# Differentiable Profit-Ordering Loss

- ID: 1989
- Canonical URL: https://synthcore.org/idea/1989/differentiable-profit-ordering-loss
- API JSON: https://synthcore.org/api/idea/1989.json
- API Markdown: https://synthcore.org/api/idea/1989.md
- Verification status: failed_benchmark
- Source: [arXiv:2608.08377](https://arxiv.org/abs/2608.08377)
- Category: training
- Solves: accuracy, sample-efficiency, generalization
- ML areas: optimizer, loss, training-dynamics, rl
- Math tags: optimization, statistics, control-theory
- Ratings: usefulness 8/10; difficulty 5/10; novelty 5/10

## Idea description

Train a neural forecaster or policy network to preserve the pairwise ordering that determines profitable charge and discharge decisions, rather than optimizing only pointwise forecast error. Combine a conventional prediction loss with a pairwise ranking loss weighted by the economic price gap, then pass the prediction through a feasibility-aware storage scheduler.

## Mathematical statement

Let Π = (π₁,...,πₙ) be the true price vector, C be observed context, S be an information signal, and Π̂ = fθ(C,S) be the neural prediction. Let x = g(Π̂,C) belong to the feasible schedule set X. Revenue is J = ⟨x,Π⟩ = Σₜ xₜπₜ. For each time pair (i,j), define the true ordering label yᵢⱼ = sign(πᵢ − πⱼ), predicted margin mᵢⱼ = π̂ᵢ − π̂ⱼ, and economic weight wᵢⱼ = |πᵢ − πⱼ|. The weighted logistic ranking loss is Lrank = (1/|P|) Σᵢⱼ wᵢⱼ log(1 + exp(−yᵢⱼmᵢⱼ/τ)), where P is the sampled pair set and τ > 0 is a temperature. The combined objective L = Lforecast + λLrank makes mistakes on large price gaps more expensive. The paper predicts that realized schedule revenue should correlate more strongly with ordering agreement than with MSE.

## Key formulas

- $$J(g,S)=E[⟨g(S,C),Π⟩],  ⟨x,Π⟩=Σₜ xₜπₜ,  x∈X$$
- $$yᵢⱼ=sign(πᵢ−πⱼ),  mᵢⱼ=π̂ᵢ−π̂ⱼ,  wᵢⱼ=|πᵢ−πⱼ|$$
- $$Lrank(θ)=(1/|P|)Σ₍ᵢ,ⱼ₎∈P wᵢⱼ log(1+exp(−yᵢⱼ(π̂ᵢ−π̂ⱼ)/τ))$$
- $$L(θ)=Lforecast(θ)+λLrank(θ)$$

## Implementation notes

Use a day-ahead price forecaster whose output is consumed by a deterministic storage scheduler. The network receives context and history, outputs n predicted prices, and the scheduler computes a feasible schedule using a linear program or differentiable optimization layer. For every training day, construct all pairs when n is small, or sample pairs containing the largest price gaps and pairs near the predicted charge/discharge threshold. Compute yᵢⱼ, wᵢⱼ, and the predicted margin, then optimize the stated loss. Pseudocode: `pred = net(context); pairs = sample_pairs(price); y = sign(price[i]-price[j]); w = abs(price[i]-price[j]); rank = mean(w * softplus(-y * (pred[i]-pred[j]) / tau)); loss = mse(pred, price) + lambda * rank; loss.backward(); optimizer.step()`. The paper supplies the ordering mechanism and predicts that rank agreement is economically dominant; estimate λ, τ, and scheduler gradient treatment empirically. First test an MLP and a small Transformer on the 939 French day-ahead days, with MSE training as baseline and identical storage simulation for all methods. Measure RMSE, Kendall rank correlation, pairwise agreement, and feasible revenue uplift. The quantitative signature is that revenue should be explained more strongly by rank agreement than by MSE; in a controlled ordering sweep, target the paper’s reported rank-agreement R² of approximately 0.99 versus 0.34 for the information-bound proxy.

## Verification

- Status: failed_benchmark
- Mechanism evidence: yes
- Mechanism confirmed: no
- Practical verdict: no_effect
- Verdict: Built a weighted pairwise logistic ranking loss, synthetic MLP forecaster, and deterministic feasible storage scheduler in run_experiment.py. The mechanism checks passed: the zero-margin gradient and equal-margin loss both scaled 8.0x for an 8.0x economic gap, while the rank-gradient contribution scaled linearly with lambda and was exactly zero at lambda=0. In the mini experiment, ranking slightly improved pairwise agreement (0.9071 vs 0.9056) but produced identical revenue and slightly worse RMSE, so there is no demonstrated economic win in this setup.

### Mechanism check

- Verdict: Built a weighted pairwise logistic ranking loss, synthetic MLP forecaster, and deterministic feasible storage scheduler in run_experiment.py. The mechanism checks passed: the zero-margin gradient and equal-margin loss both scaled 8.0x for an 8.0x economic gap, while the rank-gradient contribution scaled linearly with lambda and was exactly zero at lambda=0. In the mini experiment, ranking slightly improved pairwise agreement (0.9071 vs 0.9056) but produced identical revenue and slightly worse RMSE, so there is no demonstrated economic win in this setup.
- Confidence: 8/10
- Limitations: This is a synthetic 240-day benchmark rather than the 939-day French dataset; the scheduler is greedy rather than an LP or differentiable optimization layer, and only one ranking weight and temperature were used in the neural comparison. The toy lambda sweep verifies gradient scaling, not a guaranteed monotonic trained-ordering improvement.

### Practical benchmark

- Paired seeds: 0
- Baseline mean: 0
- Idea mean: 0
- p-value: 1
- Paired wins: 0/0
- Benchmark verdict: no measurable effect

## Artifacts

- [bench_experiment.py](https://synthcore.org/code/756/bench_experiment.py)
- [report.md](https://synthcore.org/code/756/report.md)
- [report_bench_2026-09-01T024636.md](https://synthcore.org/code/756/report_bench_2026-09-01T024636.md)
- [results.json](https://synthcore.org/code/756/results.json)
- [run_experiment.py](https://synthcore.org/code/756/run_experiment.py)
- [Download all files as ZIP](https://synthcore.org/download/756)

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.