Feature Priming in Online Linear Regression: Sparse-Regret Lower Bounds and a Tight Univariate Rate

In high-dimensional online prediction, the best predictor may depend on only a few features, so regret should scale with sparsity rather than the ambient dimension. Feature priming pursues this goal by estimating feature weights from past data and refitting a minimum-norm predictor on the rescaled design. Warmuth and Amid asked at COLT 2023 whether any of three such rules admits a competitive online regret guarantee. Using the natural Moore--Penrose protocol based only on past data, we give a negative answer to the sparse-logarithmic form of this COLT open problem. Our analysis identifies a common obstruction: cheap nuisance interpolation causes the refit to underweight the truly predictive coordinate. An exact target-mass identity and a two-sign argument turn this effect into clipped prediction loss. Hadamard constructions force $Ω(\min\{T,\sqrt{d}\})$ regret for all three rules against a zero-loss one-sparse comparator, with extensions to fixed prime powers and selectors among the rules. Conversely, regret is controlled by data rank, and a Euclidean-normalized triangular construction matches this dependence for powered univariate priming, even under nonnegative second-stage ridge regularization; a paired ridge construction also covers all three powered rules. Exploratory diagnostics on frozen language-model activations exhibit the same relation among nuisance interpolation, target weight, and loss. The exact multivariate and Pearson frontiers remain open.

Bridging the Last Mile of Prediction: Enhancing Time Series Forecasting with Conditional Guided Flow Matching

Diffusion models, a type of generative model, have shown promise in time series forecasting. But they face limitations like rigid source distributions and limited sampling paths, which hinder their performance. Flow matching offers faster generation, higher-quality outputs, and greater flexibility, while also possessing the ability to utilize valuable information from the prediction errors of prior models, which were previously inaccessible yet critically important. To address these challenges and fully unlock the untapped potential of flow matching, we propose Conditional Guided Flow Matching (CGFM). CGFM extends flow matching by incorporating the outputs of an auxiliary model, enabling a previously unattainable capability in the field: learning from the errors of the auxiliary model. For time series forecasting tasks, it integrates historical data as conditions and guidance, constructs two-sided conditional probability paths, and uses a general affine path to expand the space of probability paths, ultimately leading to improved predictions. Extensive experiments show that CGFM consistently enhances and outperforms state-of-the-art models, highlighting its effectiveness in advancing forecasting methods.