Finding Probably Approximate Optimal Solutions by Training to Estimate the Optimal Values of Subproblems

2025-11-03

AI Summary

This paper addresses black-box binary optimization—where the objective function is unknown, unlabeled, and non-differentiable. We propose an unsupervised solution framework that requires neither policy evaluation nor historical solved instances. Our core method constructs an estimator for the optimal subproblem objective value, using the expected deviation from optimality conditions as an unsupervised loss function; this estimator is trained via inequality-constrained function approximation, guiding global search without explicitly solving subproblems. Unlike conventional approaches relying on policy-value estimation or labeled data, our framework leverages in-distribution statistical properties to approximate optimal solutions. Experiments demonstrate its effectiveness and robustness under fully unsupervised settings with unknown objectives.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search
• Machine Learning: Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

The paper is about developing a solver for maximizing a real-valued function of binary variables. The solver relies on an algorithm that estimates the optimal objective-function value of instances from the underlying distribution of objectives and their respective sub-instances. The training of the estimator is based on an inequality that facilitates the use of the expected total deviation from optimality conditions as a loss function rather than the objective-function itself. Thus, it does not calculate values of policies, nor does it rely on solved instances.

Problem

• Develops solver for maximizing real-valued binary functions
• Estimates optimal values of subproblems via distributional training
• Uses deviation from optimality as loss function

Innovation

• Estimates optimal values of subproblems via training
• Uses inequality-based loss for deviation from optimality
• Avoids policy value calculation and solved instances