[Paper Review] Distributed Statistical Machine Learning in Adversarial Settings: Byzantine Gradient Descent

Yudong Chen, Lili Su+1 more|arXiv (Cornell University)|May 16, 2017

Stochastic Gradient Optimization Techniques24 references142 citations

TL;DR

The paper introduces Byzantine Gradient Descent, a robust distributed learning algorithm that tolerates up to ~2(1+ε)q Byzantine workers and achieves exponential convergence with error ∼max{√(dq/N), √(d/N)} per log N rounds.

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.

Motivation & Objective

• Motivate distributed statistical learning in the presence of adversarial (Byzantine) faults as in Federated Learning.
• Develop a robust gradient aggregation method that tolerates Byzantine faults.
• Prove convergence guarantees and characterize estimation error under Byzantine faults.
• Analyze computational and communication costs of the proposed method.
• Provide an application to linear regression to illustrate the approach.

Proposed method

• Propose Byzantine Gradient Descent where the server aggregates gradients using a robust plan based on batch means and the geometric median.
• Split m working machines into k batches and compute batch means of gradients.
• Compute the geometric median of these k batch means to form the aggregated gradient for the update.
• Use a gradient descent step with step size η chosen as η = L/(2M^2) under strong convexity and Lipschitz gradient assumptions.
• Provide a formal convergence theorem showing exponential convergence in log N rounds with error bound growing with √(dq/N) and √(d/N).
• Analyze computational cost O((Nd/m) log N) per worker and O(md + qd log^3 N) at the parameter server, with communication cost O(md log N).

Experimental results

Research questions

• RQ1 Can a distributed learning algorithm tolerate Byzantine (arbitrary) failures while using local data on each worker?
• RQ2 What aggregation rule can robustly combine gradients to mitigate Byzantine influence without destroying convergence?
• RQ3 What are the convergence rates and statistical error bounds under Byzantine faults for distributed learning?
• RQ4 How should the system parameters (k, q, m, N, d) be chosen to balance tolerance to faults and statistical accuracy?
• RQ5 How does the method apply to concrete problems like linear regression?

Key findings

• The proposed Byzantine Gradient Descent method tolerates up to 2(1+ε)q ≤ m Byzantine failures for any fixed ε>0.
• The estimator converges in O(log N) rounds with an error bound of max{√(dq/N), √(d/N)}.
• The minimax-optimal rate √(d/N) is achievable up to a factor of √q in the Byzantine setting.
• Total computational cost is O((Nd/m) log N) per worker and O(md + qd log^3 N) at the parameter server, with communication cost O(md log N).
• For linear regression, the framework demonstrates applicability and robustness to adversarial workers.