Data Encoding for Byzantine-Resilient Distributed Optimization

Abstract

We study distributed optimization in the presence of Byzantine adversaries, where both data and computation are distributed among $m$ worker machines, $t$ of which may be corrupt. The compromised nodes may collaboratively and arbitrarily deviate from their pre-specified programs, and a designated (master) node iteratively computes the model/parameter vector for generalized linear models . In this work, we primarily focus on two iterative algorithms: Proximal Gradient Descent (PGD) and Coordinate Descent (CD). Gradient descent (GD) is a special case of these algorithms. PGD is typically used in the data-parallel setting, where data is partitioned across different samples, whereas, CD is used in the model-parallelism setting, where data is partitioned across the parameter space. At the core of our solutions to both these algorithms is a method for Byzantine-resilient matrix-vector (MV) multiplication; and for that, we propose a method based on data encoding and error correction over real numbers to combat adversarial attacks. We can tolerate up to $t\leq \lfloor \frac {m-1}{2}\rfloor $ corrupt worker nodes, which is information-theoretically optimal. We give deterministic guarantees, and our method does not assume any probability distribution on the data. We develop a sparse encoding scheme which enables computationally efficient data encoding and decoding. We demonstrate a trade-off between the corruption threshold and the resource requirements (storage, computational, and communication complexity). As an example, for $t\leq \frac {m}{3}$ , our scheme incurs only a constant overhead on these resources, over that required by the plain distributed PGD/CD algorithms which provide no adversarial protection. To the best of our knowledge, ours is the first paper that connects MV multiplication with CD and designs a specific encoding matrix for MV multiplication whose structure we can leverage to make CD secure against adversarial attacks. Our encoding scheme extends efficiently to (i) the data streaming model, in which data samples come in an online fashion and are encoded as they arrive, and (ii) making stochastic gradient descent (SGD) Byzantine-resilient. In the end, we give experimental results to show the efficacy of our proposed schemes.