Decision-Aware Training for Sample-Based Generative Models

Kornelius Raeth; Nicole Ludwig

arxiv: 2607.01171 · v1 · pith:2YDE2BP2new · submitted 2026-07-01 · 💻 cs.LG · stat.ML

Decision-Aware Training for Sample-Based Generative Models

Kornelius Raeth , Nicole Ludwig This is my paper

Pith reviewed 2026-07-02 15:18 UTC · model grok-4.3

classification 💻 cs.LG stat.ML

keywords decision-aware trainingsample-based generative modelsenergy scoreproper scoring rulesprobabilistic forecastingcost-sensitive learning

0 comments

The pith

Augmenting the energy score with a differentiable decision loss trains sample-based generative models to reduce downstream decision costs.

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

Sample-based generative models for probabilistic forecasting are typically trained with strictly proper scoring rules such as the energy score, which allocate training signal according to data density alone. The paper proposes decision-aware training that adds a differentiable decision loss term penalizing the costs incurred when a decision maker acts on samples drawn from the model. This combined objective remains theoretically grounded because the decision loss itself qualifies as a proper scoring rule. Validation on one synthetic and two real-world tasks shows that the resulting models deliver lower decision costs in the regions where forecast errors matter most, while still producing full sample-based probability distributions.

Core claim

Decision-aware training augments the energy score objective with a differentiable decision loss that directly penalizes the cost incurred by acting on the model's forecast; the combined loss remains a proper scoring rule, allowing sample-based generative models to produce forecasts that are both calibrated and decision-optimal.

What carries the argument

The differentiable decision loss, which measures expected cost of optimal actions taken under samples from the model and combines with the energy score while preserving proper scoring properties.

If this is right

Training signal is reallocated toward forecast errors that carry high decision cost rather than uniform density matching.
Full probabilistic forecasts are retained because the added term does not replace the energy score.
Improvements concentrate in cost-sensitive regions of the forecast space.
The method applies to any sample-based generative model whose original objective is the energy score.

Where Pith is reading between the lines

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

The same decision-loss construction could be paired with other proper scoring rules if they admit differentiable decision terms.
Domains with known asymmetric costs, such as inventory or medical triage, could adopt the method by substituting the appropriate cost function.
Scaling behavior with decision-problem complexity remains open for empirical test on larger models.

Load-bearing premise

A decision loss reflecting the downstream cost structure can be expressed as a differentiable function of the generated samples.

What would settle it

A controlled experiment in which the decision-aware model produces strictly higher total decision costs than the standard energy-score model when both are evaluated on held-out data using the true cost function.

Figures

Figures reproduced from arXiv: 2607.01171 by Kornelius Raeth, Nicole Ludwig.

**Figure 1.** Figure 1: Method overview. In the forward pass the model hθ produces samples {yˆm} given input x. These feed two branches: the Energy Score (CRPS) computes the standard scoring rule loss given the observed target y; the Optimization layer solves Eq. (1) to obtain a ∗ , and the decision loss evaluates c(a ∗ , y). Gradients flow back from CRPS via standard autodiff (green arrows) and from decision loss through the opt… view at source ↗

**Figure 2.** Figure 2: Synthetic decision task. Left: Cost function c(a, y) for five protection levels a, overlaid with the marginal p(y) (grey). The threshold at y = 1.0 separates the two modes; mode positions are fixed across x, only the mixture weight varies. Center (two panels): Predicted a ∗ (x) (mean ± 1 std over training seeds, representative data seed) for wd = 0 and wd = 0.95. Pure CRPS (wd = 0, left) fails to track the… view at source ↗

**Figure 3.** Figure 3: Wind power dispatch results (λ = 5, implicit generative model). Left: aggregate CRPS vs. decision loss trade-off across all wd and seeds. Increasing wd trades CRPS for decision loss improvement at the aggregate level. Right (three panels): conditional metrics by power curve region vs. wd (mean ± 1 std across seeds). Improvements concentrate in the cut-off region; the rated region degrades (in CRPS); the ra… view at source ↗

**Figure 4.** Figure 4: Frost protection results (implicit generative model). Left (three panels): Aggregate CRPS, decision loss, and decision miscalibration vs. wd for α ∈ {0.2, 0.3, 0.5} (mean ± std over seeds). Decision loss remains mostly flat, with a small improvement for α ∈ {0.3, 0.5}; CRPS degrades with wd; decision miscalibration improves most visibly for α = 0.5. Right (two panels): Conditional predictive density for wd… view at source ↗

read the original abstract

Sample-based generative models are increasingly used for probabilistic forecasting in high-stakes decision settings, yet their training objectives are blind to the decision maker's cost structure. These models are commonly trained with strictly proper scoring rules, such as the energy score, which allocate their training signal in proportion to data density, with no awareness of where forecast errors are most costly for downstream decisions. We therefore propose decision-aware training for sample-based generative models, augmenting the energy score objective with a differentiable decision loss that directly penalises the cost incurred by acting on the model's forecast. This combined loss is theoretically grounded, as the decision loss is itself a proper scoring rule. We validate our method on one synthetic and two real-world tasks, showing targeted improvements in cost-sensitive regions while retaining full probabilistic forecasts.

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.

Desk Editor's Note private letter to a colleague

The paper adds a decision loss to the energy score for sample-based models but the abstract leaves differentiability and properness unshown.

read the letter

The main takeaway is that this work augments the energy score with a differentiable decision loss so that sample-based generative models pay attention to downstream decision costs instead of just matching data density.

What is new is the targeted extension to sample-based models rather than a generic application of decision-aware ideas. The paper does a clear job naming the mismatch: standard training spreads its signal according to density, which ignores where forecast errors actually hurt decisions most in high-stakes settings.

The soft spots sit in the missing technical steps. The abstract asserts the combined loss is theoretically grounded because the decision loss is itself a proper scoring rule, yet supplies no derivation or construction for how that loss stays differentiable through the generated samples. Many real decision problems use argmax or threshold operations that are not smooth, so the claim only holds if the paper restricts itself to specially chosen smooth costs or ad-hoc relaxations. No error bars, no details on the weighting hyperparameter, and no information on the three tasks appear in the provided text either.

The stress-test note on non-smooth mappings therefore looks relevant until the full paper shows otherwise.

This paper is for researchers working at the overlap of probabilistic forecasting, generative models, and decision theory. A reader already thinking about proper scoring rules and asymmetric costs could extract the core idea and try to fill in the gaps themselves. It deserves a serious referee because the underlying problem is practical and the proposed fix is direct, even if the current version needs the derivations and experimental controls written out before it can be evaluated properly.

Recommendation: send it to peer review.

Referee Report

3 major / 1 minor

Summary. The paper proposes decision-aware training for sample-based generative models by augmenting the energy score objective with a differentiable decision loss that directly penalizes the cost of acting on the model's forecast. It claims the combined loss is theoretically grounded because the decision loss is itself a proper scoring rule, and reports targeted improvements on one synthetic and two real-world tasks while retaining full probabilistic forecasts.

Significance. If the differentiability construction and proper-scoring property can be established for general downstream costs, the approach would address a genuine limitation in applying generative models to high-stakes probabilistic forecasting by aligning training gradients with decision-relevant error regions.

major comments (3)

[Abstract] Abstract, paragraph 3: the assertion that 'the decision loss is itself a proper scoring rule' is stated without derivation, reference, or explicit construction; because this property is invoked to ground the combined objective, its absence leaves the central theoretical claim unsupported.
[Abstract] Abstract, paragraph 3: the claim that the decision loss is 'differentiable' with respect to generated samples is asserted without specifying the required smoothness assumptions on the downstream optimization (e.g., handling of argmax or integer programs); this assumption is load-bearing for end-to-end training and is not automatic for arbitrary cost structures.
[Validation] Validation section (implied by 'one synthetic and two real-world tasks'): no error bars, statistical significance tests, or details on hyperparameter selection for the loss weighting are reported; without these, the claimed 'targeted improvements in cost-sensitive regions' cannot be assessed as robust.

minor comments (1)

[Abstract] The abstract does not indicate whether the method preserves the strictly proper character of the energy score after augmentation or only claims propriety for the added term.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive comments. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: [Abstract] Abstract, paragraph 3: the assertion that 'the decision loss is itself a proper scoring rule' is stated without derivation, reference, or explicit construction; because this property is invoked to ground the combined objective, its absence leaves the central theoretical claim unsupported.

Authors: We agree the abstract states the claim without supporting material. The manuscript body derives the proper scoring property of the decision loss; we will revise the abstract to reference the relevant theoretical section and add a concise justification of the property to better support the claim. revision: yes
Referee: [Abstract] Abstract, paragraph 3: the claim that the decision loss is 'differentiable' with respect to generated samples is asserted without specifying the required smoothness assumptions on the downstream optimization (e.g., handling of argmax or integer programs); this assumption is load-bearing for end-to-end training and is not automatic for arbitrary cost structures.

Authors: The referee correctly identifies that differentiability requires explicit assumptions. We will add a paragraph in the methods section specifying the smoothness conditions on the downstream problem and noting the use of differentiable relaxations (e.g., for argmax) where needed. revision: yes
Referee: [Validation] Validation section (implied by 'one synthetic and two real-world tasks'): no error bars, statistical significance tests, or details on hyperparameter selection for the loss weighting are reported; without these, the claimed 'targeted improvements in cost-sensitive regions' cannot be assessed as robust.

Authors: We agree that error bars, significance tests, and hyperparameter details are necessary for robustness assessment. We will revise the validation section to report these elements, including multiple-run statistics and the procedure used to select the loss weighting. revision: yes

Circularity Check

0 steps flagged

No load-bearing circularity; decision loss presented as external proper scoring rule property

full rationale

The paper augments the energy score with a differentiable decision loss and states that the combined loss is theoretically grounded because the decision loss is itself a proper scoring rule. This property is asserted as an independent fact rather than derived from or fitted to the same inputs within the paper. No equations or self-citations are shown that reduce the decision loss or the overall objective to a quantity defined by construction from the training data or prior fitted parameters. The central claim retains independent content from the energy score literature and the asserted properness of the added term.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the unproven assertion that the decision loss remains a proper scoring rule when added to the energy score and that the cost function is known and differentiable; no free parameters or invented entities are explicitly introduced in the abstract.

free parameters (1)

loss weighting hyperparameter
Likely required to balance energy score and decision loss but not mentioned in abstract.

axioms (1)

domain assumption The decision loss is itself a proper scoring rule
Invoked to ground the combined objective theoretically (abstract paragraph 3).

pith-pipeline@v0.9.1-grok · 5652 in / 1269 out tokens · 26324 ms · 2026-07-02T15:18:55.519433+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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arXiv preprint arXiv:2312.15796 , year=

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with weightw d ≥0. A.3. Finite-Sample Analysis of the Training Objective A.4. Derivation of per-sample gradient (Eq. (7)) Problem setup.Define f(a;{ˆyj}M j=1) := 1 M PM j=1 c(a,ˆyj); we write f(a) for brevity, making the sample dependence explicit only where the argument requires it. The optimal action solves min a f(a)s.t.a≥a min, a≤a max.(11) Since the ...

2022
[15]

Applying this withx=a,θ={ˆy j}M j=1, andF(a;{ˆyj}M j=1) :=f ′(a;{ˆyj}M j=1), the IFT requires∂F/∂a̸= 0ata ∗: ∂F ∂a a∗ = 1 M MX j=1 ∂2c ∂a2 (a∗,ˆyj) =:H >0,(17) where H >0 follows from strict convexity of f at a∗; the IFT therefore guarantees that a∗ is locally a smooth function of {ˆyj}M j=1. In our setting, a∗ is already defined as a function of {ˆyj}; t...

2025
[16]

with learning rate 10−3 to train the models over 2000 epochs. Loss scales of CRPS and decision loss are estimated once before training and fixed throughout; optionally, a short CRPS pre-training phase precedes scale estimation to ensure realistic samples and cost values. We perform 200 epochs of CRPS pre-training for the wind power dispatch task, since we...

2000
[17]

Some of the troughs (a∗ = 0, boundary case) that exist for wd = 0remain a blind spot throughout

fails to track the ground truth action at the transitions between the extremes; decision-aware training ( wd >0.0 ) improves the tracking of a∗ and gets closer to the ground truth for large wd. Some of the troughs (a∗ = 0, boundary case) that exist for wd = 0remain a blind spot throughout. 14 Decision-Aware Training for Sample-Based Generative Models −10 ...

2019

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Bonev, T

Bonev, B., Kurth, T., Mahesh, A., Bisson, M., Kossaifi, J., Kashinath, K., Anandkumar, A., Collins, W. D., Pritchard, M. S., and Keller, A. Fourcastnet 3: A geometric ap- proach to probabilistic machine-learning weather fore- casting at scale.arXiv preprint arXiv:2507.12144,

work page arXiv

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Ad- vanced strategies for wind power trading in short-term electricity markets

Bourry, F., Juban, J., Costa, L., and Kariniotakis, G. Ad- vanced strategies for wind power trading in short-term electricity markets. InEuropean Wind Energy Conference & Exhibition EWEC 2008, pp. 8–pages. EWEC,

2008

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Archesweather & archesweathergen: a de- terministic and generative model for efficient ml weather forecasting.arXiv preprint arXiv:2412.12971,

Couairon, G., Singh, R., Charantonis, A., Lessig, C., and Monteleoni, C. Archesweather & archesweathergen: a de- terministic and generative model for efficient ml weather forecasting.arXiv preprint arXiv:2412.12971,

work page arXiv

[4] [4]

S., Zhou, G., Murphy, K., Gretton, A., and Doucet, A

De Bortoli, V ., Galashov, A., Guntupatti, J. S., Zhou, G., Murphy, K., Gretton, A., and Doucet, A. Distributional diffusion models with scoring rules.arXiv preprint arXiv:2502.02483,

work page arXiv

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Derr, R., Finocchiaro, J., and Williamson, R. C. Three types of calibration with properties and their semantic and formal relationships.arXiv preprint arXiv:2504.18395,

work page arXiv

[6] [6]

Smooth calibration and decision making.arXiv preprint arXiv:2504.15582,

Hartline, J., Wu, Y ., and Yang, Y . Smooth calibration and decision making.arXiv preprint arXiv:2504.15582,

work page arXiv

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The era5 global reanalysis.Quarterly journal of the royal meteorological society, 146(730): 1999–2049,

Hersbach, H., Bell, B., Berrisford, P., Hirahara, S., Hor´anyi, A., Mu˜noz-Sabater, J., Nicolas, J., Peubey, C., Radu, R., Schepers, D., et al. The era5 global reanalysis.Quarterly journal of the royal meteorological society, 146(730): 1999–2049,

1999

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Kingma, D. P. and Ba, J. Adam: A method for stochastic optimization.arXiv preprint arXiv:1412.6980,

work page internal anchor Pith review Pith/arXiv arXiv

[9] [9]

Improved probabilistic regression using diffusion models.arXiv preprint arXiv:2510.04583,

Kneissl, C., Bulte, C., Scholl, P., and Kutyniok, G. Improved probabilistic regression using diffusion models.arXiv preprint arXiv:2510.04583,

work page arXiv

[10] [10]

C., Roberts, C., Adewoyin, R., Bouall`egue, Z

Lang, S., Alexe, M., Clare, M. C., Roberts, C., Adewoyin, R., Bouall`egue, Z. B., Chantry, M., Dramsch, J., Dueben, P. D., Hahner, S., et al. Aifs-crps: ensemble forecast- ing using a model trained with a loss function based on the continuous ranked probability score.arXiv preprint arXiv:2412.15832,

work page arXiv

[11] [11]

Learning in Implicit Generative Models

Mohamed, S. and Lakshminarayanan, B. Learn- ing in implicit generative models.arXiv preprint arXiv:1610.03483,

work page internal anchor Pith review Pith/arXiv arXiv

[12] [12]

arXiv preprint arXiv:2312.15796 , year=

Price, I., Sanchez-Gonzalez, A., Alet, F., Andersson, T. R., El-Kadi, A., Masters, D., Ewalds, T., Stott, J., Mohamed, S., Battaglia, P., et al. Gencast: Diffusion-based ensemble forecasting for medium-range weather.arXiv preprint arXiv:2312.15796,

work page arXiv

[13] [13]

Score-Based Generative Modeling through Stochastic Differential Equations

Song, Y ., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Er- mon, S., and Poole, B. Score-based generative modeling through stochastic differential equations.arXiv preprint arXiv:2011.13456,

work page internal anchor Pith review Pith/arXiv arXiv 2011

[14] [14]

with weightw d ≥0. A.3. Finite-Sample Analysis of the Training Objective A.4. Derivation of per-sample gradient (Eq. (7)) Problem setup.Define f(a;{ˆyj}M j=1) := 1 M PM j=1 c(a,ˆyj); we write f(a) for brevity, making the sample dependence explicit only where the argument requires it. The optimal action solves min a f(a)s.t.a≥a min, a≤a max.(11) Since the ...

2022

[15] [15]

Applying this withx=a,θ={ˆy j}M j=1, andF(a;{ˆyj}M j=1) :=f ′(a;{ˆyj}M j=1), the IFT requires∂F/∂a̸= 0ata ∗: ∂F ∂a a∗ = 1 M MX j=1 ∂2c ∂a2 (a∗,ˆyj) =:H >0,(17) where H >0 follows from strict convexity of f at a∗; the IFT therefore guarantees that a∗ is locally a smooth function of {ˆyj}M j=1. In our setting, a∗ is already defined as a function of {ˆyj}; t...

2025

[16] [16]

with learning rate 10−3 to train the models over 2000 epochs. Loss scales of CRPS and decision loss are estimated once before training and fixed throughout; optionally, a short CRPS pre-training phase precedes scale estimation to ensure realistic samples and cost values. We perform 200 epochs of CRPS pre-training for the wind power dispatch task, since we...

2000

[17] [17]

Some of the troughs (a∗ = 0, boundary case) that exist for wd = 0remain a blind spot throughout

fails to track the ground truth action at the transitions between the extremes; decision-aware training ( wd >0.0 ) improves the tracking of a∗ and gets closer to the ground truth for large wd. Some of the troughs (a∗ = 0, boundary case) that exist for wd = 0remain a blind spot throughout. 14 Decision-Aware Training for Sample-Based Generative Models −10 ...

2019