  Variance reduction methods such as SVRG and SpiderBoost use a mixture of
large and small batch gradients to reduce the variance of stochastic gradients.
Compared to SGD, these methods require at least double the number of operations
per update to model parameters. To reduce the computational cost of these
methods, we introduce a new sparsity operator: The random-top-k operator. Our
operator reduces computational complexity by estimating gradient sparsity
exhibited in a variety of applications by combining the top-k operator and the
randomized coordinate descent operator. With this operator, large batch
gradients offer an extra benefit beyond variance reduction: A reliable estimate
of gradient sparsity. Theoretically, our algorithm is at least as good as the
best algorithm (SpiderBoost), and further excels in performance whenever the
random-top-k operator captures gradient sparsity. Empirically, our algorithm
consistently outperforms SpiderBoost using various models on various tasks
including image classification, natural language processing, and sparse matrix
factorization. We also provide empirical evidence to support the intuition
behind our algorithm via a simple gradient entropy computation, which serves to
quantify gradient sparsity at every iteration.
