On The Memory Complexity of Uniformity Testing

TL;DR

This paper investigates the memory requirements for uniformity testing of distributions using limited memory machines, establishing bounds that show the minimal number of states needed based on distribution size and distance from uniform.

Contribution

It provides the first tight bounds on the number of states required for uniformity testing with limited memory, moving beyond collision counting and Paninski prior methods.

Findings

Upper bound on states: O(n log n / ε)

Lower bound on states: Ω(n + 1/ε)

Different techniques are needed from prior collision-based methods

Abstract

In this paper we consider the problem of uniformity testing with limited memory. We observe a sequence of independent identically distributed random variables drawn from a distribution $p$ over $[n]$, which is either uniform or is $\varepsilon$-far from uniform under the total variation distance, and our goal is to determine the correct hypothesis. At each time point we are allowed to update the state of a finite-memory machine with $S$ states, where each state of the machine is assigned one of the hypotheses, and we are interested in obtaining an asymptotic probability of error at most $0<\delta<1/2$ uniformly under both hypotheses. The main contribution of this paper is deriving upper and lower bounds on the number of states $S$ needed in order to achieve a constant error probability $\delta$, as a function of $n$ and $\varepsilon$, where our upper bound is $O(\frac{n\log n}{\varepsilon})$ and our lower bound is $\Omega (n+\frac{1}{\varepsilon})$. Prior works in the field have almost exclusively used collision counting for upper bounds, and the Paninski mixture for lower bounds. Somewhat surprisingly, in the limited memory with unlimited samples setup, the optimal solution does not involve counting collisions, and the Paninski prior is not hard. Thus, different proof techniques are needed in order to attain our bounds.