The recently introduced optimizer, Muon, has gained increasing attention due to its superior performance across a wide range of applications. However, its effectiveness in federated learning remains unexplored. To address this gap, this paper investigates the performance of Muon in the federated learning setting. Specifically, we propose a new algorithm, FedMuon, and establish its convergence rate for nonconvex problems. Our theoretical analysis reveals multiple favorable properties of FedMuon. In particular, due to its orthonormalized update direction, the learning rate of FedMuon is independent of problem-specific parameters, and, importantly, it can naturally accommodate heavy-tailed noise. The extensive experiments on a variety of neural network architectures validate the effectiveness of the proposed algorithm.

We study risk-sensitive reinforcement learning (RL) based on the entropic risk measure. Although existing works have established non-asymptotic regret guarantees for this problem, they leave open an exponential gap between the upper and lower bounds. We identify the deficiencies in existing algorithms and their analysis that result in such a gap.

In online (sequential) calibration, a forecaster predicts probability distributions over a finite outcome space $[d]$ over a sequence of $T$ days, with the goal of being calibrated. While asymptotically calibrated strategies are known to exist, they suffer from the curse of dimensionality: the best known algorithms require $\exp(d)$ days to achieve non-trivial calibration. In this work, we present the first asymptotically calibrated strategy that guarantees non-trivial calibration after a polynomial number of rounds. Specifically, for any desired accuracy $\epsilon > 0$, our forecaster becomes $\epsilon$-calibrated after $T = d^{O(1/\epsilon^2)}$ days. We complement this result with a lower bound, proving that at least $T = d^{\Omega(\log(1/\epsilon))}$ rounds are necessary to achieve $\epsilon$-calibration. Our results resolve the open questions posed by [Abernethy-Mannor'11, Hazan-Kakade'12]. Our algorithm is inspired by recent breakthroughs in swap regret minimization [Peng-Rubinstein'24, Dagan et al.'24]. Despite its strong theoretical guarantees, the approach is remarkably simple and intuitive: it randomly selects among a set of sub-forecasters, each of which predicts the empirical outcome frequency over recent time windows.

We propose a new framework for algorithmic stability in the context of multiclass classification. In practice, classification algorithms often operate by first assigning a continuous score (for instance, an estimated probability) to each possible label, then taking the maximizer -- i.e., selecting the class that has the highest score. A drawback of this type of approach is that it is inherently unstable, meaning that it is very sensitive to slight perturbations of the training data, since taking the maximizer is discontinuous. Motivated by this challenge, we propose a pipeline for constructing stable classifiers from data, using bagging (i.e., resampling and averaging) to produce stable continuous scores, and then using a stable relaxation of argmax, which we call the "inflated argmax," to convert these scores to a set of candidate labels. The resulting stability guarantee places no distributional assumptions on the data, does not depend on the number of classes or dimensionality of the covariates, and holds for any base classifier. Using a common benchmark data set, we demonstrate that the inflated argmax provides necessary protection against unstable classifiers, without loss of accuracy.