arXiv subjects

Jason Akoun

Publications and source records attributed to Jason Akoun.

2 recordsLinked to original sources

arXiv2022

SignSGD: Fault-Tolerance to Blind and Byzantine Adversaries

Distributed learning has become a necessity for training ever-growing models by sharing calculation among several devices. However, some of the devices can be faulty, deliberately or not, preventing the proper convergence. As a matter of fact, the baseline distributed SGD algorithm does not converge in the presence of one Byzantine adversary. In this article we focus on the more robust SignSGD algorithm derived from SGD. We provide an upper bound for the convergence rate of SignSGD proving that this new version is robust to Byzantine adversaries. We implemented SignSGD along with Byzantine strategies attempting to crush the learning process. Therefore, we provide empirical observations from our experiments to support our theory. Our code is available on GitHub https://github.com/jasonakoun/signsgd-fault-tolerance and our experiments are reproducible by using the provided parameters.

cs.LG↗

arXiv2020

On a New Formula for Arithmetic Functions

In this paper we establish a new formula for the arithmetic functions that verify $ f(n) = \sum_{d|n} g(d)$ where $g$ is also an arithmetic function. We prove the following identity, $$\forall n \in \mathbb{N}^*, \ \ \ f(n) = \sum_{k=1}^n \mu \left(\frac{k}{(n,k)}\right) \frac {\varphi(k)}{\varphi\left(\frac{k}{(n,k)}\right)} \sum_{l=1}^{\left\lfloor\frac{n}{k}\right\rfloor} \frac{g(kl)}{kl} $$ where $\varphi$ and $\mu$ are respectively Euler's and Mobius' functions and (.,.) is the GCD. First, we will compare this expression with other known expressions for arithmetic functions and pinpoint its advantages. Then, we will prove the identity using exponential sums' proprieties. Finally we will present some applications with well known functions such as $d$ and $\sigma$ which are respectively the number of divisors function and the sum of divisors function.

math.GM↗