[Paper Review] Five Proofs of Chernoff's Bound with Applications

Wolfgang Mulzer|arXiv (Cornell University)|Jan 10, 2018

advanced mathematical theories19 references3 citations

TL;DR

This paper presents five distinct proofs of Chernoff's bound using different mathematical techniques—moment generating functions, Chvátal's combinatorial method, the Impagliazzo-Kabanets constructive approach, the entropy method, and the method of types—each yielding unique generalizations. The key contribution is showing how each proof technique naturally extends to dependent or non-i.i.d. settings, such as hypergeometric and negatively correlated random variables, with tight tail bounds expressed via Kullback-Leibler divergence.

ABSTRACT

We discuss five ways of proving Chernoff's bound and show how they lead to different extensions of the basic bound.

Motivation & Objective

• To provide multiple, conceptually distinct proofs of Chernoff's bound to deepen understanding of its underlying principles.
• To demonstrate how each proof technique enables different generalizations beyond i.i.d. Bernoulli trials.
• To unify various concentration inequalities (e.g., Hoeffding, Bernstein) under a common framework using KL divergence.
• To extend Chernoff's bound to non-independent settings, such as sampling without replacement and negatively correlated variables.
• To offer constructive and information-theoretic interpretations of tail bounds via entropy and expectation methods.

Proposed method

• Using the moment method with exponential moment generating functions and Markov's inequality to derive the bound via optimization over a parameter λ.
• Applying Chvátal's method with a parameter τ ≥ 1 and binomial expansion to bound tail probabilities combinatorially.
• Employing the Impagliazzo-Kabanetz method via random subset selection to derive a constructive proof using conditional expectation bounds.
• Utilizing the entropy method by analyzing the entropy of the empirical distribution and applying the method of types to derive tail bounds.
• Leveraging the method of types to interpret the tail probability as a relative entropy (KL divergence) between distributions.
• Proving generalizations via probabilistic coupling and expectation inequalities, particularly for hypergeometric and negatively correlated settings.

Experimental results

Research questions

• RQ1 How do different proof techniques for Chernoff's bound lead to distinct generalizations?
• RQ2 Can Chernoff's bound be extended to sampling without replacement using combinatorial methods?
• RQ3 To what extent can the independence assumption in Chernoff's bound be relaxed while preserving the tail bound?
• RQ4 How does the KL divergence naturally emerge in the concentration bound across different proof frameworks?
• RQ5 Can constructive or information-theoretic proofs yield tighter or more interpretable bounds than classical moment-based methods?

Key findings

• The moment method yields the standard Chernoff bound via optimization of the moment generating function, resulting in a bound expressed as exp(−n D_KL(p+t||p)).
• Chvátal's method generalizes to the hypergeometric distribution, proving that Pr[H(N,P,n) ≥ (p+t)n] ≤ exp(−n D_KL(p+t||p)) for sampling without replacement.
• The Impagliazzo-Kabanetz method extends to negatively correlated random variables, where E[∏_{i∈I} X_i] ≤ ∏_{i∈I} p_i implies the same tail bound.
• The entropy method and method of types provide a clean, information-theoretic interpretation of the bound as a relative entropy between empirical and expected distributions.
• All five proof techniques yield the same KL-divergence-based tail bound, but each reveals different structural insights and enables unique extensions.
• The paper establishes that the KL divergence term D_KL(p+t||p) is the optimal exponent in the exponential tail bound, confirming tightness across all methods.