[Paper Review] Nonparametric learning for impulse control problems

Sören Christensen, Claudia Strauch|arXiv (Cornell University)|Sep 20, 2019

Advanced Bandit Algorithms Research19 references4 citations

TL;DR

This paper proposes a nonparametric learning framework for impulse control problems under partial information, integrating exploration and exploitation through adaptive sampling based on convergence rates of invariant density estimators. It establishes a uniform regret bound of $ O(T^{-1/3}) $, achieving optimal balance between learning the drift and optimizing long-term control in a stochastic harvesting context.

ABSTRACT

One of the fundamental assumptions in stochastic control of continuous time processes is that the dynamics of the underlying (diffusion) process is known. This is, however, usually obviously not fulfilled in practice. On the other hand, over the last decades, a rich theory for nonparametric estimation of the drift (and volatility) for continuous time processes has been developed. The aim of this paper is bringing together techniques from stochastic control with methods from statistics for stochastic processes to find a way to both learn the dynamics of the underlying process and control in a reasonable way at the same time. More precisely, we study a long-term average impulse control problem, a stochastic version of the classical Faustmann timber harvesting problem. One of the problems that immediately arises is an exploration-exploitation dilemma as is well known for problems in machine learning. We propose a way to deal with this issue by combining exploration and exploitation periods in a suitable way. Our main finding is that this construction can be based on the rates of convergence of estimators for the invariant density. Using this, we obtain that the average cumulated regret is of uniform order $O({T^{-1/3}})$.

Motivation & Objective

• To address the challenge of impulse control when the drift of a diffusion process is unknown, a common but unrealistic assumption in classical stochastic control.
• To resolve the exploration-exploitation trade-off inherent in learning the dynamics while simultaneously optimizing control performance.
• To develop a unified framework that combines nonparametric statistics with stochastic control, avoiding restrictive parametric assumptions on the drift.
• To derive finite-time performance guarantees in terms of regret, specifically bounding the cumulative loss due to learning.

Proposed method

• The method combines nonparametric kernel density estimation of the invariant density with a time-splitting strategy that alternates between exploration (learning) and exploitation (control) phases.
• It uses a kernel estimator $ \widehat{\rho}_{t,h}(y) $ for the invariant density, with bandwidth $ h $ chosen based on the convergence rate of the estimator.
• The exploration phase is designed to last $ t $ time units, during which the process is observed to estimate the drift nonparametrically, while the exploitation phase uses the estimated dynamics for control.
• The regret analysis relies on decomposing the estimation error into bias and martingale terms, with the latter controlled via the Burkholder–Davis–Gundy inequality.
• The convergence rate of the density estimator is linked to the Hölder continuity of the true density and the order of the kernel function $ Q $, ensuring bias control.
• The final regret bound is derived by combining bounds on the expected $ L^1 $-error of the density estimator with the time allocation between exploration and exploitation.

Experimental results

Research questions

• RQ1 How can one simultaneously learn the unknown drift of a diffusion process and perform optimal impulse control in a continuous-time setting?
• RQ2 What is the minimal regret achievable when the dynamics are learned nonparametrically in real time, without prior parametric assumptions?
• RQ3 How should exploration and exploitation be balanced in a learning-control framework to minimize cumulative regret?
• RQ4 What role do the convergence rates of nonparametric density estimators play in determining the performance of the control policy?
• RQ5 Can a uniform regret bound of order $ O(T^{-1/3}) $ be achieved in a nonparametric impulse control problem?

Key findings

• The proposed method achieves a uniform regret bound of $ O(T^{-1/3}) $, which is optimal under the given nonparametric estimation constraints.
• The regret bound is derived by balancing the exploration time $ t $ and the resulting estimation error, with the optimal bandwidth $ h \sim t^{-1/3} $.
• The bias term in the density estimator is controlled via the Hölder continuity of the true invariant density and the kernel order, yielding $ \mathcal{O}(h^\beta) $ bias.
• The martingale term in the estimation error is bounded using the Burkholder–Davis–Gundy inequality, leading to a $ \mathcal{O}(\sqrt{h}) $ contribution to the $ L^1 $-error.
• The analysis shows that the $ L^1 $-error of the density estimator is $ \mathcal{O}(1/\sqrt{t}) $, which contributes to the overall regret scaling.
• The framework successfully integrates nonparametric statistics with stochastic control, demonstrating that learning and control can be performed simultaneously with quantifiable performance guarantees.