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

翻译：新兴研究方向表明，机器学习预测有助于热启动离散优化问题（如二分图匹配）的算法。已有研究给出了与预测和最优解之间某种距离成正比的时间复杂度界，该距离可通过从历史最优解中学习预测来近似最小化。然而，当存在多个最优解时，此类保证可能失去意义。事实上，二分图匹配的对偶问题及更广泛的L-/L^♮-凸函数最小化问题均存在任意多个最优解，使得这类依赖预测的界趋于任意大。为解决这一理论关键问题，我们提出了一种面向L-/L^♮-凸函数最小化的新型带预测热启动框架。该框架给出的时间复杂度界与预测到所有最优解集合的距离成正比。主要技术难点在于学习可证明接近所有最优解集合的预测，对此我们提出一种基于在线梯度下降的方法。由此，我们首次证明了无论存在多少个最优解，能够可证明热启动算法的预测具有多项式时间可学习性。