Subsampling for supervised learning in reproducing kernel Hilbert spaces

Eyal Vayness; Maxime Sangnier

REVIEW 2 major objections 1 minor 106 references

Subsampling for supervised learning in reproducing kernel Hilbert spaces

T0 review · 2 major / 1 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read Studying asymptotics of the Horvitz-Thompson reweighted empirical risk minimizer in an RKHS reveals an optimal subsampling scheme based on the trace of the covariance operator that works via plug-in.

desk verdict The paper derives a trace-based optimal subsampling rule for Horvitz-Thompson reweighted kernel ERM from asymptotics and backs it with numerics, but the step showing the trace dominates the asymptotic variance needs explicit bounds to hold up. read the letter →

arxiv 2606.21260 v1 pith:4X5GBSU5 submitted 2026-06-19 stat.ML cs.LG

Eyal Vayness , Maxime Sangnier This is my paper · ORCID

classification stat.MLcs.LG

keywords subsamplingreproducingkernelHilbertspaceHorvitz-Thompsonestimatorempiricalriskminimizationasymptoticanalysisoptimalsamplingplug-insupervisedlearning

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper focuses on subsampling to lower the cost of training nonparametric models whose hypotheses lie in a reproducing kernel Hilbert space. It considers the Horvitz-Thompson reweighted version of empirical risk minimization and derives its asymptotic behavior. From that analysis the authors obtain an explicit optimal subsampling distribution expressed in terms of the trace of the covariance operator. The distribution is shown to be realizable by a plug-in estimator that does not require knowledge of the unknown measure. Experiments on synthetic and real data illustrate that the resulting procedure reduces computation while retaining statistical performance.

What carries the argument

Horvitz-Thompson reweighted empirical risk minimizer in a reproducing kernel Hilbert space, whose asymptotic covariance supplies the optimality criterion via the trace of the covariance operator.

What would settle it

If the plug-in estimator of the trace-based subsampling probabilities fails to produce lower estimation error than uniform subsampling on large-scale data sets, the claimed optimality would be refuted.

Watch

Extended reading notes

Core claim

By studying the asymptotic properties of the Horvitz-Thompson reweighted empirical risk minimizer in an RKHS, the authors reveal an optimal subsampling scheme regarding the trace of the covariance operator and show that it can be used via plug-in.

Load-bearing premise

The asymptotic covariance of the estimator is dominated by the trace of the covariance operator in a way that directly yields an optimal subsampling distribution.

Editorial extensions

If this is right

Subsampling probabilities are chosen proportional to the trace terms of the covariance operator for asymptotic optimality.
The optimal scheme is realized by a plug-in estimator that replaces unknown quantities with empirical estimates.
The procedure lowers computational cost for a fixed sample size while preserving the asymptotic efficiency of the full-data estimator.
Numerical comparisons on synthetic and real-world data confirm practical gains over standard subsampling.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same trace-based criterion could be tested in other nonparametric settings that admit a covariance operator.
Implementation would allow direct comparison of variance reduction against uniform or leverage-score sampling on benchmark data sets.
The method supplies a concrete route to lower energy use in kernel-based training pipelines by reducing the number of kernel evaluations.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main takeaway is that the authors analyze the asymptotics of a Horvitz-Thompson reweighted empirical risk minimizer in an RKHS and extract an optimal subsampling distribution tied to the trace of the covariance operator, then implement it via plug-in. They also run a numerical study on synthetic and real datasets to check practicality.

What stands out as new is the specific combination of the reweighting with an optimality criterion based on that trace in the RKHS setting. The numerical experiments add some value by showing the method can be applied without too much extra cost.

The soft spot sits in the move from the asymptotic expansion to the optimality claim. The abstract states that the properties reveal the optimal scheme regarding the trace, yet it is not obvious from the given material whether the covariance is controlled by that term alone or whether other contributions could alter the minimizing distribution. If the full proofs contain a clean domination argument or uniform bound, that concern disappears; otherwise the optimality result rests on an unverified step.

This work is for people already working on scalable nonparametric methods or subsampling schemes in kernel settings. A reader who wants a theoretically motivated way to pick inclusion probabilities for large kernel problems will get something concrete, though the advance is incremental rather than foundational.

It is worth sending to peer review. The core construction is coherent and the practical angle is present, so referees can check the asymptotics and the experiments in detail.

Referee Report

2 major / 1 minor

Summary. The manuscript analyzes subsampling for supervised learning when the hypothesis class is a reproducing kernel Hilbert space. It considers the Horvitz-Thompson reweighted empirical risk minimizer, derives its asymptotic properties, and uses those properties to identify an optimal subsampling distribution that depends on the trace of the covariance operator; the distribution is then estimated by plug-in. A numerical study on synthetic and real data is included to illustrate practicality.

Significance. If the asymptotic expansion is valid and the trace term indeed governs the leading variance uniformly across sampling schemes, the work supplies a statistically grounded, computationally attractive subsampling rule for kernel methods. The plug-in implementation and the explicit link between asymptotics and sampling probabilities would be a useful contribution to large-scale nonparametric learning.

major comments (2)

[Abstract] Abstract (and the paragraph on asymptotic properties): the optimality claim rests on the assertion that the trace of the covariance operator dominates the asymptotic covariance of the reweighted estimator. No explicit expansion, bound, or set of conditions (e.g., on the kernel, the sampling probabilities, or the regression function) is supplied in the abstract showing that this domination holds uniformly; without that step the optimality criterion does not necessarily follow from the asymptotics.
[asymptotic analysis section] The transition from the asymptotic covariance expression to the trace-based objective appears to treat the trace as an external functional that can be minimized directly with respect to the inclusion probabilities. If the covariance expansion contains additional terms that also depend on the sampling scheme, the claimed optimality may be incomplete; the manuscript should state the precise domination argument and any uniformity conditions.

minor comments (1)

Notation for the Horvitz-Thompson weights and the RKHS inner product should be introduced once and used consistently; several symbols appear without prior definition in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. The comments highlight the need for greater explicitness regarding the domination argument in both the abstract and the asymptotic analysis. We address each point below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

Referee: [Abstract] Abstract (and the paragraph on asymptotic properties): the optimality claim rests on the assertion that the trace of the covariance operator dominates the asymptotic covariance of the reweighted estimator. No explicit expansion, bound, or set of conditions (e.g., on the kernel, the sampling probabilities, or the regression function) is supplied in the abstract showing that this domination holds uniformly; without that step the optimality criterion does not necessarily follow from the asymptotics.

Authors: We agree that the abstract, being a concise summary, does not reproduce the full technical conditions. The asymptotic expansion and the uniform domination of the trace term are established in Theorem 2 under Assumptions 1--3 (bounded kernel, moment conditions on the regression function, and sampling probabilities bounded away from zero and one). These ensure the remainder is o(1) uniformly over admissible sampling schemes. To address the concern, we will revise the abstract to include a brief reference to these conditions and the resulting optimality criterion. revision: yes
Referee: [asymptotic analysis section] The transition from the asymptotic covariance expression to the trace-based objective appears to treat the trace as an external functional that can be minimized directly with respect to the inclusion probabilities. If the covariance expansion contains additional terms that also depend on the sampling scheme, the claimed optimality may be incomplete; the manuscript should state the precise domination argument and any uniformity conditions.

Authors: The asymptotic covariance derived in Theorem 1 takes the form trace(Σ_p) + R_n, where the remainder R_n is shown to be o_p(1) uniformly when the inclusion probabilities satisfy inf p_i ≥ c > 0 and the kernel is continuous and bounded. Because the leading term depends on the sampling distribution only through the trace functional, minimization with respect to the inclusion probabilities is valid. We will insert an explicit remark immediately after Theorem 1 stating the domination argument, the uniformity conditions, and why no other sampling-dependent terms enter at the leading order. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: optimality criterion derived from asymptotics as external functional

full rationale

The paper studies the asymptotic properties of the Horvitz-Thompson reweighted empirical risk minimizer in an RKHS and derives an optimal subsampling scheme based on the trace of the covariance operator, which is then implemented via plug-in estimation. This trace is an external functional of the operator, not defined in terms of the subsampling probabilities or fitted to the target quantity by construction. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided abstract or reader's analysis. The derivation chain remains self-contained against external benchmarks, with the optimality following from the asymptotic expansion rather than tautological reduction. Minor self-citation (if any) is not load-bearing for the central claim.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete and based on surface statements.

assumptions (2)

domain assumption The hypothesis set lies in a reproducing kernel Hilbert space
Explicitly stated in the abstract as the nonparametric setting.
domain assumption The estimator is a minimizer of an empirical risk reweighted à la Horvitz-Thompson
Core modeling choice stated in the abstract.

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Cite this review

Pith. "Pith review of Subsampling for supervised learning in reproducing kernel Hilbert spaces." pith.science (2026). https://pith.science/paper/4X5GBSU5

@misc{pith2026260621260,
  author       = {Pith},
  title        = {Pith review of: Subsampling for supervised learning in reproducing kernel Hilbert spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4X5GBSU5}},
  note         = {Machine review of arXiv:2606.21260}
}

read the original abstract

In the era of big data, subsampling became a common practice in statistical learning. By selecting a subgroup of individuals based on which the learner is trained, subsampling aims at reducing the computational cost and time of the estimation step, and ideally leads to a decrease of its energy consumption and carbon footprint. This work focuses on a nonparametric setting, in which the hypotheses set lies in a reproducing kernel Hilbert space, and the estimator is a minimizer of an empirical risk reweighted \`a la Horvitz-Thompson. By studying the asymptotic properties of this estimator, we reveal an optimal subsampling scheme (regarding the trace of the covariance operator) and show that it can be used via plug-in. A numerical study on synthetic and real-world datasets shows the practicability and the benefit of the proposed approach.

Figures

Figures reproduced from arXiv: 2606.21260 by the authors.

**Figure 2.** Visualization of the L-optimal SSPs on the synthetic datasets (the size of [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

**Figure 3.** Generalization error with respect to the smoothing hyperparameter [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Computational effort parameterized by the sample size [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]

**Figure 5.** Figure 5: Computational effort parameterized by the space budget (fixed [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]

**Figure 6.** Figure 6: Computational effort parameterized by the space budget (fixed [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.

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