The importance of convexity in learning with squared loss

We show that if the closure of a function class under the metric induced by some probability distribution is not convex, then the sample complexity for agnostically learning with squared loss (using only hypotheses in )i s where is the probability of success and is the required accuracy. In comparison, if the class is convex and has finite pseudodimension, then the sample complexity is . If a nonconvex class has finite pseudodimension, then the sample complexity for agnostically learning the closure of the convex hull of ,i s . Hence, for agnostic learning, learning the convex hull provides better approximation capabilities with little sample complexity penalty.