Learning, Space, and Cryptography

This is our final project for CS 221 (Computational Complexity), an expository report on [Raz17], which gives samples-space lower bounds (and a fortiori time-space lower bounds) for a broad class of learning problems. Specifically, it is shown that if we consider the learning problem M : Θ× X → {−1, 1} and a learner is given a stream of samples (xi, yi) where yi = Mθ(xi), and the largest singular value of M is |X | · |Θ|1/2−ε, then any learning algorithm for the corresponding learning problem requires either a memory of size at least O((εn)) or at least 2 samples, where n = log2 |Θ|. We elucidate the seemingly strange singular-value condition, showing that it has connections to the difficulty of ascertaining y from x and given a probability distribution on θ. We then strive to simplify the arguments in the paper in light of these insights.