[Paper Review] Chernoff-Hoeffding Inequality and Applications

Jeff M. Phillips|arXiv (Cornell University)|Sep 27, 2012

Algorithms and Data Compression11 references18 citations

TL;DR

This paper presents a concise derivation and application of the Chernoff-Hoeffding inequality for bounding the tail probabilities of sums of bounded, independent random variables. It demonstrates its use in proving the Johnson-Lindenstrauss Lemma for dimensionality reduction and in subset sampling for density estimation, showing that random projections preserve pairwise distances with high probability and that small random samples can accurately approximate point densities in high-dimensional spaces.

ABSTRACT

When dealing with modern big data sets, a very common theme is reducing the set through a random process. These generally work by making "many simple estimates" of the full data set, and then judging them as a whole. Perhaps magically, these "many simple estimates" can provide a very accurate and small representation of the large data set. The key tool in showing how many of these simple estimates are needed for a fixed accuracy trade-off is the Chernoff-Hoeffding inequality[Che52,Hoe63]. This document provides a simple form of this bound, and two examples of its use.

Motivation & Objective

• To provide a clear, accessible derivation of the Chernoff-Hoeffding inequality for bounded random variables.
• To demonstrate its application in proving the Johnson-Lindenstrauss Lemma for dimensionality reduction in big data.
• To show how random sampling can be used to approximate point densities in high-dimensional spaces with probabilistic guarantees.
• To illustrate the use of the union bound in extending per-pair bounds to global guarantees over multiple data pairs.
• To clarify the conditions under which the inequality applies, particularly regarding the range of the deviation parameter α.

Proposed method

• Derives the Chernoff-Hoeffding inequality for sums of independent, bounded random variables using moment generating functions and convexity.
• Applies the inequality to the Johnson-Lindenstrauss Lemma by modeling random projections as sums of inner products with random unit vectors.
• Uses the expectation and variance of individual projection components to bound the overall deviation of the projected norm from the original.
• Applies the union bound across all pairs of points to extend a per-pair probability guarantee to a global guarantee with high probability.
• Uses the exponential moment method and the inequality $ \frac{1}{2}(e^x + e^{-x}) \leq e^{x^2/2} $ to derive the concentration bound.
• Shows that Theorem 1.2 is a special case of Theorem 1.1 by setting $ \Delta_i = 2\Delta $ and $ \mathbb{E}[M] = 0 $.

Experimental results

Research questions

• RQ1 How many random projections are needed to preserve pairwise distances in high-dimensional data with high probability?
• RQ2 Can random sampling be used to accurately estimate the number of points in a high-dimensional rectangle?
• RQ3 What is the minimal number of samples required to ensure a high-probability approximation of the density of points in a region?
• RQ4 How does the union bound interact with concentration inequalities when multiple events are involved?
• RQ5 What are the tightest conditions on the deviation parameter α for which the Chernoff-Hoeffding bound remains valid?

Key findings

• For the Johnson-Lindenstrauss Lemma, $ k = (8/\varepsilon^2)\ln(n/\delta) $ random projections ensure that all pairwise distances are preserved within a factor of $ 1\pm\varepsilon $ with probability at least $ 1-\delta $.
• The expected value of each projection component $ \phi_i(p) $ is $ \mathbb{E}[X_i] = 1/k $, and the variance is bounded by $ \text{Var}[X_i] \leq 1/k^2 $.
• Using Theorem 1.3, the required number of dimensions is $ k \geq (4/\varepsilon^2)\ln(2/\gamma) $ for a single pair, and the union bound increases this to $ k \geq (8/\varepsilon^2)\ln(n/\delta) $ for all $ n^2 $ pairs.
• For subset sampling, the Chernoff-Hoeffding bound ensures that a random sample of size $ O(\log(1/\delta)/\varepsilon^2) $ provides a $ (1\pm\varepsilon) $-approximation to the true count in any axis-aligned rectangle with probability at least $ 1-\delta $.
• The union bound is nearly tight even for independent events, as shown by the asymptotic equivalence of $ \delta/t $ and $ \ln(1/(1-\delta))/t $ for small $ \delta $.
• The proof of the Chernoff-Hoeffding inequality relies on the convexity of the exponential function and the inequality $ \frac{1}{2}(e^x + e^{-x}) \leq e^{x^2/2} $, which enables the derivation of the final bound.