Time-space lower bounds for two-pass learning

A line of recent works showed that for a large class of learning problems, any learning algorithm requires either super-linear memory size or a super-polynomial number of samples [11, 7, 12, 9, 2, 5]. For example, any algorithm for learning parities of size n requires either a memory of size Ω(n2) or an exponential number of samples [11]. All these works modeled the learner as a one-pass branching program, allowing only one pass over the stream of samples. In this work, we prove the first memory-samples lower bounds (with a super-linear lower bound on the memory size and super-polynomial lower bound on the number of samples) when the learner is allowed two passes over the stream of samples. For example, we prove that any two-pass algorithm for learning parities of size n requires either a memory of size Ω(n1.5) or at least 2Ω([EQUATION]) samples. More generally, a matrix M : A x X → {-1, 1} corresponds to the following learning problem: An unknown element x ∈ X is chosen uniformly at random. A learner tries to learn x from a stream of samples, (a1, b1), (a2, b2) ..., where for every i, ai ∈ A is chosen uniformly at random and bi = M(ai, x). Assume that k, l, r are such that any submatrix of M of at least 2-k · |A| rows and at least 2-l · |X| columns, has a bias of at most 2-r. We show that any two-pass learning algorithm for the learning problem corresponding to M requires either a memory of size at least Ω(k · min[EQUATION]), or at least 2Ω(min[EQUATION]) samples.