The paper introduces a general method for creating data‑adaptive, anytime‑valid envelopes that can be applied to a wide range of iterative stochastic algorithms. By reducing error analysis to a single requirement—time‑uniform concentration for predictably weighted sums of noise—the authors provide off‑the‑shelf boundary constructions using mixtures, stitching, and betting martingales. These envelopes are shown to be applicable to algorithms such as stochastic gradient descent, temporal difference learning, and Bures‑Wasserstein SGD, and they either match or improve upon existing bounds, with practical implementation details and numerical validation.
arXiv:2608. 25551v1 Announce Type: new Abstract: Stochastic gradient descent (SGD) is typically analyzed at a deterministic horizon chosen before the algorithm is run, even though practical stopping decisions are made adaptively by inspecting the evolving trajectory.
arXiv:2610.19444v1 Announce Type: cross Abstract: Random reshuffling stochastic gradient descent (RR-SGD) is widely used for finite-sum optimization in machine learning. For convex $L$-smooth compone...
arXiv:2505.20817v3 Announce Type: replace-cross Abstract: Gradient clipping is widely used in language-model training to control heavy-tailed gradient noise and can improve convergence guarantees ove...
The paper introduces Batched SGD, a variant that groups online samples into epochs and performs a single update per epoch using a low‑variance gradient estimate. This batching approach allows a straightforward high‑probability analysis without restrictive assumptions or auxiliary sequences, yielding near‑optimal rates for both strongly convex and non‑convex objectives under standard smoothness and sub‑Gaussian noise conditions. The authors also extend the method to federated learning, providing the first high‑probability guarantees with logarithmic communication complexity, linear speedup in the number of agents, and robustness to data heterogeneity.
The paper establishes the first optimal last‑iterate convergence rates, in high probability, for Clipped‑SGD when the gradient noise is heavy‑tailed. It addresses the gap left by earlier results that required strong noise assumptions, such as exponentially decaying tails. By proving these rates, the authors provide theoretical support for the practical effectiveness of gradient clipping in stochastic convex optimization.
arXiv:2602. 01903v2 Announce Type: replace Abstract: This work studies online episodic tabular Markov decision processes (MDPs) with known transitions and develops best-of-both-worlds algorithms that achieve refined data-dependent regret bounds in the adversarial regime and variance-dependent regret bounds in the stochastic regime.
The paper introduces a contraction framework for stochastic operators that incorporates bootstrapping, where a variable is updated using a frozen copy as a target that is refreshed every $K$ steps. By modeling the sampled update as a stochastic operator, the authors derive a finite‑time bound for i.i.d. samples that applies to any target‑update period and does not require gradient structure or uniformly bounded sampling error. The framework shows that the iterates converge geometrically in root mean square to a ball around the fixed point, with the error floor scaling with the step size, and it generalizes existing deterministic and stochastic‑gradient bounds.
arXiv:2606. 03831v1 Announce Type: new Abstract: This paper investigates non-stationary online learning using the metric of interval regret, which requires an online algorithm to perform well over every time interval.
The paper investigates when it is worthwhile to pay for higher‑fidelity large language models (LLMs) in stochastic optimization tasks. It introduces two cost‑aware algorithmic frameworks that adaptively switch between mixed‑fidelity first‑order oracles, one for known bias and one for unknown bias, and demonstrates that these methods can reduce cumulative cost while maintaining accuracy. Experiments with synthetic data and LLM completions confirm that the adaptive policies skip expensive oracles and achieve comparable performance to fixed high‑fidelity approaches.
arXiv:2506.04192v4 Announce Type: replace-cross Abstract: Stochastic Frank-Wolfe is a classical optimization method for solving constrained optimization problems. On the other hand, recent optimizers...
arXiv:2608. 06182v1 Announce Type: cross Abstract: We study stochastic extragradient (SEG) methods for solving monotone variational inequality problems (VIPs) over a feasible set.
The paper studies how to allocate a fixed computational budget across the denoising steps of diffusion models to improve sample quality at deployment. It shows that the expected benefit of evaluating multiple candidates at a step can be decomposed into a step‑specific sensitivity and a universal sample‑size factor, and that the optimal allocation follows a water‑filling structure. Experiments demonstrate that this allocation achieves the same quality as a uniform strategy while reducing function evaluations by 20–50%.