Settling the Sample Complexity of Online Reinforcement Learning (2307.13586v3)

Abstract: A central issue lying at the heart of online reinforcement learning (RL) is data efficiency. While a number of recent works achieved asymptotically minimal regret in online RL, the optimality of these results is only guaranteed in a ``large-sample'' regime, imposing enormous burn-in cost in order for their algorithms to operate optimally. How to achieve minimax-optimal regret without incurring any burn-in cost has been an open problem in RL theory. We settle this problem for the context of finite-horizon inhomogeneous Markov decision processes. Specifically, we prove that a modified version of Monotonic Value Propagation (MVP), a model-based algorithm proposed by \cite{zhang2020reinforcement}, achieves a regret on the order of (modulo log factors) \begin{equation*} \min\big{ \sqrt{SAH^3K}, \,HK \big}, \end{equation*} where $S$ is the number of states, $A$ is the number of actions, $H$ is the planning horizon, and $K$ is the total number of episodes. This regret matches the minimax lower bound for the entire range of sample size $K\geq 1$, essentially eliminating any burn-in requirement. It also translates to a PAC sample complexity (i.e., the number of episodes needed to yield $\varepsilon$-accuracy) of $\frac{SAH^3}{\varepsilon^2}$ up to log factor, which is minimax-optimal for the full $\varepsilon$-range. Further, we extend our theory to unveil the influences of problem-dependent quantities like the optimal value/cost and certain variances. The key technical innovation lies in the development of a new regret decomposition strategy and a novel analysis paradigm to decouple complicated statistical dependency -- a long-standing challenge facing the analysis of online RL in the sample-hungry regime.

• The paper achieves minimax-optimal regret bounds for online reinforcement learning in finite-horizon MDPs without requiring any burn-in phase, matching the theoretical lower bound min { SAH 3 K , HK } .
• Key technical innovations include a novel "profiles" analysis paradigm to decouple statistical dependencies and an epoch-based doubling trick for efficient model updates.
• The findings provide a comprehensive framework for sample-optimal online RL and open avenues for future research into sample efficiency for various problem structures and RL frameworks.

Settling the Sample Complexity of Online Reinforcement Learning

The paper "Settling the Sample Complexity of Online Reinforcement Learning" provides significant advancements in understanding the sample efficiency of online reinforcement learning (RL) for finite-horizon inhomogeneous Markov decision processes (MDPs). One of the primary challenges in RL is to balance exploration and exploitation efficiently, particularly in online settings where data collection is continual and adaptive. The authors target the long-standing problem of achieving minimax-optimal regret in RL without incurring any burn-in costs, thereby addressing both theoretical and practical concerns in RL.

Core Findings and Contributions

1. Minimax-Optimal Regret in Online RL: The authors propose a modified version of the Monotonic Value Propagation ($\mathtt{MVP}$) algorithm, initially introduced by Zhang et al. in 2020, that is shown to achieve regret of order $\min\big\{ \sqrt{SAH^3K}, HK \big\}$ for the entire range of episode count $K \geq 1$. Here, $S$ is the number of states, $A$ is the number of actions, $H$ is the horizon length, and $K$ is the total number of episodes. This result matches the minimax lower bound, effectively eliminating any burn-in requirements.
2. PAC Sample Complexity: The paper asserts that the sample complexity, in terms of episodes needed for $\varepsilon$-accuracy, is $\frac{SAH^3}{\varepsilon^2}$, which is minimax-optimal across the full range of $\varepsilon$. This reinforces the efficiency of the algorithm in learning accurate models of an environment with fewer samples.
3. Problem-Dependent Insights: The authors extend their analysis by providing regret bounds that depend on certain problem-specific quantities, like optimal value/cost and variance metrics. These bounds allow for better performance approximation in environments where such quantities are known or can be estimated, offering considerable improvements in sample efficiency.

Technical Innovations

The paper's technical contributions include overcoming complex dependency issues between episodic samples and estimation methods. Specifically:

• Decoupling Dependencies: A novel analysis paradigm based on "profiles" is used, allowing for more efficient decoupling of statistical dependencies, which have been a significant barrier in previous analyses.
• Doubling Trick for Model Updates: By employing an epoch-based policy updating mechanism, the authors significantly reduce complexity inherent in earlier algorithms by limiting the frequency of model updates.

Implications and Future Directions

The research presents a comprehensive framework for achieving sample-optimal results in online RL, with potential applications across various fields, including robotics and adaptive systems that rely on continual learning from limited interactions. The insights extend theoretical groundwork and set a platform for further refinement of RL algorithms that could adapt to specific problem structures or alternative RL frameworks, such as model-free approaches.

The findings also open new speculative path for understanding sample complexity and optimal policy design across different RL horizons, including discounted settings or generalized policy spaces. This could stimulate further research in efficient policy learning, specifically tailored to high-dimensional problems or data-sparse environments.

In summary, the paper stands as a rigorous endeavor to resolve sample complexity challenges in RL, helping move both theoretical and applied advancements forward with its minimax-optimal approach.