138 research outputs found
Stochastic Intermediate Gradient Method for Convex Problems with Inexact Stochastic Oracle
In this paper we introduce new methods for convex optimization problems with
inexact stochastic oracle. First method is an extension of the intermediate
gradient method proposed by Devolder, Glineur and Nesterov for problems with
inexact oracle. Our new method can be applied to the problems with composite
structure, stochastic inexact oracle and allows using non-Euclidean setup. We
prove estimates for mean rate of convergence and probabilities of large
deviations from this rate. Also we introduce two modifications of this method
for strongly convex problems. For the first modification we prove mean rate of
convergence estimates and for the second we prove estimates for large
deviations from the mean rate of convergence. All the rates give the complexity
estimates for proposed methods which up to multiplicative constant coincide
with lower complexity bound for the considered class of convex composite
optimization problems with stochastic inexact oracle
Primal-dual accelerated gradient methods with small-dimensional relaxation oracle
In this paper, a new variant of accelerated gradient descent is proposed. The
pro-posed method does not require any information about the objective function,
usesexact line search for the practical accelerations of convergence, converges
accordingto the well-known lower bounds for both convex and non-convex
objective functions,possesses primal-dual properties and can be applied in the
non-euclidian set-up. Asfar as we know this is the rst such method possessing
all of the above properties atthe same time. We also present a universal
version of the method which is applicableto non-smooth problems. We demonstrate
how in practice one can efficiently use thecombination of line-search and
primal-duality by considering a convex optimizationproblem with a simple
structure (for example, linearly constrained)
Analysis of Kernel Mirror Prox for Measure Optimization
By choosing a suitable function space as the dual to the non-negative measure
cone, we study in a unified framework a class of functional saddle-point
optimization problems, which we term the Mixed Functional Nash Equilibrium
(MFNE), that underlies several existing machine learning algorithms, such as
implicit generative models, distributionally robust optimization (DRO), and
Wasserstein barycenters. We model the saddle-point optimization dynamics as an
interacting Fisher-Rao-RKHS gradient flow when the function space is chosen as
a reproducing kernel Hilbert space (RKHS). As a discrete time counterpart, we
propose a primal-dual kernel mirror prox (KMP) algorithm, which uses a dual
step in the RKHS, and a primal entropic mirror prox step. We then provide a
unified convergence analysis of KMP in an infinite-dimensional setting for this
class of MFNE problems, which establishes a convergence rate of in the
deterministic case and in the stochastic case, where is the
iteration counter. As a case study, we apply our analysis to DRO, providing
algorithmic guarantees for DRO robustness and convergence.Comment: Accepted to AISTATS 202
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