Finding Probably Approximate Optimal Solutions by Training to Estimate the Optimal Values of Subproblems
Nimrod Megiddo, Segev Wasserkrug, Orit Davidovich, Shimrit Shtern
TL;DR
The paper addresses maximizing real-valued functions of binary variables by learning to estimate the optimal values of residual subproblems from the distribution of problem instances. It replaces solving instances with a training objective based on deviation from optimality conditions, via a parameterized value function $V(f;\boldsymbol{\theta})$ and a stochastic-gradient procedure that uses smooth approximations to enable differentiation. A central result establishes that the loss $\Phi_n(V,V^*)$ is bounded above by $\Psi_n(V)$, enabling learned $V$ to approximate true subproblem optima through the recursive optimality equations, without requiring solved instances. The framework is demonstrated on Knapsack variants and canonical combinatorial problems—Maximum Weighted Satisfiability, Maximum Weighted Independent Set, and Maximum-Cut—including feasibility-enhanced formulations, suggesting potential scalability with network depth independent of problem size.
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.