Generalization of Hamiltonian algorithms
Authors: Andreas Maurer
Venue: NeurIPS 2024 poster (2024)
Keywords: Generalization
stochastic algorithms
concentration inequalities
PAC-Bayes
Primary area: learning_theory
TL;DR: The paper gives bounds for stochastic algorithms including the Gibbs algorithm

Abstract: A method to prove generalization results for a class of stochastic learning algorithms is presented. It applies whenever the algorithm generates a distribution, which is absolutely continuous distribution relative to some a-priori measure, and the logarithm of its density is exponentially concentrated about its mean. Applications include bounds for the Gibbs algorithm and randomizations of stable deterministic algorithms, combinations thereof and PAC-Bayesian bounds with data-dependent priors.

Generalization of Hamiltonian algorithms
Andreas Maurer
Computational Statistics and Machine Learning Istituto Italiano di Tecnologia, 16163 Genoa, Italy am@andreas-mauren
Abstract
The paper proves generalization results for a class of stochastic learning algorithms. The method applies whenever the algorithm generates absolutely continuous distribution relative to some a-priori measure and the Radon Nikodym derivative has subgaussian concentration. Applications are bounds for the Gibbs algorithm and randomizations of stable deterministic algorithms as well as AAC-Baaeii bounds with data-dependent priors.
Introduction
stochastic learning algorithm Q takes input a sample (X,,..,,,,) X", drawn from distribution fs on space X' of data, and outputs a probability measure Qx on a loss-class H of functions >> [0, oo). key problem in the study of these algorithms is to bound the generalization gap
A (h, = EE
Σ (X.)
(1)
between the expected and the empirical loss of hypothesis drawn from Qx. Here we want to generate h only once and seek guarantees with high probability as X n² and h ~ Qx- Alternatively one might want a bound on the expectation En-Ox A (h, X)] with high probability in X ~ ju",, corresponding to the use of a stochastic hypothesis, where a new h Qx is generated for every test point. We concentrate on the former question, but many of the techniques presented also apply to the latter, often easier problem.
From Markov's inequality follows that for X, 8 > O with probability at least 8 as X pm and ~ Qx
In Ex
E AA + In (1/6)
A (h, X
(2)
which suggests to bound the log-moment generating function In Ex [x (exp (AA (h, X))]] With such a bound at hand one can optimize X to establish generalization of the algorithm Q ; X +
Qx.
Inequality 2 is relevant to stochastic algorithms in general, and in particular to the Gibbs-algorithm, where dQx (h) x exp (- (B/n) Σ (X,)) dm (h) for some inverse temperature parameter 3 and some nonnegative a priori measure x on H. The Gibbs algorithm has its origins in statistical mechanics (Gibbs (1902)). In the context of machine learning it be viewed empirical risk minimization, to which it converges as / - 00, whenever π has full support. The distribution, often called Gibbs posterior (Catoni (007)), ), minimizer of the PAC-Bayesian bounds (McAllester] 1999)). ). It is also the limiting distribution of stochastic gradient Langevin dynamics (Raginsky al. (0117) under rather general conditions. Generalization bounds expectation are given by Raginsky et al. [007]) Kuzborskij et al. (2019). most recently by Aminian et al. [2021]
randomized version of
Bounds in probability are given by Lever 2013 implicitly by Dziugaite and Roy 2018 and
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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in Rivasplata et al. [2000] following the method of Kuzborskij et al. (2019). There is also a bound by Aminian al. [2023]. improving on the one in (Lever et al.) 2013
Bounding In Ex [ (exp (AA (h, X)]]] is also the vehicle (and principal technical obstacle) to prove PAC-Bayesian bounds with data-dependent prior Qx, pointed out by Rivasplata et al. [2020] (Theorem 1). Such bounds with data-independent prior Qx = Q. have an over twenty year old tradition in learning theory, starting with the seminal work of McAllester (McAllester (1999). ) Langford and Seeger (Langford and Seeger [200T). Seeger (2002). D. see also Guedj] 2019 the prior is data-independeet, the two expectations in In Ex E lexp (AA (h, X]] can exchanged, which reduces the analysis to classical Chernoff- or Hoeffding
sooeffding-iequalitte. inequalities. But a dominant term in these bounds, the KL-divergence LL (P. Q) := E [In (dP/dQ) (h)), will be large unless P is well aligned to Q, so the prior Q should already put more weight on good hypotheses with small loss. This motivates the use of catributiio-dependeen priors, and as the distribution is unknown, one is led to think about data-dependent priors. Catoni already considers the data-dependent Gibbs distribution as a prior in a derivation departing from the distribution deeerioddddeeeeennt Gibbs measure exp (- [h (X)]) (Lemma 6.2 in Catoni [2003] D. Dziugaite and Roy| [2017] used Gaussian prior with data-dependent width and minimized the PAC-Bayes bound for Gaussian posterior on multi-layer neural network, obtaining good classifier accompanied by non-vacuous bound. This significant advance raised interest in PAC-Bayes with data uata-dependent priors. The same authors introduced a method to control deeendeenndd priors based on differential privacy (Dziugaite and Roy| (2018). Recently Pérez- Pérez-Ortiz et al. [2021] used Gaussian and Laplace priors, whose means were trained directly from one part of the sample, the remaining part being used to evaluate the PAC-Bayes bound. These developments further motivate the search for in-sample bounds on the log-moment
generating function appearing above in
We make the following contributions:
An economical and general method to bound In Ex [EΛ-Ox [exp (AA (h, X))] whenever the logarithm of the density Qx concentrates exponentially about its mean, In particular, whenever Qx has the Hamiltonian form dQx (h) x exp (H (h, X)) dr (h), then it is sufficient that the Hamiltonian H satisfies a bounded difference condition. The method also extends to the case, when H is only sub-Gaussian in its arguments.
Applications to the Gibbs algorithm yielding competitive generalization guarantees, both for bounded and nab-Gaussian losses. Despite its simplicity and generality the method improves over existing results on this well studied problem, removing unnecessary logarithmic factors and various superfluous terms.
Generalization guarantees for hypotheses sampled once from stochastic kernels centered at the output of uniformly stable algorithms, considerably strengthening a previous result of Rivasplata et al. [2018).
2 Notation and Preliminaries
For E N. we write [m] := (1.....m). Random variables are written in upper case letters, like X, Y.Y. X etc and primes indicate iid copies, like X', X", X' etc. Vectors are bold like x.X, etc. Throughout X is measurable space of data with a probability measure μı, and X will always be the iid vector X = (X1,.., Xn) ~ 1r" and X ~ p.. H is a measurable space of measurable functions X + [0, oo), and there is a nonnegative a priori measure on H. The measure क need not be probability measure, it could be Lebesgue measure on the space IRd of parametrizations of H. Averages over æ will be written as integrals. P (H) is the set of probability measures on H. Unless otherwise specified E denotes the expectation in X j or X ~ n², All functions H. x X appearing in this paper are assumed to have exponential moments of all orders, with respect to both arguments.
For x E X", ". E {1...., n and y E X we define the substitution operator s/ acting on X and the partial difference operator D y.y acting on functions f x + R by
S x (f1, #x-1, Th Tett, Rn) and D* / = s (5t x) - (St)))
D Nay always refers to the second argument for functions on H X The generalization
gap A (h, X) is defined as in (. Sometimes write L (h) E (X)) and 1 (h,

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11/n) Ei= h (X,), so that A = (h) (h, X). A table of notation is provided in Appendix
2.1 Hamiltonian algorithms
stochastic algorithm Q x E X + Qx E (HL) will be called absolutely continuous, if Qx is for every
absolutely continuous with respect absolutely continuous algorithms in the sequel. A real function H on H. x X" is called a Hamiltonian for Q (a term taken from statistical physics), if for all h H and all X
x and vice versa. We will only consider
dQx (h) (H di (h)
with Z (x) / H(hh)) dm (h).
2 7(x)
The normalizing function is called the partition function. Every absolutely continuous Q has the canonical Hamiltonian Ho (h, x) In ((dQx/ /dxx) (h)) (logarithm of the Radon Nikodym derivative) with partition function Z = 1, but adding any function Ç 2 + R to HQ will give Hamiltonian for the same algorithm with partition function (x) = exp (C (x)). ). In practice Q is often defined by
specifying some Hamiltonian H1 so Hq (h, x) = H (h, x) - In 2 (x) in general. If i is a probability
measure, then En~Q₂ ( (h, x)] is the KL-divergence KL (Qx: .T.
Hamiltonian for the Gibbs algorithm at inverse temperature 3 is
H (h,x) - (h,
h (xx),
putting larger weights on hypotheses with small empirical loss. This is the simplest case covered by our proposed method. If there is a computational cost associated with each h, it may be in- cluded to promote hypotheses with faster execution. We could also add a negative multiple of Eiey (h (z,) - (z,))². so as to encourage hypotheses with small empirical variance. Monte Carlo methods, such as Metropolis- Hastings algorithm, can used to sample from such distribu- tions. Often these are slow to converge, which underlines the practical importance of using a single hypothesis generated only once.
If H. is parametrized by R one may also first compute a vector A (x) E R with some algorithm A and then sample from an absolutely continuous stochastic kernel к centered Hamiltonian is In k (h A (x)). In one concrete version the kernel is an isotropic gaussian, and
(x). so
H (h, = = - | A (x)] (2o²). Generalization of these methods is discussed in Section 4.2
If Q())) and (22) are absolutely continuous stochastic algorithms with Hamiltonians H₃ and H₂ respectively, then an elementary calculation shows that H₃ + Hy is a Hamiltonian for the algorithm Q obtained by sampling from (()) with the measure æ replaced by Q(2), In this way Hamiltonian algorithms of different type can be combined.
3 Main results
Let Q be an absolutely continuous stochastic algorithm, F: H. x X + some function and define
dr (h) = In E (1
Our method is based on the following proposition. Proposition 3.1. With Q, F and as above
(i) In Ex- En-ox [eF()))] < SリPдεи tF (h). (ii) Let 8 O. Then with probability at least 1 8 in X nm and h Qx we have
(b. X) S sup oF (h) + In (1/8).
WCH
(ii) Let 8 > 0. Then with probubility at least 8 in X ~ n²"h we huve
En-Qx [F (h, I sup &r (h) + In (1/5). kEW
3

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Proof. (1) With Jensen's 's inequality
[e (c)) In Ex S/x P(). Halh. (h)
In E Ex Qx
H
/ Exaun [ ((N,))+H(((x))))/eeee (z(Ho(hX)) (1))
=
JH
In / C (HH daT (h)
S In /
dn (h) In J/ ( do (h)
18
(eur(h)) sup th (h).
In E
NeW
(ii) then follows from Markov's inequality (Section since In Ex=1" [cRa-ax [P(A.))]]
(iii) follows also by Markov's inequality, [cFx-ax(P(h I In Ex= DE, nx F...)]
3. by Jensen's 'ss inequality.
To see the point of this proposition let F X), AA X). Since A (h, X) is centered. Dix is of the form In Ex- e [x(
Many concentration inequalities in the literature (McDiarmid 1998]. Boucheron et al. 2013) ) follow the classical ideas of Bernstein and Chernoff and are derived from bounds on such moment generating functions. If F' (h,X) is not centered, we can use Hölder's s inequality to separate the contributions which I' and Ho make to L/r. The F-contribbuionn can be treated separately and the contribution of HQ can again be treated with the methods of concentration inequalities.
or the Cauchy-Schwarl
The last conclusion of the proposition is to show that we can always get bounds in expectation from bounds on the exponential moment. the sequel we only state the stronger un-expected or "disintegrated" results.
Typically the exponential moment bounds for functions of independent variables depend the function's 's stability with respect to changes in its arguments, In Section 4.2 a more advanced concentration inequality will be used, but all other results below depend only on the following classical exponential moment bounds. Most of them can be found in McDiarmid [198), but since the results there are formulated as deviation inequalities, a proof is given in the appendix, Section A.2
Proposition 3.2. Let X, Xi,... Xn be iid random variables with values in X, X (X1, Xa) and f X +
R measurable.
(i) If is such that for all k E [n], x d we have Ex
s c², then
f(x)) [(x))=[((x']]] < enr
(ii) If D W (x) S e for all ls E [n), v.y/ E X and x E X", then E [[][())- S e
(iil) there is E (0,2), such that for all ls [n]] and x E X" we have f (x) - Exiny [f (S)xxx) S
then with U2 su Ex~a (f (S*x) Ex'nn [f [S{xx)) |}
Ex [oxx))
1
Σ
I exp
Uk
2
Notice that the second conclusion is instrumental in the usual proof of McDiarmid's (or bounded- difference-) inequality (McDiarmid (1998).
3.1 Bounded differences
In the simplest case the Hamiltonian satisfies a bounded difference condition as in (ii) above. Then the only minor complication is to show that the logarithm of the partition function inherits this property. This is dealt with in the following lemma.

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Lemma 3.3. Suppose H : H. x x" - R and that for all h E H. &s E [n], v. v E X and x E x we
have D H (h, I c. Let
V.y
/ ( du (h))
Hq (h, x) = H (h,
In 2 (x) with Z (x)
Then vh E H,k [n], v' E X, x have D D.yy HQ (h, 2c.
Proof. This follows from the linearity of the partial difference operator D Mayr and
2 (S In SN exp (D v. H (h, x)) exp (H (h,, stx)) dr (h)
D V'y In 2 (x)
Z (Stx)
JN exp (H (h, (1x)) dr (h)
In sup exp (Di n (h,x)) Sc. kEH
Theorem 3.4. Suppose H is a Hamiltonian for Q and that for all k E [m]. h E H. us of e X and x E Xn we have D
a.y H (h, x) S and h (1) E 10, b). Then
(i) For any X
In Ex-ge Ex [c S sup Dsa (h)
S
12
Ab
+ 2c
hEH
8
(ii) If then with probability at least
8 as X ~ 1eth and Qx we have
In (1/6)
(h, X) b c
2n
Proof. Using the previous lemma D yorr (AA (h, x) + He (h, x))
S (Ab/n) + 2c.
Since
E [ X)] = 0, Proposition 3.2 (ii) gives, for any E H.
/ Xb 2
aia (h) In [ (h.x')
+ 2c
n
and the first conclusion follows from Proposition
3.1 (i) with F xh, AA (h, X).
From Proposition 3.1 (ii) we get with probability least as X μ., Qx that
n Ab
A02
8n + be nd² / + In (1/8)
A (h,
--1
2c
+ In
+Img)
8
8)
X
Substitution of the optimal value = / (8n/b2) (nc² / + In (1/6)) and subadditivity of L + Vi give the second conclusion.
In applications, such as the Gibbs algorithm, we typically have e c c (1/n). The previous result is simple and already gives competitive bounds for several methods, but it does not take into account the properties of the hypothesis chosen from Qx. To make the method more flexible we use the Cauchy-Schware inequality to write
dr (h) In E (0 (h.x')]
<
2 In 2F(
+ In
(3)
and treat the two terms separately. The disadvantage here is an increase of constants in the bounds. The big advantage is that different types of bounds be combined.
The next result based on this idea. similar Bernstein's 's inequality for sums of independent variables.
5

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Theorem 3.5. Under the conditions of Theorem 3.4 define for each h E H. its variance v (h) ==
(B (x'))))) Then for > 0 with probability at least 1 - 5 in X ~ u²" and h ~ Qx
In (1/0) A (h S spe(()) c²
+ b 2 In (1(0))
Similar to Bernstein's inequality the above gives a better bound the chosen hypothesis has a small variance. For the proof we use the following lemma, which is direct consequence of Proposition 3.2 (iii).
Lemma 3.6. Assume that for all h E and x E X, we have (x)) E [0, bj and (h) as in Theorem 3.5 above. Let X be function X H (0, n//b) and define Fx H X"
R by
A 2h)2
v (h)
Fx X) (h) A (h,
bd (h) /n
Then for all h E H we have E [e2 1
Proof. For every h E H. we have 26 (h) /n (0,2). Also Yx E X", and wh E H.
1 S A (h, x)] (R (X') - 11 (xx)) S b
A xh,x) Ex
n2
Thus for every h E H we can apply Proposition 3.2 (iii) to f (x) = 2A (h) A (h, x) and obtain
[22(thx)) 1/2
- -11)2 v (h)
E (exp (2) (h) A (h. X)) exp
Dd (h) /n
2A (h)2
(h)
-Xhh2 v (h)
s exp
exp
1.
2 26A (h) /n
Dd (h) /n
Proof of Theorem
3.5
Let
nc + In (1/0)
A (h)
(b/n) / ne² In (1/8) +
e(h)
and let FA be as in Lemma 3.6 It follows from Lemma 3.3 and Proposition 3.2 (ii) that for all h E H
we have In 2 (Ha(()
x))
I 2nc². Then (3) and Lemma 3.6 give &r (h) S no².
Thus from Proposition 3.1 and division by A (h)
(h) v (h) nc? + In (1/0)
Pr X X-Jue Xo
(h, X
< 6.
1 (h) /n
A(h)
Inserting the definition of X (h) and simplifying completes the proof.
At the expense of larger constants the role of the variance in this result can be replaced by the empirical error, using (h) < E h (X)) < bb (X)) and simple algebraic inversion, which is given the appendix, Section
A.3
Corollary 3.7. Under the conditions of Theorem 3.5 for 5 > with probability at least ô in X nd and Qx
( c2+ In (1/0)
In (1/0))
A (h, X) S L
+ 5b c

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3.2 Subgaussian hypotheses
Some of the above extends unbounded hypotheses. real random variable is o-subgaussian for x
a > O, if E [exp (A - [Y]]] S exfo+/2 /2 for every À E R. The proof of the following result is given in the appendix (Section/A.4 and uses ideas very similar to the proofs above.
Theorem 3.8. Let Q have Hamiltonian H and assume that vh. E H there is p (h) > 0 such that
E E (eMEE((())==)M( S exp
(1)
2
Let p suPREM P (h) and suppose that AA E R in],
E [[ $ exp
( 12002
(i) Then for any E H, A
In ExxQx [e^a] S Uns (h) - (Ap //nn + 2
2
and with probability at least 8 we have as X ~ 4et and ~ Qx that
21 (1/0)
A (h, X) S p 20 +
n
(ii) With probability at least
8 we have as X 1rth and h Qx
that
(h, (h) (V320 + 4 (1/6)
The assumptions mean that every hypothesis has its own subgaussian parameter and that the Hamilto- nian is subgaussian in every argument if all other arguments are fixed. The first conclusion parallels the bound for Hamiltonians with bounded differences in Theorem 3.4 The second conclusion has larger constants, but scales with the subgaussian parameter of the hypothesis actually chosen from Qx, which can be considerably smaller, similar to the Bernstein-tyy teesteeinttype inequality Theorem 3.5
4
Applications
4.1
The Gibbs algorithm
The Gibbs distribution for sample is dQaxx (h) -- (B// (x:)) du (h), the Hamiltonian is H (h,, x) (3/n) EL= h (x;). It is the minimizer of the PAC-Bayesian bounds McAllester [1999] ) as well as the limiting distribution of stochastic gradient Langevin dynamics (Raginsky al. [017]] ), generalization bounds for the Gibbs distribution translate to guarantees for these algorithms. Let us first assume bounded hypotheses, for simplicity h X + [o. 1]. Then we can use Theorems 3.4 and 3.5 and Corollary 3.7 with 3/n. Theorem 3.4 gives with probability
at least 1 - d in X n and h ~ Q8.x that
3
In (1/8)
A A X)
11
2n
We were able find this simple bound in the literature. It improves over
46 2 in ((1 /8)
A (h, X <
Vn
obtained in (Rivasplata et al. importantly, in its dependence on the confidence parameter 8. The principal merit of (4). however, lies in the generality and simplicity its proof (compare the proof of Lemma 3 in Rivasplata et al. 2 )
2020 Sec. 2.1 and Lemma 3) not only in constants, but, more
7

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Upon the substitution c- 8/n Theorem 3.5 leads to variance dependent bound, for which we know of comparable result.
From Corollary 3.7
we get for the Gibbs algorithm with probability at least 5 5'in X and ~ Qa.x
(h, X) S 22/ 22/2 (h,
g
In (1/0) + 18 (1/0) 72
(5)
n?
n
For hypotheses with small empirical error this approximates a "fast convergence rate" of O (1 (1/n). Comparable bounds in the literature involve the so-called "little KL-divergenee". For two num- bers 8,t E [0, 1] the relative entropy of two Bernoulli variables, with means s and respec- tively, is kl t) = s snn (s/t) + (1 - s) In ((1 s) t)). Various authors give bounds on Ea-Qax hd (h, I (1)) with high probability in the sample. Rivasplata et al.
2020 give
Ex-Qax
[a (L,,,)
2/3² n
+
22n3
B n3/2
+
4vn
and there is similar bound in Dziugaite and Roy 2018 and a slightly weaker one in Lever
et al. 2013). The most useful form of these bounds is obtained using the following inversion
rule
Tolstikhin and Seldin (2013), see also Alquier [2021] D: if kl L S B then
(h, x) $ V /2 (h, x) B + 2B. If this rule is applied to the kl-bound above, it becomes clear, that it is inferior to (5). not only because of the logarithmic dependence on n, but also because of artifact terms, which are difficult to interpret, like the superfluous 3/n3/2.
If every h (X) is (h) -ubbauusssn and p supи (h), then by linearity of the subgaussian pa- rameter H (h, X) is p///n-uubaauseeeeeeee in every argument for every h, and Theorem 3.8 gives with probability at least 1 - 8 in X n² and h Q3,x
4v2pB
In (1/0)
A (h. S p (h)
n
n
2023 gave very interesting bound probability for sub- uub-gaussian hypothe- ses, which however is not quite comparable to the above, as it bounds the posterior expectation of A and relies on distribution- deredddde prior.
Recently Amminia al.
4.2 Randomization of stable algorithms
Suppose that H is parametrized by Rd, with π being Lebesgue measure. To simplify notation we identify a hypothesis h H with its parametrizing vector, so that is simultaneously a vector in Rd and function h x E X' + h (x) E R. Following RRivasplata et al. (018)) we define the hypothesis sensitivity coefficient vector valued algorithm A A
Rd as
A (x)]l D A. A
CA
max kE(m]
sup x ane/ Ex
In typical applications C = O (1/n) (compare the SVM-application in Rivasplata et al. 2018 as derived originally from [Bousquet and Elisseeff 20022). )
Consider first the algorithm arising from the Hamiltonian
H (h, x) = -G (h Ax)),
(6)
where G Rdd + [0, 0o) is any function with Lipschitz norm || | One computes A (x) and
samples h from the stochastic kernel proportional to exp (-G (h h A (x))). By the triangle inequality H satisfies the bounded difference conditions of Theorems 3.4 and 3.5 with e= ||G|| CA- If every
h E 7 (as function on X') has range in [0, 1], then, although this algorithm is of a completely different nature, we immediately recover the generalization guarantees as for the Gibbs-algorithn with 8/n replaced by ||Glllap CA. An obvious example is G (h) = |||||| /o for o > 0.
Another interesting algorithm arises from the Hamiltonian
||| A [x
H (h,x)
20
8

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for a > O, corresponding to gaussian randomization, For stochastic hypotheses there is an elegant treatment by Rivasplata al. [2018) using the PAC-Bayesian theorem, and resulting in the bound (with probability at least ò as ~ p²
nc 20
In (t) + In
2
@x kl
Since the squared norm is not Lipschitz the previous argument does not work, but with slight variation of the method we can prove the following result (proof in Section B.I.
Theorem 4.1. Lei H = Rd with Lebesgue measure н. Suppose Q has Hamiltonian H (h, X) -Ilh A (x)ll /2a², where has stability coefficient Ca. Let > and assume that 12ne S 0² and that every h E H (as function on X) has range in [0, 1]. Denote the variance of A by
(A) = || (X) - ([A(X
Then
(i) Jf > 8 then In Ex Enxex (elm)))(((():x))))))))) $ V.(A + lm
(ii) Ifn > 8 then with probubility at least 1 - 8 as X nM
kl (i (1,
(n) (2
0² V (A) In
+ 21n (1/6)
Ex-Qx
(ii) With probability at least 8 as X n1 and h Qx
(h, (3/02) (A) + In (1/8)
n
(iv) Let v (h) be the variance of h, defined as Theorem
3.5
Then with probability at least ô as
X n²d and h Qx
(3/02) V (A) + In (1/5) A X S 2/ 0 (h)
(3/0²) V (A) + In
72
The expected kl-bound (ii) is given only for direct comparison with 7 (iii) and (vi) are stronger, not only by being disintegrated, but also by avoiding the logarithmic dependence on
In comparison 7 (ii) has slightly larger constants and we require that 12nc <²0 The latter assumption is mild and holds for sufficiently large n if ne + 0 as n + oo (in applications of (7) CA 0 (1/n), but no may even remain bounded away from zero for 12nc < hold. On the other hand V (A) E
A (X)
E [A 12 is always bounded above by nc² (see the proof of
Lemma 6 in Rivasplata et al. (2018) so we recover ( from (ii), while our bound can take advantage of benign distributions. In fortunate cases (AA) can be arbitrarily close to zero, while the ne² in 7 is consequence of the use of McDiarmid's inequality in the proof, and 7 remains worst case bound.
The bound (iv) can be inverted as Corollary 3.7 to give faster rates for small empirical errors, but without the logarithmic dependence
in the inverted version of (ii).
4.3 PAC-Bayes bounds with data-dependent priors
We quote Theorem I (ii) in (Rivasplata et al. (020 ). For the convenience of the reader we give a proof in the appendix (Section B.2
Theorem 4.2. Let F H x x + R be measurable. With probability at least - 8 in the draw of
X pd we have for all P (H)
Ex-P [F (b, X)] < KL (P. Px) + In Ex Bn-ex e F(h.X)
]]
+ In (1/0).
By substitution of our bounds on In Ex FRLOx [() we obtain raw forms PAC-Bayesian bounds with prior Qx for all the Hamiltonian algorithms considered above. But since the final

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form often involves optimizations, some care is needed. In the simplest case let F (h,
(n/2) kl [ (h, X), x) (h)). substitute (i) of Theorem 4.1 above and divide by n/2, to prove the following.
Theorem 4.3. Under the conditions Theorem 4.1 we have with probability at least 1 - 5 in X 1t for all P E that
(K,(P,Qx)) + In (2/n) + (1/0)
EAWP hdl ((l,x),,(N)
It applies to the case, when the prior is an isotropic Gaussian, centered on the output of the algorithm A. a method related the methods in Dziugaaee and Roy [2018] and Pérez-Ortizz [0021] Section
B.2 sketches how PAC-Bayesian bounds analogous to 4. (11) and (iv) are obtained.
5
Conclusion and future directions
The paper presented method to bound the generalization gap for randomly generated and determin- istically exccuted hypotheses.
By using Marton's coupling method for example in Paulin 2013 one can prove an analogue to Theorem 3.4 for non-iid data generated by uniformly ergodic Markov chain.
It also appears possible apply the method to iterated stochastic algorithms, where the randomization of a stable "microalgorithm" is repeated, as with stochastic gradient Langevin dynamics (SGLD).
An obvious challenge is to give bounds for the Gibbs algorithm in the limit 3 - oo, more generally, the regime 3 > n. unlikely that the methods this paper can be successfully applied to this problem without considerable modifications.
10

---

References
Alquier. User-friendly introduction to pac-bayes bounds. arXiv preprint arXiv:2110.11216 2021.
G. Aminian, Y. Bu, L. Toni, M. Rodrigues, and G. Wornell An exact characterization of the generalization error for the gibbs algorithm. Advances in Neural Information Processing Systems, 34:8106-81 -1066-8188. 18, 2021.
G. Aminian, Y. Bu, L. Toni, M. R. Rodrigues, and G. W. Wornell. cettmmation-thoreetii characteriza- tions of generalization error for the gibbs algorithm. IEEE Transactions on Information Theory, 2023.
M. Anthony and P. Bartlett Learning Neural Networks: Theoretical Foundations. Cambridge University Press, 1999.
S. Boucheron, G. Lugosi, and P. Massart. Concentration Inequalities. Oxford University Press, 2013.
O. Bousquet and A. Elisseeff. Stability and generalization. Journal Machine Learning Research, 2:499-526, 2002.
V. V. Buldygin and Y. V. Kozachenko. Sub-gaussian random variables. Ukrainian Mathematical Journal, 32:483 489, 1980.
O. Catoni. A pac-bayesian approach to adaptive classification. preprint, 840:2, 2003.
O. Catoni. Pac-bayesian supervised classification: the thermodynamics of statistical learning. arXiv preprint arXiv:0712.00288, 2007.
G. K. Dziugaite and M. Roy. Computing nonvacuous generalization bounds for deep (stochastic) neural networks with many more parameters than training data. arXiv preprint arXiv: 1703. 1008, 2017.
G. K. Dziugaite and D. M. Roy. Data-dependent sac-bayes priors via differential privacy. Advances in neural information processing systems, 31, 2018.
W. Gibbs. Elementary principles in statistical mechanics: developed with especial reference the rational foundations of thermodynamics. C. Scribner's sons, 1902.
B. Guedj. A primer pac-bayesian learning, arXiv preprint arXiv: arXiv:191..05353 2019.
I. Kuzborskij, N. Cesa-Bianchi and C. Szepesvári. Distribution-depended analysis of gibbs-erm principle. In Conference on Learning Theory, pages
PMLR, 2019.
J. Langford and M. Seeger. Bounds for averaging classifiers. School of Computer Science, Carnegie Mellon University, 2001.
G. Lever, F. Laviolette, and J. Shawe-Taylor Tighter pac-bayes bounds through distribution-dependennt priors. Theoretical Computer Science, 473:4-28, 2013.
A. Maurer, note on the pac bayesian theorem. arXiv preprint cs/041 1099, 2004.
A. Maurer. Concentration inequalities for functions of independent variables. Random Structures Algorithms, 29(2):121-138, 2006.
D. A. McAllester. Pac-bayesian model averaging. Proceedings of the twelfth annual conference on Computational learning theory, pages 164-170, 1999.
C. McDiarmid. Concentration. In Probabilistic Methods of Algorithmic Discrete Mathematics, pages 195-248, Berlin, 1998. Springer.
D. Paulin. Concentration inequalities for markov chains by marton couplings and spectral methods. Electron. J. Probab, 20<79):1-32, 2015.
M. Pérez-Ortiz, Rivasplata, Shawe-Taae Shawe-Taylor, and C. Szepesvári. Tighter risk certificates for neural networks. The Journal of Machine Learning Research, 22(1):10326-10365 2021.

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M. Raginsky, A. Rakhlin, and M. Telgarsky. Non-convex learning via stochastic gradient langevin dynamics: nonasymptotic analysis. In Conference on Learning Theory, pages 1674-1703. PMLR, 2017.
O. Rivasplata, E. bounds for stable algorithms with instance-dependent priors. Advances Neural Information Processing Systems, 31, 2018.
J. S. Shawe-Taylor S. Sun, and C. Szepesvári. Pac-bayes
O. Rivasplata, I. Kuzborskij, C. Szepesvári, and J. Shawe-Taylor. Pac-bayes analysis beyond the usual bounds. Advances in Neural Information Processing Systems, 33: 33:11633-16845 2020.
M. Seeger. Pac-bayesian generalisation error bounds for gaussian process classification. Journal of machine learning research, 3(0ct):233-269, 2002.
I. O. Tolstikhin and Y. Seldin. poc-bayes-emmpiiial--eeeaes inequality. Advances in Neural Information Processing Systems, 26, 2013.
12

---

A Remaining proofs of Section 3
A.1 Markov's inequality
We use the following consequence of Markov's inequality.
Lemma A.l. For any real random variable Y and 8 0 we have
Pr (r > 1 [e]] In (1/6)} 5 Proof. From Markov's inequality Pr Pr(et fe" > [e 18} S
Take logarithms.
A.2 Proof of Proposition 3.2
Lemma A.2. (i) Let y (t) (e' - 1)/ if 0. Then the function up is increasing, and if the
random variable X satisfies E [X] = 0 and X - bforr b > 0, then
E [ < (
(ii) v (0)51/(22))) < 1<
Proof. Part (i) is Lemma 2.8 in McDiarmid 1998. (ii) follows from a term by term comparison of the power series
1
50 (l)
and
2)
2 t
2
k=0
k=
Proposition A.3 (Restatement of Proposition 3.2 Let X, X1, Xn be iid random variables with values in X, X = (X1,.., (i.,,,.,,,, Xn) and f X measurable.
(i) If s is such that for all E [n). x have Ex ((x) [[(x)_[[((x]]]] =nn
< e²", then
(ii) If D* D fx) < cor all ls E [n], v. E X and x E X", then
S 10
(iii) there is b (0, such that for all [m] and x E X2 we have o (x) Exxin [f [ s
then with v2 suprex" Ex- (J (S$ x) Ex [f (Sk
Ex (() [x(x)) < exp
1
2 -
Proof. (i) For S C [n]] we write Es [1| = EE [1 so Iss [.] integration over all variables in S. By independence (Es [1] S C [n]] is set of commuting projections and Es, [Es, [.]] = Es,us, [-]. Eij expectation in variables up to Xx. and Exx, is expectation only in Xx. The assumption therefore reads
[e( (xx)) B()l)X] Se"
E(s)
Using E [BN If (X)]]
1pp-11 Eug If (X)]]. We have the telescopic expansion
Σ E E [X) - E(s) [f (X)]
Bi-11 [f Eug If (X)]]
13

---

We claim that for all m. 0 m I n
B -
I mnr exp
E
Σ
B
[f (X) EE) [X)]]
k==+11
from which the proposition follows with m n. Because of above telescopic expansion the claim is true for m = 0, and we assume it to hold for m 1. Then
((x) [(xx)](((x']]]
12 1 (X) - -ess 1/ [XXX]]
<elm (m- E
Σ
exp
Bi--1| Bay D (X3]] )
=m
exp
E [m-
-
[m) [f (X) +
Σ
Amm++11
because the later terms depend only on the variables xm+1, Xn. By Jensen's inequality the last expression is bounded by
Eik- [f() Bay If (x01) ]
1)
exp
f (X) - (m) [f (X)]+
Σ
kmm+1
Bi [X( - B(a) if [(X0I)
= (m E
(( exp
k=m+1
E [f(x) Eay I/ (x)))]
(m
[mm)
[
exp
Σ
Akm++1
again because the later terms do depend Xms and by assumption the last expression is bounded
by
ER-11 (X( Eso) [s (x(xI)
e E
Σ
exp
1==++1
which completes the induction and the proof of (i).
(ii) follows from (i) and Hoeffding's 's lemma (Lemma 2.2 in Boucheron al. 2013) which says that
Expp (xx
ff (Stxx) as function of y has range set of diameter
(iii) Follows from (i) and Lemma A.2 since
-xx [[(+
Ex-
A.3 Proof of Corollary 3.7
Lemma A.4. Let L, A and assume that
L LS
Then L i. + 2V LAA + 5A
Proof.
L < L+2/WA + --
+ (VL- - VA) SL+2A = VI / L+ VA
- L5 V V VA <i+2 LA + (3+ v2) A.
The lemma follows from 3 + 5.
14

---

To get Corollary 3.7 apply this to Theorem 3.5
A.4
Subgaussian hypotheses and proof of Theorem 3.8
Lemma A.5. (From Buldygin and Kozachenko 1980 (i) if is o-subgaussian then
(Y (Y- S 0². (ii) if Y1 and Y2 are 01 and a. subgassusin respectively, the Y1 + Y2 is
01 + dz-Subgaussian.
The next lemma shows that the log-partition function In Z (X) is exponentially concentrated, when- ever the Hamiltonian H (h, X) is subgaussian uniformly in h.
Lemma A.6. Let and H () be o-subgaussian for every E H and
Z (x)
/
H((hx)) du (h).
H
(i) Then E (el- 2(x)+ (x
[|] s p2o0?
(ūi) If X" + R is p-subgaussian then
In E [118S/0-1//XX))-2222222222XX1111 s (0 ()+
Since the inequalities are given only for 2 they do not quite imply that In Z itself is subgaussian.
Proof. We only need to prove (ii), which implies (i) by setting S = O. By Jensen's 's inequality
Ex [=((xx)-1(((x)====xxxx++=00 I Exx [er ( 2(x)
Ex [enS(X) 2 (x)
(x)) -P)
Ex (e- PE Z (x)p)
Define probability measure on H by v (A) = Z e ee(u( (h) for A S H measurable. with Z, Jn (l (h). Then
(Ennu [
zB S Enroe [e(E[[((xx)]]HH00__) zz.
z ( P
by Jensen's inequality, since + 11 is convex. Similarly
Z (X)² EnaE [6](+10xx)-e((((xx))) z
Thus the above inequality can be written
2()) x(x))) Ex [e(((())
Ex En
x Ex Ennn [
The first factor can be bounded by
(EE(((X)) P(00-0)
Ex
by the subgaussian assumptions for f and and Lemma A.5 (ii), and similarly the second factor is bounded by
-
Putting the two bounds together completes the proof.
15

---

Theorem A.7 (Restatement of Theorem 3.8). Let Q have Hamiltonian H and assume that 4h E H. there is p (h) such that
VEER, (eMX)(((Mx)))((((((]]] Se
x²
Let p SuPhEH P (h) and suppose that VEE [n] h 7
[( S 2//
E
E
(i) Then for any h E H. A
In Ex~ [e^a] S Sra (M) = (A) //n + 2 and with probubility at least
2
8 we have as X μºt and h ~ Qx that
21 (1/0)
A (h, S 20 +
(ii) With probability at least we have as X ~ 4tth and h ~ Qx that
(V320 +
4ln (1/6)
A Ah, X) 5(())
11
Proof. Let E H be any fixed hypothesis. By assumption and Proposition 3.2 (i) H (h, X) is Vioo-subgaussian and XA (h, X) is Ap (h) /isuuugausss....
Using the previous lemma (ii) with p = 1 and f (X) = AA () + H (h, X) [ (h, X')], which is centered and Ap (h) //vn + Vno-subgauussin, we get that
u (h) Ix
In 100000
2
With p (g) S p we get from Proposition 3.1 that with probability least 1
(Ap/vm+ + 2 In (1/5)
A (h,X)
21
X
Xp²
2n
+ In (1/6)
2n +
+ 2po
The optimal choice of and subadditivity of V give
2 In (1/0)
A X) S p 20 +
71
(ii) We proceed as in the proof of Theorem 3.5 For HQ (h, X) H X) In Z (X) the previous
lemma yields with f (X) (g. X)=
E H (g. ) and p 2 that
Ex E((
< 8nd2
Also
Ex 2
S 2022)/01)//1
16

---

Now define
(h) \ ( (1))
(8no² + In (1/0))
(h))
and P (h, X) = X(h) (N, X) - X (h)² p (h)² /n. Then with Cauchy-Schward
IE 11 (h.xx)
VF (h)
o(N /(n))///>>>/////////////// /Bxx ( In Ex (22AS(hx))) 1/2 -2(N)
5 Sno²
Proposition 3.1 substitution of A (h) and subadditivity of f - vt then give
( (M)
8no² + In (1/6)
A X)
(h)
(h))
(32no² + +44 4111 (1/0))
n
4ln (1/6) v320 +
((h)
n
B Remaining proofs for Section
B.1 Proof of Theorem 4.1
We need the following Lemma.
Lemma B.1. Let w, E Rd and [L, 00) then
Ez-N(w,a21) e -|ll----
(222 ) w|]
Proof. We can absorb 22r in the definition of the norm. Then by translation
BenN(w.1) [e A111=== -|1 [|====1])))
Ez-N(o.1) (e M(1xx (1-)--1-=))
e wIl Em-Nio,n [22(x
Rotating u w to ||0 - w|l e1, 01, where ej the first basis vector, and using independence of the
components gives
E2-Nion) [22Mor
Ee-N(O,)) [22llo-
2 w 12/2 12 dt
V2T
00
dt 2
22T
Combination with the previous identity and taking 2or back out of the norm completes the proof.
Our proof of Theorem 4. uses the following exponential concentration inequality, implicit in the proof of Theorem 13 in (Maurer (2066) and in the proof of Theorem 6.19 in (Boucheron al. 2013
Theorem B.2. Let x + R and define operator D by
(Stx)) )²
(D2f) (x) Σ
f fxx)
inf
VEX
17

---

for some a > 0 and all x E X", D2 (x) I of (x), then for X E (0,2/a)
In [1(1113)-0((( aa [f (X)
2XE If (X)
equivalently In (x(x))
2 - as
- at
Theorem B.3 (Restatement of Theorem 4.1 Let H = R² with Lebesgue measure т. Suppose Q has Hamiltonian H (h, X) - |// - A (X)
/2a², where A has stability coefficient CA. Let 5 > O
and assume that 12nd I o² and that every h E H (as a function on X) has range in [0, 1). Denote
the variance of A by V (A) E [IA E [A (X')II
Then
(i) If 8 then In Ex
VA) + ln
(ii) n > then with probability at least 1 - 8 as X pM
kl (h, (I, S 0² V In 2lm (1/8)
Ex
(iii) With probability at least 8 as X pº and Qx
A Ah, (3/02) V (A) + In (1/0)
(iv) Let v (h) be the variance of h, defined as in Theorem 3.5
Then with probability at least 5 as
X nddh and Qx
Ah, 2 (1) ( + (1/0) (3/o²)
n
Proof of Theorem 4.7 All Gaussians with covariance a²I have the same normalizing factors, so the partition function for (h, x) - ||| A (x)]l² 2 (2a²) is also the normalizing factor of N (A (Xx)) whence, using Cauchy-Schwarz,
In Ex En-ex [(
1/r ((N.X))H2(h dx (h)
In Ex
=InEx (BLNNE(AA)))
(l..))
20
A0011
HACO!!I 2
S sup In Ex 2F(AxX) + In Ex hew
11h-31400111 2)
20
20
C.
The bound on C depends on the respective choice of and will be treated below. Using Lemma B. the second term is equal to
B In Ex e (2
2
To apply Theorem
B.2 to f (x)
| (x) - E[A(X)]] we fix x E X" and k E [n], and let y E X
be a minimizer of A [ [A (X [x) Then
(((x) inf ($tx)) DEX
(IA - EIA - |A (S EIA(X)113)
=(A(x)= - A (S A()) - -E(A + A (5*x) - EE(A (x)))
< <||A(x) - A (8xx) (x) (E || (5tx) - (x)I()) < <c
(x).
18

---

Summing over i we get D2 (x) S f (x). Since 12nc /02 1 < 2 we can apply Theorem B.2 Ancy and À 3/ to obtain
(3/0²) R | (X) -I(A())
B =
2 EE
-
2 - 12nc3/0" /02
3
3
x σz 1 [AA
- -
- ( (A).
(i) Let F Xh, (n/2)kl kl (L ( (h)). Then using
(Maurer
[2004]
C
sup, In Ex [22F(hxx] 12 (1/2) (2/1), so from the above
In Ex BA-Ox ( [(N2a[((()))))) S 0²
3 (A)+ +
1 2 (2V7),
which is (i). By Jensen's 's inequality
In Ex (n/2)/aaaxx (1(BE),))))(( S In Ex Ex-Qx [c00)))((((,:,))0000))]
so part (ii) then follows from Markov's inequality and division by n/2.
(iii) Let F (h. X) YA (h, X) for X > 0. Using Proposition 3.2 (ii) we get for all h H that
C' (1/2) [ X2/ (4m). so
X
In Ex ER-Ox
e (A((M))
v (A).
(8)
4n σ²
Markov's inequality gives with probability at least
as X ~ μm and h Qx that
3 V
(A) In (1/8)
A (h,
X
+
4n
x
Optimization of À gives (iii).
(iv) We proceed as in the proof of Theorem
3.5
Let
3. (A) + In (1/8)
(h)
(1/n) (A) In (1/0) + ((h)
and set
A (h)² (h)
FA (h, X) = =1(h) (1,
X (h) /n n
By Lemma 3.6 2C = In Ex (x.x))) $ s0 so
In Ex Ennex [ < o2 (A),
3
and with probability at least 1 1- as as ~ nm and h ~ Qx
(h) v vh)
+ oz (A) + In (1/5)
3
Pr A
(h. X) I- X(h) /n
< 8. A (h)
Inserting the definition of X (h) completes the proof.
B.2 PAC-Bayes bounds with data-dependent priors
Theorem B.4 (Restatement of Theorem 4.2 Let F H x X + R be measurable. With probability
at least 8 in the draw of X ~ pm we have for all P E P (H)
ExpP [F (h, X)] < KL
Ex
BnNex
e [( + (1/8).
19

---

Proof. Let P E P1 (HL be arbitrary. Then
BnaP [F X)) (I, Qx)
In exp (BAup [F (h, X) In (dP/dQx (k))))
In BROP lexpl (F (h,X) X) In (dP/dQx (h)))
In BNNP e( (dP/dQx (11))
(((
In @x
So only need to bound the last expression, which is independent of P. But by Markov's inequality (Lemma A.I with probability at least in ~ 1r"
(e (()))))
s In Fx exp IIn Banex (er())))) +1n (1/6)
In Rx
In Ex (eF(hx)] + In (1/6).
We close with method to deal with the problem of parameter optimization in the derivation of PAC- Bayesian bounds from our results. We apply it to the case of Gaussian priors centered on the output of stable algorithms, but analogous results can be equally derived for Hamiltonians with bounded differences or sub- gaussian Hamiltonians. Our first result a PAC-Bayesian bound analogous to part (iii) of Theorem 4.1
Theorem B.5. Under the conditions of Theorem 4.1 we have with probability at least 1 - ô in n² for all P E (H) that
3 ( 2KL (K, (I
EpoP [A (h, (),
To prove this we first establish the following intermediate result. Proposition B.6. Under the conditions of Theorem 4.1 let K >> 0 and 6 0. Then with probability at least 5 as ~ nt¹ we have for any E P (H) with KL (P, Qx) v that
(A) + + In (1/8)
P |A (h, (h,
Proof. If KL S K we get from Theorem 4.2 and inequality ( with probability least 1- 8 n
EpP [A (), K
s
Epor
DA (h, X) KL
< In Ex Ennex
[2
+ In (1/8)
X2
3 v (A) + In (1/8).
+
4n
Bring K the other side, divide by and and optimize À to complete the proof.
To get rid of K model-selection lemma from Anthony and Bartlett 1999 Lemma B.7. (Lemma 15.6 in Anthony and Bartlett 1999)) Suppose Pr is probability distribution and
{E (a1,, 022 0) aa, $1}
is a set of events, such that (i) For all < and < 1
Pr (a, 8.
20

---

(ii) For all 0 < a1 S aas S 02 S 1 and < <01
02, 01) _ E (a,
Then for 0 < a, 1,
Pr
U oE(0,1]
E (aa, a, da (1 a)) S
Proof. Define the events
3/4 (A) + +aj -1 + In (1/6)
E (01,002 8) = 3P, (P. S az EEEP (Ah,
V
n
By Proposition they satisfy (i) of Lemma B.7
and is easy to see, that (ii) also holds. we set
1/2 the conclusion of Lemma B.7 becomes
ExxP [A (h, X)) (A) 2KL (P, Qx) In (1
To get a PAC-Bayesian bound analogous to Theorem 4.1 (iv) we proceed similarly and obtain the intermediate bound that for > 0 with probability at least 1 å in X ~ n² for all P such that EpPP (h)) s V and KL (P, Qx) < K
EpOP A (h, X S v 2+ (() K In (1/6) ) (o
K +K (n/00)) of (A)
Then proceed above, except that we have to use Lemma B.7 twice, once with K and once with nV. The result of this mechanical procedure
Theorem B.8. For > with probability at least 5 in X ~ 1² for all
3 ( LV(A)+C
Ex-P [A (h, s 22 (2EmrP (c))) + 1/n)
where
= 2KL +1+ (2 + (2 (A) + 1)) /6).
21

---

C
Table of notation
space of data
1
probability data
71
sample size
X
generic member (II, Ln) E X training set X (X1, .... Xn) ~ 4t
H
loss class (loss fetn. composed with hypotheses, h X 0,)0)
P(H)
probability measures on H.
π
nonnegative a- priooi measure on H.
(hh)
Lhh) Enzy ( [.) expected loss of h EHH
X))
(h, X) (1 (1/n) Σ h (Xx), empirical loss of h H
(),X)
Lhh) (h, X), generalization gap
Q
Q:X X + Qx E P (H), stochastic algorithm
Qx (h)
density w.r.t of Qx evaluated at h E H. Qx (h) exp (Hq (h, x))
H:H X"
Hamiltonian
Z
7:X" - R, 7 (x)
exp H (h, x)) dm (h), partition function
HQ
xh, H
In Z (x) In Qx (h)
S
S (21, .... F2 2k.1 ...., In). substitution operator
D.V
(Di, D J) (x)
( (S x), partial difference operator
kl (p.g)
(p, pln 2 (1 p) In 2 re, entropy of Bernoulli variables
q
(I,
[ de dp, Ki-divergence of p.-measures and
11-11
Euclidean norm on RD.
22

---

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Answer: [NA]
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should be clear whether the error bar is the standard deviation the standard error of the mean.
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It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of enrors is not verified.
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The conference expects that many papers will be foundational research and not tied particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
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If this information is not available online, the authors are encouraged to reach out to the asset's 's creators.
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Reviews:
[{"review_id": "1LOObQYJWv", "reviewer": "Reviewer_mcjd", "summary": "This paper introduces a general method to bound the logarithm of the expecattion of the exponential of the generalization gap for stochastic learning algorithms. This method is applicable when the distribution of the algorithm concentrates exponentially around its mean, extending to cases where the Hamiltonian form satisfies a bounded difference condition or is sub-Gaussian. Further, the paper discuss applications to the gibbs algorithm, improving existing bounds, Lastly the paper extends generalization guarantees to hypotheses sampled once from stochastic kernels centered at the output of uniformly stable algorithms. This advancement strengthens previous results and improves the understanding of generalization properties for algorithms based on stochastic kernels.", "questions": "In addition to the weaknesses: \n\nIn context of the paper, and the definition of \\Delta(h,X), which we know is centered, I am not sure how Prop 3.1 is useful or applied in this paper. I understand it could be potentially useful where F and H_Q can be handled separately (assuming F being non-centered is still somehow amenable to this). But for this paper, this is not the case. \n\nThe notation S^k_X f(\\mathbf{x}) used in prop 3.2 onwards is clunky at best, and badly overloaded at worst. May I suggest using f(S^k_X \\mathbf{x}) ? \n\ntheorem 3.4 needs h(y’) \\in [0,b] ? I think you mean \\forall t \\in \\mathcal{X} h(t) \\in [0,b] ?\n\n“In practice Q is often defined by specifying some Hamiltonian H , so HQ (h, x) = H (h, x) − ln Z (x) in general” – I am not sure what this sentence means. Please expand. Are you saying the canonical H_Q can be trivially written like that? How is Q being defined through that? \nL186 “generalization guarantee of rapid convergence” .. could you expand on this please? What order is considered as “rapid”?\n\nMinor: In a couple of places some of the references seem to be hard-coded ? e.g. references to A.1 because those are not clickable links.", "limitations": "Partially.", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "The paper is very-well written. it is crisp and clear about the notations, contributions and the context of the novel results with respect to the exsiting literature. The theoretical rigor is well-presented and the results are very well supported through coherent arguments. I specially appreciate the \"self-contained\" nature of the paper, where everything including the very fundamental markov inequality is written in the appendix for convenience of the reader so there is no need to refer to other papers to understand, verify and appreciate the contributions of the paper.\n\nThe paper makes non-trivial theoretical contributions. It provides generalization guarantees for bounded hypothesis (section 3.1) and unbounded hypothesis through subgaussianity (Section 3.2). The contributions include a general method for bounding generalization gaps for the specific case of Hamiltonian algorithms, thereby identifying an important subclass of problems to bound the notorious ln E_X E_h [exp \\lambda \\Delta] which is often not amenable for general problems. This contribution is important. Further, the authors specify precisely where the bounds improve upon exisiting results (e.g. dependence on the confidence parameter \\delta is improved from previously known O(\\sqrt(1/\\delta) to O(1/\\delta) for the gibbs algorithm), often with simpler proof techniques and cleaner constants/exponents.", "weaknesses": "The paper is very notation-heavy, and likely not accessible to everyone. This is not really a critique of the paper tbh, but more of the nature of the result/paper and the short conference format. The authors have done a fair job of explaining the notations. But It is easy to get lost in keeping track. For example, the notion of canonical Hamiltonian H_Q and its usefulness (for example in the simple proof of Prop 3.1) is easy to overlook unless one really gets into the weeds. I would suggest adding a couple of sentences cementing the importance of H_Q in the analysis. Beyond helping through and simplifying some results, is there additional importance of H_Q ?\n\nThe conclusion and future directions/discussions/limitations section is severely lacking. Can the authors talk about limitations of the proposed method, and any further possible open problems that the reader might be interested in. \n\nThere is no empirical verification, even with toy setups. Can the authors comment on how the presented theory can be empirically demonstrated/verified?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Please also see the general rebuttal for planned improvements to the paper.\n\nExperiments to demonstrate the theoretical results are planned for future\nwork. For real data the costly part is the repeated\nsampling from the Gibbs distribution, because one has to\nawait the mixing time between Monte-Carlo-samples for approximate independence. Recording for each sample\nthe training error and a test error for independent data provides the\ninformation to test the predictions on both the expected and un-expected\ngeneralization gaps, and their dependence on the training error. Note that\nthe benefit of the un-expected bounds, once one is willing to accept their\ncorrectness, is that only one sample is needed in the test.\n\nQuestion 1: A separate treatment of $F$ and $H_{Q}$ is used in the proofs of\nthe Bernstein-type inequality Theorem 3.5 and Theorem 3.8 (ii). Proposition\n3.1 is essential in both of these cases.\n\nQuestion 2: The notation $S_{X}^{k}f\\left( \\mathbf{x}\\right) $ has entered\nthe paper through some copy/paste process. It is essentially a typo for\nwhich we apologize. As you say, $f\\left( S_{X}^{k}\\mathbf{x}\\right) $ should\nbe used.\n\nQuestion 3: In the statement of Theorem 3.4 the universal quantifier was\nunderstood to apply to \\textit{all} of $k,h,y,y^{\\prime }$ and $\\mathbf{x}$.\nWhile this still seems correct, it is probably clearer to state the uniform\nboundedness of the $h\\in \\mathcal{H}$ separately, as you propose.\n\nQuestion 4: \"In practice...\". An example is the Gibbs-algorithm. One\nspecifies $H\\left( h,\\mathbf{x}\\right) =-\\left( \\beta /n\\right)\n\\sum_{i}h\\left( x_{i}\\right) $. Then the density is $Q_{\\mathbf{x}}\\left(\nh\\right) =Z\\left( \\mathbf{x}\\right) ^{-1}\\exp \\left( H\\left( h,\\mathbf{x}%\n\\right) \\right) =\\exp \\left( H\\left( h,\\mathbf{x}\\right) -\\ln Z\\left( \n\\mathbf{x}\\right) \\right) =\\exp \\left( H_{Q}\\left( \\mathbf{x}\\right) \\right) \n$.\n\nQuestion 5: \"rapid convergence\". What was meant is that then $\\Delta \\left( h,%\n\\mathbf{X}\\right) $ is approximately of order $1/n$, rather than $\\sqrt{1/n}$\nas in Theorem 3.4. But, as this also depends on the order which $c$ has in $%\nn $, it really depends on the concrete application of Corollary 3.7. Since\n\"convergence\" may also be confused with the convergence of a\nMonte-Carlo-method to its limiting distribution, the text will be modified.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "5Eo27DzFN3", "reviewer": "Reviewer_16mk", "summary": "This submission is a purely theoretical work, whose main goal is to bound $\\Delta(h,\\mathbf{X})=\\mathbb{E}[h(X)]-\\frac{1}{n}\\sum^n_{i=1}h(X_i)$, i.e. equation (1). After some preliminary results, the results that show this bound under different conditions are Theorem 3.4, Theorem 3.5 and Theorem 3.8. Some possible (theoretical) applications are also shown, in the form of the Gibbs algorithm, randomisation of stable algorithms and PAC-Bayes bounds with data-dependent priors.", "questions": "Displayed equation after L89: This seems to be a pointwise statement for both $h$ and $\\mathbf{x}$, so should it be “for all $h\\in\\mathcal{H}$ and for all $\\mathbf{x}\\in\\mathcal{X}^n$? Unless the statement should be “… is called a Hamiltonian for $Q_\\mathbf{x}$” instead of “Hamiltonian for $Q$”?\n\nDisplayed equation after L162: Shouldn’t $2bc$ be $\\frac{bc}{2}$, taking into account the factor of $\\frac{1}{8}$? The $2nc^2$ in the numerator of the fraction should also be $\\frac{nc^2}{2}$. It looks like this is corrected on L163, and working through the maths, I indeed got the claimed inequality in Theorem 3.4(ii). \n\nL183: I think you are missing a $\\ln$ in front of the exponential?", "limitations": "L88: \"A function $H:\\mathcal{H}\\times\\mathcal{X}^n$\" perhaps it is better to write “A function $H$ on $\\mathcal{H}\\times\\mathcal{X}^n$? I leave this up to the authors. \n\nDisplayed equation after L153: The spacing is a bit strange on the left?\n\nL185: an -> and\n\nDisplayed equation after L195: The comma should be a full stop. \n\nL178, L389: “Assume that all” -> “Assume that for all”\n\nL391: everz -> every", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "Unfortunately, I have not worked in any of the subfields that this paper is concerned with, so my judgment on its importance should be taken with a pinch of salt, but I am convinced of its significance and the contributions that this paper makes. \n\nI have also tried to go through some of the maths in detail to check for its soundness, and barring a couple of very minor errors (see \"Questions\"), I think this paper is excellent in that regard. \n\nI really liked the way this paper was written, straight to the point with minimum fuss, and my impression is overwhelmingly positive.", "weaknesses": "I think the authors made a deliberate choice to be concise with the proofs so that the proof can be included in the main body of the paper. I really like this, but in some places, the proof is too short, so that the readers are asked to do a significant amount of algebra by themselves. If possible, it would be great if the authors could use the extra page to flesh out the proofs a little bit.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Special thanks to you for the encouraging words and the careful reading,\nwhich uncovered several typos and inaccuracies.\n\nPlease also see the general rebuttal for planned improvements to the paper.\n\nQuestion 1: L89. Yes, it should be \"for all $h\\in \\mathcal{H}$ and $\\mathbf{x%\n}\\in \\mathcal{X}^{n}$\".\n\nQuestion 2: Display after L162. You are absolutely right, thank you. No idea\nhow this got into the paper.\n\nQuestion 3: L183. You are right again, thank you.\n\nL88: Will be changed. The other typos will be corrected accordingly.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "9RuTph8iXG", "reviewer": "Reviewer_hAd4", "summary": "Summary\n-------\n\nThe papers presents a method to bound the `generalization gap', i.e the difference between\nthe expected and empirical losses for a hypothesis h drawn from a probability class \nreturned by the stochastic learning algorithm. The authors present a general-purpose \nmethod to bound the exponentiated difference, and apply this technique to Gibbs sampling \n(and other applications). \n\nAs someone who is not an expert on the topic, but still theoretically inclined, I was\nunable to appreciate the contributions of this paper. It didn't help that the paper's\npresentation was not up to scratch either (see comments below). I have given a\nmiddle-of-the-line score to reflect this but will wait to hear from expert reviewers.\n\n\n\nDetailed comments\n=================\n\nQuestions \n- Is h a hypothesis or a loss function? In 9, 10 and equation (1), it is written as a loss\n  but in line 12 it is treated as a hypothesis?\n- What is the motivation for bounding the difference between the expected loss of a \n  a *learned* hypothesis, and the error on the training set? In typical use cases, one is \n  interested in the test/generalization error after training. \n- From what I could gather from the Gibbs sampling use case in 4.1, this can be used to \n  bound the rate of convergence of Gibbs sampling. Is this correct? \n\n\nThe presentation of the paper could improve significantly.\n- It is customary (at least in CS venues), to include an outline of the results, describe \n  their significance of each result, and outline the key challenges and\n  techniques used in the paper.\n- The paper would also benefit from a dedicated related work section, where the authors \n  explicitly state the most relevant work and how their results compare to them. The most\n  interesting comparison I saw was in section 4.1 where they compare their bound to \n  Rivasplata et al, and the improvement in the \\sqrt{\\log(1/\\delta)} term. Even here, the \n  authors have not explained why this gap \\Delta is significant. Does it lead to fast\n  convergence rates for Gibbs sampling?\n  Perhaps the authors could state this result up front to prime the reader for what they\n  can expect from this paper.\n- On the same note, the authors should do a better job of highlihgting the key results in \n  this paper. Currently, there are several lemmas/theorems presented one after the other \n  and it is hard to appreciate which ones are the most significant, and why they are \n  significant. My recommendation would be to identify the ~2 main theorems of the paper", "questions": "See above", "limitations": "See above", "rating": 6, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "See above", "weaknesses": "See above", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Please also see the general rebuttal for planned improvements to the paper.\n\nQ1: $\\mathcal{H}$ is a loss-class and $h\\in \\mathcal{H}$ is a hypothesis\ncomposed with a fixed loss function. Loss-classes are a notational\nsimplification frequently used in the analysis of generalization.\n\nQ2: The bounds apply to the difference of expected and training error after\ntraining.\n\nQ3: The bounds apply to hypotheses sampled from the Gibbs distribution and\ngive convergence in the sample size $n$. They do not refer to the\nconvergence of Monte Carlo methods to the Gibbs distribution.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "EjbJHVsdDY", "reviewer": "Reviewer_Dxrq", "summary": "This work provides a novel approach to bound the generalization error (high probability bounds) of the Gibbs algorithm as an important example of Hamiltonian algorithms. The authors also applied their method to a different example of the Hamiltonian algorithm.", "questions": "Sea weaknesses.", "limitations": "Sea weaknesses.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Strengths:\n\n1- A novel approach is proposed for bounding the generalization error of the Gibbs algorithm.\n\n2- There are some improvements in comparison with other works.\n\n3- The proof approach is well-discussed.\n\n4- Section about Randomization of stable algorithms is interesting.", "weaknesses": "1- In the case of the Gibbs algorithm, the contribution of this work is incremental. In particular, the authors should elaborate on why their results for the Gibbs algorithm are novel.\n\n2- For the Gibbs algorithm, the main challenge is studying the asymptotic regime, where $\\beta$ ---> \\infty. It would be useful to study this regime or mention it as limitation of this analysis.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Please also see the general rebuttal for planned improvements to the paper.\n\n1 - The improvements for the Gibbs algorithm go beyond constants. (4) is\nsmaller than the inequality in the next display by a factor of $1/\\sqrt{\\ln\n\\left( 1/\\delta \\right) }$. If the confidence parameter $\\delta $ goes to\nzero, the quotient of the next display to (4) goes to infinity. Similarly\n(5) is better than the competing bounds by a factor of $1/\\ln \\sqrt{n}$. And\neven if logarithmic factors are considered irrelevant: does a simpler and more\ngeneral method of proof, which in some special case gives only incrementally improved results, not provide\na clearer perspective on the underlying phenomenon?\n\n2 - It is planned to use an extra page for a future-work/limitations\nsection. There will be reference to the challenge of the $\\beta \\rightarrow\n\\infty $ limit, and the $\\beta >n$ regime. These problems seem to require\ntechniques different from those developed in the paper.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
Many thanks to the reviewers, who provided many useful comments. Major
planned improvements are:

1. The revision will contain a section on future directions and
limitations. It will mention potential applications to iterated algorithms
and weakly dependent data. The limitations part will point to the remaining
challenges of the $\beta >n$ regime and the limit $\beta \rightarrow \infty $.
.

2. A glossary of notation in tabular form will be included at the
beginning of the appendix.

Meta-review: This paper introduces a method for bounding the generalization gap of stochastic learning algorithms by focusing on the exponential concentration of the algorithm's distribution around its mean. The approach is applicable to various scenarios, including Gibbs sampling and randomizations of stable algorithms, enhancing the understanding and existing bounds on generalization properties. The reviewers unanimously praised the paper for its clarity, theoretical rigor, and mathematical correctness, with concerns adequately addressed in the rebuttal. Given its strong theoretical contributions, I recommend acceptance, in agreement with all four reviewers.
Decision: Accept (poster)