  We extend the Approximate-Proximal Point (aProx) family of model-based
methods for solving stochastic convex optimization problems, including
stochastic subgradient, proximal point, and bundle methods, to the minibatch
and accelerated setting. To do so, we propose specific model-based algorithms
and an acceleration scheme for which we provide non-asymptotic convergence
guarantees, which are order-optimal in all problem-dependent constants and
provide linear speedup in minibatch size, while maintaining the desirable
robustness traits (e.g. to stepsize) of the aProx family. Additionally, we show
improved convergence rates and matching lower bounds identifying new
fundamental constants for "interpolation" problems, whose importance in
statistical machine learning is growing; this, for example, gives a
parallelization strategy for alternating projections. We corroborate our
theoretical results with empirical testing to demonstrate the gains accurate
modeling, acceleration, and minibatching provide.
