D$^2$O: A Dual Debiasing Operator for Training-Free Test-Time Adaptation of Vision–Language Models

Training-free test-time adaptation (TTA) for vision-language models (VLMs) can improve zero-shot classification under mild shifts, but often degrades under severe style/environment variation. We identify two shared failure modes in training-free pipelines: (i) retrieval confounding, where feature similarity is dominated by nuisance/style variation and corrupts retrieval evidence; and (ii) environment-biased priors, where VLM logits exhibit environment-dependent centered shifts that distort gating and prior-like terms. Therefore, we propose D$^2$O, a training-free debiasing operator that outputs three inference-time objects: a retrieval-oriented content feature for semantic matching, a style-aware routing coordinate for bias tracking, and debiased logits for corrected priors. D$^2$O composes plug-and-play with retrieval-based and closed-form Gaussian adapters in online and transductive settings. We further provide operator-to-decision guarantees: finite-difference covariance recovers a nuisance-sensitive subspace, routing-based EMA controls centered-logit bias estimates, and these errors yield bounded posterior log-odds perturbations, leading to a margin-based condition for label invariance. Extensive experiments show that D$^2$O achieves its clearest gains under style/environment-dominant shifts, with broader gains elsewhere. Code is available at https://github.com/MAiTL-Group/D2O.