Relative to the large literature on upper bounds on complexity of convex optimization, lesser attention has been paid to the fundamental hardness of these problems. Recent years have seen a surge in optimization methods tailored to sparse optimization problems. In this paper, we study the complexity of stochastic convex optimization in an oracle model of computation, when the objective is optimized at a sparse vector in a high dimensional space. Our result is matched by an appropriately tuned method of mirror descent, establishing the minimiax optimality of the result.