Scrambled Objects for Least-Squares Regression

Part of Advances in Neural Information Processing Systems 23 (NIPS 2010)

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Odalric Maillard, Rémi Munos


<p>We consider least-squares regression using a randomly generated subspace G<em>P\subset F of finite dimension P, where F is a function space of infinite dimension, e.g.~L</em>2([0,1]^d). G<em>P is defined as the span of P random features that are linear combinations of the basis functions of F weighted by random Gaussian i.i.d.~coefficients. In particular, we consider multi-resolution random combinations at all scales of a given mother function, such as a hat function or a wavelet. In this latter case, the resulting Gaussian objects are called {\em scrambled wavelets} and we show that they enable to approximate functions in Sobolev spaces H^s([0,1]^d). As a result, given N data, the least-squares estimate \hat g built from P scrambled wavelets has excess risk ||f^* - \hat g||</em>\P^2 = O(||f^<em>||^2_{H^s([0,1]^d)}(\log N)/P + P(\log N )/N) for target functions f^</em>\in H^s([0,1]^d) of smoothness order s>d/2. An interesting aspect of the resulting bounds is that they do not depend on the distribution \P from which the data are generated, which is important in a statistical regression setting considered here. Randomization enables to adapt to any possible distribution. We conclude by describing an efficient numerical implementation using lazy expansions with numerical complexity \tilde O(2^d N^{3/2}\log N + N^2), where d is the dimension of the input space.</p>