piCholesky: Polynomial Interpolation of Multiple Cholesky Factors for Efficient Approximate Cross-Validation

Abstract

The dominant cost in solving least-square problems using Newton's method is often that of factorizing the Hessian matrix over multiple values of the regularization parameter ($λ$). We propose an efficient way to interpolate the Cholesky factors of the Hessian matrix computed over a small set of $λ$ values. This approximation enables us to optimally minimize the hold-out error while incurring only a fraction of the cost compared to exact cross-validation. We provide a formal error bound for our approximation scheme and present solutions to a set of key implementation challenges that allow our approach to maximally exploit the compute power of modern architectures. We present a thorough empirical analysis over multiple datasets to show the effectiveness of our approach.