Conservative Hull-based Classifier (CHC)

Updated 28 October 2025

The Conservative Hull-based Classifier (CHC) is a geometric online classification algorithm that reduces costly expert queries by updating convex hulls based on labeled examples.
It achieves minimax optimal regret in one dimension and near-optimal bounds in higher dimensions through efficient linear or quadratic programming for point-in-hull checks.
Empirical evaluations show CHC outperforms traditional baselines in embedding-rich applications, effectively minimizing human intervention in high-dimensional question-answering tasks.

The Conservative Hull-based Classifier (CHC) is a geometric online classification algorithm designed for scenarios where queries arrive sequentially and expert labels are costly. CHC maintains sets of expert-labeled examples for each class, constructing convex hulls in the embedding space and consulting the expert only when a new query lies outside all known hulls. This allows the classifier to achieve minimax optimal regret in one dimension and nearly optimal bounds in higher dimensions, particularly when the query distribution and the geometry of class regions facilitate rapid “locking in” of the correct label. CHC is especially pertinent for embedding-rich applications such as question answering systems powered by LLMs, where each query is represented as a high-dimensional vector and human annotation is expensive.

1. Problem Context and Motivation

CHC operates in a sequential decision-making framework where an agent must assign one of $N$ labels to each $d$ -dimensional query embedding $q_t$ . At every round, the agent can either guess—with no feedback on correctness—or incur a cost to query a human expert who provides the true label. The true class region polytopes $\mathcal{C}_i \subset \mathbb{R}^d$ are unknown a priori and must be inferred over time from labeled queries.

The central motivation for CHC is to minimize cumulative regret relative to an oracle who never makes mistakes and has free expert access. Regret, in this setup, quantifies the difference in reward between the oracle and the agent, factoring both wrong guesses and the cost of consultation. CHC aims for sample-efficient learning under stringent feedback constraints, tailored to domains where mistaken guesses are penalized and information is costly to acquire (Réveillard et al., 27 Oct 2025).

2. Algorithmic Structure and Decision Rule

At time $t$ , CHC tracks

$\mathcal{Q}_{i, t}$ : the set of expert-labeled embeddings for class $i$
$\hat{\mathcal{C}}_{i, t}$ : the convex hull of $\mathcal{Q}_{i, t}$

Given a new query $q_t$ , the algorithm proceeds as follows:

Predict if inside Hull:
- If $q_t \in \hat{\mathcal{C}}_{i, t}$ for some class $i$ , predict label $i$ with certainty.
Call Expert otherwise:
- If $q_t$ is not inside any class’s hull, consult the expert, obtain label $i_t$ , and update $\mathcal{Q}_{i_t, t+1} \gets \mathcal{Q}_{i_t, t} \cup \{q_t\}$

This conservative principle ensures that CHC only makes non-expert predictions when the geometric evidence is unambiguous, eliminating online misclassifications entirely (Réveillard et al., 27 Oct 2025).

Algorithm pseudocode:

Algorithm Conservative Hull-based Classifier (CHC)
Initialize: For each label i in 1..N, set Q_i = ∅
for t = 1 to T:
    Receive query q_t
    found = False
    for i = 1 to N:
        if q_t is in convex_hull(Q_i):
            Predict label i
            found = True
            break
    if not found:
        Call expert, obtain label i_t
        Q_{i_t} = Q_{i_t} ∪ {q_t}

Convex hull membership can be checked via linear programming for

\mathbb{R}^d

and via quadratic programming for spherical embeddings.

3. Theoretical Performance and Regret Analysis

CHC’s regret is governed primarily by the rate at which convex hulls expand to cover the true class regions: $R_{\text{CHC}}(T) = (\beta-\alpha)\, \mathbb{E}[C_T] = (\beta-\alpha) \sum_{t=1}^{T} \sum_{i=1}^{N} \mathbb{E}\left[ \mu(\mathcal{C}_i \setminus \hat{\mathcal{C}}_{i, t}) \right]$ where $C_T$ is the expected number of expert queries after $T$ rounds, $\beta$ the reward for a correct guess, and $\alpha$ the cost for expert consultation.

Main regret bounds:

For queries in the hypercube $\mathcal{E}=[0,1]^d$ : $R_{\text{CHC}}(T) = O(\log^d T)$
For spherical embeddings $\mathcal{E}=S^{d-1}$ : $R_{\text{CHC}}(T) = O(\log^{d-1} T)$

When $d=1$ , CHC is provably minimax optimal for all nonatomic distributions: $R_{\text{CHC}}(T) \leq 2N(\beta-\alpha)\log(T+1)$ No algorithm can attain regret of smaller order in this setting, as demonstrated by matching lower bounds (Réveillard et al., 27 Oct 2025).

Scaling considerations: In high dimensions or with large class counts, regret increases polynomially in $\log T$ and may become substantial if the geometry is complex or classes are tightly clustered. CHC maintains certainty at the expense of slower initial coverage, which is optimal in risk-averse or high-cost domains.

4. Implementation and Computational Properties

CHC does not require explicit computation of high-dimensional hulls, which can be computationally intractable. Instead, point-in-hull checks are accomplished via efficient linear (for Euclidean) or quadratic (for spherical) programming, with complexity scaling in both dimension and the number of labeled points per class.

No parameter tuning is required. The method is inherently robust to label noise because only expert-confirmed points extend the hulls. In environments where the embedding structure is favorable—such as those induced by LLMs—hulls rapidly "lock in" the correct class for most queries, reducing intervention.

5. Empirical Validation

CHC was evaluated on synthetic data and real-world question-answering datasets:

Synthetic results: Regret follows the predicted $\log^d T$ scaling; hull expansion quickly covers familiar regions, minimizing unnecessary expert consultation.

LLM embeddings:

Datasets: Quora Question Groups, ComQA, CQADupStack
Embedding models: Nomic, E5-Large, Mistral_E5 (up to 4,096 dimensions)
Higher-dimensional embeddings (Mistral_E5) induce stronger separation, reducing regret and required expert calls.
CHC consistently outperforms center-based and $k$ -means baselines for minimizing human intervention, validating its efficacy in semantically rich geometries.

Performance table: (see paper for full details)

Property	Value
Guarantee	Never predicts incorrectly
Regret bound ( $[0,1]^d$ )	$O(\log^d T)$
Regret bound ( $S^{d-1}$ )	$O(\log^{d-1} T)$
Minimax optimality ( $d=1$ )	Yes
Best for	High-separation LLM embeddings
Computational cost	Per-query LP/QP in $d$ , hull size

A plausible implication is that CHC will be most effective in domains with well-separated semantic clusters and sufficient query volume to learn geometry. In regimes where queries are ambiguous or hulls expand slowly (e.g., very high $N$ or adverse embeddings), regret rises and expert calls become more frequent—a property analytically captured by the regret bounds.

CHC generalizes classic convex hull decision rules to the online, expert-in-the-loop setting, prioritizing zero mistakes with minimum human involvement. In contrast:

DataGrinder (Khabbaz, 2015) aggregates votes from all 2D projections and convex hulls but does not address online expert cost or regret minimization.
Nested Cavity Classifier (Mustafa et al., 2019) and Adaptive Multi Convex Hull (Chen et al., 2014) emphasize hull nesting, cluster semantics, or adaptive clustering, but lack explicit regret analyses or rigorous cost-aware query protocols.

CHC’s conservative prediction mechanism distinguishes it from multi-hull approaches, which risk error in ambiguous or overlapping polytopes. Its strategy of updating only from expert-corroborated points ensures strict geometric fidelity and optimality in the intended regime.

7. Practical Impact and Limitations

CHC offers a rigorously conservative protocol for online classification with expensive human supervision. Its minimax optimality in one-dimensional settings and nearly optimal scaling in higher dimensions make it uniquely suitable for applications where false positives incur heavy costs and geometric learning is feasible. Limitations arise in scenarios with low query volumes relative to dimensionality or high class counts, where hulls require extensive sampling to achieve adequate coverage. In such cases, hybrid classifiers or more aggressive thresholding (as in Generalized Hull-based Classifier) may be preferable.

In summary, the Conservative Hull-based Classifier embodies a powerful geometric paradigm for online decision-making under uncertainty, characterized by strong regret guarantees, zero-error prediction, and efficient use of expert consultation, especially in high-dimensional semantic embedding spaces (Réveillard et al., 27 Oct 2025).