Refining Minimax Regret for Unsupervised Environment Design (2402.12284v2)

Abstract: In unsupervised environment design, reinforcement learning agents are trained on environment configurations (levels) generated by an adversary that maximises some objective. Regret is a commonly used objective that theoretically results in a minimax regret (MMR) policy with desirable robustness guarantees; in particular, the agent's maximum regret is bounded. However, once the agent reaches this regret bound on all levels, the adversary will only sample levels where regret cannot be further reduced. Although there are possible performance improvements to be made outside of these regret-maximising levels, learning stagnates. In this work, we introduce Bayesian level-perfect MMR (BLP), a refinement of the minimax regret objective that overcomes this limitation. We formally show that solving for this objective results in a subset of MMR policies, and that BLP policies act consistently with a Perfect Bayesian policy over all levels. We further introduce an algorithm, ReMiDi, that results in a BLP policy at convergence. We empirically demonstrate that training on levels from a minimax regret adversary causes learning to prematurely stagnate, but that ReMiDi continues learning.

• The paper introduces Bayesian level-perfect Minimax Regret (BLP) to overcome RL stagnation when standard MMR halts further learning.
• It proposes the ReMiDi algorithm that iteratively refines agent policies via adversarial engagements and environment sampling.
• Empirical results in tabular and Minigrid scenarios demonstrate that ReMiDi outperforms PLR by maintaining performance gains beyond high-regret environments.

Refining Minimax Regret for Unsupervised Environment Design

In the paper presented, the authors explore the challenges and mitigate limitations of the Minimax Regret (MMR) decision rule employed in Unsupervised Environment Design (UED) within reinforcement learning contexts. The crux of their contribution is the introduction of a refined objective—Bayesian level-perfect Minimax Regret (BLP)—to resolve the pitfalls of conventional MMR when faced with environments possessing high irreducible regret.

Key Contributions and Methodology

The authors identify a crucial problem with standard MMR in that it potentially leads to stagnation. Specifically, once a reinforcement learning (RL) agent achieves the worst-case bound on regret across a set of high-regret environments, further learning is halted even if performance can still be optimized outside these environments.

In response, the authors propose BLP as a refinement over MMR. This method strategically constrains the agent's policy space such that, during iterative phases, policies align closely with a Bayesian framework by respecting learnt optimal actions based on previously encountered environments. The theoretical elegance lies in leveraging a Perfect Bayesian policy approach within an MMR context, achieved by iteratively refining across subsets of environments rather than across the full space—a process explicitly formalized as a succession of two-player zero-sum games where the adversary's level selection is counterbalanced by the agent's constrained policy refinement.

Additionally, the authors develop ReMiDi (Refining Minimax Regret Distributions), a proof-of-concept algorithm embodying the BLP framework. ReMiDi iteratively refines agent policy through a sequence of adversarial engagements and environment sampling, thereby ensuring that the RL agent not only achieves but retains minimax regret guarantees while improving on environments distinguishable from previously targeted regret-optimal scenarios.

Empirical Demonstration

The effectiveness of ReMiDi is empirically validated across various setups, including a tabular setting illustrating MMR limitations and Minigrid experiments involving T-mazes and blindfolded scenarios. Intriguingly, these experiments underline that while $\text{PLR}^\perp$, a popular regret-based UED approach, adeptly zeroes in on high-regret environments, it fails to adapt once high-regret levels are mastered. ReMiDi, in contrast, demonstrates robust learning by maintaining performance gains on both high-regret and additional non-maximal environments.

In another tested setting, the lever game, the authors illustrate how ReMiDi surpasses $\text{PLR}^\perp$ in achieving both visible and invisible optimal solutions, leveraging its inherently better adaptability by sidestepping MMR-induced learning stagnation.

Theoretical Implications and Future Research

This work brings to light an insightful discourse on decision rules and their applicability in RL, spotlighting how a theoretically sound construct like MMR can be reformulated through BLP for refined real-world utility in UED. The findings imply that further exploration into more sophisticated game-theoretic refinements and computationally efficient implementations of BLP could yield even broader application landscapes for RL agents, ideal for larger and open-ended domains where irreducible regret is prevalent.

Future work might aim to reconcile computational feasibility with theoretical robustness, particularly in aligning learned belief systems for trajectory realization in more complex or stochastic environments.

Conclusion

In sum, this paper marks a significant step in reinforcing the theoretical depth and practical execution of unsupervised environment design. By introducing and empirically validating BLP, the work propels the field towards developing more universally adaptable, robust RL systems, capable of transcending the intrinsic limits posed by high-regret environments while aligning closer with ideal Bayesian decision-making paradigms.