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Meta Reinforcement Learning with Finite Training Tasks -- a Density Estimation Approach

Zohar Rimon, Aviv Tamar, Gilad Adler

TL;DR

The paper tackles the challenge of achieving near Bayes-optimal performance in meta-reinforcement learning with a finite set of training tasks by first learning the task distribution over MDPs via kernel density estimation and then training a policy that is Bayes-optimal for this estimated distribution. It provides PAC-style generalization bounds that scale linearly with the horizon $T$ and exponentially with the dimension $d$ of the task-parameter space, and shows that when the task distribution lies on a low-dimensional manifold, PCA can reduce the effective dimension to $d'$, yielding substantially tighter bounds. The approach is instantiated in a practical algorithm, VariBAD Dream, where KDE regularization is applied to the VariBAD latent space to generate dream environments, improving in-distribution generalization for moderate training-task budgets. The work demonstrates both theoretical guarantees and empirical improvements, offering a principled, scalable way to regularize meta-RL through learned task distributions and dimensionality reduction.

Abstract

In meta reinforcement learning (meta RL), an agent learns from a set of training tasks how to quickly solve a new task, drawn from the same task distribution. The optimal meta RL policy, a.k.a. the Bayes-optimal behavior, is well defined, and guarantees optimal reward in expectation, taken with respect to the task distribution. The question we explore in this work is how many training tasks are required to guarantee approximately optimal behavior with high probability. Recent work provided the first such PAC analysis for a model-free setting, where a history-dependent policy was learned from the training tasks. In this work, we propose a different approach: directly learn the task distribution, using density estimation techniques, and then train a policy on the learned task distribution. We show that our approach leads to bounds that depend on the dimension of the task distribution. In particular, in settings where the task distribution lies in a low-dimensional manifold, we extend our analysis to use dimensionality reduction techniques and account for such structure, obtaining significantly better bounds than previous work, which strictly depend on the number of states and actions. The key of our approach is the regularization implied by the kernel density estimation method. We further demonstrate that this regularization is useful in practice, when `plugged in' the state-of-the-art VariBAD meta RL algorithm.

Meta Reinforcement Learning with Finite Training Tasks -- a Density Estimation Approach

TL;DR

The paper tackles the challenge of achieving near Bayes-optimal performance in meta-reinforcement learning with a finite set of training tasks by first learning the task distribution over MDPs via kernel density estimation and then training a policy that is Bayes-optimal for this estimated distribution. It provides PAC-style generalization bounds that scale linearly with the horizon and exponentially with the dimension of the task-parameter space, and shows that when the task distribution lies on a low-dimensional manifold, PCA can reduce the effective dimension to , yielding substantially tighter bounds. The approach is instantiated in a practical algorithm, VariBAD Dream, where KDE regularization is applied to the VariBAD latent space to generate dream environments, improving in-distribution generalization for moderate training-task budgets. The work demonstrates both theoretical guarantees and empirical improvements, offering a principled, scalable way to regularize meta-RL through learned task distributions and dimensionality reduction.

Abstract

In meta reinforcement learning (meta RL), an agent learns from a set of training tasks how to quickly solve a new task, drawn from the same task distribution. The optimal meta RL policy, a.k.a. the Bayes-optimal behavior, is well defined, and guarantees optimal reward in expectation, taken with respect to the task distribution. The question we explore in this work is how many training tasks are required to guarantee approximately optimal behavior with high probability. Recent work provided the first such PAC analysis for a model-free setting, where a history-dependent policy was learned from the training tasks. In this work, we propose a different approach: directly learn the task distribution, using density estimation techniques, and then train a policy on the learned task distribution. We show that our approach leads to bounds that depend on the dimension of the task distribution. In particular, in settings where the task distribution lies in a low-dimensional manifold, we extend our analysis to use dimensionality reduction techniques and account for such structure, obtaining significantly better bounds than previous work, which strictly depend on the number of states and actions. The key of our approach is the regularization implied by the kernel density estimation method. We further demonstrate that this regularization is useful in practice, when `plugged in' the state-of-the-art VariBAD meta RL algorithm.
Paper Structure (26 sections, 9 theorems, 54 equations, 6 figures, 1 algorithm)

This paper contains 26 sections, 9 theorems, 54 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Markov Decision Processes
  4. Kernel Density Estimation
  5. Principal Component Analysis
  6. Problem Statement
  7. The Learning Problem
  8. Generalization Bounds
  9. Bounds for Parametric Spaces with Low Dimensional Structure
  10. Related Work
  11. Experiments
  12. Discussion & Future Work
  13. Appendix
  14. Limitations
  15. Regret Bounds Using Prior Estimation
  16. ...and 11 more sections

Key Result

Theorem 1

[Theorem 2 of KDEBounds] Under Assumptions less than inf - Holder, there exists a positive constant $C^{\prime} \equiv C^{\prime}\left(d, \left\lVert f\right\rVert_{\infty}, C_{\alpha}, \alpha, K\right)$ such that the following holds with probability at least $1-1 / n$, uniformly in $h>(\log n / n)^ where $\sigma_{min}$ is the smallest eigenvalue of $\mathbf{H_0}$.

Figures (6)

  • Figure 1: The HalfCircle domain (taken from offlineBRL): the task is to navigate to a goal position that can be anywhere on the half-circle (light blue). A Bayes-optimal agent first searches along the half circle for the goal, and once found, moves directly towards it.
  • Figure 2: Average return on HalfCircle with KDE and mixup dream environments and without dream environments. The average is shown in dashed lines, with the 95% confidence intervals (15 random seeds). We do not show the intervals for the mixup run for visualization clarity; mixup obtained similar intervals as without dream. The full comparison is in Section \ref{['sec:mixupComp']} of the supplementary.
  • Figure 3: Average return on half-circle of the original VariBad and the VariBAD Dream variant with access to the MDP parametric space. The average is shown in dashed lines, with the 95% confidence intervals (using 6 random seeds).
  • Figure 4: Average return on HalfCircle with KDE and mixup dream environments. The average is shown in dashed lines, with the 95% confidence intervals (15 random seeds).
  • Figure 5: Comparison of dream environments' reward maps, generated during the training using 20 real environments. On the top row - our suggested KDE method. On the bottom row - the Mixup method. Each column corresponds to a different training iteration with a 1000 iteration interval between each one. The trajectory of the policy for the sampled dream environment is plotted on top of the reward map as well.
  • ...and 1 more figures

Theorems & Definitions (23)

  • Example 1
  • Theorem 1
  • Theorem 2
  • Example 2
  • Example 3
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Remark 1
  • Theorem 6
  • ...and 13 more