Efficient Unbiased Sparsification
This provides a theoretical foundation for efficient compression in applications like federated learning, though it appears incremental as it extends known concepts to specific divergence classes.
The paper tackles the problem of unbiased sparsification, which compresses vectors without bias while minimizing a divergence measure, and finds that for permutation-invariant divergences, the optimal sparsifications are robust across different divergence functions like squared Euclidean distance and Kullback-Leibler divergence.
An unbiased $m$-sparsification of a vector $p\in \mathbb{R}^n$ is a random vector $Q\in \mathbb{R}^n$ with mean $p$ that has at most $m<n$ nonzero coordinates. Unbiased sparsification compresses the original vector without introducing bias; it arises in various contexts, such as in federated learning and sampling sparse probability distributions. Ideally, unbiased sparsification should also minimize the expected value of a divergence function $\mathsf{Div}(Q,p)$ that measures how far away $Q$ is from the original $p$. If $Q$ is optimal in this sense, then we call it efficient. Our main results describe efficient unbiased sparsifications for divergences that are either permutation-invariant or additively separable. Surprisingly, the characterization for permutation-invariant divergences is robust to the choice of divergence function, in the sense that our class of optimal $Q$ for squared Euclidean distance coincides with our class of optimal $Q$ for Kullback-Leibler divergence, or indeed any of a wide variety of divergences.