Geometry-Aware Offline-to-Online Learning in Linear Contextual Bandits

We study offline-to-online learning in linear contextual bandits with biased offline regression data: the offline parameter need not match the online one, so history should not be treated as a single warm start. We model directional transfer with a shift certificate $(M_{\mathrm{shift}},ρ)$ and offline ridge estimation, yielding a geometry-aware confidence region for the online parameter rather than an isotropic radius. We propose \emph{Ellipsoidal-MINUCB}, which combines a standard online branch with an offline-informed pooled branch and uses offline information only when it tightens uncertainty. With high probability, regret is bounded by the minimum of a standard SupLinUCB-style fallback and a pooled term that separates statistical width from a certificate-weighted shift penalty. Under a simple alignment condition, the pooled term further simplifies to a rate governed by an effective dimension induced by the offline geometry. We also show that a purely Euclidean (scalar) shift bound, by itself, does not determine which feature directions are transferable. Beyond this fixed certificate, we show how to learn a data-driven certificate from data at finitely many refresh times and establish a high-probability regret bound for Ellipsoidal-MINUCB with epoch-wise learned certificates. Experiments match the main prediction: gains are strongest at intermediate horizons when offline coverage and transferability align, while the method otherwise tracks the safe online baseline.

Turb-L1: Achieving Long-term Turbulence Tracing By Tackling Spectral Bias

Accurately predicting the long-term evolution of turbulence is crucial for advancing scientific understanding and optimizing engineering applications. However, existing deep learning methods face significant bottlenecks in long-term autoregressive prediction, which exhibit excessive smoothing and fail to accurately track complex fluid dynamics. Our extensive experimental and spectral analysis of prevailing methods provides an interpretable explanation for this shortcoming, identifying Spectral Bias as the core obstacle. Concretely, spectral bias is the inherent tendency of models to favor low-frequency, smooth features while overlooking critical high-frequency details during training, thus reducing fidelity and causing physical distortions in long-term predictions. Building on this insight, we propose Turb-L1, an innovative turbulence prediction method, which utilizes a Hierarchical Dynamics Synthesis mechanism within a multi-grid architecture to explicitly overcome spectral bias. It accurately captures cross-scale interactions and preserves the fidelity of high-frequency dynamics, enabling reliable long-term tracking of turbulence evolution. Extensive experiments on the 2D turbulence benchmark show that Turb-L1 demonstrates excellent performance: (I) In long-term predictions, it reduces Mean Squared Error (MSE) by $80.3\%$ and increases Structural Similarity (SSIM) by over $9\times$ compared to the SOTA baseline, significantly improving prediction fidelity. (II) It effectively overcomes spectral bias, accurately reproducing the full enstrophy spectrum and maintaining physical realism in high-wavenumber regions, thus avoiding the spectral distortions or spurious energy accumulation seen in other methods.