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An emerging line of work has shown that machine-learned predictions areuseful to warm-start algorithms for discrete optimization problems, such asbipartite matching. Previous studies have shown time complexity boundsproportional to some distance between a prediction and an optimal solution,which we can approximately minimize by learning predictions from past optimalsolutions. However, such guarantees may not be meaningful when multiple optimalsolutions exist. Indeed, the dual problem of bipartite matching and, moregenerally, $\text{L}$-/$\text{L}^\natural$-convex function minimization havearbitrarily many optimal solutions, making such prediction-dependent boundsarbitrarily large. To resolve this theoretically critical issue, we present anew warm-start-with-prediction framework for$\text{L}$-/$\text{L}^\natural$-convex function minimization. Our frameworkoffers time complexity bounds proportional to the distance between a predictionand the set of all optimal solutions. The main technical difficulty lies inlearning predictions that are provably close to sets of all optimal solutions,for which we present an online-gradient-descent-based method. We thus give thefirst polynomial-time learnability of predictions that can provably warm-startalgorithms regardless of multiple optimal solutions.

Rethinking Warm-Starts with Predictions: Learning Predictions Close to  Sets of Optimal Solutions for Faster $\text{L}$-/$\text{L}^\natural$-Convex  Function Minimization