Time-Space Lower Bounds for Two-Pass Learning

Abstract

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(n 2 ) 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(n 1.5 ) or at least 2 Ω(√n) samples. More generally, a matrix M : A × 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, `, r are such that any submatrix of M of at least 2 −k · |A| rows and at least 2 −` · |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{k, √ `} , or at least 2 Ω(min{k,√ `,r}) samples.