Novel Change of Measure Inequalities with Applications to PAC-Bayesian Bounds and Monte Carlo Estimation

Yuki Ohnishi Department of Statistics, Purdue University Jean Honorio Department of Computer Science, Purdue University

Abstract

We introduce several novel change of measure inequalities for two families of divergences: $f$ -divergences and $\alpha$ -divergences. We show how the variational representation for $f$ -divergences leads to novel change of measure inequalities. We also present a multiplicative change of measure inequality for $\alpha$ -divergences and a generalized version of Hammersley-Chapman-Robbins inequality. Finally, we present several applications of our change of measure inequalities, including PAC-Bayesian bounds for various classes of losses and non-asymptotic intervals for Monte Carlo estimates.

1 Introduction

The Probably Approximate Correct (PAC) Bayesian inequality was introduced by [32] and [23]. This framework allows us to produce PAC performance bounds (in the sense of a loss function) for Bayesian-flavored estimators [12], and several extensions have been proposed to date (see e.g., [30, 24, 21, 31, 2]). The core of these theoretical results is summarized by a change of measure inequality. The change of measure inequality is an expectation inequality involving two probability measures where the expectation with respect to one measure is upper-bounded by the divergence between the two measures and the moments with respect to the other measure. The change of measure inequality also plays a major role in information theory. For instance, [17] derived robust uncertainty quantification bounds for statistical estimators of interest with change of measure inequalities. Recent research efforts have been put into more generic perspectives to get PAC-Bayes bounds and to get rid of assumptions such as boundedness of the loss function (e.g., [19, 34, 25, 13, 11]). All PAC-Bayesian bounds contained in these works massively rely on one of the most famous change of measure inequalities, the Donsker-Varadhan representation for the KL-divergence. Several change of measure inequalities had been proposed along with PAC-Bayes bounds lately. [20] proposed a proof scheme of PAC-Bayesian bounds based on the Rényi divergence. [15] proposed an inequality for the $\chi^{2}$ divergence and derived a PAC-Bayesian bound for linear classification. [1] proposed a novel change of measure inequality and PAC-Bayesian bounds based on the $\alpha$ -divergence. The aforementioned works were proposed for specific purposes. A comprehensive study on change of measure inequalities has not been performed yet. Our work proposes several novel and general change of measure inequalities for two families of divergences: $f$ -divergences and $\alpha$ -divergences. It is a well-known fact that the $f$ -divergence can be variationally characterized as the maximum of an optimization problem rooted in the convexity of the function $f$ . This variational representation has been recently used in various applications of information theory, such as $f$ -divergence estimation [26] and quantification of the bias in adaptive data analysis [16]. Recently, [29] showed that the variational representation of the $f$ -divergence can be tightened when constrained to the space of probability densities.

Our main contributions are as follows:

•

By using the variational representation for $f$ -divergences, we derive several change of measure inequalities. We perform the analysis for the constrained regime (to the space of probability densities) as well as the unconstrained regime.
•

We present a multiplicative change of measure inequality for the family of $\alpha$ -divergences. This generalizes the previous results [1, 15] for the $\alpha$ -divergence, which in turns apply to PAC-Bayes inequalities for types of losses not considered before.
•

We also generalize prior results for the Hammersley-Chapman-Robbins inequality [18] from the particular $\chi^{2}$ divergence, to the family of $\alpha$ -divergences.
•

We provide new PAC-Bayesian bounds with the $\alpha$ -divergence and the $\chi^{2}$ -divergence from our novel change of measure inequalities for bounded, sub-Gaussian, sub-exponential and bounded-variance loss functions. Our results are either novel, or have a tighter complexity term than existing results in the literature, and pertain to important machine learning prediction problems, such as regression, classification and structured prediction.
•

We provide a new scheme for estimation of non-asymptotic intervals for Monte Carlo estimates. Our results indicate that the empirical mean over a sampling distribution concentrates around an expectation with respect to any arbitrary distribution.

2 Change of Measure Inequalities

Table 1: Summary of the change of measure inequalities. For simplicity, we denote

\mathbb{E}_{P}[\cdot]\equiv\mathbb{E}_{h\sim P}[\cdot]

and

\phi\equiv\phi(h)

Bound Type	Divergence	Uppper-Bound for Every $Q$ and a Fixed $P$	Reference
Constrained	KL	$\mathbb{E}_{Q}[\phi]\leq KL(Q\\|P)+\log(\mathbb{E}_{P}[e^{\phi}])$	[23]
Variational	Pearson $\chi^{2}$	$\mathbb{E}_{Q}[\phi]\leq\chi^{2}(Q\\|P)+\mathbb{E}_{P}[\phi]+\frac{1}{4}\mathbb{V}{\rm ar}_{P}[\phi]$	Lemma 2
Representation	Total Variation	$\mathbb{E}_{Q}[\phi]\leq TV(Q\\|P)+\mathbb{E}_{P}[\phi]$ for $\phi\in[0,1]$	Lemma 4
Unconstrained	KL	$\mathbb{E}_{Q}[\phi]\leq KL(Q\\|P)+(\mathbb{E}_{P}[e^{\phi}]-1)$	[23]
Variational	Pearson $\chi^{2}$	$\mathbb{E}_{Q}[\phi]\leq\chi^{2}(Q\\|P)+\mathbb{E}_{P}[\phi]+\frac{1}{4}\mathbb{E}_{P}[\phi^{2}]$	Lemma 3
Representation	Total Variation	$\mathbb{E}_{Q}[\phi]\leq TV(Q\\|P)+\mathbb{E}_{P}[\phi]$ for $\phi\in[0,1]$
	$\alpha$	$\mathbb{E}_{Q}[\phi]\leq D_{\alpha}(Q\\|P)+\frac{(\alpha-1)^{\frac{\alpha}{\alpha-1}}}{\alpha}\mathbb{E}_{P}[\phi^{\frac{\alpha}{\alpha-1}}]+\frac{1}{\alpha(\alpha-1)}$	Lemma 5
	Squared Hellinger	$\mathbb{E}_{Q}[\phi]\leq H^{2}(Q\\|P)+\mathbb{E}_{P}[\frac{\phi}{1-\phi}]$ for $\phi\ <1$	Lemma 6
	Reverse KL	$\mathbb{E}_{Q}[\phi]\leq\overline{KL}(Q\\|P)+\mathbb{E}_{P}[\log(\frac{1}{1-\phi})]$ for $\phi\ <1$	Lemma 7
	Neyman $\chi^{2}$	$\mathbb{E}_{Q}[\phi]\leq\overline{\chi^{2}}(Q\\|P)+2-2\mathbb{E}_{P}[\sqrt{1-\phi}]$ for $\phi\ <1$	Lemma 8
Multiplicative	Pearson $\chi^{2}$	$\mathbb{E}_{Q}[\phi]\leq\sqrt{(\chi^{2}(Q\\|P)+1)\mathbb{E}_{P}[\phi^{2}]}$	[15]
	$\alpha$	$\mathbb{E}_{Q}[\phi]\leq(\alpha(\alpha-1)D_{\alpha}(Q\\|P)+1)^{\frac{1}{\alpha}}(\mathbb{E}_{P}[\|\phi\|^{\frac{\alpha}{\alpha-1}}])^{\frac{\alpha-1}{\alpha}}$	[1]
Generalized	Pearson $\chi^{2}$	$\chi^{2}(Q\\|P)\geq\frac{(\mathbb{E}_{Q}[\phi]-\mathbb{E}_{P}[\phi])^{2}}{\mathbb{V}{\rm ar}_{P}[\phi]}$	[18]
HCR	Pseudo $\alpha$	$\|\mathbb{E}_{Q}[\phi]-\mathbb{E}_{P}[\phi]\|\leq\widetilde{\mathcal{D_{\alpha}}}(Q\\|P)^{\frac{1}{\alpha}}(\mathbb{E}_{P}[\|\phi-\mu_{P}\|^{\frac{\alpha}{\alpha-1}}])^{\frac{\alpha-1}{\alpha}}$	Lemma 12

Table 2: Some common

f

-divergences with corresponding generator.

Divergence	Formula with probability measures $P$ and $Q$ defined on a common space $\mathcal{H}$	Corresponding Generator $f(t)$
KL	$KL(Q\\|P)=\int_{\mathcal{H}}\log\frac{dQ}{dP}dQ$	$t\log t-t+1$
Reverse KL	$\overline{KL}(Q\\|P)=\int_{\mathcal{H}}\log\frac{dP}{dQ}dP$	$-\log t$
Pearson $\chi^{2}$	$\chi^{2}(Q\\|P)=\int_{\mathcal{H}}(\frac{dQ}{dP}-1)^{2}dP$	$(t-1)^{2}$
Neyman $\chi^{2}$	$\overline{\chi^{2}}(Q\\|P)=\int_{\mathcal{H}}(\frac{dP}{dQ}-1)^{2}dQ$	$\frac{(1-t)^{2}}{t}$
Total Variation	$TV(Q\\|P)=\frac{1}{2}\int_{\mathcal{H}}\|\frac{dQ}{dP}-1\|dP$	$\frac{1}{2}\|t-1\|$
Squared Hellinger	$H^{2}(Q\\|P)=\int_{\mathcal{H}}(\sqrt{\frac{dQ}{dP}}-1)^{2}dP$	$(\sqrt{t}-1)^{2}$
$\alpha$	$D_{\alpha}(Q\\|P)=\frac{1}{\alpha(\alpha-1)}\int_{\mathcal{H}}(\|\frac{dQ}{dP}\|^{\alpha}-1)dP$	$\frac{t^{\alpha}-1}{\alpha(\alpha-1)}$
Pseudo $\alpha$	$\widetilde{\mathcal{D_{\alpha}}}(Q\\|P)=\int_{\mathcal{H}}\|\frac{dQ}{dP}-1\|^{\alpha}dP$	$\|t-1\|^{\alpha}$
$\phi_{p}$ [1]	$D_{\phi_{p}-1}(Q\\|P)=\int_{\mathcal{H}}(\|\frac{dQ}{dP}\|^{p}-1)dP$	$t^{p}-1$

In this section, we formalize the definition of $f$ -divergences and present the constrained representation (to the space of probability measures) as well as the unconstrained representation. Then, we provide different change of measure inequalities for several divergences. We also provide multiplicative bounds as well as a generalized Hammersley-Chapman-Robbins bound. Table 1 summarizes our results.

2.1 Change of Measure Inequality from the Variational Representation of $f$ -divergences

Let $f:(0,+\infty)\rightarrow\mathbb{R}$ be a convex function. The convex conjugate $f^{*}$ of $f$ is defined by:

f^{*}(y)=\sup_{x\in\mathbb{R}}(xy-f(x)).

(1)

The definition of $f^{*}$ yields the following Young-Fenchel inequality

f(x)\geq xy-f^{*}(y)

which holds for any $y$ . Using the notation of convex conjugates, the $f$ -divergence and its variational representation is defined as follows.

Definition 1 ( $f$ -divergence and its variational representation).

Let $\mathcal{H}$ be any arbitrary domain. Let $P$ and $Q$ denote the probability measures over the Borel $\sigma$ -field on $\mathcal{H}$ . Additionally, let $f$ : $[0,\infty)\rightarrow\mathbb{R}$ be a convex and lower semi-continuous function that satisfies $f(1)=0$ .

D_{f}(Q\|P):=\mathbb{E}_{P}\bigg{[}f\bigg{(}\frac{dQ}{dP}\bigg{)}\bigg{]}

For simplicity, we denote $\mathbb{E}_{P}[\cdot]\equiv\mathbb{E}_{h\sim P}[\cdot]$ in the sequel. Many common divergences, such as the KL-divergence and the Hellinger divergence, are members of the family of $f$ -divergences, coinciding with a particular choice of $f(t)$ . Table 2 presents the definition of each divergence with the corresponding generator $f(t)$ . It is well known that the $f-$ divergence can be characterized as the following variational representation.

Lemma 1 (Variational representation of $f-$ divergence, Lemma 1 in [27]).

Let $\mathcal{H}$ , $P$ , $Q$ and $f$ be defined as in Definition 1. Let $\phi$ : $\mathcal{H}\rightarrow\mathbb{R}$ be a real-valued function. The $f$ -divergence from $P$ to $Q$ is characterized as

D_{f}(Q\|P)\geq\sup_{\phi}\mathbb{E}_{Q}[\phi]-\mathbb{E}_{P}[f^{*}(\phi)]

[29] shows that this variational representation for $f$ -divergences can be tightened.

Theorem 1 (Change of measure inequality from the constrained variational representation for $f$ -divergences [29]).

Let $\mathcal{H}$ , $P$ , $Q$ and $f$ be defined as in Definition 1. Let $\phi$ : $\mathcal{H}\rightarrow\mathbb{R}$ be a real-valued function. Let $\Delta(\mu):=\{g:\mathcal{H}\rightarrow\mathbb{R}:g\geq 0,\|g\|_{1}=1\}$ denote the space of probability densities with respect to $\mu$ , where the norm is defined as $\|g\|_{1}:=\int_{\mathcal{H}}|g|d\mu$ , given a measure $\mu$ over $\mathcal{H}$ . The general form of the change of measure inequality for f-divergences is given by

\begin{split}\mathbb{E}_{Q}[\phi]&\leq D_{f}(Q\|P)+(\mathbb{I}^{R}_{f,P})^{*}(\phi)\\ (\mathbb{I}^{R}_{f,P})^{*}(\phi)&=\sup_{p\in\Delta(P)}{\mathbb{E}_{P}[\phi p]-\mathbb{E}_{P}[f(p)]}\end{split}

where p is constrained to be a probability density function.

The famous Donsker-Varadhan representation for the KL-divergence, which is used in most PAC-Bayesian bounds, can be actually derived from this tighter representation by setting $f(t)=t\log(t)$ . However, it is not always easy to find a closed-form solution for Theorem 1, as it requires to resort to variational calculus, and in some cases, there is no closed-form solution. In such a case, we can use the following corollary to obtain looser bounds, but only requires to find a convex conjugate.

Corollary 1 (Change of measure inequality from the unconstrained variational representation for $f$ -divergences).

Let $P$ , $Q$ , $f$ and $\phi$ be defined as in Theorem 1. By Definition 1, we have

\begin{split}\forall Q\text{ on }\mathcal{H}:\ \mathbb{E}_{Q}[\phi]\leq D_{f}(Q\|P)+\mathbb{E}_{P}[f^{*}(\phi)]\end{split}

Detailed proofs can be found in Appendix A. By choosing a right function $f$ and deriving the constrained maximization term $(\mathbb{I}^{R}_{f,P})^{*}(\phi)$ with the help of variational calculus, we can create an upper-bound based on the corresponding divergence $D_{f}(Q\|P)$ . Next, we discuss the case of the $\chi^{2}$ divergence.

Lemma 2 (Change of measure inequality from the constrained representation of the $\chi^{2}$ -divergence).

Let $P$ , $Q$ and $\phi$ be defined as in Theorem 1, we have

\forall Q\text{ on }\mathcal{H}:\ \mathbb{E}_{Q}[\phi]\leq\chi^{2}(Q\|P)+\mathbb{E}_{P}[\phi]+\frac{1}{4}\mathbb{V}{\rm ar}_{P}[\phi]

The bound in Lemma 2 is slightly tighter than the one without the constraint. The change of measure inequality without the constraint is given as follows.

Lemma 3 (Change of measure inequality from the unconstrained representation of the Pearson $\chi^{2}$ -divergence).

Let $P$ , $Q$ and $\phi$ be defined as in Theorem 1, we have

\forall Q\text{ on }\mathcal{H}:\ \mathbb{E}_{Q}[\phi]\leq\chi^{2}(Q\|P)+\mathbb{E}_{P}[\phi]+\frac{1}{4}\mathbb{E}_{P}[\phi^{2}]

As might be apparent, the bound in Lemma 2 is tighter than the one in Lemma 3 by $(\mathbb{E}_{P}[\phi])^{2}$ because $\mathbb{V}{\rm ar}_{P}[\phi]\leq\mathbb{E}_{P}[\phi^{2}]$ . Next, we discuss the case of the total variation divergence.

Lemma 4 (Change of measure inequality from the constrained representation of the total variation divergence).

Let $\phi$ : $\mathcal{H}\rightarrow[0,1]$ be a real-valued function. Let $P$ and $Q$ be defined as in Theorem 1, we have

\begin{split}\forall Q\text{ on }\mathcal{H}:\ \mathbb{E}_{Q}[\phi]\leq TV(Q\|P)+\mathbb{E}_{P}[\phi]\end{split}

Interestingly, we can obtain the same bound on the total variation divergence even if we use the unconstrained variational representation. Next, we state our result for $\alpha$ -divergences.

Lemma 5 (Change of measure inequality from the unconstrained representation of the $\alpha$ -divergence).

Let $P$ , $Q$ and $\phi$ be defined as in Theorem 1. For $\alpha>1$ , we have

\begin{split}\forall Q\text{ on }\mathcal{H}:\ \mathbb{E}_{Q}[\phi]\leq D_{\alpha}(Q\|P)+\frac{(\alpha-1)^{\frac{\alpha}{\alpha-1}}}{\alpha}\mathbb{E}_{P}[\phi^{\frac{\alpha}{\alpha-1}}]+\frac{1}{\alpha(\alpha-1)}\end{split}

We can obtain the bounds based on the squared Hellinger divergence $H^{2}(Q\|P)$ , the reverse KL-divergence $\overline{KL}(Q\|P)$ and the Neyman $\chi^{2}$ -divergence $\overline{\chi^{2}}(Q\|P)$ in a similar fashion.

Lemma 6 (Change of measure inequality from the unconstrained representation of the squared Hellinger divergence).

Let $\phi$ : $\mathcal{H}\rightarrow(-\infty,1)$ be a real-valued function. Let $P$ and $Q$ be defined as in Theorem 1, we have

\begin{split}\forall Q\text{ on }\mathcal{H}:\ \mathbb{E}_{Q}[\phi]\leq H^{2}(Q\|P)+\mathbb{E}_{P}\bigg{[}\frac{\phi}{1-\phi}\bigg{]}\end{split}

Similarly, we obtain the following bound for the reverse-KL divergence.

Lemma 7 (Change of measure inequality from the unconstrained representation of the reverse KL-divergence).

Let $\phi$ : $\mathcal{H}\rightarrow(-\infty,1)$ be a real-valued function. Let $P$ and $Q$ be defined as in Theorem 1, we have

\begin{split}\forall Q\text{ on }\mathcal{H}:\ \mathbb{E}_{Q}[\phi]\leq\overline{KL}(Q\|P)+\mathbb{E}_{P}\bigg{[}\log\bigg{(}\frac{1}{1-\phi}\bigg{)}\bigg{]}\end{split}

Finally, we prove our result for the Neyman $\chi^{2}$ divergence based on a similar approach.

Lemma 8 (Change of measure inequality from the unconstrained representation of the Neyman $\chi^{2}$ -divergence).

Let $\phi$ : $\mathcal{H}\rightarrow(-\infty,1)$ be a real-valued function. Let $P$ and $Q$ be defined as in Theorem 1, we have

\begin{split}\forall Q\text{ on }\mathcal{H}:\ \mathbb{E}_{Q}[\phi]\leq\overline{\chi^{2}}(Q\|P)+2-2\mathbb{E}_{P}[\sqrt{1-\phi}]\end{split}

2.2 Multiplicative Change of Measure Inequality for $\alpha$ -divergences

First, we state a known result for the $\chi^{2}$ divergence.

Lemma 9 (Multiplicative change of measure inequality for the $\chi^{2}$ -divergence [15]).

Let $P$ , $Q$ and $\phi$ be defined as in Theorem 1, we have

\begin{split}\forall Q\text{ on }\mathcal{H}:\ \mathbb{E}_{Q}[\phi]\leq\sqrt{(\chi^{2}(Q\|P)+1)\mathbb{E}_{P}[\phi^{2}]}\end{split}

First, we note that the $\chi^{2}$ divergence is an $\alpha$ -divergence for $\alpha=2$ . Next, we generalize the above bound for any $\alpha$ -divergence.

Lemma 10 (Multiplicative change of measure inequality for the $\alpha$ -divergence).

Let $P$ , $Q$ and $\phi$ be defined as in Theorem 1. For any $\alpha>1$ , we have

\begin{split}\forall Q\text{ on }\mathcal{H}:\ \mathbb{E}_{Q}[\phi]\leq\big{(}\alpha(\alpha-1)D_{\alpha}(Q\|P)+1\big{)}^{\frac{1}{\alpha}}\big{(}\mathbb{E}_{P}[|\phi|^{\frac{\alpha}{\alpha-1}}]\big{)}^{\frac{\alpha-1}{\alpha}}\end{split}

Our bound is stated in the form of $\alpha$ -divergence. By choosing $\alpha=2$ , we have the same bound as Lemma 9 where $\chi^{2}(Q\|P)=2D_{\alpha}(Q\|P)$ . We will later apply the above $\alpha$ -divergence change of measure to obtain PAC-Bayes inequalities for types of losses not considered before [1, 15].

2.3 A Generalized Hammersley-Chapman-Robbins (HCR) Inequality

The HCR inequality is a famous information theoretic inequality for the $\chi^{2}$ -divergence.

Lemma 11 (HCR inequality [18]).

Let $P$ , $Q$ and $\phi$ be defined as in Theorem 1, we have

\begin{split}\forall Q\text{ on }\mathcal{H}:\ \chi^{2}(Q\|P)\geq\frac{(E_{Q}[\phi]-\mathbb{E}_{P}[\phi])^{2}}{\mathbb{V}{\rm ar}_{P}[\phi]}\end{split}

Next, we generalize the above bound for $\alpha$ -divergence.

Lemma 12 (The generalization of HCR inequality.).

Let $P$ , $Q$ and $\phi$ be defined as in Theorem 1. For any $\alpha>1$ , we have

\begin{split}\forall Q\text{ on }\mathcal{H}:\ \big{|}\mathbb{E}_{Q}[\phi]-\mathbb{E}_{P}[\phi]\big{|}\leq\widetilde{\mathcal{D_{\alpha}}}(Q\|P)^{\frac{1}{\alpha}}\big{(}\mathbb{E}_{P}[|\phi-\mu_{P}|^{\frac{\alpha}{\alpha-1}}]\big{)}^{\frac{\alpha-1}{\alpha}}\end{split}

where $\widetilde{\mathcal{D_{\alpha}}}(Q\|P)=\int_{\mathcal{H}}|\frac{dQ}{dP}-1|^{\alpha}dP$ and $\mu_{P}=\mathbb{E}_{P}[\phi]$ .

We call $\widetilde{\mathcal{D_{\alpha}}}(Q\|P)$ a pseudo $\alpha$ -divergence, which is a member of the family of $f$ -divergences with $f(t)=|t-1|^{\alpha}$ for $\alpha>1$ . See Appendix B for the formal proof. Straightforwardly, we can obtain Lemma 11 by choosing $\alpha=2$ .

3 Application to PAC-Bayesian Theory

Table 3: Summary of PAC-Bayesian bounds with

\alpha

-divergence and

\chi^{2}

-divergence.

Loss & Divergence		Generalization upper bound for $R_{D}(G_{Q})-R_{S}(G_{Q})$ , for every $Q$ , and a fixed $P$	Reference
Bounded Loss	$\alpha$	$O\big{(}R[\frac{1}{m}\log(\frac{1}{\delta})]^{\frac{1}{2}}D_{\alpha}(Q\\|P)^{\frac{1}{2\alpha}}\big{)}$	Proposition 5
	$\alpha$	$O\big{(}(\frac{1}{m})^{\frac{1}{2}}[D_{\alpha}(Q\\|P)+(R^{2}\log(\frac{1}{\delta}))^{\frac{\alpha}{\alpha-1}}]^{\frac{1}{2}}\big{)}$	Proposition 5
	$\chi^{2}$	$O\big{(}R[\frac{1}{m}\log(\frac{1}{\delta})]^{\frac{1}{2}}\chi^{2}(Q\\|P)^{\frac{1}{4}}\big{)}$	Corollary 3
	$\chi^{2}$	$O\big{(}(\frac{1}{m})^{\frac{1}{2}}[\chi^{2}(Q\\|P)+(R^{2}\log(\frac{1}{\delta}))^{2}]^{\frac{1}{2}}\big{)}$	Corollary 3
0-1 loss	$\chi^{2}$	$O\big{(}(\frac{1}{m\delta})^{\frac{1}{2}}\chi^{2}(Q\\|P)^{\frac{1}{2}}\big{)}$	[15, 20]
Sub-Gaussian	$\alpha$	$O\big{(}\sigma[\frac{1}{m}\log(\frac{1}{\delta})]^{\frac{1}{2}}D_{\alpha}(Q\\|P)^{\frac{1}{2\alpha}}\big{)}$	Proposition 6
	$\alpha$	$O\big{(}(\frac{1}{m})^{\frac{1}{2}}[D_{\alpha}(Q\\|P)+(\sigma^{2}\log(\frac{1}{\delta}))^{\frac{\alpha}{\alpha-1}}]^{\frac{1}{2}}\big{)}$	Proposition 6
	$\alpha$	$O\big{(}\sigma(\frac{1}{m})^{\frac{1}{2}}(\frac{1}{\delta})^{\frac{\alpha-1}{\alpha}}D_{\alpha}(Q\\|P)^{\frac{1}{\alpha}}\big{)}$	Theorem 1 & Proposition 6 in [1]
	$\chi^{2}$	$O\big{(}\sigma[\frac{1}{m}\log(\frac{1}{\delta})]^{\frac{1}{2}}\chi^{2}(Q\\|P)^{\frac{1}{4}}\big{)}$	Corollary 4
	$\chi^{2}$	$O\big{(}(\frac{1}{m})^{\frac{1}{2}}[\chi^{2}(Q\\|P)+(\sigma^{2}\log(\frac{1}{\delta}))^{2}]^{\frac{1}{2}}\big{)}$	Corollary 4
	$\chi^{2}$	$O\big{(}\sigma(\frac{1}{m\delta})^{\frac{1}{2}}\chi^{2}(Q\\|P)^{\frac{1}{2}}\big{)}$	Theorem 1 & Proposition 6 in [1]
Sub-exponential	$\alpha$	$O\big{(}\frac{\beta}{m}\log(\frac{1}{\delta})D_{\alpha}(Q\\|P)^{\frac{1}{2\alpha}}\big{)}$	Proposition 7
For	$\alpha$	$O\big{(}(\frac{1}{m})^{\frac{1}{2}}[D_{\alpha}(Q\\|P)+m^{\frac{-\alpha}{\alpha-1}}(\beta\log(\frac{1}{\delta}))^{\frac{2\alpha}{\alpha-1}}]^{\frac{1}{2}}\big{)}$	Proposition 7
$m<\frac{2\beta^{2}\log(\frac{2}{\delta})}{\sigma^{2}}$	$\chi^{2}$	$O\big{(}\frac{\beta}{m}\log(\frac{1}{\delta})\chi^{2}(Q\\|P)^{\frac{1}{4}}\big{)}$	Corollary 5
	$\chi^{2}$	$O\big{(}(\frac{1}{m})^{\frac{1}{2}}[(\chi^{2}(Q\\|P)+\frac{1}{m^{2}}(\beta\log(\frac{1}{\delta}))^{4})]^{\frac{1}{2}}\big{)}$	Corollary 5
Bounded	$\alpha$	$O\big{(}\sigma(\frac{1}{m\delta})^{\frac{1}{2}}D_{\alpha}(Q\\|P)^{\frac{1}{2\alpha}}\big{)}$	Proposition 1
Variance	$\alpha$	$O\big{(}(\frac{1}{m})^{\frac{1}{2}}[D_{\alpha}(Q\\|P)+(\frac{\sigma^{2}}{\delta})^{\frac{\alpha}{\alpha-1}}]^{\frac{1}{2}}\big{)}$	Proposition 1
	$\alpha$	$O\big{(}\sigma(\frac{1}{m})^{\frac{1}{2}}(\frac{D_{\alpha}(Q\\|P)}{\delta^{\alpha-1}})^{\frac{1}{\alpha}}\big{)}$	Proposition 4 in [1]
	$\chi^{2}$	$O\big{(}\sigma(\frac{1}{m\delta})^{\frac{1}{2}}\chi^{2}(Q\\|P)^{\frac{1}{4}}\big{)}$	Corollary 2
	$\chi^{2}$	$O\big{(}(\frac{1}{m})^{\frac{1}{2}}[\chi^{2}(Q\\|P)+(\frac{\sigma^{2}}{\delta})^{2}]^{\frac{1}{2}}\big{)}$	Corollary 2
	$\chi^{2}$	$O\big{(}\sigma(\frac{1}{m\delta})^{\frac{1}{2}}\chi^{2}(Q\\|P)^{\frac{1}{2}}\big{)}$	Corollary 1 in [1]

In this section, we will explore the applications of our change of measure inequalities. We consider an arbitrary input space $\mathcal{X}$ and a output space $\mathcal{Y}$ . The samples $(x,y)\in\mathcal{X}\times\mathcal{Y}$ are input-output pairs. Each example $\mathnormal{(x,y)}$ is drawn i.i.d. according to a fixed, but unknown, distribution $D$ on $\mathcal{X}\times\mathcal{Y}$ . Let $\ell:\mathcal{Y}\times\mathcal{Y}\to\mathbb{R}$ denote a generic loss function. The risk $R_{D}(h)$ of any predictor $\mathnormal{h}:\mathcal{X}\rightarrow\mathcal{Y}$ is defined as the expected loss induced by samples drawn according to $D$ . Given a training set $S$ of $m$ samples, the empirical risk $R_{S}(h)$ of any predictor $h$ is defined by the empirical average of the loss. That is

R_{D}(h)=\mathop{\mathbb{E}}_{(x,y)\sim D}\ell(h(x),y),\quad R_{S}(h)=\frac{1}{|S|}\sum_{(x,y)\in S}\ell(h(x),y)

In the PAC-Bayesian framework, we consider a hypothesis space $\mathcal{H}$ of predictors, a prior distribution $P$ on $\mathcal{H}$ , and a posterior distribution $Q$ on $\mathcal{H}$ . The prior is specified before exploiting the information contained in $S$ , while the posterior is obtained by running non a learning algorithm on $S$ . The PAC-Bayesian theory usually studies the stochastic Gibbs predictor $G_{Q}$ . Given a distribution $Q$ on $\mathcal{H}$ , $G_{Q}$ predicts an example $x$ by drawing a predictor $h$ according to $Q$ , and returning $h(x)$ . The risk of $G_{Q}$ is then defined as follows. For any probability distribution $Q$ on a set of predictors, the Gibbs risk $R_{D}(G_{Q})$ is the expected risk of the Gibbs predictor $G_{Q}$ relative to $D$ . Hence,

R_{D}(G_{Q})=\mathop{\mathbb{E}}_{(x,y)\sim D}\mathop{\mathbb{E}}_{h\sim Q}\ell(h(x),y)

(2)

Usual PAC-Bayesian bounds give guarantees on the generalization risk $R_{D}(G_{Q})$ . Typically, these bounds rely on the empirical risk $R_{S}(G_{Q})$ defined as follows.

R_{S}(G_{Q})=\frac{1}{|S|}\sum_{(x,y)\in S}\mathop{\mathbb{E}}_{h\sim Q}\ell(h(x),y)

(3)

Due to space constraints, we fully present PAC-Bayes generalization bounds for losses with bounded variance. Other results for bounded losses, sub-Gaussian losses and sub-exponential losses are included in Appendix C. Still, we briefly discuss our new results in Section 3.2.

3.1 Loss Function with Bounded Variance

In this section, we present our PAC-Bayesian bounds for the loss functions with bounded variance. Assuming any arbitrary distribution with bounded variance on the loss function $\ell$ (i.e., $\mathop{\mathbb{V}{\rm ar}}_{(x,y)\sim D}[\ell(h(x),y)]\leq\sigma^{2}$ for any $h\in\mathcal{H}$ ), we have the following PAC-Bayesian bounds.

Proposition 1 (The PAC-Bayesian bounds for loss function with bounded variance).

Let $P$ be a fixed prior distribution over a hypothesis space any $h\in\mathcal{H}$ . For a given posterior distribution $Q$ over an infinite hypothesis space $\mathcal{H}$ , let $R_{D}(G_{Q})$ and $R_{S}(G_{Q})$ be the Gibbs risk and the empirical Gibbs risk as in Equation (2) and (3) respectively. For the sample size $m$ and $\alpha>1$ , with probability at least $1-\delta$ , simultaneously for all posterior distributions $Q$ , we have

\begin{split}R_{D}(G_{Q})\leq R_{S}(G_{Q})+\sqrt{\frac{\sigma^{2}}{m\delta}\big{(}\alpha(\alpha-1)D_{\alpha}(Q\|P)+1\big{)}^{\frac{1}{\alpha}}}\end{split}

\begin{split}R_{D}(G_{Q})\leq R_{S}(G_{Q})+\sqrt{\frac{1}{m}\bigg{(}D_{\alpha}(Q\|P)+\frac{1}{\alpha(\alpha-1)}\bigg{)}+\frac{1}{m\alpha}\bigg{(}\frac{\sigma^{2}(\alpha-1)}{\delta}\bigg{)}^{\frac{\alpha}{\alpha-1}}}\end{split}

(4)

By setting $\alpha=2$ in Proposition 2, we have the following claim.

Corollary 2 (The PAC-Bayesian bounds with $\chi^{2}$ -divergence for bounded variance loss function).

Let $P$ be any prior distribution over an infinite hypothesis space $\mathcal{H}$ . For a given posterior distribution $Q$ over an infinite hypothesis space $\mathcal{H}$ , let $R_{D}(G_{Q})$ and $R_{S}(G_{Q})$ be the Gibbs risk and the empirical Gibbs risk as in Equation (2) and (3) respectively. For the sample size $m>0$ , with probability at least $1-\delta$ , simultaneously for all posterior distributions $Q$ , we have

\begin{split}R_{D}(G_{Q})&\leq R_{S}(G_{Q})+\sqrt{\frac{\sigma^{2}}{m\delta}\sqrt{\chi^{2}(Q\|P)+1}}\end{split}

(5)

\begin{split}R_{D}(G_{Q})\leq R_{S}(G_{Q})+\sqrt{\frac{1}{2m}\bigg{(}\chi^{2}(Q\|P)+1+\bigg{(}\frac{\sigma^{2}}{\delta}\bigg{)}^{2}\bigg{)}}\end{split}

Now, note that choosing $\Delta(q,p)=|q-p|$ for Equation (5) results in

\begin{split}R_{D}(G_{Q})&\leq R_{S}(G_{Q})+\sqrt{\frac{\sigma^{2}}{m\delta}(\chi^{2}(Q\|P)+1)}\end{split}

(6)

which was shown in Corollary 1 of [1]. We can easily see that (5) is tighter than (6) due to the choice of $\Delta(q,p)=(q-p)^{2}$ . Please see Table 3 for details.

3.2 Discussion

Table 3 presents various PAC-Bayesian bounds based on our change of measure inequalities depending on different assumptions on the loss function $\ell$ , and compares them with existing results in the literature. The importance of the various types of PAC-Bayes bounds is justified by the connection between PAC-Bayes bounds and regularization in a learning problem. PAC-Bayesian theory provides a guarantee that upper-bounds the risk of Gibbs predictors simultaneously for all posterior distributions $Q$ . PAC-Bayes bounds enjoy this property due to change of measure inequalities. It is a well-known fact that KL-regularized objective functions are obtained from PAC-Bayes risk bounds, in which the minimizer $\hat{Q}$ is guaranteed to exist since the risk bounds hold simultaneously for all posterior distributions $Q$ . The complexity term, partially controlled by the divergence, serves as a regularizer [10, 9, 3]. Additionally, the link between Bayesian inference techniques and PAC-Bayesian risk bounds was shown by [8], that is, the minimization of PAC-Bayesian risk bounds maximizes the Bayesian marginal likelihood. Our results pertain to important machine learning prediction problems, such as regression, classification and structured prediction and indicate which regularizer to use. All PAC-Bayes bounds presented here are either novel, or have a tighter complexity term than existing results in the literature. Since our bounds are based on either $\chi^{2}$ -divergence or $\alpha$ -divergence, we excluded the comparison with the bounds based on the KL-divergence ([30, 4, 13, 25, 8, 11, 33]). Our results for sub-exponential losses are entirely novel. For the other cases, our bounds are tighter than exisiting bounds in terms of the complexity term. For instance, our bound for bounded losses is tighter than those of [15, 20] since our bound has the complexity term $\chi^{2}(Q\|P)^{1/4}$ and $\log(1/\delta)$ , while [15, 20] have $\chi^{2}(Q\|P)^{1/2}$ and $1/\delta$ . For sub-Gaussian loss functions, our bound has the complexity term $D_{\alpha}(Q\|P)^{1/2\alpha}$ and $\log(1/\delta)$ , whereas [1] has $D_{\alpha}(Q\|P)^{1/\alpha}$ and $1/\delta$ respectively. In addition, our additive bounds, such as Equation (4), have better rates than the existing bounds in [1], since in our bound, $D_{\alpha}(Q\|P)$ and $1/\delta$ are added, while in [1], $D_{\alpha}(Q\|P)$ and $1/\delta$ are multiplied.

4 Non-Asymptotic Interval for Monte Carlo Estimates

Table 4: Summary of non-asymptotic interval for Monte Carlo estimates

Divergence	Non-asymptotic interval for Monte Carlo estimates $\big{\|}\mathbb{E}_{Q}[\phi(X)]-\frac{1}{n}\sum^{n}_{i=1}\phi(X_{i})\big{\|}\leq\frac{4L^{2}\log(\frac{2}{\delta})}{n\gamma}+\mathcal{K}$	Reference
Pseudo $\alpha$	$\mathcal{K}=\frac{2^{\frac{2\alpha-1}{\alpha}}L\widetilde{\mathcal{D_{\alpha}}}(Q\\|P)^{\frac{1}{\alpha}}}{\sqrt{\gamma}}\Gamma(\frac{3\alpha-2}{2(\alpha-1)})^{\frac{\alpha-1}{\alpha}}$	Proposition 2
$\chi^{2}$	$\mathcal{K}=\begin{cases}\sqrt{\chi^{2}(Q\\|P)\{\frac{1}{n}\sum_{i=1}^{n}\phi^{2}(x_{i})+\frac{16L^{2}}{\gamma}\sqrt{\frac{1}{n}\log\frac{2}{\delta}}\}}$ if $\log(\frac{2}{\delta})\leq n\\ \sqrt{\chi^{2}(Q\\|P)\{\frac{1}{n}\sum_{i=1}^{n}\phi^{2}(x_{i})+\frac{16L^{2}}{n\gamma}\log\frac{2}{\delta}\}}$ if $n<\log(\frac{2}{\delta})\end{cases}$	Proposition 3
$KL$	$\mathcal{K}=KL(Q\\|P)+\frac{L^{2}}{n\gamma}$	Proposition 4

We now turn to another application of change of measure inequalities. We will introduce a methodology that enables us to find a non-asymptotic interval for Monte Calro (MC) estimate. All results shown in this section are entirely novel and summarized in Table 4. Under some circumstances, our methodology could be a promising alternative to existing MC methods (e.g., Importance Sampling and Rejection Sampling [28]) because their non-asymptotic intervals are hard to analyze. In this section, we consider a problem to estimate an expectation of an Lipschitz function $\phi:\mathbb{R}^{d}\rightarrow\mathbb{R}$ with respect to a complicated distribution $Q$ , namely $\mathbb{E}_{Q}[\phi(X)]$ by the sample mean $\frac{1}{n}\sum^{n}_{i=1}\phi(X_{i})$ over a distribution $P$ , where $Q$ is any distribution we are not able to sample from. For a motivating application, consider $Q$ being a probabilistic graphical model (see e.g., [14]). Assume that we have a strongly log-concave distribution $P$ with a parameter $\gamma$ for a sampling distribution (see Appendix D for definition). Under the above conditions, we claim the following proposition.

Proposition 2 (A general expression of non-asymptotic interval for Monte Carlo estimates).

Let $X$ be a d-dimensional random vector. Let $P$ and $Q$ be the probability measures induced by $X$ . Let $P$ be a strongly log-concave distribution with parameter $\gamma>0$ . Let $\phi:\mathbb{R}^{d}\rightarrow\mathbb{R}$ be any L-Lipschitz function with respect to the Euclidean norm. Suppose we draw i.i.d. samples $X_{1},X_{2},...,X_{n}\sim P$ . For $\alpha>1$ , with probability at least $1-\delta$ , we have

\bigg{|}\mathbb{E}_{Q}[\phi(X)]-\frac{1}{n}\sum^{n}_{i=1}\phi(X_{i})\bigg{|}\leq\frac{4L^{2}\log(\frac{2}{\delta})}{n\gamma}+\frac{2^{\frac{2\alpha-1}{\alpha}}L\widetilde{\mathcal{D_{\alpha}}}(Q\|P)^{\frac{1}{\alpha}}}{\sqrt{\gamma}}\Gamma\bigg{(}\frac{3\alpha-2}{2(\alpha-1)}\bigg{)}^{\frac{\alpha-1}{\alpha}}

The second term on the right hand side indicates a bias of an empirical mean $\frac{1}{n}\sum^{n}_{i=1}\phi(X_{i})$ under the sampling distribution $P$ . Next, we present a more informative bound.

Proposition 3 ( $\chi^{2}$ -based expression of non-asymptotic interval for Monte Carlo estimates).

Let $X,P,Q,L,\gamma$ and $\phi$ be defined as in Proposition 2 . Suppose we have i.i.d. samples $X_{1},X_{2},...,X_{n}\sim P$ . Then, with probability at least $(1-\delta)^{2}$ , we have

\bigg{|}\mathbb{E}_{Q}[\phi(X)]-\frac{1}{n}\sum^{n}_{i=1}\phi(X_{i})\bigg{|}\leq\frac{4L^{2}\log(\frac{2}{\delta})}{n\gamma}+\mathcal{K}

where

\begin{split}\mathcal{K}=\begin{cases}\sqrt{\chi^{2}(Q\|P)\{\frac{1}{n}\sum_{i=1}^{n}\phi^{2}(X_{i})+\frac{16L^{2}}{\gamma}\sqrt{\frac{1}{n}\log\frac{2}{\delta}}\}}\quad\text{if }\log(\frac{2}{\delta})\leq n\\ \sqrt{\chi^{2}(Q\|P)\{\frac{1}{n}\sum_{i=1}^{n}\phi^{2}(X_{i})+\frac{16L^{2}}{n\gamma}\log\frac{2}{\delta}\}}\quad\text{if }n<\log(\frac{2}{\delta})\end{cases}\end{split}

This result provides an insight into how good a proposal distribution $P$ is, meaning that the effect of the deviation between $P$ and $Q$ , namely $\chi^{2}(Q\|P)$ , is inflated by the empirical variance. This implication might support some results in the literature ([5, 6]). So far we have considered the pseudo $\alpha$ -divergence as well as $\chi^{2}$ divergence. [7] presented an approximation for an arbitrary distribution $Q$ by distributions with log-concave density with respect to the KL divergence, which motivates the following result.

Proposition 4 (KL-based expression of non-asymptotic interval for Monte Carlo estimates).

Let $X,P,Q,L,\gamma$ and $\phi$ be defined as in Proposition 2 . Suppose we have i.i.d. samples $X_{1},X_{2},...,X_{n}\sim P$ . Then, with probability at least $1-\delta$ , we have

\bigg{|}\mathbb{E}_{Q}[\phi(X)]-\frac{1}{n}\sum^{n}_{i=1}\phi(X_{i})\bigg{|}\leq\frac{4L^{2}\log(\frac{2}{\delta})}{n\gamma}+KL(Q\|P)+\frac{L^{2}}{n\gamma}

This result shows that, under the assumption of Proposition 4, the empirical mean $\frac{1}{n}\sum^{n}_{i=1}\phi(X_{i})$ over the sampling distribution $P$ aymptotically differs from $\mathbb{E}_{Q}[\phi(X)]$ by $KL(Q\|P)$ at most.

References

[1] Pierre Alquier and Benjamin Guedj. Simpler pac-bayesian bounds for hostile data. Machine Learning, 107(5):887–902, May 2018.
[2] Amiran Ambroladze, Emilio Parrado-hernández, and John S. Shawe-taylor. Tighter pac-bayes bounds. In B. Schölkopf, J. C. Platt, and T. Hoffman, editors, Advances in Neural Information Processing Systems 19, pages 9–16. MIT Press, 2007.
[3] Olivier Bousquet and André Elisseeff. Stability and generalization. Journal of Machine Learning Research, 2:499–526, 06 2002.
[4] Olivier Catoni. Pac-bayesian supervised classification: The thermodynamics of statistical learning. 01 2007.
[5] Julien Cornebise, Éric Moulines, and Jimmy Olsson. Adaptive methods for sequential importance sampling with application to state space models. Statistics and Computing, 18(4):461–480, 2008.
[6] Adji Bousso Dieng, Dustin Tran, Rajesh Ranganath, John Paisley, and David Blei. Variational inference via chi upper bound minimization. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems 30, pages 2732–2741. Curran Associates, Inc., 2017.
[7] Lutz Duembgen, Richard Samworth, and Dominic Schuhmacher. Approximation by log-concave distributions, with applications to regression. The Annals of Statistics, 39, 02 2010.
[8] Pascal Germain, Francis Bach, Alexandre Lacoste, and Simon Lacoste-Julien. Pac-bayesian theory meets bayesian inference. In D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett, editors, Advances in Neural Information Processing Systems 29, pages 1884–1892. Curran Associates, Inc., 2016.
[9] Pascal Germain, Alexandre Lacasse, François Laviolette, and Mario Marchand. A pac-bayes risk bound for general loss functions. In B. Schölkopf, J. C. Platt, and T. Hoffman, editors, Advances in Neural Information Processing Systems 19, pages 449–456. MIT Press, 2007.
[10] Pascal Germain, Alexandre Lacasse, Mario Marchand, Sara Shanian, and François Laviolette. From pac-bayes bounds to kl regularization. In Y. Bengio, D. Schuurmans, J. D. Lafferty, C. K. I. Williams, and A. Culotta, editors, Advances in Neural Information Processing Systems 22, pages 603–610. Curran Associates, Inc., 2009.
[11] Peter D. Grünwald and Nishant A. Mehta. A tight excess risk bound via a unified PAC-Bayesian–Rademacher–Shtarkov–MDL complexity. In Aurélien Garivier and Satyen Kale, editors, Proceedings of the 30th International Conference on Algorithmic Learning Theory, volume 98 of Proceedings of Machine Learning Research, pages 433–465, Chicago, Illinois, 22–24 Mar 2019. PMLR.
[12] Benjamin Guedj. A primer on pac-bayesian learning. Preprint arXiv:1901.05353, 05 2019.
[13] Matthew Holland. Pac-bayes under potentially heavy tails. In Advances in Neural Information Processing Systems 32, pages 2715–2724. Curran Associates, Inc., 2019.
[14] Jean Honorio. Lipschitz parametrization of probabilistic graphical models. pages 347–354. Conference on Uncertainty in Artificial Intelligence, 07 2011.
[15] Jean Honorio and Tommi Jaakkola. Tight bounds for the expected risk of linear classifiers and pac-bayes finite-sample guarantees. In International Conference on Artificial Intelligence and Statistics, page 384–392, 04 2014.
[16] J. Jiao, Y. Han, and T. Weissman. Dependence measures bounding the exploration bias for general measurements. In 2017 IEEE International Symposium on Information Theory (ISIT), pages 1475–1479, 2017.
[17] Markos A. Katsoulakis, Luc Rey-Bellet, and Jie Wang. Scalable information inequalities for uncertainty quantification. Journal of Computational Physics, 336:513 – 545, 2017.
[18] E.L. Lehmann and G. Casella. Theory of Point Estimation. Springer Verlag, 1998.
[19] Guy Lever, Francois Laviolette, and John Shawe-Taylor. Tighter pac-bayes bounds through distribution-dependent priors. Theor. Comput. Sci., 473:4–28, 02 2013.
[20] François Laviolette Jean-Francis Roy. Luc Bégin, Pascal Germain. PAC-Bayesian Bounds based on the Rényi Divergence. International Conference on Artificial Intelligence and Statistics, 2016.
[21] David McAllester. Simplified pac-bayesian margin bounds. In Bernhard Schölkopf and Manfred K. Warmuth, editors, Learning Theory and Kernel Machines, pages 203–215, Berlin, Heidelberg, 2003. Springer Berlin Heidelberg.
[22] David A. McAllester. Some pac-bayesian theorems. In Machine Learning, pages 230–234. ACM Press, 1998.
[23] David A. McAllester. Pac-bayesian model averaging. In Proceedings of the Twelfth Annual Conference on Computational Learning Theory, COLT ’99, pages 164–170, New York, NY, USA, 1999. ACM.
[24] David A. McAllester. Pac-bayesian stochastic model selection. Machine Learning, 51(1):5–21, Apr 2003.
[25] Zakaria Mhammedi, Peter Grünwald, and Benjamin Guedj. Pac-bayes un-expected bernstein inequality. In Advances in Neural Information Processing Systems 32, pages 12202–12213. Curran Associates, Inc., 2019.
[26] XuanLong Nguyen, Martin J. Wainwright, and Michael I. Jordan. Estimating divergence functionals and the likelihood ratio by convex risk minimization. IEEE Trans. Inf. Theor., 56(11):5847–5861, November 2010.
[27] XuanLong Nguyen, M.J. Wainwright, and Michael Jordan. Estimating divergence functionals and the likelihood ratio by convex risk minimization. Information Theory, IEEE Transactions on, 56:5847 – 5861, 12 2010.
[28] Christian P. Robert and George Casella. Monte Carlo Statistical Methods (Springer Texts in Statistics). Springer-Verlag, Berlin, Heidelberg, 2005.
[29] Avraham Ruderman, Mark D. Reid, Darío García-García, and James Petterson. Tighter variational representations of f-divergences via restriction to probability measures. In Proceedings of the 29th International Coference on International Conference on Machine Learning, pages 1155–1162, USA, 2012. Omnipress.
[30] Matthias Seeger. Pac-bayesian generalisation error bounds for gaussian process classification. J. Mach. Learn. Res., 3:233–269, March 2003.
[31] Y. Seldin, F. Laviolette, N. Cesa-Bianchi, J. Shawe-Taylor, and P. Auer. Pac-bayesian inequalities for martingales. IEEE Transactions on Information Theory, 58(12):7086–7093, Dec 2012.
[32] John Shawe-Taylor and Robert C. Williamson. A pac analysis of a bayesian estimator. In Proceedings of the Tenth Annual Conference on Computational Learning Theory, COLT ’97, pages 2–9, New York, NY, USA, 1997. ACM.
[33] Rishit Sheth and Roni Khardon. Excess risk bounds for the bayes risk using variational inference in latent gaussian models. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems 30, pages 5151–5161. Curran Associates, Inc., 2017.
[34] Ilya O Tolstikhin and Yevgeny Seldin. Pac-bayes-empirical-bernstein inequality. In C. J. C. Burges, L. Bottou, M. Welling, Z. Ghahramani, and K. Q. Weinberger, editors, Advances in Neural Information Processing Systems 26, pages 109–117. Curran Associates, Inc., 2013.
[35] Martin J. Wainwright. High-Dimensional Statistics: A Non-Asymptotic Viewpoint. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, 2019.

Appendix A Detailed Proofs

A.1 Proof of Corollary 1

Proof.

Removing the supremum and rearranging the terms in Lemma 1, we prove our claim. ∎

A.2 Proof of Lemma 2

Proof.

By Theorem 1, we want to find

\begin{split}(\mathbb{I}^{R}_{f,P})^{*}(\phi)&=\sup_{p\in\Delta(P)}{\int_{\mathcal{H}}\phi p-(p-1)^{2}dP}\\ &=\sup_{p\in\Delta(P)}{\mathbb{E}_{P}[\phi p-(p-1)^{2}]}\end{split}

(7)

where $\phi$ is a measurable function on $\mathcal{H}$ . In order to find the supremum on the right hand side, we consider the following Lagrangian:

\begin{split}L(p,\lambda)&=\mathbb{E}_{P}[\phi p-(p-1)^{2}]+\lambda(\mathbb{E}_{P}[p]-1)\\ &=\int_{\mathcal{H}}\phi p-(p-1)^{2}dP+\lambda\bigg{(}\int_{\mathcal{H}}pdP-1\bigg{)}\end{split}

where $\lambda\in\mathbb{R}$ and $p$ is constrained to be a probability density over $\mathcal{H}$ . By talking the functional derivative with respect to $p$ and renormalize by dropping the $dP$ factor multiplying all terms and setting it to zero, we have $\frac{\partial L}{\partial pdP}=\phi-2(p-1)+\lambda=0$ . Thus, we have $p=\frac{\phi+\lambda}{2}+1$ . Since $p$ is constrained to be $\int_{\mathcal{H}}pdP=1$ , we have $\int_{\mathcal{H}}(\frac{\phi+\lambda}{2}+1)dP=1$ . Then, we obtain $\lambda=-\mathbb{E}_{P}[\phi]$ and the optimum $p=\frac{\phi-\mathbb{E}_{P}[\phi]}{2}+1$ . Plugging it in Equation (7), we have

\begin{split}(\mathbb{I}^{R}_{f,P})^{*}(\phi)&={E_{P}\bigg{[}\phi\bigg{(}\frac{\phi-\mathbb{E}_{P}[\phi]}{2}+1\bigg{)}-\bigg{(}\frac{\phi-\mathbb{E}_{P}[\phi]}{2}\bigg{)}^{2}\bigg{]}}\end{split}

which simplifies to $\mathbb{E}_{P}[\phi]+\frac{1}{4}\mathbb{V}{\rm ar}_{P}[\phi]$ . ∎

A.3 Proof of Lemma 3

Proof.

Notice that the $\chi^{2}$ -divergence is obtained by setting $f(x)=(x-1)^{2}$ . In order to find the convex conjugate $f^{*}(y)$ from Equation (1), let $g(x,y)=xy-(x-1)^{2}$ . We need to find the supremum of $g(x,y)$ with respect to $x$ . By using differentiation with respect to $x$ and setting the derivative to zero, we have $\frac{\partial g}{\partial x}=y-2(x-1)=0$ . Thus, plugging $x=\frac{y}{2}+1$ for $g(x,y)=xy-(x-1)^{2}$ , we obtain $f^{*}(y)=y+\frac{y^{2}}{4}$ . Plugging $f(x)$ and $f^{*}(y)$ in Corollary 1, we prove our claim. ∎

A.4 Proof of Lemma 4

Proof.

By Theorem 1, we want to find

\begin{split}(\mathbb{I}^{R}_{f,P})^{*}(\phi)&=\sup_{p\in\Delta(P)}{\mathbb{E}_{P}[\phi p-\frac{1}{2}|p-1|]}\end{split}

In order to find the supremum on the right hand side, we consider the following Lagrangian

\begin{split}L(p,\lambda)&=\mathbb{E}_{P}[\phi p-\frac{1}{2}|p-1|)]+\lambda(\mathbb{E}_{P}[p]-1)\\ &=\begin{cases}\mathbb{E}_{P}[\phi p-\frac{1}{2}p+\frac{1}{2}]+\lambda(\mathbb{E}_{P}[p]-1)&\text{if }1\leq p\\ \mathbb{E}_{P}[\phi p+\frac{1}{2}p-\frac{1}{2}]+\lambda(\mathbb{E}_{P}[p]-1)&\text{if }0<p<1\\ \end{cases}\end{split}

Then, it is not hard to see that if $|\phi|\leq\frac{1}{2}$ , $L(p,\lambda)$ is maximized at $p=1$ , otherwise $L\rightarrow\infty$ as $p\rightarrow 1$ . Thus, Lemma 4 holds for $|\phi|\leq\frac{1}{2}$ . If we add $\frac{1}{2}$ on the both sides, then $\phi$ is bounded between 0 and 1, as we claimed in the corollary. ∎

A.5 Proof of Lemma 5

Proof.

The corresponding convex function $f(x)$ for the $\alpha$ -divergence is defined as $f(x)=\frac{x^{\alpha}-1}{\alpha(\alpha-1)}$ . Applying the same procedure as in Lemma 3, we get the convex conjugate $f^{*}(y)=\frac{(\alpha-1)^{\frac{\alpha}{\alpha-1}}}{\alpha}y^{\frac{\alpha}{\alpha-1}}+\frac{1}{\alpha(\alpha-1)}$ for $\alpha>1$ . Plugging $f(x)$ and $f^{*}(y)$ in Corollary 1, we prove our claim. ∎

A.6 Proof of Lemma 6

Proof.

The corresponding convex function $f(x)$ for the squared Hellinger divergence is defined as $f(x)=(\sqrt{x}-1)^{2}$ . Let $g(x,y)=xy-(\sqrt{x}-1)^{2}$ . We have $\frac{\partial g}{\partial x}=y-1+\frac{1}{\sqrt{x}}=0$ . Since we consider $x>0$ , $y<1$ . Applying the same procedure as in Lemma 3, we get the convex conjugate $f^{*}(y)=\frac{y}{1-y}$ . Plugging $f(x)$ and $f^{*}(y)$ in Corollary 1, we prove our claim. ∎

A.7 Proof of Lemma 7

Proof.

The corresponding convex function $f(x)$ for the reverse KL-divergence is defined as $f(x)=-\log(x)$ . Applying the same procedure as in Lemma 6, we get the convex conjugate $f^{*}(y)=\log(-\frac{1}{y})-1$ . Plugging $f(x)$ and $f^{*}(y)$ in Corollary 1, we have

\mathbb{E}_{Q}[\psi]\leq\overline{KL}(Q\|P)+\mathbb{E}_{P}\bigg{[}\log\bigg{(}-\frac{1}{\psi}\bigg{)}\bigg{]}-1

where $\psi<0$ . Letting $\psi=\phi-1$ , we prove our claim. ∎

A.8 Proof of Lemma 8

Proof.

The corresponding convex function $f(x)$ for the Neyman $\chi^{2}$ -divergence is defined as $f(x)=-\frac{(x-1)^{2}}{x}$ . Applying the same procedure as in Lemma 6, we get the convex conjugate $f^{*}(y)=2-2\sqrt{1-y}$ . Plugging $f(x)$ and $f^{*}(y)$ in Corollary 1, we prove our claim. ∎

A.9 Proof of Lemma 10

Proof.

\begin{split}\mathbb{E}_{Q}[\phi]&=\int_{\mathcal{H}}\phi dQ\\ &=\int_{\mathcal{H}}\phi\frac{dQ}{dP}dP\\ &\leq\bigg{(}\int_{\mathcal{H}}\bigg{|}\frac{dQ}{dP}\bigg{|}^{\alpha}dP\bigg{)}^{\frac{1}{\alpha}}\bigg{(}\int_{\mathcal{H}}|\phi|^{\frac{\alpha}{\alpha-1}}dP\bigg{)}^{\frac{\alpha-1}{\alpha}}\\ &=\bigg{(}\alpha(\alpha-1)D_{\alpha}(Q\|P)+1\bigg{)}^{\frac{1}{\alpha}}\bigg{(}\mathbb{E}_{P}[|\phi|^{\frac{\alpha}{\alpha-1}}]\bigg{)}^{\frac{\alpha-1}{\alpha}}\end{split}

The third line is due to the Hölder’s inequality. ∎

A.10 Proof of Lemma 12

Proof.

Consider the covariance of $\phi$ and $\frac{dQ}{dP}$ .

\begin{split}\bigg{|}Cov_{P}\bigg{(}\phi,\frac{dQ}{dP}\bigg{)}\bigg{|}&=\bigg{|}\int_{\mathcal{H}}\phi\frac{dQ}{dP}dP-\int_{\mathcal{H}}\phi dP\int_{\mathcal{H}}\frac{dQ}{dP}dP\bigg{|}\\ &=\bigg{|}\mathbb{E}_{Q}[\phi]-\mathbb{E}_{P}[\phi]\bigg{|}\end{split}

On the other hand,

\begin{split}&\bigg{|}Cov_{P}\bigg{(}\phi,\frac{dQ}{dP}\bigg{)}\bigg{|}\\ &=\bigg{|}\mathbb{E}_{P}\bigg{[}(\phi-\mu_{P})\bigg{(}\frac{dQ}{dP}-\mathbb{E}_{P}\bigg{[}\frac{dQ}{dP}\bigg{]}\bigg{)}\bigg{]}\bigg{|}\\ &\leq\mathbb{E}_{P}\bigg{[}\bigg{|}(\phi-\mu_{P})\bigg{(}\frac{dQ}{dP}-\mathbb{E}_{P}\bigg{[}\frac{dQ}{dP}\bigg{]}\bigg{)}\bigg{|}\bigg{]}\\ &=\int_{\mathcal{H}}\bigg{|}(\phi-\mu_{P})\bigg{(}\frac{dQ}{dP}-1\bigg{)}\bigg{|}dP\\ &\leq\bigg{(}\int_{\mathcal{H}}\bigg{|}\frac{dQ}{dP}-1\bigg{|}^{\alpha}dP\bigg{)}^{\frac{1}{\alpha}}\bigg{(}\int_{\mathcal{H}}|\phi-\mu_{P}|^{\frac{\alpha}{\alpha-1}}dP\bigg{)}^{\frac{\alpha-1}{\alpha}}\\ &\leq\widetilde{\mathcal{D_{\alpha}}}(Q\|P)^{\frac{1}{\alpha}}\bigg{(}\mathbb{E}_{P}[|\phi-\mu_{P}|^{\frac{\alpha}{\alpha-1}}]\bigg{)}^{\frac{\alpha-1}{\alpha}}\end{split}

which proves our claim. ∎

A.11 Proof of Proposition 1

Proof.

Suppose the convex function $\Delta:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ is defined as in Proposition 5. By Chebyshev’s inequality, for any $\epsilon>0$ and any $h\in\mathcal{H}$ ,

\begin{split}&\mathop{Pr}_{(x,y)\sim D}(\phi_{D}(h)\geq\epsilon)\\ &=\mathop{Pr}_{(x,y)\sim D}\bigg{(}(R_{S}(h)-R_{D}(h))^{2}\geq\frac{\epsilon}{t}\bigg{)}\\ &=\mathop{Pr}_{(x,y)\sim D}\bigg{(}\bigg{|}\frac{1}{m}\sum_{i=1}^{m}\ell(h(x_{i}),y_{i})-\mathop{\mathbb{E}}_{(x,y)\sim D}\ell(h(x),y)\bigg{|}\geq\sqrt{\frac{\epsilon}{t}}\bigg{)}\\ &\leq\frac{t\sigma^{2}}{m\epsilon}\end{split}

Setting $\delta=\frac{t\sigma^{2}}{m\epsilon}$ , we have

\begin{split}&\phi_{D}(h)\mathop{\leq}_{1-\delta}\frac{t\sigma^{2}}{m\delta}\end{split}

Now, we have the upper bound for $\phi_{D}(h)$ so we can apply the same procedure as in Proposition 5. ∎

A.12 Proof of Proposition 2

Proof.

Note that we have Lemma 12. Now, it remains to bound $\mathbb{E}_{P}[\phi(X)]$ and $\mathbb{E}_{P}[|\phi(X)-\mathbb{E}_{P}[\phi(X)]|^{\frac{\alpha}{\alpha-1}}]$ . Since $P$ is a strongly log-concave distribution and $\phi$ is an L-Lipschitz function, Theorem 3.16 in [35] implies $\phi(X)-\mathbb{E}_{P}[\phi(X)]$ is a sub-Gaussian random variable with parameter $\sigma^{2}=\frac{2L^{2}}{\gamma}$ . Therefore, we have

\begin{split}Pr\bigg{(}\bigg{|}\frac{1}{n}\sum^{n}_{i=1}\phi(X_{i})-\mathbb{E}_{P}[\phi(X)]\bigg{|}\geq\epsilon\bigg{)}\leq 2e^{\frac{n\gamma\epsilon^{2}}{4L^{2}}}\end{split}

Thus,

\bigg{|}\mathbb{E}_{P}[\phi(X)]-\frac{1}{n}\sum^{n}_{i=1}\phi(X_{i})\bigg{|}\mathop{\leq}_{1-\delta}\frac{4L^{2}}{n\gamma}\log(\frac{2}{\delta})

On the other hand, letting $U=\phi(X)-\mathbb{E}_{P}[\phi(X)]$ , then we have

\begin{split}\mathbb{E}_{P}[|U|^{\frac{\alpha}{\alpha-1}}]&=\int_{0}^{\infty}Pr\{|U|^{\frac{\alpha}{\alpha-1}}>u\}du\\ &=\int_{0}^{\infty}Pr\{|U|>u^{\frac{\alpha-1}{\alpha}}\}du\\ &=\frac{\alpha}{\alpha-1}\int_{0}^{\infty}u^{\frac{1}{\alpha-1}}Pr\{|U|>u\}du\\ &=\frac{\alpha}{\alpha-1}\int_{0}^{\infty}u^{\frac{1}{\alpha-1}}Pr\{|\phi(X)-\mathbb{E}_{P}[\phi(X)]|>u\}du\\ &\leq\frac{\alpha}{\alpha-1}\int_{0}^{\infty}u^{\frac{1}{\alpha-1}}2e^{-\frac{\gamma u^{2}}{4L^{2}}}du\\ &=\frac{\alpha}{\alpha-1}\int_{0}^{\infty}u^{\frac{\alpha}{2(\alpha-1)}-1}e^{-\frac{\gamma u}{4L^{2}}}du\\ &=\frac{\alpha}{\alpha-1}\Gamma\bigg{(}\frac{\alpha}{2(\alpha-1)}\bigg{)}\bigg{(}\frac{4L^{2}}{\gamma}\bigg{)}^{\frac{\alpha}{2(\alpha-1)}}\\ &=2\Gamma\bigg{(}\frac{3\alpha-2}{2(\alpha-1)}\bigg{)}\bigg{(}\frac{4L^{2}}{\gamma}\bigg{)}^{\frac{\alpha}{2(\alpha-1)}}\\ \end{split}

The inequality is due to sub-Gaussianity. Putting everything together with Lemma 12, we prove our claim. ∎

A.13 Proof of Proposition 3

Proof.

By Lemma 11, we have

\begin{split}|E_{Q}[\phi(X)]-\mathbb{E}_{P}[\phi(X)]|\leq\sqrt{\chi^{2}(Q\|P)\mathbb{E}_{P}[\phi^{2}(X)]}\end{split}

where $\mathbb{E}_{P}[\phi(X)]$ is bounded as in Proposition 2.

\bigg{|}\mathbb{E}_{P}[\phi(X)]-\frac{1}{n}\sum^{n}_{i=1}\phi(X_{i})\bigg{|}\mathop{\leq}_{1-\delta}\frac{4L^{2}}{n\gamma}\log(\frac{2}{\delta})

It remains to bound $E_{P}[\phi^{2}(X)]$ . Note that $U^{2}$ is a sub-exponential random variable with parameters $(4\sqrt{2}\sigma^{2},4\sigma^{2})$ when $U$ is a sub-Gaussian random variable with a parameter $\sigma$ (See Appendix B in [15] for details). Therefore,

\begin{split}Pr\bigg{(}\bigg{|}\frac{1}{n}\sum^{n}_{i=1}\phi^{2}(X_{i})-\mathbb{E}_{P}[\phi^{2}(X)]\bigg{|}\geq\epsilon\bigg{)}\leq\begin{cases}2e^{-\frac{n\gamma^{2}\epsilon^{2}}{256L^{4}}}\quad\text{if }\log\frac{2}{\delta}\leq n\\ 2e^{-\frac{n\gamma^{2}\epsilon^{2}}{256L^{4}}}\quad\text{if }\log\frac{2}{\delta}>n\end{cases}\end{split}

Thus,

\begin{split}\bigg{|}\mathbb{E}_{P}[\phi^{2}(X)]-\frac{1}{n}\sum^{n}_{i=1}\phi^{2}(X_{i})\bigg{|}\mathop{\leq}_{1-\delta}\begin{cases}\frac{16L^{2}}{n}\sqrt{\frac{1}{n}\log(\frac{2}{\delta})}\quad\text{if }\log\frac{2}{\delta}\leq n\\ \frac{16L^{2}}{n\gamma}\log(\frac{2}{\delta})\quad\text{if }\log\frac{2}{\delta}>n\end{cases}\end{split}

Putting everything together, we prove our claim. ∎

A.14 Proof of Proposition 4

Proof.

Note that we have the following change of measure inequality for any function $\psi:\mathbb{R}\rightarrow\mathbb{R}$ from the Donsker-Varadhan representation for the KL-divergence.

\mathbb{E}_{Q}[\psi(X)]\leq KL(Q\|P)+\log(\mathbb{E}_{P}[e^{\psi(X)}])

(8)

Suppose $\psi_{1}:\mathbb{R}\rightarrow\mathbb{R}$ satisfies the following condition.

\psi_{1}(X)=\phi(X)-\mathbb{E}_{P}[\phi(X)]

By plugging in Equation (8), we have

\begin{split}\mathbb{E}_{Q}[\phi(X)]-\mathbb{E}_{P}[\phi(X)]&\leq KL(Q\|P)+\log(\mathbb{E}_{P}[e^{\phi(X)-\mathbb{E}_{P}[\phi(X)]}])\end{split}

On the other hand, suppose $\psi_{2}:\mathbb{R}\rightarrow\mathbb{R}$ satisfies the following condition.

\psi_{2}(X)=\mathbb{E}_{P}[\phi(X)]-\phi(X)

By plugging in Equation (8), we have

\begin{split}\mathbb{E}_{P}[\phi(X)]-\mathbb{E}_{Q}[\phi(X)]&\leq KL(Q\|P)+\log(\mathbb{E}_{P}[e^{\mathbb{E}_{P}[\phi(X)]-\phi(X)}])\end{split}

By putting all together, we have

\begin{split}&|\mathbb{E}_{Q}[\phi(X)]-\mathbb{E}_{P}[\phi(X)]|\leq KL(Q\|P)+\log(\mathbb{E}_{P}[e^{\phi(X)-\mathbb{E}_{P}[\phi(X)]}])\vee\log(\mathbb{E}_{P}[e^{\mathbb{E}_{P}[\phi(X)]-\phi(X)}])\end{split}

Suppose $X_{1},X_{2},\dots,X_{n}$ are i.i.d. random variables.We can easily see that we have

\begin{split}|\mathbb{E}_{Q}[\phi(X)]-\mathbb{E}_{P}[\phi(X)]|&\leq KL(Q\|P)\\ &+\log(\mathbb{E}_{P}[e^{\frac{1}{n}\sum_{i=1}^{n}\phi(X_{i})-\mathbb{E}_{P}[\phi(X)]}])\vee\log(\mathbb{E}_{P}[e^{\mathbb{E}_{P}[\phi(X)]-\frac{1}{n}\sum_{i=1}^{n}\phi(X_{i})}])\end{split}

(9)

$\mathbb{E}_{P}[\phi(X)]$ is bounded as in Proposition 2.

\bigg{|}\mathbb{E}_{P}[\phi(X)]-\frac{1}{n}\sum^{n}_{i=1}\phi(X_{i})\bigg{|}\mathop{\leq}_{1-\delta}\frac{4L^{2}}{n\gamma}\log(\frac{2}{\delta})

(10)

As shown in Proposition 2, $\phi(X)-\mathbb{E}_{P}[\phi(X)]$ is a sub-Gaussian random variable with parameter $\sigma^{2}=\frac{2L^{2}}{\gamma}$ . Therefore, by Definition 2,

\log(\mathbb{E}_{P}[e^{\frac{1}{n}\sum_{i=1}^{n}\phi(X_{i})-\mathbb{E}_{P}[\phi(X)]}])\vee\log(\mathbb{E}_{P}[e^{\mathbb{E}_{P}[\phi(X)]-\frac{1}{n}\sum_{i=1}^{n}\phi(X_{i})}])\leq\frac{L^{2}}{n\gamma}

(11)

since both values in the maximum are bounded by $\frac{L^{2}}{n\gamma}$ . By (9), (10) and (11), we prove our claim. ∎

Appendix B Pseudo $\alpha$ -divergence is a member of the family of $f$ -divergences.

Proof.

Let $f(t)=|t-1|^{\alpha}$ and $\lambda\in[0,1]$ . Let $\frac{1}{\alpha}+\frac{1}{\beta}=1$ for $\alpha\geq 1$ .

\begin{split}&f(\lambda x+(1-\lambda)y)\\ &=|\lambda x+(1-\lambda)y-1|^{\alpha}\\ &=|\lambda(x-1)+(1-\lambda)(y-1)|^{\alpha}\\ &\leq\{\lambda|x-1|+(1-\lambda)|y-1|\}^{\alpha}\\ &=\{\lambda^{\frac{1}{\alpha}}|x-1|\lambda^{\frac{1}{\beta}}+(1-\lambda)^{\frac{1}{\alpha}}|y-1|(1-\lambda)^{\frac{1}{\beta}}\}^{\alpha}\\ &\leq(\lambda|x-1|^{\alpha}+(1-\lambda)|y-1|^{\alpha})(\lambda+(1-\lambda))^{\frac{\alpha}{\beta}}\\ &=\lambda|x-1|^{\alpha}+(1-\lambda)|y-1|^{\alpha}\\ &=\lambda f(x)+(1-\lambda)f(y)\end{split}

The first inequality is due to the triangle inequality and the second inequality is due to Hölder’s inequality. By the definition of convexity, $f(t)$ is convex. Also, $f(1)=0$ . By the definition of $f$ -divergence, we prove our claim. ∎

Appendix C PAC-Bayesian bounds for bounded, sub-Gaussian and sub-exponential losses

Here we complement our results in Section 3, by showing our PAC-Bayesian generalization bounds for bounded, sub-Gaussian and sub-exponential losses.

C.1 Bounded Loss Function

First, let us assume here that the loss function is bounded, i.e., for any $h\in\mathcal{H}$ and $(x,y)\in\mathcal{X}\times\mathcal{Y}$ , $\ell(h(x),y)\in[0,R]$ for $R>0$ . Note that, for $R>1$ , we cannot use the total variation, squared Hellinger, Reverse KL and Neyman $\chi^{2}$ divergence because $\phi$ is constrained to be in $[0,1]$ .

Proposition 5 (The PAC-Bayesian bounds for bounded loss function).

Let $P$ be any prior distribution over an infinite hypothesis space $\mathcal{H}$ . For a given posterior distribution $Q$ over an infinite hypothesis space $\mathcal{H}$ , let $R_{D}(G_{Q})$ and $R_{S}(G_{Q})$ be the Gibbs risk and the empirical Gibbs risk as in Equation (2) and (3) respectively. For the sample size $m>0$ and $\alpha>1$ , with probability at least $1-\delta$ , simultaneously for all posterior distributions $Q$ , we have

\begin{split}R_{D}(G_{Q})\leq R_{S}(G_{Q})+\sqrt{\frac{R^{2}}{2m}\log(\frac{2}{\delta})\big{(}\alpha(\alpha-1)D_{\alpha}(Q\|P)+1\big{)}^{\frac{1}{\alpha}}}\end{split}

(12)

\begin{split}R_{D}(G_{Q})\leq R_{S}(G_{Q})+\sqrt{\frac{1}{m}\bigg{(}D_{\alpha}(Q\|P)+\frac{1}{\alpha(\alpha-1)}\bigg{)}+\frac{1}{m\alpha}\bigg{(}\frac{R^{2}(\alpha-1)}{2}\log(\frac{2}{\delta})\bigg{)}^{\frac{\alpha}{\alpha-1}}}\end{split}

(13)

Proof.

Suppose that we have a convex function $\Delta:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ , that measures the discrepancy between the observed empirical Gibbs risk $R_{S}(G_{Q})$ and the true Gibbs risk $R_{D}(G_{Q})$ on distribution $Q$ . Given that, the purpose of the PAC-Bayesian theorem is to upper-bound the discrepancy $t\Delta(R_{D}(G_{Q}),R_{S}(G_{Q}))$ for any $t>0$ . Let $\phi_{D}(h):=t\Delta(R_{D}(h),R_{S}(h))$ , where the subscript of $\phi_{D}$ shows the dependency on the data distribution $D$ . Let $\Delta(q,p)=(q-p)^{2}$ . By applying Jensen’s inequality on convex function $\Delta$ for the first step,

\begin{split}t\Delta(R_{D}(G_{Q}),R_{S}(G_{Q}))&=t\Delta(\mathop{\mathbb{E}}_{h\sim Q}R_{D}(h),\mathop{\mathbb{E}}_{h\sim Q}R_{S}(h))\\ &\leq\mathop{\mathbb{E}}_{h\sim Q}t\Delta(R_{D}(h),R_{S}(h))\\ &=\mathop{\mathbb{E}}_{h\sim Q}\phi_{D}(h)\end{split}

(14)

where $\phi_{D}(h)=t\Delta(R_{D}(h),R_{S}(h))=t(R_{S}(h)-R_{D}(h))^{2}$ . By Hoeffding’s inequality, for any $\epsilon>0$ and any $h\in\mathcal{H}$ ,

\begin{split}&\mathop{Pr}_{(x,y)\sim D}(\phi_{D}(h)\geq\epsilon)\\ &=\mathop{Pr}_{(x,y)\sim D}\bigg{(}\big{(}R_{S}(h)-R_{D}(h)\big{)}^{2}\geq\frac{\epsilon}{t}\bigg{)}\\ &=\mathop{Pr}_{(x,y)\sim D}\bigg{(}\bigg{|}\frac{1}{m}\sum_{i=1}^{m}\ell(h(x_{i}),y_{i})-\mathop{\mathbb{E}}_{(x,y)\sim D}\ell(h(x),y)\bigg{|}\geq\sqrt{\frac{\epsilon}{t}}\bigg{)}\\ &\leq 2e^{-\frac{2m\epsilon}{R^{2}t}}\end{split}

Setting $\delta=2e^{-\frac{2m\epsilon}{R^{2}t}}$ , we have

\begin{split}\phi_{D}(h)\mathop{\leq}_{1-\delta}\frac{tR^{2}}{2m}\log(\frac{2}{\delta})\end{split}

The symbol $\displaystyle{\mathop{\leq}_{1-\delta}}$ denotes that the inequality holds with probability at least $1-\delta$ . The second line holds due to the Hoeffding’s inequality. For $\alpha>1$ , we have

\begin{split}(\phi_{D}(h))^{\frac{\alpha}{\alpha-1}}\mathop{\leq}_{1-\delta}\bigg{(}\frac{tR^{2}}{2m}\log(\frac{2}{\delta})\bigg{)}^{\frac{\alpha}{\alpha-1}}\end{split}

Also note that

\begin{split}\mathop{\mathbb{E}}_{h\sim P}[(\phi_{D}(h))^{\frac{\alpha}{\alpha-1}}]&\leq\sup_{(x,y)\sim D}\bigg{\{}\mathop{\mathbb{E}}_{h\sim P}[(\phi_{D}(h))^{\frac{\alpha}{\alpha-1}}]\bigg{\}}\\ &\leq\mathop{\mathbb{E}}_{h\sim P}\bigg{[}\sup_{(x,y)\sim D}(\phi_{D}(h))^{\frac{\alpha}{\alpha-1}}\bigg{]}\\ &\mathop{\leq}_{1-\delta}\bigg{(}\frac{tR^{2}}{2m}\log(\frac{2}{\delta})\bigg{)}^{\frac{\alpha}{\alpha-1}}\end{split}

(15)

By applying Equations (14) and (15) to Lemma 10,

\begin{split}t(R_{D}(G_{Q})-R_{S}(G_{Q}))^{2}\mathop{\leq}_{1-\delta}\frac{tR^{2}}{2m}\log(\frac{2}{\delta})\bigg{(}\alpha(\alpha-1)D_{\alpha}(Q\|P)+1\bigg{)}^{\frac{1}{\alpha}}\end{split}

which proves Equation (12). By applying Equations (14) and (15) to Lemma 5 and setting $t=m$ , we prove Equation (13). ∎

Equation (12) has a tighter bound than Proposition 2 in [1]. Also, Equation (13) has a novel expression of PAC-Bayesian theorem with $\chi^{2}$ -divergence. The same arguments apply to the bounds in Proposition 6, 7 and 1.

The following corollary is an immediate consequence of this proposition for $\alpha=2$ .

Corollary 3 (The PAC-Bayesian bounds with $\chi^{2}$ -divergence for bounded loss function).

\begin{split}R_{D}(G_{Q})&\leq R_{S}(G_{Q})+\sqrt{\frac{R^{2}}{2m}\log(\frac{2}{\delta})\sqrt{\chi^{2}(Q\|P)+1}}\end{split}

(16)

\begin{split}R_{D}(G_{Q})\leq R_{S}(G_{Q})+\sqrt{\frac{1}{2m}\bigg{(}\chi^{2}(Q\|P)+1+\bigg{(}\frac{R^{2}}{2}\log(\frac{2}{\delta})\bigg{)}^{2}\bigg{)}}\end{split}

(17)

These are novel PAC-Bayesian bounds for bounded loss functions based on $\chi^{2}$ -divergence. Most PAC-Bayesian bounds for bounded loss functions are based on the KL-divergence for the complexity term (e.g., [4, 30, 22]). [15] and [20] contain bounds for bounded loss functions with $\chi^{2}$ -divergence. Compared to their bounds, the bound (16) is tighter due to the power of $\frac{1}{4}$ on the complexity term and $\log(\frac{1}{\delta})$ instead of $\frac{1}{\delta}$ . Also, PAC-Bayes bound (17) has a unique characteristics; $\chi^{2}(Q\|P)$ and $\frac{1}{\delta}$ are independent since $\chi^{2}(Q\|P)$ is not multiplied by a factor of $\frac{1}{\delta}$ . Please see Table 3 for details.

C.2 Sub-Gaussian Loss Function

In some contexts, such as regression, considering bounded loss functions is restrictive. Next, we relax the restrictions on the loss function to deal with unbounded losses. We assume that, for any $h\in\mathcal{H}$ , $\ell(h(x),y)$ is sub-Gaussian. First, we mention the definition of sub-Gaussian random variable [35].

Definition 2.

A random variable $Z$ is said to be sub-Gaussian with the expectation $\mathbb{E}[Z]=\mu$ and variance proxy $\sigma^{2}$ if for any $\lambda\in\mathbb{R}$ ,

\begin{split}\mathbb{E}[e^{\lambda(Z-\mu)}]\leq e^{\frac{\lambda^{2}\sigma^{2}}{2}}\end{split}

Next, we present our PAC-Bayesian bounds.

Proposition 6 (The PAC-Bayesian bounds for sub-Gaussian loss function).

Let $P$ be a fixed prior distribution over an infinite hypothesis space $\mathcal{H}$ . For a given posterior distribution $Q$ over an infinite hypothesis space $\mathcal{H}$ , let $R_{D}(G_{Q})$ and $R_{S}(G_{Q})$ be the Gibbs risk and the empirical Gibbs risk as in Equation (2) and (3) respectively. For the sample size $m$ and $\alpha>1$ , with probability at least $1-\delta$ , simultaneously for all posterior distributions $Q$ , we have

\begin{split}R_{D}(G_{Q})\leq R_{S}(G_{Q})+\sqrt{\frac{2\sigma^{2}}{m}\log(\frac{2}{\delta})\big{(}\alpha(\alpha-1)D_{\alpha}(Q\|P)+1\big{)}^{\frac{1}{\alpha}}}\end{split}

\begin{split}R_{D}(G_{Q})\leq R_{S}(G_{Q})+\sqrt{\frac{1}{m}\bigg{(}D_{\alpha}(Q\|P)+\frac{1}{\alpha(\alpha-1)}\bigg{)}+\frac{1}{m\alpha}\bigg{(}2\sigma^{2}(\alpha-1)\log(\frac{2}{\delta})\bigg{)}^{\frac{\alpha}{\alpha-1}}}\end{split}

Proof.

Suppose the convex function $\Delta:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ is defined as in Proposition 5. Employing Chernoff’s bound, the tail bound probability for sub-Gaussian random variables [35] is given as follows

\begin{split}Pr(|\bar{Z}-\mu|\geq\epsilon)\leq 2e^{-\frac{m\epsilon^{2}}{2\sigma^{2}}}\end{split}

(18)

Setting $\overline{Z}=R_{S}(h)$ , $\mu=R_{D}(h)$ and $\delta=2e^{-\frac{m\epsilon^{2}}{2\sigma^{2}}}$ in the tail bound in Equation (18), for any $h\in\mathcal{H}$ , we have

\begin{split}&\mathop{Pr}_{(x,y)\sim D}(\phi_{D}(h)\geq\epsilon)\\ &=\mathop{Pr}_{(x,y)\sim D}\bigg{(}(R_{S}(h)-R_{D}(h))^{2}\geq\frac{\epsilon}{t}\bigg{)}\\ &=\mathop{Pr}_{(x,y)\sim D}\bigg{(}\bigg{|}\frac{1}{m}\sum_{i=1}^{m}\ell(h(x_{i}),y_{i})-\mathop{\mathbb{E}}_{(x,y)\sim D}\ell(h(x),y)\bigg{|}\geq\sqrt{\frac{\epsilon}{t}}\bigg{)}\\ &\leq 2e^{-\frac{m\epsilon}{2\sigma^{2}t}}\end{split}

Setting $\delta=2e^{-\frac{m\epsilon}{2\sigma^{2}t}}$ , we have

\phi_{D}(h)\mathop{\leq}_{1-\delta}\frac{2t\sigma^{2}}{m}\log(\frac{2}{\delta})

where $\phi_{D}(h)$ is defined as in Equation (14). By Equation (15), we have

\mathop{\mathbb{E}}_{h\sim P}[(\phi_{D}(h))^{\frac{\alpha}{\alpha-1}}]\mathop{\leq}_{1-\delta}\bigg{(}\frac{2t\sigma^{2}}{m}\log(\frac{2}{\delta})\bigg{)}^{\frac{\alpha}{\alpha-1}}

(19)

On the other hand, we may upper-bound $\mathop{\mathbb{E}}_{h\sim P}[(\phi_{D}(h))^{\frac{\alpha}{\alpha-1}}]$ in the following way (as in Proposition 6. in [1]).

\begin{split}\mathop{\mathbb{E}}_{h\sim P}[(\phi_{D}(h))^{\frac{\alpha}{\alpha-1}}]&\mathop{\leq}_{1-\delta}\frac{t^{\frac{\alpha}{\alpha-1}}}{\delta}\mathop{\mathbb{E}}_{(x,y)\sim D}\mathop{\mathbb{E}}_{h\sim P}[(R_{S}(h)-R_{D}(h))^{\frac{2\alpha}{\alpha-1}}]\\ &=\frac{t^{\frac{\alpha}{\alpha-1}}}{\delta}\mathop{\mathbb{E}}_{h\sim P}\mathop{\mathbb{E}}_{(x,y)\sim D}[(R_{S}(h)-R_{D}(h))^{\frac{2\alpha}{\alpha-1}}]\\ \end{split}

Let $U=R_{S}(h)-R_{D}(h)$ and $q=\frac{\alpha}{\alpha-1}$ . Then, $U$ is a sub-Gaussian random variable from our assumption.

\begin{split}&\mathop{\mathbb{E}}_{(x,y)\sim D}[(R_{S}(h)-R_{D}(h))^{\frac{2\alpha}{\alpha-1}}]=\mathop{\mathbb{E}}_{(x,y)\sim D}[U^{2q}]\\ &=\int_{0}^{\infty}Pr\{|U|^{2q}>u\}du=2q\int_{0}^{\infty}u^{2q-1}Pr\{|U|>u\}du\\ &\leq 4q\int_{0}^{\infty}u^{2q-1}e^{-\frac{mu^{2}}{2\sigma^{2}}}du\end{split}

By setting $u=\sqrt{t}$ , the previous inequality becomes

\begin{split}&\mathop{\mathbb{E}}_{(x,y)\sim D}[(R_{S}(h)-R_{D}(h))^{\frac{2\alpha}{\alpha-1}}]\leq 2q\int_{0}^{\infty}t^{q-1}e^{-\frac{mt}{2\sigma^{2}}}dt\\ &=2q\Gamma(q)\bigg{(}\frac{m}{2\sigma^{2}}\bigg{)}^{-q}=2\frac{\alpha}{\alpha-1}\Gamma\bigg{(}\frac{\alpha}{\alpha-1}\bigg{)}\bigg{(}\frac{2\sigma^{2}}{m}\bigg{)}^{\frac{\alpha}{\alpha-1}}\\ \end{split}

Therefore, we have

\begin{split}\mathop{\mathbb{E}}_{h\sim P}[(\phi_{D}(h))^{\frac{\alpha}{\alpha-1}}]\mathop{\leq}_{1-\delta}\frac{2\alpha t^{\frac{\alpha}{\alpha-1}}}{\delta(\alpha-1)}\Gamma\bigg{(}\frac{\alpha}{\alpha-1}\bigg{)}\bigg{(}\frac{2\sigma^{2}}{m}\bigg{)}^{\frac{\alpha}{\alpha-1}}\end{split}

(20)

Although one has choices to use either Equation (19) or (20), we can easily see that Equation (19) is always tighter than (20). Putting everything together with Lemma 5, we prove our claim. ∎

The following corollary is an immediate consequnece of Proposition 6 for $\alpha=2$ .

Corollary 4 (The PAC-Bayesian bounds with $\chi^{2}$ -divergence for sub-Gaussian loss function).

\begin{split}R_{D}(G_{Q})&\leq R_{S}(G_{Q})+\sqrt{\frac{2\sigma^{2}}{m}\log(\frac{2}{\delta})\sqrt{\chi^{2}(Q\|P)+1}}\end{split}

(21)

\begin{split}R_{D}(G_{Q})\leq R_{S}(G_{Q})+\sqrt{\frac{1}{2m}\bigg{(}\chi^{2}(Q\|P)+1+\bigg{(}2\sigma^{2}\log(\frac{2}{\delta})\bigg{)}^{2}\bigg{)}}\end{split}

(22)

[1] proved PAC-Bayes bound for sub-Gaussian loss function with $\chi^{2}$ -divergence. It is noteworthy that $\Delta$ may be any convex function and a different choice of $\Delta$ leads us to various bounds. For instance, choosing $\Delta(q,p)=|q-p|$ for Lemma 10 results in

\begin{split}R_{D}(G_{Q})&\leq R_{S}(G_{Q})+\sqrt{\frac{2\sigma^{2}}{m}\log(\frac{2}{\delta})(\chi^{2}(Q\|P)+1)}\end{split}

which is quite similar to the bounds in Proposition 6 in [1] and looser than Equation (21). We obtained a tighter bound due to our choice of $\Delta(q,p)=(q-p)^{2}$ .

C.3 Sub-Exponential Loss Function

We now turn to a more general class where $\ell(h(x),y)$ is sub-exponential for any $h\in\mathcal{H}$ . First, we define sub-exponentiality [35].

Definition 3.

A random variable $Z$ is said to be sub-exponential with the expectation $\mathbb{E}[Z]=\mu$ and parameters $\sigma^{2}$ and $\beta>0$ , if for any $\lambda\in\mathbb{R}$ ,

\begin{split}\mathbb{E}[e^{\lambda(Z-\mu)}]\leq e^{\frac{\lambda^{2}\sigma^{2}}{2}},\forall:|\lambda|<\frac{1}{\beta}\end{split}

Next, we provide our PAC-Bayesian bounds.

Proposition 7 (The PAC-Bayesian bounds for sub-exponential loss function).

\begin{split}R_{D}(G_{Q})\leq R_{S}(G_{Q})+\sqrt{\mathcal{K}^{1}_{\delta}\big{(}\alpha(\alpha-1)D_{\alpha}(Q\|P)+1\big{)}^{\frac{1}{\alpha}}}\end{split}

\begin{split}R_{D}(G_{Q})\leq R_{S}(G_{Q})+\sqrt{\frac{1}{t}\bigg{(}D_{\alpha}(Q\|P)+\frac{1}{\alpha(\alpha-1)}\bigg{)}+\mathcal{K}^{2}_{\alpha,\delta}}\end{split}

where

\begin{split}\mathcal{K}^{1}_{\delta}=\begin{cases}\frac{2\sigma^{2}}{m}\log(\frac{2}{\delta}),\quad\frac{2\beta^{2}\log(\frac{2}{\delta})}{\sigma^{2}}\leq m\\ (\frac{2\beta}{m}\log\frac{2}{\delta})^{2},\quad 0<m<\frac{2\beta^{2}\log(\frac{2}{\delta})}{\sigma^{2}},\end{cases}\mathcal{K}^{2}_{\alpha,\delta}=\frac{m^{\frac{1}{\alpha-1}}(\alpha-1)^{\frac{\alpha}{\alpha-1}}}{\alpha}(\mathcal{K}^{1}_{\delta})^{\frac{\alpha}{\alpha-1}}\end{split}

Proof.

Suppose the convex function $\Delta:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ is defined as in Proposition 5. For any random variables satisfying Definition 3, we have the following concentration inequality [35].

\begin{split}Pr(|\overline{Z}-\mu|\geq\epsilon)\leq\begin{cases}2e^{-\frac{m\epsilon^{2}}{2\sigma^{2}}},\quad 0<\epsilon\leq\frac{\sigma^{2}}{\beta}\\ 2e^{-\frac{m\epsilon}{2\beta}},\quad\frac{\sigma^{2}}{\beta}<\epsilon\end{cases}\end{split}

(23)

Following the proof of Proposition 6, for $0<\sqrt{\frac{\epsilon}{t}}\leq\frac{\sigma^{2}}{\beta}$ , we have

\phi_{D}(h)\mathop{\leq}_{1-\delta}\frac{2t\sigma^{2}}{m}\log(\frac{2}{\delta})

For $\frac{\sigma^{2}}{\beta}<\sqrt{\frac{\epsilon}{t}}$ , we have

\begin{split}&\mathop{Pr}_{(x,y)\sim D}(\phi_{D}(h)\geq\epsilon)\\ &=\mathop{Pr}_{(x,y)\sim D}\bigg{(}(R_{S}(h)-R_{D}(h))^{2}\geq\frac{\epsilon}{t}\bigg{)}\\ &=\mathop{Pr}_{(x,y)\sim D}\bigg{(}\bigg{|}\frac{1}{m}\sum_{i=1}^{m}\ell(h(x_{i}),y_{i})-\mathop{\mathbb{E}}_{(x,y)\sim D}\ell(h(x),y)\bigg{|}\geq\sqrt{\frac{\epsilon}{t}}\bigg{)}\\ &\leq 2e^{-\frac{m}{2\beta}\sqrt{\frac{\epsilon}{t}}}\end{split}

Setting $\delta=2e^{-\frac{m}{2\beta}\sqrt{\frac{\epsilon}{t}}}$ , we have, for $\frac{\sigma^{2}}{\beta}<\sqrt{\frac{\epsilon}{t}}$ ,

\phi_{D}(h)\mathop{\leq}_{1-\delta}t\bigg{(}\frac{2\beta}{m}\log\frac{2}{\delta}\bigg{)}^{2}

Thus,

\begin{split}\phi_{D}(h)\mathop{\leq}_{1-\delta}\begin{cases}\frac{2t\sigma^{2}}{m}\log(\frac{2}{\delta}),\quad\frac{2\beta^{2}\log(\frac{2}{\delta})}{\sigma^{2}}\leq m\\ t(\frac{2\beta}{m}\log\frac{2}{\delta})^{2},\quad 0<m<\frac{2\beta^{2}\log(\frac{2}{\delta})}{\sigma^{2}}\end{cases}\end{split}

where $\phi_{D}(h)$ is defined as in Equation (14). Now, we have the upper bound for $\phi_{D}(h)$ so we can apply the same procedure as in Proposition 5 and 7. ∎

Note that, for $\frac{2\beta^{2}\log(\frac{2}{\delta})}{\sigma^{2}}\leq m$ , $\phi_{D}(h)$ behaves like sub-Gaussian. However, when the sample size is small, a tighter bound (i.e., $t(\frac{2\beta}{m}\log\frac{2}{\delta})^{2}$ ) can be obtained. This shows the advantage of assuming sub-exponentiality over sub-Gaussianity. By setting $\alpha=2$ in Proposition 7, we have the following corollary.

Corollary 5 (The PAC-Bayesian bounds with $\chi^{2}$ -divergence for sub-exponential loss function).

\begin{split}R_{D}(G_{Q})&\leq R_{S}(G_{Q})+\sqrt{\mathcal{K}^{1}_{\delta}\sqrt{\chi^{2}(Q\|P)+1}}\end{split}

\begin{split}R_{D}(G_{Q})&\leq R_{S}(G_{Q})+\sqrt{\frac{1}{2m}\big{(}\chi^{2}(Q\|P)+1+(m\mathcal{K}^{1}_{\delta})^{2}\big{)}}\end{split}

where

\begin{split}&\mathcal{K}^{1}_{\delta}=\begin{cases}\frac{2\sigma^{2}}{m}\log(\frac{2}{\delta}),\quad\frac{2\beta^{2}\log(\frac{2}{\delta})}{\sigma^{2}}\leq m\\ (\frac{2\beta}{m}\log\frac{2}{\delta})^{2},\quad 0<m<\frac{2\beta^{2}\log(\frac{2}{\delta})}{\sigma^{2}}\end{cases}\end{split}

To the best of our knowledge, our bounds for sub-exponential losses are entirely novel.

Appendix D Log-concave distribution

We say that a distribution $P$ with a density $p$ (with respect to the Lebesgue measure) is a strongly log-concave distribution if the function $\log p$ is strongly concave. Equivalently stated, this condition means that the density can be expressed as $p(x)=\exp(-\psi(x))$ , where the function $\psi:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is strongly convex, meaning that there is some $\gamma>0$ such that

\lambda\psi(x)+(1-\lambda)\psi(y)-\psi(\lambda x+(1-\lambda)y)\geq\frac{\gamma}{2}\lambda(1-\lambda)\|x-y\|_{2}^{2}

for all $\lambda\in[0,1]$ , and $x,y\in\mathbb{R}^{d}$ .

Novel Change of Measure Inequalities with Applications to PAC-Bayesian Bounds and Monte Carlo Estimation

Abstract

1 Introduction

2 Change of Measure Inequalities

2.1 Change of Measure Inequality from the Variational Representation of ff-divergences

Definition 1 (ff-divergence and its variational representation).

Lemma 1 (Variational representation of f−f-divergence, Lemma 1 in [27]).

Theorem 1 (Change of measure inequality from the constrained variational representation for ff-divergences [29]).

Corollary 1 (Change of measure inequality from the unconstrained variational representation for ff-divergences).

Lemma 2 (Change of measure inequality from the constrained representation of the χ2\chi^{2}-divergence).

Lemma 3 (Change of measure inequality from the unconstrained representation of the Pearson χ2\chi^{2}-divergence).

Lemma 4 (Change of measure inequality from the constrained representation of the total variation divergence).

Lemma 5 (Change of measure inequality from the unconstrained representation of the α\alpha-divergence).

Lemma 6 (Change of measure inequality from the unconstrained representation of the squared Hellinger divergence).

Lemma 7 (Change of measure inequality from the unconstrained representation of the reverse KL-divergence).

Lemma 8 (Change of measure inequality from the unconstrained representation of the Neyman χ2\chi^{2}-divergence).

2.2 Multiplicative Change of Measure Inequality for α\alpha-divergences

Lemma 9 (Multiplicative change of measure inequality for the χ2\chi^{2}-divergence [15]).

Lemma 10 (Multiplicative change of measure inequality for the α\alpha-divergence).

2.3 A Generalized Hammersley-Chapman-Robbins (HCR) Inequality

Lemma 11 (HCR inequality [18]).

Lemma 12 (The generalization of HCR inequality.).

3 Application to PAC-Bayesian Theory

3.1 Loss Function with Bounded Variance

Proposition 1 (The PAC-Bayesian bounds for loss function with bounded variance).

Corollary 2 (The PAC-Bayesian bounds with χ2\chi^{2}-divergence for bounded variance loss function).

3.2 Discussion

4 Non-Asymptotic Interval for Monte Carlo Estimates

Proposition 2 (A general expression of non-asymptotic interval for Monte Carlo estimates).

Proposition 3 (χ2\chi^{2}-based expression of non-asymptotic interval for Monte Carlo estimates).

Proposition 4 (KL-based expression of non-asymptotic interval for Monte Carlo estimates).

References

Appendix A Detailed Proofs

A.1 Proof of Corollary 1

Proof.

A.2 Proof of Lemma 2

Proof.

A.3 Proof of Lemma 3

Proof.

A.4 Proof of Lemma 4

Proof.

A.5 Proof of Lemma 5

Proof.

A.6 Proof of Lemma 6

Proof.

A.7 Proof of Lemma 7

Proof.

A.8 Proof of Lemma 8

Proof.

A.9 Proof of Lemma 10

Proof.

A.10 Proof of Lemma 12

Proof.

A.11 Proof of Proposition 1

Proof.

A.12 Proof of Proposition 2

Proof.

A.13 Proof of Proposition 3

Proof.

A.14 Proof of Proposition 4

Proof.

Appendix B Pseudo α\alpha-divergence is a member of the family of ff-divergences.

Proof.

Appendix C PAC-Bayesian bounds for bounded, sub-Gaussian and sub-exponential losses

C.1 Bounded Loss Function

Proposition 5 (The PAC-Bayesian bounds for bounded loss function).

Proof.

Corollary 3 (The PAC-Bayesian bounds with χ2\chi^{2}-divergence for bounded loss function).

C.2 Sub-Gaussian Loss Function

Definition 2.

Proposition 6 (The PAC-Bayesian bounds for sub-Gaussian loss function).

Proof.

Corollary 4 (The PAC-Bayesian bounds with χ2\chi^{2}-divergence for sub-Gaussian loss function).

C.3 Sub-Exponential Loss Function

Definition 3.

Proposition 7 (The PAC-Bayesian bounds for sub-exponential loss function).

Proof.

Corollary 5 (The PAC-Bayesian bounds with χ2\chi^{2}-divergence for sub-exponential loss function).

Appendix D Log-concave distribution

2.1 Change of Measure Inequality from the Variational Representation of $f$ -divergences

Definition 1 ( $f$ -divergence and its variational representation).

Lemma 1 (Variational representation of $f-$ divergence, Lemma 1 in [27]).

Theorem 1 (Change of measure inequality from the constrained variational representation for $f$ -divergences [29]).

Corollary 1 (Change of measure inequality from the unconstrained variational representation for $f$ -divergences).

Lemma 2 (Change of measure inequality from the constrained representation of the $\chi^{2}$ -divergence).

Lemma 3 (Change of measure inequality from the unconstrained representation of the Pearson $\chi^{2}$ -divergence).

Lemma 5 (Change of measure inequality from the unconstrained representation of the $\alpha$ -divergence).

Lemma 8 (Change of measure inequality from the unconstrained representation of the Neyman $\chi^{2}$ -divergence).

2.2 Multiplicative Change of Measure Inequality for $\alpha$ -divergences

Lemma 9 (Multiplicative change of measure inequality for the $\chi^{2}$ -divergence [15]).

Lemma 10 (Multiplicative change of measure inequality for the $\alpha$ -divergence).

Corollary 2 (The PAC-Bayesian bounds with $\chi^{2}$ -divergence for bounded variance loss function).

Proposition 3 ( $\chi^{2}$ -based expression of non-asymptotic interval for Monte Carlo estimates).

Appendix B Pseudo $\alpha$ -divergence is a member of the family of $f$ -divergences.

Corollary 3 (The PAC-Bayesian bounds with $\chi^{2}$ -divergence for bounded loss function).

Corollary 4 (The PAC-Bayesian bounds with $\chi^{2}$ -divergence for sub-Gaussian loss function).

Corollary 5 (The PAC-Bayesian bounds with $\chi^{2}$ -divergence for sub-exponential loss function).