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Test-Time Regret Minimization in Meta Reinforcement Learning

Mirco Mutti, Aviv Tamar

TL;DR

This work analyzes test-time regret minimization in meta reinforcement learning with a finite set of MDPs under a perfect training assumption. It proves a fundamental lower bound under a separation condition, showing that regret must scale at least as $\Omega\left( \frac{T M \log(H)}{\lambda} \right)$, and that prior approaches achieving $O(M^2 \log(MH))$ are nearly optimal within this regime. To overcome the inherent linear-in-$M$ barrier, the paper introduces strong identifiability and presents three concrete structural regimes—clustering, tree structure, and revealing policies—that enable fast, $\log(H)$-type rates and sublinear dependence on $M$ (up to polylog factors). These results deepen the understanding of when meta-RL can outperform standard RL at test time and provide a blueprint for algorithm design in structured multitask environments. The findings illuminate when structured meta-learning offers practical efficiency gains and outline several directions for extending these insights to broader, possibly infinite task sets.

Abstract

Meta reinforcement learning sets a distribution over a set of tasks on which the agent can train at will, then is asked to learn an optimal policy for any test task efficiently. In this paper, we consider a finite set of tasks modeled through Markov decision processes with various dynamics. We assume to have endured a long training phase, from which the set of tasks is perfectly recovered, and we focus on regret minimization against the optimal policy in the unknown test task. Under a separation condition that states the existence of a state-action pair revealing a task against another, Chen et al. (2022) show that $O(M^2 \log(H))$ regret can be achieved, where $M, H$ are the number of tasks in the set and test episodes, respectively. In our first contribution, we demonstrate that the latter rate is nearly optimal by developing a novel lower bound for test-time regret minimization under separation, showing that a linear dependence with $M$ is unavoidable. Then, we present a family of stronger yet reasonable assumptions beyond separation, which we call strong identifiability, enabling algorithms achieving fast rates $\log (H)$ and sublinear dependence with $M$ simultaneously. Our paper provides a new understanding of the statistical barriers of test-time regret minimization and when fast rates can be achieved.

Test-Time Regret Minimization in Meta Reinforcement Learning

TL;DR

This work analyzes test-time regret minimization in meta reinforcement learning with a finite set of MDPs under a perfect training assumption. It proves a fundamental lower bound under a separation condition, showing that regret must scale at least as , and that prior approaches achieving are nearly optimal within this regime. To overcome the inherent linear-in- barrier, the paper introduces strong identifiability and presents three concrete structural regimes—clustering, tree structure, and revealing policies—that enable fast, -type rates and sublinear dependence on (up to polylog factors). These results deepen the understanding of when meta-RL can outperform standard RL at test time and provide a blueprint for algorithm design in structured multitask environments. The findings illuminate when structured meta-learning offers practical efficiency gains and outline several directions for extending these insights to broader, possibly infinite task sets.

Abstract

Meta reinforcement learning sets a distribution over a set of tasks on which the agent can train at will, then is asked to learn an optimal policy for any test task efficiently. In this paper, we consider a finite set of tasks modeled through Markov decision processes with various dynamics. We assume to have endured a long training phase, from which the set of tasks is perfectly recovered, and we focus on regret minimization against the optimal policy in the unknown test task. Under a separation condition that states the existence of a state-action pair revealing a task against another, Chen et al. (2022) show that regret can be achieved, where are the number of tasks in the set and test episodes, respectively. In our first contribution, we demonstrate that the latter rate is nearly optimal by developing a novel lower bound for test-time regret minimization under separation, showing that a linear dependence with is unavoidable. Then, we present a family of stronger yet reasonable assumptions beyond separation, which we call strong identifiability, enabling algorithms achieving fast rates and sublinear dependence with simultaneously. Our paper provides a new understanding of the statistical barriers of test-time regret minimization and when fast rates can be achieved.
Paper Structure (30 sections, 15 theorems, 70 equations, 2 figures, 1 table, 7 algorithms)

This paper contains 30 sections, 15 theorems, 70 equations, 2 figures, 1 table, 7 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Markov Decision Processes and RL
  4. Meta Reinforcement Learning
  5. Test-Time Regret Minimization
  6. Previous Fast Rates for Test-Time Regret
  7. A Lower Bound for Test-Time Regret Minimization under Separation
  8. Proof Sketch
  9. Strong Identifiability: Beyond Separation for Faster Rates
  10. Meta RL with Clustering
  11. Meta RL with a Tree Structure
  12. Meta RL with a few Revealing Policies
  13. Related Works
  14. Conclusion
  15. Missing Proofs
  16. ...and 15 more sections

Key Result

theorem 3.0

Let $\mathcal{M}$ be a set of MDPs for which Assumption ass:separation_mdp, ass:reachability hold. For any $\mathcal{M}_i \in \mathcal{M}$, we have where $\mathbb{A}$ is Algorithm alg:mdp_separation with inputs $\mathcal{D} = \mathcal{M}$ and $n = \frac{c \log^2 (S M H / \lambda) \log (MH)}{\lambda^4}$ for a sufficiently large constant $c$.

Figures (2)

  • Figure 1: Visualization of the MDP $\mathcal{M}_i$ in the lower bound instance. Note that the role of state $s_i$ and $s_{M + i}, \ldots, s_{\frac{3M}{2} + i}$ change for every MDP in $\mathcal{M}$. Also note that $s_L, s_H$ on the left and right refer to the same pair of states, which are reported twice only to ease inspection. The bottom chart report the specification of the transition probabilities. The values of $\Delta_1, \Delta_2$ are designed to be small enough to make the optimal policy hard to identify playing only slightly sub-optimal policies and large enough to penalize easy identification, respectively.
  • Figure 2: Visualization of the $\mathcal{M}_i$ bandit in the problem instance designed to derive the lower bound. The optimal action $a_i$ and the identifying actions $a \in \mathcal{A}_1 \cup \mathcal{A}_2$ change for every $\mathcal{M}_i$.

Theorems & Definitions (26)

  • definition 1
  • theorem 3.0: chen2021understanding
  • theorem 4.0: Lower bound
  • definition 2: Strong identifiability
  • theorem 5.0
  • theorem 5.0
  • theorem 5.0
  • theorem 1.0: chen2021understanding
  • proof
  • lemma 1.1: chen2021understanding
  • ...and 16 more