Extractor-Based Time-Space Lower Bounds for Learning

TL;DR

This paper establishes tight lower bounds on the memory and sample complexity for learning problems represented by matrices with certain bias properties, advancing understanding of the fundamental limits of learning algorithms.

Contribution

It introduces a new lower bound technique that shows learning algorithms require either large memory or many samples, improving previous bounds to be tight and broadly applicable.

Findings

Any learning algorithm needs either large memory or exponential samples.

The bounds are tight and improve upon previous quadratic bounds.

The results unify and extend prior memory-sample lower bounds.

Abstract

A matrix $M: A \times X \rightarrow \{-1,1\}$ corresponds to the following learning problem: An unknown element $x \in X$ is chosen uniformly at random. A learner tries to learn $x$ from a stream of samples, $(a_1, b_1), (a_2, b_2) \ldots$, where for every $i$, $a_i \in A$ is chosen uniformly at random and $b_i = M(a_i,x)$. Assume that $k,\ell, r$ are such that any submatrix of $M$ of at least $2^{-k} \cdot |A|$ rows and at least $2^{-\ell} \cdot |X|$ columns, has a bias of at most $2^{-r}$. We show that any learning algorithm for the learning problem corresponding to $M$ requires either a memory of size at least $\Omega\left(k \cdot \ell \right)$, or at least $2^{\Omega(r)}$ samples. The result holds even if the learner has an exponentially small success probability (of $2^{-\Omega(r)}$). In particular, this shows that for a large class of learning problems, any learning algorithm requires either a memory of size at least $\Omega\left((\log |X|) \cdot (\log |A|)\right)$ or an exponential number of samples, achieving a tight $\Omega\left((\log |X|) \cdot (\log |A|)\right)$ lower bound on the size of the memory, rather than a bound of $\Omega\left(\min\left\{(\log |X|)^2,(\log |A|)^2\right\}\right)$ obtained in previous works [R17,MM17b]. Moreover, our result implies all previous memory-samples lower bounds, as well as a number of new applications. Our proof builds on [R17] that gave a general technique for proving memory-samples lower bounds.