# Byzantine Stochastic Gradient Descent

## TL;DR

This paper introduces a Byzantine-resilient stochastic gradient descent method for distributed optimization, capable of tolerating adversarial machine failures and achieving near-optimal convergence rates.

## Contribution

It proposes a new Byzantine-tolerant SGD algorithm with proven optimality in both sample and time complexity.

## Key findings

- Achieves $ ilde{O}(1/( ext{epsilon}^2 m) + ext{alpha}^2/ ext{epsilon}^2)$ convergence iterations.
- Tolerates an $ ext{alpha}$-fraction of Byzantine machines in distributed setting.
- Proves lower bounds showing near-optimality of the proposed method.

## Abstract

This paper studies the problem of distributed stochastic optimization in an adversarial setting where, out of the $m$ machines which allegedly compute stochastic gradients every iteration, an $\alpha$-fraction are Byzantine, and can behave arbitrarily and adversarially. Our main result is a variant of stochastic gradient descent (SGD) which finds $\varepsilon$-approximate minimizers of convex functions in $T = \tilde{O}\big( \frac{1}{\varepsilon^2 m} + \frac{\alpha^2}{\varepsilon^2} \big)$ iterations. In contrast, traditional mini-batch SGD needs $T = O\big( \frac{1}{\varepsilon^2 m} \big)$ iterations, but cannot tolerate Byzantine failures. Further, we provide a lower bound showing that, up to logarithmic factors, our algorithm is information-theoretically optimal both in terms of sampling complexity and time complexity.