Research map / Time series / Long-term forecasting
Loss Shaping Constraints for Long-Term Time Series Forecasting
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Proposes loss shaping constraints for multi-step time series forecasting, bounding per-step expected loss with a resilient (relaxed) primal-dual algorithm to flatten error distribution across the horizon while keeping average MSE competitive.
Problem
Multi-step long-term time series forecasting optimizes window-averaged loss, which can leave very uneven errors across forecasting steps for transformer models.
Approach
Impose per-time-step upper bounds on expected loss as constraints (P-LS), relax them with learnable resilient slacks that penalize deviation, and solve the empirical dual with a primal-dual loss shaping algorithm; duality gap bounds are provided for the nonconvex setting.
Key results
- Across experiments, loss shaping constrained models reduced Window STD by 3.47% on average while MSE decreased by 4.47% on average relative to ERM.
- Resilient FiLM achieves the lowest Window STD compared to constrained and ERM-trained models.
- On ECL with Exchange-rate-style exponential constraints, the exponential constraint achieves the lowest MSE across all windows for two of three models.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| apply constant loss-shaping constraints | transformer forecasting models · 288 experiment settings across forecasting datasets and horizons | Window STD: STD decreases by 3.47% | Helped |
| apply constant loss-shaping constraints | transformer forecasting models · 288 experiment settings across forecasting datasets and horizons | MSE: MSE by 4.47% | Helped |
| use resilient constraints instead of hard constraints (ablation) | constrained forecasting models · forecasting benchmarks | Window STD | Helped |
| use resilient constraints instead of hard constraints (ablation) | iTransformer · ETTh2, ETTm1, and ETTm2 | MSE | Helped |
| use exponential rather than constant constraints (ablation) | Nonstationary Transformer · ECL, 96 | MSE: 0.178 -> 0.170 | Helped |
| use exponential rather than constant constraints (ablation) | Pyraformer · ECL, 96 | MSE: 0.287 -> 0.274 | Helped |
| use exponential rather than constant constraints (ablation) | Reformer · ECL, 96 | MSE: 0.299 -> 0.298 | Mixed |
| use exponential rather than constant constraints (ablation) | forecasting models · ECL, prediction windows 96, 192, 336, and 720 | Window STD | Mixed |
| add resilient relaxation to hard constraints (ablation) | Autoformer · ETTh1, 96 | MSE: 0.644 -> 0.606 | Helped |
| add resilient relaxation to hard constraints (ablation) | iTransformer · ETTh2, 96 | MSE: 0.203 -> 0.190 | Helped |
| add resilient relaxation to hard constraints (ablation) | FiLM · Exchange Rate, 96 | MSE: 0.094 -> 0.100 | Hurt |
| apply loss-shaping constraints | FiLM · Traffic, 96 | MSE: 0.411 -> 0.652 | Hurt |
Surprising findings
- Constraint violation is not used as a metric because constraints are often infeasible at test time due to generalization gap, making loss shaping ineffective on test data for some setups.
- Relative effect of the method on MSE and Window STD stays roughly constant as prediction length grows, so absolute gains increase with horizon.
Limitations
- Constraints are often infeasible at training end or fail to transfer to test data due to generalization gap.
- Effect varies significantly across architectures, with some models resisting the imposed loss landscape.
- Only constant and exponential-style constraint choices explored; grid search over six epsilon values per setting.
Experimental setup
Datasets: ECL Weather Exchange Rate Traffic ETT ILI
Benchmarks: ECL Weather Exchange Rate Traffic ETT ILI
Metrics: MSE Window STD
Baselines: ERM Autoformer FiLM Informer Nonstationary Transformer Pyraformer Reformer Transformer iTransformer
Builds on: resilient constrained learning Autoformer iTransformer Informer
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