Statistical Query Algorithms for Mean Vector Estimation and Stochastic Convex Optimization

• 成果类型: Article
• 署名作者: Feldman, Vitaly; Guzman, Cristobal; Vempala, Santosh
• 署名单位: Apple Inc; Pontificia Universidad Catolica de Chile; Pontificia Universidad Catolica de Chile; University System of Georgia; Georgia Institute of Technology
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2020.1111
• 发表日期: 2021
• 页码: 912-945
• 关键词: lower bounds; oracle complexity; perceptron; noise; MODEL

摘要

Stochastic convex optimization, by which the objective is the expectation of a random convex function, is an important and widely used method with numerous applications in machine learning, statistics, operations research, and other areas. We study the complexity of stochastic convex optimization given only statistical query (SQ) access to the objective function. We show that well-known and popular first-order iterative methods can be implemented using only statistical queries. For many cases of interest, we derive nearly matching upper and lower bounds on the estimation (sample) complexity, including linear optimization in the most general setting. We then present several consequences for machine learning, differential privacy, and proving concrete lower bounds on the power of convex optimization-based methods. The key ingredient of our work is SQ algorithms and lower bounds for estimating the mean vector of a distribution over vectors supported on a convex body in R d . This natural problem has not been previously studied, and we show that our solutions can be used to get substantially improved SQ versions of Perception and other online algorithms for learning halfspaces.