[Paper Review] Chernoff's Inequality - A very elementary proof

Nathan Linial, Zur Luria|arXiv (Cornell University)|Mar 30, 2014

Graph theory and applications1 references11 citations

TL;DR

This paper presents a simplified, elementary proof of a strengthened version of Chernoff's inequality using combinatorial weighting and indicator variable analysis. It establishes a tight bound on the tail probability of sums of indicator random variables under weak dependence assumptions, yielding an exponential decay rate of the form $ e^{-D(\beta\|\alpha)n} $, where $ D(\beta\|\alpha) $ is the Kullback-Leibler divergence.

ABSTRACT

Most types of messages we transmit (e.g., video, audio, images, text) are not fully compressed, since they do not have known efficient and information theoretically optimal compression algorithms. When transmitting such messages, standard error correcting codes fail to take advantage of the fact that messages are not fully compressed. We show that in this setting, it is sub-optimal to use standard error correction. We consider a model where there is a set of "valid messages" which the sender may send that may not be efficiently compressible, but where it is possible for the receiver to recognize valid messages. In this model, we construct a (probabilistic) encoding procedure that achieves better tradeoffs between data rates and error-resilience (compared to just applying a standard error correcting code). Additionally, our techniques yield improved efficiently decodable (probabilistic) codes for fully compressed messages (the standard setting where the set of valid messages is all binary strings) in the high-rate regime.

Motivation & Objective

• To provide a simple, self-contained proof of a strengthened Chernoff-type inequality.
• To derive tail bounds for sums of dependent indicator random variables under minimal assumptions.
• To generalize classical Chernoff bounds by relaxing independence and identical distribution requirements.
• To establish a tight bound using combinatorial identities and entropy-based estimates.

Proposed method

• Define weights $ w_S = \Pr(X_i = 1 \iff i \in S) $ to represent joint outcomes of indicator variables.
• Express the tail probability $ \Pr(\sum X_i \geq \beta n) $ as a sum over sets $ T $ with $ |T| \geq \beta n $.
• Use the inequality $ \binom{|T|}{k} \geq \binom{\beta n}{k} $ for $ |T| \geq \beta n $, leveraging the increasing nature of binomial coefficients.
• Apply double counting over sets $ S $ of size $ k $ to relate $ \sum_T w_T $ to $ \sum_{|S|=k} \Pr(\bigwedge_{i \in S} X_i = 1) $.
• Use standard binomial coefficient estimates to bound the ratio $ \binom{n}{k} / \binom{\beta n}{k} $ by $ e^{-D(\beta\|\alpha)n} $ under the assumption $ \Pr(\bigwedge_{i \in S} X_i = 1) \leq \alpha^k $.
• Establish tightness by showing equality holds iff $ \Pr(\sum X_i = \beta n \text{ or } \sum X_i < k) = 1 $.

Experimental results

Research questions

• RQ1 Can Chernoff's inequality be proven with minimal assumptions on the dependence structure of indicator variables?
• RQ2 What is the tightest possible bound on the tail probability of a sum of dependent indicator variables?
• RQ3 How does the bound change when the joint probability of any $ k $-set of variables being 1 is bounded by $ \alpha^k $?
• RQ4 Under what conditions does the bound $ e^{-D(\beta\|\alpha)n} $ become tight?

Key findings

• The inequality $ \Pr(\sum X_i \geq \beta n) \leq \frac{1}{\binom{\beta n}{k}} \sum_{|S|=k} \Pr(\bigwedge_{i \in S} X_i = 1) $ holds under no independence assumptions.
• When $ \Pr(\bigwedge_{i \in S} X_i = 1) \leq \alpha^k $ for all $ S $ of size $ k = \left(\frac{\beta - \alpha}{1 - \alpha}\right)n $, the tail probability is bounded by $ e^{-D(\beta\|\alpha)n} $.
• The bound $ e^{-D(\beta\|\alpha)n} $ is derived via standard entropy estimates of binomial coefficients.
• The inequality is tight: equality holds if and only if $ \Pr(\sum X_i = \beta n \text{ or } \sum X_i < k) = 1 $.
• The result generalizes classical Chernoff bounds by relaxing the i.i.d. assumption to weaker moment-like constraints on joint probabilities.