Multivariate Convex Regression at Scale
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We present new large-scale algorithms for fitting a multivariate convex regression function to n samples in d dimensions—a key problem in shape constrained nonparametric regression with widespread applications in engineering and the applied sciences. The infinite-dimensional learning task can be expressed via a convex quadratic program (QP) with O(nd) decision variables and O(n^2) constraints. While instances with n in the lower thousands can be addressed with current algorithms within reasonable runtimes, solving larger problems (e.g., n≈ 10^4 or 10^5) are computationally challenging. To this end, we present an active set type algorithm on the Lagrangian dual (of a perturbation) of the primal QP. For computational scalability, we perform approximate optimization of the reduced sub-problems; and propose a variety of randomized augmentation rules for expanding the active set. Although the dual is not strongly convex, we present a novel linear convergence rate of our algorithm on the dual. We demonstrate that our framework can solve instances of the convex regression problem with n=10^5 and d=10—a QP with 10 billion variables—within minutes; and offers significant computational gains (e.g., in terms of memory and runtime) compared to current algorithms.
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