Algorithmic Connections Between Active Learning and Stochastic Convex Optimization

TL;DR

This paper establishes new connections between active learning and stochastic convex optimization, introducing algorithms that adapt to unknown parameters and achieve optimal rates using minimal feedback.

Contribution

It presents novel algorithms for active learning and stochastic optimization that leverage their theoretical links, enabling adaptation to unknown parameters and efficient minimax performance.

Findings

Active learning algorithm for 1D thresholds achieves minimax rates.

Coordinate descent with noisy gradient signs matches real gradient convergence.

Combined method adapts to unknown convexity and smoothness parameters.

Abstract

Interesting theoretical associations have been established by recent papers between the fields of active learning and stochastic convex optimization due to the common role of feedback in sequential querying mechanisms. In this paper, we continue this thread in two parts by exploiting these relations for the first time to yield novel algorithms in both fields, further motivating the study of their intersection. First, inspired by a recent optimization algorithm that was adaptive to unknown uniform convexity parameters, we present a new active learning algorithm for one-dimensional thresholds that can yield minimax rates by adapting to unknown noise parameters. Next, we show that one can perform $d$-dimensional stochastic minimization of smooth uniformly convex functions when only granted oracle access to noisy gradient signs along any coordinate instead of real-valued gradients, by using a simple randomized coordinate descent procedure where each line search can be solved by $1$-dimensional active learning, provably achieving the same error convergence rate as having the entire real-valued gradient. Combining these two parts yields an algorithm that solves stochastic convex optimization of uniformly convex and smooth functions using only noisy gradient signs by repeatedly performing active learning, achieves optimal rates and is adaptive to all unknown convexity and smoothness parameters.