Hoeffding Serfling Bound

Easy:

Imagine you have a bag of marbles, but you don’t know what color they are. You want to guess the average color of all marbles in the bag by taking out a few marbles and checking their colors. Now, you want to know how confident you can be that your guess is close to the real average color of all marbles in the bag.

Hoeffding-Serfling bound is like a rule that helps you figure out how accurate your guess is likely to be based on the marbles you’ve already checked. It tells you that if you randomly pick a few marbles from the bag and find their average color, then the real average color of all marbles in the bag is likely to be close to your guess, within a certain range.

For example, if you pick 10 marbles and find their average color to be blue, the Hoeffding-Serfling bound might tell you that there’s a high probability that the real average color of all marbles in the bag is somewhere close to blue, say within a range of blue-ish colors. But if you picked only 2 marbles and found their average color to be blue, the bound might tell you that your guess could be less accurate because you haven’t checked enough marbles yet.

So, it helps you understand how confident you can be about your guess based on the sample you’ve taken from a larger population.

Moderate:

Hoeffding’s inequality provides an upper bound on the probability that the sum of bounded independent random variables deviates from its expected value by a certain amount. In simpler terms, it helps us understand how likely it is for the average of a set of numbers to be far from the true average. Imagine you have a group of numbers, and you want to know how close their average is to the actual average of all numbers. Hoeffding’s inequality gives us a way to estimate this difference. It’s like having a safety net that tells us the maximum possible error in our estimation. So, if you’re trying to figure out how close your guess is to the real answer, Hoeffding’s inequality helps you understand the chances of being significantly off.

Hard:

The Hoeffding-Serfling bound is a mathematical result that provides a tail bound on the maximum deviation between the empirical mean of a set of independent and identically distributed (iid) random variables and their expected value. In simpler terms, it gives a probabilistic guarantee about how close the average value of a set of observations is likely to be to the true underlying mean.

The bound states that for a sequence X = {X1, … , Xn} of n iid random variables taking values in [a, b], and let Mu be the true mean of these variables, then for any positive t, the following inequality holds with probability at least 1 — e^(-2\*t² /(b-a)²):

| (1/n)\*(sum from i=1 to n of Xi) — Mu | <= sqrt((b-a)² \* ln(1/delta)/(2n)) + (b-a)/(2n)

Here, delta is a free parameter that determines the level of confidence in the bound, which means that by adjusting its value, we can make the right-hand side arbitrarily small, thereby increasing the strength of the bound. However, doing so also decreases the probability on the left-hand side, meaning that the bound becomes less tight.

This bound is often used in statistics and machine learning to analyze the performance of algorithms that rely on sampling data randomly. For example, it can be used to upper-bound the generalization error of a model trained on a finite dataset, providing guarantees about how well the model is likely to perform on new unseen data. Additionally, it can be applied to various types of sequential decision making problems such as multi-armed bandits and reinforcement learning, where agents need to balance exploration and exploitation based on observed rewards over time.

A few books on deep learning that I am reading:

Book 1

Book 2

Book 3