Regularized Top-$k$: A Bayesian Framework for Gradient Sparsification
This work addresses convergence issues in distributed machine learning training, offering a method to improve efficiency with gradient sparsification, though it appears incremental as it builds upon existing Top-k techniques.
The paper tackles the problem of error accumulation in gradient sparsification for distributed settings, which can deteriorate convergence, and proposes a novel algorithm called regularized Top-k (RegTop-k) that controls learning rate scaling; in experiments, RegTop-k converges to the global optimum in distributed linear regression at high compression ratios and outperforms Top-k in training ResNet-18 on CIFAR-10.
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 algorithm regularized Top-$k$ (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 numerical experiments. In distributed linear regression, it is observed that while Top-$k$ remains at a fixed distance from the global optimum, RegTop-$k$ converges to the global optimum at significantly higher compression ratios. We further demonstrate the generalization of this observation by employing RegTop-$k$ in distributed training of ResNet-18 on CIFAR-10, where it noticeably outperforms Top-$k$.