Title: Generalization Bounds via Conditional $f$-Information
Venue: NeurIPS 2024
Accepted: yes
Keywords: information-theoretic generalization bound

generalization

f-divergence
Abstract: In this work, we introduce novel information-theoretic generalization bounds using the conditional $f$-information framework, an extension of the traditional conditional mutual information (MI) framework. We provide a generic approach to derive generalization bounds via $f$-information in the supersample setting, applicable to both bounded and unbounded loss functions. Unlike previous MI-based bounds, our proof strategy does not rely on upper bounding the cumulant-generating function (CGF) in the variational formula of MI. Instead, we set the CGF or its upper bound to zero by carefully selecting  the measurable function invoked in the variational formula. Although some of our techniques are partially inspired by recent advances in the coin-betting framework (e.g., Jang et al. (2023)), our results are independent of any previous findings from regret guarantees of online gambling algorithms. Additionally, our newly derived MI-based bound recovers many previous results and improves our understanding of their potential limitations. Finally, we empirically compare various $f$-information measures for generalization, demonstrating the improvement of our new bounds over the previous bounds.

Generalization Bounds via Conditional f -Information

Abstract

In this work, we introduce novel information-theoretic generalization bounds using the conditional f -information framework, an extension of the traditional conditional mutual information (MI) framework. We provide a generic approach to derive generalization bounds via f -information in the supersample setting, applicable to both bounded and unbounded loss functions. Unlike previous MIbased bounds, our proof strategy does not rely on upper bounding the cumulantgenerating function (CGF) in the variational formula of MI. Instead, we set the CGF or its upper bound to zero by carefully selecting the measurable function invoked in the variational formula. Although some of our techniques are partially inspired by recent advances in the coin-betting framework (e.g., Jang et al. (2023)), our results are independent of any previous findings from regret guarantees of online gambling algorithms. Additionally, our newly derived MI-based bound recovers many previous results and improves our understanding of their potential limitations. Finally, we empirically compare various f -information measures for generalization, demonstrating the improvement of our new bounds over the previous bounds.

1 Introduction

Understanding generalization is a longstanding topic of machine learning research. Recently, information-theoretic generalization measures, e.g., mutual information (MI) between the input (i.e. data) and output (i.e. hypothesis) of a learning algorithm, have been proposed by [2-4]. While original MI-based information-theoretic generalization bounds have made significant progress in analyzing the generalization of stochastic gradient-based algorithms in nonconvex learning problems (e.g., deep learning) [5-13], they are found to have several limitations. One of the most severe limitations is their unboundedness, where the MI measure can be infinite while the true generalization error is small [14]. To mitigate this issue, various bound-tightening techniques have been proposed [15, 6, 7]. Among them, [16] introduces a conditional mutual information (CMI) framework, which allows for the derivation of a CMI-based bound that is strictly upper-bounded by a constant. In the CMI framework, an additional 'ghost sample' is introduced alongside the original sample drawn from the unknown data distribution. A Bernoulli random variable sequence is then used to determine the membership of training data from these two samples. Ultimately, the CMI between these Bernoulli random variables and the output hypothesis of the learning algorithm, conditioned on these two samples, characterizes generalization. This setup, known as the 'supersample' setting, ensures the

boundedness of the CMI bound due to the constant entropy of a Bernoulli random variable. Intuitively, this CMI quantity leaks less information between data and the hypothesis, resulting in tighter bounds compared to the original MI quantity, as theoretically justified in [17]. Furthermore, several studies have aimed to further tighten CMI-based bounds. For example, previous tightening techniques in [15, 6, 7] are also applicable to CMI bounds [17-21]. Notably, another way to tighten the CMI bound is by utilizing variants of the original CMI quantity, such as functional CMI ( f -CMI) [22], evaluated CMI (e-CMI) [16, 23], and loss-difference CMI (ld-CMI) [24]. As the output hypothesis is replaced by the predictions, loss pairs, and loss differences of the two samples, these CMI quantities are tighter than the original hypothesis-based CMI due to the data-processing inequality. While most information-theoretic bounds in the supersample setting focus on (conditional) mutual information measures, it is natural to ask whether there is a general way to extend this CMI framework to other statistical distance/divergence measures, such as other f -divergences.

Using alternative dependence measures to replace MI in information-theoretic generalization bounds has been studied in several existing works [25-30], with some of them considering the supersample setting. In particular, [26] provides some Wasserstein distance-based and total variation-based bounds in the supersample setting. Furthermore, [29] proposes a general convex analysis-based framework that replaces the input-output MI quantity in [2-4] with any strongly convex function. While [29] itself does not discuss the supersample setting, the further extension in [30] indeed considers invoking the supersample setting to strengthen the bounds.

In this work, we present a generic approach to derive generalization bounds based on conditional f -information, a natural extension from mutual information to other f -dvergence-based dependence measures. Our proof strategy is significantly different from the original CMI framework [16] and the convex analysis (or regret analysis) based framework [29, 30]. Specifically, our development starts from the variational representation of f -divergence, which involves the convex conjugate of a convex function used to define the f -divergence. While the previous CMI framework focuses on upper-bounding the cumulant generating function (CGF) of a selected measurable function (which relates to the generalization error), we make a particular choice of such a measurable function, namely the inverse of that convex conjugate function. This choice makes CGF equal to (or upper bounded by) zero, hence eliminated from the variational representation. The remaining task is to lower bound this inverse of the convex conjugate function under a joint distribution. In the case of MI (or KL), the expectation of this inverse function is close to the log wealth quantity used in the coin-betting framework for obtaining concentration inequalities [31, 1, 32]. Indeed, the lower bounding techniques here are inspired by some inequalities used in the coin-betting framework [31, 1, 32], and we extend them to the cases of other f -divergence in the supersample setting. Notably, unlike [30, 1], our conditional f -information generalization bounds do not rely on any existing regret guarantees of online learning algorithms. We discuss the connection with the coin-betting framework in Appendix E.

Given that f -divergences all obey the data-processing inequality [33], we focus on using loss difference (ld)-based f -information, an extension of ld-CMI in [24], in this work. Specifically, our main contributions are summarized as follows: 1) In the case where the loss difference between a data pair is bounded, we first provide a novel variational formula (cf. Lemma 3.1) for f -information, enabling us to derive conditional f -information-based generalization bounds; 2) For the KL case, we first provide an 'oracle' CMI bound (cf. Theorem 3.1). This bound recovers many previous CMI bounds, including the square-root bound and fast-rate bounds in low empirical risk settings. It also helps us understand the potential looseness of previous square-root CMI bounds, where some quantities that can vanish as the sample size n increases are simply ignored. Additionally, some novel fast-rate bounds are implied (cf. Corollary 3.1-3.2). Particularly, Corollary 3.2 contains a term in the form of the product of total variation and KL divergence, a similar quantity also appearing in a recent PAC-Bayesian bound [34]; 3) We present several other f -information-based bounds, including the looser measure, χ 2 -information (cf. Theorem B.2) and tighter measures, squared Hellinger (SH)information (cf. Theorem 3.2) and Jensen-Shannon (JS)-information (cf. Theorem 3.3). Due to the similarity in expressions, most arguments for our CMI bounds are also applicable to these bounds based on f -information measures; 4) We extend our framework to the unbounded loss difference case, where we provide a refined variational formula for f -information (cf. Lemma 4.1), enabling us to provide a novel f -information-based bound (cf. Theorem 4.1 for the case of KL). The obtained bound is no worse than a previous information-theoretic generalization bound in [25] (adapted to the same supersample setting); 5) Empirical results show that the new bounds in our framework, particularly the squared Hellinger-information bound, outperform previous results.

2 Preliminaries

Notation Throughout this paper, we adopt a convention where capitalized letters denote random variables, and their corresponding lowercase letters represent specific realizations. Let P X denote the distribution of a random variable X , and P X | Y represent the conditional distribution of X given Y . When conditioning on a specific realization, we use P X | Y = y or simply P X | y . Additionally, we use E X and E P interchangeably to denote the expectation over X ∼ P , whenever it is clear from context. Moreover, E X | Y = y (or E X | y ) denotes the expectation over X ∼ P X | Y = y .

Generalization Error Let Z be the domain of the instances, and let W be the domain of the hypotheses. We denote the unknown distribution of the instances by µ , and let S = { Z i } n i =1 i.i.d ∼ µ be the training sample. A learning algorithm A , characterized by a conditional distribution P W | S , takes the training sample S as the input and outputs a hypothesis W ∈ W , i.e. A : Z n →W . To evaluate the quality of the output hypothesis W , we use a loss function ℓ : W × Z → R + 0 . For any hypothesis w , the population risk is defined as L µ ( w ) ≜ E Z ′ [ ℓ ( w,Z ′ )] , where Z ′ ∼ µ is an independent testing instance. In practice, since µ is unknown, we use the empirical risk, which is defined as L S ( w ) ≜ 1 n ∑ n i =1 ℓ ( w,Z i ) , as a proxy for the population risk of w . The expected generalization error is then defined as E µ ( A ) ≜ E W [ L µ ( W )] -E W,S [ L S ( W )] .

Supersample Setting Wefollow the traditional supersample setting introduced by [16]. Let an n × 2 matrix ˜ Z ∈ Z n × 2 be the supersample, where each entry is drawn i.i.d. from the data distribution µ . We index the columns of ˜ Z by { 0 , 1 } , and denote the i th row of ˜ Z as ˜ Z i with entries ( ˜ Z i, 0 , ˜ Z i, 1 ) . We use U = { U i } n i =1 ∼ Unif( { 0 , 1 } n ) , independent of ˜ Z , to determine the training sample membership from ˜ Z . Specifically, when U i = 0 , ˜ Z i, 0 in ˜ Z is included in the training set S , and ˜ Z i, 1 is used for testing; U i = 1 dictates the opposite case. Let U i = 1 -U i , the training sample S is then equivalent to ˜ Z U = { ˜ Z i,U i } n i =1 , and the testing sample is ˜ Z U = { ˜ Z i,U i } n i =1 . Additionally, we let L i, 0 ≜ ℓ ( W, ˜ Z i, 0 ) and define L i, 1 similarly. Let ∆ L i = L i, 1 -L i, 0 be the loss difference in the i th row of ˜ Z . To avoid complicated subscripts, we also use superscripts + and -to replace the subscripts 0 and 1 , respectively, namely, ˜ Z + i = ˜ Z i, 0 , ˜ Z -i = ˜ Z i, 1 , L + i = L i, 0 and L -i = L i, 1 . Furthermore, we define G i ≜ ( -1) U i ∆ L i , i.e. the testing loss minus the training loss at position i . Clearly, 1 n ∑ n i =1 G i is an unbiased estimator of E µ ( A ) . In other words, E µ ( A ) = 1 n ∑ n i =1 E ∆ L i ,U i [ G i ] .

f -Divergence and f -Information The family of f -divergence is defined as follows. Definition 2.1 ( f -divergence [33]) . Let P and Q be two distributions on Θ , and let ϕ : R + → R be a convex function with ϕ (1) = 0 . If P ≪ Q , then the f -divergence between P and Q is defined as

where dP dQ is a Radon-Nikodym derivative.

A variational representation for f -divergence, as provided below, has been independently investigated in previous works [35, 25, 36, 37, 33]. Recently, this variational representation has also been applied to PAC-Bayesian generalization theory [38] and domain adaptation theory [39, 40].

Lemma 2.1 ([37, Corollary 3.5]) . Let ϕ ∗ be the convex conjugate 1 of ϕ , and G = { g : Θ → dom( ϕ ∗ ) } . The following, known as variational formula of f -divergence, holds:

Following [33, Section 7.8], let I ϕ ( X ; Y ) ≜ D ϕ ( P X,Y || P X P Y ) be the f -information, which extends the standard mutual information (MI) defined in terms of KL divergence.

We now set θ = (∆ L i , U i ) and g (∆ L i , U i ) = tG i in Lemma 2.1, where t ∈ R , P = P ∆ L i ,U i is the joint distribution of (∆ L i , U i ) and Q = P ∆ L i P U i is the product of the marginal distributions. Consequently, Lemma 2.1 directly implies that

where we let λ = 0 in Eq. (2). Notice that if ϕ ( x ) = x log x + c ( x -1) for any constant c , then I ϕ ( X ; Y ) becomes MI. In this case, the optimal λ ∗ = -log E Q [ e g ( θ ) ] in Lemma 2.1, and Lemma 2.1 recovers the Donsker and Varadhan's variational formula (cf. Lemma A.1). Here, the second term in Eq. (1) represents the cumulant generating function of (∆ L i , U ′ i ) . By analyzing the tail behavior of (∆ L i , U ′ i ) and applying Eq. (1), one can derive the final MI-based generalization bound. For example, if the loss is bounded, Eq. (1) recovers [24, Theorem 3.2] by using Hoeffding's Lemma.

On one hand, while Eq. (1) is tighter than Eq. (2), the parameter λ may not have an analytic form for divergences beyond KL and χ 2 . In this sense, Eq. (2) has its own merits. On the other hand, previous MI-based generalization bounds use concentration results to further upper bound the second term in Eq. (1). While this strategy has been successful for the MI case, for instance by using Hoeffding's Lemma, it may be challenging to obtain similar concentration results for other f -information. In this work, we introduce a novel approach to prove generalization bounds for f -information.

3 Conditional f -Information Bounds: Bounded Loss Difference Case

We begin by presenting a general lemma that will serve as a main recipe for obtaining generalization bounds in this section. This lemma introduces a new variational formula for f -divergence, which may be of independent interest beyond the context of generalization.

Lemma 3.1. Given random variables X , Y , and a measurable function f for ( X,Y ) . For every t ∈ [ b 1 , b 2 ] ⊆ R , assume that ϕ ∗ is invertible within the range of t · f , which we denote by Γ t . Let ϕ ∗-1 be the inverse of ϕ ∗ on ∪ t ∈ [ b 1 ,b 2 ] Γ t , and let Y ′ be an independent copy of Y . If E X,Y ′ [ f ( X,Y ′ )] = 0 , then

Although Lemma 3.1 is a simple and straightforward result derived from Lemma 2.1 by setting g = ϕ ∗-1 ◦ ( tf ) , it is quite powerful for deriving f -information-based generalization bounds. Specifically, in the supersample setting, due to the symmetric properties of two columns of ˜ Z , we set X = ∆ L i , Y = U i and f (∆ L i , U ′ i ) = ( -1) U ′ i ∆ L i (where U ′ i is an independent copy of U i ), the condition in Lemma 3.1, namely E X,Y ′ [ f ( X,Y ′ )] = 0 , is clearly met. Then we have

After carefully choosing b 1 , b 2 such that ϕ ∗-1 is well defined, the main focus is to find a lower bound for the function ϕ ∗-1 over a certain interval. We will start with the standard MI case.

3.1 Mutual Information (KL-based) Generalization Bound

Consider ϕ ( x ) = x log x + x -1 . Its convex conjugate function is ϕ ∗ ( y ) = e y -1 with the inverse ϕ ∗-1 ( z ) = log(1 + z ) . Building upon Lemma 3.1, we obtain the following bound:

Theorem 3.1. Assume the loss difference ℓ ( w,z 1 ) -ℓ ( w,z 2 ) is bounded in [ -1 , 1] for any w ∈ W and z 1 , z 2 ∈ Z , we have

We remark that a bounded loss difference is a less restrictive assumption than a strictly bounded loss. For example, if ℓ is L -Lipschitz in the second argument, then ℓ ( w,z 1 ) -ℓ ( w,z 2 ) ≤ L || z 1 -z 2 || .

Thus, as long as || z 1 -z 2 || is bounded, the loss difference is bounded even when ℓ itself is unbounded. Additionally, we refer to Theorem 3.1 as the ' oracle ' CMI bound because the upper bound itself includes the expected generalization error at position i , E [ G i ] , which is precisely the quantity we aim to bound.

From Theorem 3.1, first, if we simply upper bound the loss difference ∆ L i (and G i ) by one, Theorem 3.1 recovers [24, Theorem 3.2] up to a constant. More importantly, Theorem 3.1 indicates that in the case of bounded loss difference or bounded loss, solely using I (∆ L i ; U i ) as the generalization measure for algorithm A is insufficient to accurately characterize its generalization error. Specifically, when 2 ( E [ ∆ L 2 i ] + | E [ G i ] | ) vanishes as n increases, relying on the MI-based measure alone, i.e., √ I (∆ L i ; U i ) , will always result in a slow convergence rate for |E µ ( A ) | . While similar observations have been made in recent studies [41-43], where it is found that the sub-Gaussian variance proxy in previous MI bounds may also vanish in some examples, our Theorem 3.1 provides a more straightforward understanding of the potential looseness of O ( √ I (∆ L i ; U i )) . Additionally, further discussion comparing our oracle CMI bound with the MI-based results in [29, 30] is provided in Appendix B.4.

To highlight the differences in our proof techniques compared to previous information-theoretic generalization bounds, we provide a proof sketch below.

Proof Sketch of Theorem 3.1. Lemma 3.1 gives us I (∆ L i ; U i ) ≥ sup t E [ log ( 1+ t ( -1) U i ∆ L i )] . Let f ( x ) = log(1+ x ) -x + ax 2 and set a = | E [ G i ] | 2 E [ G 2 i ] + 1 2 . By demonstrating that f ( x ) ≥ 0 holds when a ≥ 1 2 and | x | ≤ 1 -1 2 a , we have sup t> -1 E [log (1 + tG i )] ≥ sup t ∈ [ 1 2 a -1 , 1 -1 2 a ] E [ tG i -at 2 G 2 i ] . The supremum is attained when t ∗ = E [ G i ] 2 a E [ G 2 i ] = E [ G i ] | E [ G i ] | + E [ G 2 i ] , which is achievable. Therefore, I (∆ L i ; U i ) ≥ sup t ∈ ( -1 , + ∞ ) E ∆ L i ,U i [ log ( 1 + t ( -1) U i ∆ L i )] ≥ E 2 [ G i ] 4 a E [ G 2 i ] , which simplifies to

The remaining steps are straightforward. See Appendix B.1 for the complete proof.

It is also worth noting that by letting g = ϕ ∗-1 and E Q [ g ] = 0 in the case of KL divergence, we obtain λ ∗ = 0 in Lemma 2.1. As a result, Eq. (2) becomes equivalent to Eq. (1). In this sense, starting from Eq. (2) does not compromise the tightness of Eq. (1). Moreover, the proof provides deeper insights into the tightness of Theorem 3.1, which we will now discuss.

Small Empirical Risk Case or Realizable Setting Previously, fast-rate information-theoretic generalization bounds in the realizable setting-where the square-root function is removed-have been derived by demonstrating that the CGF can be negative [16, 44, 23, 24], or by invoking the channel capacity result of the binary channel [44, 24].

Based on Theorem 3.1, the square-root function can be directly removed in the realizable setting. Specifically, if the learning algorithm A is an interpolating algorithm, i.e., the training loss is always minimized to zero, then G i ∈ [0 , 1] always holds. In this case, | E [ G i ] | = E [ G i ] and E [ G 2 i ] ≤ E [ G i ] , allowing us to obtain E [ G i ] ≤ 4 I (∆ L i ; U i ) from Eq. (3). Consequently, Theorem 3.1 simplifies to E µ ( A ) ≤ 4 n ∑ n i =1 I (∆ L i ; U i ) . Note that for this bound to hold, the condition of zero training loss can be relaxed to the training loss being always no larger than the testing loss.

Furthermore, to achieve a better constant in the bound, observe that the proof of our Theorem 3.1 utilizes the inequality log(1 + x ) ≥ x -ax 2 for | x | ≤ 1 -1 2 a . If x ∈ [0 , 1] always holds, one can use the inequality log(1 + x ) ≥ x log 2 instead, where equality holds when the loss is zero-one loss (See Figure 1(Right) for a visualized illustration).

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As a result, we obtain E µ ( A ) ≤ ∑ n i =1 I (∆ L i ; U i ) n log 2 , which is tighter than the previous bound and will never be vacuous (i.e. the bound is always smaller than 1 ) since I (∆ L i ; U i ) ≤ H ( U i ) = log 2 . In fact, this bound can exactly characterize the generalization error (i.e., equality holds) if the loss is the zero-one loss, as demonstrated in [24, Theorem 3.3].

New Fast-Rate Bounds Theorem 3.1 not only recovers the previous fast-rate ld-CMI bound in the realizable setting but also introduces new fast-rate bounds.

By solving Eq. (3) for | E [ G i ] | , we obtain the following bound.

Corollary 3.1. Under the conditions of Theorem 3.1, we have

This fast-rate bound has not been proved in [24], yet it resembles [24, Eq. (2)] in structure. Specifically, the single-loss MI term, 2 I ( L + i ; U i ) in [24, Eq. (2)], is now substituted with I (∆ L i ; U i ) , and the empirical risk term, E [ L S ( W )] , is replaced by E [ ∆ L 2 i ] . Here, ∆ L i = L -i -L + i , and notice that both L + i and L -i follow the same marginal distribution due to the symmetric construction of the supersample, so we can obtain that E [ ∆ L 2 i ] = E [ ( L -i -L + i ) 2 ] ≤ 4 E [ ( L + i -E [ L + i ] ) 2 ] = 4Var( L + i ) . Consequently, Corollary 3.1 further implies the following bound:

Notably, Eq. (4) hints that if the variance of the single loss in the supersample is sufficiently small, the bound will decay at the same rate as in the realizable setting, namely O ( I (∆ L i ; U i )) , as the second term vanishes.

Instead of solving Eq. (3), if we directly upper-bound | E [ G i ] | in Theorem 3.1, we can also obtain the following bound from Theorem 3.1.

Corollary 3.2. Under the conditions of Theorem 3.1, we have

Note that Corollary 3.2 is tighter than Corollary 3.1 by Jensen's inequality and Pinsker's inequality. While we focus on the MI-based or KL-based generalization bound in this section, the inclusion of total variation (TV) in the second term of Corollary 3.2 expands the scope of the bound. Notably, in a recent study by [34], a component of the form √ D TV D KL is also present in their generalization bounds. Specifically, they derive PAC-Bayesian generalization bounds using the Zhang-CutkoskyPaschalidis (ZCP) divergence, initially explored in [45]. Interestingly, the ZCP divergence can be further upper bounded by O ( √ D TV D KL +D TV ) . Although it remains uncertain whether the first term in Corollary 3.2 is tighter than D TV or not, we demonstrate that the component √ D TV D KL can emerge directly in the derivation of KL-based bounds, without using the ZCP divergence. Additionally, note that the term E [ ∆ L 2 i ] in Corollary 3.2 can likewise be replaced by 4Var( L + i ) term.

3.2 Other f -Information-based Generalization Bound

Based on Lemma 3.1, it is straightforward to apply similar techniques to many well-known f -divergences or f -information measures to obtain generalization bounds. In this section, we discuss several other f -information-based bounds. Before we study the specific f -divergence, we remark that, in Section 3.1, our focus was on the unconditional mutual information I (∆ L i ; U i ) . As illustrated in [24], using the data-processing inequality (DPI) and the chain rule, we have I (∆ L i ; U i ) ≤ I ( L + i , L -i ; U i ) ≤ I ( W ; U i | ˜ Z i ) . This makes it easy to derive other CMI variantbased bounds. While other f -divergences also satisfy DPI, they may not adhere to the chain rule. Accordingly, for these cases, we use the disintegrated conditional f -information, defined as I z ϕ ( X ; Y ) ≜ D ϕ ( P XY | z || P X | z P Y | z ) (noting that I ( X ; Y | Z ) = E Z [ I Z ϕ ( X ; Y )) ] ), instead of the unconditional quantity. In this context, at least I ϕ (∆ L i ; U i | ˜ Z i ) ≤ I ϕ ( W ; U i | ˜ Z i ) still holds, ensuring that the hypothesis-based conditional f -information bound can be directly derived.

X 2 -based Generalization Bound Consider ϕ ( x ) = ( x -1) 2 . Its convex conjugate is ϕ ∗ ( y ) = y 2 4 + y . For y ≥ -2 , the inverse of conjugate function ϕ ∗-1 ( z ) = 2( √ z +1 -1) exists. Since 2( √ z +1 -1) ≥ log(1+ z ) , any lower bound that holds for log(1+ z ) will also hold for 2( √ z +1 -1) . See Figure 1(Left) for a visualization. In other words, all the bounds in Section 3.1 will also hold when using χ 2 -divergence. This can also be immediately noticed by the inequality D KL ( P || Q ) ≤ χ 2 ( P || Q ) [33]. We defer the formal disintegrated χ 2 -information bound in Appendix B.7.

Squared Hellinger (SH) Distance Generalization Bound Consider ϕ ( x ) = ( √ x -1) 2 , and its convex conjugate is ϕ ∗ ( y ) = y 1 -y . For y ∈ ( -∞ , 1) , the inverse function ϕ ∗-1 ( z ) = z 1+ z exists for z ∈ ( -1 , + ∞ ) . We then have the following oracle conditional squared Hellinger (SH)-information bound.

Theorem 3.2. Under the same conditions in Theorem 3.1, we have

where I ˜ z i H 2 (∆ L i ; U i ) = D H 2 ( P ∆ L i ,U i | ˜ z i || P ∆ L i | ˜ z i P U i ) is the (disintegrated) SH-information.

Given that D H 2 ( P || Q ) ≤ D TV ( P, Q ) , the TV-based bound can be derived as a corollary from above.

Jensen-Shannon (JS) Divergence Generalization Bound Consider ϕ ( x ) = x log 2 x 1+ x +log 2 1+ x , with the convex conjugate ϕ ∗ ( y ) = -log(2 -e y ) . For y < log 2 , the inverse function ϕ ∗-1 ( z ) = log(2 -e -z ) exists for z > -log 2 . This leads to the following oracle JS-information bound.

Theorem 3.3. Under the same conditions in Theorem 3.1, we have

where I ˜ z i JS (∆ L i ; U i ) = D JSD ( P ∆ L i ,U i | ˜ z i || P ∆ L i | ˜ z i P U i ) is the (disintegrated) JS-information.

Jeffrey's divergence [46], which is defined as D Jeffrey ( P || Q ) ≜ D KL ( P || Q ) + D KL ( Q || P ) , is an f -divergence with the convex function ϕ ( x ) = ( x -1) log x . Since D JSD ( P || Q ) ≤ 1 4 D Jeffrey ( P || Q ) [47], a Jeffrey's divergence-based bound can be derived as a corollary from Theorem 3.3 (and clearly, also from Theorem 3.1 due to the non-negativity of KL divergence).

Notably, Theorems 3.2-3.3 share similar expressions with Theorem 3.1, and deriving analogous corollaries to Corollaries 3.1-3.2 is feasible (see Appendix B.7).

In the proofs of Theorems 3.2 and Theorem 3.3, we continue to use x -ax 2 to lower bound the inverse of conjugate functions x 1+ x and log(2 -e -x ) for SH-information and JS-information, respectively. However, due to the complexity of handling log(2 -e -x ) in the context of JS-information, we opted for a much rougher selection of the parameter a than in the proof of Theorem 3.1 for simplicity. Consequently, the constants in Theorem 3.3 are not optimal and can be refined with a more finegrained analysis.

4 Extension to Unbounded Loss Difference

Previous bounds are typically applicable when dealing with bounded loss differences (albeit not necessarily within the range of [ -1 , 1] ). This limitation arises from the fact that the function ϕ ∗-1 may fail to exist in Lemma 3.1 when f is unbounded, regardless of any non-trivial adjustments made to the range of t . To overcome this limitation, we now extend our analysis to the case of unbounded loss differences by refining Lemma 3.1. One crucial modification to Lemma 3.1 involves defining the measurable function g = tϕ ∗-1 · ✶ | f |≤ C for some C ≥ 0 , where ✶ is the indicator function. We present the following lemma tailored for unbounded functions.

Lemma 4.1. Let X be an arbitrary random variable, ε be a Rademacher variable, and t ∈ ( -b, b ) . Assume that there exists a constant C ≥ 0 such that ϕ ∗ is invertible in ( -bC, bC ) . If ϕ ∗ (0) = 0 , then

Note that ϕ ∗ (0) = 0 is satisfied by all divergences discussed in Section 3. Armed with Lemma 4.1, our proof strategy hinges on separately bounding the truncated terms E [ G i · ✶ | X |≤ C ] and E [ G i · ✶ | X | >C ] . To achieve this, we utilize Lemma 4.1 along with similar lower-bounding techniques as detailed in Section 3 to bound the first term. For the second term, we invoke Hölder's inequality, drawing inspiration from prior works such as [25, 27].

Before presenting the refined bound for unbounded loss difference, we introduce some additional nota- tions. Define the L p norm of a random variable X ∼ P as || X || p ≜ { ( E P | X | p ) 1 p for p ∈ [1 , ∞ ) , ess sup | X | for p = ∞ , where ess sup | X | ≜ inf { M : P ( | X | > M ) = 0 } is the essential supremum. Furthermore, let the f -divergence generated by the convex function ϕ α ( x ) = | x -1 | α be D ϕ α ( P || Q ) ≜ E Q [( dP dQ -1 ) α ] . This divergence, which takes χ 2 -divergence and total variation as special cases (with α = 2 and α = 1 , respectively), and it has been utilized in previous information-theoretic generalization bounds [25, 29, 30]. We denote the corresponding f -information of D ϕ α ( P || Q ) as I ϕ α .

Now, we are ready to present a MI-based bound, with straightforward extensions to other f -information measures.

1 1

Since U i is a Bernoulli random variable, according to [25, Lemma 1], we know that I ϕ α (∆ L i ; U i ) < 1 + 2 α -1 for α ∈ [1 , 2] and remains bounded for other values of α as well. Thus, Theorem 4.1 generally preserves the boundedness property of the original CMI bound unless ∆ L i has an infinite L qβ -norm. Additionally, due to truncation, ζ 1 ≤ √ 2 C for any given C , while ζ 2 heavily depends on the tail behavior of ∆ L i ∼ P ∆ L i (or G i ∼ P U ′ i P ∆ L i equivalently). We now discuss several common cases.

Bounded Loss Notably, Theorem 4.1 also covers the bounded loss cases. For instance, if ℓ is bounded in [0 , 1] a.s., setting C = 1 in Theorem 4.1 leads to ζ 2 = 0 , directly recovering Theorem 3.1 for bounded loss. In fact, the choice of C = 1 might not necessarily be the optimal for ℓ ∈ [0 , 1] . Particularly, for certain C ∈ (0 , 1) , we let α = 1 and β = ∞ , then ζ 2 = || ∆ L i || ∞ and ( I ϕ α (∆ L i ; U i )) 1 /α = I TV (∆ L i ; U i ) . Since D TV ≲ √ D KL by Pinsker's inequality, this alternative choice holds potential for even tighter bounds than Theorem 3.1.

'Almost Bounded' Loss When the loss function exhibits sub-Gaussian or sub-Gamma tail behaviors, the probability P ( | G i | > C ) decays rapidly. By carefully selecting the threshold C , the dominance of the first term in the bound can be expected. Assume the loss difference ∆ L i is sub-Gaussian with certain variance proxy, say, σ . Setting C = σ and q = 2 , we have P ( | ∆ L i | > C ) ≤ 2 e -1 and || ∆ L i || qβ ≲ √ qβσ [48]. Consequently, each term in the summation of Theorem 4.1 simplifies to O ( σ √ I (∆ L i ; U i ) + σ (2 e ) 1 -α 2 α √ α α -1 α √ I ϕ α (∆ L i ; U i ) ) . However, the relationship between I (∆ L i ; U i ) and I ϕ α (∆ L i ; U i ) is not clear beyond the cases of α = 1 and α = 2 (corresponding to total variation and χ 2 -information, respectively). Studying the overall behavior of the bound represents an intriguing direction for future research. Furthermore, consider the alternative case with C = 0 and q = 1 , where the first term in Theorem 4.1 becomes zero. The second term, using || ∆ L i || β ≲ √ βσ for sub-gaussian random variables, becomes O ( σ √ α α -1 α √ I ϕ α (∆ L i ; U i )) . Compared to existing MI bounds, e.g., O ( σ √ I (∆ L i ; U i )) , we know from D TV ( P, Q ) ≲ √ D KL ( P || Q ) and D KL ( P || Q ) ≤ χ 2 ( P || Q ) that I ϕ 1 (∆ L i ; U i ) ≲ √ I (∆ L i ; U i ) ≲ √ I ϕ 2 (∆ L i ; U i ) , suggesting that some α ∈ (1 , 2) may yield a tighter bound than MI.

Heavy-tailed Loss Heavy-tailed losses, although lacking universally agreed definitions, typically refer to the cases where the moment-generating function (MGF) does not exists (away from 0 ) [49].

Remarkably, our Theorem 4.1 remains meaningful for such losses since it does not rely on the existence of the MGF of the loss function. Specifically, without any additional knowledge about the tail behavior of ∆ L i , we have the following result from Theorem 4.1.

Corollary 4.1. Under the conditions in Theorem 4.1, let γ = q -1 qβ , we have

It's worth noting that as γ → 0 (i.e. q → 1 ), we can see that ( γ 1 γ +1 + γ -γ γ +1 ) → 1 . Consequently, in Corollary 4.1, the bound simplifies to 1 n ∑ n i =1 ( || ∆ L i || β α √ I ϕ α (∆ L i ; U i ) ) . This is an extension of [25, Theorem 3], incorporating individual techniques [6] and loss-difference methods [24] in the supersample setting. Notably, our Corollary 4.1 improves upon [25, Theorem 3], as γ → 0 may not necessarily be the optimal choice. Furthermore, it is also intriguing to investigate whether our Corollary 4.1 can recover [29, Corollary 5].

5 Numerical Results

In this section, we conduct an empirical comparison between our novel conditional f -information generalization bounds and several existing information-theoretic generalization bounds. Our experimental setup closely aligns with the settings in [24]. In particular, we undertake two distinct prediction tasks as follows: 1) Linear Classifier on Synthetic Gaussian Dataset , where we train a simple linear classifier using a synthetic Gaussian dataset; 2) CNN and ResNet-50 on Real-World Datasets . The second task follows the same deep learning training protocol as in [22], involves training a 4 -layer CNN on a binary MNIST dataset (' 4 vs 9 ') [50] and fine-tuning a ResNet-50 model [51], pretrained on ImageNet [52], on CIFAR10 [53]. In both tasks, we assess prediction error as our performance metric. That is, we utilize the zero-one loss function to compute generalization error. Note that during training, we use the cross-entropy loss as a surrogate to enable optimization with gradient-based methods.

0

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Binary kl

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The comparison of generalization bounds mainly focuses on those presented in Section 3. Specifically, we consider the oracle bounds, given in Theorem 3.1, Theorem 3.2, and Theorem 3.3, which we denote as CMI(Oracle) , CSHI(Oracle) , and CJSI(Oracle) , respectively. Given that the

squared Hellinger-information is a tighter measure than MI, and the constants in Theorem 3.2 exhibit less pessimism compared to those in the JS-information bound, we introduce two additional squared Hellinger-information bounds for comparison. The first one, akin to Corollary 3.2, is E µ ( A ) ≤ 1 n ∑ n i =1 E √ ( 4 E [∆ L 2 i ] + 2 E [ D TV ( P ∆ L i | U i , ˜ Z i , P ∆ L i | ˜ Z i )]) I ˜ Z i H 2 (∆ L i ; U i ) (cf. Corollary B.1 in Appendix B.7), denoted as CSHI(Var) . The second one adopts a more pessimistic choice by replacing E [∆ L 2 i ] with the constant 1 , labeled as CSHI(Worst) . Additionally, we include three previous information-theoretic generalization bounds as baselines: the original ld-CMI bound in [24, Theorem3.1] ( ld-CMI ), the sharpness-based single-loss MI bound in [24, Theorem 4.5] ( Sharpness ), and the binary KL-based CMI bound in [23, Theorem 5] ( Binary kl ). Further details on experimental settings are available in the Appendix D.

The final results are shown in Figure 2. First, we observe that for the linear classifier on the two-class data (Figure 2a), most bounds are close to the exact generalization error, with Sharpness being the tightest. Note that the sharpness-based MI bound in [24, Theorem 4.5] is applicable solely for zero-one loss. As the prediction task becomes more complex (Figures 2b-2f), our squared Hellingerinformation-based bounds are the tightest. Specifically, the squared Hellinger-information-based bound is always tighter than the CMI bound in the oracle type. Notably, CSHI(Var) also outperforms all other f -information-based bounds in these cases, and even the pessimistic CSHI(Worst) bound outperforms them, except for Sharpness , in Figures 2c-2d. Additionally, by comparing CMI(Oracle) and ld-CMI , we see significant potential for improving the original ld-CMI bound. Finally, although the JS-information bound contains relatively pessimistic constants, it still outperforms some MI-based bounds in many cases (e.g., Figures 2c-2e). These observations suggest that considering alternative f -information rather than MI is indeed necessary for studying generalization.

6 Other Related Works and Limitations

In addition to the studies mentioned previously, there have been recent developments in informationtheoretic generalization bounds in the supersample setting. For example, [54, 44] propose a 'leaveone-out' (LOO) construction of supersamples, reducing the size of the supersample from 2 n to n + 1 . Additionally, [55] constructs a 'neighboring-hypothesis' matrix, enabling the derivation of a hypothesis-conditioned CMI bound. The key component in this bound is the product of the new CMI term and an algorithmic stability parameter. Furthermore, significant progress has been made in understanding MI or CMI as generalization measures in stochastic convex optimization (SCO) problems. Specifically, [56-58] all point out that MI or CMI bounds can be non-vanishing in some SCO examples, while the exact generalization error is vanishing. Notably, [58] discovers a 'CMI-accuracy tradeoff' phenomenon in certain scenarios: For the generalization error to be upper bounded by a vanishing parameter, the original hypothesis-based CMI must be lower bounded by the reciprocal of this parameter (or even the reciprocal of its square), indicating that the CMI bound cannot characterize the learnability of such problems. Although this limitation is currently only observed for hypothesis-based CMI, it is anticipated to hold in other variants of CMI, such as ld-CMI.

Our work also contributes to understanding the potential looseness of previous CMI generalization bounds. However, the current state of our bounds cannot explain or resolve the limitation presented in [58] in a non-trivial way. Nonetheless, our framework provides some intuition that, as long as a gap exists between ϕ ∗-1 and x -ax 2 (see Figure 1(Right)), there will always be room for constructing counterexamples that challenge information-theoretic measures, although such examples may be rare. Another limitation of this work is the lack of a high-probability generalization guarantee. There are two potential directions to address this: either by applying similar techniques to the framework in [59], which is the PAC-Bayesian counterpart of the CMI bound, or by combining it with recent information-bottleneck generalization bounds [60, 61].

Acknowledgments and Disclosure of Funding

This work is supported partly by an NSERC Discovery grant. The authors would like to thank the anonymous AC and reviewers for their careful reading and valuable suggestions.

References

Kyoungseok Jang, Kwang-Sung Jun, Ilja Kuzborskij, and Francesco Orabona. Tighter pac-bayes bounds through coin-betting. In The Thirty Sixth Annual Conference on Learning Theory , pages 2240-2264. PMLR, 2023.

Daniel Russo and James Zou. Controlling bias in adaptive data analysis using information theory. In Artificial Intelligence and Statistics . PMLR, 2016.

Daniel Russo and James Zou. How much does your data exploration overfit? controlling bias via information usage. IEEE Transactions on Information Theory , 66(1):302-323, 2019.

Aolin Xu and Maxim Raginsky. Information-theoretic analysis of generalization capability of learning algorithms. Advances in Neural Information Processing Systems , 2017.

Ankit Pensia, Varun Jog, and Po-Ling Loh. Generalization error bounds for noisy, iterative algorithms. In 2018 IEEE International Symposium on Information Theory (ISIT) . IEEE, 2018.

Yuheng Bu, Shaofeng Zou, and Venugopal V Veeravalli. Tightening mutual information based bounds on generalization error. In 2019 IEEE International Symposium on Information Theory (ISIT) , pages 587-591. IEEE, 2019.

Jeffrey Negrea, Mahdi Haghifam, Gintare Karolina Dziugaite, Ashish Khisti, and Daniel M Roy. Information-theoretic generalization bounds for sgld via data-dependent estimates. Advances in Neural Information Processing Systems , 2019.

Hao Wang, Yizhe Huang, Rui Gao, and Flavio Calmon. Analyzing the generalization capability of sgld using properties of gaussian channels. Advances in Neural Information Processing Systems , 34:24222-24234, 2021.

Bohan Wang, Huishuai Zhang, Jieyu Zhang, Qi Meng, Wei Chen, and Tie-Yan Liu. Optimizing information-theoretical generalization bound via anisotropic noise of sgld. Advances in Neural Information Processing Systems , 2021.

Gergely Neu, Gintare Karolina Dziugaite, Mahdi Haghifam, and Daniel M Roy. Informationtheoretic generalization bounds for stochastic gradient descent. In Conference on Learning Theory . PMLR, 2021.

Ziqiao Wang and Yongyi Mao. On the generalization of models trained with SGD: Informationtheoretic bounds and implications. In International Conference on Learning Representations , 2022.

Ziqiao Wang and Yongyi Mao. Two facets of sde under an information-theoretic lens: Generalization of sgd via training trajectories and via terminal states. arXiv preprint arXiv:2211.10691 , 2022.

Futoshi Futami and Masahiro Fujisawa. Time-independent information-theoretic generalization bounds for sgld. Advances in Neural Information Processing Systems , 36, 2024.

Raef Bassily, Shay Moran, Ido Nachum, Jonathan Shafer, and Amir Yehudayoff. Learners that use little information. In Algorithmic Learning Theory . PMLR, 2018.

Amir Asadi, Emmanuel Abbe, and Sergio Verdú. Chaining mutual information and tightening generalization bounds. Advances in Neural Information Processing Systems , 31, 2018.

Thomas Steinke and Lydia Zakynthinou. Reasoning about generalization via conditional mutual information. In Conference on Learning Theory . PMLR, 2020.

Mahdi Haghifam, Jeffrey Negrea, Ashish Khisti, Daniel M Roy, and Gintare Karolina Dziugaite. Sharpened generalization bounds based on conditional mutual information and an application to noisy, iterative algorithms. Advances in Neural Information Processing Systems , 2020.

Hassan Hafez-Kolahi, Zeinab Golgooni, Shohreh Kasaei, and Mahdieh Soleymani. Conditioning and processing: Techniques to improve information-theoretic generalization bounds. Advances in Neural Information Processing Systems , 33:16457-16467, 2020.

Borja Rodríguez-Gálvez, Germán Bassi, Ragnar Thobaben, and Mikael Skoglund. On random subset generalization error bounds and the stochastic gradient langevin dynamics algorithm. In 2020 IEEE Information Theory Workshop (ITW) , pages 1-5. IEEE, 2021.

Ruida Zhou, Chao Tian, and Tie Liu. Individually conditional individual mutual information bound on generalization error. IEEE Transactions on Information Theory , 68(5):3304-3316, 2022.

Ruida Zhou, Chao Tian, and Tie Liu. Stochastic chaining and strengthened information-theoretic generalization bounds. In 2022 IEEE International Symposium on Information Theory (ISIT) . IEEE, 2022.

Hrayr Harutyunyan, Maxim Raginsky, Greg Ver Steeg, and Aram Galstyan. Informationtheoretic generalization bounds for black-box learning algorithms. In Advances in Neural Information Processing Systems , 2021.

Fredrik Hellström and Giuseppe Durisi. A new family of generalization bounds using samplewise evaluated CMI. In Advances in Neural Information Processing Systems , 2022.

Ziqiao Wang and Yongyi Mao. Tighter information-theoretic generalization bounds from supersamples. In International Conference on Machine Learning . PMLR, 2023.

Jiantao Jiao, Yanjun Han, and Tsachy Weissman. Dependence measures bounding the exploration bias for general measurements. In 2017 IEEE International Symposium on Information Theory (ISIT) , pages 1475-1479. IEEE, 2017.

Borja Rodríguez Gálvez, Germán Bassi, Ragnar Thobaben, and Mikael Skoglund. Tighter expected generalization error bounds via wasserstein distance. Advances in Neural Information Processing Systems , 34, 2021.

Amedeo Roberto Esposito, Michael Gastpar, and Ibrahim Issa. Robust generalization via α -mutual information. In International Zurich Seminar on Information and Communication (IZS 2020) , page 96, 2020.

Amedeo Roberto Esposito and Michael Gastpar. From generalisation error to transportation-cost inequalities and back. In 2022 IEEE International Symposium on Information Theory (ISIT) , pages 294-299. IEEE, 2022.

Gábor Lugosi and Gergely Neu. Generalization bounds via convex analysis. In Conference on Learning Theory , pages 3524-3546. PMLR, 2022.

Gábor Lugosi and Gergely Neu. Online-to-pac conversions: Generalization bounds via regret analysis. arXiv preprint arXiv:2305.19674 , 2023.

Kwang-Sung Jun and Francesco Orabona. Parameter-free online convex optimization with sub-exponential noise. In Conference on learning theory , pages 1802-1823. PMLR, 2019.

Francesco Orabona and Kwang-Sung Jun. Tight concentrations and confidence sequences from the regret of universal portfolio. IEEE Transactions on Information Theory , 70(1):436-455, 2024.

Yury Polyanskiy and Yihong Wu. Information Theory: From Coding to Learning . Cambridge university press, 2023.

Ilja Kuzborskij, Kwang-Sung Jun, Yulian Wu, Kyoungseok Jang, and Francesco Orabona. Better-than-kl pac-bayes bounds. arXiv preprint arXiv:2402.09201 , 2024.

Aharon Ben-Tal and Marc Teboulle. An old-new concept of convex risk measures: The optimized certainty equivalent. Mathematical Finance , 17(3):449-476, 2007.

Jeremiah Birrell, Markos A Katsoulakis, and Yannis Pantazis. Optimizing variational representations of divergences and accelerating their statistical estimation. IEEE Transactions on Information Theory , 68(7):4553-4572, 2022.

Rohit Agrawal and Thibaut Horel. Optimal bounds between f-divergences and integral probability metrics. In International Conference on Machine Learning , pages 115-124. PMLR, 2020.

Paul Viallard, Maxime Haddouche, Umut ¸ Sim¸ sekli, and Benjamin Guedj. Tighter generalisation bounds via interpolation. arXiv preprint arXiv:2402.05101 , 2024.

Ziqiao Wang and Yongyi Mao. On f-divergence principled domain adaptation: An improved framework. arXiv preprint arXiv:2402.01887 , 2024.

Wenliang Liu, Guanding Yu, Lele Wang, and Renjie Liao. An information-theoretic framework for out-of-distribution generalization. arXiv preprint arXiv:2403.19895 , 2024.

Xuetong Wu, Jonathan H Manton, Uwe Aickelin, and Jingge Zhu. Fast rate generalization error bounds: Variations on a theme. In 2022 IEEE Information Theory Workshop (ITW) , pages 43-48. IEEE, 2022.

Xuetong Wu, Jonathan H Manton, Uwe Aickelin, and Jingge Zhu. On the tightness of information-theoretic bounds on generalization error of learning algorithms. arXiv preprint arXiv:2303.14658 , 2023.

Ruida Zhou, Chao Tian, and Tie Liu. Exactly tight information-theoretic generalization error bound for the quadratic gaussian problem. 2023 IEEE International Symposium on Information Theory (ISIT) , pages 903-908, 2023.

Mahdi Haghifam, Shay Moran, Daniel M Roy, and Gintare Karolina Dziugiate. Understanding generalization via leave-one-out conditional mutual information. In 2022 IEEE International Symposium on Information Theory (ISIT) , pages 2487-2492. IEEE, 2022.

Zhiyu Zhang, Ashok Cutkosky, and Yannis Paschalidis. Optimal comparator adaptive online learning with switching cost. Advances in Neural Information Processing Systems , 35:2393623950, 2022.

Harold Jeffreys. An invariant form for the prior probability in estimation problems. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences , 186(1007): 453-461, 1946.

Gavin E. Crooks. Inequalities between the jenson-shannon and jeffreys divergences. In Tech. Note 004 , 2008.

Roman Vershynin. High-dimensional probability: An introduction with applications in data science , volume 47. Cambridge university press, 2018.

Fredrik Hellström, Giuseppe Durisi, Benjamin Guedj, and Maxim Raginsky. Generalization bounds: Perspectives from information theory and pac-bayes. arXiv preprint arXiv:2309.04381 , 2023.

Yann LeCun, Corinna Cortes, and CJ Burges. Mnist handwritten digit database. ATT Labs [Online]. Available: http://yann.lecun.com/exdb/mnist , 2, 2010.

Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition , pages 770-778, 2016.

Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A largescale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition , pages 248-255, 2009.

Alex Krizhevsky. Learning multiple layers of features from tiny images. Technical report, University of Toronto, 2009.

Mohamad Rida Rammal, Alessandro Achille, Aditya Golatkar, Suhas Diggavi, and Stefano Soatto. On leave-one-out conditional mutual information for generalization. In Advances in Neural Information Processing Systems , 2022.

Ziqiao Wang and Yongyi Mao. Sample-conditioned hypothesis stability sharpens informationtheoretic generalization bounds. In Thirty-seventh Conference on Neural Information Processing Systems , 2023.

Mahdi Haghifam, Borja Rodríguez-Gálvez, Ragnar Thobaben, Mikael Skoglund, Daniel M Roy, and Gintare Karolina Dziugaite. Limitations of information-theoretic generalization bounds for gradient descent methods in stochastic convex optimization. In International Conference on Algorithmic Learning Theory , pages 663-706. PMLR, 2023.

Roi Livni. Information theoretic lower bounds for information theoretic upper bounds. arXiv preprint arXiv:2302.04925 , 2023.

Idan Attias, Gintare Karolina Dziugaite, Mahdi Haghifam, Roi Livni, and Daniel M. Roy. Information complexity of stochastic convex optimization: Applications to generalization, memorization, and tracing. In Forty-first International Conference on Machine Learning , 2024.

Peter Grunwald, Thomas Steinke, and Lydia Zakynthinou. Pac-bayes, mac-bayes and conditional mutual information: Fast rate bounds that handle general vc classes. In Conference on Learning Theory . PMLR, 2021.

Kenji Kawaguchi, Zhun Deng, Xu Ji, and Jiaoyang Huang. How does information bottleneck help deep learning? In International Conference on Machine Learning , pages 16049-16096. PMLR, 2023.

Yuxin Dong, Tieliang Gong, Hong Chen, Shujian Yu, and Chen Li. Rethinking informationtheoretic generalization: Loss entropy induced PAC bounds. In The Twelfth International Conference on Learning Representations , 2024.

Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar. Foundations of machine learning . MIT press, 2018.

F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine learning in Python. Journal of Machine Learning Research , 12:2825-2830, 2011.

Francesco Orabona and Kwang-Sung Jun. Tight concentrations and confidence sequences from the regret of universal portfolio. IEEE Transactions on Information Theory , 2023.

Appendices

A Technical Lemmas

Lemma A.1 (Donsker and Varadhan's variational formula) . Let Q , P be probability measures on Θ , for any bounded measurable function f : Θ → R , we have D KL ( Q || P ) = sup f E θ ∼ Q [ f ( θ )] -log E θ ∼ P [exp f ( θ )] .

Lemma A.2. Let P and Q be two probability measures on the same metric space X . Let f be some measurable function associated with X . The total variation distance between P and Q can be written as

Lemma A.3. The following inequalities hold:

Proof. We prove these inequalities separately.

We first prove Eq. (5) We analyze the function f ( x ) = log(1 + x ) -x + ax 2 . By computing the first derivative f ′ ( x ) :

Let f ′ ( x ) = 0 , we have x (2 a -1 + 2 ax ) = 0 , which gives two critical points:

Note that x = 1 -2 a 2 a is within the interval [ 1 2 a -1 , 1 -1 2 a ] . Then computing the second derivative f ′′ ( x ) :

Since a ≥ 1 2 and -1 < x < 1 , we have f ′′ ( x ) ≥ 0 , indicating that f ( x ) is either concave upwards or linear at these critical points. It is easy to check that the minimum value f (0) = 0 , so f ( x ) ≥ 0 in this interval. Hence, log(1 + x ) ≥ x -ax 2 holds for a ≥ 1 2 and x ∈ [ 1 2 a -1 , 1 -1 2 a ] .

We then prove Eq. (6) We analyze the function f ( x ) = x 1+ x -x + ax 2 . By simplifying this function, we have:

Notice that a -1 ≥ 0 and 1 -a ≤ ax ≤ a -1 , then it is easy to see that the numerator x 2 ( a -1+ ax ) is always non-negative for x ∈ [ 1 a -1 , 1 -1 a ] and a ≥ 1 . In addition, the denominator 1 + x is always positive for x ∈ [ 1 a -1 , 1 -1 a ] . Therefore, f ( x ) = x 2 (( a -1)+ ax ) 1+ x ≥ 0 .

Consequently, we have shown that x 1+ x ≥ x -ax 2 for a ≥ 1 and x ∈ [ 1 a -1 , 1 -1 a ] .

We then prove Eq. (7) We analyze the function f ( x ) = log(2 -e -x ) -x + ax 2 . By taking its derivative, we have

We now demonstrate that the function f ′ ( x ) > 0 for x ∈ [ 0 , 1 2 ] and f ′ ( x ) ≤ 0 for x ∈ [ -1 2 , 0 ] .

The second derivative of f ( x ) is

Let t = 2 e x , since x ∈ [ -1 2 , 1 2 ] , we know that t ∈ [2 e -1 2 , 2 e 1 2 ] . Hence, we reformulate f ′′ ( x ) as

Notice that t + 1 t is monotonically increasing when t > 1 , so f ′′ ( t ) is monotonically increasing in t ∈ [2 e -1 2 , 2 e 1 2 ] . Consequently,

Additionally, it is clear that f ′′ ( x = 0) = f ′′ ( t = 2 e 0 ) = -2 + 2 a > 0 for a ≥ 4 . WWe now consider two cases:

Case 1. If a > 13 . 35 , then f ′′ ( t ) ≥ f ′′ ( t = 2 e -1 2 ) ≥ 0 . Hence, f ′ ( x ) is monotonically increasing in x ∈ [ -1 2 , 1 2 ] . Since f ′ ( x = 0) = 0 , we demonstrate that f ′ ( x ) > 0 for x ∈ [ 0 , 1 2 ] and f ′ ( x ) ≤ 0 for x ∈ [ -1 2 , 0 ] .

Case 2. If a ≤ 13 . 35 , then there exists an x 0 such that f ′′ ( x 0 ) ≤ 0 . Recall that f ′′ ( x = 0) = f ′′ ( t = 2 e 0 ) = -2 + 2 a > 0 and f ′′ ( x ) is monotonically increasing in x ∈ [ -1 2 , 1 2 ] , we know that such x 0 ∈ [ -1 2 , 0] and f ′ ( x ) ≥ 0 for all x ∈ [ 0 , 1 2 ] . Thus, f ′ ( x ) first decreases in x ∈ [ -1 2 , x 0 ] then increases in x ∈ [ x 0 , 1 2 ] .

Since f ′ ( x = -1 2 ) = 1 2 e -1 2 -1 -a ≈ 3 . 69 -a < 0 for a ≥ 4 , we conclude that f ′ ( x ) > 0 for x ∈ [ 0 , 1 2 ] and f ′ ( x ) ≤ 0 for x ∈ [ -1 2 , 0 ] .

Consequently, we know that f ( x ) decreases in x ∈ [ -1 2 , 0] and increases in [0 , 1 2 ] . By checking f ( x ) ≥ f (0) = 0 , the inequality log(2 -e -x ) ≥ x -ax 2 holds for a ≥ 4 and x ∈ [ -1 2 , 1 2 ] . This completes the proof.

B Omitted Proofs in Section 3 and Additional Results

B.1 Proof of Theorem 3.1

Proof. Lemma 3.1 indicates that

Recall that G i = ( -1) U i ∆ L i , and let a = | E [ G i ] | 2 E [ G 2 i ] + 1 2 . Since a ≥ 1 2 and 1 -1 2 a ≥ 0 , we can apply Eq. (5) in Lemma A.3 to have f ( x ) = log(1 + x ) -x + ax 2 ≥ 0 for x ∈ [ 1 2 a -1 , 1 -1 2 a ] .

Hence, by G i ∈ [ -1 , 1] , we restrict t ∈ [ 1 2 a -1 , 1 -1 2 a ] and let x = tG i to have the following inequality,

where the last equality holds when t ∗ = E [ G i ] 2 a E [ G 2 i ] .

We now show that t ∗ can indeed be reached within t ∈ [ 1 2 a -1 , 1 -1 2 a ] .

Substituting a = | E [ G i ] | 2 E [ G 2 i ] + 1 2 , we can see that t ∗ = E [ G i ] | E [ G i ] | + E [ G 2 i ] . In addition, notice that

Clearly, either E [ G i ] = -| E [ G i ] | or E [ G i ] = | E [ G i ] | holds. Consequently, t ∗ can be achieved at one of the endpoints of [ 1 2 a -1 , 1 -1 2 a ] .

Therefore,

which is equivalent to

Then, by Jensen's inequality,

where the last inequality is by Eq. (9).

This completes the proof.

B.2 Proof of Corollary 3.1

Proof. Recall Eq. (3) (or Eq. (9) equivalently),

Notice that | E [ G i ] | ≥ 0 , we now solve the inequality above for | E [ G i ] | . Let A 1 = E [ G 2 i ] and A 2 = I (∆ L i ; U i ) , then by the quadratic-root formula, the solution to E 2 [ G i ] -2 A 2 | E [ G i ] | -2 A 1 A 2 ≤ 0 is

Therefore, we have

The remaining steps are routine, which will complete the proof.

B.3 Proof of Corollary 3.2

Proof. Based on Theorem 3.1, by √ x + y ≤ √ x + √ y , we have

Let G ′ i = ( -1) U i ∆ L ′ i , where ∆ L ′ i is an independent copy of ∆ L i . Notice that E [ G ′ i ] = 0 . Then,

where the last inequality is by Jensen's inequality.

Notice that G i ≤ 1 , by the dual form of total variation (cf. Lemma A.2), we have

Plugging the above into Eq. (10) will complete the proof.

B.4 Comparison with 'Online-to-PAC' Generalization Framework

We now discuss the comparison with [29, 30], consider the case of the KL or mutual information. The only explicit expected generalization bound provided in [30] is their Corollary 21, which recovers the square-root bound of O ( √ I ( W ; S ) /n ) . This bound is clearly weaker than our fast-rate bound in our Corollary 3.1-3.2, due to the omission of vanishing terms in our oracle bound in Theorem 3.1. In fact, a more refined MI bound is presented in the earlier version of [30], namely [29, Corollary 4]. This bound takes the form

which can indeed be derived from [30] due to the generality of their framework. Recall that our Theorem 3.1 gives the bound

Notably, because we apply individual and loss-difference techniques, our averaged MI term is always tighter than that of [29, 30], as 1 n ∑ n i =1 √ I (∆ L i ; U i ) ≤ √ I ( W ; S ) n generally holds. To fairly compare our framework with theirs, we ignore the difference between MI terms and only focus on the novel components of each bound, specifically E Z ∥ ℓ ( · , Z ) -E Z [ ℓ ( · , Z )] ∥ 2 ∞ in their work and E [∆ L 2 i ] + | E [ G i ] | in ours, let's consider the following simple example:

Example 1. Let W ∈ [ -1 , 1] , and let the input space be Z = { 1 , -1 } . Assume µ = Unif ( Z ) , i.e. Z is a Rademacher variable. Consider a convex and 1 -Lipschitz loss function ℓ ( w,z ) = -w · z .

Under the ERM algorithm, W = A ( S ) = 1 n ∑ n i =1 Z i . Notice that for any w ∈ W , E Z [ ℓ ( w,Z )] = 1 2 ( -w · (1 -1)) = 0 , hence E Z ∥ ℓ ( · , Z ) -E Z [ ℓ ( · , Z )] ∥ 2 ∞ = E Z ∥ ℓ ( · , Z ) ∥ 2 ∞ = 1 . In contrast, since ∆ L i ∈ [ -1 , 1] in this case, E [∆ L 2 i ] ≤ E [ | ∆ L i | ] and | E [ G i ] | ≤ E [ | ∆ L i | ] , we have E [∆ L 2 i ] + | E [ G i ] | ≤ 2 E [ | ∆ L i | ] . Moreover, E [ | ∆ L i | ] = E [ | W · ( Z + i -Z -i ) | ] ≤ 2 n E [ | ∑ n i =1 Z i | ] ≤ 2 √ n , where the last step is by the Khintchine-Kahane inequality [62, Theorem D.9].

Thus, in this example, E Z ∥ ℓ ( · , Z ) -E Z [ ℓ ( · , Z )] ∥ 2 ∞ = 1 , while our bound E [∆ L 2 i ] + | E [ G i ] | ≤ 2 E [ | ∆ L i | ] ≤ O ( 1 √ n ) shows a tighter convergence rate. Consequently, even without using the individual trick and loss-difference technique, our bound (Theorem 3.1) can still be tighter than [29, 30] in terms of convergence rate in this example. This suggests that, MI-based bounds in [29, 30] can still encounter limitations (e.g., slow convergence rate), as with previous MI-based bounds.

Finally, to clarify the comparison, we only demonstrate that our oracle CMI bound could be tighter. It is worth noting that the unpublished version of [30] represents an initial exploration of their 'online-to-PAC' generalization framework. As such, the results in their current version may primarily aim to recover previous findings, and their framework likely has room for further advancements.

B.5 Proof of Theorem 3.2

Proof. Lemma 3.1 indicates that

Let a = | E [ G i | ˜ z i ] | 2 E [ G 2 i | ˜ z i ] + 1 . Consider the function f ( x ) = x 1+ x -x + ax 2 . Notice that a ≥ 1 and -1 ≤ 1 a -1 ≤ 1 -1 a ≤ 1 . By Eq. (6) in Lemma A.3, we know that when x ∈ [ 1 a -1 , 1 -1 a ] , f ( x ) ≥ 0 . Hence, recall that G i ∈ [ -1 , 1] , we restrict t ∈ [ 1 a -1 , 1 -1 a ] to have the following inequality,

where the last equality holds when t ∗ = E [ G i | ˜ z i ] 2 a E [ G 2 i | ˜ z i ] .

We now show that t ∗ can indeed be reached within t ∈ [ 1 a -1 , 1 -1 a ] . Substituting a , we can see that

In addition, 1 -1 a = | E [ G i | ˜ z i ] | | E [ G i | ˜ z i ] | +2 E [ G 2 i | ˜ z i ] and 1 a -1 = -| E [ G i | ˜ z i ] | | E [ G i | ˜ z i ] | +2 E [ G 2 i | ˜ z i ] . Clearly, either E [ G i | ˜ z i ] = -| E [ G i | ˜ z i ] | or E [ G i | ˜ z i ] = | E [ G i | ˜ z i ] | holds. Consequently, t ∗ can be achieved in [ 1 a -1 , 1 -1 a ] . Therefore,

which is equivalent to

Then, by Jensen's inequality,

Plugging Eq. (12) into the above will complete the proof.

B.6 Proof of Theorem 3.3

Proof. Lemma 3.1 indicates that

Let a = | E [ G i | ˜ z i ] | E [ G 2 i | ˜ z i ] +4 . Consider the function f ( x ) = log(2 -e -x ) -x + ax 2 . Notice that a ≥ 4 , and by Eq. (7) in Lemma A.3, we know that when x ∈ [ -1 2 , 1 2 ] , f ( x ) ≥ 0 . Hence, recall that G i ∈ [ -1 , 1] , we restrict t ∈ [ -1 2 , 1 2 ] to have the following inequality,

as the same with the previous proofs, the last equality holds when t ∗ = E [ G i | ˜ z i ] 2 a E [ G 2 i | ˜ z i ] .

We then check the conditions that t ∗ ∈ [ -1 2 , 1 2 ] :

which is clearly satisfied by a = | E [ G i | ˜ z i ] | E [ G 2 i | ˜ z i ] +4 . Consequently, t ∗ can be achieved in [ -1 2 , 1 2 ] .

Therefore,

With additional arrangements, we have

Then, by Jensen's inequality,

Plugging Eq. (14) into the above will complete the proof.

B.7 Additional f -Information Disintegrated Bounds

The following bounds can be derived by first employing a similar approach as in the proof of Theorem 3.1 to upper bound E [ G i ] . Then, we incorporate this individual bound into

∆ L i ,U i | ˜ Z i ∣ ∣ ∣ ∣

Theorem B.1. Under the same conditions in Theorem 3.1, we have

Theorem B.2. Under the same conditions in Theorem 3.1, we have

where I ˜ z i χ 2 (∆ L i ; U i ) = D χ 2 ( P ∆ L i ,U i | ˜ z i || P ∆ L i | ˜ z i P U i ) is the (disintegrated) χ 2 -information.

The following bound is a corollary from Theorem 3.2, which is used in our experiments in Section 5. Corollary B.1. Under the same conditions in Theorem 3.1, we have

The proof is nearly the same to Corollary 3.2 except that we do not apply √ x + y ≤ √ x + √ y to unpack the terms.

C Omitted Proof in Section 4

C.1 Proof of Lemma 4.1

Proof. By the variational representation of f -divergence, we have

For the second term above, we have

C.2 Proof of Theorem 4.1

Proof. Since ( -1) U i is a Rademacher variable and ϕ ∗ (0) = e 0 -1 = 0 for KL divergence, we can apply Lemma 4.1 first then follow the similar developments in Theorem 3.1 by controlling tG i ∈ [ -1 , 1] . In particular, for any C ≥ 0 ,

✶ where a = C ∣ ∣ ∣ E [ G i ✶ | Gi | ≤ C ]∣ ∣ ∣ 2 E [ G 2 i ✶ | Gi | ≤ C ] + 1 2 . Notice that t ∗ = E [ G i ✶ | Gi | ≤ C ] 2 a E [ G 2 i ✶ | Gi | ≤ C ] = E [ G i ✶ | Gi | ≤ C ] C E [ G i ✶ | Gi | ≤ C ] + E [ G 2 i ✶ | Gi | ≤ C ] and it is easy to verify t ∗ ∈ [ 1 -2 a 2 aC , 2 a -1 2 aC ] , so the last equality can be achieved. Consequently,

Furthermore, let P = P ∆ L i ,U i and Q = P ∆ L i P U ′

where we use the fact that E ∆ L i ,U ′ i [ G i ✶ | ∆ L i | >C ] = E ∆ L i [ ∆ L i ✶ | ∆ L i | >C -∆ L i ✶ | ∆ L i | >C ] 2 = 0 . Then, by using Hölder's inequality twice,

Moreover, notice that

Plugging Eq. (15) and Eq. (16) into the above will complete the proof.

C.3 Proof of Corollary 4.1

Proof. First, by Markov's inequality, we have

In addition, recall that ζ 1 ≤ 2 C , we let

Then, define f ( C ) = A 1 C + A 2 C γ . Solving min C> 0 f ( C ) , we have

Plugging C ∗ into f ( C ) , we have

From the proof of Theorem 4.1, we notice that

Substituting A 1 and A 2 into the expression of f ( C ∗ ) , we will finally obtain

This completes the proof.

D Experimental Details

Our experimental setup largely follows [24], and the code of our experiments can be found at https://github.com/ZiqiaoWangGeothe/Conditional-f-Information-Bound.

Linear Classifier on Synthetic Gaussian Dataset In this experiment, similar to [24], we use the popular Python package scikit-learn [63] to generate synthetic Gaussian data. Each feature of data is drawn independently from a Gaussian distribution, ensuring that all features are informative for the class labels. Specifically, we set the dimension of feature vector to 5 and create different classes of points, which are normally distributed. We train the linear classifier using full-batch gradient descent with a fixed learning rate of 0 . 01 for a total of 300 epochs, employing early stopping when the training error falls below a threshold (e.g., < 0 . 5% ). To ensure robustness and statistical significance, we generate 50 different supersamples for each experiment. Within each supersample, we create 100 different mask random variables, resulting in a total of 5 , 000 runs for each experimental setting. This thorough setup allows us to compare and the disintegrated f -information-based bounds.

CNN and ResNet-50 on Real-World Datasets In these experiments, the same setup is initially given in [22], and is also used in [23, 24, 61]. Specifically, we draw k 1 samples of ˜ Z and k 2 samples of U for each given ˜ z . For the CNN on the binary MNIST dataset, we set k 1 = 5 and k 2 = 30 . The 4-layer CNN model is trained using the Adam optimizer with a learning rate of 0 . 001 and a momentum coefficient of β 1 = 0 . 9 . The training process spans 200 epochs with a batch size of 128 . For ResNet-50 on CIFAR10, we set k 1 = 2 and k 2 = 40 . The ResNet model is trained using stochastic gradient descent (SGD) with a learning rate of 0 . 01 and a momentum coefficient of 0 . 9 for a total of 40 epochs. The batch size for this experiment is set to 64 . In the SGLD experiment, we train a 4-layer CNN on the binary MNIST dataset with a batch size of 100 for 40 epochs. The initial learning rate is 0 . 01 and decays by a factor of 0 . 9 after every 100 iterations. Let t be the iteration index; the inverse temperature of SGLD is given by min { 4000 , max { 100 , 10 e t/ 100 }} . We set the training sample size to n = 4000 , and k 1 = 5 and k 2 = 30 . Checkpoints are saved every 4 epochs.

All these experiments are conducted using NVIDIA A100 GPUs with 40 GB of memory. For more comprehensive details, including model architectures, we recommend referring to [22-24].

E CMI Framework from the Perspective of Coin-Betting Sequences

Deriving generalization bounds from a coin-betting perspective is inspired by [64, 1, 34]. We now adapt this strategy to derive CMI bounds, invoking inequalities similar to those in Lemma A.3.

Assume ℓ ∈ [0 , 1] . Let ∆ ℓ i ∈ [ -1 , 1] be a sequence of 'continuous coin' outcomes chosen arbitrarily, and at each round i , a player bets c i ≥ 0 money on the sign of the outcome ε i ∈ {-1 , 1 } . Then, ∆ ℓ i is revealed, and the bettor wins or loses c i ε i ∆ ℓ i money.

Let the initial wealth Wealth 0 = 1 . The wealth at the end of round n is:

Notice that c i ∈ [0 , Wealth i -1 ] . Now consider a strategy where the player always bets a fixed fraction t ∈ [0 , 1] of the current money, i.e., c i = t Wealth i -1 . Then:

It follows that

Unlike the traditional online gambling game, we incorporate the individual technique from [6] by analyzing the expected log-wealth of the i th bet 2 , given by

There are two kinds of players: those guessing the coin outcome randomly and those with side information. Let ε = ( -1) U i where U i ∼ Bern(1 / 2) . The average log-wealth of the random guesser is E U ′ i , ∆ L i log(1 + t ( -1) U ′ i ∆ L i ) , and the average log-wealth of the informed player is W i ( t ) = E U i , ∆ L i log(1 + t ( -1) U i ∆ L i ) .

Clearly, the value of side information impacts the second player's income. To quantify this impact, by the DV-representation of KL divergence (cf. Lemma A.1), we have

where the last equality holds because E U ′ i , ∆ L i t ( -1) U ′ i ∆ L i = E ∆ L i [ E U ′ i t ( -1) U ′ i ∆ L i ] = E ∆ L i [ t ∆ L i -t ∆ L i 2 ] = 0 . Notice that the log -wealth of the comparator, i.e. the random-guesser, appears in Eq. (17) in the form of inf λ E ∆ L i ,U ′ i [ ϕ ∗ ( log(1 + t ( -1) U ′ i ∆ L i ) + λ )] -λ .

Similar to [1], by the inequality log(1 + x ) ≥ x -x 2 for x ≥ -0 . 68 (see Figure 1(Right) for a visualization of this inequality), we have

We then have

Moreover, we now assume that the side information is 'perfect' such that ( -1) U i ∆ L i ≥ 0 (i.e. an interpolating or overfitting algorithm) always holds for each i , namely the income is always positive. Under this condition, the players can safely set t = 1 , that is, the players 'all-in' their money. Then, by log (1 + x ) ≥ x (1+ x ) for x > -1 , we have

Consequently, recovering several existing ld-CMI bounds (up to a constant) from Eq. (19) and Eq. (18) is straightforward.

which gives us

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Peer reviews:
comments: SUMMARY:
This paper introduces novel generalization bounds using the conditional f-information framework. It first derives conditional f-information-based generalization bounds for bounded loss, then examines mutual information generalization bounds as a specific example of their general theorem. This analysis helps to highlight the potential looseness of previous CMI bounds. Additionally, the paper presents several other f-information-based generalization bounds and extends the framework to handle unbounded loss cases. Finally, the tightness of the obtained bounds is demonstrated through empirical comparisons with other existing bounds.

STRENGTHS:
- **Clear presentation**: The structure of this paper is clear and easy to follow, with definitions and theorems clearly presented.
- **Empirical validation**: Although it is a theoretical work, the paper includes empirical studies to demonstrate the utility of the proposed generalization bounds.

WEAKNESSES:
As the authors mention, the results pertain to the expected generalization error and therefore lack the high-probability generalization guarantee, which is more critical.

QUESTIONS:
- For the definition of the expected generalization error (line 101 on page 3), should the correct definition be $\\mathbb{E}\_S[\\mathbb{E}\_W[L_{\\mu}(W)-L\_S(W)]]$ instead of the current formula in the paper? If I understand correctly, $\\mathbb{E}\_W$ refers to taking the expectation with respect to $P\_{W|S}$, which depends on $S$.
- In the “Other Fast-Rate Bound Cases” section, could the authors provide some examples where $\\mathbb{E}[\\Delta L^2\_i] \lesssim \text{Var}(L\_i^{+})$ holds?

LIMITATIONS:
See weaknesses and questions.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-06-15

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a general framework for generalization bounds using a careful application of the Donsker-Vardaran representation of the conditional $f$-information, and show that a suitable quadratic Taylor expansion of the bounds, for various $f$-divergences, improves over the state of the art.\n\nSTRENGTHS:\nThis paper has a number of strengths:\n(1) The technique appears generic and \"fundamental:\" it works regardless of $f$ divergence, and unifies the perspectives on quite a few of them. Moreover, it seems to improve over prior art, consistently, and in many instantiations. Finally, Lemma 3.1 does seem useful in its own right. From what I understand, the use of the variational representation also seems novel. \n(2) The bound admits extension to unbounded losses, even those which do not admit MGFs\n(3) The bounds satisfy an oracle property, in that they adapt to problem-dependent quantities. \n(4) For a paper focused on theoretical contribution, the experiments are rather compelling: across toy domains and CIFAR 10, the Hellinger-based oracle confidence interval consistency improves upon prior art. \n(5) The authors go to great lengths to explain and differentiate their contributions from prior work.\n\nWEAKNESSES:\nI am not an expert in the state-of-art guarantees for generalization, so please take these concerns with a grain of salt. However, my only concern would be that these guarantees do not feel \"field changing\" in the sense that they appear to be an extension of and elaboration upon an accepted framework: construct a super-sample, evaluate its ($f$-)information, and evaluate the ensuing consequences. The idea of \"generalize Shannon information to $f$-information\" feels like a natural extension, and therefore its hard for me to find that some of these results are surprising. Moreover, the extension to unbounded losses appears to be based on standard truncation techniques. \n\nThat is to say, form an outsider's perspective, this appears a good, solid work with nice fundamental insights. But it doesn't feel groundbreaking enough for me to say, advocate for an award. \n\nThere are also a few typos here and there, and the preliminaries read a bit dense at times. This is probably standard in the community, and does not trouble me that much. \n\nPerhaps one suggestion: regarding the presentation, the authors could do well to offer a table in the appendix where they compare the bounds to prior art in a more structured and systematic way. The comparison in the paper often assumes familiarity with prior work, and it would be more effective to re-present the formal statements upon which the authors are improving an an Appendix. Then the authors can more clearly delineate the key advantages of their bounds over past art.\n\nQUESTIONS:\nHad any other works in the literature considered bounds based on $f$-divergences before? Did the original bounds, based on Shannon information, using the variational characterization in any meaningful way? It would be quite helpful for me to understand this better to fully gauge novelty.\n\nLIMITATIONS:\nThe authors do an adequate job of explaining the limitations of their work: lack of high probability guarantees, failure to account for the tradeoff in source [52], and perhaps the general limitations of purely information-theoretic approaches to generalization.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-04"}

comments: SUMMARY:
Understanding generalization using information-theoritic measures of dependence between input and output of a learning algorithm is an important area of study. The main focus of this line of work is using the KL divergence as a measure of dependence and providing generalization bounds. In this work, the authors propose a more general approach that replaces the KL divergence with arbitrary f-divergences. The main contribution of this work is presenting an approach to obtain such a generalisation bounds.

STRENGTHS:
The main strength of this work is showing KL divergence is not necessary for obtaining information-theoritic generalization bounds. Also, compared to other work showing this fact, the proof strategy in this paper is much simpler.

WEAKNESSES:
I think the weakness of the work are the following:

1- The point that one can replace KL with an arbitrary f-divergence was shown in the following paper:

Lugosi G, Neu G. Online-to-PAC conversions: Generalization bounds via regret analysis. arXiv preprint arXiv:2305.19674. 2023 May 31.

I could not find a clear comparison of the approach proposed in this work and the the paper by Lugosi et.

2- More general question is that by replacing KL with f-divergence what sort of new insights can we obtain? I am trying to understand the real advantage of replacing KL with f-divergences.

3- It is difficult to parse Theorem 4.1. for me. How one can find a good truncation value?

QUESTIONS:
1- Write down the definition of convex conjugate?
2- One main question here is that what is the gain of using f-divergence compared to the usual mutual information?
3- Typo in Lemma 3.1. It should be b_1 and b_2.

LIMITATIONS:
N/A
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-15

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper extends conditional mutual information bounds to other $f$-divergences. A list of bounds involving various $f$-information terms are established. The results are derived by evaluating a previously established variational formula for f-divergences at a specific function. Analysis tailored to various choices of $f$ is then conducted to extract expected generalization bounds.\n\nSTRENGTHS:\nThe paper is clear in the assumptions made to derive its results. The buildup to each theorem is well detailed making it easy for the reader to follow.\n\nWEAKNESSES:\nThe main weakness of this work is in the motivation and the reason for being for the results. The derived extensions have the quantity to be bounded appearing within the bound itself. The leads to strange tautological statements like in line 165 where it is said that if each term in the expected generalization decays fast with $n$ then so does the expected generalization. The $\\chi^2$, Hellinger, Jensen-Shannon results have increasingly complicated terms that also involve the very quantity trying to be bound. It is unclear to me why such bounds would be of interest, especially since any instantiation requires falling back to a previously established result. \n\nFor example, the sub-gaussian assumption seems to be necessary to have any hope of controlling the quantities appearing in theorem 4.1, and when the assumption is made, the result yields a bound that is worse than existing bounds. The heavy-tail corollary 4.2 is even more impenetrable, the very same quantities we wish to bound appear on the right hand side within $L_p$ norms. The authors should provide justifications as to why these true inequalities are not circular and of limited interest.\n\nMinor note: There is some prior work establishing expected generalization bounds using different $f$-divergences [1].\n[1] Esposito, Amedeo Roberto, and Michael Gastpar. \"From generalisation error to transportation-cost inequalities and back.\" 2022 IEEE International Symposium on Information Theory (ISIT). IEEE, 2022.\n\nQUESTIONS:\nHow exactly are the experimental plots made ? The theorems involve expectations and mutual information terms, are the authors estimating those? Can the authors add more information on what is plotted?\n\nLIMITATIONS:\nThe limitations are discussed.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-16"}