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Unsupervised Conformal Inference: Bootstrapping and Alignment to Control LLM Uncertainty

Conference: ICLR 2026 arXiv: 2509.23002 Code: None Area: Image Generation Keywords: Unsupervised conformal inference, bootstrapping, LLM hallucination detection, Gram matrix, conformal alignment

TL;DR

This paper proposes an unsupervised conformal inference framework (BB-UCP) that achieves distribution-free finite-sample coverage guarantees for LLM generation under label-free, API-compatible conditions, via Gram matrix interaction energy scoring, batch bootstrap calibration, and conformal alignment, effectively detecting and filtering hallucinated outputs.

Background & Motivation

State of the Field

Background: Uncertainty quantification (UQ) for LLMs is critical for trustworthy AI. In black-box API settings, gradients, logits, and hidden states are inaccessible, requiring decisions based solely on sampled outputs. Existing methods include semantic entropy, self-consistency, and embedding-geometry-based approaches.

Limitations of Prior Work: (1) Conformal prediction (CP) provides distribution-free finite-sample guarantees, but generative tasks break the classical supervised setup — text prompts are not quantifiable covariates; (2) Full-UCP requires recomputation for each candidate (computationally intensive), while Split-UCP is data-inefficient (high splitting cost); (3) Existing methods fail to align low-cost signals with high-cost quality metrics through calibration.

Key Challenge: A label-free, API-compatible, theoretically grounded test-time filtering mechanism is needed, yet existing CP methods are ill-suited for unsupervised generative settings.

Goal: How to provide theoretically guaranteed quality control for LLM generation under a label-free, black-box-API-only setting?

Key Insight: Leveraging the geometric signal of the Gram matrix of response embeddings as a consistency score, and designing batched, bootstrap-enhanced unsupervised conformal prediction, with conformal alignment to calibrate cheap signals against quality predicates.

Core Idea: Gram matrix interaction energy quantifies response typicality; bootstrap calibration stabilizes thresholds; conformal alignment binds geometric signals to factuality objectives.

Method

Overall Architecture

A three-layer framework: (1) Gram matrix atypicality scoring — measuring the alignment of each response with others in the same group; (2) BB-UCP — batch bootstrap calibration for exact quantile estimation; (3) Conformal alignment — calibrating a single strictness parameter \(\tau\) so that a user-defined predicate holds on unseen batches with probability \(\geq 1-\alpha\).

Key Designs

  1. Atypicality Scoring:

    • Interaction energy \(e(i;G) = \|G_{:,i}\|_2 = \|Vv_i\|_2\); under unit-norm embeddings, \(e(i;G) = (\sum_j \cos^2\theta_{ij})^{1/2}\)
    • Bounds: \(1 \leq e(i;G) \leq \sqrt{n}\) (Theorem 2.1)
    • Atypicality score \(\Phi(i;G) = 1 - {e(i;G)}/{B_E}\); higher values indicate greater novelty/atypicality

  2. Batch-Bootstrapped UCP (BB-UCP):

    • B-UCP: within-batch leave-one-out residuals \(R_{j,i} = \phi(Y_{j,i}; \mathcal{B}_{j,-i})\), aggregated across batches and thresholded at an adjusted quantile
    • BB-UCP: applies bootstrapping over per-batch residuals on top of B-UCP to stabilize empirical quantiles and reduce the influence of outlier batches
    • Coverage guarantees (Theorems 3.1–3.2): under batch exchangeability, \(\Pr\{Y_{n+1} \in C_n\} \geq 1-\alpha\)

  3. Conformal Alignment:

    • Parameterized strictness \(\tau \in [0,1]\); filtered retention set \(\hat{J}_j(\tau) = \{i: Q_{j,i} > \tau\}\)
    • Predicate \(\mathcal{P}_j^{\text{CVAR}}(\tau)\) evaluates the quality gap between the retained and discarded sets based on CVaR difference
    • Calibrated \(\hat{\tau}\) is the \(K = \lceil(1-\alpha)(J+1)\rceil\)-th order statistic of minimum passing strictness over historical batches
    • Alignment guarantee (Theorem 3.3): \(\Pr\{\mathcal{P}_{J+1}(\hat{\tau}) = 1\} \geq 1-\alpha\)

Evaluation Metrics

Factuality severity \(\text{FS}(a) = 1 - \max_{r \in \mathcal{R}_q} \text{BERTScoreF1}(\text{head}(a), r)\), evaluated only on the answer head (first sentence or Final field), truncated to 16 tokens.

Loss & Training

The model is trained end-to-end with an objective combining task loss and regularization terms.

Key Experimental Results

Experiment 1: Single-Query Calibration (S-UCP vs. BB-UCP)

  • BB-UCP yields smaller interval lengths across all datasets and all \(\alpha\) levels
  • BB-UCP coverage is more conservative (above target), yet produces tighter thresholds

Experiment 2: Cross-Query Calibration

  • LOQO empirical coverage is close to the \(1-\alpha\) target across all datasets
  • \(\Delta\text{FS} > 0\) holds across all datasets and all \(\alpha\) levels (retained sets exhibit better factuality)
  • \(\Delta\text{FS} \approx 0.209\) on NQ-Open (largest improvement)

Experiment 3: Conformal Alignment (CVaR-gap)

  • Reductions in factuality severity are consistently positive across all datasets and risk levels
  • NQ-Open and NQ-Open-Vend exhibit the largest median improvements

Key Findings

  • The bootstrap mechanism effectively stabilizes threshold estimation, producing tighter intervals while maintaining conservative coverage
  • Gram matrix geometric signals are highly correlated with factuality without requiring logits or gradients
  • In low-pool/low-diversity settings (NQ-Open), coverage is slightly below target, yet factuality gains remain significant
  • Conformal alignment effectively bridges cheap geometric signals and expensive factuality metrics

Highlights & Insights

  • Fully label-free and API-compatible: requires no token probabilities, gradients, or annotated data — only sampled outputs and embeddings
  • Seamless integration of theory and practice: distribution-free finite-sample guarantees (without model-specific assumptions) combined with practical hallucination detection
  • Elegant three-layer progressive design: scoring → calibration → alignment, each layer addressing one core problem
  • Generality of conformal alignment: adaptable to any non-decreasing right-continuous predicate, enabling broad applicability

Limitations & Future Work

  • The batch exchangeability assumption may be violated in real-world deployment
  • Lightweight embedding models (MiniLM) may lose subtle semantic distinctions
  • Gram matrix discriminability degrades in low-diversity generation scenarios (e.g., simple factual questions)
  • Applying this framework to hallucination detection in multimodal LLMs is a promising direction
  • vs. Semantic Entropy: semantic entropy requires defining equivalence classes; the proposed method directly leverages embedding geometry
  • vs. Conformal Risk Control: this work extends to fully unsupervised batch settings
  • vs. QRM/URM: those methods require preference data for training; the proposed method requires no annotation whatsoever

Supplementary Details

  • Datasets: ASQA (ambiguous), NQ-Open (single-hop factual), HotpotQA (multi-hop compositional), AmbigQA (aliases and answer sets)
  • Embedding model: all-MiniLM-L6-v2 (lightweight sentence encoder)
  • BERTScore uses roberta-large with baseline rescaling
  • Ablations include decoding entropy stress tests and vendor/model switching
  • Supports multi-vendor APIs including OpenAI, Together, and Gemini
  • Bootstrap resampling incurs low computational overhead, preserves exchangeability, and is directly deployable

Rating

  • Novelty: ⭐⭐⭐⭐⭐ The combination of unsupervised conformal prediction, Gram matrix geometry, and conformal alignment is highly original
  • Experimental Thoroughness: ⭐⭐⭐⭐ Four QA datasets, three experimental dimensions, two ablation analyses
  • Writing Quality: ⭐⭐⭐⭐ Theoretically rigorous but notation-heavy; readers are expected to have a CP background
  • Value: ⭐⭐⭐⭐⭐ Provides a practically needed quality control tool for LLM deployment