Refereed Learning Protocols

• Refereed Learning Protocols are formal frameworks where a resource-limited learner interacts with two competing provers (one honest) to assess opaque models efficiently.
• They utilize techniques like certifiable sampling and histogram-based decompositions to achieve near-optimal loss with significantly reduced communication and query complexity.
• These protocols are pivotal in model selection, AI safety, and secure delegation, offering robust, verifiable alternatives to single-prover assessment methods.

Refereed learning protocols are formal procedures in computational learning theory and cryptography in which a learner leverages interaction with multiple competing provers—of which only one is assumed honest—in order to solve learning or assessment tasks about possibly opaque or black-box models. The core idea is to enable a learner, possibly with limited computational or oracle access, to efficiently and verifiably assess properties (such as the relative quality or correctness) of models or outputs in the presence of potentially adversarial or unreliable agents. This framework generalizes interactive proof methods and refereed delegation from computational complexity to the assessment and verification of machine learning and statistical tasks. Refereed learning protocols produce strong guarantees even in high-precision regimes, achieving levels of reliability and loss unattainable at comparable cost in non-interactive or single-prover settings (Canetti et al., 6 Oct 2025).

1. Conceptual Framework and Definitions

Refereed learning is formalized as a three-party interaction involving a learner (with potentially limited resources) and two competing provers, only one of which is honest (Canetti et al., 6 Oct 2025). The essential task is to use interaction with these provers (who output models or answer queries about models) to reach conclusions about the quality or properties of black-box models relative to ground truth, while making significantly fewer direct queries to the ground truth function than would otherwise be required.

A typical refereed learning protocol specifies the following components:

• Learner: The entity with limited computational or oracle access, wishing to assess properties of a function, model, or distribution.
• Provers: Two agents, at least one honest, who provide models, answers, or sampling access with the intent to convince the learner. Dishonest provers may attempt to mislead, but the protocol is designed to be robust to at most one honest participant.
• Interaction Protocol: The procedure that interleaves queries, model outputs, and verification steps, possibly making limited use of the ground truth function.

The critical notion is that in the high-precision regime and under minimal ground truth access (e.g., only a single query), refereed learning protocols enable the learner to select a model whose error is within a multiplicative $(1+\varepsilon)$ factor of the best model’s loss, while communicating only $(1+\frac{1}{\varepsilon^2})\cdot \mathrm{poly}(d)$ bits (with $d$ the ambient dimension), which is provably unattainable in the single-prover or prover-less setting (Canetti et al., 6 Oct 2025).

2. Core Protocols and Methodologies

A basic refereed learning protocol for model selection proceeds by having each prover submit a candidate model. The learner then leverages certifiable sampling or simulation procedures, guided by the competing provers, to sample from distributions that may not be efficiently samplable otherwise. In particular, Birgé’s 1987 decomposition [Birgé 1987]—partitioning the domain using exponentially growing intervals and constructing histograms—enables efficient approximation of decreasing densities or monotone distributions. This technique permits the learner, via interaction with the provers, to estimate losses robustly with very few ground truth samples, even when the full domain is large or high-dimensional (Canetti et al., 6 Oct 2025).

The protocol’s tight error and communication bounds are achieved through a combination of competition between provers (ensuring at least one is incentivized to reveal correct information) and carefully structured queries that exploit differences between the models to force informative responses. When disagreements arise, certifiable sampling is used to isolate inputs on which the models differ; the learner then queries the ground truth on these points to determine the relative performance of the models (Canetti et al., 6 Oct 2025).

An illustration of the protocol’s efficiency appears in the high-precision regime: to achieve a loss within factor $(1+\varepsilon)$ of optimal, the learner makes one ground truth query and communicates only $(1+\frac{1}{\varepsilon^2})\mathrm{poly}(d)$ bits. In contrast, with a single prover, the learner would need to query the ground truth at almost every point in the domain as the dimension $d$ grows (Canetti et al., 6 Oct 2025).

3. Delegation, Verification, and Complexity-Theoretic Foundations

Refereed learning protocols are rooted in a tradition of multiparty interactive proofs and refereed delegation [Kilian 1992; Canetti et al. 2011]. The key technical novelty is leveraging economic or adversarial competition between provers so that any attempt by a dishonest prover to mislead the learner can be exposed by the honest prover—this mirrors the “noisy oracle” and multi-prover interactive proofs framework introduced by Feige, Shamir, and Tennenholtz (1988) and further refined in later works on refereed delegation and interactive proofs [Lund et al. 1992; Goldwasser, Tauman Kalai, and Rothblum 2015].

In certain distribution testing and model selection problems, lower bounds for sampling and verification (e.g., exponential lower bounds for evaluating the permanent or the Tutte polynomial [Dell et al. 2014]) demonstrate that passive or single-prover methods must incur exponential costs. Refereed protocol designs circumvent these lower bounds in many practical cases by invoking competition and limited ground truth queries, reducing the learner’s computational burden without sacrificing high-confidence guarantees (Canetti et al., 6 Oct 2025).

The use of certifiable sampling—enabled by histogram flattening across exponentially sized intervals—further tightens the communication and computational complexity, leveraging results from distribution testing and density estimation [Birgé 1987; Canonne 2020].

4. Applications to Model Selection and Black-Box Assessment

The principal application developed in refereed learning protocols is the model selection problem: the learner, given opaque access to two black-box models, must determine which model better matches the ground truth. The protocol is designed to return a model whose error is at most $(1+\varepsilon)$ times that of the optimal model, using minimal ground truth queries.

This capability is of special importance in large-scale model validation, AI safety, and regulatory or auditing contexts, where access to proprietary or complex models is limited and running ground truth evaluations (e.g., through human labeling or costly simulation) is expensive (Canetti et al., 6 Oct 2025). By interrogating two or more possibly adversarial models, the learner can—using just one or a handful of ground truth queries—achieve precision unattainable with brute-force or non-competitive methods.

Moreover, such protocols underlie emerging practices in AI safety debates [Irving et al. 2018; Guo et al. 2024], multi-agent evaluations, and secure delegation in cryptographically verifiable computation [Arun et al. 2025].

5. Lower Bounds and Optimality

Refereed learning protocols have been shown to achieve optimality with respect to several key resource measures:

Resource Type	Single-Prover Lower Bound	Refereed Protocol Upper Bound
Ground truth queries	$\Omega(N)$ (for $N$-point domain)	$1$ (in high-precision regime)
Communication	$\Omega(N)$ bits	$(1+\frac{1}{\varepsilon^2})\cdot\text{poly}(d)$ bits
Prover complexity	$1$	$2$ (one assumed honest)

These bounds are demonstrated to be tight in the sense that reducing the number of provers, increasing adversarial capacity, or relaxing the trust assumption leads to a strict degradation in achievable accuracy or efficiency [(Canetti et al., 6 Oct 2025); Dell et al. 2014]. The technical machinery securing these results includes certifiable sampling (using flattened histograms and oblivious decompositions), adversarial noise models, and competitive interaction to reveal model disparities.

6. Broader Connections and Relevance

The refereed learning paradigm synthesizes and extends several lines of research:

• Interactive Proofs and Delegation: The foundational interactive proof frameworks of Feige et al. (1988), Kilian (1992), Lund et al. (1992), and Canetti et al. (2011, 2013) motivate the use of multiple provers and refereed delegation in settings previously reserved for purely computational verification.
• Statistical Learning Theory: Results from density estimation and statistical risk minimization (e.g., works of Birgé 1987, Canonne 2020) provide the theoretical underpinnings for sampling and approximation steps in refereed learning protocols.
• AI and Machine Learning Verification: Recent applications by Goldwasser et al. (2021) and Arun et al. (2025) illustrate the practical deployment of refereed protocols in ML verification, while new research on AI safety debates and multi-agent evaluations highlights the conceptual utility of competition-based verification.
• Zero-Knowledge and Complexity Theory: Developments in interactive proof systems, zero-knowledge proofs, and complexity lower bounds (e.g., Walfish and Blumberg 2015; Canetti and Karchmer 2021) inform the design of protocols that are both efficient and resistant to strategic manipulation.

A notable implication is that in learning-theoretic and computational tasks where efficient verification with limited resources is otherwise provably impossible, the use of refereed learning protocols offers a path to practical, robust, and high-precision outcomes.

7. Limitations and Open Challenges

Despite their strong guarantees, refereed learning protocols are subject to certain limitations:

• Reliance on (at least) one honest prover: The soundness guarantee typically hinges on the assumption that among the provers, only one need be honest, but if all provers act in concert adversarially, correctness can be compromised.
• Communication and computation with provers: The protocol shifts some overhead (though polylogarithmic in $1/\varepsilon$ or polynomial in $d$) to the interaction and communication between learner and provers.
• Complexity lower bounds: For certain settings and domains, e.g., those connected to difficult counting problems or exponentially large support, exponential time or communication may be unavoidable even in the refereed setting [Dell et al. 2014].
• Applicability to wider learning paradigms: Extending these protocols beyond model selection and equivalence testing to richer classes of learning problems remains an active research area.

A plausible implication is that future research will further delineate the precise scope in which refereed learning protocols dominate single-prover or proverless baselines, as well as their potential in scalable deployment for model auditing and safety assessment.

• (Canetti et al., 6 Oct 2025) Refereed Learning
• Birgé 1987, "On the Risk of Histograms for Estimating Decreasing Densities"
• Dell et al. 2014, "Exponential Time Complexity of the Permanent and the Tutte Polynomial"
• Canetti et al. 2011, 2013
• Goldwasser, Tauman Kalai, and Rothblum 2015
• Goldwasser et al. 2021
• Arun et al. 2025
• Irving et al. 2018
• Canonne 2020
• Walfish and Blumberg 2015
• Canetti and Karchmer 2021

For a detailed technical account, see (Canetti et al., 6 Oct 2025) and citations therein.