On the Taylor Expansion of Value Functions

• 成果类型: Article
• 署名作者: Braverman, Anton; Gurvich, Itai; Huang, Junfei
• 署名单位: Northwestern University; Chinese University of Hong Kong
• 刊物名称: OPERATIONS RESEARCH
• ISSN/ISSBN: 0030-364X
• DOI: 10.1287/opre.2019.1903
• 发表日期: 2020
• 页码: 631-654
• 关键词: approximations; optimality

摘要

We introduce a framework for approximate dynamic programming that we apply to discrete-time chains on Z(+)(d) with countable action sets. The framework is grounded in the approximation of the (controlled) chain's generator by that of another Markov process. In simple terms, our approach stipulates applying a second-order Taylor expansion to the value function, replacing the Bellman equation with one in continuous space and time in which the transition matrix is reduced to its first and second moments. In some cases, the resulting equation can be interpreted as a Hamilton-Jacobi-Bellman equation for a Brownian control problem. When tractable, the Taylored equation serves as a useful modeling tool. More generally, it is a starting point for approximation algorithms. We develop bounds on the optimality gap-the suboptimality introduced by using the control produced by the Taylored equation. These bounds can be viewed as a conceptual underpinning, analytical rather than relying on weak convergence arguments, for the good performance of controls derived from Brownian approximations. We prove that under suitable conditions and for suitably large initial states, (1) the optimality gap is smaller than a 1-alpha fraction of the optimal value, with which alpha is an element of (0, 1) is the discount factor, and (2) the gap can be further expressed as the infinite-horizon discounted value with a lowerorder per-period reward. Computationally, our framework leads to an aggregation approach with performance guarantees. Although the guarantees are grounded in partial differential equation theory, the practical use of this approach requires no knowledge of that theory.