Tight Time-Space Lower Bounds for Constant-Pass Learning

In his breakthrough paper, Raz showed that any parity learning algorithm requires either quadratic memory or an exponential number of samples [FOCS’16, JACM’19]. A line of work that followed extended this result to a large class of learning problems. Until recently, all these results considered learning in the streaming model, where each sample is drawn independently, and the learner is allowed a single pass over the stream of samples. Garg, Raz, and Tal [CCC’19] considered a stronger model, allowing multiple passes over the stream. In the 2-pass model, they showed that learning parities of size n requires either a memory of size $n^{1.5}$ or at least $2^{\sqrt{n}}$ samples. (Their result also generalizes to other learning problems.) In this work, for any constant q, we prove tight memory-sample lower bounds for any parity learning algorithm that makes q passes over the stream of samples. We show that such a learner requires either $\Omega\left(n^{2}\right)$ memory size or at least $2^{\Omega(n)}$ samples. Beyond establishing a tight lower bound, this is the first nontrivial lower bound for q-pass learning for any $q \geq 3$. Similar to prior work, our results extend to any learning problem with many nearly-orthogonal concepts.We complement the lower bound with an upper bound, showing that parity learning with q passes can be done efficiently with $O\left(n^{2} / \log q\right)$ memory.