We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let @math be a Boolean function and consider a function @math obtained by applying @math to conjunctions of possibly overlapping subsets of @math variables. If @math has quantum query complexity @math , we give an algorithm for evaluating @math using @math quantum queries. This improves on the bound of @math that follows by treating each conjunction independently, and is tight for worst-case choices of @math . Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of @math . By recursively applying our composition theorems, we obtain a nearly optimal @math upper bound on the quantum query complexity and approximate degree of linear-size depth- @math AC @math circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC @math circuits. We also show that any substantially faster learning algorithm will require fundamentally new techniques.
 We give a computationally efficient algorithm that learns (under distributional assumptions) a halfspace in the difficult agnostic framework of Kearns, Schapire, and Sellie [Mach. Learn., 17 (1994), pp. 115-141], where a learner is given access to a distribution on labelled examples but where the labelling may be arbitrary (similar to malicious noise). It constructs a hypothesis whose error rate on future examples is within an additive @math of the optimal halfspace, in time poly @math for any constant @math , for the uniform distribution over @math or unit sphere in @math as well as any log-concave distribution in @math . It also agnostically learns Boolean disjunctions in time @math with respect to any distribution. Our algorithm, which performs @math polynomial regression, is a natural noise-tolerant arbitrary-distribution generalization of the well-known “low-degree” Fourier algorithm of Linial, Mansour, and Nisan. We observe that significant improvements on the running time of our algorithm would yield the fastest known algorithm for learning parity with noise, a challenging open problem in computational learning theory. Humans appear to be able to learn new concepts without needing to be programmed explicitly in any conventional sense. In this paper we regard learning as the phenomenon of knowledge acquisition in the absence of explicit programming. We give a precise methodology for studying this phenomenon from a computational viewpoint. It consists of choosing an appropriate information gathering mechanism, the learning protocol, and exploring the class of concepts that can be learnt using it in a reasonable (polynomial) number of steps. We find that inherent algorithmic complexity appears to set serious limits to the range of concepts that can be so learnt. The methodology and results suggest concrete principles for designing realistic learning systems. We consider efficient agnostic learning of linear combinations of basis functions when the sum of absolute values of the weights of the linear combinations is bounded. With the quadratic loss function, we show that the class of linear combinations of a set of basis functions is efficiently agnostically learnable if and only if the class of basis functions is efficiently agnostically learnable. We also show that the sample complexity for learning the linear combinations grows polynomially if and only if a combinatorial property of the class of basis functions, called the fat-shattering function, grows at most polynomially. We also relate the problem to agnostic learning of f0 1g-valued function classes by showing that if a class of f0 1g-valued functions is efficiently agnostically learnable (using the same function class) with the discrete loss function, then the class of linear combinations of functions from the class is efficiently agnostically learnable with the quadratic loss function.