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Finite Lp moments on losses suffice for sharp high-probability generalization bounds via an extension of McDiarmid's inequalities.

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2026-06-27 21:00 UTC pith:NERZRIHH

load-bearing objection The paper extends McDiarmid concentration to finite Lp moments and applies it to generalization bounds in ERM, transductive regression, and meta-learning.

arxiv 2606.06855 v1 pith:NERZRIHH submitted 2026-06-05 stat.ML cs.LGmath.STstat.TH

Stability beyond Bounded Differences: Sharp Generalization Bounds under Finite L_p Moments

Qianqian Lei , Soham Bonnerjee , Yuefeng Han , Wei Biao Wu This is my paper
classification stat.ML cs.LGmath.STstat.TH
keywords algorithmic stabilitygeneralization boundsconcentration inequalitiesLp momentsMcDiarmid inequalityempirical risk minimizationheavy-tailed losses
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a stability framework that needs only finite Lp moments rather than uniform boundedness or sub-Gaussian tails. It first proves sharp concentration inequalities for functions of independent random variables under these moment constraints, then applies them to obtain high-probability generalization guarantees for empirical risk minimization, transductive regression, and meta-learning. A sympathetic reader would care because many modern losses are unbounded or heavy-tailed, so classical stability results do not apply. The work shows that Lp stability alone is enough to guarantee robust generalization when boundedness fails.

Core claim

We develop a stability-based framework that requires only a finite Lp moment condition. Our first contribution is sharp concentration inequalities for functions of independent random variables under Lp constraints, extending McDiarmid's bounded-differences techniques beyond the classical regime. Leveraging these results, we derive sharp high-probability generalization bounds across a range of learning paradigms, including empirical risk minimization, transductive regression, and meta-learning. These guarantees show that Lp stability suffices for robust generalization even when boundedness fails.

What carries the argument

The new concentration inequalities that extend McDiarmid's bounded-differences technique to the finite Lp moment regime, which then produce the stability-based generalization bounds.

Load-bearing premise

The derived concentration inequalities continue to hold under finite Lp moments without any extra boundedness or stronger tail conditions.

What would settle it

A counter-example consisting of independent random variables with finite Lp moments for which the deviation probability of some function exceeds the bound stated by the new concentration inequality.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • High-probability generalization bounds hold for empirical risk minimization when only Lp stability is assumed.
  • The same bounds apply directly to transductive regression and meta-learning under the Lp moment condition.
  • Lp stability is sufficient for robust generalization even if losses are unbounded.
  • The concentration results are presented as sharp under the stated Lp constraints.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The moment-based approach might allow stability analysis for algorithms whose losses exhibit power-law tails.
  • Similar moment conditions could replace tail assumptions in concentration results for other dependent-data settings.
  • One could test the bounds by constructing explicit Lp-stable learners on synthetic data with controlled moments and checking whether observed gaps stay inside the predicted probability.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The paper develops a stability-based framework requiring only finite Lp moment conditions rather than boundedness or sub-Gaussian tails. It derives sharp concentration inequalities for functions of independent random variables under Lp constraints by extending McDiarmid's bounded-differences approach, then applies these to obtain high-probability generalization bounds for empirical risk minimization, transductive regression, and meta-learning.

Significance. If the claimed sharpness and extension hold, the work meaningfully broadens the scope of stability arguments to heavy-tailed and unbounded loss settings common in modern learning, weakening a standard restrictive assumption in the literature. The parameter-free character of the Lp-based bounds (no ad-hoc parameters or invented entities) is a strength.

minor comments (2)
  1. [Abstract] Abstract and §1: the term 'sharp' is used repeatedly for the concentration inequalities and generalization bounds; the manuscript should explicitly state the sense in which sharpness is achieved (e.g., matching constants with known lower bounds or tightness under specific Lp regimes).
  2. [§3–§5] Theorems in §3–§5: verify that the dependence on the moment order p and the stability parameter appears explicitly in the final bounds, and that no hidden boundedness or tail conditions are introduced in the proofs.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives concentration inequalities from finite Lp moment assumptions by extending McDiarmid's bounded-differences method, then applies those inequalities to obtain generalization bounds for ERM, transductive regression, and meta-learning. No quoted step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional renaming; the central claims remain mathematically independent of the target generalization statements.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the central claim rests on the unverified validity of the new Lp concentration inequalities.

pith-pipeline@v0.9.1-grok · 5686 in / 1115 out tokens · 20780 ms · 2026-06-27T21:00:08.224582+00:00 · methodology

0 comments
read the original abstract

While algorithmic stability is a central tool for understanding generalization of learning algorithms, existing high-probability guarantees typically rely on uniform boundedness or sub-Gaussian/sub-Weibull tail assumptions, which can be overly restrictive for modern settings with heavy-tailed or unbounded losses. We develop a stability-based framework that requires only a finite $L_p$ moment condition. Our first contribution is sharp concentration inequalities for functions of independent random variables under $L_p$ constraints, extending McDiarmid's bounded-differences techniques beyond the classical regime. Leveraging these results, we derive sharp high-probability generalization bounds across a range of learning paradigms, including empirical risk minimization, transductive regression, and meta-learning. These guarantees show that $L_p$ stability suffices for robust generalization even when boundedness fails, substantially weakening the standard assumptions in the stability literature.

Figures

Figures reproduced from arXiv: 2606.06855 by Qianqian Lei, Soham Bonnerjee, Wei Biao Wu, Yuefeng Han.

Figure 2
Figure 2. Figure 2: Plot of p(y) versus y for the neural network regression experiment in Section D.3. 5. Conclusion and Future Works In this work, we provide a systematic treatment of stability analysis under weakened Lp assumptions by establishing a new large-deviation inequality, which may be of indepen￾dent interest. Theorems 2.2 and 2.6 broaden the scope of stability-based generalization by accommodating set￾tings where … view at source ↗
Figure 1
Figure 1. Figure 1: Plot of p(y) versus y; both curves stabilize around C ν/2 0 for large y. exhibits initial exponential growth before slowing down and stabilizing near C ν/2 0 , further vindicating the importance of the polynomial-in-y term in Theorem 3.4. For larger ν (e.g., ν = 4.4), the Gaussian tail dominates the polynomial tail more strongly at smaller values of y. Consequently, p(y) may exceed the threshold C ν/2 0 in… view at source ↗
Figure 3
Figure 3. Figure 3: Plot of p(y) versus y; both curves stabilize around C ν/2 0 for large y. the ratio p(y) exhibits an initial exponential growth before slowing down and stabilizing near C ν/2 0 , further vindicating the behavior typified in (106) in light of Theorem 3.4. For larger ν (e.g., ν = 4.4), the Gaussian tail dominates the polynomial tail more strongly at smaller values of y. Consequently, p(y) may exceed the thres… view at source ↗
Figure 4
Figure 4. Figure 4: Plot of p(y) versus m. Note that here βˆ is trained via ridge regression with λ = 1.0 only on the training sample S. Similar to Section D.1, tail probabilities are empirically estimated via 50, 000 Monte Carlo draws [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Plot of p(y) versus y for the transductive regression problem in Section 3.2. D.3. Additional experiments: a modern application. In principle, any data with heavy tail would exhibit behavior similar to what we predict in our theory. Nevertheless, as baby steps towards more grounded stability theory, we perform a similar experiment on two layer neural-network for a regression problem. Specifically, we consi… view at source ↗
Figure 6
Figure 6. Figure 6: Plot of p(y) versus y for the neural network regression experiment in Section D.3. through in distinct characterizations of Gaussian and polynomial tails. As predicted by our theory and can also be seen for the simpler ridge-regression example, p(y) stabilizes at around C ν/2 0 ≈ 1.562. This corroborates our theory on a prototype of modern ML algorithm. Note that here, the number of parameters being optimi… view at source ↗

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