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Validate a machine-learning proof with three complementary checks: formalize and mechanically check the precise theorem, design numerical experiments that try to falsify its consequences, and reproduce the published result before replicating the method in an independent implementation. Each check answers a different question. A proof assistant establishes only the encoded statement under its encoded assumptions; finite experiments cannot prove a universal claim; and agreement between two programs is meaningful only when their assumptions, data, randomness, and numerical conditions are made comparable.
The strongest validation record therefore connects the mathematics to the executable system, tests the boundaries where the claim could fail, and documents enough provenance for another team to repeat every step.
What exactly are you validating?
“A machine-learning proof” can refer to several different claims. Before choosing a checker or writing an experiment, identify the object whose validity is at issue:
- A mathematical identity, such as the unbiasedness of a stochastic-gradient estimator.
- A bound, such as an expected generalization-error guarantee.
- A property of an algorithm under explicit data, distribution, precision, or convergence assumptions.
- The correspondence between a formal algorithm and the code used to train or evaluate a model.
- An empirical result reported for a particular dataset, implementation, hardware stack, and random protocol.
These are not interchangeable. A theorem about an idealized optimizer does not automatically certify a production training pipeline. A successful run does not establish the theorem. Write the claim with definitions, quantifiers, assumptions, and scope before collecting evidence.
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Turn an informal claim into a testable statement
Record what is quantified and what is fixed. For example, “the estimator is unbiased” should identify the random variable, the conditioning information, the mathematical gradient, and the expectation. “The model has a generalization guarantee” should identify the loss, sample space, confidence or expectation statement, hypothesis class, and training assumptions. State whether the claim concerns exact real arithmetic, a finite-precision implementation, or an approximation.
Separate theorem scope from system scope
Make a boundary list. The theorem may cover a gradient formula while excluding data loading, preprocessing, random-number generation, floating-point behavior, optimizer state, checkpointing, and deployment. An experiment may cover an end-to-end script while saying nothing about all possible inputs. Keeping these scopes separate prevents a local result from being presented as a certificate for the entire system.
What a proof assistant can and cannot establish
A proof assistant checks a formal statement by reducing it to rules accepted by its kernel and the declared libraries, definitions, and axioms. This can expose errors that ordinary review misses: if the program and theorem do not fit, the proof does not type-check. The check is powerful precisely because it is narrow.
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- Record the proof assistant, version, libraries, build configuration, and any axioms or trusted extensions relevant to the theorem.
- Read the formal definitions of data, distributions, model, loss, randomness, and arithmetic. Check that each matches the intended mathematical object.
- Compare every theorem assumption with the assumptions used by the proposed algorithm and experiment. Look for hidden requirements such as independence, boundedness, differentiability, or exact probability distributions.
- Identify the theorem’s quantifiers and conclusion. Determine whether it covers all inputs, an expectation, a high-probability event, or one fixed dataset.
- Trace the correspondence between formal functions and executable code. A formally verified function that is not the function called by the training script does not certify that script.
Selsam, Liang, and Dill’s Certigrad work illustrates this approach: the authors formalized a machine-learning system and proved that sampled gradients are unbiased estimates of the mathematical gradients. Their point is not that formalization makes every ML concern disappear, but that implementation errors capable of breaking the stated theorem are exposed because the proof fails. The result still depends on the encoded definitions and assumptions.
Formal results with bounded scope
Bagnall and Stewart’s MLCERT uses Coq to check generalization bounds and demonstrates the framework on neural networks trained with TensorFlow on extended MNIST. That example supports confidence in the specified bound and formal model; it does not prove that every TensorFlow operation, data-preparation choice, hardware behavior, or deployment decision satisfies the same property. Treat the verified theorem as a contract with explicit boundaries, not as a blanket approval of the pipeline.
How to design numerical experiments that can find flaws
Numerical work is most useful when designed as a falsification attempt rather than a demonstration. Ask which observation would contradict the claim, then construct tests likely to produce that observation if an assumption, derivation, or implementation is wrong.
Build a test matrix from the theorem
- Small exact cases: use dimensions and values for which a hand calculation or high-precision reference is practical.
- Boundary cases: test zero, one, empty, maximal, minimal, and equality conditions that the assumptions permit.
- Limiting cases: vary a parameter toward a stated limit and check whether the predicted behavior appears.
- Stress cases: use ill-conditioned matrices, extreme magnitudes, long sequences, rare classes, or near-degenerate distributions when numerical instability could matter.
- Assumption violations: deliberately break one assumption at a time to verify that the conclusion is no longer being claimed outside its domain.
- Reference comparisons: compare outputs with independently calculated values, symbolic results, higher precision, or a second numerical method.
For every test, record the input identity, expected relation, observed value, tolerance, arithmetic precision, and whether the tolerance was chosen before seeing the result. A plot without a declared tolerance can hide systematic error.
Test stochastic claims correctly
When a theorem concerns an expectation or probability, one run is not the target. Specify the random-number generator and seed policy, number of repetitions, estimator of the quantity of interest, and uncertainty summary. Compare the observed average or confidence interval with the theorem’s prediction under the same sampling assumptions. A finite sample that agrees with an expectation is evidence, not proof of equality.
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Look for numerical instability
Machine-learning systems combine noisy data, non-convex objectives, model misspecification, and finite precision. Run sensitivity checks that vary precision, ordering, batch size, and conditioning where those factors are relevant. If two mathematically equivalent expressions diverge numerically, report the divergence and determine whether the theorem assumes exact arithmetic or provides a stability bound. Do not silently replace a failed computation with a more convenient result.
Reproduction, replication, and reevaluation are different checks
Use the terms precisely because each changes a different source of uncertainty. Werner and coauthors describe a sequence that reproduces a result with the available implementation, replicates it with an alternative implementation, and reevaluates it on different datasets.
| Activity | What changes | Question answered | Typical evidence |
|---|---|---|---|
| Reproduction | Nothing essential: use the published code, data, configuration, and conditions. | Can the reported result be obtained from the artifacts as described? | Matching metrics, logs, checkpoints, and documented environment, with discrepancies explained. |
| Replication | Implement the core method independently while matching the original inputs and assumptions. | Does the finding survive a separate implementation rather than relying on one codebase? | Comparable outputs, controlled randomness, and an account of every implementation choice. |
| Reevaluation | Change datasets or evaluation settings while retaining the method and a justified protocol. | How sensitive is the finding to data or task choice? | Results on the new data, with the changed conditions clearly separated from replication. |
Do not call a new-dataset experiment a replication, and do not treat rerunning the same script as evidence of implementation independence. Each activity should have its own report and failure analysis.
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A step-by-step validation workflow
1. Freeze the claim and version
Create a short claim record containing the theorem or empirical proposition, definitions, quantifiers, assumptions, intended algorithm, and scope. Assign a version or commit identifier to the formal statement and to the code that is supposed to implement it. If the statement changes, open a new validation record rather than overwriting the old one.
2. Map assumptions to executable checks
For each assumption, state how it is established or monitored. A distributional assumption may require a data-generation procedure; a bounded-input assumption may require a validation assertion; a differentiability assumption may require restricting the model class. Mark assumptions that are merely asserted. This map is often where an informal proof and an actual training script diverge.
3. Run the formal checker
Build the formal development from a clean environment and record the checker output. Inspect dependencies and axioms that materially affect the theorem. A successful build confirms that the encoded statement follows in that formal system; it does not confirm that the encoding is the intended mathematics. Review the definitions and the code-to-theorem correspondence separately.
4. Create falsification experiments
Derive tests from each conclusion and assumption. Include exact toy cases, boundaries, limiting behavior, stress conditions, and independent references. Predefine tolerances and failure criteria. Keep a test that fails: a falsification-oriented suite is valuable even when it finds no counterexample.
5. Reproduce the published result
Use the available implementation and data first. Record source revision, data checksums or unambiguous identities, configuration files, software versions, hardware-relevant settings, random seeds or stochastic protocol, numerical precision, and the exact invocation. Compare intermediate artifacts as well as final scores; a matching final number can conceal a different execution path.
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6. Replicate independently
Implement the core method without copying the original implementation’s internals. Match the theorem’s assumptions, input transformations, objective, stopping rule, and evaluation metric. Where randomness is intrinsic, compare distributions of outcomes or confidence intervals rather than demanding identical trajectories. Investigate any systematic difference before deciding whether it is an implementation bug, an unstated assumption, or an expected numerical effect.
7. Reevaluate on changed data
Only after reproduction and replication should you change datasets or tasks to test robustness. Preserve the original protocol as a control. Explain why the new data are relevant and which theorem assumptions remain valid. A changed-data result is a new empirical observation, not confirmation that the original experiment was reproduced.
8. Publish discrepancies, not just successes
Report failed builds, missing artifacts, tolerance breaches, unstable runs, and deviations from the published protocol. Classify each discrepancy as mathematical, formalization, implementation, environment, data, or reporting related. A transparent failure can identify a defect more reliably than a polished success.
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The NeurIPS reproducibility program describes reproducibility as obtaining similar results with the same code and data when available, and it promotes code submission, a community challenge, and a reproducibility checklist. Use that spirit to make the evidence inspectable:
- The exact theorem text, formal definitions, proof-assistant and library versions, dependencies, and relevant axioms.
- Source-code revision identifiers for the original, reproduced, and independent implementations.
- Data identities, preprocessing steps, train-validation-test partitions, and any generated-data seeds.
- Configuration values, command or script entry points, stopping criteria, evaluation metrics, and checkpoint-selection rules.
- Operating-system, language, framework, accelerator, compiler, and numerical-precision information where they can affect results.
- Random-number generator, seed policy, number of runs, aggregation method, and uncertainty or tolerance calculations.
- Expected outputs, observed outputs, logs, failure traces, and explanations for every change from the stated protocol.
These fields are practical reporting guidance rather than a claim that one checklist is a universal standard. The goal is to let an independent reader distinguish a mathematical disagreement from an environment mismatch.
How to compare the evidence from each method
Use the following questions when deciding whether a validation record is strong enough for its intended claim:
| Axis | Formal proof | Numerical experiment | Independent implementation |
|---|---|---|---|
| Claim coverage | Checks the formal theorem and its encoded assumptions. | Checks selected consequences and inputs. | Checks whether the method survives a separate executable realization. |
| Independence | Depends on the formal definitions, checker, libraries, and trusted base. | Can use independent references or precision, but may share code with the method under test. | Provides the strongest separation from the original code when implemented without reuse. |
| Sensitivity | Can expose logical inconsistency in the formal development. | Can expose counterexamples, edge-case failures, and numerical instability within tested ranges. | Can expose hidden implementation choices and undocumented assumptions. |
| Reproducibility | Requires a buildable formal artifact and dependency record. | Requires inputs, protocol, tolerances, and environment details. | Requires a clear specification of the algorithm and matched evaluation conditions. |
| Remaining scope | Unformalized code, data pipelines, hardware, and deployment behavior may remain unchecked. | Untested inputs and universal claims remain outside the evidence. | Agreement does not prove the theorem or guarantee behavior on new data. |
No row substitutes for another. Select the combination that matches the risk of the claim: a theorem about an estimator needs formal and numerical checks; a published benchmark needs reproduction and replication; a safety-critical deployment also needs testing of the surrounding data and software pipeline.
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Common failure modes and how to correct them
Calling numerical agreement a proof
Symptom: hundreds of test cases match the prediction, so the report says the theorem is confirmed.
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Correction: describe the tests as evidence against selected counterexamples. State the tested domain, tolerance, precision, and untested cases, then rely on the formal proof for the universal statement if the formalization is faithful.
Proving the wrong theorem
Symptom: the checker succeeds, but the formal model omits preprocessing, uses a different loss, or assumes data properties the experiment does not satisfy.
Correction: perform the definition-and-assumption audit and publish a correspondence table between formal objects and executable components. Narrow the claim when the mapping cannot be established.
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Symptom: an independent-looking script copies the original formula or utility code and obtains the same result.
Correction: derive the implementation from the written specification, use separate numerical references where possible, and compare intermediate values. Independence is about the path to the result, not merely a different filename.
Ignoring stochastic variation
Symptom: one seed is treated as representative, or two runs are judged inconsistent because their trajectories differ.
Correction: specify the stochastic protocol, repeat runs, report an appropriate summary, and distinguish expected variation from systematic disagreement.
Hiding numerical instability
Symptom: tolerances are widened after failures, precision is changed without disclosure, or unstable cases are removed.
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Correction: set tolerances in advance, report precision and conditioning, retain failed cases, and explain whether the theorem includes a numerical-stability guarantee.
Confusing reevaluation with replication
Symptom: a different dataset is presented as proof that the original experiment was reproduced.
Correction: report same-condition reproduction and alternative-code replication first, then label changed-data results as reevaluation with their new assumptions.
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Interpreting a disagreement
A mismatch is a diagnostic signal, not an immediate verdict. Triage it in this order:
- Check data identity, preprocessing, labels, and partitioning.
- Check source revisions, configuration values, evaluation code, and stopping rules.
- Check random-number generators, seeds, parallelism, nondeterministic kernels, and run counts.
- Check arithmetic precision, overflow or underflow, operation ordering, and conditioning.
- Check whether both implementations satisfy the theorem’s assumptions.
- Check the formal definitions and proof dependencies for a mismatch with the intended algorithm.
For each discovered cause, rerun the smallest test that isolates it. Keep the original failing artifact and the corrected result so readers can see what changed. If no cause is found, report the disagreement with its uncertainty and scope rather than selecting the more convenient number.
What a defensible final claim looks like
A defensible report states three conclusions separately. First, the proof assistant checked a named formal theorem under named definitions, dependencies, and assumptions. Second, numerical tests found no counterexample in a declared set of exact, boundary, limiting, and stress cases, within declared tolerances. Third, the published result was reproduced where artifacts allowed, and the method was or was not replicated independently under matched conditions. Any reevaluation on different data is presented as additional evidence with a separate scope.
This wording is deliberately narrower than “the system is proven correct.” Formal mathematics, experiments, and independent implementations reinforce one another only when their boundaries are explicit. The combination gives readers a way to inspect the claim, challenge it, and repeat it rather than asking them to trust a single type of evidence.
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