The Procrustean Bed of Time Series: The Optimization Bias in Point-wise Loss Functions
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2025
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| author | Cai, Rongyao Wan, Yuxi Zhang, Kexin Jin, Ming Ge, Zhiqiang Dong, Daoyi Yu, Hang Liu, Yong Wen, Qingsong |
| author_facet | Cai, Rongyao Wan, Yuxi Zhang, Kexin Jin, Ming Ge, Zhiqiang Dong, Daoyi Yu, Hang Liu, Yong Wen, Qingsong |
| contents | Intuitively, a more deterministic time series should be easier to forecast. However, point-wise loss functions (e.g., MSE and MAE), serving as differentiable surrogates for the ideal optimization target, score each timestamp independently and therefore disregard temporal dependence. This mismatch induces a systematic optimization bias that cannot be eliminated merely by improving model expressiveness or optimizer. To formalize this issue, we define the Expectation of Optimization Bias (EOB) as the Kullback--Leibler divergence between the true joint distribution and the factorized i.i.d. surrogate induced by the point-wise paradigm. Under covariance-stationary Gaussian assumptions, we derive closed-form expressions for the stochastic component of EOB, establishing it as an irreducible lower bound on the total bias in linear systems, and further extend it to nonlinear regimes through a Gaussian mixture model lower bound. Crucially, we prove this bias is governed intrinsically by two data properties, i.e., sequence length and Structural Signal-to-Noise Ratio (SSNR), regardless of specific model architecture, optimizer, or point-wise loss forms. This theory motivates a principled debiasing program based on sequence length reduction and structural orthogonalization, which we instantiate through DFT/DWT combined with a novel harmonized $\ell_p$ norm. Extensive experiments validate the predicted SSNR--horizon dynamics, resolve the classic trigonometric fitting failure as an objective-induced pathology, and demonstrate substantial plug-and-play gains. Notably, on iTransformer, our proposed objective reduces average MSE/MAE by 5.2%/5.0% in forecasting across 11 datasets and by 27.4%/19.4% in imputation across 9 datasets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_18610 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Procrustean Bed of Time Series: The Optimization Bias in Point-wise Loss Functions Cai, Rongyao Wan, Yuxi Zhang, Kexin Jin, Ming Ge, Zhiqiang Dong, Daoyi Yu, Hang Liu, Yong Wen, Qingsong Machine Learning Intuitively, a more deterministic time series should be easier to forecast. However, point-wise loss functions (e.g., MSE and MAE), serving as differentiable surrogates for the ideal optimization target, score each timestamp independently and therefore disregard temporal dependence. This mismatch induces a systematic optimization bias that cannot be eliminated merely by improving model expressiveness or optimizer. To formalize this issue, we define the Expectation of Optimization Bias (EOB) as the Kullback--Leibler divergence between the true joint distribution and the factorized i.i.d. surrogate induced by the point-wise paradigm. Under covariance-stationary Gaussian assumptions, we derive closed-form expressions for the stochastic component of EOB, establishing it as an irreducible lower bound on the total bias in linear systems, and further extend it to nonlinear regimes through a Gaussian mixture model lower bound. Crucially, we prove this bias is governed intrinsically by two data properties, i.e., sequence length and Structural Signal-to-Noise Ratio (SSNR), regardless of specific model architecture, optimizer, or point-wise loss forms. This theory motivates a principled debiasing program based on sequence length reduction and structural orthogonalization, which we instantiate through DFT/DWT combined with a novel harmonized $\ell_p$ norm. Extensive experiments validate the predicted SSNR--horizon dynamics, resolve the classic trigonometric fitting failure as an objective-induced pathology, and demonstrate substantial plug-and-play gains. Notably, on iTransformer, our proposed objective reduces average MSE/MAE by 5.2%/5.0% in forecasting across 11 datasets and by 27.4%/19.4% in imputation across 9 datasets. |
| title | The Procrustean Bed of Time Series: The Optimization Bias in Point-wise Loss Functions |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2512.18610 |