arXiv research

Many classical algorithms are found until several years later to outlive the confines in which they were conceived, and continue to be relevant in unforeseen settings. In this paper, we show that SVRG is one such method: being originally designed for strongly convex objectives, it is also very robust in non-strongly co…

A generalized optimistic method for saddle point problems with improved complexity.

Stochastic gradient descent in continuous time (SGDCT) provides a computationally efficient method for the statistical learning of continuous-time models, which are widely used in science, engineering, and finance. The SGDCT algorithm follows a (noisy) descent direction along a continuous stream of data. The parameter …

Boosting is a popular way to derive powerful learners from simpler hypothesis classes. Following previous work (Mason et al., 1999; Friedman, 2000) on general boosting frameworks, we analyze gradient-based descent algorithms for boosting with respect to any convex objective and introduce a new measure of weak learner p…

DFFL tackles federated learning with heterogeneous objectives and constraints.

This paper analyzes SGD with increasingly weighted averaging for optimization and generalization.

In this paper, we introduce various mechanisms to obtain accelerated first-order stochastic optimization algorithms when the objective function is convex or strongly convex. Specifically, we extend the Catalyst approach originally designed for deterministic objectives to the stochastic setting. Given an optimization me…

SUSTAIN algorithm tackles stochastic bilevel optimization with near-optimal complexity.

A new method solves convex optimization problems on manifolds efficiently.

We study dual-based algorithms for distributed convex optimization problems over networks, where the objective is to minimize a sum $\sum_{i=1}^{m}f_i(z)$ of functions over in a network. We provide complexity bounds for four different cases, namely: each function $f_i$ is strongly convex and smooth, each function is ei…

We present a stochastic setting for optimization problems with nonsmooth convex separable objective functions over linear equality constraints. To solve such problems, we propose a stochastic Alternating Direction Method of Multipliers (ADMM) algorithm. Our algorithm applies to a more general class of nonsmooth convex …

Paper tackles fast convergence for non-convex strongly-concave min-max problems.

Uniform diffusion approximation for SGD in non-convex settings.

Paper tackles Hessian/Jacobian-free stochastic bilevel optimization with ${O}(ε^{-1.5})$ complexity.

Study large deviations rates for SGD with strongly convex functions.

RR algorithm improves convergence rate without strong convexity assumptions.

The paper tackles minimax optimality in continuum contextual bandits with Hölder continuity.

We develop and analyze an asynchronous algorithm for distributed convex optimization when the objective writes a sum of smooth functions, local to each worker, and a non-smooth function. Unlike many existing methods, our distributed algorithm is adjustable to various levels of communication cost, delays, machines compu…

Majorization-minimization algorithms consist of successively minimizing a sequence of upper bounds of the objective function. These upper bounds are tight at the current estimate, and each iteration monotonically drives the objective function downhill. Such a simple principle is widely applicable and has been very popu…

The incremental aggregated gradient algorithm is popular in network optimization and machine learning research. However, the current convergence results require the objective function to be strongly convex. And the existing convergence rates are also limited to linear convergence. Due to the mathematical techniques, th…

A new method for distributed optimization reduces communication rounds without minibatches.

In this work we introduce a new optimisation method called SAGA in the spirit of SAG, SDCA, MISO and SVRG, a set of recently proposed incremental gradient algorithms with fast linear convergence rates. SAGA improves on the theory behind SAG and SVRG, with better theoretical convergence rates, and has support for compos…

We consider a wide range of regularized stochastic minimization problems with two regularization terms, one of which is composed with a linear function. This optimization model abstracts a number of important applications in artificial intelligence and machine learning, such as fused Lasso, fused logistic regression, a…

We introduce a generic scheme for accelerating gradient-based optimization methods in the sense of Nesterov. The approach, called Catalyst, builds upon the inexact accelerated proximal point algorithm for minimizing a convex objective function, and consists of approximately solving a sequence of well-chosen auxiliary p…

The main goal of this work is equipping convex and nonconvex problems with Barzilai-Borwein (BB) step size. With the adaptivity of BB step sizes granted, they can fail when the objective function is not strongly convex. To overcome this challenge, the key idea here is to bridge (non)convex problems and strongly convex …

Zeroth-order optimization methods lack inherent privacy guarantees.

A multiobjective optimization problem is simplicial if the Pareto set and front are homeomorphic to a simplex and, under the homeomorphisms, each face of the simplex corresponds to the Pareto set and front of a subproblem. In this paper, we show that strongly convex problems are simplicial under a mild assumption on th…

In this paper, we develop a new accelerated stochastic gradient method for efficiently solving the convex regularized empirical risk minimization problem in mini-batch settings. The use of mini-batches is becoming a golden standard in the machine learning community, because mini-batch settings stabilize the gradient es…

Frank-Wolfe algorithm (FW) and its variants have gained a surge of interests in machine learning community due to its projection-free property. Recently people have reduced the gradient evaluation complexity of FW algorithm to $\log(\frac{1}ε)$ for the smooth and strongly convex objective. This complexity result is esp…

We consider the problem of minimizing the sum of an average function of a large number of smooth convex components and a general, possibly non-differentiable, convex function. Although many methods have been proposed to solve this problem with the assumption that the sum is strongly convex, few methods support the non-…

We consider the problem of minimizing the sum of two convex functions: one is the average of a large number of smooth component functions, and the other is a general convex function that admits a simple proximal mapping. We assume the whole objective function is strongly convex. Such problems often arise in machine lea…

In this paper, we revisit the convergence of the Heavy-ball method, and present improved convergence complexity results in the convex setting. We provide the first non-ergodic O(1/k) rate result of the Heavy-ball algorithm with constant step size for coercive objective functions. For objective functions satisfying a re…

Optimizes CM for stochastic convex optimization with progressive precision.

In this paper, we consider first-order convergence theory and algorithms for solving a class of non-convex non-concave min-max saddle-point problems, whose objective function is weakly convex in the variables of minimization and weakly concave in the variables of maximization. It has many important applications in mach…

New methods boost first-order optimization with faster rates.

The Adam algorithm has become extremely popular for large-scale machine learning. Under convexity condition, it has been proved to enjoy a data-dependant $O(\sqrt{T})$ regret bound where $T$ is the time horizon. However, whether strong convexity can be utilized to further improve the performance remains an open problem…

New study shows acceleration in hyperbolic spaces is impossible for strongly geodesically convex functions.

Random extrapolation speeds up coordinate descent for sparse and dense data.

Paper solves robust convex problems with heavy-tailed noise.

The paper analyzes the efficiency of unlearning methods and establishes bounds for minimax computation times.

GenFlow optimizes faster, avoiding saddle points in fixed time.

This paper studies the lower bound complexity for the optimization problem whose objective function is the average of $n$ individual smooth convex functions. We consider the algorithm which gets access to gradient and proximal oracle for each individual component. For the strongly-convex case, we prove such an algorith…

We study the minimization of a convex function $f(X)$ over the set of $n\times n$ positive semi-definite matrices, but when the problem is recast as $\min_U g(U) := f(UU^\top)$, with $U \in \mathbb{R}^{n \times r}$ and $r \leq n$. We study the performance of gradient descent on $g$---which we refer to as Factored Gradi…

We consider stochastic strongly convex optimization with a complex inequality constraint. This complex inequality constraint may lead to computationally expensive projections in algorithmic iterations of the stochastic gradient descent~(SGD) methods. To reduce the computation costs pertaining to the projections, we pro…

We show that Newton's method converges globally at a linear rate for objective functions whose Hessians are stable. This class of problems includes many functions which are not strongly convex, such as logistic regression. Our linear convergence result is (i) affine-invariant, and holds even if an (ii) approximate Hess…

Paper shows linear convergence of ISTA and FISTA for ill-conditioned images.

Epoch gradient descent method (a.k.a. Epoch-GD) proposed by Hazan and Kale (2011) was deemed a breakthrough for stochastic strongly convex minimization, which achieves the optimal convergence rate of $O(1/T)$ with $T$ iterative updates for the {\it objective gap}. However, its extension to solving stochastic min-max pr…

Almost all local minima in neural networks are strongly convex.