Stochastic Dual Ascent for Solving Linear Systems

TL;DR

This paper introduces stochastic dual ascent (SDA), a novel randomized iterative algorithm for projecting vectors onto linear system solutions, unifying and improving upon existing methods with broad applicability and fast convergence.

Contribution

The paper develops SDA, a new dual-based randomized algorithm for linear systems, with convergence guarantees and applicability to various existing methods and distributed consensus.

Findings

Primal iterates converge exponentially fast in expectation.

The convergence rate applies to dual and primal function values and the duality gap.

Many existing randomized methods are special cases of SDA, often with improved rates.

Abstract

We develop a new randomized iterative algorithm---stochastic dual ascent (SDA)---for finding the projection of a given vector onto the solution space of a linear system. The method is dual in nature: with the dual being a non-strongly concave quadratic maximization problem without constraints. In each iteration of SDA, a dual variable is updated by a carefully chosen point in a subspace spanned by the columns of a random matrix drawn independently from a fixed distribution. The distribution plays the role of a parameter of the method. Our complexity results hold for a wide family of distributions of random matrices, which opens the possibility to fine-tune the stochasticity of the method to particular applications. We prove that primal iterates associated with the dual process converge to the projection exponentially fast in expectation, and give a formula and an insightful lower bound for the convergence rate. We also prove that the same rate applies to dual function values, primal function values and the duality gap. Unlike traditional iterative methods, SDA converges under no additional assumptions on the system (e.g., rank, diagonal dominance) beyond consistency. In fact, our lower bound improves as the rank of the system matrix drops. Many existing randomized methods for linear systems arise as special cases of SDA, including randomized Kaczmarz, randomized Newton, randomized coordinate descent, Gaussian descent, and their variants. In special cases where our method specializes to a known algorithm, we either recover the best known rates, or improve upon them. Finally, we show that the framework can be applied to the distributed average consensus problem to obtain an array of new algorithms. The randomized gossip algorithm arises as a special case.