  We introduce a novel optimization problem formulation that departs from the
conventional way of minimizing machine learning model loss as a black-box
function. Unlike traditional formulations, the proposed approach explicitly
incorporates an initially pre-trained model and random sketch operators,
allowing for sparsification of both the model and gradient during training. We
establish insightful properties of the proposed objective function and
highlight its connections to the standard formulation. Furthermore, we present
several variants of the Stochastic Gradient Descent (SGD) method adapted to the
new problem formulation, including SGD with general sampling, a distributed
version, and SGD with variance reduction techniques. We achieve tighter
convergence rates and relax assumptions, bridging the gap between theoretical
principles and practical applications, covering several important techniques
such as Dropout and Sparse training. This work presents promising opportunities
to enhance the theoretical understanding of model training through a
sparsification-aware optimization approach.
