1,720 research outputs found
Fast Objective & Duality Gap Convergence for Nonconvex-Strongly-Concave Min-Max Problems
This paper focuses on stochastic methods for solving smooth non-convex
strongly-concave min-max problems, which have received increasing attention due
to their potential applications in deep learning (e.g., deep AUC maximization,
distributionally robust optimization). However, most of the existing algorithms
are slow in practice, and their analysis revolves around the convergence to a
nearly stationary point. We consider leveraging the Polyak-\L ojasiewicz (PL)
condition to design faster stochastic algorithms with stronger convergence
guarantee. Although PL condition has been utilized for designing many
stochastic minimization algorithms, their applications for non-convex min-max
optimization remain rare. In this paper, we propose and analyze a generic
framework of proximal epoch-based method with many well-known stochastic
updates embeddable. Fast convergence is established in terms of both {\bf the
primal objective gap and the duality gap}. Compared with existing studies, (i)
our analysis is based on a novel Lyapunov function consisting of the primal
objective gap and the duality gap of a regularized function, and (ii) the
results are more comprehensive with improved rates that have better dependence
on the condition number under different assumptions. We also conduct deep and
non-deep learning experiments to verify the effectiveness of our methods
A Primal-Dual Algorithmic Framework for Constrained Convex Minimization
We present a primal-dual algorithmic framework to obtain approximate
solutions to a prototypical constrained convex optimization problem, and
rigorously characterize how common structural assumptions affect the numerical
efficiency. Our main analysis technique provides a fresh perspective on
Nesterov's excessive gap technique in a structured fashion and unifies it with
smoothing and primal-dual methods. For instance, through the choices of a dual
smoothing strategy and a center point, our framework subsumes decomposition
algorithms, augmented Lagrangian as well as the alternating direction
method-of-multipliers methods as its special cases, and provides optimal
convergence rates on the primal objective residual as well as the primal
feasibility gap of the iterates for all.Comment: This paper consists of 54 pages with 7 tables and 12 figure
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
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