We analyze nonlinearly preconditioned gradient methods for solving smooth minimization problems. We introduce a generalized smoothness property, based on the notion of abstract convexity, that is broader than Lipschitz smoothness and provide sufficient first- and second-order conditions. Notably, our framework encapsulates algorithms associated with the gradient clipping method and brings out novel insights for the class of $(L_0,L_1)$-smooth functions that has received widespread interest recently, thus allowing us to extend beyond already established methods. We investigate the convergence of the proposed method in both the convex and nonconvex setting.

We study nonlinearly preconditioned gradient methods for smooth nonconvex optimization problems, focusing on sigmoid preconditioners that inherently perform a form of gradient clipping akin to the widely used gradient clipping technique. Building upon this idea, we introduce a novel heavy ball-type algorithm and provide convergence guarantees under a generalized smoothness condition that is less restrictive than traditional Lipschitz smoothness, thus covering a broader class of functions. Additionally, we develop a stochastic variant of the base method and study its convergence properties under different noise assumptions. We compare the proposed algorithms with baseline methods on diverse tasks from machine learning including neural network training.

We show that adaptive proximal gradient methods for convex problems are not restricted to traditional Lipschitzian assumptions. Our analysis reveals that a class of linesearch-free methods is still convergent under mere local Hölder gradient continuity, covering in particular continuously differentiable semi-algebraic functions. To mitigate the lack of local Lipschitz continuity, popular approaches revolve around $\varepsilon$-oracles and/or linesearch procedures. In contrast, we exploit plain Hölder inequalities not entailing any approximation, all while retaining the linesearch-free nature of adaptive schemes. Furthermore, we prove full sequence convergence without prior knowledge of local Hölder constants nor of the order of Hölder continuity. Numerical experiments make comparisons with baseline methods on diverse tasks from machine learning covering both the locally and the globally Hölder setting.

In this work, we propose a new framework for solving the optimal consensus problem of agents with continuous dynamics. Motivated by the EXTRA algorithm, we introduce novel auxiliary continuous variables that utilize only sampled measurements of the neighbor outputs. The use of suitable smoothing functions, which allow us to gradually incorporate the samples in the new variables within a sampling period, guarantees the continuity of these variables. It is shown that the regulation of these EXTRA-based smoothed optimal consensus variables to a neighborhood of zero ensures approximate optimal consensus of the agent outputs. Taking this property into account, adaptive fuzzy distributed controllers are then designed for fully actuated Lagrangian agents that approximate the unknown system nonlinearities and guarantee approximate optimal consensus of all agent outputs. Moreover, all closed-loop signals are proven to remain bounded. Simulations on a swarm of wheeled robots validate the effectiveness of the proposed approach.