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Analysing Multi-Task Regression via Random Matrix Theory with Application to Time Series Forecasting
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Derives a closed-form, random-matrix-theory analysis of a regularized linear multi-task regression (shared plus task-specific hyperplanes), estimates train/test risk and optimal lambda, and applies the regularizer as an add-on loss to univariate forecasters (PatchTST, DLinear) for multivariate time series.
Problem
Multi-task regression in the high-dimensional regime lacks precise performance prediction and principled hyperparameter selection for the coupling parameter between shared and task-specific components; multivariate forecasting often relies on univariate models that ignore cross-channel information.
Approach
Formulates multi-task regression as a regularized objective with a shared hyperplane W0 and task deviations V_t, uses random matrix theory (deterministic equivalents for concentrated random vectors) to derive asymptotic train/test risks and a closed-form optimal lambda, estimates the needed quantities from data, and adds the resulting shared-transformation term to univariate forecasting losses.
Key results
- On ETTh1, the MTL-regularized PatchTST achieves MSE 0.385 at horizon 96 versus 0.387 without regularization, and MTL-regularized DLinearU reaches 0.367 vs 0.397 (horizon 96).
- Theory predictions of train and test MSE versus lambda closely match empirical curves on synthetic two-task data, including the optimal lambda location.
- MTL-regularized PatchTST and DLinearU are reported to outperform SAMformer and ITransformer on several ETT and Weather horizons, with ITransformer consistently beaten by at least one regularized method.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add MTL regularization to PatchTST | PatchTST · ETTh1, H=96/192/336/720 | MSE: 0.387 -> 0.385; 0.424 -> 0.422; 0.442 -> 0.433; 0.451 -> 0.430 | Helped |
| Add MTL regularization to PatchTST | PatchTST · ETTh2, H=96/192/336/720 | MSE: 0.295 -> 0.291; 0.351 -> 0.346; 0.342 -> 0.332; 0.393 -> 0.384 | Helped |
| Add MTL regularization to PatchTST | PatchTST · Weather, H=96/192/336/720 | MSE: 0.149 -> 0.148; 0.193 -> 0.190; 0.246 -> 0.242; 0.322 -> 0.316 | Helped |
| Add MTL regularization to DLinearU | DLinearU · ETTh1, H=96/192/336/720 | MSE: 0.397 -> 0.367; 0.422 -> 0.405; 0.431 -> 0.431; 0.428 -> 0.454 | Mixed |
| Add MTL regularization to DLinearU | DLinearU · ETTh2, H=96/192/336/720 | MSE: 0.294 -> 0.267; 0.361 -> 0.331; 0.361 -> 0.367; 0.395 -> 0.412 | Helped |
| Add MTL regularization to DLinearU | DLinearU · Weather, H=96/192/336/720 | MSE: 0.196 -> 0.149; 0.243 -> 0.206; 0.283 -> 0.249; 0.339 -> 0.326 | Helped |
| Add MTL regularization to Transformer | Transformer · ETTh1, H=96/192/336/720 | MSE: 0.370 -> 0.368; 0.411 -> 0.407; 0.437 -> 0.433; 0.470 -> 0.455 | Mixed |
| Add MTL regularization to Transformer | Transformer · ETTh2, H=96/192/336/720 | MSE: 0.273 -> 0.270; 0.339 -> 0.337; 0.369 -> 0.366; 0.428 -> 0.405 | Mixed |
| Add MTL regularization to Transformer | Transformer · Weather, H=96/192/336/720 | MSE: 0.170 -> 0.154; 0.214 -> 0.198; 0.260 -> 0.258; 0.326 -> 0.331 | Mixed |
| Use MTL regularization with univariate forecasting models | PatchTST, DLinearU, Transformer · ETTh1, ETTh2, Weather | MSE | Mixed |
Surprising findings
- Univariate models with the added shared-transformation regularizer are often competitive with or better than multivariate state-of-the-art models such as SAMformer and ITransformer.
Limitations
- The theory is restricted to linear models with concentrated random features and a growth regime with T=O(1) tasks.
- The forecasting extension is a heuristic add-on whose theoretical link to the linear analysis is not fully established, and the authors note the linear approach may have limited practical applicability.
- Experiments use only three seeds and a small set of benchmarks, with results reported on limited horizons.
Experimental setup
Datasets: Appliance Energy ETTh1 ETTh2 Weather
Benchmarks: ETTh1 ETTh2 Weather Appliance Energy
Metrics: MSE
Baselines: PatchTST DLinear SAMformer ITransformer
Builds on: random matrix theory deterministic equivalents multi-task regression framework PatchTST DLinear
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