### A Method for Animating Children’s Drawings of the Human Figure

Harrison Jesse Smith, Qingyuan Zheng, Yifei Li, Somya Jain, Jessica K. Hodgins

Conference on Learning Theory (COLT)

Gaussian processes (GP) are a popular Bayesian approach for the optimization of black-box functions. Despite their effectiveness in simple problems, GP-based algorithms hardly scale to complex high-dimensional functions, as their per-iteration time and space cost is at least *quadratic* in the number of dimensions *d* and iterations *t*. Given a set of *A* alternative to choose from, the overall runtime *O*(*t*^{3}*A*) quickly becomes prohibitive. In this paper, we introduce BKB (*budgeted kernelized bandit*), a novel approximate GP algorithm for optimization under bandit feedback that achieves near-optimal regret (and hence near-optimal convergence rate) with near-constant per-iteration complexity and no assumption on the input space or covariance of the GP.

Combining a kernelized linear bandit algorithm (GP-UCB) with randomized matrix sketching technique (i.e., leverage score sampling), we prove that selecting inducing points based on their posterior variance gives an accurate low-rank approximation of the GP, preserving variance estimates and confidence intervals. As a consequence, BKB does not suffer from variance starvation, an important problem faced by many previous sparse GP approximations. Moreover, we show that our procedure selects at most *Õ*(*d*_{eff}) points, where deff is the effective dimension of the explored space, which is typically much smaller than both d and t. This greatly reduces the dimensionality of the problem, thus leading to a *O*(*T**Ad*^{2}_{eff}) runtime and *O*(*Ad*_{eff}) space complexity.

Harrison Jesse Smith, Qingyuan Zheng, Yifei Li, Somya Jain, Jessica K. Hodgins

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