  Sparsity promoting regularizers are widely used to impose low-complexity
structure (e.g. l1-norm for sparsity) to the regression coefficients of
supervised learning. In the realm of deterministic optimization, the sequence
generated by iterative algorithms (such as proximal gradient descent) exhibit
"finite activity identification", namely, they can identify the low-complexity
structure in a finite number of iterations. However, most online algorithms
(such as proximal stochastic gradient descent) do not have the property owing
to the vanishing step-size and non-vanishing variance. In this paper, by
combining with a screening rule, we show how to eliminate useless features of
the iterates generated by online algorithms, and thereby enforce finite
activity identification. One consequence is that when combined with any
convergent online algorithm, sparsity properties imposed by the regularizer can
be exploited for computational gains. Numerically, significant acceleration can
be obtained.
