Prediction Loss Guided Decision-Focused Learning

Chanyeong Kim; Haeun Jeon; Hyunglip Bae; Woo Chang Kim; Yongjae Lee

REVIEW 3 major objections 5 minor 1 cited by

Prediction Loss Guided Decision-Focused Learning

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Blending prediction and decision gradients makes decision-focused learning converge to a Pareto-stationary point with lower regret.

desk verdict The angle-bisecting update is a neat idea, but the main convergence theorem as written is not proven—the proof only shows lim inf of a stationarity measure is zero, assumes boundedness without saying so, and the claimed O(1/√T) rate is contradicted by the proof's own bound. read the letter →

arxiv 2509.08359 v1 pith:YK7J5OTM submitted 2025-09-10 cs.LG

Haeun Jeon , Hyunglip Bae , Chanyeong Kim , Yongjae Lee , Woo Chang Kim This is my paper

classification cs.LG

keywords decision-focusedlearningpredict-then-optimizegradientperturbationParetostationaritymulti-taskconflictend-to-endstochasticoptimizationregret

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

Decision-focused learning (DFL) trains a predictor by pushing gradients through a downstream optimizer, but its loss surface is flat in parts and sharp elsewhere, so plain DFL is unstable. This paper proposes to fix that by perturbing the decision-loss gradient with the prediction-loss gradient: early in training the update direction bisects the angle between the two gradients, then gradually shifts toward the decision loss as training proceeds. The merge uses a geometric-mean scale so neither gradient dominates, and a sigmoid-like decay parameter controls the handoff. The authors prove that with the decay switched off (kappa=0), the update converges to a point where neither loss can be improved without hurting the other, at an O(1/sqrt T) rate under smoothness assumptions. On knapsack, budget-allocation, and portfolio problems, the method reports the lowest normalized regret in most settings with consistently small standard errors.

What carries the argument

The central object is the merged gradient g = m * (alpha * u_pred + u_dec) / ||alpha * u_pred + u_dec||, where u_pred and u_dec are unit directions of the prediction and decision loss gradients, m is the geometric mean of their norms, and alpha is a sigmoid-like decaying parameter with an inflection point and steepness kappa. The bisecting-plus-decay construction guarantees, by Proposition 3.2, that the update never conflicts with the decision-loss gradient; with kappa=0 it never conflicts with either gradient. The geometric-mean scaling prevents the usually much larger prediction gradient from dominating the sum. The proof of Theorem 3.4 uses the aggregate loss L_pred + L_dec and shows that

What would settle it

For a smooth problem with exact decision gradients (portfolio optimization is one), run the kappa=0 update and record at every epoch the quantity the proof calls m_k psi_k: the combined progress toward both losses along the update direction. Theorem 3.4 predicts the minimum of this quantity over the first T epochs shrinks as O(T^-(1-alpha)) and tends to zero. If it stops decreasing while both individual gradient norms are clearly nonzero and their angle is below pi, the Pareto-stationary guarantee is falsified. A separate quick check: the paper's Proposition 3.2 says the merged update never ha

Watch

Extended reading notes

Core claim

On its own terms, the paper claims that the instability of vanilla DFL can be removed without surrogate losses and without extra training: simply replace the raw decision-loss gradient with a geometric-mean-scaled vector that bisects the angle between the prediction-loss and decision-loss gradients, then optionally let a decaying weight pull the direction toward the decision loss over epochs. With the decay switched off (kappa=0), the update never conflicts with either gradient, and Theorem 3.4 states that under Lipschitz-smooth losses and standard step-size conditions the iterates reach a Pareto-stationary point of L_pred and L_dec at rate O(1/sqrt T). The empirical claim is comparative: ac

Load-bearing premise

The convergence guarantee assumes exact, differentiable decision-loss gradients, but the experiments that showcase the method—knapsack and budget allocation—have discrete decisions and use approximate surrogate gradients, so the theorem does not cover the settings where the method is demonstrated.

Editorial extensions

If this is right

Any existing differentiable DFL solver can be wrapped with this update rule; no surrogate loss model or extra training loop is needed.
The kappa=0 variant carries a convergence guarantee to a Pareto-stationary point at O(1/sqrt T), matching the guarantees of gradient-surgery multi-task methods.
Because the update never conflicts with the decision-loss gradient, even when prediction and decision gradients disagree strongly and differ in scale by 10 to 1000 times, the decision loss can still improve.
On knapsack, budget allocation, and portfolio problems the method reports lower normalized regret with lower standard error than PFL, DFL, convex combinations, and gradient-perturbation baselines.
When either PFL or DFL underperforms, the blended update provides a fallback that maintains low regret.

Reading between the lines

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

An untested consequence: because the method works on discrete problems despite the exact-gradient assumption, it may act as a regularizer on surrogate-gradient DFL more broadly; a controlled study varying solver approximation quality would separate the method's geometric effect from the solver's noise.
The geometric-mean scale means the update magnitude shrinks toward zero when either gradient vanishes; near a point where one loss is flat, the method may slow down while a convex combination would still move. This late-training behavior is not analyzed in the paper.
The same angle-bisecting rule could be applied to other multi-objective training settings where one gradient is more reliable early and another is the true target, such as physics-informed or safety-constrained training; the paper only demonstrates predict-then-optimize problems.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know this paper for its method, not for its theorem. The update rule in Eqs. (4)–(6)—unit-gradient bisection, geometric-mean magnitude, sigmoid-shaped decay—is genuinely new in the DFL context. It is simple, plug-and-play, adds no training overhead, and the paper correctly positions it against PCGrad, MGDA, and DCGD. The geometric analysis of why naive convex combinations fail (near-zero decision gradients with conflicting directions) is useful, and the code is linked. So there is real substance here.

The soft spots are in the theory, and they are load-bearing. Theorem 3.4 says κ=0 converges to a Pareto stationary point and gives O(1/√T). The proof only establishes lim inf (m_k ψ_k)=0. Passing from that to a Pareto stationarity claim needs bounded iterates; the proof never shows the sequence is bounded. It also silently assumes m_k ≤ A, which is not in the theorem statement. The rate claim is worse: the proof's own bound is O(T^{-(1-α)}) with α∈(1/2,1), and since 1−α < 1/2, T^{-(1−α)} is asymptotically larger than T^{-1/2}. Writing “< O(1/√T)” is backwards. These are fixable if you add a compactness or boundedness assumption, but as stated the main guarantee does not hold.

On the empirical side, the method wins in most settings but the wins are frequently within standard error, and it loses on unweighted knapsack. The phrase “consistently outperforms” is too strong. Also, the theory assumes exact decision-loss gradients, while the knapsack and budget experiments use surrogate gradients from discrete solvers; the authors acknowledge this in the conclusion, so it is not hidden, but it does mean the theory does not cover the empirical setting. The recommended κ=1 variant has no convergence guarantee at all, which is worth saying plainly.

Who is this for? People working on predict-then-optimize and decision-focused learning. The method is worth discussing and probably worth citing as related work, even though the theoretical justification needs repair. I would send it to peer review—a serious referee can ask for the theorem to be fixed, the rate restated, and the empirical claims toned down. The idea is simple enough that this could become a useful part of the DFL toolbox once the analysis is cleaned up.

Referee Report

3 major / 5 minor

Summary. The paper proposes a gradient-perturbation method for decision-focused learning (DFL) in which the prediction-loss gradient is used to guide the decision-loss gradient. The update is g = m (α u_pred + u_dec)/||α u_pred + u_dec||, with α a sigmoid-like decay in the epoch number and m the geometric mean of the two gradient norms. The authors claim that the update never conflicts with either gradient when κ=0, that the κ=0 variant converges to a Pareto stationary point, and that the method improves training stability and regret across knapsack, budget allocation, and portfolio optimization problems. The theory is proved only for κ=0, while the experiments also use the decaying κ=1 variant. The manuscript includes a code link and reports normalized test regret with standard errors for five settings.

Significance. If the convergence claim were correct, the method would be a simple, plug-and-play addition to existing DFL pipelines: it requires no surrogate loss, no additional training, and only one extra gradient evaluation. The empirical study covers standard benchmarks and the code is made available. However, the central theoretical contribution is not established in the submitted form: the proof of Theorem 3.4 contains a false rate comparison and an unjustified convergence step, and the theorem covers only the κ=0 case while the method is also used with κ=1. The empirical gains over strong baselines are often small and not statistically verified. The idea is still worth pursuing, but the current manuscript overstates both its theoretical and empirical support.

major comments (3)

[Appendix A, proof of Theorem 3.4] The proof does not establish convergence to a Pareto stationary point. It shows only lim inf_{k→∞}(m_k ψ_k)=0, then concludes that along a subsequence one of three conditions holds and that this implies Pareto stationarity. This inference requires a convergent subsequence of iterates; Lipschitz continuity of gradients does not imply boundedness of the iterates, so no limit point is guaranteed. In addition, the proof introduces the assumption m_k ≤ A without stating it in the theorem. The rate claim is also reversed: for α∈(1/2,1), T^{-(1−α)} is asymptotically larger, not smaller, than T^{-1/2}, so the statement '< O(1/√T)' is false. The theorem as written is therefore unsupported.
[Sections 3.3–3.4 and 5] Theorem 3.4 covers only κ=0, i.e., a fixed bisector direction. The decayed variant κ=1, which is used in the experiments and motivated by the method description, has no convergence guarantee. As t grows, α tends to 0 and the update direction approaches u_dec; the behavior of this time-varying direction is not analyzed. The conclusion's statement that 'our method' converges to a Pareto stationary point is therefore overbroad. Please state precisely which variant is covered by which guarantee.
[Table 1 and Section 4.2] The empirical claim 'our approach is the only method that consistently outperforms across all tasks' is not supported. In the unweighted knapsack setting, the best regret is 0.063±0.040 (convex combination, PCGrad, MGDA), while the proposed method achieves 0.066±0.047 (κ=0) and 0.068±0.047 (κ=1), so the proposed method is not the best in that task. Moreover, the standard errors overlap for many entries, and no statistical tests are reported. Separately, the theoretical assumptions require differentiable exact decision-loss gradients, but the knapsack and budget-allocation experiments use binary decisions and surrogate gradients; this is acknowledged in the limitations, but it means the experiments do not directly validate the convergence theorem.

minor comments (5)

[Definition 3.1] Typo: 'gradeints' should be 'gradients'.
[Equation (4) and Appendix A] The symbol α is used both for the sigmoid-like decay parameter in Eq. (4) and for the step-size exponent in the proof of Theorem 3.4. This is confusing; please use different symbols.
[Appendix A, proof of Theorem 3.4] The proof says to stop if u_pred + u_dec = 0, but the update is undefined if either individual gradient is zero. Even if a zero gradient is Pareto stationary, the proof should state a convention for handling this case.
[Table 1] The table header contains 'PCGR A D' (should be 'PCGrad'). The β values for the convex-combination rows are not aligned with the row labels in the rendered table, making it hard to read.
[Section 5] 'An edge case worth exploring further.' is a sentence fragment; consider rephrasing as a complete sentence.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the update rule, convergence theorem, and experiments are self-contained, and no fitted parameter or self-citation is passed off as a prediction.

full rationale

The paper's central derivation is the definition of the merged gradient g in Eq. (6), g = m·(α u_pred + u_dec)/||α u_pred + u_dec||, with m the geometric mean of the two gradient norms and α a sigmoid-like decay. Theorem 3.4 then proves convergence to Pareto stationarity for κ=0 from standard descent-lemma arguments (Eq. 8-11) under Lipschitz continuity and step-size conditions. This is not circular: the update direction is not defined as a Pareto-stationary point, and the theorem derives the stationarity measure's liminf from the update rule and step-size assumptions rather than assuming it. The algorithm's hyperparameters (κ, c=50) are fixed before evaluation, and the test regret is computed from a held-out decision-loss comparison, not used to fit the update rule. There are no load-bearing self-citations; the cited works are external (Amos & Kolter, Wilder et al., Désidéri, Yu et al., etc.) and are used as prior method baselines, not as the justification for the paper's own guarantee. The explicit limitation in Section 5 that the theory assumes exact decision-loss gradients while experiments use surrogate gradients for discrete problems is a theory-experiment gap, not a circularity. The proof does contain rigor issues (e.g., the unstated assumption m_k≤A, extracting a convergent subsequence without boundedness of x_k, and the rate claim O(T^{-(1-α)}) < O(1/√T) is arithmetically reversed since 1-α<1/2), but these are correctness concerns, not instances of a claim reducing to its own input. Therefore the circularity score is 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central method introduces two hand-chosen hyperparameters (κ, c) but no fitted constants. The theoretical results depend on standard smoothness and step-size assumptions. The key domain assumption is differentiability of the decision loss, which is explicitly acknowledged as a limitation and is violated in the discrete experiments.

free parameters (2)

κ (decay steepness) = 0 or 1
Chosen by hand from {0,1}; controls how quickly the prediction-loss gradient influence decays. Not fitted to data.
c (inflection point) = 50
Fixed at epoch 50 for all experiments; controls the midpoint of the sigmoid-like decay schedule. Chosen a priori, not tuned per problem.

assumptions (5)

domain assumption L_pred and L_dec are differentiable with M_i-Lipschitz continuous gradients
Assumed in Theorem 3.4 for the convergence proof. Does not hold for discrete decision problems (knapsack, budget allocation) where the mapping a*(y) is piecewise constant.
domain assumption Exact decision loss gradients are available
Stated in the Conclusion as a limitation: 'the current formulation assumes access to exact decision loss gradients.' This is needed to define ∇L_dec in Eqs (3) and (6).
domain assumption The decision loss L_dec and aggregate L = L_pred + L_dec are bounded below
Used in the proof of Theorem 3.4 to apply the telescoping sum argument.
standard math Step sizes satisfy sum η_k = infinity and sum η_k^2 < infinity
Standard Robbins-Monro step-size condition for nonconvex stochastic optimization, stated in Theorem 3.4.
domain assumption Differentiable solvers provide usable gradients for discrete optimization in the experiments
The experiments use differentiable solvers (Agrawal et al. 2019; Wilder et al. 2019) to backpropagate through discrete optimization, which provides surrogate gradients whose relationship to the true L_dec is not formally analyzed.

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Cite this review

Pith. "Pith review of Prediction Loss Guided Decision-Focused Learning." pith.science (2026). https://pith.science/paper/YK7J5OTM

@misc{pith2026250908359,
  author       = {Pith},
  title        = {Pith review of: Prediction Loss Guided Decision-Focused Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YK7J5OTM}},
  note         = {Machine review of arXiv:2509.08359}
}

read the original abstract

Decision-making under uncertainty is often considered in two stages: predicting the unknown parameters, and then optimizing decisions based on predictions. While traditional prediction-focused learning (PFL) treats these two stages separately, decision-focused learning (DFL) trains the predictive model by directly optimizing the decision quality in an end-to-end manner. However, despite using exact or well-approximated gradients, vanilla DFL often suffers from unstable convergence due to its flat-and-sharp loss landscapes. In contrast, PFL yields more stable optimization, but overlooks the downstream decision quality. To address this, we propose a simple yet effective approach: perturbing the decision loss gradient using the prediction loss gradient to construct an update direction. Our method requires no additional training and can be integrated with any DFL solvers. Using the sigmoid-like decaying parameter, we let the prediction loss gradient guide the decision loss gradient to train a predictive model that optimizes decision quality. Also, we provide a theoretical convergence guarantee to Pareto stationary point under mild assumptions. Empirically, we demonstrate our method across three stochastic optimization problems, showing promising results compared to other baselines. We validate that our approach achieves lower regret with more stable training, even in situations where either PFL or DFL struggles.

Figures

Figures reproduced from arXiv: 2509.08359 by the authors.

**Figure 2.** Comparison of prediction loss gradients ∇Lpred and decision loss gradients ∇Ldec during training on the budget allocation problem. Each subplot represents: (a) the cosine similarity between ∇Lpred and ∇Ldec, (b) the gradient norm ratio ∥∇Ldec∥/∥∇Lpred∥ on a log scale, and (c) an illustrative case where two gradients with characteristics of (a) and (b) are naively added. In subplot (a), cosine similarity values near … view at source ↗

**Figure 3.** This figure illustrates the update directions used by our method and the baselines. From top-left to bottom-right, [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

End-to-End Fairness Optimization with Fair Decision-Focused Learning
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Jointly optimizing prediction accuracy, prediction disparity, and decision regret during training yields fairer prediction-informed resource allocations, with closed-form decision Jacobians for α-fair allocation problems.

Reference graph

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

, " * write output.state after.block = add.period write newline

ENTRY address archivePrefix author booktitle chapter edition editor eid eprint howpublished institution isbn journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.a...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

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Agrawal, A.; Amos, B.; Barratt, S.; Boyd, S.; Diamond, S.; and Kolter, J. Z. 2019. Differentiable convex optimization layers. Advances in neural information processing systems, 32

work page 2019

[4] [4]

Amos, B.; and Kolter, J. Z. 2017. Optnet: Differentiable optimization as a layer in neural networks. In International Conference on Machine Learning, 136--145. PMLR

work page 2017

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Amos, B.; Koltun, V.; and Kolter, J. Z. 2019. The limited multi-label projection layer. arXiv preprint arXiv:1906.08707

work page Pith review arXiv 2019

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T.; Mukadam, M.; and Amos, B

Bansal, D.; Chen, R. T.; Mukadam, M.; and Amos, B. 2024. Taskmet: Task-driven metric learning for model learning. Advances in Neural Information Processing Systems, 36

work page 2024

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Berthet, Q.; Blondel, M.; Teboul, O.; Cuturi, M.; Vert, J.-P.; and Bach, F. 2020. Learning with differentiable pertubed optimizers. Advances in neural information processing systems, 33: 9508--9519

work page 2020

[8] [8]

D \'e sid \'e ri, J.-A. 2012. Multiple-gradient descent algorithm (MGDA) for multiobjective optimization. Comptes Rendus Mathematique, 350(5-6): 313--318

work page 2012