No abstract

We present polynomial families complete for the well-studied algebraic complexity classes VF, VBP, VP, and VNP. The polynomial families are based on the homomorphism polynomials studied in the recent works of Durand et al. (2014) and Mahajan et al. (2018). We consider three different variants of graph homomorphisms, namely injective homomorphisms , directed homomorphisms , and injective directed homomorphisms , and obtain polynomial families complete for VF, VBP, VP, and VNP under each one of these. The polynomial families have the following properties: • The polynomial families complete for VF, VBP, and VP are model independent, i.e., they do not use a particular instance of a formula, algebraic branching programs, or circuit for characterising VF, VBP, or VP, respectively. • All the polynomial families are hard under p -projections.

We consider stochastic approximation for the least squares regression problem in the non-strongly convex setting. We present the first practical algorithm that achieves the optimal prediction error rates in terms of dependence on the noise of the problem, as O(d/t) while accelerating the forgetting of the initial conditions to O(d/t 2 ). Our new algorithm is based on a simple modification of the accelerated gradient descent. We provide convergence results for both the averaged and the last iterate of the algorithm. In order to describe the tightness of these new bounds, we present a matching lower bound in the noiseless setting and thus show the optimality of our algorithm.