Local and Global Recursive Ridge Regression for Online Residual Correction of Frozen Multivariate Forecasters under Label Delay

Completed forecasts provide residual feedback, but retaining full residual blocks increases auxiliary state. We propose Endpoint-Preserving Online Correction (EPOC) with a compressed residual state. It stores low-order discrete cosine transform (DCT) coefficients and the final value of the preceding residual block. Each channel shares the endpoint across component-wise online ridge regressions using current-forecast coefficients. We evaluate eight multivariate series, DLinear and PatchTST, three seeds, and two training variants: 96 matched fixed-base conditions at a 24-step horizon. EPOC achieves mean condition-wise reductions in mean squared error (MSE) and mean absolute error (MAE) of 15.40% and 9.35% from the uncorrected base, respectively, with a median of 6,352 B in retained auxiliary arrays. EPOC also outperforms the $δ$-Adapter, COSA, FAC, and OMPB in paired MSE on most conditions while retaining less state. Full ELF achieves the largest mean MSE reduction, 19.29%, but its median retained state is 474,048 B ($\times$75 relative to EPOC). Equal-size summary controls favor the endpoint by 1.65--2.20% in paired MSE; a coefficient-reconstructed endpoint yields similar accuracy to the observed endpoint, highlighting its role as a shared input. Increasing the retained DCT component count from 4 to 8 adds 1.00 percentage point of MSE reduction for 5,728 B. On jointly trained bases, EPOC lowers MSE by 16.69--20.15% relative to globally blended TEFL-style adapters applied to the same base. Code and numerical records are available at https://github.com/keiotakmin/endpoint-preserving-residual-correction.

Post-hoc residual correction can reduce forecasting error, but always-on updates often fail to generalize: when residual signals are weak or noisy, expressive correctors can overfit and even increase the overall forecasting errors. We introduce CRC, a robust residual correction framework that couples a hybrid corrector with a validation-calibrated selective deployment rule. CRC proposes corrections via a conservative ridge "floor" plus conditional nonlinear refinement, but deploys them only when held-out evidence indicates net error reduction over a frozen baseline forecaster. CRC treats when to revise a forecast as a validation-calibrated deployment decision, not only how to fit residuals. Across standard long-sequence benchmarks and multiple backbones, CRC improves horizon-averaged MSE and MAE in most dataset-backbone pairs, with particularly large gains on complex datasets (e.g., over 20% relative MSE reduction on Traffic). We additionally report the non-degradation rate (NDR) as a stability diagnostic of harmful updates and ablate the validation-calibrated "firewall" that stabilizes MSE gains.

Retrieval-augmented generation (RAG) complements parametric models with retrieved external evidence. The same idea is attractive for continuous-output regression, but directly reusing retrieved target values is often not robust when samples differ in output level, numerical scale, or local dynamics. Moreover, conventional forecasting pipelines generally use residuals for model optimization and error diagnosis, but do not retain individual historical residual examples as memory that can be accessed at inference time.For multivariate time-series forecasting, we propose RATL, a plug-in residual-retrieval and feedback-correction method. RATL freezes a base forecaster to construct retrieval keys and turns its historical forecast residuals into a train-only memory specific to that base model. At inference time, RATL retrieves residual trajectories from similar historical contexts subject to causal availability constraints, then uses a set-aware router operating over forecast blocks and variables to select and combine these trajectories. Experiments show that historical residuals matched to the current context contain reusable forecasting information and that RATL improves frozen base forecasters in most experimental settings. Ablations further show that learned routing strengthens raw residual feedback, while validation-based correction-strength selection limits residual over-injection.On real-world benchmarks, we use iTransformer as the primary frozen base forecaster, compare against multiple strong forecasting baselines, and test transferability across backbones. The results show that RATL can further improve base-forecaster performance in most settings.Overall, RATL shifts the retrieved object from historical target values to base-model-specific historical forecast errors, providing a plug-in, residual-memory-based paradigm for learned feedback correction in continuous-output forecasting.