2,200 research outputs found
On the convergence of mirror descent beyond stochastic convex programming
In this paper, we examine the convergence of mirror descent in a class of
stochastic optimization problems that are not necessarily convex (or even
quasi-convex), and which we call variationally coherent. Since the standard
technique of "ergodic averaging" offers no tangible benefits beyond convex
programming, we focus directly on the algorithm's last generated sample (its
"last iterate"), and we show that it converges with probabiility if the
underlying problem is coherent. We further consider a localized version of
variational coherence which ensures local convergence of stochastic mirror
descent (SMD) with high probability. These results contribute to the landscape
of non-convex stochastic optimization by showing that (quasi-)convexity is not
essential for convergence to a global minimum: rather, variational coherence, a
much weaker requirement, suffices. Finally, building on the above, we reveal an
interesting insight regarding the convergence speed of SMD: in problems with
sharp minima (such as generic linear programs or concave minimization
problems), SMD reaches a minimum point in a finite number of steps (a.s.), even
in the presence of persistent gradient noise. This result is to be contrasted
with existing black-box convergence rate estimates that are only asymptotic.Comment: 30 pages, 5 figure
Linear Coupling: An Ultimate Unification of Gradient and Mirror Descent
First-order methods play a central role in large-scale machine learning. Even
though many variations exist, each suited to a particular problem, almost all
such methods fundamentally rely on two types of algorithmic steps: gradient
descent, which yields primal progress, and mirror descent, which yields dual
progress.
We observe that the performances of gradient and mirror descent are
complementary, so that faster algorithms can be designed by LINEARLY COUPLING
the two. We show how to reconstruct Nesterov's accelerated gradient methods
using linear coupling, which gives a cleaner interpretation than Nesterov's
original proofs. We also discuss the power of linear coupling by extending it
to many other settings that Nesterov's methods cannot apply to.Comment: A new section added; polished writin
A Stochastic Interpretation of Stochastic Mirror Descent: Risk-Sensitive Optimality
Stochastic mirror descent (SMD) is a fairly new family of algorithms that has
recently found a wide range of applications in optimization, machine learning,
and control. It can be considered a generalization of the classical stochastic
gradient algorithm (SGD), where instead of updating the weight vector along the
negative direction of the stochastic gradient, the update is performed in a
"mirror domain" defined by the gradient of a (strictly convex) potential
function. This potential function, and the mirror domain it yields, provides
considerable flexibility in the algorithm compared to SGD. While many
properties of SMD have already been obtained in the literature, in this paper
we exhibit a new interpretation of SMD, namely that it is a risk-sensitive
optimal estimator when the unknown weight vector and additive noise are
non-Gaussian and belong to the exponential family of distributions. The
analysis also suggests a modified version of SMD, which we refer to as
symmetric SMD (SSMD). The proofs rely on some simple properties of Bregman
divergence, which allow us to extend results from quadratics and Gaussians to
certain convex functions and exponential families in a rather seamless way
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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