Bayesian Modeling and Uncertainty Quantification for Learning to Optimize: What, Why, and How

Optimizing an objective function with uncertainty awareness is well-known to improve the accuracy and confidence of optimization solutions. Meanwhile, another relevant but very different question remains yet open: how to model and quantify the uncertainty of an optimization algorithm (a.k.a., optimizer) itself? To close such a gap, the prerequisite is to consider the optimizers as sampled from a distribution, rather than a few prefabricated and fixed update rules. We first take the novel angle to consider the algorithmic space of optimizers, and provide definitions for the optimizer prior and likelihood, that intrinsically determine the posterior and therefore uncertainty. We then leverage the recent advance of learning to optimize (L2O) for the space parameterization, with the end-to-end training pipeline built via variational inference, referred as uncertainty-aware L2O (UA-L2O). Our study represents the first effort to recognize and quantify the uncertainty of the optimization algorithm. The extensive numerical results show that, UA-L2O achieves superior optimization performances in two out of three test functions, the loss function in data privacy attack, and the energy function in docking four out of five protein pairs, with the most accurate posterior estimation. Besides, the optimized solutions close to the real solution are of a larger population and tighter confidence intervals, demonstrating a better calibration capability. All our codes will be publicly released upon acceptance.