0.7 Chernoff's bound and hoeffding's inequality

Introduction

Motivation

In the last lecture we consider a learning problem in which the optimal function belonged to a finite class of functions.Specifically, for some collection of functions $\mathcal{F}$ with finite cardinality $\left|\mathcal{F}\right|\le \infty$ , we have

This is almost always not the situation in the real-world learning problems. Let us suppose we have a finite collection ofcandidate functions $\mathcal{F}$ . Furthermore, we do not assume that the optimal function $f^{*}$ , which satisfies

where the $inf$ is taken over all measurable functions, is a member of $\mathcal{F}$ . That is, we make few, if any, assumptions about $f^{*}$ . This situation is sometimes termed as Agnostic Learning . The root of the word agnostic literally means not known . The term agnostic learning is used to emphasize the fact that often, perhaps usually, we may have no prior knowledgeabout $f^{*}$ . The question then arises about how we can reasonably select an $f\in \mathcal{F}$ in this setting.

The problem

The PAC style bounds discussed in the previous lecture , offer some help. Since we are selecting a function based on the empirical risk,the question is how close is ${\widehat{R}}_n\left(f\right)$ to $R\left(f\right)$ $\forall f\in \mathcal{F}$ . In other words, we wish that the empirical risk is a good indicator of the true risk for every function in $\mathcal{F}$ . If this is case, the selection of $f$ that minimizes the empirical risk

should also yield a small true risk, that is, $R\left(\widehat{f_n}\right)$ should be close to ${min}_{f\in \mathcal{F}}R\left(f\right)$ . Finally, we can thus state our desired situation as

for small values of $ϵ$ and $\delta$ . In other words, with probability at least $1-\delta$ , $|\widehat{R_n}\left(f\right)-R\left(f\right)|>ϵ$ , $\forall f\in \mathcal{F}$ . In this lecture, we will start to develop bounds of this form. First we will focus on bounding $P\left(\right|\widehat{R_n}\left(f\right)-R\left(f\right)|>ϵ)$ for one fixed $f\in \mathcal{F}$ .

Developing initial bounds

To begin, let us recall the definition of empirical risk for ${\{X_i,Y_i\}}_{i=1}^n$ be a collection of training data. Then the empirical risk is defined as

Note that since the training data ${\{X_i,Y_i\}}_{i=1}^n$ are assumed to be i.i.d. pairs, the terms in the sum are i.i.d random variables.

Let

The collection of losses ${\left\{L_i\right\}}_{i=1}^n$ is i.i.d according to some unknown distribution (depending on the unknown joint distribution of (X,Y) and the loss function). Theexpectation of $L_i$ is $E\left[\ell (f\left(X_i\right),Y_i)\right]=E\left[\ell (f\left(X\right),Y)\right]=R\left(f\right)$ , the true risk of $f$ . For now, let's assume that $f$ is fixed.

We know from the strong law of large numbers that the average (or empirical mean) $\widehat{R_n}\left(f\right)$ converges almost surely to the true mean $R\left(f\right).$ That is, $\widehat{R_n}\left(f\right)\to R\left(f\right)$ almost surely as $n\to \infty$ . The question is how fast.

Concentration of measure inequalities

Concentration inequalities are upper bounds on how fast empirical means converge to their ensemble counterparts, in probability. The areaof the shaded tail regions in Figure 1 is $P\left(\right|\widehat{R_n}\left(f\right)-R\left(f\right)|>ϵ)$ . We are interested in finding out how fast this probability tends to zero as $n\to \infty$ .

At this stage, we recall Markov's Inequality . Let $Z$ be a nonnegative random variable.