Chernoff bound

Chernoff bound is a probabilistic inequality that provides an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than a certain amount. It is a powerful tool in probability theory and statistics, especially in the context of random variables, probability theory, and statistical inference. The Chernoff bound is particularly useful in areas such as algorithm analysis, machine learning, and information theory, where it helps in quantifying the tail behaviors of sums of random variables.

Definition

Given a set of independent random variables \(X_1, X_2, \ldots, X_n\), each bounded by an interval, the Chernoff bound provides an exponential upper limit on the probability that the sum of these variables deviates from its expected value. Formally, if \(X = \sum_{i=1}^{n}X_i\) where each \(X_i\) is a random variable with expected value \(\mu_i\), then for any \(\delta > 0\), the probability that \(X\) deviates from its expected sum \(\mu = \sum_{i=1}^{n}\mu_i\) by a factor of \(\delta\) can be bounded above as follows:

\[ P(X \geq (1 + \delta)\mu) \leq e^{-\frac{\delta^2 \mu}{2 + \frac{2}{3}\delta}} \]

and

\[ P(X \leq (1 - \delta)\mu) \leq e^{-\frac{\delta^2 \mu}{2}} \]

These inequalities are known as the Chernoff bounds.

Applications

Chernoff bounds are widely used in various fields to provide guarantees on the performance of algorithms and systems. Some of the key applications include:

• Algorithm Analysis: In the analysis of algorithms, especially those involving randomized algorithms, Chernoff bounds are used to show that the probability of the algorithm deviating significantly from its expected behavior is extremely low.

• Machine Learning: In machine learning, Chernoff bounds help in bounding the error rates of learning algorithms and in the design of algorithms with provable guarantees on their generalization error.

• Network Theory: In network theory, Chernoff bounds are applied to analyze the reliability and performance of network protocols under stochastic traffic conditions.

• Information Theory: Chernoff bounds are used in information theory to analyze the error probabilities in communication systems and coding theory.

History

The Chernoff bound is named after Herman Chernoff, who introduced these inequalities in a seminal paper in 1952. However, similar bounds were known and used in various forms before Chernoff's work. The significance of Chernoff's contribution lies in the generalization and refinement of these bounds, making them more widely applicable and easier to use in practice.

See Also

• Hoeffding's inequality
• Markov's inequality
• Chebyshev's inequality
• Law of large numbers
• Central limit theorem