SHADE: Shrinkage via Halted Alpha-stable Denoising for Heavy-Tailed Probabilistic Time Series Forecasting

Time-series observations are particularly susceptible to noise, and many real-world series have low signal-to-noise ratios (SNRs) together with heavy-tailed and time-varying disturbances. When Gaussian diffusion models are used for forecasting, reversing all the way to the zero-noise endpoint may yield an overly narrow and overconfident predictive distribution. We propose SHADE, an alpha-stable L\'evy diffusion forecaster that reads out at a nonzero depth so that the residual stable kernel acts as distribution-level shrinkage. By decomposing halting risk into smoothing bias and estimation error, we develop a scale-calibrated halt for estimating the optimal shrinkage $\tau^*$ and a context-amortized halt that requires no explicit noise labels. Gains concentrate in the extreme heavy-tailed regime $\alpha\leq1.3$: Across synthetic noise settings, median CRPS improves by 21.7\%, while the NLL falls significantly; Cryptocurrency and foreign-exchange data provide further evidence of improved tail calibration in real-data implement. On near-light-tailed or high-SNR data, the model reverts to Gaussian diffusion or full denoising. SHADE maintains overall forecast error comparable to strong baselines while providing more honest tail uncertainty for heavy-tailed series.