PILOT: Probe-guided Imputation via Local Subspace Optimization for Time-Series Forecasting

Missing values can substantially degrade the forecasting performance of time series foundation models (TSFMs), while conventional impute-then-forecast pipelines optimize reconstruction rather than downstream forecasting. End-to-end task-aware training could address this mismatch but requires differentiating through the TSFM, which may be too large or accessible only as a black box. We propose Probe-guided Imputation via Local Subspace Optimization for Time-Series Forecasting (PILOT), a black-box-compatible method that improves imputation using only forward queries to the TSFM. PILOT estimates forecast-loss changes in a local optimization subspace using finite differences on the current batch and a probe sample, and applies a robust correction when the proposed update may conflict with the full-training forecasting objective. We establish conflict-inner-product preservation, second-order finite-difference accuracy, finite-population estimation guarantees, and directional protection under confidence-set coverage. Controlled experiments confirm the predicted $90^{\circ }$ conflict boundary and ${\epsilon }^2,{\eta }^2$ scaling, while experiments on five real-world time series show that PILOT improves test forecast MSE in 77 of 100 paired configurations.