Approximate Vanishing Ideal Computations at Scale

The approximate vanishing ideal of a set of points $X = \{\mathbf{x}_1, \ldots, \mathbf{x}_m\}\subseteq [0,1]^n$ is the set of polynomials that approximately evaluate to $0$ over all points $\mathbf{x} \in X$ and admits an efficient representation by a finite set of polynomials called generators. Algorithms that construct this set of generators are extensively studied but ultimately find little practical application because their computational complexities are thought to be superlinear in the number of samples $m$. In this paper, we focus on scaling up the Oracle Approximate Vanishing Ideal algorithm (OAVI), one of the most powerful of these methods. We prove that the computational complexity of OAVI is not superlinear but linear in the number of samples $m$ and polynomial in the number of features $n$, making OAVI an attractive preprocessing technique for large-scale machine learning. To further accelerate OAVI's training time, we propose two changes: First, as the name suggests, OAVI makes repeated oracle calls to convex solvers throughout its execution. By replacing the Pairwise Conditional Gradients algorithm, one of the standard solvers used in OAVI, with the faster Blended Pairwise Conditional Gradients algorithm, we illustrate how OAVI directly benefits from advancements in the study of convex solvers. Second, we propose Inverse Hessian Boosting (IHB): IHB exploits the fact that OAVI repeatedly solves quadratic convex optimization problems that differ only by very little and whose solutions can be written in closed form using inverse Hessian information. By efficiently updating the inverse of the Hessian matrix, the convex optimization problems can be solved almost instantly, accelerating OAVI's training time by up to multiple orders of magnitude. We complement our theoretical analysis with extensive numerical experiments on data sets whose sample numbers are in the millions.