Finite-Sample Concentration of the Multinomial in Relative Entropy

TL;DR

This paper derives finite-sample concentration bounds for the divergence of empirical distributions from the true distribution in multinomial settings, improving existing bounds and providing moment estimates.

Contribution

It introduces a gamma distribution-based bound for the moment generating function of the KL divergence in multinomial samples, enhancing finite-sample analysis.

Findings

Exponential concentration inequality improves over standard bounds for large divergence values.

Finite-sample bounds on moments are close to asymptotic values, within constant factors.

Reduction to binary case allows for straightforward generalization to multinomial distributions.

Abstract

We show that the moment generating function of the Kullback-Leibler divergence (relative entropy) between the empirical distribution of $n$ independent samples from a distribution $P$ over a finite alphabet of size $k$ (i.e. a multinomial distribution) and $P$ itself is no more than that of a gamma distribution with shape $k - 1$ and rate $n$. The resulting exponential concentration inequality becomes meaningful (less than 1) when the divergence $\varepsilon$ is larger than $(k-1)/n$, whereas the standard method of types bound requires $\varepsilon > \frac{1}{n} \cdot \log{\binom{n+k-1}{k-1}} \geq (k-1)/n \cdot \log(1 + n/(k-1))$, thus saving a factor of order $\log(n/k)$ in the standard regime of parameters where $n\gg k$. As a consequence, we also obtain finite-sample bounds on all the moments of the empirical divergence (equivalently, the discrete likelihood-ratio statistic), which are within constant factors (depending on the moment) of their asymptotic values. Our proof proceeds via a simple reduction to the case $k = 2$ of a binary alphabet (i.e. a binomial distribution), and has the property that improvements in the case of $k = 2$ directly translate to improvements for general $k$. In particular, we conjecture a bound on the binomial moment generating function that would almost close the quadratic gap between our finite-sample bound and the asymptotic moment generating function bound from Wilks' theorem (which does not hold for finite samples).

Equations

$$V_{n,k,P}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\operatorname{\operatorname{D}}\Big{(}\MT@delim@Swapped\of@star{X_{1}/n,\dotsc,X_{k}/n}\;\Big{\|}\;\mathopen{}\MT@delim@Swapped\of@star{p_{1},\dotsc,p_{k}}\Big{)}$$

$$\displaystyle\operatorname{\operatorname{D}}\Big{(}\MT@delim@Swapped\of@star{q_{1},\dotsc,q_{k}}\;\Big{\|}\;\mathopen{}\MT@delim@Swapped\of@star{p_{1},\dotsc,p_{k}}\Big{)}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\sum_{i=1}^{k}q_{i}\log\frac{q_{i}}{p_{i}}$$

$$\mathrm{Pr}\text{MT@delim@Swapped}\text{brof@star}{V}_{n,k,P}\ge \epsilon \le e^{-n\epsilon }\cdot \text{MT@delim@Swapped}\text{of@star}{\frac{e\epsilon n}{k-1}}^{k-1}.$$

$$\operatorname*{E}\MT@delim@Swapped\brof@star{\exp\MT@delim@Swapped\of@nostar[\big{]}{t\cdot V_{n,k,P}}}\leq\MT@delim@Swapped\of@star{\frac{1}{1-t/n}}^{k-1}.$$

$$E\text{MT@delim@Swapped}\text{brof@star}\exp \text{MT@delim@Swapped}\text{of@star}t\cdot V_{n,2,(p,1-p)}\le f(t/n).$$

$$E\text{MT@delim@Swapped}\text{brof@star}\exp \text{MT@delim@Swapped}\text{of@star}t\cdot V_{n,k,P}\le f(t/n{)}^{k-1}.$$

$$V_{n,k,P}$$

$$\displaystyle=\operatorname{\operatorname{D}}\big{(}\MT@delim@Swapped\of@star{X_{k}/n,1-X_{k}/n}\;\big{\|}\;\mathopen{}\MT@delim@Swapped\of@star{p_{k},1-p_{k}}\big{)}+\frac{n-X_{k}}{n}\cdot V_{n-X_{k},k-1,P^{\prime}}$$

$$V_{n-X_k,k-1,P^{\prime }}=D\left(\text{MT@delim@Swapped}\text{of@star}\frac{X_1}{n-X_k},\dots ,\frac{X_{k-1}}{n-X_k}\Vert \text{MT@delim@Swapped}\text{of@star}{p}_1^{\prime },\dots ,p_{k-1}^{\prime }\right)$$

$$\displaystyle\operatorname*{E}\MT@delim@Swapped\brof@nostar[\bigg{]}{&\exp\MT@delim@Swapped\of@nostar[\big{]}{t\cdot V_{n,k,P}}}$$

$$\displaystyle=\operatorname*{E}\MT@delim@Swapped\brof@nostar[\bigg{]}{\operatorname*{E}\Big{[}\exp\MT@delim@Swapped\of@nostar[\big{]}{t\cdot V_{n,k,P}}\,\Big{|}\,\mathopen{}X_{k}\Big{]}}$$

$$\displaystyle=\operatorname*{E}\MT@delim@Swapped\brof@nostar[\Bigg{]}{\exp\MT@delim@Swapped\of@nostar[\Big{]}{t\cdot\operatorname{\operatorname{D}}\big{(}\MT@delim@Swapped\of@star{X_{k}/n,1-X_{k}/n}\;\big{\|}\;\mathopen{}\MT@delim@Swapped\of@star{p_{k},1-p_{k}}\big{)}}\cdot\operatorname*{E}\left[\exp\MT@delim@Swapped\of@star{t\cdot\frac{n-X_{k}}{n}\cdot V_{n-X_{k},k-1,P^{\prime}}}\,\middle|\,\mathopen{}X_{k}\right]}.$$

$$\displaystyle\leq\operatorname*{E}\MT@delim@Swapped\brof@star{\exp\MT@delim@Swapped\of@nostar[\Big{]}{t\cdot\operatorname{\operatorname{D}}\big{(}\MT@delim@Swapped\of@star{X_{k}/n,1-X_{k}/n}\;\big{\|}\;\mathopen{}\MT@delim@Swapped\of@star{p_{k},1-p_{k}}\big{)}}\cdot f\MT@delim@Swapped\of@star{\frac{t(n-X_{k})/n}{n-X_{k}}}^{k-2}}$$

$$\displaystyle=f(t/n)^{k-2}\cdot\operatorname*{E}\MT@delim@Swapped\brof@star{\exp\MT@delim@Swapped\of@nostar[\Big{]}{t\cdot\operatorname{\operatorname{D}}\big{(}\MT@delim@Swapped\of@star{X_{k}/n,1-X_{k}/n}\;\big{\|}\;\mathopen{}\MT@delim@Swapped\of@star{p_{k},1-p_{k}}\big{)}}}.$$

$$f(x)=\frac{1}{1-x}$$

$$\exp{\MT@delim@Swapped\of@nostar[\Big{]}{n\cdot\operatorname{\operatorname{D}}\big{(}\MT@delim@Swapped\of@star{i/n,1-i/n}\;\big{\|}\;\mathopen{}\MT@delim@Swapped\of@star{p,1-p}\big{)}}}=\frac{\Pr\MT@delim@Swapped\brof@star{B_{n,i/n}=i}}{\Pr\MT@delim@Swapped\brof@star{B_{n,p}=i}}$$

$$E\text{MT@delim@Swapped}\text{brof@star}\exp \text{MT@delim@Swapped}\text{of@star}nx\cdot D\left(\text{MT@delim@Swapped}\text{of@star}\frac{B_{n,p}}{n},1-\frac{B_{n,p}}{n}\Vert \text{MT@delim@Swapped}\text{of@star}p,1-p\right)$$

$$G_n(p,x)\stackrel{\text{makebox}[0.0pt]\text{mbox}def}{=}\sum _{i=0}^n\mathrm{Pr}\text{MT@delim@Swapped}\text{brof@star}{B}_{n,(1-x)p+ix/n}=i$$

$$\frac{\partial }{\partial q}{R}_n(q,x)$$

$$=\sum _{i=0}^n\left(\genfrac{}{}{0ex}{}{n}{i}\right)\text{MT@delim@Swapped}\text{of@star}i(q+ix/n{)}^{i-1}(1-q-ix/n{)}^{n-i}-(n-i)(q+ix/n{)}^i(1-q-ix/n{)}^{n-i-1}$$

$$=n\sum _{i=1}^n\left(\genfrac{}{}{0ex}{}{n-1}{i-1}\right)\text{MT@delim@Swapped}\text{of@star}{q+x/n+\frac{i-1}{n-1}\cdot \frac{x(n-1)}{n}}^{i-1}\text{MT@delim@Swapped}\text{of@star}{1-q-x/n-\frac{i-1}{n-1}\cdot \frac{x(n-1)}{n}}^{n-1-(i-1)}$$

$$-n\sum _{i=0}^{n-1}\left(\genfrac{}{}{0ex}{}{n-1}{i}\right)\text{MT@delim@Swapped}\text{of@star}{q+\frac{i}{n-1}\cdot \frac{x(n-1)}{n}}^i\text{MT@delim@Swapped}\text{of@star}{1-q-\frac{i}{n-1}\cdot \frac{x(n-1)}{n}}^{n-1-i}$$

$$=n\sum _{i=0}^{n-1}\left(\genfrac{}{}{0ex}{}{n-1}{i}\right)\text{MT@delim@Swapped}\text{of@star}{q+x/n+\frac{i}{n-1}\cdot \frac{x(n-1)}{n}}^i\text{MT@delim@Swapped}\text{of@star}{1-q-x/n-\frac{i}{n-1}\cdot \frac{x(n-1)}{n}}^{n-1-i}$$

$$-n\sum _{i=0}^{n-1}\left(\genfrac{}{}{0ex}{}{n-1}{i}\right)\text{MT@delim@Swapped}\text{of@star}{q+\frac{i}{n-1}\cdot \frac{x(n-1)}{n}}^i\text{MT@delim@Swapped}\text{of@star}{1-q-\frac{i}{n-1}\cdot \frac{x(n-1)}{n}}^{n-1-i}$$

$$=n\text{MT@delim@Swapped}\text{of@star}{R}_{n-1}\text{MT@delim@Swapped}\text{of@star}q+\frac{x}{n},\frac{x(n-1)}{n}-R_{n-1}\text{MT@delim@Swapped}\text{of@star}q,\frac{x(n-1)}{n}$$

$$\displaystyle=n\MT@delim@Swapped\of@nostar[\big{]}{R_{n-1}\MT@delim@Swapped\of@star{0,x(n-1)/n}-R_{n-1}\MT@delim@Swapped\of@star{0,x(n-1)/n}}=0$$

$$\sum _{j=0}^i\left(\genfrac{}{}{0ex}{}{n}{j}\right)\text{MT@delim@Swapped}\text{of@star}{\frac{j}{n}}^j\cdot \left(\genfrac{}{}{0ex}{}{n-j}{i-j}\right)\text{MT@delim@Swapped}\text{of@star}{-\frac{j}{n}}^{i-j}$$

$$=\frac{n!}{n^i(n-i)!}\cdot \frac{1}{i!}\sum _{j=0}^i\left(\genfrac{}{}{0ex}{}{i}{j}\right)j^i(-1{)}^{i-j}$$

$$\frac{n!}{n^i(n-i)!}$$

$$\mathrm{Pr}\text{MT@delim@Swapped}\text{brof@star}{V}_{n,k,P}\ge \epsilon$$

$$E\text{MT@delim@Swapped}\text{brof@star}\text{MT@delim@Swapped}\text{of@star}{2n{V}_{n,k,P}}^m$$

$$\underset{n\to \infty }{\lim }E\text{MT@delim@Swapped}\text{brof@star}(2n{V}_{n,k,P}{)}^m$$

$$\underset{n\to \infty }{\lim }E\text{MT@delim@Swapped}\text{brof@star}\text{MT@delim@Swapped}\text{of@star}{2n{V}_{n,k,P}-E\text{MT@delim@Swapped}\text{brof@star}2n{V}_{n,k,P}}^m$$

$$\underset{n\to \infty }{\lim }E\text{MT@delim@Swapped}\text{brof@star}\exp \text{MT@delim@Swapped}\text{of@star}s\cdot 2n{V}_{n,k,P}$$

$$\underset{n\to \infty }{\lim }E\text{MT@delim@Swapped}\text{brof@star}\exp \text{MT@delim@Swapped}\text{of@star}s\cdot \text{MT@delim@Swapped}\text{of@star}2n{V}_{n,k,P}-E\text{MT@delim@Swapped}\text{brof@star}2n{V}_{n,k,P}$$

$$f(x)=\frac{2}{\sqrt{1-x}}-1$$

$$\operatorname{\operatorname{D}}_{>}\left(p\;\middle\|\;\mathopen{}q\right)\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\begin{cases}0&p\leq q\\ \operatorname{\operatorname{D}}\Big{(}\MT@delim@Swapped\of@star{p,1-p}\;\Big{\|}\;\mathopen{}\MT@delim@Swapped\of@star{q,1-q}\Big{)}&p>q\end{cases}$$

$$\operatorname*{E}\MT@delim@Swapped\brof@star{\exp\MT@delim@Swapped\of@nostar[\Big{]}{t\cdot\operatorname{\operatorname{D}}_{>}\left(B/n\;\middle\|\;\mathopen{}p\right)}}\leq\frac{1}{\sqrt{1-t/n}}$$

Full text

**Keywords: ** Concentration inequalities, empirical distributions, Kullback–Leibler divergence, likelihood-ratio test, binomial tail bounds

I Introduction

A key problem in statistics is to understand the rate of convergence of an empirical distribution of independent samples to the true underlying distribution. Indeed, this convergence is the basis of hypothesis testing and statistical inference in general [1]. For the case of discrete distributions over a finite alphabet, the Neyman–Pearson lemma [2] shows that for optimal hypothesis testing it is important to consider the likelihood-ratio statistic, or equivalently [3], the Kullback–Leibler divergence (relative entropy) from the true distribution to the empirical distribution, as formally defined in I.1:

Definition I.1**.**

Let $X=(X_{1},\dotsc,X_{k})$ be distributed according to a multinomial distribution with $n$ samples and probabilities $P=(p_{1},\dotsc,p_{k})$, and define

[TABLE]

where

[TABLE]

is the Kullback–Leibler divergence between two probability distributions on a finite set $\left\{1,\dotsc,k\right\}$ (represented as probability mass functions), and $\log$ is in the natural base (as are all logarithms and exponentials in this work). The likelihood-ratio statistic is $2nV_{n,k,P}$ [3].

In this language, the Neyman–Pearson lemma states that the uniformly most powerful hypothesis test for significance $\alpha$ rejects a hypothesis $P=(p_{1},\dots,p_{k})$ if and only if $V_{n,k,P}$ is at least $\varepsilon_{\alpha}$, where $\varepsilon_{\alpha}$ is such that $\Pr\MT@delim@Swapped\brof@star{V_{n,k,P}\geq\varepsilon_{\alpha}}\leq\alpha$. To apply this test in practice an upper bound on $\varepsilon_{\alpha}$ is needed, so to maximize the power of a provably correct finite-sample test we seek upper bounds on $\Pr\MT@delim@Swapped\brof@star{V\geq\varepsilon}$ which are meaningful (less than $1$) for $\varepsilon$ as small as possible. Equivalently, tight control on $\varepsilon$ reduces the number of samples needed to obtain a given level of significance, which is of importance in areas as disparate as high-dimensional statistics [4], combinatorial constructions in complexity theory [5], and private machine learning [6].

In this work, we focus on tail bounds for $\Pr\MT@delim@Swapped\brof@star{V_{n,k,P}\geq\varepsilon}$ which decay exponentially for small $\varepsilon$, ideally when $\varepsilon\approx\operatorname*{E}\MT@delim@Swapped\brof@star{V_{n,k,P}}$. Paninski [7] showed that $\operatorname*{E}\MT@delim@Swapped\brof@star{V_{n,k,P}}\leq\log\MT@delim@Swapped\of@star{1+\frac{k-1}{n}}\leq\frac{k-1}{n}$, and conversely Jiao et al. [8] showed that for $P$ the uniform distribution and large enough $n$ that $\operatorname*{E}\MT@delim@Swapped\brof@star{V_{n,k,U_{k}}}\geq\frac{k-1}{n}\cdot\frac{1}{2}$, so in general the smallest $\varepsilon$ for which one can expect a meaningful bound is of order $(k-1)/n$. In this work, we derive the first tail bound decaying exponentially in $\varepsilon$ for $\varepsilon$ as small as $(k-1)/n$, whereas existing bounds either require $\varepsilon$ to be at least order $(k-1)/n\cdot\log(n/k)$ when $k<n$ ([9, 10]) or work only for the uniform distribution and decay exponentially in $\varepsilon^{2}$ ([11]), which when $\varepsilon<1$ is significantly weaker than decay in $\varepsilon^{1}$. Formally, our result is as follows:

Theorem I.2**.**

Let $V_{n,k,P}$ be as in I.1. Then for all $\varepsilon>\frac{k-1}{n}$, it holds that

[TABLE]

Theorem I.2 is in fact an immediate corollary of our main technical result, which is a bound on the moment generating function of $V_{n,k,P}$.

Theorem I.3**.**

Let $V_{n,k,P}$ be as in I.1. Then for all $0\leq t<n$ it holds that

[TABLE]

Note that this is also the moment generating function of a gamma distribution with shape $k-1$ and rate $n$. Bounding the moment generating function is a standard technique to obtain concentration bounds (see e.g. [12]), but to the best of our knowledge Theorem I.3 is the first to give a finite bound on $\operatorname*{E}\MT@delim@Swapped\brof@star{\exp\MT@delim@Swapped\of@star{s\cdot 2nV_{n,k,P}}}$ independent of $n$ for any constant $s>0$. As a consequence, we are able to give the first (to the best of our knowledge) upper bounds on the $m$’th moments of $2nV_{n,k,P}$ which do not depend on $n$ for all $m>2$. Using Wilks’ theorem [13] on the asymptotic distribution of the likelihood-ratio statistic, we are then able to compute the asymptotic moments of $2nV_{n,k,P}$ for fixed $k$ and $P$ as $n$ goes to infinity. Furthermore, our finite sample bounds on the $m$’th non-central moment are within constant factors (with the constant depending on $m$) of the asymptotic value.

The rest of this work is organized as follows. In Section II we prove Theorems I.3 and I.2, with the proof divided into two parts: in Section II.A we show Theorem I.3 can be derived from bounds for the special case of a binary alphabet ($k=2$), e.g. a binomial distribution, and in Section II.B we give a bound for this simpler case. In Section III we use Theorem I.3 to derive moment bounds and asymptotic results. Finally, in Section IV we compare our bounds to existing results in the literature and suggest possible directions for future research, and in particular conjecture an improvement to Theorem I.3 which would nearly close the quadratic gap between our finite-sample bound and the bound of Wilks’ theorem on the asymptotic distribution of likelihood-ratio statistic (which does not hold in general for finite $n$).

II Proof of Finite-Sample Bounds

In this section we prove our main technical result, the moment generating function bound of Theorem I.3, and use it to derive our new tail bound Theorem I.2.

II.A Reducing the Multinomial to the Binomial

We first show that the moment generating function of the empirical relative entropy for arbitrary finite alphabets of size $k$ can be bounded in terms of the special case $k=2$. Formally, this requires the bound to be of a particular form:

Definition II.1**.**

A function $f:[0,1)\to\mathbb R$ is a sample-independent MGF bound for the binomial KL if for every positive integer $n$, real $t\in[0,n)$, and $p\in[0,1]$ it holds that

[TABLE]

Remark 1*.*

Recalling that $2nV_{n,k,P}$ is the likelihood-ratio statistic, II.1 is equivalent to requiring bounds on the moment generating function $\operatorname*{E}\MT@delim@Swapped\brof@star{\exp\MT@delim@Swapped\of@star{s\cdot 2nV_{n,2,(p,1-p)}}}$ for $0\leq s<1/2$ which do not depend on $n$ or $p$.

We can now state our reduction.

Proposition II.2**.**

Let $P=(p_{1},\dotsc,p_{k})$ be a distribution on a set of size $k$ for $k\geq 2$. Then for every sample-independent MGF bound for the binomial KL $f:[0,1)\to\mathbb R$ and $0\leq t<n$, the moment generating function of $V_{n,k,P}$ satisfies

[TABLE]

Proof.

This is a simple induction on $k$. The base case $k=2$ holds by definition of sample-independent MGF bound for the binomial KL.

For the inductive step, we compute conditioned on the value of $X_{k}$. Note that if $p_{k}=1$ then the inductive step is trivial since $V_{n,k,P}=0$ with probability $1$, so assume that $p_{k}<1$. For each $i\in\left\{1,\dotsc,k-1\right\}$ define $p^{\prime}_{i}=p_{i}/(1-p_{k})$, so that conditioned on $X_{k}=m$, the variables $(X_{1},\dotsc,X_{k-1})$ are distributed multinomially with $n-m$ samples and probabilities $P^{\prime}=(p^{\prime}_{1},\dotsc,p^{\prime}_{k-1})$. Simple rearranging (using the chain rule) implies that

[TABLE]

where

[TABLE]

and where we treat the second term of Eq. II.1 as [math] if $X_{k}=n$. Now for every $0\leq t<n$ we have

[TABLE]

By definition of a sample-independent MGF bound for the binomial KL, the second term is at most $f(t/n)$, so we get a bound of $f(t/n)^{k-1}$ as desired. ∎

Remark 2*.*

Mardia et al. [10] use the same chain rule decomposition of the multinomial KL to inductively bound the (non-exponential) moments.

II.B Bounding the Binomial

It remains to give a sample-independent MGF bound for the binomial KL:

Proposition II.3**.**

The function

[TABLE]

is a sample-independent MGF bound for the binomial KL.

Remark 3*.*

Hoeffding’s inequality [14] can be used to give a simple proof of the weaker claim that $2^{x}/(1-x)$ is a sample-independent MGF bound for the binomial KL.

Proof.

Let $B_{n,p}$ denote a random variable with $\operatorname{Binomial}(n,p)$ distribution. Using the fact that

[TABLE]

for any integers $0\leq i\leq n$, we can expand the moment generating function as

[TABLE]

For every $n$ and $i$, the function $q\mapsto\Pr\MT@delim@Swapped\brof@star{B_{n,q}=i}=\binom{n}{i}q^{i}(1-q)^{n-i}$ is easily seen to be log-concave over $[0,1]$, so we can upper bound the moment generating function by

[TABLE]

It turns out $G_{n}$ does not depend on $p$ and can be simplified significantly, which we prove in the following two lemmas.

Lemma II.4**.**

For all non-negative integers $n$ and real numbers $x$ and $p$ we have $G_{n}(p,x)=G_{n}(0,x)$.

Proof.

Define $R_{n}(q,x)=\sum_{i=0}^{n}\binom{n}{i}\MT@delim@Swapped\of@star{q+ix/n}^{i}\MT@delim@Swapped\of@star{1-q-ix/n}^{n-i}$ (where when $i=n=0$ we treat $0/0=1$) so that $G_{n}(p,x)=R_{n}\MT@delim@Swapped\of@star{(1-x)p,x}$ and it suffices to prove that $R_{n}(q,x)=R_{n}(0,x)$. We prove this by induction on $n$: the base case of $n=0$ holds since $R_{n}(q,x)=1$ always, and for the inductive step we have

[TABLE]

where the last line is by the inductive hypothesis. ∎

Lemma II.5**.**

For all non-negative integers $n$ we have $G_{n}(p,x)=\displaystyle\sum_{i=0}^{n}\frac{n!}{n^{i}(n-i)!}\cdot x^{i}$.

Proof.

By Lemma II.4 we have that $G_{n}(p,x)=G_{n}(0,x)=\sum_{i=0}^{n}\MT@delim@Swapped\of@star{\frac{ix}{n}}^{i}\MT@delim@Swapped\of@star{1-\frac{ix}{n}}^{n-i}$ is a polynomial in $x$ of degree at most $n$. For any non-negative integer $i\leq n$ we can compute the coefficient of $x^{i}$ in $G_{n}(0,x)$ by summing over the power of $x$ contributed by the $(jx/n)^{j}$ term for each $j$:

[TABLE]

where $\frac{1}{i!}\sum_{j=0}^{i}\binom{i}{j}j^{i}(-1)^{i-j}$ is by definition the Stirling number of the second kind $\genfrac{\{}{\}}{0.0pt}{}{i}{i}$ and is equal to $1$ (see e.g. [15, Chapter 6.1]), so that we can simplify this to

[TABLE]

as desired. ∎

Putting together Lemma II.4 and Lemma II.5, we have that the moment generating function is at most $G_{n}(p,x)=\sum_{i=0}^{n}\frac{n!}{n^{i}(n-i)!}x^{i}$, where $\frac{n!}{n^{i}(n-i)!}=\prod_{j=0}^{i-1}\MT@delim@Swapped\of@star{1-j/n}\leq 1$ and thus for each $x\in[0,1)$ we have $G_{n}(p,x)\leq\sum_{i=0}^{n}x^{i}\leq\sum_{i=0}^{\infty}x^{i}=1/(1-x)$. ∎

Together, Propositions II.2 and II.3 imply our moment generating function bound (Theorem I.3), and thus a Chernoff bound implies our tail bound:

Proof of Theorem I.2.

By Theorem I.3, we know for every $t\in[0,n)$ that $\operatorname*{E}\MT@delim@Swapped\brof@star{\exp\MT@delim@Swapped\of@star{t\cdot V_{n,k,P}}}\leq\MT@delim@Swapped\of@star{\frac{1}{1-t/n}}^{k-1}$, so by a Chernoff bound

[TABLE]

The result follows by making the optimal choice $t/n=1-(k-1)/(\varepsilon n)$ when $\varepsilon>\MT@delim@Swapped\of@star{k-1}/{n}$. ∎

III Moment and Asymptotic Bounds

In this section we use Theorem I.3 to give finite-sample and asymptotic bounds on the moments of $V_{n,k,P}$. We will need some basic facts about subexponential random variables, for which we follow the textbook of Vershynin [16].

Lemma III.1** ([16, Definition 2.7.5, Proposition 2.7.1]).**

There is a universal constant $C>0$ such that every real-valued random variable $X$ with finite subexponential norm $\MT@delim@Swapped\norm@star{X}_{\psi_{1}}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny def}}}{=}}\inf\left\{t>0:\operatorname*{E}\MT@delim@Swapped\brof@star{\exp\MT@delim@Swapped\of@star{\MT@delim@Swapped\abs@star{X}/t}}\leq 2\right\}$ satisfies $\operatorname*{E}\MT@delim@Swapped\brof@star{\MT@delim@Swapped\abs@star{X}^{m}}^{1/m}\leq Cm\MT@delim@Swapped\norm@star{X}_{\psi_{1}}$ for all $m\geq 1$.

Lemma III.1 allows us to bound the moments of $2nV_{n,k,P}$ uniformly for all $n$.

Theorem III.2**.**

For every $n$, $k$, and $P$, it holds that $\MT@delim@Swapped\norm@star{2nV_{n,k,P}}_{\psi_{1}}\!\leq 4(k-1)$ and $\MT@delim@Swapped\norm@star{2nV_{n,k,P}-\operatorname*{E}\MT@delim@Swapped\brof@star{2nV_{n,k,P}}}_{\psi_{1}}\allowbreak\leq 8(k-1)$. In particular, there exist universal constants $C_{1},C_{2}>0$ such that for all $n$, $k$, $P$ and $m\geq 1$

[TABLE]

Proof.

Theorem I.3 implies for all $n$, $k$, and $P$ that $\operatorname*{E}\MT@delim@Swapped\brof@star{\exp\MT@delim@Swapped\of@star{\frac{1}{4(k-1)}\cdot 2nV_{n,k,P}}}\leq\MT@delim@Swapped\of@star{1-\frac{1}{2(k-1)}}^{-(k-1)}\leq 2$, so by Lemma III.1 we have that $\MT@delim@Swapped\norm@star{2nV_{n,k,P}}_{\psi_{1}}\leq 4(k-1)$. By the triangle inequality and convexity of norms, this lets us bound the norm of the centered random variable as $\MT@delim@Swapped\norm@star{2nV_{n,k,P}-\operatorname*{E}\MT@delim@Swapped\brof@star{2nV_{n,k,P}}}_{\psi_{1}}\leq 2\MT@delim@Swapped\norm@star{2nV_{n,k,P}}_{\psi_{1}}\leq 8(k-1)$. ∎

Our asymptotic results rely on Wilks’ theorem [13] on the asymptotic behavior of the likelihood ratio test, which for fixed $k$ and $P$ implies that the random variable $2nV_{n,k,P}$ converges in distribution to the chi-squared distribution with $k-1$ degrees of freedom as $n$ goes to infinity (see also [17, Theorem 4.2]). Though in general convergence in distribution does not imply convergence of moments or of the moment generating function [18], it turns out that the bounds from Theorem III.2 are strong enough for convergence in distribution to imply convergence of the moments.

Theorem III.3**.**

Let $k\geq 2$ be an integer and $P=(p_{1},\dots,p_{k})$ be a probability distribution over a finite alphabet of size $k$ with $p_{i}\neq 0$ for every $i\in\left\{1,\dots,k\right\}$. Then for every $m\geq 1$ we have

[TABLE]

and for every $s\in[0,1/2)$ we have

[TABLE]

Remark 4*.*

[10] prove the one-sided lower bound that $\liminf_{n\to\infty}\operatorname{Var}\MT@delim@Swapped\of@star{2nV_{n,k,P}}\geq\operatorname{Var}\MT@delim@Swapped\of@star{\chi^{2}_{k-1}}$, which is a special case of the second equality above.

Proof.

Given a sequence of random variables $\MT@delim@Swapped\of@star{X_{n}}_{n\in\mathbb N}$ which convergence in distribution to a random variable $X$, a sufficient condition for $\lim_{n\to\infty}\operatorname*{E}\MT@delim@Swapped\brof@star{X_{n}}=\operatorname*{E}\MT@delim@Swapped\brof@star{X}$ is that $\sup_{n}\operatorname*{E}\MT@delim@Swapped\brof@star{\MT@delim@Swapped\abs@star{X_{n}}^{1+\alpha}}<\infty$ for some $\alpha>0$ (see e.g. [18]).

Wilks’ theorem [13] shows that $2nV_{n,k,P}$ converges in distribution to $\chi^{2}_{k-1}$, and thus the continuous mapping theorem implies that $\MT@delim@Swapped\of@star{2nV_{n,k,P}}^{m}$ converges in distribution to $\MT@delim@Swapped\of@star{\chi^{2}_{k-1}}^{m}$ for every $m\geq 1$. Theorem III.2 implies $\sup_{n}\operatorname*{E}\MT@delim@Swapped\brof@star{\MT@delim@Swapped\abs@star{\MT@delim@Swapped\of@star{2nV_{n,k,P}}^{m}}^{2}}\leq(Cm(k-1))^{2m}<\infty$, which establishes the first claim. In particular, for $m=1$ we have $\lim_{n\to\infty}\operatorname*{E}\MT@delim@Swapped\brof@star{2nV_{n,k,P}}=\operatorname*{E}\MT@delim@Swapped\brof@star{\chi^{2}_{k-1}}$, so Slutsky’s theorem implies that $2nV_{n,k,P}-\operatorname*{E}\MT@delim@Swapped\brof@star{2nV_{n,k,P}}$ converges in distribution to $\chi^{2}_{k-1}-\operatorname*{E}\MT@delim@Swapped\brof@star{\chi^{2}_{k-1}}$. Again by the continuous mapping theorem we thus have that $\MT@delim@Swapped\of@star{2nV_{n,k,P}-\operatorname*{E}\MT@delim@Swapped\brof@star{2nV_{n,k,P}}}^{m}$ converges in distribution to $\MT@delim@Swapped\of@star{\chi^{2}_{k-1}-\operatorname*{E}\MT@delim@Swapped\brof@star{\chi^{2}_{k-1}}}^{m}$, so since Theorem III.2 implies $\sup_{n}\operatorname*{E}\MT@delim@Swapped\brof@star{\MT@delim@Swapped\abs@star{\MT@delim@Swapped\of@star{2nV_{n,k,P}-\operatorname*{E}\MT@delim@Swapped\brof@star{2nV_{n,k,P}}}^{m}}^{2}}\leq(Cm(k-1))^{2m}<\infty$, we also get the second claim.

For the moment generating function claims, first note that they are trivial for $s=0$, as both sides are always $1$, and for $s\in(0,1/2)$ we have $1/2>1/4+s/2>s$. Now, since the continuous mapping theorem implies $\exp(s\cdot 2nV_{n,k,P})$ converges in distribution to $\exp\MT@delim@Swapped\of@star{s\cdot\chi^{2}_{k-1}}$, and Theorem I.3 implies $\sup_{n}\operatorname*{E}\MT@delim@Swapped\brof@star{\MT@delim@Swapped\abs@star{\exp\MT@delim@Swapped\of@star{(1/4+s/2)\cdot 2nV_{n,k,P}}}}\leq\MT@delim@Swapped\of@star{1/2-s}^{k-1}<\infty$, we get the third claim. Finally, for the last claim, we again have that $\exp\MT@delim@Swapped\of@star{s\cdot\MT@delim@Swapped\of@star{2nV_{n,k,P}-\operatorname*{E}\MT@delim@Swapped\brof@star{2nV_{n,k,P}}}}$ converges in distribution to $\exp\MT@delim@Swapped\of@star{s\cdot\MT@delim@Swapped\of@star{\chi^{2}_{k-1}-\operatorname*{E}\MT@delim@Swapped\brof@star{\chi^{2}_{k-1}}}}$ by the continuous mapping theorem, and since $V_{n,k,P}\geq 0$ we have $\exp\MT@delim@Swapped\of@star{(1/4+s/2)\cdot\MT@delim@Swapped\of@star{2nV_{n,k,P}-\operatorname*{E}\MT@delim@Swapped\brof@star{2nV_{n,k,P}}}}\leq\exp\MT@delim@Swapped\of@star{(1/4+s/2)\cdot 2nV_{n,k,P}}$ and we conclude as for the third claim. ∎

IV Discussion

In this section we compare our bounds to existing results in the literature and discuss possible directions for future work.

IV.A Moment generating function bounds

To the best of our knowledge, this work is the first to explicitly consider the moment generating function of the empirical divergence, and existing tail bounds do not give finite bounds on $\sup_{n}\operatorname*{E}\MT@delim@Swapped\brof@star{\exp\MT@delim@Swapped\of@star{x\cdot nV_{n,k,P}}}=\sup_{n}\int_{0}^{\infty}\Pr\MT@delim@Swapped\brof@star{nV_{n,k,P}>\frac{\log t}{x}}\,dt$ for any $k\geq 3$ or constant $x>0$. Thus, we focus on comparing our finite sample bound (Theorem I.3) to the asymptotic one (Theorem III.3).

In Theorem III.3 we showed for all $x\in[0,1)$ that $\lim_{n\to\infty}\operatorname*{E}\MT@delim@Swapped\brof@star{\exp\MT@delim@Swapped\of@star{x\cdot nV_{n,k,P}}}=\MT@delim@Swapped\of@star{1-x}^{-(k-1)/2}$, whereas our finite sample bound of Theorem I.3 instead gave the upper bound $\operatorname*{E}\MT@delim@Swapped\brof@star{\exp\MT@delim@Swapped\of@star{x\cdot nV_{n,k,P}}}\leq\MT@delim@Swapped\of@star{1-x}^{-(k-1)}$, which is quadratically worse. This loss arises from our binomial bound from Proposition II.3 of $(1-x)^{-1}$ for the case $k=2$, where the correct asymptotic bound is $\MT@delim@Swapped\of@star{1-x}^{-1/2}$. Unfortunately, it is not the case that this latter asymptotic bound holds for all $n$, $p$, and $0\leq x<1$: indeed, this is violated even for $(n,p,x)=(2,1/2,1/2)$. Nevertheless, we conjecture that Proposition II.3 can be improved to something closer to the asymptotic bound:

Conjecture IV.1**.**

The function

[TABLE]

is a sample-independent MGF bound for the binomial KL.

Remark 5*.*

$1/\sqrt{1-x}\leq 2/\sqrt{1-x}-1\leq 1/(1-x)$ for all $x\in[0,1)$.

IV.1 would follow from the following more natural conjecture, which looks at a single branch of the KL divergence and is supported by numerical evidence:

Conjecture IV.2**.**

Letting

[TABLE]

it holds for every positive integer $n$, real $t\in[0,n)$, and $p\in[0,1]$ that

[TABLE]

where $B\sim\operatorname{Binomial}(n,p)$.

Remark 6*.*

We believe the results (or techniques) of Zubkov and Serov [19] and Harremoës [20] strengthening Hoeffding’s inequality may be of use in proving these conjectures.

Proof of IV.1 given IV.2.

We have that

[TABLE]

so for every $i\in\left\{0,1,\dotsc,n\right\}$

[TABLE]

Letting x=\exp\MT@delim@Swapped\of@nostar[\Big{]}{t\cdot\operatorname{\operatorname{D}}_{>}\left(i/n\;\middle\|\;\mathopen{}p\right)} and y=\exp\MT@delim@Swapped\of@nostar[\Big{]}{t\cdot\operatorname{\operatorname{D}}_{>}\left(1-i/n\;\middle\|\;\mathopen{}1-p\right)}, we have that at least one of $x$ and $y$ is equal to $1$, so that

[TABLE]

and thus by taking expectations over $i=B$ for $B\sim\operatorname{Binomial}(n,p)$, we get

[TABLE]

We conclude by bounding both terms using IV.2, since $n-B$ is distributed as $\operatorname{Binomial}(n,1-p)$. ∎

IV.B Moment bounds

The moments of $V_{n,k,P}$ have seen some study in the literature. Most notably, Paninski [7] showed by comparison to the $\chi^{2}$-statistic that $\operatorname*{E}\MT@delim@Swapped\brof@star{V_{n,k,P}}\leq\log\MT@delim@Swapped\of@star{1+\frac{k-1}{n}}\leq\frac{k-1}{n}$. In the reverse direction, [8] showed that if $n\geq 15k$ then for the uniform distribution it holds that $\operatorname*{E}\MT@delim@Swapped\brof@star{V_{n,k,U_{k}}}\geq\frac{k-1}{2n}$, complementing the asymptotic result that $\lim_{n\to\infty}\operatorname*{E}\MT@delim@Swapped\brof@star{nV_{n,k,U_{k}}}=\frac{k-1}{2}$, which follows from Theorem III.3 (and can also be derived from [10]). For higher moments, [10] showed that $\operatorname{Var}(V_{n,k,P})\leq Ck/n^{2}$ for some constant $C$, and asymptotically that $\liminf_{n\to\infty}\operatorname{Var}(2nV_{n,k,P})\geq\operatorname{Var}(\chi^{2}_{k-1})=2(k-1)$. To the best of our knowledge, no bounds on the higher moments have appeared in the literature.

In Theorem III.2 we showed for every $m\geq 1$ that $\operatorname*{E}\MT@delim@Swapped\brof@star{\MT@delim@Swapped\of@star{2nV_{n,k,P}}^{m}}\leq\MT@delim@Swapped\of@star{Cm(k-1)}^{m}$ for some universal constant $C>0$, and we showed in Theorem III.3 the asymptotic equality $\lim_{n\to\infty}\operatorname*{E}\MT@delim@Swapped\brof@star{\MT@delim@Swapped\of@star{2nV_{n,k,P}}^{m}}=2^{m}\frac{\Gamma\MT@delim@Swapped\of@star{m+\frac{k-1}{2}}}{\Gamma\MT@delim@Swapped\of@star{\frac{k-1}{2}}}=\MT@delim@Swapped\of@star{C^{\prime}m(k-1)}^{m}$ where $C^{\prime}$ is bounded in a constant range. Thus, our finite-sample bound is asymptotically optimal up to the universal constant $C$.

However, the situation is quite different for the central moments $\operatorname*{E}\MT@delim@Swapped\brof@star{\MT@delim@Swapped\of@star{2nV_{n,k,P}-\operatorname*{E}\MT@delim@Swapped\brof@star{2nV_{n,k,P}}}^{m}}$, where we again showed the finite sample bound $\MT@delim@Swapped\of@star{Cm(k-1)}^{m}$, but asymptotically from Theorem III.3 the bound is $\MT@delim@Swapped\of@star{C^{\prime}m(k-1)}^{\MT@delim@Swapped\floor@star{m/2}}$ for $m\geq 2$ and some $C^{\prime}$ in a constant range. For $m=2$, [10] were able to achieve this bound up to constant factors, but it is an intriguing open question to get finite sample central moment bounds with the asymptotically correct power for $m>2$.

IV.C Tail bound

To understand our tail bound (Theorem I.2), we compare our result to existing bounds in the literature. Antos and Kontoyiannis [11] used McDiarmid’s bounded differences inequality [21] to give a concentration bound for the empirical entropy, which in the case of the uniform distribution implies the bound

[TABLE]

This bound has the advantage of providing subgaussian concentration around the expectation, but for the case of small $\varepsilon<1$ it is preferable to have a bound with linear dependence on $\varepsilon$. Unfortunately, existing tail bounds which decay like $e^{-n\varepsilon}$ are not, in the common regime of parameters where $n\gg k$, meaningful for $\varepsilon$ close to $\operatorname*{E}\MT@delim@Swapped\brof@star{V_{n,k,P}}\leq(k-1)/n$. For example, the method of types [9] is used to prove the standard bound

[TABLE]

which is commonly used in proofs of Sanov’s theorem (see e.g. [22]). However, this bound is meaningful only for $\varepsilon>\frac{1}{n}\cdot\log\binom{n+k-1}{k-1}\geq\frac{k-1}{n}\cdot\log\MT@delim@Swapped\of@star{1+\frac{n}{k-1}}$, which is off by a factor of order $\log\MT@delim@Swapped\of@star{1+\frac{n}{k-1}}$. A recent bound due to Mardia et al. [10] improved on the method of types bound for all settings of $k$ and $n$, but for $3\leq k\leq\frac{e^{2}}{2\pi}\cdot n$ still requires $\varepsilon>\frac{k}{n}\cdot\log\MT@delim@Swapped\of@star{\sqrt{\frac{e^{3}n}{2\pi k}}}>\frac{k-1}{n}\cdot\log\MT@delim@Swapped\of@star{1+\frac{n-1}{k}}/2$, which again has dependence on $\log\MT@delim@Swapped\of@star{1+\frac{n-1}{k}}$.

Thus, if $k\leq n$, then our bound is meaningful for $\varepsilon$ smaller than what is needed for the method of types bound or the bound of [10] by a factor of order $\log(n/k)$, which for $k$ as large as $n^{0.99}$ is still $\log(n)$, and for $k$ as large as $n/\log n$ is of order $\log\log n$. However, Theorem I.2 has slightly worse dependence on $\varepsilon$ than the other bounds, so for example it is better than the method of types bound if and only if

[TABLE]

In particular, when $n\geq e(k-1)$, our bound is better for $\varepsilon$ up to order $\frac{n}{k-1}$ times larger than $\frac{k-1}{n}$. However, we can also see that our bound can be better only when $\sqrt[k-1]{\binom{n+k-1}{k-1}}\geq e$, which asymptotically is equivalent to $k-1\leq Cn$, where $C\approx 1.84$ is the solution to the equation $(1+C)/C\cdot H(C/(1+C))=1$ for $H$ the binary entropy function in nats. From a finite-sample perspective, note that the condition is always satisfied in the standard setting of parameters where $n\geq k$, that is, the number of samples is larger than the size of the alphabet. In this regime, we can also compare to the “interpretable” upper bound of [10, Theorem 3], to see that Theorem I.2 is better if

[TABLE]

so that in particular our bound is better for $\varepsilon$ up to order $\sqrt{\frac{n}{k}}^{1+1/(k-1)}\geq\sqrt{\frac{n}{k}}$ times larger than $\frac{k-1}{n}$.

V Acknowledgements

The author thanks Flavio du Pin Calmon and the anonymous reviewers for their helpful comments and suggestions.