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Use three complementary checks: formalize the precise claim and verify its proof where practical; run numerical experiments designed to expose counterexamples and implementation errors; and reproduce important results before testing an independent implementation. Each checks something different. Experiments can increase confidence or reveal a flaw, but finite tests do not prove a universal theorem. A machine-checked proof establishes its formal statement under its encoded assumptions—not automatically that those assumptions, definitions, or code match the intended machine-learning method.
What validation can—and cannot—establish
A machine-learning result often has several layers: a mathematical claim, a formal proof of that claim, an algorithm intended to realize it, and experimental code that produces reported results. Treating those layers as interchangeable leaves important gaps. A proof assistant checks a formalized statement; an experiment checks selected cases; and a second implementation can test whether a result survives a change in code. Together, these checks provide stronger and more varied evidence than any one alone.
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| Check | What it can establish | What it does not establish by itself |
|---|---|---|
| Machine-checked proof | The encoded theorem follows from the definitions, assumptions, and proof environment accepted by the proof assistant. | That the formalization captures the intended theorem or algorithm, or that every surrounding training, data-processing, and deployment component satisfies the theorem. |
| Numerical experiments | Whether selected test cases behave as predicted, and whether those cases expose counterexamples, implementation defects, or numerical problems. | A universal theorem, even if every tested input agrees with the prediction. |
| Reproduction | Whether the available implementation and data can produce similar results under documented conditions. | That the result is independent of the original code or robust to changed datasets. |
| Independent replication | Whether an alternative implementation of the core method agrees under matched assumptions and conditions. | That every implementation shares no assumptions or defects, or that the result generalizes to different datasets. |
| Reevaluation | How the method performs when assessed using different datasets. | A direct same-condition replication: changing the dataset changes what is being tested. |
This separation is practical, not merely terminological. Selsam, Liang, and Dill’s 2017 paper, Developing Bug-Free Machine Learning Systems With Formal Mathematics, presents Certigrad, a workflow that implements a system and states its correctness theorem in an interactive proof assistant. Its formal result concerns a particular property: sampled gradients are unbiased estimates of the mathematical gradients. That does not make it a proof of every property of an end-to-end machine-learning pipeline.
Bagnall and Stewart’s 2019 paper, Certifying the True Error: Machine Learning in Coq with Verified Generalization Guarantees, checks generalization bounds in Coq and demonstrates them on neural networks trained with TensorFlow on extended MNIST. The example illustrates a valuable boundary: checking a bound is not the same as proving every detail of the training pipeline or the behavior of a deployed model.
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Start by stating exactly what the proof claims
Before selecting tests, write down the claim in a form that can be checked. “The model works” or “the gradients are correct” is too broad to validate. Identify the mathematical object, the property, the quantifiers, and the conditions under which the property is meant to hold. Keep the theorem separate from the experimental claim that a particular run produced a particular result.
- Object: Name the item covered, such as a gradient estimator, a bound on expected error, or a specific algorithm.
- Property: State what is asserted about it: for example, unbiasedness of a sampled-gradient estimator or a generalization guarantee.
- Scope and quantifiers: Specify whether the claim is universal over inputs or parameters, applies only to a defined class, or concerns an expectation over a stated source of randomness.
- Assumptions: Record the conditions on data, distributions, model, objective, or other mathematical objects required by the theorem.
- Excluded components: Note which parts of the actual system are not in the statement, such as data preparation, numerical execution, or deployment behavior, unless they are explicitly modeled.
This step prevents a common scope error: testing a convenient implementation detail while treating the result as evidence for a broader theorem. If the theorem is about an expected quantity, for example, a single output is not itself the theorem; the experiment must be designed to examine the relevant consequence without claiming that finite observations settle the expectation universally.
Audit the formalization, not just the proof assistant’s green check
A proof assistant checks a formal statement, not the author’s informal intention. The critical review therefore asks both whether the proof is accepted and whether the formal objects express the intended mathematics. Record the proof assistant and checker used, the theorem and definitions under review, and any dependencies or axioms relevant to the result. Inspect how the algorithm, data assumptions, and claimed property are represented.
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- Locate the exact theorem. Identify the statement that the checker accepts. Do not infer that a nearby informal claim is covered merely because it appears in the same paper or codebase.
- Read its definitions and assumptions. Check what counts as an input, output, distribution, gradient, error, or valid model in the formal development. An assumption omitted from the theorem cannot be supplied by an experiment after the fact.
- Trace the correspondence to the intended method. Ask whether the formal definitions represent the algorithm the paper describes and the implementation is meant to execute. This correspondence is a separate question from proof acceptance.
- Check the trusted formal context. Record the checker and relevant dependencies or axioms, and review how they bear on the theorem. A successful check is meaningful within that formal context.
- State the coverage boundary. Describe what the result does not cover, including any unformalized portions of training or deployment that matter to the reader’s interpretation.
Selsam, Liang, and Dill describe interactive theorem proving as a way to expose implementation errors in the system represented by their formal development: a program error there would prevent the proof from succeeding. That strength depends on the program and theorem actually representing the intended system and claim. It should not be extended into a blanket guarantee about code or pipeline elements outside the formalization.
Design numerical experiments to try to falsify the claim
Numerical experiments are most informative when they are chosen to challenge the assumptions and predicted consequences, rather than merely to produce a reassuring example. Translate the theorem’s conditions into test cases, and make a failure informative: retain the input, settings, expected value, observed value, and tolerance so another person can inspect what happened.
Build a test set around the assumptions
- Small hand-checkable cases: Use examples for which the expected result can be calculated independently and inspected without relying on the implementation under test.
- Boundary and limiting cases: Probe values at the edges of the stated domain and cases that approach a limit relevant to the claim. These can reveal mistaken conditions or incorrect handling of special cases.
- Assumption-sensitive cases: Include cases that satisfy the theorem’s assumptions and, where useful, deliberately violate one assumption to check whether the test or implementation distinguishes the two. A violation is not a counterexample to a theorem whose assumptions do not hold.
- Numerically difficult cases: Where relevant, test conditions likely to stress precision or conditioning. Selsam, Liang, and Dill identify numerical instability among practical challenges in machine-learning systems, alongside noisy data, non-convex objectives, and model misspecification.
Compare against an independent reference
For each experiment, identify what prediction is being checked and how the reference value is obtained. Prefer a hand calculation or a separate calculation path when feasible; reusing the same function to generate both the result and its “expected” value can make the check circular. If randomness is involved, define the stochastic protocol and compare quantities appropriate to the claim rather than treating one random draw as a universal result.
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Set and report a tolerance that matches the quantity and numerical precision being tested. Record the observed discrepancy as well as whether it falls within tolerance. A pass under a chosen tolerance means only that the tested case met that criterion; the tolerance does not convert a finite test suite into a proof.
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A failed test is evidence to investigate, not an inconvenient result to discard. Preserve the case and determine whether it reveals a counterexample under the theorem’s assumptions, an implementation mismatch, a numerical issue, or a test that did not meet the stated conditions. If a test violates an assumption, report that fact rather than presenting it as a refutation of the theorem. The cited work does not prescribe a universal suite of numerical tests; the cases should follow from the particular claim.
Reproduce first, then replicate independently
Use the available implementation and data to reproduce a published result where possible. This tests whether the reported outcome can be obtained under the original materials and documented conditions. Then implement the core method separately and compare its outputs with the original under matched assumptions, inputs, and randomness controls. An alternative implementation makes the check more independent of the original code, though agreement still does not amount to a proof.
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Keep reevaluation distinct: testing on different datasets asks how the method behaves under changed data, not whether the original experiment can be reproduced under its own conditions. Werner and coauthors’ 2024 paper, Reproduce, Replicate, Reevaluate. The Long but Safe Way to Extend Machine Learning Methods, explicitly distinguishes reproduction using the available implementation, replication using an alternative implementation, and reevaluation using different datasets. Their framing treats the sequence as a way to detect deficiencies and strengthen machine-learning research.
- Reproduce the reported run. Obtain the available code and data where possible, follow the described conditions, and record the result—including failure to run or material disagreement.
- Identify the core method to replicate. Separate the algorithm or claimed result from incidental choices in the published code. Document any interpretation needed to implement it.
- Build a separate implementation. Avoid copying the original implementation’s critical logic wholesale. The goal is to create a meaningful alternative path, not a second wrapper around the same code.
- Match the comparison conditions. Use matched assumptions and inputs, and make randomness controls comparable. Explain unavoidable differences rather than silently treating outputs from unlike settings as direct agreement.
- Compare at the right level. Check intermediate outputs where they help localize divergence, as well as the final reported quantity. State the comparison criterion and report disagreements, not only the final summary.
- Reevaluate separately. If using different datasets, label the result as reevaluation and explain the change in data. Do not present it as same-condition replication.
Compare validation methods by the evidence they provide
No single method is best for every claim. Choose and interpret checks by asking what part of the result they cover, how independent they are, whether they can expose the failure modes that matter, and what another team would need to rerun them.
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|---|---|---|
| Claim coverage | Does the check address the theorem itself, a specific implementation, or only selected outputs? | A proof, test, and rerun cover different objects; say which one the evidence concerns. |
| Independence | Does the check reuse the original code, or use an alternative implementation and potentially different tools? | Reusing code supports reproduction; independent code adds a distinct implementation path for replication. |
| Sensitivity | Can the experiments probe edge cases, numerical instability, or counterexamples within the stated assumptions? | Test design determines which failures are likely to be noticed; untested cases remain unexamined. |
| Reproducibility | Are code, data, conditions, and environment described well enough for another team to rerun the result? | Missing provenance makes disagreement harder to diagnose and successful reruns harder to interpret. |
| Scope limits | Which assumptions or pipeline components lie outside the formal theorem or experiments? | State those boundaries so readers do not infer a system-wide guarantee from narrower evidence. |
These are complementary criteria, not a numerical score. A formal proof can provide strong assurance about a precisely encoded theorem while leaving implementation correspondence unresolved. A replication can challenge dependence on the original code while leaving universal mathematical validity unresolved. The useful question is not which label sounds strongest, but which uncertainty each check reduces.
Make the evidence auditable
Another team needs enough information to distinguish a real disagreement from a changed setup. Record the provenance of the theorem, experiment, and implementation together, not just the final metric. NeurIPS reproducibility work has included a code submission policy, a community reproducibility challenge, and a reproducibility checklist. Its report describes reproducibility as obtaining similar results using the same code and data when available. The following artifact record is practical guidance; it is not a claim that every field is a mandatory NeurIPS checklist item.
- Formal result: theorem version, assumptions, formal definitions, proof assistant and checker, and relevant dependencies or axioms.
- Implementation: source-code revision and which code is original versus independently implemented.
- Data: dataset identity and the version or selection used, distinguishing original data from data used for reevaluation.
- Execution: software environment, numerical precision, random seeds or other stochastic protocol, and relevant run conditions.
- Evaluation: exact commands or scripts, expected and observed quantities, tolerance, and any failures or deviations from the published procedure.
Report a failed reproduction or a disagreement between implementations as part of the evidence. Explain the conditions actually achieved and where they differed from the intended comparison. A reproducibility checklist is useful only when it makes the work inspectable; a checklist entry is not a substitute for sharing enough detail to understand what was run.
How to state a defensible validation conclusion
Describe each result at the level it supports. For example, say that the specified theorem was checked under its formal assumptions; that a stated set of numerical probes agreed within a reported tolerance; and that an alternative implementation agreed on matched inputs. Do not compress those into “the system is proven correct” unless the formal claim genuinely covers the system property being asserted.
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1Fix the driver behind crashes, sound loss and screen glitches2Repair Windows errors before they cause bigger problems3Scan for outdated or missing drivers - takes under a minuteA strong validation account makes gaps visible: which conditions are assumptions rather than tested facts, which tests were finite probes, which code was reused, and which system components sit outside the theorem. The resulting conclusion may be narrower than the original informal claim, but it is more useful because a reader can see exactly what has—and has not—been checked.
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