Dual-Calibrated Online Conformal Inference with Hybrid Quantile Loss

Conformal inference provides rigorous coverage guarantees for uncertainty quantification, yet its dynamic coverage accuracy is highly susceptible to distribution shifts in non-stationary environments. Existing approaches construct prediction sets via dynamically estimated conditional quantiles but suffer from two key limitations. First, most algorithms passively correct for coverage deviations, which often results in excessively conservative prediction intervals. Second, prevailing update strategies rely on the standard quantile loss, whose inherent drawbacks undermine coverage adaptability. To address these issues, we introduce an $\epsilon$-insensitive Hybrid Quantile ($\epsilon$-HQ) loss with a bounded quadratic component, formally establish its Lipschitz continuity and generalized weak pseudo-convexity (GWPC), and design a gain-balancing mechanism to match gain magnitudes between the two components. Building upon this loss, we propose a Dual-Calibrated Online Conformal Inference framework, $\epsilon$-HQ‑DCOCI. The first calibration refines gradient signals via adaptive residual mapping and insensitive-zone tuning, enabling continuous feedback correction. The second calibration accelerates threshold updates under persistent drift via momentum-based smoothing of historical subgradient directions. Theoretically, we prove that $\epsilon$-HQ‑DCOCI achieves sublinear regret bounds and approximately valid coverage in non-stationary settings. Experiments on real-world datasets, including financial time series and medical image classification, demonstrate that our framework consistently yields tighter and more efficient prediction sets compared to state-of-the-art baselines.