vqSGD: Vector Quantized Stochastic Gradient Descent

In this work, we present a family of vector quantization schemes vqSGD (Vector-Quantized Stochastic Gradient Descent) that provide an asymptotic reduction in the communication cost with convergence guarantees in first-order distributed optimization. In the process we derive the following fundamental information theoretic fact: ⊖(_R^d₂ ) bits are necessary and sufficient (up to an additive O(log d) term) to describe an unbiased estimator ĝ(g) for any g in the ddimensional unit sphere, under the constraint that kĝ(g)k₂ ≤ R almost surely. In particular, we consider a randomized scheme based on the convex hull of a point set, that returns an unbiased estimator of a d-dimensional gradient vector with almost surely bounded norm. We provide multiple efficient instances of our scheme, that are near optimal, and require only o(d) bits of communication at the expense of tolerable increase in error. The instances of our quantization scheme are obtained using the properties of binary error-correcting codes and provide a smooth tradeoff between the communication and the estimation error of quantization. Furthermore, we show that vqSGD also offers some automatic privacy guarantees.