Identified-Set Geometry of Distributional Model Extraction under Top-K Censored API Access

Indistinguishability properties such as differential privacy bounds or low empirically measured membership inference are widely treated as proxies to show a model is sufficiently protected against broader memorization risks. However, we show that indistinguishability properties are neither sufficient nor necessary for preventing data extraction in LLM APIs. We formalize a privacy-game separation between extraction and indistinguishability-based privacy, showing that indistinguishability and inextractability are incomparable: upper-bounding distinguishability does not upper-bound extractability. To address this gap, we introduce $(l, b)$-inextractability as a definition that requires at least $2^b$ expected queries for any black-box adversary to induce the LLM API to emit a protected $l$-gram substring. We instantiate this via a worst-case extraction game and derive a rank-based extraction risk upper bound for targeted exact extraction, as well as extensions to cover untargeted and approximate extraction. The resulting estimator captures the extraction risk over multiple attack trials and prefix adaptations. We show that it can provide a tight and efficient estimation for standard greedy extraction and an upper bound on the probabilistic extraction risk given any decoding configuration. We empirically evaluate extractability across different models, clarifying its connection to distinguishability, demonstrating its advantage over existing extraction risk estimators, and providing actionable mitigation guidelines across model training, API access, and decoding configurations in LLM API deployment. Our code is publicly available at: https://github.com/Emory-AIMS/Inextractability.

We present Caliber, an output-perturbation defense against model extraction that formulates noise selection as a calibration problem: how much the defense degrades the supervision signal used to train a surrogate, and the provable per-input query cost of recovering the clean logits. To defend against an attacker that uses returned scores for knowledge distillation, Caliber adds independent and identically distributed Gaussian noise to the internal logits. We establish two properties of the resulting perturbed predictions. Monotone agreement degradation: When the clean logits have a unique maximizer, agreement with the clean prediction decreases strictly with the noise scale, so every target in $(1/K,1)$ corresponds to a unique positive scale; task accuracy is bounded by computable lower and upper envelopes. Per-input recovery cost: We derive a closed-form minimax lower bound on the repeated queries needed to recover the clean logits for a fixed input. Caliber normalizes noise variance by the squared median top-two logit margin and fits the resulting noise-utility relationship with a logistic curve, either per model or shared within a task. Across more than thirty model-dataset combinations, per-model calibration achieves mean absolute relative errors of 0.6-1.4%. End-to-end experiments show that surrogate performance generally tracks the configured degradation, while fixed-input averaging follows the expected variance reduction.

Knowledge distillation (KD) transfers capabilities from large language models (LLMs) to smaller students, yet it can fail unpredictably and also underpins model leakage risks. Our analysis revealed several distillation traps: tail noise, off-policy instability, and, most fundamentally, the teacher-student gap, that distort training signals. These traps manifest as overconfident hallucinations, self-correction collapse, and local decoding degradation, causing distillation to fail. Motivated by these findings, we propose a post-hoc calibration method that, to the best of our knowledge, for the first time enables control over a teacher's distillability via reinforcement fine-tuning (RFT). Our objective combines task utility, KL anchor, and across-tokenizer calibration reward. This makes distillability a practical safety lever for foundation models, connecting robust teacher-student transfer with deployment-aware model protection. Experiments across math, knowledge QA, and instruction-following tasks show that students distilled from distillable calibrated teachers outperform SFT and KD baselines, while undistillable calibrated teachers retain their task performance but cause distilled students to collapse, offering a practical knob for both better KD and model IP protection.