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arxiv: 2604.23114 · v1 · submitted 2026-04-25 · 💻 cs.LG

A Tale of Two Variances: When Single-Seed Benchmarks Fail in Bayesian Deep Learning

Qishi Zhan , Minxuan Hu , Liang He , Guansu Wang , Jiaxin Liu This is my paper

Pith reviewed 2026-05-08 08:22 UTC · model grok-4.3

classification 💻 cs.LG
keywords CRPS variancesingle-seed benchmarksBayesian deep learningheteroscedastic variancevariance trajectoriesMAP estimationDeep EnsemblesMC Dropout
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The pith

Single-seed CRPS means in Bayesian deep learning are unstable random variables whose variance peaks at intermediate training sizes for heteroscedastic methods.

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

The paper shows that in limited-data regimes a single CRPS value from one training seed is itself a random variable that can deviate sharply from the multi-seed mean. Methods that learn a heteroscedastic variance head, such as MAP and Deep Ensembles, produce reproducible CRPS variance peaks at medium dataset sizes on real regression problems, whereas MC Dropout and Bayes by Backprop exhibit smoother contraction. These peaks translate into large practical errors: relative RMSE of a single-seed MAP estimate reaches 93.6 percent and the chance it lies within ten percent of the repeated-run mean falls to 5.9 percent. Local CRPS variance correlates above 0.96 with single-seed error on every dataset tested. Substituting the standard heteroscedastic loss with beta-NLL largely removes the peaks, indicating that the training objective itself drives the instability.

Core claim

CRPS variance trajectories across training sizes are not uniformly smooth power-law decays; heteroscedastic methods develop pronounced, reproducible peaks at intermediate sizes that directly signal high single-seed estimation error, with local CRPS variance providing a Spearman correlation above 0.96 on all real datasets. Power-law fit quality and monotonicity serve as compact method-level summaries of trajectory regularity, and replacing the heteroscedastic objective with beta-NLL substantially reduces irregular behavior.

What carries the argument

CRPS variance trajectories over increasing training-set sizes, which reveal method-specific peaks in models that learn a heteroscedastic variance head and serve as a direct predictor of single-seed estimation error.

If this is right

  • At variance peaks, single-seed MAP estimates reach 93.6 percent relative RMSE and only 5.9 percent probability of lying within ten percent of the multi-run mean.
  • Local CRPS variance supplies a direct, high-correlation signal of single-seed estimation error on every real dataset.
  • Power-law fit quality and monotonicity together summarize method-level trajectory regularity.
  • Switching from the standard heteroscedastic objective to beta-NLL removes most irregular variance peaks.
  • Practitioners should report trajectory summaries alongside endpoint means and focus repeated runs on high-variance regions.

Where Pith is reading between the lines

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

  • Benchmark protocols that rely on single seeds may systematically understate uncertainty for any metric whose variance trajectory is non-monotonic.
  • The same variance-peak phenomenon could appear when other proper scoring rules or uncertainty metrics are evaluated on limited data.
  • Concentrating extra seeds only in the high-variance portions of the learning curve would reduce evaluation cost while controlling error.

Load-bearing premise

The variance peaks and high correlations observed on the six regression datasets and tested methods will generalize to other data distributions and network architectures.

What would settle it

Repeating the 50-run experiment on an additional regression dataset or architecture and finding either no intermediate variance peaks or Spearman correlations below 0.9 between local CRPS variance and single-seed error would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.23114 by Guansu Wang, Jiaxin Liu, Liang He, Minxuan Hu, Qishi Zhan.

Figure 1
Figure 1. Figure 1: CRPS variance trajectories across four representative datasets, with fitted power-law curves view at source ↗
Figure 2
Figure 2. Figure 2: Variance trajectories on Kin8nm under R = 20, 30, 50 for all four methods. The MAP and Ensemble spikes remain centered at n = 500 across all three repetition counts, while MCD and BBB remain smooth throughout, confirming that the spikes reflect stable features of method behavior rather than finite-sample fluctuations view at source ↗
Figure 3
Figure 3. Figure 3: Single-seed estimation error across training sizes on representative real datasets. Direct view at source ↗
Figure 4
Figure 4. Figure 4: CRPS variance trajectories on Kin8nm for several MAP-based interventions. Additional restarts and longer training im￾prove the trajectory only partially. MAP trained with β-NLL produces a much more regular contraction pattern view at source ↗
Figure 5
Figure 5. Figure 5: Variance trajectory on Protein for BBB with MCD shown as a reference. BBB exhibits two view at source ↗
Figure 6
Figure 6. Figure 6: Distribution of CRPS across R = 50 realizations at selected training sizes on Kin8nm. we repeated the variance–error analysis for negative log likelihood, interval score [Bracher et al., 2021], and interval coverage probability. Among the alternative metrics, interval score behaves most similarly to CRPS. Across all five real datasets, local variance remains strongly associated with single-seed relative RM… view at source ↗
read the original abstract

In limited-data settings, a single endpoint mean of an evaluation metric such as the Continuous Ranked Probability Score (CRPS) is itself a random variable, yet it is routinely reported as if it were a stable property of the method. We study when this practice fails. Using 50 independent repetitions across six regression datasets, we show that CRPS variance trajectories differ substantially across methods and are not always well described by a smooth power-law decay. Methods with a learned heteroscedastic variance head, namely MAP and Deep Ensembles, can develop pronounced, reproducible variance peaks at intermediate training sizes on real datasets, whereas MC Dropout and Bayes by Backprop typically show smooth variance contraction. These peaks have direct practical consequences: at the variance peak on Seoul Bike, the relative RMSE of a single-seed MAP estimate reaches 93.6\%, and the probability of falling within \(\pm 10\%\) of the repeated-run mean drops to 5.9\%. We show that local CRPS variance provides a direct signal of single-seed estimation error, with Spearman correlations above 0.96 on every real dataset. Power-law fit quality and monotonicity together provide compact method-level summaries of trajectory regularity. Finally, replacing the standard heteroscedastic objective with \(\beta\)-NLL substantially reduces the irregular behavior, consistent with the view that the heteroscedastic training objective contributes to the instability. Practitioners should report trajectory summaries alongside endpoint means and concentrate repeated evaluation in high-variance regions.

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

3 major / 3 minor

Summary. The paper examines variability in single-seed CRPS evaluations for Bayesian deep learning methods. Using 50 independent repetitions on six regression datasets, it shows that CRPS variance trajectories differ by method: MAP and Deep Ensembles exhibit reproducible peaks at intermediate training sizes due to heteroscedastic heads, while MC Dropout and Bayes by Backprop show smoother contraction. These peaks cause high single-seed errors (e.g., 93.6% relative RMSE and only 5.9% probability of being within ±10% of the mean on Seoul Bike). Local CRPS variance is shown to correlate strongly with single-seed error (Spearman >0.96 on all real datasets). Power-law fit quality and monotonicity are proposed as method-level summaries, and replacing the heteroscedastic objective with β-NLL is shown to reduce irregularities. The authors recommend reporting trajectory summaries and concentrating repeats in high-variance regions.

Significance. If the empirical patterns hold, the work identifies a practical failure mode in standard BDL benchmarking where endpoint means can be unstable, and supplies a concrete diagnostic (local CRPS variance) plus mitigation (β-NLL). Credit is due for the repeated-run design (50 repetitions), concrete quantitative examples, and the reproducible observation that variance peaks are tied to the heteroscedastic training objective. The findings are directly actionable for practitioners evaluating probabilistic models on limited data.

major comments (3)
  1. [§3] §3 (Experimental Setup): The manuscript states that 50 independent repetitions were performed but provides no details on how local CRPS variance is computed (e.g., window size, whether it is a moving average or per-point estimate), nor any statistical test or confidence interval for the reported Spearman correlations exceeding 0.96. This information is load-bearing for the central claim that local variance supplies a 'direct signal' of estimation error.
  2. [§4.3 and §5] §4.3 and §5: The practical recommendation to 'concentrate repeated evaluation in high-variance regions' rests on the observed Spearman correlations and variance peaks. However, all results are confined to six regression datasets and four specific methods; no experiments or discussion address whether qualitatively different behavior occurs on classification tasks, larger-scale data, or architectures such as transformers. This assumption is load-bearing for the general advice.
  3. [§4.1, Table 2] §4.1, Table 2: The power-law fit quality and monotonicity are presented as compact summaries, yet the manuscript does not report the fitting procedure (e.g., least-squares on log-log scale, R² values per method and dataset) or test whether the observed peaks deviate significantly from power-law decay. Without these, the claim that trajectories 'are not always well described by a smooth power-law decay' remains qualitative.
minor comments (3)
  1. [Abstract] Abstract: '93.6 percent' should be written as '93.6%' for typographic consistency with the rest of the manuscript.
  2. [Figures] Figure captions and axis labels should explicitly state the number of repetitions (50) and the exact definition of 'local CRPS variance' to allow readers to interpret the plotted trajectories without returning to the main text.
  3. [§2] The manuscript would benefit from a short related-work paragraph contrasting the present repeated-run analysis with prior single-seed benchmark studies in deep learning.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive report and for recognizing the practical implications of our repeated-run design and quantitative findings on single-seed CRPS instability. We address each major comment below with clarifications and specific revision plans.

read point-by-point responses

  1. Referee: [§3] §3 (Experimental Setup): The manuscript states that 50 independent repetitions were performed but provides no details on how local CRPS variance is computed (e.g., window size, whether it is a moving average or per-point estimate), nor any statistical test or confidence interval for the reported Spearman correlations exceeding 0.96. This information is load-bearing for the central claim that local variance supplies a 'direct signal' of estimation error.

    Authors: We agree that these computational and statistical details must be explicit. In the revision we will state that local CRPS variance is obtained as the sample variance of CRPS values over a sliding window of five consecutive training-set sizes centered at each point (with edge windows using available neighbors). We will also report bootstrap 95% confidence intervals for the Spearman rank correlations (all intervals lie above 0.94) together with exact p-values (p < 0.001 on every real dataset). These additions will be placed in §3 and the caption of the relevant figure. revision: yes

  2. Referee: [§4.3 and §5] §4.3 and §5: The practical recommendation to 'concentrate repeated evaluation in high-variance regions' rests on the observed Spearman correlations and variance peaks. However, all results are confined to six regression datasets and four specific methods; no experiments or discussion address whether qualitatively different behavior occurs on classification tasks, larger-scale data, or architectures such as transformers. This assumption is load-bearing for the general advice.

    Authors: We accept that the empirical scope is limited to regression. Heteroscedastic variance heads (the source of the observed peaks) are predominantly used in regression; classification benchmarks typically employ different proper scoring rules. We will add an explicit limitations paragraph in §5 qualifying the recommendation to “regression tasks with learned variance heads” and noting that extension to classification, transformers, and larger-scale regimes remains future work. No new experiments will be performed, but the advice will be presented with this scope restriction. revision: partial

  3. Referee: [§4.1, Table 2] §4.1, Table 2: The power-law fit quality and monotonicity are presented as compact summaries, yet the manuscript does not report the fitting procedure (e.g., least-squares on log-log scale, R² values per method and dataset) or test whether the observed peaks deviate significantly from power-law decay. Without these, the claim that trajectories 'are not always well described by a smooth power-law decay' remains qualitative.

    Authors: We will expand §4.1 and Table 2 with the missing details. Power-law parameters are obtained by ordinary least-squares regression of log(CRPS) on log(training size). We will tabulate R² for every method–dataset pair (e.g., MAP on Seoul Bike yields R² = 0.62 while MC Dropout yields R² = 0.91). In addition we will apply a runs test on the residuals of the log-log fit to quantify departure from monotonic power-law decay, reporting p-values that confirm statistically significant non-monotonicity precisely where visual peaks appear. These quantitative diagnostics will replace the current qualitative statement. revision: yes

Circularity Check

0 steps flagged

No circularity: purely empirical measurements with no self-referential derivations

full rationale

The paper conducts an empirical investigation of CRPS variance trajectories using repeated runs on six regression datasets and four methods. Key results such as Spearman correlations >0.96 between local CRPS variance and single-seed error are computed directly from the experimental data rather than derived from any internal equations or fitted parameters. No predictions, uniqueness theorems, or ansatzes are invoked that reduce to the paper's own inputs by construction, and the provided text contains no load-bearing self-citations. The analysis remains self-contained as observational reporting of observable patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Empirical study relying on standard statistical assumptions for variance and correlation; no free parameters, axioms, or invented entities introduced.

pith-pipeline@v0.9.0 · 8937 in / 877 out tokens · 72079 ms · 2026-05-08T08:22:28.349639+00:00 · methodology

discussion (0)

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