# Computational Concentration of Measure: Optimal Bounds, Reductions, and More

## TL;DR

This paper presents a polynomial-time algorithm for computationally realizing measure concentration in product spaces, with applications to learning vulnerabilities, Gaussian distributions, and coin-tossing protocols.

## Contribution

It introduces MUCIO, an algorithm for computational measure concentration, and establishes a reduction framework, extending results to weighted Hamming, Gaussian, and discrete processes.

## Key findings

- Polynomial-time algorithm for measure concentration in product spaces.
- Applications to cryptographic vulnerabilities and learning attacks.
- Extensions to weighted Hamming, Gaussian, and coin-tossing protocols.

## Abstract

Product measures of dimension $n$ are known to be concentrated in Hamming distance: for any set $S$ in the product space of probability $\epsilon$, a random point in the space, with probability $1-\delta$, has a neighbor in $S$ that is different from the original point in only $O(\sqrt{n\ln(1/(\epsilon\delta))})$ coordinates. We obtain the tight computational version of this result, showing how given a random point and access to an $S$-membership oracle, we can find such a close point in polynomial time. This resolves an open question of [Mahloujifar and Mahmoody, ALT 2019]. As corollaries, we obtain polynomial-time poisoning and (in certain settings) evasion attacks against learning algorithms when the original vulnerabilities have any cryptographically non-negligible probability. We call our algorithm MUCIO ("MUltiplicative Conditional Influence Optimizer") since proceeding through the coordinates, it decides to change each coordinate of the given point based on a multiplicative version of the influence of that coordinate, where influence is computed conditioned on previously updated coordinates. We also define a new notion of algorithmic reduction between computational concentration of measure in different metric probability spaces. As an application, we get computational concentration of measure for high-dimensional Gaussian distributions under the $\ell_1$ metric. We prove several extensions to the results above: (1) Our computational concentration result is also true when the Hamming distance is weighted. (2) We obtain an algorithmic version of concentration around mean, more specifically, McDiarmid's inequality. (3) Our result generalizes to discrete random processes, and this leads to new tampering algorithms for collective coin tossing protocols. (4) We prove exponential lower bounds on the average running time of non-adaptive query algorithms.