AltGDmin: Alternating GD and Minimization for Partly-Decoupled (Federated) Optimization
This addresses optimization bottlenecks in federated learning and machine learning tasks like recommender systems and computer vision, but it is incremental as it builds on existing alternating minimization methods.
The paper introduces AltGDmin, an optimization framework for problems where one variable subset is quickly minimized and the other is differentiable, tackling issues like slow convergence in alternating minimization. It demonstrates faster performance than AltMin in applications such as low-rank compressive sensing and matrix completion, though specific numerical gains are not provided.
This article describes a novel optimization solution framework, called alternating gradient descent (GD) and minimization (AltGDmin), that is useful for many problems for which alternating minimization (AltMin) is a popular solution. AltMin is a special case of the block coordinate descent algorithm that is useful for problems in which minimization w.r.t one subset of variables keeping the other fixed is closed form or otherwise reliably solved. Denote the two blocks/subsets of the optimization variables Z by Za, Zb, i.e., Z = {Za, Zb}. AltGDmin is often a faster solution than AltMin for any problem for which (i) the minimization over one set of variables, Zb, is much quicker than that over the other set, Za; and (ii) the cost function is differentiable w.r.t. Za. Often, the reason for one minimization to be quicker is that the problem is ``decoupled" for Zb and each of the decoupled problems is quick to solve. This decoupling is also what makes AltGDmin communication-efficient for federated settings. Important examples where this assumption holds include (a) low rank column-wise compressive sensing (LRCS), low rank matrix completion (LRMC), (b) their outlier-corrupted extensions such as robust PCA, robust LRCS and robust LRMC; (c) phase retrieval and its sparse and low-rank model based extensions; (d) tensor extensions of many of these problems such as tensor LRCS and tensor completion; and (e) many partly discrete problems where GD does not apply -- such as clustering, unlabeled sensing, and mixed linear regression. LRCS finds important applications in multi-task representation learning and few shot learning, federated sketching, and accelerated dynamic MRI. LRMC and robust PCA find important applications in recommender systems, computer vision and video analytics.