Black-Box Assisted Regression: Phase Transitions and Minimax Optimality

• URL: http://arxiv.org/abs/2606.25743v1
• Date: Wed, 24 Jun 2026 12:11:59 GMT
• Title: Black-Box Assisted Regression: Phase Transitions and Minimax Optimality
• Abstract summary: Foundation models are often used as fixed black-box predictors for downstream tasks with limited labeled data.<n>We give a finite-sample minimax characterization showing a phase transition at $_c(n) asymp n-/(2+d)$, with leading risk $min2, n-2/(2+d)$.<n>We analyze a Safe Residual Estimator: it learns a correction around $f_0$, initializes the residual head at zero so the initial predictor equals $f_0$, and uses holdout selection
• Score: 13.030603690976202
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Foundation models are often used as fixed black-box predictors for downstream tasks with limited labeled data, but their predictions may be biased and unsafe to trust blindly. We study this setting through black-box assisted nonparametric regression: a learner observes labeled samples and can query a fixed predictor $f_0$, while the target $f^*$ is close to $f_0$ in $L_2(P_X)$ up to an unknown radius $δ$. We give a finite-sample minimax characterization showing a phase transition at $δ_c(n) \asymp n^{-β/(2β+d)}$, with leading risk $\min\{δ^2, n^{-2β/(2β+d)}\}$. We then analyze a Safe Residual Estimator: it learns a correction around $f_0$, initializes the residual head at zero so the initial predictor equals $f_0$, and uses holdout selection to revert to $f_0$ when the learned correction is not supported by validation data. Here, "safe" means avoiding negative transfer, i.e., performing worse than the black-box predictor alone. The estimator matches the leading minimax term up to an additive validation-selection cost. Synthetic regression experiments verify the predicted phase transition, while CIFAR-100 with CLIP and AG News with Qwen3-8B provide practice-facing evidence that the same residual-correction tradeoff is useful beyond the formal squared-loss regression setting.

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