Concentration Inequalities for Multinoulli Random Variables

Jian Qian Sequel Team - Inria Lille Ronan Fruit Sequel Team - Inria Lille Matteo Pirotta Sequel Team - Inria Lille Alessandro Lazaric Facebook AI Research

Abstract

We investigate concentration inequalities for Dirichlet and Multinomial random variables.

1 Problem Formulation

We analyse the concentration properties of the random variable $Z_{n} \geq 0$ defined as:

Z_{n} := max v \in [0, D]^{S} {{({ˆ p}_{n} - p)}^{T} v}

(1)

where ${ˆ p}_{n} \in Δ^{S}$ is a random vector, $p \in Δ^{S}$ is deterministic and $Δ^{S} = {x \in R^{S} : \sum_{i = 1}^{S} x_{i} = 1 \land x_{i} \geq 0}$ is the $(S - 1)$ -dimensional simplex. It is easy to show that the maximum in Eq. 1 is equivalent to computing the (scaled) $ℓ_{1}$ -norm of the vector ${ˆ p}_{n} - p$ :

(2)

where we have used the fact that $\frac{D}{2} ({ˆ p}_{n} - p)^{T} e = 0$ . As a consequence, $Z_{n}$ is a bounded random variable in $[0, D]$ . While the following discussion apply to Dirichlet distributions, we focus on ${ˆ p}_{n} \sim \frac{1}{n} Multinomial (n, p)$ . The results previously available in the literature are summarized in the following.

The literature has analysed the concentration of the $ℓ_{1}$ -discrepancy of the true distribution and the empirical one in this setting.

Proposition 1.

(Weissman et al., 2003) Let $p \in Δ^{S}$ and $ˆ p \sim \frac{1}{n} Multinomial (n, p)$ . Then, for any $S \geq 2$ and $δ \in [0, 1]$ :

P(∥ˆp−p∥1≥√2Sln(\sfrac2δ)n)≤P(∥ˆp−p∥1≥ ⎷2ln(\sfrac(2S−2)δ)n)≤δ

(3)

This concentration inequality is at the core of the proof of UCRL, see (Jaksch et al., 2010, App. C.1). Another inequality is provided in (Devroye, 1983, Lem. 3).

Proposition 2.

(Devroye, 1983) Let $p \in Δ^{S}$ and $ˆ p \sim \frac{1}{n} Multinomial (n, p)$ . Then, for any $0 \leq δ \leq 3 exp (- 4 S / 5)$ :

P(∥ˆpn−p∥1≥5√ln(\sfrac3δ)n)≤δ

(4)

While Prop. 1 shows an explicit dependence on the dimension of the random variable, such dependence is hidden in Prop. 2 by the constraint on $δ$ . Note that for any $0 \leq δ \leq 3 exp (- 4 S / 5)$ , $√ln(\sfrac3δ)n>√4S5n$ . This shows that the $ℓ_{1}$ -deviation always scales proportionally to the dimension of the random variable, i.e., as $\sqrt{S}$ .

A better inequality. The natural question is whether is possible to derive a concentration inequality independent from the dimension of $p$ by exploiting the correlation between $ˆ p$ and the maximizer vector $v^{*}$ . This question has been recently addressed in (Agrawal and Jia, 2017, Lem. C.2):

Lemma 3.

(Agrawal and Jia, 2017) Let $p \in Δ^{S}$ and $ˆ p \sim \frac{1}{n} Multinomial (n, p)$ . Then, for any $δ \in [0, 1]$ :

P(∥ˆpn−p∥1≥√2ln(\sfrac1δ)n)≤δ

Their results resemble the one in Prop. 2 but removes the constraint on $δ$ . As a consequence, the implicit or explicit dependence on the dimension $S$ is removed. In the following, we will show that Lem. 3 may not be correct.

2 Theoretical Analysis (the asymptotic case)

In this section, we provide a counter-argument to the Lem. 3 in the asymptotic regime (i.e., $n \to + \infty$ ). The overall idea is to show that the expected value of $Z_{n}$ asymptotically grows as $O (\sqrt{S})$ and $Z_{n}$ itself is well concentrated around its expectation. As a result, we can deduce that all quantiles of $Z_{n}$ grow as $O (\sqrt{S})$ as well.

We consider the true vector $p$ to be uniform, i.e., $p = (\frac{1}{S}, \dots, \frac{1}{S})$ and $ˆ p \sim \frac{1}{n} Multinomial (n, p)$ .¹ The following lemma provides a characterization of the variable $Z_{S} := {lim}_{n \to + \infty} \sqrt{n} Z_{n}$ .

Lemma 4.

Consider $S \in N$ , $S = {1, \dots, S}$ and $p = (\frac{1}{S}, \dots, \frac{1}{S})$ be the uniform distribution on $S$ . Let $e_{S}$ be the vector of ones of dimension $S$ . Define $Y \sim N (0, I_{S} - \frac{1}{S - 1} N)$ where $N = e_{S} e_{S}^{T} - I_{S}$ is the matrix with $0$ in all the diagonal entry and $1$ elsewhere, and $Y^{+} = (max (Y_{i}, 0))_{i \in S}$ . Then:

Z_{S} = lim n \to + \infty \sqrt{n} Z_{n} \sim ∥ Y^{+} ∥_{1} D \sqrt{\frac{S - 1}{S^{2}}} .

Furthermore,

E [Z_{S}] = \sqrt{\frac{S - 1}{S^{2}}} \cdot E [S \sum i = 1 Y_{i}^{+}] = \sqrt{S - 1} \cdot E [Y_{1}^{+}] = \sqrt{\frac{S - 1}{2 π}} .

While the previous lemma may already suggest that $Z_{S}$ should grow as $O (\sqrt{S})$ as its expectation, it is still possible that a large part of the distribution is concentrated around a value independent from $S$ , with limited probability assigned to, e.g., values growing as $O (S)$ , which could justify the $O (\sqrt{S})$ growth of the expectation. Thus, in order to conclude the analysis, we need to show that $Z_{S}$ is concentrated “enough” around its expectation.

Since the random variables $Y_{i}$ are correlated, it is complicated to directly analyze the deviation of $Z_{S}$ from its mean. Thus we first apply an orthogonal transformation on $Y$ to obtain independent r.v. (recall that jointly normally distributed variables are independent if uncorrelated).

Lemma 5.

Consider the same settings of Lem. 4 and recall that $Y \sim N (0, I_{S} - \frac{1}{S - 1} N)$ . There exists an orthogonal transformation $U \in O_{S} (R)$ , s.t.

W = \sqrt{\frac{S - 1}{S}} U Y \sim N (0, [\begin{matrix} I_{S - 1} & 0 0 & 0 \end{matrix}]) .

By exploiting the transformation $U$ we can write that $Z_{S} \sim g (W) := \frac{1}{\sqrt{S}} e_{S}^{T} {(U^{T} W)}^{+}$ . Since $W_{i}$ are i.i.d. standard Gaussian random variables and $g$ is $1$ -Lipschitz, we can finally characterize the mean and the deviations of $Z_{S}$ and derive the following anticoncentration inequality for $Z_{S}$ .

Theorem 6.

Let $p \in Δ^{S} = (\frac{1}{S}, \dots, \frac{1}{S})$ and ${ˆ p}_{n} \sim \frac{1}{n} Multinomial (n, p)$ . Define $Z_{n} = {max}_{v \in [0, D]} {({ˆ p}_{n} - p)^{T} v}$ and $Z_{S} = {lim}_{n \to + \infty} \sqrt{n} Z_{n}$ . Then, for any $δ \in (0, 1)$ :

This result shows that every quantile of $Z_{S}$ is dependent on the dimension of the random variable, i.e., $\sqrt{S}$ . Similarly to Lem. 2, it is possible to lower bound the quantile by a dimension-free quantity at the price of having an exponential dependence on $S$ in $δ$ .

Appendix A Proof for the asymptotic scenario

In this section we report the proofs of lemmas and theorem stated in Sec. 2.

a.1 Proof of Lem. 4

Let $Y_{n, i} = \frac{1}{\sqrt{n} \sqrt{\frac{S - 1}{S^{2}}}} n \sum j = 1 (X_{i}^{j} - \frac{1}{S})$ and $Y_{n} = (Y_{n, i})_{i \in S}$ . Then:

	$\sqrt{n} Z_{n}$	$= \sqrt{n} max v \in [0, D]^{S} (ˆ p - p)^{T} v = \sqrt{n} max v \in [0, D]^{S} S \sum i = 1 \frac{v_{i}}{n} n \sum j = 1 (X_{i}^{j} - \frac{1}{S})$
		$= max v \in [0, D]^{S} S \sum i = 1 Y_{n, i} v_{i} \sqrt{\frac{S - 1}{S^{2}}} = D \sqrt{\frac{S - 1}{S^{2}}} \cdot e^{T} Y_{n}^{+},$

where we used the fact that the $v$ maximizing $Z_{n}$ takes the largest value $D$ for all positive components $Y_{n, i}$ and is equal to $0$ otherwise. We recall that the covariance of the normalized multinoulli variable $Y_{n, i}$ with probabilities $p_{i} = 1 / S$ is $I_{S} - \frac{1}{S - 1} N$ . As a result, a direct application of the central limit theorem gives $Yn\lx@stackrelD→Y∼N(0,IS−1S−1N)$ . Then we can apply the functional CLT and obtain , where $Y^{+}$ is a random vector obtained by truncating from below at $0$ the multi-variate Gaussian vector $Y$ . Since the marginal distribution of each random variable $Y_{i}$ is $N (0, 1)$ , i.e., are identically distributed (see definition in Lem. 4), $Y_{i}^{+}$ has a distribution composed by a Dirac distribution in $0$ and a half normal distribution, and its expected value is $E [Y_{i}^{+}] = 1 / \sqrt{2 π}$ , while leads to the final statement on the expectation.

a.2 Proof of Lem. 5

Denote $λ (A)$ the set of eigenvalues of square matrix $A$ . Let $B \in R^{S \times S}$ such that $B = [\begin{matrix} 0_{S, S - 1} & e_{S} \end{matrix}]$ , where $0_{S, S - 1} \in R^{S \times (S - 1)}$ is a matrix full of zeros. Then, we can write the eigenvalues of the covariance matrix of $Y$ as

	$λ (I_{S} - \frac{1}{S - 1} N)$	$= λ (\frac{S}{S - 1} I_{S} - \frac{1}{S - 1} e_{S} e_{S}^{T}) = λ (\frac{S}{S - 1} I_{S} - \frac{1}{S - 1} B B^{T})$
		$= \frac{S}{S - 1} λ (I_{S} - \frac{1}{S - 1} B^{T} B) = \frac{S}{S - 1} λ (I_{S} - [\begin{matrix} 0_{S - 1} & 0 0 & 1 \end{matrix}]),$

where we use the fact that $λ (I - A^{T} A) = λ (I - A A^{T})$ . As a result, the covariance of $Y$ has one eigenvalue at $0$ and eigenvalues equal to $\frac{S}{S - 1}$ with multiplicity $S - 1$ . As a result, we can diagonalize it with an orthogonal matrix $U \in O_{S} (R)$ (obtained using the normalized eigenvectors) and obtain

U (I_{S} - \frac{1}{S - 1} N) U^{T} = [\begin{matrix} \frac{S}{S - 1} I_{S - 1} & 0 0 & 0 \end{matrix}] .

Define $W = \sqrt{\frac{S - 1}{S}} U Y$ , then:

	$C o v (W, W)$	$= \frac{S - 1}{S} C o v (U Y, U Y) = \frac{S - 1}{S} U C o v (Y, Y) U^{T}$
		$= \frac{S - 1}{S} U (I_{S} - \frac{1}{S - 1} N) U^{T} = [\begin{matrix} I_{S - 1} & 0 0 & 0 \end{matrix}] .$

Thus $W \sim N (0, [\begin{matrix} I_{S - 1} & 0 0 & 0 \end{matrix}])$ .

a.3 Proof of Thm. 6

By exploiting Lem. 4 and Lem. 5 we can write:

Z_{S}

\sim e_{S}^{T} Y^{+} \cdot \sqrt{\frac{S - 1}{S^{2}}} = e_{S}^{T} {(\sqrt{\frac{S}{S - 1}} U^{T} W)}^{+} \cdot \sqrt{\frac{S - 1}{S^{2}}} = e_{S}^{T} {(U^{T} W)}^{+} \cdot \frac{1}{\sqrt{S}}

Let $g (\cdot) = e_{S}^{T} {(U^{T} \cdot)}^{+} \frac{1}{\sqrt{S}}$ . Then $g$ is $1$ -Lipschitz:

| g (x) - g (y) |

\leq L i p (e_{s}^{T} \cdot) L i p (U^{T} \cdot) L i p ((\cdot)^{+}) \frac{1}{\sqrt{S}} ∥ x - y ∥_{2} = \sqrt{S} \cdot 1 \cdot 1 \cdot \frac{1}{\sqrt{S}} ∥ x - y ∥_{2}

where $L i p (f)$ denotes the Lipschitz constant of a function $f$ and we exploit the fact that $U$ is an orthonormal matrix.

We can now study the concentration of the variable $Z_{S}$ . Given that $W$ is a vector of i.i.d. standard Gaussian variables² and $g$ is $1$ -Lipschitz, we can use (Wainwright, 2017, Thm. 2.4) to prove that for all $t > 0$ :

P (Z_{S} \geq E [Z_{S}] - t)

\geq 1 - P (| Z_{S} - E [Z_{S}] | \geq t) \geq 1 - 2 e^{- \frac{t^{2}}{2}} .

Substituting the value of $E [Z_{S}]$ and inverting the bound gives the desired statement.

Footnotes

The analysis holds also in the case $ˆ p \sim Direchlet (n p)$ , see (Osband and Roy, 2017).
Note that we can drop the last component of $W$ since it is deterministically zero.

References

Shipra Agrawal and Randy Jia. Optimistic posterior sampling for reinforcement learning: worst-case regret bounds. In NIPS, pages 1184–1194, 2017.
Luc Devroye. The equivalence of weak, strong and complete convergence in $ℓ_{1}$ for kernel density estimates. The Annals of Statistics, 11(3):896–904, 09 1983. doi: 10.1214/aos/1176346255.
Thomas Jaksch, Ronald Ortner, and Peter Auer. Near-optimal regret bounds for reinforcement learning. Journal of Machine Learning Research, 11:1563–1600, 2010.
Ian Osband and Benjamin Van Roy. Why is posterior sampling better than optimism for reinforcement learning? In ICML, volume 70 of Proceedings of Machine Learning Research, pages 2701–2710. PMLR, 2017.
Wainwright. High-dimensional statistics: A non-asymptotic viewpoint. 2017.
Tsachy Weissman, Erik Ordentlich, Gadiel Seroussi, Sergio Verdu, and Marcelo J Weinberger. Inequalities for the l1 deviation of the empirical distribution. Technical Report HPL-2003-97R1, Hewlett-Packard Labs, 2003.

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