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arxiv: 1906.09247 · v1 · pith:WKEWDMQLnew · submitted 2019-06-21 · 💻 cs.LG · stat.ML

Learning from weakly dependent data under Dobrushin's condition

Yuval Dagan , Constantinos Daskalakis , Nishanth Dikkala , Siddhartha Jayanti This is my paper

Pith reviewed 2026-05-25 18:51 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords generalization boundsdependent dataDobrushin's conditionRademacher complexityVC dimensionlearning theoryweak dependencespatial data
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The pith

Under Dobrushin's condition, standard Gaussian and Rademacher complexities and VC dimension suffice to bound generalization error for weakly dependent data.

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

The paper shows that data drawn from a joint distribution satisfying Dobrushin's condition allows the same complexity measures used in the independent case to control generalization and learning. Generalization bounds hold up to constant factors relative to i.i.d. analogs, while learnability rates suffer only logarithmic degradation in sample size. This matters for network or spatial sampling where observations are dependent yet not arbitrarily coupled, so existing tools apply without new complexity notions.

Core claim

When the data distribution satisfies Dobrushin's condition, the Gaussian complexity, Rademacher complexity, and VC dimension of a hypothesis class are sufficient to bound its generalization error and to obtain learnability rates; these bounds degrade by only constant factors for generalization and by logarithmic factors in the training-set size compared with the corresponding i.i.d. bounds.

What carries the argument

Dobrushin's condition, a quantitative bound on the dependence between any pair of variables, which lets standard complexity measures carry the argument without modification.

If this is right

  • Generalization error bounds for any hypothesis class remain within constant factors of their i.i.d. versions.
  • Learnability rates degrade by at most logarithmic factors in the number of training points.
  • The same Gaussian, Rademacher, and VC-dimension quantities control performance for network and spatial data.
  • No new complexity measures are required beyond those already used for independent samples.

Where Pith is reading between the lines

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

  • Existing learning algorithms whose analysis relies only on Rademacher or VC bounds can be transferred to graph or lattice data without change, once Dobrushin's condition is verified.
  • Empirical checks of pairwise dependence strength on real sensor or social-network datasets could immediately certify that standard bounds apply.
  • The result suggests a route to relax Dobrushin's condition further while retaining the same complexity measures, by replacing the uniform pairwise bound with a decaying influence graph.

Load-bearing premise

The joint distribution of the observed variables must satisfy Dobrushin's condition that limits pairwise dependence.

What would settle it

A concrete counter-example would be a hypothesis class and a distribution obeying Dobrushin's condition for which the observed generalization gap grows faster than a constant multiple of the i.i.d. bound given by its Rademacher complexity.

read the original abstract

Statistical learning theory has largely focused on learning and generalization given independent and identically distributed (i.i.d.) samples. Motivated by applications involving time-series data, there has been a growing literature on learning and generalization in settings where data is sampled from an ergodic process. This work has also developed complexity measures, which appropriately extend the notion of Rademacher complexity to bound the generalization error and learning rates of hypothesis classes in this setting. Rather than time-series data, our work is motivated by settings where data is sampled on a network or a spatial domain, and thus do not fit well within the framework of prior work. We provide learning and generalization bounds for data that are complexly dependent, yet their distribution satisfies the standard Dobrushin's condition. Indeed, we show that the standard complexity measures of Gaussian and Rademacher complexities and VC dimension are sufficient measures of complexity for the purposes of bounding the generalization error and learning rates of hypothesis classes in our setting. Moreover, our generalization bounds only degrade by constant factors compared to their i.i.d. analogs, and our learnability bounds degrade by log factors in the size of the training set.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper extends statistical learning theory beyond i.i.d. samples to data drawn from distributions satisfying Dobrushin's condition (a quantitative pairwise dependence bound), motivated by network or spatial sampling rather than time series. It claims that the standard i.i.d. complexity measures—Gaussian complexity, Rademacher complexity, and VC dimension—remain sufficient to control generalization error and learning rates, with generalization bounds degrading only by constant factors relative to their i.i.d. counterparts and learnability (sample-complexity) bounds degrading by logarithmic factors in the training-set size.

Significance. If the central claims hold, the work is significant for providing a direct reduction from weakly dependent spatial/network data to existing i.i.d. complexity measures under an explicit, standard dependence condition, without introducing new dependence-adjusted complexities. This yields clean, usable bounds that differ from the i.i.d. case only by universal constants or log n factors, which is a strong positive for applications in spatial statistics and graph-structured data.

minor comments (2)
  1. The abstract states that the bounds 'only degrade by constant factors' and 'log factors in the size of the training set,' but does not indicate whether these constants or log factors are made explicit or left as existential; if the former, a short table or remark comparing the precise prefactors to the i.i.d. case would strengthen the contribution.
  2. The motivation section contrasts the setting with ergodic time-series work; a one-sentence pointer to the specific Dobrushin parameter range that excludes typical mixing conditions used in that literature would clarify the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. The referee's summary accurately captures the paper's focus on generalization and learnability under Dobrushin's condition for weakly dependent spatial/network data, with bounds that degrade only by constants or log factors relative to the i.i.d. case.

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard complexity measures under explicit quantitative assumption

full rationale

The paper's central claim is that Dobrushin's condition (a quantitative bound on pairwise dependence) allows the standard i.i.d. Gaussian/Rademacher complexities and VC dimension to control generalization and learning rates, with only constant-factor degradation in bounds and log-factor degradation in sample complexity. The abstract states this extension directly without any self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations. No equations or derivation steps reduce the claimed bounds to the inputs by construction; the dependence control is an external assumption that propagates to the empirical process. The result is therefore self-contained against external benchmarks and receives score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard concentration and complexity tools from learning theory plus the domain assumption of Dobrushin's condition; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Dobrushin's condition holds for the data-generating distribution
    Invoked to control dependence so that standard complexity measures apply.

pith-pipeline@v0.9.0 · 5740 in / 1074 out tokens · 26155 ms · 2026-05-25T18:51:45.485045+00:00 · methodology

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