We develop a family of reformulations of an arbitrary consistent linear system into a stochastic problem. The reformulations are governed by two user-defined parameters: a positive definite matrix defining a norm, and an arbitrary discrete or continuous distribution over random matrices. Our reformulation has several equivalent interpretations, allowing for researchers from various communities to leverage their domain specific insights. In particular, our reformulation can be equivalently seen as a stochastic optimization problem, stochastic linear system, stochastic fixed point problem and a probabilistic intersection problem. We prove sufficient, and necessary and sufficient conditions for the reformulation to be exact. 

Further, we propose and analyze three stochastic algorithms for solving the reformulated problem---basic, parallel and accelerated methods---with global linear convergence rates. The rates can be interpreted as condition numbers of a matrix which depends on the system matrix and on the reformulation parameters. This gives rise to a new phenomenon which we call stochastic preconditioning, and which refers to the problem of finding parameters (matrix and distribution) leading to a sufficiently small condition number. Our basic method can be equivalently interpreted as stochastic gradient descent, stochastic Newton method, stochastic proximal point method, stochastic fixed point method, and stochastic projection method, with fixed stepsize (relaxation parameter), applied to the reformulations.

I will also comment on extensions of these ideas to convex feasibility. Time permitting, I may comment on the implications this work has for quasi-Newton methods, and on modified methods utilizing the heavy ball momentum.

The talk is mainly based on the following papers:

[1] Peter Richtárik and Martin Takáč , Stochastic reformulations of linear systems: algorithms and convergence theory, arXiv:1706.01108, 2017

[2] Ion Necoara, Andrei Patrascu and Peter Richtárik . Randomized projection methods for convex feasibility problems: conditioning and convergence rates, arXiv:1801.04873, 2018

Further related papers:

[3] Robert M. Gower and Peter Richtárik. Randomized iterative methods for linear systems, SIAM Journal on Matrix Analysis and Applications 36(4), 1660-1690, 2015

[4] Robert M. Gower and Peter Richtárik. Stochastic dual ascent for solving linear systems, arXiv:1512.06890, 2015

[5] Nicolas Loizou and Peter Richtárik. Momentum and stochastic momentum for stochastic gradient, Newton, proximal point and subspace descent methods, arXiv:1712.09677, 2017

[6] Robert M. Gower and Peter Richtárik. Randomized quasi-Newton updates are linearly convergent matrix inversion algorithms, SIAM Journal on Matrix Analysis and Applications 38(4), 1380-1409, 2017