The Procrustean Bed of Time Series: The Optimization Bias in Point-wise Loss Functions

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 ${\mathrm{\ell }}_p$ 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.