Distributed Statistical Machine Learning in Adversarial Settings: Byzantine Gradient Descent (1705.05491v2)

Abstract: We consider the problem of distributed statistical machine learning in adversarial settings, where some unknown and time-varying subset of working machines may be compromised and behave arbitrarily to prevent an accurate model from being learned. This setting captures the potential adversarial attacks faced by Federated Learning -- a modern machine learning paradigm that is proposed by Google researchers and has been intensively studied for ensuring user privacy. Formally, we focus on a distributed system consisting of a parameter server and $m$ working machines. Each working machine keeps $N/m$ data samples, where $N$ is the total number of samples. The goal is to collectively learn the underlying true model parameter of dimension $d$. In classical batch gradient descent methods, the gradients reported to the server by the working machines are aggregated via simple averaging, which is vulnerable to a single Byzantine failure. In this paper, we propose a Byzantine gradient descent method based on the geometric median of means of the gradients. We show that our method can tolerate $q \le (m-1)/2$ Byzantine failures, and the parameter estimate converges in $O(\log N)$ rounds with an estimation error of $\sqrt{d(2q+1)/N}$, hence approaching the optimal error rate $\sqrt{d/N}$ in the centralized and failure-free setting. The total computational complexity of our algorithm is of $O((Nd/m) \log N)$ at each working machine and $O(md + kd \log^3 N)$ at the central server, and the total communication cost is of $O(m d \log N)$. We further provide an application of our general results to the linear regression problem. A key challenge arises in the above problem is that Byzantine failures create arbitrary and unspecified dependency among the iterations and the aggregated gradients. We prove that the aggregated gradient converges uniformly to the true gradient function.

• The paper proposes a Byzantine-resilient gradient descent algorithm using the geometric median to mitigate adversarial machine behavior in distributed learning.
• It guarantees logarithmic convergence with bounded estimation errors even when a significant fraction of workers are compromised.
• The method is efficient in computation and communication, making it suitable for Federated Learning and other decentralized applications.

Distributed Statistical Machine Learning in Adversarial Settings: Byzantine Gradient Descent

This paper addresses the challenge of distributed statistical learning in environments susceptible to adversarial attacks, specifically focusing on the scenario known as Byzantine faults. The paper is situated within the context of decentralized systems like Federated Learning, where security concerns arise from the potential for recruited devices to behave maliciously.

Problem Setting and Motivation

The problem under consideration involves a parameter server and multiple working machines, where each machine holds a subset of the total dataset. The core challenge arises from Byzantine failures, where a fraction of machines can be compromised by an adversary and act against the system's interests. These faulty machines have full knowledge of the system but cannot alter the local data. This work aims to develop robust algorithms for effective learning despite such adversities.

Proposed Solution

The authors introduce a variant of the gradient descent algorithm that leverages the geometric median of means of gradients to tackle Byzantine failures. This robust method can endure up to $2(1+\epsilon)q \leq m$ Byzantine failures, with $\epsilon > 0$. The proposed method guarantees convergence within $O(\log N)$ rounds and results in an estimation error scaling as $\max\{\sqrt{\frac{dq}{N}}, \sqrt{\frac{d}{N}}\}$. Although this error rate includes an additional factor relative to the failure-free setting, the algorithm remains computationally efficient with complexity $O((Nd/m) \log N)$ at each machine and $O(md + qd \log^3 N)$ at the central server, with total communication cost $O(md \log N)$.

Implications and Results

The robustness of the proposed gradient descent method is particularly suited to settings where data is scarce, and failures are expected, making it applicable to Federated Learning scenarios. An analysis shows that the algorithm manages the arbitrary behavior of Byzantine nodes by ensuring the aggregate gradient converges uniformly to the true gradient. Key results include theoretical guarantees of exponential convergence and a bounded estimation error in adversarial conditions.

The authors extend their results to linear regression as a specific application, showcasing that the proposed algorithm performs robustly in estimating model parameters under adversarial influence.

Critical Analysis and Future Directions

The main contribution of this paper is the establishment of a practical, robust distributed learning algorithm that tolerates a substantial number of Byzantine faults while maintaining communication efficiency. The trade-off between fault tolerance and statistical accuracy is an area that might be further refined through improved algorithms or refined analysis.

The authors outline opportunities for future investigations: enhancing privacy in Federated Learning, adapting methods to asynchronous environments, and exploring alternative aggregation rules for gradient selection that might simplify the approach against weaker adversaries.

Overall, the paper provides valuable insights and advancements in distributed learning, particularly for applications requiring resilience to malicious disruptions, and sets the stage for further exploration in the balance between fault tolerance, efficiency, and accuracy.