🤖 AI Summary
This paper addresses drift optimization for stochastic processes under Lipschitz-continuous controller constraints—i.e., optimizing the drift term subject to path-dependent regulatory constraints to enhance system performance. To tackle this infinite-dimensional, nonconvex, path-constrained optimization problem, we first formulate a novel drift control framework incorporating regulated path constraints. We then propose a sample-average approximation (SAA) method integrating path discretization, function-space discretization, and Monte Carlo sampling, and derive a computationally tractable path-guided directional derivative. A recursive mirror-descent-based optimization algorithm is further designed. Theoretically, we establish consistency guarantees for the SAA estimator and quantify its convergence complexity, explicitly characterizing the computational trade-offs among discretization accuracy, sample size, and iteration count. This work provides a new implementable paradigm for high-dimensional controlled stochastic systems.
📝 Abstract
This paper introduces a drift optimization model of stochastic optimization problems driven by regulated stochastic processes. A broad range of problems across operations research, machine learning, and statistics can be viewed as optimizing the"drift"associated with a process by minimizing a cost functional, while respecting path constraints imposed by a Lipschitz continuous regulator. Towards an implementable solution to such infinite-dimensional problems, we develop the fundamentals of a Sample Average Approximation (SAA) method that incorporates (i) path discretization, (ii) function-space discretization, and (iii) Monte Carlo sampling, and that is solved using an optimization recursion such as mirror descent. We start by constructing pathwise directional derivatives for use within the SAA method, followed by consistency and complexity calculations. The characterized complexity is expressed as a function of the number of optimization steps, and the computational effort involved in (i)--(iii), leading to guidance on how to trade-off the computational effort allocated to optimization steps versus the"dimension reduction"steps in (i)--(iii).