We propose a conversion scheme that turns regret minimizing algorithms into fixed point iterations, with convergence guarantees following from regret bounds. The resulting iterations can be seen as a grand extension of the classical Krasnoselskii--Mann iterations, as the latter are recovered by converting the Online Gradient Descent algorithm. This approach yields new simple iterations for finding fixed points of non-self operators. We also focus on converting algorithms from the AdaGrad family of regret minimizers, and thus obtain fixed point iterations with adaptive guarantees of a new kind. Numerical experiments on various problems demonstrate faster convergence of AdaGrad-based fixed point iterations over Krasnoselskii--Mann iterations.

In this work, we address the following question: What minimal structural assumptions are needed to prevent the degradation of statistical learning bounds with increasing dimensionality? We investigate this question in the classical statistical setting of signal estimation from $n$ independent linear observations $Y_i = X_i^{\top}θ+ ε_i$. Our focus is on the generalization properties of a broad family of predictors that can be expressed as linear combinations of the training labels, $f(X) = \sum_{i=1}^{n} l_{i}(X) Y_i$. This class -- commonly referred to as linear prediction rules -- encompasses a wide range of popular parametric and non-parametric estimators, including ridge regression, gradient descent, and kernel methods. Our contributions are twofold. First, we derive non-asymptotic upper and lower bounds on the generalization error for this class under the assumption that the Bayes predictor $θ$ lies in an ellipsoid. Second, we establish a lower bound for the subclass of rotationally invariant linear prediction rules when the Bayes predictor is fixed. Our analysis highlights two fundamental contributions to the risk: (a) a variance-like term that captures the intrinsic dimensionality of the data; (b) the noiseless error, a term that arises specifically in the high-dimensional regime. These findings shed light on the role of structural assumptions in mitigating the curse of dimensionality.

Motivated by the emergence of large language models (LLMs) and the importance of democratizing their training, we propose GWTF, the first crash tolerant practical decentralized training framework for LLMs. Differently from existing distributed and federated training frameworks, GWTF enables the efficient collaborative training of a LLM on heterogeneous clients that volunteer their resources. In addition, GWTF addresses node churn, i.e., clients joining or leaving the system at any time, and network instabilities, i.e., network links becoming unstable or unreliable. The core of GWTF is a novel decentralized flow algorithm that finds the most effective routing that maximizes the number of microbatches trained with the lowest possible delay. We extensively evaluate GWTF on GPT-like and LLaMa-like models and compare it against the prior art. Our results indicate that GWTF reduces the training time by up to 45% in realistic and challenging scenarios that involve heterogeneous client nodes distributed over 10 different geographic locations with a high node churn rate.

Developing efficient multi-objective optimization methods to compute the Pareto set of optimal compromises between conflicting objectives remains a key challenge, especially for large-scale and expensive problems. To bridge this gap, we introduce SPREAD, a generative framework based on Denoising Diffusion Probabilistic Models (DDPMs). SPREAD first learns a conditional diffusion process over points sampled from the decision space and then, at each reverse diffusion step, refines candidates via a sampling scheme that uses an adaptive multiple gradient descent-inspired update for fast convergence alongside a Gaussian RBF-based repulsion term for diversity. Empirical results on multi-objective optimization benchmarks, including offline and Bayesian surrogate-based settings, show that SPREAD matches or exceeds leading baselines in efficiency, scalability, and Pareto front coverage.

Training deep neural networks in the presence of noisy labels and data heterogeneity is a major challenge. We introduce Lightweight Learnable Adaptive Weighting (LiLAW), a novel method that dynamically adjusts the loss weight of each training sample based on its evolving difficulty level, categorized as easy, moderate, or hard. Using only three learnable parameters, LiLAW adaptively prioritizes informative samples throughout training by updating these weights using a single mini-batch gradient descent step on the validation set after each training mini-batch, without requiring excessive hyperparameter tuning or a clean validation set. Extensive experiments across multiple general and medical imaging datasets, noise levels and types, loss functions, and architectures with and without pretraining demonstrate that LiLAW consistently enhances performance, even in high-noise environments. It is effective without heavy reliance on data augmentation or advanced regularization, highlighting its practicality. It offers a computationally efficient solution to boost model generalization and robustness in any neural network training setup.