Related papers: Accelerated Objective Gap and Gradient Norm Conver…
The analysis of gradient descent-type methods typically relies on the Lipschitz continuity of the objective gradient. This generally requires an expensive hyperparameter tuning process to appropriately calibrate a stepsize for a given…
In this work, we study the iteration complexity of gradient methods for minimizing convex quadratic functions regularized by powers of Euclidean norms. We show that, due to the uniform convexity of the objective, gradient methods have…
Subgradient methods are the natural extension to the non-smooth case of the classical gradient descent for regular convex optimization problems. However, in general, they are characterized by slow convergence rates, and they require…
We consider the minimization of non-convex quadratic forms regularized by a cubic term, which exhibit multiple saddle points and poor local minima. Nonetheless, we prove that, under mild assumptions, gradient descent approximates the…
Based on a result by Taylor, Hendrickx, and Glineur (J. Optim. Theory Appl., 178(2):455--476, 2018) on the attainable convergence rate of gradient descent for smooth and strongly convex functions in terms of function values, an elementary…
In this paper, we study the convergence rate of the gradient (or steepest descent) method with fixed step lengths for finding a stationary point of an $L$-smooth function. We establish a new convergence rate, and show that the bound may be…
Classical optimisation theory guarantees monotonic objective decrease for gradient descent (GD) when employed in a small step size, or ``stable", regime. In contrast, gradient descent on neural networks is frequently performed in a large…
We analyze the last-iterate convergence of the Anchored Gradient Descent Ascent algorithm for smooth convex-concave min-max problems. While previous work established a last-iterate rate of $\mathcal{O}(1/t^{2-2p})$ for the squared gradient…
We provide a concise, self-contained proof that the Silver Stepsize Schedule proposed in Part I directly applies to smooth (non-strongly) convex optimization. Specifically, we show that with these stepsizes, gradient descent computes an…
Due to the non-smoothness of optimization problems in Machine Learning, generalized smoothness assumptions have been gaining a lot of attention in recent years. One of the most popular assumptions of this type is $(L_0,L_1)$-smoothness…
The subgradient method is one of the most fundamental algorithmic schemes for nonsmooth optimization. The existing complexity and convergence results for this method are mainly derived for Lipschitz continuous objective functions. In this…
Distributed optimization has received a lot of interest in recent years due to its wide applications in various fields. In this work, we revisit the convergence property of the decentralized gradient descent [A. Nedi{\'c}-A.Ozdaglar (2009)]…
We provide larger step-size restrictions for which gradient descent based algorithms (almost surely) avoid strict saddle points. In particular, consider a twice differentiable (non-convex) objective function whose gradient has Lipschitz…
Selecting an effective step-size is a fundamental challenge in first-order optimization, especially for problems with non-Euclidean geometries. This paper presents a novel adaptive step-size strategy for optimization algorithms that rely on…
This paper proposes a novel proximal-gradient algorithm for a decentralized optimization problem with a composite objective containing smooth and non-smooth terms. Specifically, the smooth and nonsmooth terms are dealt with by gradient and…
This work introduces a two-step stepsize schedule for stochastic gradient methods minimizing smooth strongly convex functions. We consider the setting where only stochastic gradient approximations, which are unbiased, of bounded variance,…
Stochastic gradient descent (SGD) gives an optimal convergence rate when minimizing convex stochastic objectives $f(x)$. However, in terms of making the gradients small, the original SGD does not give an optimal rate, even when $f(x)$ is…
A vast literature on convergence guarantees for gradient descent and derived methods exists at the moment. However, a simple practical situation remains unexplored: when a fixed step size is used, can we expect gradient descent to converge…
This paper considers the distributed optimization problem over a network, where the objective is to optimize a global function formed by a sum of local functions, using only local computation and communication. We develop an Accelerated…
We consider the problem of optimizing the sum of a smooth convex function and a non-smooth convex function using proximal-gradient methods, where an error is present in the calculation of the gradient of the smooth term or in the proximity…