Explicit Iteration Complexity of Exact Data-Driven Inverse Optimization for Integer Linear Programs

2026-07-27 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

The paper by Akira Kitaoka presents explicit iteration complexity bounds for exact data-driven inverse optimization problems (DDIOPs) for Integer Linear Programs (ILPs). Previously, the required number of iterations for gradient-based methods to achieve exact consistency was bounded by T=O(1/γ(ℓsub)²) in terms of a problem-dependent geometric constant γ(ℓsub), but this constant lacked an explicit lower bound as a function of problem size. This work derives explicit lower bounds for γ(ℓsub) based on ILP structure, including total unimodularity, Graver bases, and M-convexity/M♮-convexity. These bounds allow the number of iterations for projected subgradient descent to be expressed as a fully explicit function of sample size N, feature dimension d, feature ranges, and constraint matrix structure, up to polynomial factors in basic constants like the diameter of the weight set, step-size, and Lipschitz constant. It also extends to the Prediction Loss of Features (PLF) minimum.

Key takeaway

For Research Scientists developing or applying inverse optimization for ILPs, this work provides crucial quantitative insights into algorithm termination. You can now determine an explicit iteration budget before running gradient-based methods like PSGD, especially when exploiting discrete structures like M-convexity, which guarantees polynomial iteration bounds. For general ILPs, be aware that iteration counts can grow exponentially with dimension d, aligning with NP-hardness.

Key insights

Explicit lower bounds for γ(ℓsub) quantify DDIOP iteration complexity for ILPs.

Principles

• ILP structure (e.g., M-convexity) can yield polynomial iteration bounds.
• General ILPs may exhibit exponential iteration complexity in dimension d.
• γ(ℓsub) lower bounds enable a priori iteration budget determination.

Method

The method involves applying projected subgradient descent to the suboptimality loss ℓsub(θ), then substituting derived explicit lower bounds for the geometric constant γ(ℓsub) into existing iteration upper bounds.

In practice

• Use M-convexity properties for better iteration estimates.
• Quantify computational value of discrete structure.
• Determine iteration budget before running DDIOP algorithms.

Topics

• Inverse Optimization
• Integer Linear Programming
• Subgradient Descent
• Iteration Complexity
• M-convexity
• Graver Basis

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.