Reinforcement Learning in Feature Space: Matrix Bandit, Kernels, and Regret Bound (1905.10389v2)

Abstract: Exploration in reinforcement learning (RL) suffers from the curse of dimensionality when the state-action space is large. A common practice is to parameterize the high-dimensional value and policy functions using given features. However existing methods either have no theoretical guarantee or suffer a regret that is exponential in the planning horizon $H$. In this paper, we propose an online RL algorithm, namely the MatrixRL, that leverages ideas from linear bandit to learn a low-dimensional representation of the probability transition model while carefully balancing the exploitation-exploration tradeoff. We show that MatrixRL achieves a regret bound ${O}\big(H^2d\log T\sqrt{T}\big)$ where $d$ is the number of features. MatrixRL has an equivalent kernelized version, which is able to work with an arbitrary kernel Hilbert space without using explicit features. In this case, the kernelized MatrixRL satisfies a regret bound ${O}\big(H^2\widetilde{d}\log T\sqrt{T}\big)$, where $\widetilde{d}$ is the effective dimension of the kernel space. To our best knowledge, for RL using features or kernels, our results are the first regret bounds that are near-optimal in time $T$ and dimension $d$ (or $\widetilde{d}$) and polynomial in the planning horizon $H$.

• The paper presents MatrixRL, which learns a low-dimensional representation of MDP transitions using linear bandit techniques to minimize regret.
• The approach achieves regret bounds of O(H²d log T√T) in both feature and kernel spaces, enhancing exploration in high-dimensional state-action environments.
• The study provides key theoretical insights that advance scalable reinforcement learning strategies by establishing near-optimal regret guarantees.

Reinforcement Learning in High Dimensions: Matrix Bandit, Kernels, and Regret Analysis

The paper presents a paper on reinforcement learning (RL) focusing on improving exploration efficiency in high-dimensional state-action spaces. The challenge addressed is the "curse of dimensionality," which traditional RL approaches face when dealing with large state-action spaces. The authors propose MatrixRL, an online RL algorithm that utilizes ideas from linear bandits to provide a theoretically grounded approach to regret minimization in episodic Markov decision processes (MDPs).

Key Contributions

The main contributions of the paper are as follows:

1. MatrixRL Algorithm: This new algorithm systematically learns a low-dimensional representation of the MDP's transition probabilities using a set of given features. It leverages linear bandit techniques to balance exploration and exploitation, thereby allowing the agent to learn the environment efficiently. MatrixRL's regret bound is shown to be $O(H^2d\log T\sqrt{T})$, where $d$ represents the number of features, $H$ is the planning horizon, and $T$ denotes the number of time steps.
2. Kernelized MatrixRL: A kernelized version of MatrixRL is proposed, which extends its applicability to arbitrary reproducing kernel Hilbert spaces without relying on explicit feature mappings. This version achieves a similar regret bound of $O(H^2d\log T\sqrt{T})$, where $d$ represents the effective dimension of the kernel space.
3. Theoretical Insights: The paper provides the first regret bounds for RL using features or kernels that are near-optimal in time ($T$) and dimension ($d$), and polynomial in the planning horizon $H$. This result positions MatrixRL as a substantial theoretical advance in the RL literature.

Analytical Framework

The framework is built on the premise that the transition probabilities within the MDP can be embedded in a suitably chosen feature space. This assumption enables the approximation of the transition dynamics using a core matrix, $M^*$, which the algorithm seeks to learn. The authors employ ridge regression to estimate this core matrix, constructing confidence intervals that inform the action-selection strategy to ensure optimism in the face of uncertainty.

Regret Analysis

MatrixRL's regret is analyzed under the lens of feature-based RL, leveraging regularity conditions for feature embeddings to bound regret. Notably, a distinction is drawn between feature spaces with different regularity conditions, leading to different regret bounds. The extension to kernel spaces involves establishing a notion of effective dimension, which correlates with function approximation capacity and informs the regret analysis.

Practical and Theoretical Implications

The work has profound implications both practically and theoretically:

• Practical: MatrixRL and its kernelized variant can be applied to a broader class of problems where the state-action space is prohibitively large, including robotics and complex decision-making scenarios in partially observable environments.
• Theoretical: The results advance the understanding of RL in high-dimensional spaces, potentially influencing future studies on developing more sophisticated exploration strategies that focus on leveraging structural properties of the underlying MDP.

Future Directions

Future work may address the relaxation of regularity conditions, exploration of alternative feature selection methods, and development of adaptive algorithms that dynamically construct feature spaces. Moreover, empirical validation and application of MatrixRL in real-world settings could further highlight its practical utility.

In conclusion, the paper marks a significant step towards more efficient RL strategies in high-dimensional environments by providing robust theoretical guarantees while maintaining computational feasibility. As the field progresses, these insights into feature and kernel-based exploration could considerably impact the development of scalable RL algorithms.