Tight Bound for Sum of Heterogeneous Random Variables: Application to Chance Constrained Programming

TL;DR

This paper introduces a tight Bennett-type concentration inequality for sums of heterogeneous independent variables, providing computationally efficient confidence bounds and applications to chance-constrained programming, improving over classical bounds.

Contribution

It develops a computationally tractable, tight Bennett-type inequality for heterogeneous variables and applies it to improve solutions in chance-constrained programming problems.

Findings

Outperforms classical inequalities like Chebyshev-Cantelli and Hoeffding in knapsack problems.

Provides a robust SVM hyperplane with improved accuracy.

Develops a polynomial-time algorithm for confidence bounds.

Abstract

We study a tight Bennett-type concentration inequality for sums of heterogeneous and independent variables, defined as a one-dimensional minimization. We show that this refinement, which outperforms the standard known bounds, remains computationally tractable: we develop a polynomial-time algorithm to compute confidence bounds, proved to terminate with an epsilon-solution. From the proposed inequality, we deduce tight distributionally robust bounds to Chance-Constrained Programming problems. To illustrate the efficiency of our approach, we consider two use cases. First, we study the chance-constrained binary knapsack problem and highlight the efficiency of our cutting-plane approach by obtaining stronger solution than classical inequalities (such as Chebyshev-Cantelli or Hoeffding). Second, we deal with the Support Vector Machine problem, where the convex conservative approximation we obtain improves the robustness of the separation hyperplane, while staying computationally tractable.