Learning to steer with Brownian noise (2410.03221v1)

Abstract: This paper considers an ergodic version of the bounded velocity follower problem, assuming that the decision maker lacks knowledge of the underlying system parameters and must learn them while simultaneously controlling. We propose algorithms based on moving empirical averages and develop a framework for integrating statistical methods with stochastic control theory. Our primary result is a logarithmic expected regret rate. To achieve this, we conduct a rigorous analysis of the ergodic convergence rates of the underlying processes and the risks of the considered estimators.

• The paper derives logarithmic regret rates through rigorous ergodic convergence analysis applied to both explore-first and adaptive algorithms.
• It employs a bang-bang control strategy based on the Hamilton-Jacobi-Bellman equation to effectively balance exploration and exploitation.
• The study demonstrates that model-based reinforcement learning leads to enhanced interpretability and reduced data requirements in continuous-time settings.

Learning to Steer with Brownian Noise

The paper "Learning to Steer with Brownian Noise" examines an ergodic form of the bounded velocity follower problem within stochastic control, specifically considering situations where the decision-maker must learn unknown system parameters while simultaneously maintaining control. This research is positioned at the intersection of stochastic control and model-based reinforcement learning (RL), aiming to devise strategies that provide improved learning efficiency and interpretability.

Problem Definition and Theoretical Framework

The paper focuses on a process driven by standard Brownian motion, where the decision-maker controls the drift component within a bounded interval. The research adopts an ergodic criterion, employing a long-term average cost function that aims to keep the process as close as possible to a target value, typically zero. The optimal control strategy, when system parameters are known, is derived using the Hamilton-Jacobi-BeLLMan (HJB) equation, and is shown to be of a bang-bang type.

Learning Under Uncertainty

When decision-makers lack knowledge of system parameters, direct application of optimal control is infeasible. Therefore, the authors propose two algorithms:

1. Explore-First Algorithm: This algorithm initially explores the parameter space before switching to a control strategy based on the estimated parameters. The paper proves it achieves a regret rate of order $\sqrt{T}$ for a learning interval of $\sqrt{T}$.
2. Adaptive Position Averaging with Clipping (APAC): This algorithm constantly updates the parameter estimate, leveraging adaptive strategies which yield a convergence rate of order $\log(T)$. This improved convergence is attributed to continuous learning and adaptation.

Both methodologies address the fundamental need to balance exploration and exploitation, common within reinforcement learning paradigms.

Analytical and Numerical Results

The paper's primary contribution is the demonstration of logarithmic expected regret rates, achievable through a rigorous analysis of ergodic convergence rates and estimator risks. The results underscore the efficacy of integrating statistical methods with stochastic control theory, offering theoretical guarantees and enhancing RL strategies' interpretability and efficiency. Additionally, numerical comparisons indicate that model-based approaches such as the ones proposed significantly reduce required data compared to model-free methods like deep-Q learning.

Implications and Future Directions

This research showcases the value of model-based reinforcement learning, particularly in scenarios with unknown dynamics. The proposed algorithms provide structured methodologies for developing efficient and interpretable control strategies in continuous-time settings. The implications are broad, especially in areas where drift parameters are challenging to estimate directly, such as finance or autonomous systems.

Future research could explore extending these methods to more complex models, including those with higher-dimensional state spaces or non-linear dynamics. Additionally, the integration of deep learning architectures could potentially enhance the adaptability of the models while maintaining theoretical rigor.

In conclusion, the paper presents significant advancements in the field of stochastic control when dealing with uncertainties, offering robust strategies that bridge the gap between theory and practical application in reinforcement learning contexts.