Strong Memory Lower Bounds for Learning Natural Models

TL;DR

This paper establishes fundamental memory lower bounds for streaming algorithms learning natural models, showing that minimal example usage requires space proportional to the product of data dimension and model complexity.

Contribution

It provides the first broad range memory lower bounds for natural learning problems in streaming settings, extending beyond specific cases like parity.

Findings

Memory lower bounds match the ambient space dimension

Bounds apply across a wide range of input sizes

Space complexity scales with data and model parameters

Abstract

We give lower bounds on the amount of memory required by one-pass streaming algorithms for solving several natural learning problems. In a setting where examples lie in $\{0,1\}^d$ and the optimal classifier can be encoded using $\kappa$ bits, we show that algorithms which learn using a near-minimal number of examples, $\tilde O(\kappa)$, must use $\tilde \Omega( d\kappa)$ bits of space. Our space bounds match the dimension of the ambient space of the problem's natural parametrization, even when it is quadratic in the size of examples and the final classifier. For instance, in the setting of $d$-sparse linear classifiers over degree-2 polynomial features, for which $\kappa=\Theta(d\log d)$, our space lower bound is $\tilde\Omega(d^2)$. Our bounds degrade gracefully with the stream length $N$, generally having the form $\tilde\Omega\left(d\kappa \cdot \frac{\kappa}{N}\right)$. Bounds of the form $\Omega(d\kappa)$ were known for learning parity and other problems defined over finite fields. Bounds that apply in a narrow range of sample sizes are also known for linear regression. Ours are the first such bounds for problems of the type commonly seen in recent learning applications that apply for a large range of input sizes.