7,882 research outputs found
Hybrid Deterministic-Stochastic Methods for Data Fitting
Many structured data-fitting applications require the solution of an
optimization problem involving a sum over a potentially large number of
measurements. Incremental gradient algorithms offer inexpensive iterations by
sampling a subset of the terms in the sum. These methods can make great
progress initially, but often slow as they approach a solution. In contrast,
full-gradient methods achieve steady convergence at the expense of evaluating
the full objective and gradient on each iteration. We explore hybrid methods
that exhibit the benefits of both approaches. Rate-of-convergence analysis
shows that by controlling the sample size in an incremental gradient algorithm,
it is possible to maintain the steady convergence rates of full-gradient
methods. We detail a practical quasi-Newton implementation based on this
approach. Numerical experiments illustrate its potential benefits.Comment: 26 pages. Revised proofs of Theorems 2.6 and 3.1, results unchange
Block stochastic gradient iteration for convex and nonconvex optimization
The stochastic gradient (SG) method can minimize an objective function
composed of a large number of differentiable functions, or solve a stochastic
optimization problem, to a moderate accuracy. The block coordinate
descent/update (BCD) method, on the other hand, handles problems with multiple
blocks of variables by updating them one at a time; when the blocks of
variables are easier to update individually than together, BCD has a lower
per-iteration cost. This paper introduces a method that combines the features
of SG and BCD for problems with many components in the objective and with
multiple (blocks of) variables.
Specifically, a block stochastic gradient (BSG) method is proposed for
solving both convex and nonconvex programs. At each iteration, BSG approximates
the gradient of the differentiable part of the objective by randomly sampling a
small set of data or sampling a few functions from the sum term in the
objective, and then, using those samples, it updates all the blocks of
variables in either a deterministic or a randomly shuffled order. Its
convergence for both convex and nonconvex cases are established in different
senses. In the convex case, the proposed method has the same order of
convergence rate as the SG method. In the nonconvex case, its convergence is
established in terms of the expected violation of a first-order optimality
condition. The proposed method was numerically tested on problems including
stochastic least squares and logistic regression, which are convex, as well as
low-rank tensor recovery and bilinear logistic regression, which are nonconvex
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