Exploration-Exploitation in Constrained MDPs (2003.02189v1)

Abstract: In many sequential decision-making problems, the goal is to optimize a utility function while satisfying a set of constraints on different utilities. This learning problem is formalized through Constrained Markov Decision Processes (CMDPs). In this paper, we investigate the exploration-exploitation dilemma in CMDPs. While learning in an unknown CMDP, an agent should trade-off exploration to discover new information about the MDP, and exploitation of the current knowledge to maximize the reward while satisfying the constraints. While the agent will eventually learn a good or optimal policy, we do not want the agent to violate the constraints too often during the learning process. In this work, we analyze two approaches for learning in CMDPs. The first approach leverages the linear formulation of CMDP to perform optimistic planning at each episode. The second approach leverages the dual formulation (or saddle-point formulation) of CMDP to perform incremental, optimistic updates of the primal and dual variables. We show that both achieves sublinear regret w.r.t.\ the main utility while having a sublinear regret on the constraint violations. That being said, we highlight a crucial difference between the two approaches; the linear programming approach results in stronger guarantees than in the dual formulation based approach.

• The paper introduces four algorithms that achieve sublinear regret in CMDPs by balancing exploration with strict safety constraints.
• It adapts UCRL2 techniques and integrates exploration bonuses through extended linear programming for managing state-action occupancy.
• The study highlights a trade-off between robust theoretical guarantees and computational efficiency in safe reinforcement learning.

Exploration-Exploitation in Constrained MDPs

The paper entitled "Exploration-Exploitation in Constrained MDPs" authored by Yonathan Efroni, Shie Mannor, and Matteo Pirotta provides a comprehensive analysis of methods for tackling the exploration-exploitation trade-off in Constrained Markov Decision Processes (CMDPs). Within the scope of sequential decision-making under constraints, the paper introduces and evaluates four algorithms designed to optimize utility while adhering to constraints, contributing to the field of safe reinforcement learning.

CMDP Framework and Challenges

CMDPs extend the traditional MDP framework to integrate constraints on policies over a finite horizon. This integration is crucial for applications requiring guaranteed safety or adherence to specific requirements, such as robotics or autonomous driving. In CMDPs, the complexity arises from the dual objectives: maximizing cumulative rewards and satisfying multiple constraints. The learning process involves discovering optimal policies within state-action spaces that comply with these constraints.

Theoretical Contributions

The paper details two primary approaches: UCRL-like optimism-based methods and Lagrangian-based dual and primal-dual algorithms.

1. Optimistic Model-Based Approaches:
   • CUCRL: This algorithm adapts UCRL2 to CMDPs, employing optimistic planning over plausible CMDPs reconstructed from observed samples. It uses extended linear programming to handle state-action-state occupancy measures, achieving sublinear regret on utility and constraints.
   • CUCBVI: Building on optimistic model principles, CUCBVI incorporates exploration bonuses directly into CMDPs. Despite its computational efficiency due to a reduced LP complexity, its theoretical guarantees bear less favorable constant terms compared to CUCRL.
2. Lagrangian-Based Approaches:
   • OptDual-CMDP: Utilizing a dual projected sub-gradient method, this algorithm iteratively updates Lagrange multipliers based on estimated constraint violations. It achieves sublinear regret but only provides bounds on cumulative regrets, allowing error cancellations.
   • OptPrimalDual-CMDP: Featuring incremental updates within a primal-dual framework, this approach optimizes both primal and dual variables iteratively. Its computational simplicity comes at the cost of theoretical guarantees similar to OptDual-CMDP, with bounds allowing for suboptimal policy convergence over learning stages.

Practical and Theoretical Implications

The UCRL-like algorithms present robustness in theoretical guarantees with defined regret forms suitable for applications needing assured adherence to constraints during learning. However, the practical volume of computation required by these methods may prove challenging for larger state-action spaces.

In contrast, the Lagrangian approaches, driven by their relatively lighter computational requirements, present themselves as attractive options for real-world deployment where computational resources may be limiting. Nonetheless, the weaker theoretical guarantees (e.g., allowing for regret cancellations) highlight an important consideration for deployments where strict adherence to constraints is critical throughout learning.

Future Directions

The paper points to several avenues for future research. Ensuring tighter bounds on Lagrangian-based methods is one such direction, potentially bridging the gap between computational efficiency and theoretical performance guarantees. Additionally, investigating hybrid methods that balance between the proposed approaches may further boost performance while maintaining lower computational overhead.

The analysis and findings derived from this paper are crucial for advancing reinforcement learning techniques capable of operating safely within constrained environments. As AI continues to permeate various sectors, the need for reliable and efficient algorithms capable of managing exploration and exploitation within constraint-rich settings becomes increasingly critical.