There is extensive literature on accelerating first-order optimization methods in an Euclidean setting. Under which conditions such acceleration is feasible in Riemannian optimization problems is an active area of research. Motivated by the recent success of silver stepsize methods in the Euclidean setting, we undertake a study of such algorithms in the Riemannian setting. We provide the new class of algorithms determined by the choice of vector transport that allows the silver stepsize acceleration on Riemannian manifolds for the function classes associated with the corresponding vector transport. As a core application, we show that our algorithm recovers the standard Wasserstein gradient descent on the 2-Wasserstein space and, as a result, provides the first provable accelerated gradient method for potential functional optimization problems in the Wasserstein space.

Real-world time series analysis, such as healthcare, autonomous driving, and solar energy, faces unique challenges arising from the scarcity of labeled data, highlighting the need for effective semi-supervised learning methods. While the Virtual Adversarial Training (VAT) method has shown promising performance in leveraging unlabeled data for smoother predictive distributions, straightforward extensions of VAT often fall short on time series tasks as they neglect the temporal structure of the data in the adversarial perturbation. In this paper, we propose the framework of functional Virtual Adversarial Training (f-VAT) that can incorporate the functional structure of the data into perturbations. By theoretically establishing a duality between the perturbation norm and the functional model sensitivity, we propose to use an appropriate Sobolev (H s) norm to generate structured functional adversarial perturbations for semi-supervised time series classification. Our proposed f-VAT method outperforms recent methods and achieves superior performance in extensive semi-supervised time series classification tasks (e.g., up to 9% performance improvement). We also provide additional visualization studies to offer further insights into the superiority of f-VAT.

We study generalized smoothness in nonconvex optimization, focusing on (L0,L1)smoothness and anisotropic smoothness. The former was empirically derived from practical neural network training examples, while the latter arises naturally in the analysis of nonlinearly preconditioned gradient methods. We introduce a new sufficient condition that encompasses both notions, reveals their close connection, and holds in key applications such as phase retrieval and matrix factorization. Leveraging tools from dynamical systems theory, we then show that nonlinear preconditioning - including gradient clipping - preserves the saddle point avoidance property of classical gradient descent. Crucially, the assumptions required for this analysis are actually satisfied in these applications, unlike in classical results that rely on restrictive Lipschitz smoothness conditions. We further analyze a perturbed variant that efficiently attains second-order stationarity with only logarithmic dependence on dimension, matching similar guarantees of classical gradient methods.

We study the problem of regression with interval targets, where only upper and lower bounds on target values are available in the form of intervals. This problem arises when the exact target label is expensive or impossible to obtain, due to inherent uncertainties. In the absence of exact targets, traditional regression loss functions cannot be used. First, we study the methodology of using a loss function compatible with interval targets, for which we establish non-asymptotic generalization bounds based on smoothness of the hypothesis class that significantly relax prior assumptions. Second, we propose a novel minmax learning formulation: minimize against the worst-case (maximized) target labels within the provided intervals. The maximization problem in the latter is non-convex, but we show that good performance can be achieved by incorporating smoothness constraints. Finally, we perform extensive experiments on real-world datasets and show that our methods achieve state-of-the-art performance.

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.

Gradient clipping is increasingly important in centralized learning (CL) and federated learning (FL). Many works focus on its optimization properties under strong assumptions involving Gaussian noise and standard smoothness. However, practical machine learning tasks often only satisfy weaker conditions, such as heavy-tailed noise and (L0,L1)-smoothness. To bridge this gap, we propose a high-probability analysis for clipped Stochastic Gradient Descent (SGD) under these weaker assumptions. Our findings show a better convergence rate than existing ones can be achieved, and our high-probability analysis does not rely on the bounded gradient assumption. Moreover, we extend our analysis to FL, where a gap remains between expected and high-probability convergence, which the naive clipped SGD can not bridge. Thus, we design a new Federated Clipped Batched Gradient (FedCBG) algorithm, and prove the convergence and generalization bounds with high probability for the first time. Our analysis reveals the trade-offs between the optimization and generalization performance. Extensive experiments demonstrate that FedCBG can generalize better to unseen client distributions than state-of-the-art baselines.

Aircraft manufacturing is the jewel in the crown of industry, in which generating high-fidelity airfoil geometries with controllable and editable representations remains a fundamental challenge. Existing deep learning methods, which typically rely on predefined parametric representations (e.g., Bézier curves) or discrete point sets, face an inherent trade-off between expressive power and resolution adaptability. To tackle this challenge, we introduce FuncGenFoil, a novel functionspace generative model that directly reconstructs airfoil geometries as function curves. Our method inherits the advantages of arbitrary-resolution sampling and smoothness from parametric functions, as well as the strong expressiveness of discrete point-based representations. Empirical evaluations demonstrate that FuncGenFoil improves upon state-of-the-art methods in airfoil generation, achieving a relative 74.4% reduction in label error and a 23.2% increase in diversity on the AF-200K dataset. Our results highlight the advantages of function-space modeling for aerodynamic shape optimization, offering a powerful and flexible framework for high-fidelity airfoil design.

Smoothness is known to be crucial for acceleration in offline optimization, and for gradient-variation regret minimization in online learning. Interestingly, these two problems are actually closely connected -- accelerated optimization can be understood through the lens of gradient-variation online learning. In this paper, we investigate online learning with Hölder smooth functions, a general class encompassing both smooth and non-smooth (Lipschitz) functions, and explore its implications for offline optimization.

We study online adversarial regression with convex losses against a rich class of continuous yet highly irregular competitor functions,% prediction rules, modeled by Besov spaces $B_{pq}^s$ with general parameters $1 \leq p,q \leq \infty$ and smoothness $s > \tfrac{d}{p}$. We introduce an adaptive wavelet-based algorithm that performs sequential prediction without prior knowledge of $(s,p,q)$, and establish minimax-optimal regret bounds against any comparator in $B_{pq}^s$.

We study generalized smoothness in nonconvex optimization, focusing on $(L_0, L_1)$-smoothness and anisotropic smoothness. The former was empirically derived from practical neural network training examples, while the latter arises naturally in the analysis of nonlinearly preconditioned gradient methods. We introduce a new sufficient condition that encompasses both notions, reveals their close connection, and holds in key applications such as phase retrieval and matrix factorization. Leveraging tools from dynamical systems theory, we then show that nonlinear preconditioning--including gradient clipping--preserves the saddle point avoidance property of classical gradient descent. Crucially, the assumptions required for this analysis are actually satisfied in these applications, unlike in classical results that rely on restrictive Lipschitz smoothness conditions. We further analyze a perturbed variant that efficiently attains second-order stationarity with only logarithmic dependence on dimension, matching similar guarantees of classical gradient methods.