  Stochastic model-based methods have received increasing attention lately due
to their appealing robustness to the stepsize selection and provable efficiency
guarantee. We make two important extensions for improving model-based methods
on stochastic weakly convex optimization. First, we propose new minibatch
model-based methods by involving a set of samples to approximate the model
function in each iteration. For the first time, we show that stochastic
algorithms achieve linear speedup over the batch size even for non-smooth and
non-convex (particularly, weakly convex) problems. To this end, we develop a
novel sensitivity analysis of the proximal mapping involved in each algorithm
iteration. Our analysis appears to be of independent interests in more general
settings. Second, motivated by the success of momentum stochastic gradient
descent, we propose a new stochastic extrapolated model-based method, greatly
extending the classic Polyak momentum technique to a wider class of stochastic
algorithms for weakly convex optimization. The rate of convergence to some
natural stationarity condition is established over a fairly flexible range of
extrapolation terms.
  While mainly focusing on weakly convex optimization, we also extend our work
to convex optimization. We apply the minibatch and extrapolated model-based
methods to stochastic convex optimization, for which we provide a new
complexity bound and promising linear speedup in batch size. Moreover, an
accelerated model-based method based on Nesterov's momentum is presented, for
which we establish an optimal complexity bound for reaching optimality.
