Regularized Top-$ k $: A Bayesian Framework for Gradient Sparsification
Ali Bereyhi, Ben Liang, Gary Boudreau, Ali Afana · IEEE Transactions on Signal Processing · 2025
Error accumulation is effective for gradient sparsification in distributed settings: initially-unselected gradient entries are eventually selected as their accumulated error exceeds a certain level. The accumulation essentially behaves as a scaling of the learning rate for the selected entries. Although this property prevents the slow-down of lateral movements in distributed gradient descent, it can deteriorate convergence in some settings. This work proposes a novel sparsification scheme that controls the learning rate scaling of error accumulation. The development of this scheme follows two major steps: first, gradient sparsification is formulated as an inverse probability (inference) problem, and the Bayesian optimal sparsification mask is derived as a maximum-a-posteriori estimator. Using the prior distribution inherited from TOP-k, we derive a new sparsification algorithm which can be interpreted as a regularized form of TOP-k. We call this algorithmregularizedTOP-k(REGTOP-k). It utilizes past aggregated gradients to evaluate posterior statistics of the next aggregation. It then prioritizes the local accumulated gradient entries based on these posterior statistics. We validate our derivation through various numerical experiments. In distributed linear regression, it is observed that while TOP-kremains at a fixed distance from the global optimum, REGTOP-kconverges to the global optimum at significantly higher compression ratios. We further demonstrate the generalization of this observation by employing REGTOP-kin distributed training of ResNet-18 on CIFAR-10, as well as fine-tuning of multiple computer vision models on the ImageNette dataset. Our numerical results confirm that as the compression ratio increases, REGTOP-ksparsification noticeably outperforms TOP-k.