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 Fwith finite cardinality |F|≤∞, we have min f ∈ F R ( f ) = 0 ⇒ f * ∈ F . This is almost always not the situation in the real-world learning problems. Let us suppose we have a finite collection of candidate functions F. Furthermore, we do not assume that the optimal function f*, which satisfies R ( f * ) = inf f R ( f ) where the inf is taken over all measurable functions, is a member of 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 knowledge about f*. The question then arises about how we can reasonably select an f∈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 R^n(f) to R(f)∀f∈F. In other words, we wish that the empirical risk is a good indicator of the true risk for every function in F. If this is case, the selection of f that minimizes the empirical risk f n ^ = arg min f ∈ F n R ^ n ( f ) should also yield a small true risk, that is, R(fn^) should be close to minf∈FR(f). Finally, we can thus state our desired situation as P max f ∈ F n | R n ^ ( f ) - R ( f ) | > ϵ < δ , for small values of ϵ and δ. In other words, with probability at least 1-δ, |Rn^(f)-R(f)|>ϵ, ∀f∈F. In this lecture, we will start to develop bounds of this form. First we will focus on bounding P(|Rn^(f)-R(f)|>ϵ) for one fixed f∈F.

Developing Initial Bounds To begin, let us recall the definition of empirical risk for {Xi,Yi}i=1n be a collection of training data. Then the empirical risk is defined as R ^ n ( f ) = 1 n ∑ i = 1 n ℓ ( f ( X i ) , Y i ) . Note that since the training data {Xi,Yi}i=1n are assumed to be i.i.d. pairs, the terms in the sum are i.i.d random variables. Let L i = ℓ ( f ( X i ) , Y i ) . The collection of losses {Li}i=1n is i.i.d according to some unknown distribution (depending on the unknown joint distribution of (X,Y) and the loss function). The expectation of Li is E[ℓ(f(Xi),Yi)]=E[ℓ(f(X),Y)]=R(f), the true risk of f. For now, let's assume that f is fixed. E [ R n ^ ( f ) ] = 1 n ∑ i = 1 n E [ ℓ ( f ( X i ) , Y i ) ] = 1 n ∑ i = 1 n E [ L i ] = R ( f ) We know from the strong law of large numbers that the average (or empirical mean) Rn^(f) converges almost surely to the true mean R(f). That is, Rn^(f)→R(f) almost surely as n→∞. 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 area of the shaded tail regions in Figure 1 is P(|Rn^(f)-R(f)|>ϵ). We are interested in finding out how fast this probability tends to zero as n→∞.

At this stage, we recall Markov's Inequality. Let Z be a nonnegative random variable. E [ Z ] = ∫ 0 ∞ z p ( z ) d z = ∫ 0 t z p ( z ) d z + ∫ u ∞ z p ( z ) d z ≥ 0 + t ∫ t ∞ z p ( z ) d z = t P ( Z ≥ t ) ⇒ P ( Z ≥ t ) ≤ E [ Z ] t ⇒ P ( Z 2 ≥ t 2 ) ≤ E [ Z 2 ] t 2 Take Z = | R n ( f ) ^ - R ( f ) | and t = ϵ P ( | R n ^ ( f ) - R ( f ) | ≥ ϵ ) ≤ E [ | R n ( f ) ^ - R ( f ) | 2 ] ϵ 2 ≤ var ( R ^ n ( f ) ) ϵ 2 = ∑ i = 1 n var ( L i n ) ϵ 2 = var ( ℓ ( X ) , Y ) n ϵ 2 = σ L 2 n ϵ 2 . So, the probability goes to zero at a rate of at least n-1. However, it turns out that this is an extremely loose bound. According to the Central Limit Theorem R n ^ ( f ) = 1 n ∑ i = 1 n L i → N R ( f ) , σ L 2 n as n → ∞ in distribution. This suggests that for large values of n, P ( | R n ^ ( f ) - R ( f ) | ≥ ϵ ) ≈ O e - n ϵ 2 2 σ L 2 . That is, the Gaussian tail probability is tending to zero exponentially fast.

Chernoff's Bound Note that for any nonnegative random variable Z and t>0, P ( Z ≥ t ) = P ( e s Z ≥ e s t ) ≤ E [ e s Z ] e s t , ∀ s > 0 by Markov's inequality . Chernoff's bound is based on finding the value of s that minimizes the upper bound. If Z is a sum of independent random variables. For example, say Z = ∑ i = 1 n ℓ ( f ( X i ) , Y i ) - R ( f ) = n R ^ n ( f ) - R ( f ) then the bound becomes P ∑ i = 1 n ( L i - E [ L i ] ) ≥ t ≤ e - s t E [ e s ∑ i = 1 n ( L i - E [ L i ] ) ] ≤ e - s t ∏ i = 1 n E [ e s ( L i - E [ L i ] ) ] , from independence. Thus, the problem of finding a tight bound boils down to finding a good bound for E[ss(Li-E[Li])]. Chernoff ('52), first studied this situation for binary random variables. Then, Hoeffding ('63) derived a more general result for arbitrary bounded random variables.

Hoeffding's Indequality Theorem Hoeffding's Inequality Let Z1,Z2,...,Zn be independent bounded random variables such that Zi∈[ai,bi] with probability 1. Let Sn=∑i=1nZi. Then for any t>0, we have P ( | S n - E [ S n ] | ≥ t ) ≤ 2 e - 2 t 2 ∑ i = 1 n ( b i - a i ) 2 . The key to proving Hoeffding's inequality is the following upper bound: if Z is a random variable with E[Z]=0 and a≤Z≤b, then E [ e s Z ] ≤ e s 2 ( b - a ) 2 8 . This upper bound is derived as follows. By the convexity of the exponential function, e s z ≤ z - a b - a e s b + b - z b - a e s a , for a ≤ z ≤ b .

Thus, E [ e s Z ] ≤ E Z - a b - a e s b + E b - Z b - a e s a = b b - a e s a - a b - a e s b , since E [ Z ] = 0 = ( 1 - θ + θ e s ( b - a ) ) e - θ s ( b - a ) , where θ = - a b - a . Now let u = s ( b - a ) and define φ ( u ) ≡ - θ u + log ( 1 - θ + θ e u ) . Then we have E [ e s Z ] ≤ ( 1 - θ + θ e s ( b - a ) ) e - θ s ( b - a ) = e φ ( u ) . To minimize the upper bound let's express φ(u) in a Taylor's series with remainder : φ ( u ) = φ ( 0 ) + u φ ' ( 0 ) + u 2 2 φ ' ' ( v ) for some v ∈ [ 0 , u ] φ ' ( u ) = - θ + θ e u 1 - θ + θ e u ⇒ φ ' ( u ) = 0 φ ' ' ( u ) = θ e u 1 - θ + θ e u - θ e u ( 1 - θ + θ e u ) 2 = θ e u 1 - θ + θ e u ( 1 - θ e u 1 - θ + θ e u ) = ρ ( 1 - ρ ) . Now, φ''(u) is maximized by ρ = θ e u 1 - θ + θ e u = 1 2 ⇒ φ ' ' ( u ) ≤ 1 4 . So, φ ( u ) ≤ u 2 8 = s 2 ( b - a ) 2 8 ⇒ E [ e s Z ] ≤ e s 2 ( b - a ) 2 8 . Now, we can apply this upper bound to derive Hoeffding's inequality. P ( S n - E [ S n ] ≥ t ) ≤ e - s t ∏ i = 1 n E [ e s ( L i - E [ L i ] ) ] ≤ e - s t ∏ i = 1 n e s 2 ( b i - a i ) 2 8 = e - s t e s 2 ∑ i = 1 n ( b i - a i ) 2 8 = e - 2 t 2 ∑ i = 1 n ( b i - a i ) 2 by choosing s = 4 t ∑ i = 1 n ( b i - a i ) 2 Similarly, P(E[Sn]-Sn≥t)≤e-2t2∑i=1n(bi-ai)2. This completes the proof of the Hoeffding's theorem. ApplicationLet Zi=1f(Xi)≠Yi-R(f), as in the classification problem. Then for a fixed f, it follows from Hoeffding's inequality (i.e., Chernoff's bound in this special case) that P ( | R n ^ ( f ) - R ( f ) | ≥ ϵ ) = P 1 n | S n - E [ S n ] | ≥ ϵ = P ( | S n - E [ S n ] | ≥ n ϵ ) ≤ 2 e - 2 ( n ϵ ) 2 n = 2 e - 2 n ϵ 2 . Now, we want a bound like this to hold uniformly for all f∈F. Assume that F is a finite collection of models and let |F| denote its cardinality. We would like to bound the probability that maxf∈F|Rn^(f)-R(f)|≥ϵ. Note that the event max f ∈ F | R n ^ ( f ) - R ( f ) | ≥ ϵ ≡ ⋃ f ∈ F | R n ^ ( f ) - R ( f ) | ≥ ϵ . Therefore P max f ∈ F | R n ^ ( f ) - R ( f ) | ≥ ϵ = P ⋃ f ∈ F | R n ^ ( f ) - R ( f ) | ≥ ϵ ≤ ∑ f ∈ F P ( | R n ^ ( f ) - R ( f ) | ≥ ϵ ) , the `` union of events '' bound ≤ 2 | F | e - 2 n ϵ 2 , by Hoeffding's inequality. Thus, we have shown that with probability at least 1-2|F|e-2nϵ2, ∀f∈F | R n ^ ( f ) - R ( f ) | < ϵ . And accordingly, we can be reasonably confident in selecting f from F based on the empirical risk function R^n.