Unifying Stochastic Convex Optimization and Active Learning

First order stochastic convex optimization is an extremely well-studied area with a rich history of over a century of optimization research. Active learning is a relatively newer discipline that grew independently of the former, gaining popularity in the learning community over the last few decades due to its promising improvements over passive learning. Over the last year, we have uncovered concrete theoretical and algorithmic connections between these two fields, due to their inherently sequential nature and decision-making based on feedback of earlier choices, that have yielded new methods and proofs techniques in both fields. Here, we summarize the foundations of these connections and summarize our recent advances, with special focus on the implications for stochastic optimization. Specifically, we get an interesting lower bound technique that is quite transparent and are simultaneously tight for derivative-free optimization and first-order optimization for both point and function error. We show that a randomized coordinate descent algorithm with an active learning line search can achieve minimax optimal rates while being adaptive to unknown uniform convexity parameters. This procedure only relies on unidirectional noisy gradient signs as opposed to real valued gradient vectors as a result, rounding errors and other errors that preserve the sign of the gradient lead to deterministic (non-stochastic) rates of convergence.