A complete characterization of statistical query learning with applications to evolvability

Abstract

Statistical query (SQ) learning model of Kearns is a natural restriction of the PAC learning model in which a learning algorithm is allowed to obtain estimates of statistical properties of the examples but cannot see the examples themselves (Kearns, 1998 [29]). We describe a new and simple characterization of the query complexity of learning in the SQ learning model. Unlike the previously known bounds on SQ learning (Blum, et al., 1994; Bshouty and Feldman, 2002; Yang, 2005; Balcázar, et al., 2007; Simon, 2007 [9,11,42,3,37]) our characterization preserves the accuracy and the efficiency of learning. The preservation of accuracy implies that our characterization gives the first characterization of SQ learning in the agnostic learning framework of Haussler (1992) [23] and Kearns, Schapire and Sellie (1994) [31]. The preservation of efficiency is achieved using a new boosting technique and allows us to derive a new approach to the design of evolution algorithms in Valiantʼs model of evolvability (Valiant, 2009 [40]). We use this approach to demonstrate the existence of a large class of monotone evolution algorithms based on square loss performance estimation. These results differ significantly from the few known evolution algorithms and give evidence that evolvability in Valiantʼs model is a more versatile phenomenon than there had been previous reason to suspect.