Sign Operator for Coping with Heavy-Tailed Noise: High Probability Convergence Bounds with Extensions to Distributed Optimization and Comparison Oracle

The growing popularity of AI optimization problems involving severely corrupted data has increased the demand for methods capable of handling heavy-tailed noise, i.e., noise with bounded $\kappa$-th moment, $\kappa \in (1,2]$. For the widely used clipping technique, effectiveness heavily depends on the careful tuning of clipping levels throughout training. In this paper, we demonstrate that using only the sign of the input, without introducing additional hyperparameters, is sufficient to cope with heavy-tailed noise effectively. For smooth non-convex functions, we prove that SignSGD achieves optimal sample complexity $\tilde{O}\left(\varepsilon^{-\frac{3\kappa - 2}{\kappa - 1}}\right)$ with high probability for attaining an average gradient norm accuracy of $\varepsilon$. Under the assumption of symmetric noise, we use SignSGD with Majority Voting to extend this bound to the distributed optimization or reduce the sample complexity to $\tilde{O}(\varepsilon^{-4})$ in the case of a single worker with arbitrary parameters. Furthermore, we explore the application of the sign operator in zeroth-order optimization with an oracle that can only compare function values at two different points. We propose a novel method, MajorityVote-CompsSGD, and provide the first-known high-probability bound $\tilde{O}(\varepsilon^{-6})$ for the number of comparisons under symmetric noise assumption. Our theoretical findings are supported by the superior performance of sign-based methods in training Large Language Models.

Diffusion & Adversarial Schrödinger Bridges via Iterative Proportional Markovian Fitting

The Iterative Markovian Fitting (IMF) procedure based on iterative reciprocal and Markovian projections has recently been proposed as a powerful method for solving the Schr\"odinger Bridge problem. However, it has been observed that for the practical implementation of this procedure, it is crucial to alternate between fitting a forward and backward time diffusion at each iteration. Such implementation is thought to be a practical heuristic, which is required to stabilize training and obtain good results in applications such as unpaired domain translation. In our work, we show that this heuristic closely connects with the pioneer approaches for the Schr\"odinger Bridge based on the Iterative Proportional Fitting (IPF) procedure. Namely, we find that the practical implementation of IMF is, in fact, a combination of IMF and IPF procedures, and we call this combination the Iterative Proportional Markovian Fitting (IPMF) procedure. We show both theoretically and practically that this combined IPMF procedure can converge under more general settings, thus, showing that the IPMF procedure opens a door towards developing a unified framework for solving Schr\"odinger Bridge problems.

Optimal Flow Matching: Learning Straight Trajectories in Just One Step

Over the several recent years, there has been a boom in development of flow matching methods for generative modeling. One intriguing property pursued by the community is the ability to learn flows with straight trajectories which realize the optimal transport (OT) displacements. Straightness is crucial for fast integration of the learned flow's paths. Unfortunately, most existing flow straightening methods are based on non-trivial iterative procedures which accumulate the error during training or exploit heuristic minibatch OT approximations. To address this issue, we develop a novel optimal flow matching approach which recovers the straight OT displacement for the quadratic cost in just one flow matching step.

Implicitly normalized forecaster with clipping for linear and non-linear heavy-tailed multi-armed bandits

The Implicitly Normalized Forecaster (INF) algorithm is considered to be an optimal solution for adversarial multi-armed bandit (MAB) problems. However, most of the existing complexity results for INF rely on restrictive assumptions, such as bounded rewards. Recently, a related algorithm was proposed that works for both adversarial and stochastic heavy-tailed MAB settings. However, this algorithm fails to fully exploit the available data. In this paper, we propose a new version of INF called the Implicitly Normalized Forecaster with clipping (INF-clip) for MAB problems with heavy-tailed reward distributions. We establish convergence results under mild assumptions on the rewards distribution and demonstrate that INF-clip is optimal for linear heavy-tailed stochastic MAB problems and works well for non-linear ones. Furthermore, we show that INF-clip outperforms the best-of-both-worlds algorithm in cases where it is difficult to distinguish between different arms.