On the convergence of adaptive first order methods: proximal gradient and alternating minimization algorithms
This work addresses incremental improvements in optimization algorithms for researchers and practitioners in machine learning and numerical computing, focusing on adaptive methods for non-smooth and strongly convex problems.
The paper tackles the problem of improving convergence guarantees and stepsize policies for adaptive first-order optimization methods, specifically proximal gradient and alternating minimization algorithms, by proposing a unified framework (adaPG^{q,r}) that provides larger stepsizes and better lower bounds, and demonstrates its efficacy through numerical simulations.
Building upon recent works on linesearch-free adaptive proximal gradient methods, this paper proposes adaPG$^{q,r}$, a framework that unifies and extends existing results by providing larger stepsize policies and improved lower bounds. Different choices of the parameters $q$ and $r$ are discussed and the efficacy of the resulting methods is demonstrated through numerical simulations. In an attempt to better understand the underlying theory, its convergence is established in a more general setting that allows for time-varying parameters. Finally, an adaptive alternating minimization algorithm is presented by exploring the dual setting. This algorithm not only incorporates additional adaptivity, but also expands its applicability beyond standard strongly convex settings.