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We obtain almost sure bounds for the weighted sum $\sum_{n \leq t} \frac{f(n)}{\sqrt{n}}$, where $f(n)$ is a Steinhaus random multiplicative function. Specifically, we obtain the bounds predicted by exponentiating the law of the iterated…

Number Theory · Mathematics 2025-11-10 Seth Hardy

In the context of bounding probability of small deviation, there are limited general tools. However, such bounds have been widely applied in graph theory and inventory management. We introduce a common approach to substantially sharpen such…

Optimization and Control · Mathematics 2020-03-09 Jiayi Guo , Simai He , Zi Ling , Yicheng Liu

In this paper, we obtain the Berry-Esseen bound for multivariate normal approximation for the Polyak-Ruppert averaged iterates of the linear stochastic approximation (LSA) algorithm with decreasing step size. Moreover, we prove the…

Machine Learning · Statistics 2025-02-04 Sergey Samsonov , Eric Moulines , Qi-Man Shao , Zhuo-Song Zhang , Alexey Naumov

We consider an \emph{approximate} version of the trace reconstruction problem, where the goal is to recover an unknown string $s\in\{0,1\}^n$ from $m$ traces (each trace is generated independently by passing $s$ through a probabilistic…

Data Structures and Algorithms · Computer Science 2021-07-21 Diptarka Chakraborty , Debarati Das , Robert Krauthgamer

We propose a linear contextual bandit algorithm with $O(\sqrt{dT\log T})$ regret bound, where $d$ is the dimension of contexts and $T$ isthe time horizon. Our proposed algorithm is equipped with a novel estimator in which exploration is…

Machine Learning · Statistics 2023-03-30 Wonyoung Kim , Myunghee Cho Paik , Min-hwan Oh

In this work, we study the phase estimation problem. We show an alternative, simpler and self-contained proof of query lower bounds. Technically, compared to the previous proofs [NW99, Bes05], our proof is considerably elementary.…

Quantum Physics · Physics 2023-04-06 Yao-Ting Lin

We consider the problem of prediction with expert advice for ``easy'' sequences. We show that a variant of NormalHedge enjoys a second-order $\epsilon$-quantile regret bound of $O\big(\sqrt{V_T \log(V_T/\epsilon)}\big) $ when $V_T > \log…

Machine Learning · Computer Science 2026-02-10 Yoav Freund , Nicholas J. A. Harvey , Victor S. Portella , Yabing Qi , Yu-Xiang Wang

This paper considers the problem of minimizing the time average of a controlled stochastic process subject to multiple time average constraints on other related processes. The probability distribution of the random events in the system is…

Optimization and Control · Mathematics 2016-12-20 Xiaohan Wei , Hao Yu , Michael J. Neely

Consider the classical problem of predicting the next bit in a sequence of bits. A standard performance measure is {\em regret} (loss in payoff) with respect to a set of experts. For example if we measure performance with respect to two…

Machine Learning · Computer Science 2013-04-30 Rina Panigrahy , Preyas Popat

For an irrational number $\alpha\in\mathbb{R}$ we consider its irrationality measure function $$ \psi_\alpha(x) = \min_{1\le q\le x,\, q\in\mathbb{Z}} \| q\alpha \|. $$ It is known for all irrational numbers $\alpha$ and $\beta$ satisfying…

Number Theory · Mathematics 2023-08-24 Viktoria Rudykh , Nikita Shulga

The push-sum based subgradient is an important method for distributed convex optimization over unbalanced directed graphs, which is known to converge at a rate of $O(\ln t/\sqrt{t})$. This paper shows that the subgradient-push algorithm…

Optimization and Control · Mathematics 2023-08-03 Yixuan Lin , Ji Liu

We consider linear two-time-scale stochastic approximation algorithms driven by martingale noise. Recent applications in machine learning motivate the need to understand finite-time error rates, but conventional stochastic approximation…

Machine Learning · Computer Science 2025-12-12 Seo Taek Kong , Sihan Zeng , Thinh T. Doan , R. Srikant

Two-time-scale stochastic approximation (SA) is an algorithm with coupled iterations which has found broad applications in reinforcement learning, optimization and game control. In this work, we derive mean squared error bounds for…

Machine Learning · Computer Science 2026-02-24 Siddharth Chandak

In this paper, we refine the Berry-Esseen bounds for the multivariate normal approximation of Polyak-Ruppert averaged iterates arising from the linear stochastic approximation (LSA) algorithm with decreasing step size. We consider the…

Machine Learning · Statistics 2025-10-15 Bogdan Butyrin , Eric Moulines , Alexey Naumov , Sergey Samsonov , Qi-Man Shao , Zhuo-Song Zhang

We provide tools to help automate the error analysis of algorithms that evaluate simple functions over the floating-point numbers. The aim is to obtain tight relative error bounds for these algorithms, expressed as a function of the unit…

Numerical Analysis · Mathematics 2024-05-07 Jean-Michel Muller , Bruno Salvy

Computing ratios of normalizing constants plays an important role in statistical modeling. Two important examples are hypothesis testing in latent variables models, and model comparison in Bayesian statistics. In both examples, the…

Applications · Statistics 2024-08-26 Tom Guédon , Charlotte Baey , Estelle Kuhn

Writing for a general mathematical audience, we provide elementary upper and lower bounds on the growth (as a function of N) of the sum \sum_{n=1}^N (-1)^{\floor{n x}} for various fixed x. For example, if x is a quadratic irrational, then…

Number Theory · Mathematics 2007-05-23 Kevin O'Bryant , Bruce Reznick , Monika Serbinowska

We develop a new reduction that converts any online convex optimization algorithm suffering $O(\sqrt{T})$ regret into an $\epsilon$-differentially private stochastic convex optimization algorithm with the optimal convergence rate $\tilde…

Machine Learning · Computer Science 2022-10-14 Qinzi Zhang , Hoang Tran , Ashok Cutkosky

In this work, we provide a non-asymptotic convergence analysis of projected gradient descent for physics-informed neural networks for the Poisson equation. Under suitable assumptions, we show that the optimization error can be bounded by…

Optimization and Control · Mathematics 2025-05-13 Jonas Nießen , Johannes Müller

The estimation of a probability p from repeated Bernoulli trials is considered in this paper. A sequential approach is followed, using a simple stopping rule. A closed-form expression and an upper bound are obtained for the mean absolute…

Statistics Theory · Mathematics 2018-12-19 Luis Mendo