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Learning to Give Checkable Answers with Prover-Verifier Games

Cem Anil, Guodong Zhang, Yuhuai Wu, Roger Grosse

arXiv:2108.12099v1cs.LGcs.AIcs.GT

TL;DR

Machine-learning systems can be powerful yet unreliable, motivating methods for producing decisions that humans or simpler systems can check. The paper introduces Prover-Verifier Games, analyzes formulations with desirable equilibria, and demonstrates robust verification protocols that extract useful information from untrusted provers. The approach remains effective against optimized prover messages, while unresolved equilibrium and scaling issues limit deployment.

  • Problem

    Machine-learning systems often lack robustness and reliability, creating a need for outputs that humans or cheaper, better-understood systems can check.

    Source: §1 · p. 1

  • Method

    Prover-Verifier Games train a trusted verifier to choose correct answers against an untrusted prover that seeks to persuade it, while comparing simultaneous and sequential game formulations.

    Source: §1 · p. 1, §3.2 · p. 4

  • Results

    PVG-trained verifiers learn robust decision rules that use reliable information from untrusted provers, including perfect precision and recall against optimized provers and messages in one evaluation.

    Source: §1 · p. 2, §5.3 · p. 9, §6 · p. 10

  • Takeaways & Limitations

    Reliable proof-verification protocols can emerge from appropriately formulated learning games, even when the verifier is frozen and prover messages are directly optimized.

    Source: §5.3 · p. 10, §6 · p. 10

    Deployment before confirming equilibrium, and scaling to harder problems or settings without known verification protocols, may lead to unexpected behavior.

    Source: §3.3 · p. 6, §6 · p. 10

Abstract

from arXiv · show

Our ability to know when to trust the decisions made by machine learning systems has not kept up with the staggering improvements in their performance, limiting their applicability in high-stakes domains. We introduce Prover-Verifier Games (PVGs), a game-theoretic framework to encourage learning agents to solve decision problems in a verifiable manner. The PVG consists of two learners with competing objectives: a trusted verifier network tries to choose the correct answer, and a more powerful but untrusted prover network attempts to persuade the verifier of a particular answer, regardless of its correctness. The goal is for a reliable justification protocol to emerge from this game. We analyze variants of the framework, including simultaneous and sequential games, and narrow the space down to a subset of games which provably have the desired equilibria. We develop instantiations of the PVG for two algorithmic tasks, and show that in practice, the verifier learns a robust decision rule that is able to receive useful and reliable information from an untrusted prover. Importantly, the protocol still works even when the verifier is frozen and the prover's messages are directly optimized to convince the verifier.

1 Introduction

The paper introduces Prover-Verifier Games to make machine-learning decisions checkable despite robustness and reliability problems. It analyzes which game formulations yield desirable equilibria and shows that learned verifiers can extract reliable information from untrusted provers.

  • Prover-Verifier Games pair a trusted verifier seeking correct decisions with an untrusted prover seeking to persuade it regardless of correctness.The intended outcome is a learned proof protocol that improves reliability beyond unaided verification.Source: §1 · p. 1
  • Theoretical analysis identifies simultaneous and verifier-first sequential formulations as having desirable equilibria, while prover-first sequential games produce dysfunctional solutions.The analysis connects simultaneous play to Nash equilibria and verifier-first play to verifier-leading Stackelberg equilibria.Source: §1 · pp. 1–2
  • Experiments on two algorithmic tasks show that PVG-trained verifiers learn sensible protocols for receiving useful, reliable information from untrusted provers.Collaboratively trained verifiers are easily deceived, unlike PVG-trained verifiers.Source: §1 · p. 2

2 Background

The background frames verification as an interaction between a trusted, limited verifier and a powerful prover, then introduces differentiable-game equilibrium concepts. A learned example illustrates how training can produce a coordinate-based verification protocol.

2.1 Interactive Proof Systems (IPS)

  • Interactive proof systems require a trusted computationally constrained verifier to determine a decision answer through communication with a potentially adversarial prover.Their defining properties are completeness and soundness.Source: §2.1 · p. 2

2.2 Differentiable Game Optimization

  • A differentiable two-player game gives each agent parametrized strategies and loss functions, with an equilibrium specifying the strategies adopted by both players.A Nash equilibrium occurs when neither player can unilaterally improve its objective.Source: §2.2 · p. 2
  • A Stackelberg equilibrium designates one player as leader and requires the follower to optimize conditionally while the leader accounts for that response.The paper assumes unique equilibria for simplicity.Source: §2.2 · p. 2
  • The learned blue-plus protocol uses prover-supplied coordinates as certificates that the verifier accepts for true positives and rejects for maximally convincing false-image patches.This provides a concrete coordinate-based verification protocol discovered through training.Source: Figure 1 · p. 3

3 Prover-Verifier Game

The Prover-Verifier Game trains competing prover and verifier agents to produce decision rules with completeness and soundness-like guarantees. The analysis identifies game formulations with desirable equilibria, explains failure modes of others, and shows gradient-based convergence in a restricted task.

3.1 Desiderata and Prover-Verifier Incentives

  • The verifier seeks correct answers while the prover argues for a fixed answer, encouraging recall on positive instances and precision against adversarial proofs.The desired protocol permits perfect recall with an appropriate prover and guarantees perfect precision against every prover.Source: §3.1 · p. 3

3.2 Different Prover-Verifier Game Formulations

  • PVGs combine prover and verifier policies with simultaneous or Stackelberg play, while varying when instances are revealed and restricting interaction to one message without feedback.The framework uses probability distributions over prover messages and verifier labels, with cross-entropy-style losses whose gradients vanish when incentives are satisfied.Source: §3.2 · p. 4

3.3 PVG Formulations with Desirable Equilibria

  • The abstract analysis assumes deterministic labels, a prover defending label 1, constant policies in both policy classes, and an available sound verifier policy.The conclusions do not depend on details such as Boolean-circuit implementations or complexity-theoretic assumptions.Source: §3.3 · p. 5
  • Only three of eight PVG formulations have desirable equilibria: delayed-instance simultaneous play and both verifier-first sequential setups.These equilibria correspond to complete and sound proof-verification protocols; Table 1 summarizes necessity, sufficiency, failure modes, and Binary Erasure Channel convergence.Source: §3.3 · p. 5, Table 1 · p. 5
  • Prover-leading Stackelberg equilibria are neither necessary nor sufficient for complete and sound protocols, whereas delayed-instance verifier-leading equilibria are both necessary and sufficient.The prover-leading setting can produce flood-the-zone behavior, while revealing the instance before the verifier chooses its policy permits a trivial verifier failure mode.Source: §3.3 · pp. 5–6
  • Nash equilibria can still ignore the prover entirely, creating a coordination problem in which neither player can unilaterally adopt a useful protocol.Thus, convergence to equilibrium alone does not guarantee a complete and sound proof-verification protocol in the simultaneous formulation.Source: §3.3 · p. 6

3.4 Finding Proof-Verification Strategies Using Gradient-Based Optimizers

  • Simultaneous gradient descent converges to an informative Binary Erasure Channel strategy, but prover-first training converges to an undesirable input-independent message strategy.The ideal strategy sends the correct proof token and makes the verifier predict one only for that token; entropy regularization avoids degenerate verifier behavior.Source: §3.4 · p. 7, C Proofs for Equilibrium Analysis · p. 16

4 Related Work

The paper situates Prover-Verifier Games among proof systems, interpretability games, model verification, and AI-safety debates, while distinguishing its goal of learning proof-verification protocols.

Computational Complexity

  • Proof systems frame prover-verifier interaction, and PVGs most directly resemble vanilla NP proofs while potentially encoding network weights for interactive-proof-like protocols.The related-work discussion connects PVGs to interactive, zero-knowledge, and probabilistically checkable proofs.Source: §4 · p. 7

Interpretability

  • Game-theoretic interpretability methods use competing or collaborative agents to select rationales, classify inputs, or generate arguments for fact-checking.Examples include selective rationalization games [20] [37] [4] and debate dynamics for knowledge-graph fact verification [13].Source: §4 · p. 7

Verifying correctness and functional properties of models

  • Model-verification research studies interactive proof systems, PAC verification, and methods for testing, training, or formally verifying adherence to specifications.The paper characterizes these approaches as orthogonal because they verify model properties rather than learn proof-verification protocols [32] [28] [36] [7] [17].Source: §4 · pp. 7–8

AI Safety

  • AI-safety work uses adversarial communication to assess correctness, but PVGs use a single prover and focus on discovering reliable proof protocols rather than relying on adversarial debaters.The comparison is made with debate games [15] and the broader goal of identifying argumentation forms that lead reliably to true conclusions [3].Source: §4 · p. 8
  • Deep Verifier Networks detect unreliable inputs or predictions with a hard-coded verification structure, whereas PVGs learn the verification protocol and target a different application.Che et al. focus on out-of-distribution detection applications.Source: §4 · p. 8

5 Simulations

The simulations use gradient-based training, robustness stress tests, and frozen-verifier evaluations across two algorithmic tasks. PVG-trained verifiers remain reliable against stronger optimized provers and messages, unlike collaboratively trained baselines.

5.1 Practical Design Decisions

  • Verifier capacity is constrained so it cannot independently solve difficult search problems while remaining able to check proofs.The authors identify subset sum, SAT, and NP problems as settings where suitable architectural constraints are easier to obtain.Source: §5.1 · p. 8
  • Practical training uses alternating gradient updates and solves for Nash equilibria because Stackelberg response gradients are often computationally difficult.All Stackelberg equilibria are also Nash equilibria, and the simultaneous setup has favorable convergence properties according to Theorem 4.Source: §5.1 · p. 8

5.2 Evaluating Success

  • The evaluation freezes the verifier and strengthens the prover by continued training or direct L-BFGS optimization of messages, approximating verifier-leading Stackelberg robustness tests.Precision and recall are measured against both forms of prover augmentation.Source: §5.2 · pp. 8–9

5.3 Results

  • The BEC experiment uses 16 communication tokens, balanced labels, Adam optimization, batch size 2000, learning rate 0.0003, and five verifier updates per prover update.The prover receives one-hot input bits and the verifier receives one-hot communication tokens.Source: §5.3 · p. 9
  • Perfect precision and recall hold for the frozen PVG-trained BEC verifier against both an optimized prover and directly optimized messages, while the collaborative verifier has zero precision.These evaluations test robustness after removing the verifier’s opportunity to rely on a prover-specific tell.Source: §5.3 · p. 9, Table 2 · p. 9, §5.2 · p. 9
  • Training dynamics show the verifier quickly identifies the correct BEC protocol, with the desired protocol emerging around iteration 400.Figure 2 tracks prover and verifier logits alongside verifier accuracy.Source: Figure 2 · p. 9, §5.3 · p. 10
  • The FindThePlus task constrains the verifier to inspect only part of each 10 × 10 image, requiring prover advice about where to look.Images contain a single 3 × 3 plus whose color determines the label, and the verifier may ignore the advice.Source: §5.3 · p. 10
  • Near-perfect specificity is achieved by the frozen PVG-trained FindThePlus verifier against optimized provers and messages, while the collaborative baseline has zero specificity.Directly optimized proof vectors yield frozen-verifier precision of 0.99, and the verifier learns to use prover-supplied coordinates to locate relevant patches.Source: §5.3 · p. 10, Table 3 · p. 10

6 Discussion

The discussion concludes that PVGs can learn robust proof-verification protocols, while identifying scaling and equilibrium-convergence issues for future work.

6 Discussion

  • PVGs provide a game-theoretic framework for solving decision problems in a verifiable manner.Source: §6 · p. 10
  • The paper demonstrates theoretically and empirically that some PVG formulations learn robust proof-verification protocols.Source: §6 · p. 10
  • Scaling PVGs to more challenging problems and problems without known verification protocols remains future work.Source: §6 · p. 10
  • Deploying a PVG-trained system before equilibrium is reached could lead to unexpected behaviour.Source: §6 · p. 10
  • PVGs could support interpretable proofs, useful downstream features, and safer interaction with powerful untrusted systems.Source: §6 · p. 10

A When Log-Loss Gradients Vanish

B Proofs of Equilibrium Theorems

The equilibrium proofs distinguish dysfunctional prover-leading sequential games from verifier-leading and simultaneous formulations with desirable equilibrium properties.

B Proofs of Equilibrium Theorems

C Proofs for Equilibrium Analysis

The equilibrium analysis shows that simultaneous and Stackelberg formulations can converge to sharply different prover and verifier behaviours.

C Proofs for Equilibrium Analysis

D Training Heuristics

Several training choices improve consistency or convergence speed without being essential for success.

D Training Heuristics

  • Updating the verifier multiple times per prover update helps prevent it from ignoring prover messages early in training.Source: D Training Heuristics · p. 17
  • Increasing prover-update frequency after verifier accuracy exceeds 90 percent speeds the transition to fully robust protocols.Source: D Training Heuristics · p. 17
  • Pretraining the prover to solve and autoencode the task encourages label-correlated initial messages and reduces verifier message neglect.Source: D Training Heuristics · p. 17

E.1 Binary Erasure Channel Experiments

The Binary Erasure Channel experiments tune neural prover-verifier training and report convergence to a complete, sound verification protocol. Training stability depended on optimizer settings, batch size, network width, and verifier update frequency.

  • The prover used a two-hidden-layer fully connected LeakyReLU network with width 100, LayerNorm, and Gumbel-softmax communication tokens.Source: §E.1 · p. 17
  • The verifier used a one-hidden-layer fully connected LeakyReLU network with width 100 and LayerNorm.Source: §E.1 · p. 17
  • The verifier learned a complete and sound proof-verification protocol, with frozen precision and recall reaching 1.Source: §E.1 · p. 17
  • Hyperparameter tuning compared Adam with SGD, learning rates from 1e-3 to 1e-4, batch sizes of 500 or 2000, and network widths starting at 32.Source: §E.1 · p. 18
  • Adam with batch size 2000, network width 100, learning rate 3e-4, and five verifier steps stabilized training.Source: §E.1 · pp. 17–18

E.2 FindThePlus Experiments:

FindThePlus tests whether a spatially constrained verifier can use prover messages to locate and classify colored plus patterns. The verifier’s learned gaze and optimized-prover evaluation support a reliable protocol, while pretraining and tuning materially affect training.

E.2 FindThePlus Experiments:

  • The task uses 10 × 10 single-channel binary images containing exactly one 3 × 3 plus pattern, formed using either 0-pixels or 1-pixels, with balanced labels.Source: §E.2 · p. 18
  • The verifier uses four Spatial Transformer heads to convert prover messages into 2 × 3 transformation matrices, then classifies transformed images.Source: §E.2 · p. 18
  • Classification pretraining accelerated convergence, whereas autoencoding had little impact; prover steps increased adaptively up to 15 when verifier accuracy stayed above 75%.Source: §E.2 · p. 18
  • The verifier achieved perfect frozen precision against an optimized prover before the reported best-frozen-precision checkpoint.Source: §E.2 · p. 19
  • Pretraining the prover to convergence hurt performance, which the authors relate to Neural Collapse and leave for future study.Source: §E.2 · p. 19

E.3 Additional Plus Visualizations

  • The visualizations show four verifier heads, with the first consistently locating blue pluses or convincing image patches communicated by the prover.Source: §E.3 · p. 19, Figure 4 · p. 20

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