This work can specifically ensure, without any smoothness assumptions, convergence to Mordukhovich stationarity as long as the base directions asymptotically revert to the negative gradient for small stepsizes.
Abstract
Nonlinear optimization problems with complicated, nonconvex, yet geometrically structured constraints can be tackled by projected-gradient methods: under weak regularity assumptions, these approaches were recently proved to possess convergence properties to the strongest stationarity conditions. In this work, we show how momentum terms, commonly used in nonlinear optimization to speed up the convergence process, can be integrated within this algorithmic framework without harming convergence guarantees. Preliminarily, we highlight an intrinsic issue induced by the direct replacement of the negative gradient with a general descent direction within the projected approach. Then, we present suitable backtracking mechanisms for the pre-projection step, allowing us to integrate momentum terms in the direction. By this technique, we can specifically ensure, without any smoothness assumptions, convergence to Mordukhovich stationarity as long as the base directions asymptotically revert to the negative gradient for small stepsizes; moreover, if the base search direction reverts exactly to the negative gradient for the smallest steps, the algorithm is proved to converge to Bouligand and Proximally stationary points, with and without (local) smoothness assumptions respectively. Finally, the proposed procedure is numerically tested on some classes of problems, namely, sparsity and bounded-rank constrained problems; the results indicate that the proposed method is computationally effective, taking advantage of the additional information provided by the momentum term.
This paper proposes a general line-search Newton framework for unconstrained optimization that avoids repeated Hessian regularization by exploiting the Newton direction only when it is well-defined and suitable and provides the first Newton-type algorithm together with a comprehensive convergence analysis for this important class of nonconvex optimization problems.
Gradient-based learning under non-convex constraints exhibits a notable phenomenon: Despite the existence of many equivalent global minimizers, optimization algorithms consistently converge to a small subset of structured solutions. This behavior, known as implicit bias, remains insufficiently understood in constrained and non-convex settings. In this article, we investigate the mechanism of implicit bias induced by projected gradient-based optimization over general non-convex feasible sets. By modeling projected gradient descent as a continuous-time dynamical system, we derive a projected gradient flow characterized by tangent and normal cone decompositions, which capture the local geometry of the constraint set. Based on this formulation, we show that constraint geometry continuously filters gradient directions along the optimization trajectory, leading to a trajectory-dependent implicit regularization effect without modifying the objective function. We further formalize this effect through a cumulative normal projection energy functional and prove that the optimization dynamics converge to solutions minimizing both empirical risk and geometric incompatibility with the constraint set. Extensive experiments on synthetic and real-world datasets validate the theoretical predictions, demonstrating consistent alignment between solution geometry, optimization trajectories, and generalization performance. These results provide a unified geometric and dynamical explanation of implicit bias in constrained learning systems.
Generalized smoothness, such as (L0, L1)-smoothness, have recently attracted considerable attention due to their ability to model optimization problems arising in modern machine and deep learning, where the classical Lipschitz assumptions of the gradient is often violated. At the same time, computing exact gradients may be impractical or computationally expensive in many applications. In this work, we study convex (L0, L1)-smooth optimization (for normalized gradient method we consider quasi-convex problems too) under access only to a normalized approximation recently proposed Comparison Oracle, which returns an inexact normalized gradient in linear time with a bounded absolute error. Within this framework, we develop comparison-oracle variants of Normalized Gradient Descent and Gradient Descent with Polyak stepsizes. We establish explicit upper bounds on the approximation error that guarantee convergence and derive convergence rates for all proposed methods. Unlike existing analyses, our results require neither classical smoothness assumptions nor access to exact gradients or their exact normalized counterparts. Finally, numerical experiments corroborate the theoretical findings.
We study adaptive gradient descent for continuously differentiable, possibly nonconvex objectives under one-sided H\"older regularity. Unlike classical H\"older- or Lipschitz-gradient assumptions, which control the full gradient variation, our condition bounds only the directional term appearing in the descent inequality. This can allow less conservative step sizes when large gradient changes are orthogonal to, or favorable along, the update direction. We propose an adaptive scalar-step method based on an estimate of positive one-sided H\"older curvature, combined with a simple sufficient-decrease safeguard. For nonconvex objectives on a convex region containing the accepted update segments, we prove an explicit best-iterate stationarity bound with a rate determined by the H\"older exponent. Unlike predetermined diminishing step-size schemes, the method adapts to the local descent geometry. We evaluate the approach on two full-batch benchmarks designed to separate directional curvature from full gradient variation. On a binary classification problem, the method achieves the lowest final cross-entropy, objective value, and gradient norm, together with the largest classification margin among the compared scalar gradient methods. On a nonconvex H\"older regression problem, it attains the lowest final objective gap and gradient norm. These results indicate that one-sided H\"older curvature is an effective adaptive step-size signal when full-gradient variation is inflated by directions that do not hinder descent.
This work revisits a variant of the Polyak step-size based on Bregman projections due to Kiwiel (1997), and shows that mirror Polyak enjoys guarantees similar to its Euclidean counterpart, automatically adapting to relative notions of smoothness, Lipschitz continuity, or strong convexity.
Frederik Kunstner, Ryan D'Orazio, V. S. Portella et al.· 0 citations