# A Data Efficient and Feasible Level Set Method for Stochastic Convex Optimization with Expectation Constraints

## TL;DR

This paper introduces a stochastic feasible level set method (SFLS) for solving large-scale stochastic convex optimization problems with expectation constraints, emphasizing early feasibility and low data complexity.

## Contribution

The paper develops a novel SFLS algorithm that maintains feasibility at each step and improves data efficiency over existing methods for stochastic convex optimization with constraints.

## Key findings

- SFLS achieves high-probability feasibility at each iteration.
- SFLS demonstrates lower data complexity than existing methods.
- Numerical results show faster feasible solution attainment with small optimality gaps.

## Abstract

Stochastic convex optimization problems with expectation constraints (SOECs) are encountered in statistics and machine learning, business, and engineering. In data-rich environments, the SOEC objective and constraints contain expectations defined with respect to large datasets. Therefore, efficient algorithms for solving such SOECs need to limit the fraction of data points that they use, which we refer to as algorithmic data complexity. Recent stochastic first order methods exhibit low data complexity when handling SOECs but guarantee near-feasibility and near-optimality only at convergence. These methods may thus return highly infeasible solutions when heuristically terminated, as is often the case, due to theoretical convergence criteria being highly conservative. This issue limits the use of first order methods in several applications where the SOEC constraints encode implementation requirements. We design a stochastic feasible level set method (SFLS) for SOECs that has low data complexity and emphasizes feasibility before convergence. Specifically, our level-set method solves a root-finding problem by calling a novel first order oracle that computes a stochastic upper bound on the level-set function by extending mirror descent and online validation techniques. We establish that SFLS maintains a high-probability feasible solution at each root-finding iteration and exhibits favorable iteration complexity compared to state-of-the-art deterministic feasible level set and stochastic subgradient methods. Numerical experiments on three diverse applications validate the low data complexity of SFLS relative to the former approach and highlight how SFLS finds feasible solutions with small optimality gaps significantly faster than the latter method.