25 research outputs found
A First-order Method for Monotone Stochastic Variational Inequalities on Semidefinite Matrix Spaces
Motivated by multi-user optimization problems and non-cooperative Nash games
in stochastic regimes, we consider stochastic variational inequality (SVI)
problems on matrix spaces where the variables are positive semidefinite
matrices and the mapping is merely monotone. Much of the interest in the theory
of variational inequality (VI) has focused on addressing VIs on vector
spaces.Yet, most existing methods either rely on strong assumptions, or require
a two-loop framework where at each iteration, a projection problem, i.e., a
semidefinite optimization problem needs to be solved. Motivated by this gap, we
develop a stochastic mirror descent method where we choose the distance
generating function to be defined as the quantum entropy. This method is a
single-loop first-order method in the sense that it only requires a
gradient-type of update at each iteration. The novelty of this work lies in the
convergence analysis that is carried out through employing an auxiliary
sequence of stochastic matrices. Our contribution is three-fold: (i) under this
setting and employing averaging techniques, we show that the iterate generated
by the algorithm converges to a weak solution of the SVI; (ii) moreover, we
derive a convergence rate in terms of the expected value of a suitably defined
gap function; (iii) we implement the developed method for solving a
multiple-input multiple-output multi-cell cellular wireless network composed of
seven hexagonal cells and present the numerical experiments supporting the
convergence of the proposed method
Stochastic proximal splitting algorithm for composite minimization
Supported by the recent contributions in multiple branches, the first-order
splitting algorithms became central for structured nonsmooth optimization. In
the large-scale or noisy contexts, when only stochastic information on the
smooth part of the objective function is available, the extension of proximal
gradient schemes to stochastic oracles is based on proximal tractability of the
nonsmooth component and it has been deeply analyzed in the literature. However,
there remained gaps illustrated by composite models where the nonsmooth term is
not proximally tractable anymore. In this note we tackle composite optimization
problems, where the access only to stochastic information on both smooth and
nonsmooth components is assumed, using a stochastic proximal first-order scheme
with stochastic proximal updates. We provide the iteration complexity (in expectation of squared distance to the
optimal set) under the strong convexity assumption on the objective function.
Empirical behavior is illustrated by numerical tests on parametric sparse
representation models
A damped forward-backward algorithm for stochastic generalized Nash equilibrium seeking
We consider a stochastic generalized Nash equilibrium problem (GNEP) with
expected-value cost functions. Inspired by Yi and Pavel (Automatica, 2019), we
propose a distributed GNE seeking algorithm by exploiting the forward-backward
operator splitting and a suitable preconditioning matrix. Specifically, we
apply this method to the stochastic GNEP, where, at each iteration, the
expected value of the pseudo-gradient is approximated via a number of random
samples. Our main contribution is to show almost sure convergence of our
proposed algorithm if the sample size grows large enough
A Stochastic forward-backward splitting method for solving monotone inclusions in Hilbert spaces
We propose and analyze the convergence of a novel stochastic forward-backward
splitting algorithm for solving monotone inclusions given by the sum of a
maximal monotone operator and a single-valued maximal monotone cocoercive
operator. This latter framework has a number of interesting special cases,
including variational inequalities and convex minimization problems, while
stochastic approaches are practically relevant to account for perturbations in
the data. The algorithm we propose is a stochastic extension of the classical
deterministic forward-backward method, and is obtained considering the
composition of the resolvent of the maximal monotone operator with a forward
step based on a stochastic estimate of the single-valued operator. Our study
provides a non-asymptotic error analysis in expectation for the strongly
monotone case, as well as almost sure convergence under weaker assumptions. The
approach we consider allows to avoid averaging, a feature critical when
considering methods based on sparsity, and, for minimization problems, it
allows to obtain convergence rates matching those obtained by stochastic
extensions of so called accelerated methods. Stochastic quasi Fejer's sequences
are a key technical tool to prove almost sure convergence.Comment: 20 page
A Proximal-Point Algorithm with Variable Sample-sizes (PPAWSS) for Monotone Stochastic Variational Inequality Problems
We consider a stochastic variational inequality (SVI) problem with a
continuous and monotone mapping over a closed and convex set. In strongly
monotone regimes, we present a variable sample-size averaging scheme (VS-Ave)
that achieves a linear rate with an optimal oracle complexity. In addition, the
iteration complexity is shown to display a muted dependence on the condition
number compared with standard variance-reduced projection schemes. To contend
with merely monotone maps, we develop amongst the first proximal-point
algorithms with variable sample-sizes (PPAWSS), where increasingly accurate
solutions of strongly monotone SVIs are obtained via (VS-Ave) at every step.
This allows for achieving a sublinear convergence rate that matches that
obtained for deterministic monotone VIs. Preliminary numerical evidence
suggests that the schemes compares well with competing schemes
Distributed projected-reflected-gradient algorithms for stochastic generalized Nash equilibrium problems
We consider the stochastic generalized Nash equilibrium problem (SGNEP) with
joint feasibility constraints and expected-value cost functions. We propose a
distributed stochastic projected reflected gradient algorithm and show its
almost sure convergence when the pseudogradient mapping is monotone and the
solution is unique. The algorithm is based on monotone operator splitting
methods tailored for SGNEPs when the expected-value pseudogradient mapping is
approximated at each iteration via an increasing number of samples of the
random variable. Finally, we show that a preconditioned variant of our proposed
algorithm has convergence guarantees when the pseudogradient mapping is
cocoercive.Comment: arXiv admin note: text overlap with arXiv:1910.1177
Stochastic Proximal Gradient Algorithm with Minibatches. Application to Large Scale Learning Models
Stochastic optimization lies at the core of most statistical learning models.
The recent great development of stochastic algorithmic tools focused
significantly onto proximal gradient iterations, in order to find an efficient
approach for nonsmooth (composite) population risk functions. The complexity of
finding optimal predictors by minimizing regularized risk is largely understood
for simple regularizations such as norms. However, more complex
properties desired for the predictor necessitates highly difficult regularizers
as used in grouped lasso or graph trend filtering. In this chapter we develop
and analyze minibatch variants of stochastic proximal gradient algorithm for
general composite objective functions with stochastic nonsmooth components. We
provide iteration complexity for constant and variable stepsize policies
obtaining that, for minibatch size , after
iterations suboptimality is
attained in expected quadratic distance to optimal solution. The numerical
tests on regularized SVMs and parametric sparse representation
problems confirm the theoretical behaviour and surpasses minibatch SGD
performance
A Stochastic Subgradient Method for Nonsmooth Nonconvex Multi-Level Composition Optimization
We propose a single time-scale stochastic subgradient method for constrained
optimization of a composition of several nonsmooth and nonconvex functions. The
functions are assumed to be locally Lipschitz and differentiable in a
generalized sense. Only stochastic estimates of the values and generalized
derivatives of the functions are used. The method is parameter-free. We prove
convergence with probability one of the method, by associating with it a system
of differential inclusions and devising a nondifferentiable Lyapunov function
for this system. For problems with functions having Lipschitz continuous
derivatives, the method finds a point satisfying an optimality measure with
error of order , after executing iterations with constant
stepsize
A Single Time-Scale Stochastic Approximation Method for Nested Stochastic Optimization
We study constrained nested stochastic optimization problems in which the
objective function is a composition of two smooth functions whose exact values
and derivatives are not available. We propose a single time-scale stochastic
approximation algorithm, which we call the Nested Averaged Stochastic
Approximation (NASA), to find an approximate stationary point of the problem.
The algorithm has two auxiliary averaged sequences (filters) which estimate the
gradient of the composite objective function and the inner function value. By
using a special Lyapunov function, we show that NASA achieves the sample
complexity of for finding an -approximate
stationary point, thus outperforming all extant methods for nested stochastic
approximation. Our method and its analysis are the same for both unconstrained
and constrained problems, without any need of batch samples for constrained
nonconvex stochastic optimization. We also present a simplified variant of the
NASA method for solving constrained single level stochastic optimization
problems, and we prove the same complexity result for both unconstrained and
constrained problems
Nonasymptotic convergence of stochastic proximal point algorithms for constrained convex optimization
A very popular approach for solving stochastic optimization problems is the
stochastic gradient descent method (SGD). Although the SGD iteration is
computationally cheap and the practical performance of this method may be
satisfactory under certain circumstances, there is recent evidence of its
convergence difficulties and instability for unappropriate parameters choice.
To avoid these drawbacks naturally introduced by the SGD scheme, the stochastic
proximal point algorithms have been recently considered in the literature. We
introduce a new variant of the stochastic proximal point method (SPP) for
solving stochastic convex optimization problems subject to (in)finite
intersection of constraints satisfying a linear regularity type condition. For
the newly introduced SPP scheme we prove new nonasymptotic convergence results.
In particular, for convex and Lipschitz continuous objective functions, we
prove nonasymptotic estimates for the rate of convergence in terms of the
expected value function gap of order , where is the
iteration counter. We also derive better nonasymptotic bounds for the rate of
convergence in terms of expected quadratic distance from the iterates to the
optimal solution for smooth strongly convex objective functions, which in the
best case is of order . Since these convergence rates can be
attained by our SPP algorithm only under some natural restrictions on the
stepsize, we also introduce a restarting variant of SPP method that overcomes
these difficulties and derive the corresponding nonasymptotic convergence
rates. Numerical evidence supports the effectiveness of our methods in
real-world problems