spaces.
weaknesses: - This work opens an important question: why is a PPO-style loss insufficient for Vlearn, which thus requires TRPL? Figure 3 shows that changing the objective impacts performance in a drastic way. The answer provided ("the heuristic trust region provided by PPO is insufficient for the off-policy case", as well as lines 290-295) is not entirely satisfactory. A more detailed empirical or formal analysis would be very interesting.
- The contribution of this paper is largely in bringing ideas from the linear setting to deep RL. While this implies that the objective and the guarantees are not exactly novel, verifying that formal ideas can work in practice remains very valuable. In particular, the application of WIS in deep RL is, to the best of my knowledge, novel.

---

confidence: 4
contribution: 3
initial_rating: 8
presentation: 3
questions: I am curious about the distinction between “high-dimensional action space” and “low-dimensional action space.” In Section 4.3, experiments indicate that Vlearn performs less effectively than SAC. Could you elaborate on why Vlearn may not perform as well in low-dimensional action spaces?
soundness: 3
strengths: - Vlearn demonstrated promising performance compared to existing reinforcement learning algorithms in high-dimensional action spaces.
- Extensive experiments and ablation studies confirmed the effectiveness of Vlearn.
- Theoretical analysis provided insight into the motivation and design choices behind Vlearn.
summary: This paper introduces Vlearn, a novel off-policy policy gradient method for reinforcement learning. Extensive experimental results demonstrate that Vlearn achieves promising performance compared to existing approaches.
weaknesses: Please refer to the questions part.

---

confidence: 4
contribution: 2
initial_rating: 3
presentation: 2
questions: I am having trouble wrapping my head around the fact that since each transition sample in the replay buffer has different associated log probabilities, learning will not be unstable. Typically, in importance sampling, the target and reference distributions are fixed. In this case, the target distribution is changing (since the policy is learning) and the reference is also changing since different transition samples in the buffer have different log probabilities. Do the authors have a sense of why this is not affecting learning? I would have expected instability. It would be interesting if there was an experiment to test this out.
soundness: 3
strengths: - The technique is simple and relatively straightforward to adopt based on the presented pseudo-code.
- It is theoretically justified. Their proposed loss function upper-bounds the loss we actually want to minimize, and it shares the same solution as the loss we care about.
- Experiments are run over multiple seeds and results present confidence intervals. 
- The presented ablations show the importance of different components of the paper’s algorithm.
- Figure 1 nicely illustrates some intuition about the method.
summary: The paper proposes to use state-only value functions to model the critic during off-policy learning instead of using a state-action critic. It introduces a method to use importance sampling to correct for the distribution mismatch between data in the replay buffer and the policy being learned. The method has a theoretical justification and is applied to various high-dimensional domains.
weaknesses: - The exact motivation of the paper is unclear to me. There are different thoughts scattered throughout, but I think these need to be consolidated to better understand what is actually being solved here. For example there are several seemingly related and unrelated comments that I think need to be consolidated: a) why are high-dimensional actions the actual issue? Our methods already handle high-dimensional states. If high-dimensional actions are an issue, it should be made clearer why. b) “Q-functions dependency on actions prevents it from updating actions that were not used while acting” (line 40); I agree this is an issue, but its only mentioned in passing. c) or is the main goal to scale off-policy algorithms?; if so there, are many other ideas and algorithms that need to be considered. d) or is the idea to just make an algorithmic improvement to V-trace? (line 173).
- Related to the above point. Since the motivation is unclear, I am not sure if the proposed baselines in the experiments are fair. The comparison to V-trace makes sense, but all the other algorithms are base/original implementations without any other techniques, so I would expect them to perform poorly anyway. For example, if the goal is to stabilize off-policy learning then there are plenty of algorithms that can be evaluated such as using layernorm [1] or batch norm [2], or removing the target network [1, 2] etc.
- The abstract makes a claim that the method improves upon exploration and exploitation. I dont think this is necessarily accurate. The experiments do not explicitly test for say exploration. It would be safer to say that it yields high-return agents.
- Lines 119 to 121 need a citation.
- The paper needs to fix its use of “target”. The RL literature typically uses it to refer to the target network and the target that the current estimate is trying to regress to. However, the paper uses target to refer to the current estimate as well (line 177), which makes parsing the paper challenging.
- Line 163 onwards discusses essentially the double sampling problem. A citation is needed here (see https://koppelresearch.wordpress.com/2018/09/13/double-sampling-a-quirk-of-reinforcement-learning/) and the term double sampling should be used. Also the way double sampling is presented here makes it seem that its an issue with equation (2) only and not equation (1), but its an issue for (1) as well.
- In my opinion, if the goal is for improved off-policy learning, the proposed algorithm introduces many more complicated pieces such as twin value function networks that are more complicated than methods that improve off-policy learning with simpler techniques (such as removing the target network and simply using layer norm [1]).

[1] SIMPLIFYING DEEP TEMPORAL DIFFERENCE LEARNING. Gallici et al. 2025.

[2] CrossQ: Batch Normalization in Deep Reinforcement Learning for Greater Sample Efficiency and Simplicity. Bhatt et al. 2024.

---

confidence: 4
contribution: 2
initial_rating: 5
presentation: 3
questions: - I am surprised by the effectiveness of IS in a setting with high dimensional actions, where the ratios may have particularly high variance. Clipping can help avoid overly large values, but what about overly small values? Do the ratios remain close to one especially as the replay buffer grows and the behavior and current policies become more distinct?

- DMC also has a humanoid environment with a high dimensional action space. Does Vlearn also achieve a high performance in that task?
soundness: 3
strengths: - Vlearn provides a modern version of an old idea (weighted IS), including techniques from recent deep RL algorithms and showing competitive performance with deep RL algorithms. 

- This is a (mostly) new flavor of algorithm in the context of deep RL and may open up the design space for further algorithms.
summary: The authors introduce Vlearn, an algorithm based on weighted importance sampling with state-value functions V, rather than state-action value functions Q. The authors argue this approach eliminates the complexity of learning action-values, particularly in domains with high dimensional action spaces and demonstrate performance benefits over SAC in MuJoCo environments.
weaknesses: - Conceptual novelty is not high since the algorithm is based on well-known ideas in the literature. I think the “modern” version of these ideas has value, however it’s unclear how much value that is exactly. 

- While the results are strong against SAC, SAC is definitely not state of the art in any of the tested domains. For example, TQC or TD7 for gym and BRO or iQRL for DMC are all model-free RL algorithms with much stronger performance than SAC. Consequently, it’s hard to get excited about these results since they underperform more modern algorithms, and I worry that the results may have overly benefited from environment specific tuning (dog) and may not generalize to more complex algorithms. For me to be convinced, I would like to see stronger results or covering more tasks/domains, to convince me that this algorithm is generally useful and not over-tuned.

Meta-review: The paper proposes a new technique for offline RL that learns a state value function instead of state-action Q-function.  Combined with importance sampling, the new technique is a policy gradient technique that can leverage off-policy data.  The strength of this work is the simplicity of the proposed technique with a clear description, a theoretical analysis and good empirical evidence.  The weaknesses of this work include limited novelty since the proposed technique is an adaptation of prior work for linear settings to deep learning and comparison to baselines that are not the state of the art.  Nevertheless, the proposed technique will be of interest to the community and represents a valuable contribution.

Decision: Accept (Poster)