Logarithmic-Free Moment and Generalization Bounds for Uniformly Stable Algorithms

Logarithmic-Free Moment and Generalization Bounds for Uniformly Stable Algorithms Thanh Nguyen-Cung∗

Binh T. Nguyen∗

arXiv:2608.09870v1 [stat.ML] 10 Aug 2026

Abstract Uniform stability is a classical tool for controlling the generalization error of a learning algorithm. Bousquet, Klochkov, and Zhivotovskiy [2] showed that the problem can be reduced to a moment inequality for a sum of weakly interacting functions of independent random variables. Their bound contains an additional factor log n, and they asked whether this factor can be removed. We answer this upper-bound question affirmatively. More specifically, let Z = (Z1 , . . . , Zn ) have independent coordinates and let gi (Z) satisfy E[gi (Z) | Z−i ] = 0,

|E[gi (Z) | Zi ]| ≤ M,

∀i = 1, n

while changing any coordinate Zj , j = ̸ i, changes gi by at most β and Z−i denotes all coordinates except Zi . We prove that, for every p ≥ 2, n X i=1

≤ 16pnβ + M

gi (Z)

p

2pn.

p

This removes the log n factor from the previous bound and matches the lower bound of Bousquet, Klochkov, and Zhivotovskiy up to universal constants in the range covered by their construction. Our proof first establishes the required estimate on the Rademacher cube, then transfers it to arbitrary product distributions by a two-copy randomization argument.

1

Introduction

Let S = (Z1 , . . . , Zn ) be an i.i.d. training sample from a distribution P , and let AS be the predictor returned by a learning algorithm A. For a loss function ℓ, define the population risk and empirical risk by n 1X R(AS ) = EZ∼P ℓ(AS , Z), Remp (AS ) = ℓ(AS , Zi ). n i=1 A basic problem in learning theory is to control the generalization error R(AS ) − Remp (AS ). The empirical risk is computed from the training sample, while the population risk depends on the unknown distribution. Generalization bounds quantify when the former is a reliable estimate of the latter. ∗

VinUniversity. Email: 24thanh.nc@vinuni.edu.vn, binh.nt2@vinuni.edu.vn

1

A classical approach is based on stability. Following Bousquet and Elisseeff [1], an algorithm is γ-uniformly stable if replacing one observation in the training sample changes the loss at every test point by at most γ. More precisely, if S and S ′ differ in one coordinate, then sup |ℓ(AS , z) − ℓ(AS ′ , z)| ≤ γ. z

Thus a stable algorithm does not depend too strongly on any single training example, and this weak sensitivity can be used to control its generalization error. Assume that the loss is bounded by L. The classical stability argument of Bousquet and Elisseeff gives, up to universal constants, a high-probability bound of the form √ L |R(AS ) − Remp (AS )| ≲ nγ + √ n 

r

1 log . δ

This is sharp in the regime γ ≲ 1/n, but it does not capture the useful regime in which γ can be as √ large as order 1/ n. Feldman and Vondrák obtained substantially sharper high-probability bounds [3, 4]. Bousquet, Klochkov, and Zhivotovskiy [2] then gave a moment-based analysis that connects the different concentration regimes and yields, up to constants, 1 |R(AS ) − Remp (AS )| ≲ γ log n log + L δ

s

log(1/δ) . n

The first term has sub-exponential dependence on the confidence parameter, while the second has the usual sub-Gaussian form. The factor log n in the first term comes from a more general probabilistic problem. Let Z = (Z1 , . . . , Zn ) have independent coordinates and consider F (Z) =

n X

gi (Z).

i=1

The function gi may depend strongly on its own coordinate Zi , but its dependence on every other coordinate is assumed to be weak. More precisely, suppose that E[gi (Z) | Z−i ] = 0,

|E[gi (Z) | Zi ]| ≤ M,

and that changing Zj , for j ̸= i, changes gi by at most β. Bousquet, Klochkov, and Zhivotovskiy proved that for every p ≥ 2, n X i=1

√ √ ≤ 12 2 pnβ⌈log2 n⌉ + 4M pn.

gi (Z)

(1)

p

This inequality is the main probabilistic input in their stability analysis. Their lower-bound construction shows that, for an absolute constant κ > 0 and κ ≤ p ≤ n, the left-hand side can be of order √ pnβ + M pn. Thus the factor log n in (1) is the only gap, up to universal constants, between their general upper and lower bounds. In Section 6 of [2], they ask whether this factor can be removed for p > 2, both from the moment inequality and from the resulting high-probability bound for uniformly stable algorithms. 2

Our result. We remove this logarithmic factor. Under the same product-space assumptions, we prove n X i=1

≤ 16pnβ + M

gi (Z)

p

p ≥ 2.

2pn,

(2)

p

Hence the dependence on p, n, β, and M agrees, up to universal constants, with the lower bound of [2] in the range covered by their construction. The learning consequence follows from the reduction of [2]. For a γ-uniformly stable algorithm with loss bounded by L, their construction gives M = L and β = 2γ, together with a deterministic comparison error. We obtain r

∥R(AS ) − Remp (AS )∥p ≤ 33pγ + L

2p . n

Taking p of order log(1/δ) gives a high-probability bound of order 1 γ log + L δ

s

log(1/δ) , n

with no additional log n factor. This answers the upper-bound question posed in [2]. It is separate from the lower-bound question of whether the full tail behavior can be realized by a uniformly stable learning algorithm under a uniform loss bound. Proof idea. The proof has two main ingredients. First, on the unbiased Rademacher cube, double centering forces every summand to have the form hi (ε) = εi qi (ε−i ). A direct exponential-moment argument then controls their sum; its main ingredient is a fixed-point counting estimate for a weakly dependent product. Second, a two-copy randomization transfers the cube estimate to arbitrary product distributions. The remaining conditional-centering defect is handled by bounded differences. Neither step introduces logarithmic dependence on n. Organization. Section 2 states the main product-space inequality and proves it in three steps: the Rademacher-cube estimate, the two-copy transference argument, and the final decomposition. Section 3 then derives the consequence for uniformly stable algorithms.

2

Main result

Let Z = (Z1 , . . . , Zn ) have independent coordinates, where Zi takes values in a measurable space Zi . We denote Z−i = (Z1 , . . . , Zi−1 , Zi+1 , . . . , Zn ). For a real-valued function f on the product space, define the oscillation in coordinate j by oscj (f ) = sup {|f (z) − f (ze)| : zk = zek for all k ̸= j} . If the assumptions below hold only almost surely, the corresponding essential suprema may be used. Theorem 1. Let g1 , . . . , gn be real-valued measurable functions of Z. Assume that, for every i ∈ [n], |E[gi (Z) | Zi ]| ≤ M

E[gi (Z) | Z−i ] = 0

a.s., 3

a.s.,

and oscj (gi ) ≤ β,

j ̸= i.

Then, for every p ≥ 2, n X

≤ 16pnβ + M

gi (Z)

i=1

p

2pn.

(3)

p

For κ ≤ p ≤ n, Proposition 9 of [2] gives examples on the Rademacher cube for which the left-hand side is bounded below, up to a universal constant, by √ pnβ + M pn. Thus, Theorem 1 has the correct dependence on all four parameters, up to universal constants, in this range. We now prove Theorem 1. Let ε = (ε1 , . . . , εm ) be uniformly distributed on {−1, 1}m . For a function u : {−1, 1}m → R, write Ei u = E[u | ε−i ], Di u = u − Ei u. For x ∈ {−1, 1}m , let x(j) denote the vector obtained by flipping its jth coordinate. Then Ej u(x) =

u(x) + u(x(j) ) , 2

Dj u(x) =

u(x) − u(x(j) ) . 2

(4)

Proposition 1. Let hi : {−1, 1}m → R, i ∈ [m], satisfy Ei hi = 0 and E[hi | εi ] = 0; and assume that for every i ̸= j, sup hi (x) − hi (x(j) ) ≤ γ. (5) x∈{−1,1}m

Then, for every m ≥ 1 and p ≥ 2, m X i=1

≤ 7mpγ.

hi

(6)

p

We first record the antisymmetric representation forced by the two centering conditions. Lemma 1. For each i ∈ [m], there is a function qi : {−1, 1}m−1 → R, independent of εi , such that hi (ε) = εi qi (ε−i ), Moreover, for every j ̸= i,

∥qi ∥∞ ≤

(7)

γ , 2

(8)

m−1 γ. 2

(9)

∥Dj qi ∥∞ ≤ and

Eqi = 0.

Proof. Fix i and ε−i . The identity Ei hi = 0 gives hi (1, ε−i ) + hi (−1, ε−i ) = 0. Define qi (ε−i ) = hi (1, ε−i ), then hi (ε) = εi qi (ε−i ). Since qi is independent of εi , 0 = E[hi | εi ] = εi Eqi , so Eqi = 0.

4

If j ̸= i, flipping coordinate j does not change εi , and therefore qi (ε) − qi (ε(j) ) = hi (ε) − hi (ε(j) ) ≤ γ. Equation (4) gives (8). For the supremum bound, fix x ∈ {−1, 1}m and let Y be an independent uniform point of the cube. Changing one coordinate at a time gives |qi (x) − qi (Y )| ≤ γ dH (x−i , Y−i ), where dH is Hamming distance. Since Eqi (Y ) = 0, |qi (x)| ≤ E |qi (x) − qi (Y )| ≤ γ EdH (x−i , Y−i ) =

m−1 γ. 2

Taking the supremum over x proves (9). The next estimate is the fixed-point counting ingredient used in the moment-generating-function argument. Lemma 2. Set c⋆ =

1 , 2e + 2

C0 = 2e + 1.

(10)

Suppose that ai : {−1, 1}m → R does not depend on coordinate i and satisfies ∥ai ∥∞ ≤ c⋆ , Then E

∥Dj ai ∥∞ ≤

m Y

c⋆ m

(j ̸= i).

(1 + εi ai (ε)) ≤ C0 .

(11)

(12)

i=1

Proof. Let U1 , . . . , Um be independent and uniform on [−1, 1]. For u = (u1 , . . . , um ), define Φu : {−1, 1}m → {−1, 1}m coordinatewise by (

1, −1,

Φu (x)i =

ui ≤ ai (x), ui > ai (x).

Let N (u) = #{x ∈ {−1, 1}m : Φu (x) = x}, Independence of the Ui gives P(Ax ) =

m Y 1 + xi ai (x) i=1

Consequently, EN = 2

−m

X

m Y

2

Ax = {ΦU (x) = x}.

.

(1 + xi ai (x)) = E

x∈{−1,1}m i=1

m Y

(1 + εi ai (ε)).

(13)

i=1

It remains to bound EN . Fix x ∈ {−1, 1}m . Let S ⊆ [m], |S| = s, and set y = xS , meaning that all coordinates in S are flipped. Conditional on Ax , the variables Ui remain independent, and each is uniform on an interval of length at least 1 − c⋆ . For i ∈ S, the signs xi and yi are opposite. For both the ith coordinate 5

conditions in Ax and Ay to hold, Ui must lie between ai (x) and ai (y). Since ai does not depend on coordinate i, the points relevant to ai (x) and ai (y) differ in only the s − 1 coordinates of S \ {i}. Therefore 2c⋆ (s − 1) |ai (x) − ai (y)| ≤ . (14) m Ignoring the additional restrictions from coordinates outside S, 

P(Ay | Ax ) ≤

2c⋆ (s − 1) (1 − c⋆ )m

s

.

(15)

For s = 1, the right-hand side is zero. Using (15), E[N (N − 1)] =

X

X

P(Ax )

x

P(Ay | Ax )

y̸=x

≤ K EN, where

!  m X m 2c⋆ (s − 1) s

K=

s=2

Using

m
s s ≤ (em/s) ,

K≤

(16)

(1 − c⋆ )m

s

  ∞  X 2ec⋆ s s=2

Put q=

1 − c⋆

1 1− s

.

s

.

2ec⋆ 2e = < 1. 1 − c⋆ 2e + 1

Since (1 − 1/s)s ≤ e−1 , K≤

∞ 1 q2 4e 1X qs = = < 2. e s=2 e1−q 2e + 1

(17)

Every integer r ≥ 0 satisfies r ≤1+

r(r − 1) . 2

Therefore

K 1 EN ≤ 1 + E[N (N − 1)] ≤ 1 + EN. 2 2 Since 1 − K/2 ≥ 1/(2e + 1), we get EN ≤ 2e + 1 = C0 . Equation (13) proves the result. Define the square function m X

H=

!1/2

h2i

m X

=

i=1

!1/2

qi2

.

(18)

i=1

Lemma 3. For m ≥ 2, EH ≤

mγ , 2

(19)

and, for every s ≥ 0, H 1 2 P ≥ + s ≤ e−2s . mγ 2 



(20)

Consequently, for every 0 < λ < 1, H2 E exp λ 2 2 m γ 6

!

≤

eλ/2 . 1−λ

(21)

Proof. The discrete Poincaré inequality on the Rademacher cube states that Var(u) ≤

m X

E |Dj u|2 .

(22)

j=1

Since Eqi = 0, Di qi = 0, and ∥Dj qi ∥∞ ≤ γ/2 for j ̸= i, Eqi2 ≤

E |Dj qi |2 ≤

X j̸=i

Hence 2 1/2

EH ≤ (EH )

!1/2

m X

=

m−1 2 γ . 4

Eqi2

p

m(m − 1) mγ γ≤ . 2 2

≤

i=1

For a flip of coordinate j, qj (ε) = qj (ε(j) ), while qi (ε) − qi (ε(j) ) ≤ γ

(i ̸= j).

The reverse triangle inequality in Euclidean space gives H(ε) − H(ε(j) ) ≤

√ m − 1 γ.

(23)

McDiarmid’s inequality therefore yields 2t2 P{H − EH ≥ t} ≤ exp − m(m − 1)γ 2 Using (19) with t = mγs gives (20). Set



Y =

1 H − mγ 2

!

.



. +

2

Then P(Y > s) ≤ e−2s and 

H mγ

2

For 0 < λ < 1, 2

Ee2λY = 1 + Therefore

1 ≤ Y + 2 

Z ∞ 0

2

≤

1 + 2Y 2 . 2

2

4λse2λs P(Y > s) ds ≤

H2 E exp λ 2 2 m γ

!

2

≤ eλ/2 Ee2λY ≤

1 . 1−λ

eλ/2 . 1−λ

Proof of Proposition 1. The case m = 1 follows immediately from E[h1 | ε1 ] = 0. If γ = 0, Lemma 1 shows that every qi is constant and mean zero, so again all hi vanish. Assume m ≥ 2 and γ > 0. We use the elementary estimate 2

eu ≤ (1 + 2u)1/2 e4u ,

7

1 |u| ≤ . 4

(24)

Indeed, for 1 log(1 + 2u) + 4u2 − u, 2

ψ(u) = we have ψ(0) = 0 and

ψ ′ (u) =

2u(3 + 8u) . 1 + 2u

Thus ψ decreases on [−1/4, 0] and increases on [0, 1/4], so ψ(u) ≥ 0 on this interval. Set m X c⋆ F = hi , t0 = . mγ i=1

(25)

By (9), c⋆ (m − 1) 1 < . 2m 4 For either t = t0 or t = −t0 , applying (24) to each thi gives |t0 hi | = |t0 qi | ≤

tF

≤

e

m Y

(1 + 2thi )

1/2

exp 4t

2

i=1

=

m X

!

h2i

i=1

m Y

!1/2

2

2

e4t H .

(1 + 2thi )

(26)

i=1

By Cauchy–Schwarz, tF

Ee

≤

E

m Y

!1/2



1/2

2

2

∥Dj ai ∥∞ ≤

c⋆ m

(1 + 2thi )

Ee8t H

.

(27)

i=1

Let ai = 2tqi . Since hi = εi qi , 1 + 2thi = 1 + εi ai . Lemma 1 gives ∥ai ∥∞ ≤ c⋆

m−1 < c⋆ , m

Lemma 2 therefore gives

m Y

(j ̸= i).

(1 + 2thi ) ≤ C0 .

(28)

λ = 8t20 m2 γ 2 = 8c2⋆ < 1.

(29)

E

i=1

Put Lemma 3 implies 2

2

Ee8t H ≤

eλ/2 . 1−λ

(30)

Combining (27), (28), and (30), we obtain

Ee

t0 F

≤ M0 ,

−t0 F

Ee

≤ M0 ,

M0 =

eλ/2 C0 1−λ

!1/2

.

(31)

Thus Eet0 |F | ≤ 2M0 .

8

(32)

For x ≥ 0, t > 0, and p > 0, 

p

x ≤

p et

p

etx .

(33)

Applying this with x = |F | and t = t0 , then using (32), gives p (2M0 )1/2 (2M0 )1/p ≤ mpγ, et0 ec⋆

∥F ∥p ≤

p ≥ 2.

It remains only to bound the numerical constant. Using e > 8/3, e < 3, and ex ≤ (1 − x)−1 for 0 ≤ x < 1, 11 81 1 M02 < . < , ec⋆ 4 8 Therefore

(2M0 )1/2 33 < < 7. ec⋆ 4 21/4

This proves (6). Lemma 4 (Bounded-difference moment bound). Let X = f (W1 , . . . , WN ), where W1 , . . . , WN are independent. Suppose EX = 0 and changing Wj can change f by at most cj . Then, for every p ≥ 2, 1/2



N pX c2  ∥X∥p ≤  2 j=1 j

Proof. McDiarmid’s inequality gives, with V =

.

2 j cj ,

P

2t2 P{|X| > t} ≤ 2 exp − V Therefore p

E |X| ≤ 2p

Z ∞

p−1 −2t2 /V

t

e

0

(34)

V dt = 2 2 

p/2

!

.

p Γ +1 ≤ 2 





Vp 2

p/2

.

Taking pth roots proves the claim. Lemma 5 (Euclidean aggregation). For arbitrary random variables B1 , . . . , Bn and p ≥ 2, n X

!1/2

Bi2

i=1

≤

!1/2

∥Bi ∥2p

.

(35)

i=1

p

Proof. Apply the triangle inequality in Lp/2 to

n X

P

2 i Bi and then take square roots.

Proposition 2. Let h1 , . . . , hn be functions of arbitrary independent coordinates Z1 , . . . , Zn satisfying E[hi | Zi ] = 0,

E[hi | Z−i ] = 0,

oscj (hi ) ≤ β

(j ̸= i).

(36)

Then, for every p ≥ 2, n X i=1

≤ 16pnβ.

hi (Z) p

9

(37)

Proof. Let Z ′ = (Z1′ , . . . , Zn′ ) be an independent copy of Z and set Wi = (Zi+ , Zi− ) := (Zi , Zi′ ), W = (W1 , . . . , Wn ). Let ε1 , . . . , εn be independent unbiased signs, independent of W , and define Zi+ , Zi− ,

(

Ziε =

εi = 1, εi = −1.

If ε(i) is obtained by flipping εi , define, conditionally on W , Qi (ε; W ) =

i 1h (i) hi (Z ε ) − hi (Z ε ) . 2

(38)

By the second centering identity in (36), hi (Z) = EZi′ hi (Z) − hi (Zi′ , Z−i ) | Z . 



Jensen’s inequality gives n X i=1

n X 

hi (Z) − hi (Zi′ , Z−i )

≤

hi (Z)



i=1

p

=2

p

n X

Qi (1; W )

i=1

.

(39)

p

For a deterministic e ∈ {−1, 1}n , swap the two entries of Wj whenever ej = −1. The law of W is invariant under this operation, and Qi (1; W ) becomes Qi (e; W ). Averaging over an independent uniform e = ε gives X

Qi (1; W )

i

=

X

Qi (ε; W )

i

p

.

(40)

p

For fixed W , Qi (ε(i) ; W ) = −Qi (ε; W ). Hence there is a function ri (ε−i ; W ) such that Qi (ε; W ) = εi ri (ε−i ; W ). Define e i (ε; W ) = Qi (ε; W ) − εi bi (W ). Q

bi (W ) = Eε−i ri (ε−i ; W ),

(41)

Conditionally on W , e i | ε−i , W ] = 0, E[Q

e i | εi , W ] = 0. E[Q

Moreover, for j ̸= i, Qi (ε; W ) − Qi (ε(j) ; W ) ≤ β. Subtracting εi bi (W ) does not change any cross-coordinate oscillation. Proposition 1, conditionally on W , therefore gives n X

≤ 7pnβ.

ei Q

i=1

Lp (ε|W )

It remains to control the centering defect. For a fixed pair Wi = (u, v), set di (x−i ; u, v) =

1 [hi (u, x−i ) − hi (v, x−i )] . 2

10

(42)

Then ε bi (W ) = Eε−i di (Z−i ; Zi+ , Zi− ).

(43)

Averaging over W−i and ε−i produces an independent draw from the original law of Z−i . The first centering identity in (36) therefore implies E[bi (W ) | Wi ] = 0.

(44)

If j ̸= i, replacing the pair Wj by another pair changes the right-hand side of (43) by at most β. Lemma 4, conditionally on Wi , gives s

∥bi ∥p ≤ β Let B(W ) =

(n − 1)p . 2

(45)

2 1/2 . By Lemma 5, i bi (W )

P

s

∥B∥p ≤ β Conditionally on W , the Rademacher sum application of Lemma 4 gives X

n(n − 1)p ≤ nβ 2

r

p . 2

(46)

i εi bi (W ) has coordinate oscillations 2 |bi (W )|. Another

P

≤

εi bi (W )

i

p

2p B(W ).

Lp (ε|W )

Using (46), X

≤ pnβ.

εi bi (W )

i

(47)

p

Combining (42) and (47), X

≤ 8pnβ.

Qi (ε; W )

i

p

Equations (39) and (40) complete the proof. Proof of Theorem 1. Define ai (Zi ) = E[gi (Z) | Zi ],

hi (Z) = gi (Z) − ai (Zi ).

Since E[gi | Z−i ] = 0, we have Egi = 0 and hence Eai = 0. Independence gives E[hi | Zi ] = 0,

E[hi | Z−i ] = 0.

Subtracting ai (Zi ) does not change any oscillation in a coordinate j ̸= i. Proposition 2 therefore gives X

≤ 16pnβ.

hi (Z)

i

p

The variables ai (Zi ) are independent, centered, and bounded in absolute value by M . Their sum has coordinate oscillation at most 2M in each coordinate, so Lemma 4 gives X i

≤M

ai (Zi ) p

11

p

2pn.

Finally,

n X

gi (Z) =

i=1

n X

hi (Z) +

i=1

n X

ai (Zi ),

i=1

and the triangle inequality yields n X

≤ 16pnβ + M

gi (Z)

i=1

p

2pn.

p

This proves (3).

3

Uniformly stable algorithms

We now state the consequence for learning algorithms. Let A be γ-uniformly stable and suppose that the loss satisfies 0 ≤ ℓ ≤ L. Corollary 1. For every p ≥ 2, r

∥R(AS ) − Remp (AS )∥p ≤ 33pγ + L

2p . n

(48)

Consequently, for every δ ∈ (0, 1), with probability at least 1 − δ, 

|R(AS ) − Remp (AS )| ≤ e 33γ log

e2 δ

s

!

+L



e2  2 log . n δ 



(49)

Proof. The reduction of [2, Lemma 7] gives functions satisfying Theorem 1 with M = L and β = 2γ, together with a deterministic comparison error at most 2γn for the rescaled generalization gap. Applying Theorem 1, adding this error, and dividing by n gives r

∥R(AS ) − Remp (AS )∥p ≤ (32p + 2)γ + L

r

2p ≤ 33pγ + L n

2p . n

For the tail bound, set q = log(e2 /δ) ≥ 2. Markov’s inequality gives n

o

P |X| > e ∥X∥q ≤ e−q ≤ δ. Applying this to the generalization error proves (49).

References [1] Olivier Bousquet and André Elisseeff. Stability and generalization. Journal of Machine Learning Research, 2:499–526, 2002. [2] Olivier Bousquet, Yegor Klochkov, and Nikita Zhivotovskiy. Sharper bounds for uniformly stable algorithms. In Proceedings of the Thirty-Third Conference on Learning Theory, volume 125 of Proceedings of Machine Learning Research, pages 610–626. PMLR, 2020. [3] Vitaly Feldman and Jan Vondrák. Generalization bounds for uniformly stable algorithms. In Advances in Neural Information Processing Systems, volume 31, 2018. [4] Vitaly Feldman and Jan Vondrák. High probability generalization bounds for uniformly stable algorithms with nearly optimal rate. In Proceedings of the Thirty-Second Conference on Learning Theory, volume 99 of Proceedings of Machine Learning Research, pages 1270–1279. PMLR, 2019.

12

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