Stochastic first-order methods: non-asymptotic and computer-aided analyses via potential functions
This work addresses the challenge of systematically analyzing and tuning optimization algorithms for researchers and practitioners in machine learning and optimization, though it is incremental as it builds on existing potential function methods with a computational twist.
The paper tackles the problem of analyzing first-order optimization methods, particularly for sublinear convergence and stochastic oracles, by introducing a computer-assisted technique using semidefinite programming and potential functions, which provides worst-case guarantees and parameter tuning assistance with tightness assurances.
We provide a novel computer-assisted technique for systematically analyzing first-order methods for optimization. In contrast with previous works, the approach is particularly suited for handling sublinear convergence rates and stochastic oracles. The technique relies on semidefinite programming and potential functions. It allows simultaneously obtaining worst-case guarantees on the behavior of those algorithms, and assisting in choosing appropriate parameters for tuning their worst-case performances. The technique also benefits from comfortable tightness guarantees, meaning that unsatisfactory results can be improved only by changing the setting. We use the approach for analyzing deterministic and stochastic first-order methods under different assumptions on the nature of the stochastic noise. Among others, we treat unstructured noise with bounded variance, different noise models arising in over-parametrized expectation minimization problems, and randomized block-coordinate descent schemes.