In this paper, we revisit a classical adaptive stepsize strategy for gradient descent: the Polyak stepsize (PolyakGD), originally proposed in Polyak (1969). We study the convergence behavior of PolyakGD from two perspectives: tight worst-case analysis and universality across function classes. As our first main result, we establish the tightness of the known convergence rates of … Read more
We develop and study a subsampled cubic regularization method for finite-sum composite optimization problems, in which the function, gradient, and Hessian are estimated using possibly different sample sizes. By allowing each quantity to have its own sampling strategy, the proposed method offers greater flexibility to control the accuracy of the model components and to better … Read more
This paper considers nonsmooth convex optimization with either a subgradient or proximal operator oracle. In both settings, we identify algorithms that achieve the recently introduced game-theoretic optimality notion for algorithms known as subgame perfection. Subgame perfect algorithms meet a more stringent requirement than just minimax optimality. Not only must they provide optimal uniform guarantees on … Read more
Composite optimization problems involve minimizing the composition of a smooth map with a convex function. Such objectives arise in numerous data science and signal processing applications, including phase retrieval, blind deconvolution, and collaborative filtering. The subgradient method achieves local linear convergence when the composite loss is well-conditioned. However, if the smooth map is, in a … Read more
The Nesterov’s accelerated gradient (NAG) method generalizes the classical gradient descent algorithm by improving the convergence rate from $\mathcal{O}\left(\frac{1}{t}\right)$ to $\mathcal{O}\left(\frac{1}{t^2}\right)$ in convex optimization. This study examines the proximal gradient framework for additively separable composite functions with smooth and non-smooth components. We demonstrate that Nesterov’s accelerated proximal gradient (NAPG$_\alpha$) method attains a convergence rate of … Read more
Consider composite nonconvex optimization problems where the objective function consists of a smooth nonconvex term (with Lipschitz-continuous gradient) and a convex (possibly nonsmooth) term. Existing parameter-free methods for such problems often rely on complex multi-loop structures, require line searches, or depend on restrictive assumptions (e.g., bounded iterates). To address these limitations, we introduce a novel … Read more
In this paper, we address composite optimization problems on Hadamard manifolds, where the objective function is given by the sum of a smooth term (not necessarily convex) and a convex term (not necessarily differentiable). To solve this problem, we develop a proximal gradient method defined directly on the manifold, employing a strategy that enforces monotonicity … Read more
Preconditioning (scaling) is essential in many areas of mathematics, and in particular in optimization. In this work, we study the problem of finding an optimal diagonal preconditioner. We focus on minimizing two different notions of condition number: the classical, worst-case type, \(\kappa\)-condition number, and the more averaging motivated \(\omega\)-condition number. We provide affine based pseudoconvex … Read more
In this work, we are concerned with the study of optimization problems featuring so-called probability or chance constraints. Probability constraints measure the level of satisfaction of an underlying random inequality system and ensure that this level is high enough. Such an underlying inequality system could be expressed by an abstractly known or perhaps costly to … Read more