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Abstract

This paper concerns asynchrony in iterative processes, focusing on gradient
descent and tatonnement, a fundamental price dynamic.
Gradient descent is an important class of iterative algorithms for minimizing
convex functions. Classically, gradient descent has been a sequential and
synchronous process, although distributed and asynchronous variants have been
studied since the 1980s. Coordinate descent is a commonly studied version of
gradient descent. In this paper, we focus on asynchronous coordinate descent on
convex functions $F:\mathbb{R}^n\rightarrow\mathbb{R}$ of the form $F(x) = f(x)
+ \sum_{k=1}^n \Psi_k(x_k)$, where $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a
smooth convex function, and each $\Psi_k:\mathbb{R}\rightarrow\mathbb{R}$ is a
univariate and possibly non-smooth convex function. Such functions occur in
many data analysis and machine learning problems.
We give new analyses of cyclic coordinate descent, a parallel asynchronous
stochastic coordinate descent, and a rather general worst-case parallel
asynchronous coordinate descent. For all of these, we either obtain sharply
improved bounds, or provide the first analyses. Our analyses all use a common
amortized framework. The application of this framework to the asynchronous
stochastic version requires some new ideas, for it is not obvious how to ensure
a uniform distribution where it is needed in the face of asynchronous actions
that may undo uniformity. We believe that our approach may well be applicable
to the analysis of other iterative asynchronous stochastic processes.
We extend the framework to show that an asynchronous version of tatonnement,
a fundamental price dynamic widely studied in general equilibrium theory,
converges toward a market equilibrium for Fisher markets with CES utilities or
Leontief utilities, for which tatonnement is equivalent to coordinate descent.