3,479 research outputs found
A first-order stochastic primal-dual algorithm with correction step
We investigate the convergence properties of a stochastic primal-dual
splitting algorithm for solving structured monotone inclusions involving the
sum of a cocoercive operator and a composite monotone operator. The proposed
method is the stochastic extension to monotone inclusions of a proximal method
studied in {\em Y. Drori, S. Sabach, and M. Teboulle, A simple algorithm for a
class of nonsmooth convex-concave saddle-point problems, 2015} and {\em I.
Loris and C. Verhoeven, On a generalization of the iterative soft-thresholding
algorithm for the case of non-separable penalty, 2011} for saddle point
problems. It consists in a forward step determined by the stochastic evaluation
of the cocoercive operator, a backward step in the dual variables involving the
resolvent of the monotone operator, and an additional forward step using the
stochastic evaluation of the cocoercive introduced in the first step. We prove
weak almost sure convergence of the iterates by showing that the primal-dual
sequence generated by the method is stochastic quasi Fej\'er-monotone with
respect to the set of zeros of the considered primal and dual inclusions.
Additional results on ergodic convergence in expectation are considered for the
special case of saddle point models
Switch and Conquer: Efficient Algorithms By Switching Stochastic Gradient Oracles For Decentralized Saddle Point Problems
We consider a class of non-smooth strongly convex-strongly concave saddle
point problems in a decentralized setting without a central server. To solve a
consensus formulation of problems in this class, we develop an inexact primal
dual hybrid gradient (inexact PDHG) procedure that allows generic gradient
computation oracles to update the primal and dual variables. We first
investigate the performance of inexact PDHG with stochastic variance reduction
gradient (SVRG) oracle. Our numerical study uncovers a significant phenomenon
of initial conservative progress of iterates of IPDHG with SVRG oracle. To
tackle this, we develop a simple and effective switching idea, where a
generalized stochastic gradient (GSG) computation oracle is employed to hasten
the iterates' progress to a saddle point solution during the initial phase of
updates, followed by a switch to the SVRG oracle at an appropriate juncture.
The proposed algorithm is named Decentralized Proximal Switching Stochastic
Gradient method with Compression (C-DPSSG), and is proven to converge to an
-accurate saddle point solution with linear rate. Apart from
delivering highly accurate solutions, our study reveals that utilizing the best
convergence phases of GSG and SVRG oracles makes C-DPSSG well suited for
obtaining solutions of low/medium accuracy faster, useful for certain
applications. Numerical experiments on two benchmark machine learning
applications show C-DPSSG's competitive performance which validate our
theoretical findings.Comment: arXiv admin note: substantial text overlap with arXiv:2205.1445
Stochastic Variance Reduction Methods for Saddle-Point Problems
We consider convex-concave saddle-point problems where the objective
functions may be split in many components, and extend recent stochastic
variance reduction methods (such as SVRG or SAGA) to provide the first
large-scale linearly convergent algorithms for this class of problems which is
common in machine learning. While the algorithmic extension is straightforward,
it comes with challenges and opportunities: (a) the convex minimization
analysis does not apply and we use the notion of monotone operators to prove
convergence, showing in particular that the same algorithm applies to a larger
class of problems, such as variational inequalities, (b) there are two notions
of splits, in terms of functions, or in terms of partial derivatives, (c) the
split does need to be done with convex-concave terms, (d) non-uniform sampling
is key to an efficient algorithm, both in theory and practice, and (e) these
incremental algorithms can be easily accelerated using a simple extension of
the "catalyst" framework, leading to an algorithm which is always superior to
accelerated batch algorithms.Comment: Neural Information Processing Systems (NIPS), 2016, Barcelona, Spai
A Primal Dual Smoothing Framework for Max-Structured Nonconvex Optimization
We propose a primal dual first-order smoothing framework for solving a class
of nonsmooth nonconvex optimization problems with max-structure. We analyze the
primal and dual oracle complexities of the framework via two approaches, i.e.,
the dual-then-primal and primal-the-dual smoothing approaches. Our framework
improves the best-known oracle complexities of the existing methods, even in
the restricted problem setting. As the cornerstone of our framework, we propose
a conceptually simple primal dual method for solving a class of convex-concave
saddle-point problems with primal strong convexity, which is based on a newly
developed non-Hilbertian inexact accelerated proximal gradient algorithm. This
primal dual method has a dual oracle complexity that is significantly better
than the previous ones, and a primal oracle complexity that matches the
best-known, up to logarithmic factor. Finally, we extend our framework to the
stochastic case, and demonstrate that the oracle complexities of this extension
indeed match the state-of-the-art.Comment: 37 page
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