CoE: Collaborative Entropy for Uncertainty Quantification in Agentic Multi-LLM Systems¶

🕒 Published (v1): 2026-03-30 12:28 UTC · Source: Arxiv · Venue: ICLR · link

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ICLR; collaborative entropy for uncertainty quantification across multi-LLM systems

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TL;DR¶

Collaborative Entropy (CoE) is an information-theoretic metric that quantifies system-level semantic uncertainty in multi-LLM ensembles by decomposing it into intra-model aleatoric uncertainty and inter-model epistemic divergence. Unlike existing single-model-centric methods, CoE operates over a shared semantic cluster space and uses asymmetric KL divergence to the ensemble mean to capture directional cross-model disagreement. On TriviaQA and SQuAD, CoE outperforms semantic entropy and other baselines, with gains that grow as more heterogeneous models are added.

Problem¶

Existing uncertainty quantification (UQ) methods for LLMs are single-model-centric: they measure a single model's internal confidence but cannot characterize semantic disagreement across heterogeneous models in multi-LLM ensembles. Naively averaging per-model semantic entropy treats all models as independent and conflates two qualitatively different uncertainty sources—intra-model aleatoric uncertainty and inter-model epistemic divergence—into a single scalar, discarding actionable diagnostic information about why the system is uncertain.

Method¶

CoE is defined over a shared semantic cluster space constructed via bidirectional entailment. Each LLM \(M_i\) generates candidate responses that are partitioned into semantic clusters \(\{c_j\}\), yielding a cluster-level distribution \(p_i(\cdot|x)\). The ensemble mean is \(\bar{p}(\cdot|x) = \sum_i w_i p_i(\cdot|x)\). CoE decomposes system-level uncertainty as:

\[U_{\text{CoE}}(K) = \underbrace{\frac{1}{K}\sum_{i\in K} \text{SE}(x_i)}_{U_A: \text{ intra-model}} + \underbrace{\sum_{i\in K} w_i \cdot D_{\text{KL}}(p_i(\cdot|x) \,\|\, \bar{p}(\cdot|x))}_{U_E: \text{ inter-model}}\]

The intra-model term \(U_A\) is the mean semantic entropy across models; the inter-model term \(U_E\) measures asymmetric KL divergence from each model's distribution to the ensemble mean. Asymmetric KL is used (vs. Jensen-Shannon or Hellinger) because its reference-directed asymmetry is sensitive to directional epistemic divergence. A training-free post-hoc coordination heuristic uses CoE-derived signals to adjust model weights at inference time without parameter updates. Three formal properties are proved: non-negativity (\(U_{\text{CoE}} \geq 0\)), zero-value certainty (\(U_{\text{CoE}} = 0\) iff all models concentrate on the same cluster), and the behavior when individual models collapse to delta distributions (aleatoric component vanishes but inter-model epistemic divergence may persist).

Key Contributions¶

First system-level, closed-form uncertainty metric for heterogeneous multi-LLM ensembles, decomposing uncertainty into interpretable aleatoric (\(U_A\)) and epistemic (\(U_E\)) components over a shared semantic cluster space.
Formal proofs of Non-Negativity, Zero Value Certainty, and Delta-Distribution Behaviour, establishing CoE's theoretical guarantees and distinguishing when per-model uncertainty reduction suffices for system certainty.
A training-free, inference-time CoE-guided coordination heuristic that adjusts ensemble model weights using CoE signals.
Empirical validation on TriviaQA and SQuAD with LLaMA-3.1-8B, Qwen-2.5-7B, and Mistral-7B showing CoE outperforms single-model baselines and single-component ablations, with scaling gains as ensemble size grows.

Results¶

TriviaQA, 2-model (Llama+Qwen): CoE AUROC 0.683 vs. Semantic Entropy 0.670, \(U_E\) 0.661, \(U_A\) 0.666, \(P_\text{false}\) 0.586, Regular Entropy 0.475.
TriviaQA, 3-model (Llama+Qwen+Mistral): CoE AUROC 0.772 vs. Semantic Entropy 0.687, \(U_E\) 0.716, \(U_A\) 0.673—margin over baselines widens with ensemble size.
SQuAD: CoE AUROC up to 0.878 (3-model setting).
CoE-guided coordination heuristic: +39.0% accuracy gain under asymmetric KL, substantially exceeding symmetric alternatives (JS, Wasserstein, Hellinger).
Gains scale monotonically with heterogeneous model count, consistent with Theorem 3.

Limitations¶

Intra-model semantic entropy (\(U_A\)) conflates aleatoric and epistemic uncertainty at the single-model level: a model hallucinating diverse but incorrect answers inflates SE beyond pure aleatoric uncertainty ("epistemic contamination").
Inter-model KL divergence (\(U_E\)) may capture stylistic differences (verbosity, formatting) induced by differing RLHF procedures rather than genuine semantic disagreement.
Experiments limited to 7–8B parameter open-source models; applicability to larger (70B) or closed-source models (where token probabilities are unavailable) is unvalidated.
Evaluated only on 200-sample subsets of TriviaQA and SQuAD; coverage of diverse task types (multi-step reasoning, generation) is absent.

Relevance to Agentic AI / LLM Agents¶

Multi-LLM collaboration is a core architecture in agentic systems, where heterogeneous agents cross-check reasoning and vote on actions; knowing when the collective is uncertain—and why—is critical for safe deployment. CoE provides a principled, training-free mechanism to detect the failure mode where agents are each individually confident but semantically inconsistent, a regime invisible to existing single-model UQ tools. The two-component decomposition (\(U_A\) vs. \(U_E\)) directly informs intervention policy: high \(U_E\) calls for inter-model alignment or routing, not simply better prompting. The extension to sequential multi-agent settings (mentioned as future work) is a natural next step for agentic pipelines where uncertainty accumulates across reasoning trajectories.