Q-Learning on FrozenLake

Formulate the problem before choosing the machinery

Q-Learning on FrozenLake begins with a decision problem, not a framework. Train a tabular Q-learning agent with epsilon-greedy exploration and greedy evaluation. Restate that sentence as an observable input, a desired output, and a criterion for preferring one output over another. Identify who or what supplies supervision, whether feedback is immediate or delayed, and whether examples can be considered independent. These choices determine what can be learned and what remains an assumption. The implementation is honest only when those assumptions are visible near the data contract rather than buried in training code.

The raw material becomes a state transition. Representation decides which distinctions the system can express and which distinctions disappear. List categorical domains, numerical units, missing-value semantics, sequence or spatial axes, masks, player or client perspective, and precision. Then consider invariances: should translation, permutation, rescaling, token position, client identity, or board symmetry change the answer? An architecture that ignores the required invariance wastes data; one that imposes the wrong invariance makes the target impossible to represent.

Finally define the baseline and the abstention point. A baseline can be a constant predictor, random policy, linear rule, naive kernel, synchronous algorithm, or human heuristic. It anchors complexity in evidence. The abstention point describes inputs for which the system lacks support and should decline, defer, or fall back. Together they prevent Q-Learning on FrozenLake from being judged only by an impressive end-to-end demonstration while basic correctness, calibration, robustness, or operational usefulness remains unknown.

Connect the objective to the behavior you actually want

An objective compresses preferences into a scalar, but no scalar captures every product or scientific goal. For Q-Learning on FrozenLake, distinguish the training objective from the evaluation metric and the deployment utility. The training objective must provide a usable signal to parameters or state; evaluation must estimate generalization under a controlled protocol; deployment utility includes latency, cost, safety, and the consequence of errors. When these three disagree, optimization can succeed while the system becomes less useful.

Study each term dimensionally and statistically. Ask what happens if one term is multiplied by ten, one class becomes rare, a sequence becomes longer, a client contributes more samples, or rewards are shifted. Determine whether averages are per token, example, client, action, spatial position, or batch. Regularization is not decorative: it encodes a preference over solutions and changes units unless normalized consistently. A correct derivation names the population quantity of interest, its finite-sample estimator, and the approximation introduced by minibatches, replay, sampling, or surrogate losses.

Identifiability is the deeper constraint. Data may not contain enough information to separate competing explanations. Bellman update, Epsilon greedy, Q-table can improve computation or inductive bias, but they cannot manufacture missing evidence. State causal assumptions, observability limits, support conditions, and equivalence classes of solutions. Use sensitivity analysis and targeted interventions where possible. When identification is impossible, report uncertainty or a set of plausible answers rather than converting an arbitrary modeling choice into unwarranted confidence.

Make mathematical equivalence survive finite precision

Paper algebra assumes exact real numbers; the implementation uses finite precision, bounded memory, and discrete execution order. In Q-Learning on FrozenLake, audit exponentials, logarithms, divisions, reductions, norms, probabilities, recursive values, and accumulated updates. Rewrite unstable expressions with max subtraction, log-sum-exp, compensated accumulation, safe denominators, or higher-precision reductions. Track where a mathematically harmless reordering changes rounding and where mixed precision needs scaling or master copies.

Shapes are part of the proof. Annotate each intermediate with semantic axes rather than only dimensions: batch, token, head, channel, client, action, expert, feature, row, column, or sample. Broadcasting should be intentional and verified with asymmetric dimensions so an accidental match cannot hide. Record contiguous layout and stride assumptions when performance code depends on them. For every reshape or transpose, write both the precondition and the inverse operation needed during backward, decoding, aggregation, or reconstruction.

Build a numerical ladder: scalar example, tiny vector or matrix example, batched reference, optimized path, then realistic workload. At each rung compare values and invariants before increasing scale. This catches defects while they are still interpretable. The acceptance test should specify absolute and relative error, exceptional values, deterministic modes, and the hardware or library versions used. Numerical stability is not a final cleanup task; it is part of the algorithm’s definition.

Design evidence that can falsify the implementation

Evaluation is an experiment. For Q-Learning on FrozenLake, specify the unit of analysis, split strategy, temporal boundary, randomization, baseline, metric, and uncertainty before viewing final results. Prevent duplicates, transformed copies, future information, opponent leakage, and shared-client information from crossing the boundary. A single aggregate score can hide subgroup collapse, unstable seeds, poor calibration, tail latency, or rare catastrophic behavior, so pair it with distributions and stratified slices.

Ablations connect outcomes to mechanisms. Remove or replace Bellman update, Epsilon greedy, Q-table one at a time while controlling data, compute, and evaluation. Compare equal wall-clock or equal resource budgets when efficiency is part of the claim. Repeat stochastic runs and report variation rather than selecting the best seed. Inspect learning curves and intermediate metrics because two systems with the same final score may differ radically in sample efficiency, stability, or cost.

The test suite and the benchmark answer different questions. Unit and property tests prove local contracts; integration tests prove components agree; benchmarks estimate behavior at scale; task evaluation estimates usefulness. Preserve all four. A benchmark that bypasses validation or uses a different code path from production is weak evidence. The strongest release gate reruns the exact packaged implementation with recorded configuration and produces an artifact that another person can inspect.

Turn the learning artifact into an operable system

Production structure separates pure computation from orchestration, configuration, persistence, and interfaces. Package the core of Q-Learning on FrozenLake behind typed contracts. Keep data loading, model or state construction, training, evaluation, serialization, and serving independently invocable. Configuration should be validated, versioned, and printable. Random seeds, data identifiers, source commit, dependency lock, hardware, and metric definitions belong in the run record so an apparent regression can be reproduced instead of guessed at.

Capacity planning follows the critical path. Measure environment interactions, exploration budget, replay reuse, and evaluation games across representative input sizes and concurrency. Report warm-up separately, distinguish throughput from latency, and include tail percentiles. Define memory ownership and lifetime so caches, activations, buffers, replay, or optimizer state cannot grow without a bound. Backpressure and admission control are preferable to unpredictable collapse. Where hardware-specific acceleration exists, preserve a portable reference path for correctness and degraded operation.

Observability must explain decisions and failures without exposing sensitive content. Log stable identifiers, shapes, versions, summary statistics, timings, and error categories. Monitor input drift, output distribution, task quality, saturation, retries, and fallback rate. Establish rollback and shadow-evaluation procedures before the first risky change. A production-grade implementation is not merely more abstract than a notebook; it makes dependencies, state, failure, and evidence explicit enough for another engineer to operate safely.

Read claims as reproducible hypotheses

The research surrounding Q-Learning on FrozenLake improves representations, objectives, algorithms, systems, or evaluation protocols. Classify each paper by which lever it changes. Then identify the comparison budget: data, parameters, tokens, environment steps, hardware, communication, wall-clock time, and tuning effort. A claimed improvement may disappear when budgets are normalized or when the baseline receives equal tuning. Read methods and appendices for details that determine reproducibility, not only the abstract and headline table.

Reproduction begins with the smallest claim. Recreate one table row or ablation before attempting the entire system. Preserve the authors’ preprocessing and metric definitions, then deliberately vary one assumption. Document deviations, failed attempts, and environment details. When a result does not reproduce, distinguish an implementation defect from missing procedural knowledge, stochastic uncertainty, and genuine sensitivity. Negative evidence is useful when it narrows the conditions under which the method works.

Extension should start from a mechanism and a falsifiable prediction. The skills developed here—Q-learning, Exploration, Evaluation—suggest multiple directions, but change one major factor at a time. Predict which metric and intermediate signal should move if the explanation is correct. Use confidence intervals and preregistered stopping rules for expensive experiments where possible. Publish code, configuration, data provenance, and failure cases so the work contributes more than another isolated score.

Maintain a chain of evidence from equation to outcome

A proof ledger for Q-Learning on FrozenLake links each important claim to the smallest evidence that could disprove it. For a mathematical claim, keep a hand-worked example and a high-precision reference. For a software contract, keep unit and property tests. For an optimization claim, keep profiler traces and equal-budget baselines. For a learning claim, keep per-seed results, confidence intervals, and ablations. For a production claim, keep load tests, failure injection, monitoring queries, and rollback evidence. This structure prevents one successful end-to-end run from being treated as proof of every layer beneath it.

Record evidence beside the versioned artifact it evaluates. A metric without its dataset revision, configuration, dependency lock, hardware, and commit cannot reliably settle a regression. Likewise, a screenshot or generated sample is qualitative evidence, not a distribution. Name the claim, evidence type, acceptance threshold, owner, and date. When the implementation changes, rerun the smallest affected evidence first and then the downstream integration gates. The ledger becomes a map of confidence: it shows what is known, what is assumed, what has become stale, and where another experiment is required.

Use the ledger during review. Ask whether each test would fail for a realistic defect, whether each benchmark measures the packaged code path, whether every aggregate retains inspectable raw values, and whether uncertainty is reported at the correct independent unit. Include counterexamples and failed experiments because they define the boundary of the method. Over time this habit turns Q-learning, Exploration, Evaluation from isolated implementation skills into a reproducible engineering practice that survives new data, new hardware, new collaborators, and changing product constraints.

Init Q Table — from contract to production evidence

Init Q Table is the construction at milestone 1 of Q-Learning on FrozenLake. Its purpose is not merely to make the next function run. It establishes a contract between q-table setup and every downstream stage. Begin by naming the accepted inputs, their axes, units, legal ranges, ownership rules, and whether mutation is permitted. Then name the output with the same precision. In this project the surrounding ideas—Bellman update, Epsilon greedy, Q-table—only compose correctly when this boundary preserves those invariants. A useful implementation note records one representative shape, one smallest valid example, one boundary example, and one invalid example before any optimization is attempted.

From first principles, treat Init Q Table as a mapping from available information to a new state transition. Ask which information is genuinely known at this point and which information would leak from the future, evaluation set, opposing player, held-out client, or later pipeline stage. Write the transformation symbolically before translating it into array operations. Every reduction must state its axis; every probability must state its normalization set; every random choice must state its distribution and seed; every learned quantity must state the objective that changes it. This discipline turns an appealing formula into an executable specification that can be challenged with small counterexamples.

The reference implementation should favor clarity over cleverness. Separate validation, the mathematical core, and state updates so each can be tested independently. Use explicit intermediate names that correspond to the derivation rather than compressing the work into one expression. Confirm dtype promotion, broadcasting, device placement, and empty-input behavior. If Init Q Table depends on randomness, pass a generator instead of reading hidden global state. If it owns mutable state, return or document the updated state explicitly. The optimized implementation may later fuse operations or reuse buffers, but it must remain numerically comparable with this small version on deterministic fixtures.

Verification for Init Q Table needs more than a happy-path assertion. Prove a hand-computable normal case, a boundary case, an invalid case, and at least one invariant. Compare against seeded rollouts, learning curves, confidence intervals, and baseline matches. Add metamorphic tests when an exact answer is awkward: permutation, scaling, symmetry, conservation, monotonicity, or equivalence under a harmless representation change. Run the test repeatedly under fixed seeds to distinguish deterministic defects from statistical variation. When floating-point arithmetic is involved, justify tolerances from expected rounding error instead of choosing a loose threshold simply because the test passes.

Failure analysis asks how Init Q Table can look plausible while being wrong. Inspect reward leakage, unstable bootstrapping, invalid actions, and optimistic evaluation. Trace one example through every intermediate value and preserve enough logging to reproduce it. Distinguish a contract violation from an optimization failure and from an evaluation-design failure; each requires a different repair. A numerical answer within range is not automatically meaningful, and a rising training metric is not proof that the intended signal is being learned. The strongest debugging move is usually to shrink the input until the complete computation fits on paper, then compare the paper trace with the program line by line.

Productionizing Init Q Table changes the question from “does it work once?” to “does it remain trustworthy under load and change?” Measure environment interactions, exploration budget, replay reuse, and evaluation games. Define observability for inputs, outputs, latency, failures, drift, and resource saturation. Decide what happens on malformed data, cancellation, partial worker failure, unavailable accelerators, or a distribution outside the training envelope. Version configuration and schemas with the code, preserve reproducible seeds where appropriate, and expose a safe fallback. Optimization is accepted only when the reference tests, numerical comparisons, and task-level metrics remain within an explicitly documented budget.

• Part: Q-Table Setup. Initialize the tabular value store and basic lookups over states and actions.
• Normal case: choose the smallest input that exercises the intended transformation.
• Boundary case: use an empty, singleton, saturated, masked, terminal, or maximum-size input as appropriate.
• Invariant: verify shape, range, conservation, normalization, symmetry, immutability, or monotonicity.
• Production evidence: record correctness, latency, memory or cost, and the exact configuration.

Max Q Value — from contract to production evidence

Max Q Value is the measurement at milestone 2 of Q-Learning on FrozenLake. Its purpose is not merely to make the next function run. It establishes a contract between q-table setup and every downstream stage. Begin by naming the accepted inputs, their axes, units, legal ranges, ownership rules, and whether mutation is permitted. Then name the output with the same precision. In this project the surrounding ideas—Bellman update, Epsilon greedy, Q-table—only compose correctly when this boundary preserves those invariants. A useful implementation note records one representative shape, one smallest valid example, one boundary example, and one invalid example before any optimization is attempted.

From first principles, treat Max Q Value as a mapping from available information to a new state transition. Ask which information is genuinely known at this point and which information would leak from the future, evaluation set, opposing player, held-out client, or later pipeline stage. Write the transformation symbolically before translating it into array operations. Every reduction must state its axis; every probability must state its normalization set; every random choice must state its distribution and seed; every learned quantity must state the objective that changes it. This discipline turns an appealing formula into an executable specification that can be challenged with small counterexamples.

The reference implementation should favor clarity over cleverness. Separate validation, the mathematical core, and state updates so each can be tested independently. Use explicit intermediate names that correspond to the derivation rather than compressing the work into one expression. Confirm dtype promotion, broadcasting, device placement, and empty-input behavior. If Max Q Value depends on randomness, pass a generator instead of reading hidden global state. If it owns mutable state, return or document the updated state explicitly. The optimized implementation may later fuse operations or reuse buffers, but it must remain numerically comparable with this small version on deterministic fixtures.

Verification for Max Q Value needs more than a happy-path assertion. Prove a hand-computable normal case, a boundary case, an invalid case, and at least one invariant. Compare against seeded rollouts, learning curves, confidence intervals, and baseline matches. Add metamorphic tests when an exact answer is awkward: permutation, scaling, symmetry, conservation, monotonicity, or equivalence under a harmless representation change. Run the test repeatedly under fixed seeds to distinguish deterministic defects from statistical variation. When floating-point arithmetic is involved, justify tolerances from expected rounding error instead of choosing a loose threshold simply because the test passes.

Failure analysis asks how Max Q Value can look plausible while being wrong. Inspect reward leakage, unstable bootstrapping, invalid actions, and optimistic evaluation. Trace one example through every intermediate value and preserve enough logging to reproduce it. Distinguish a contract violation from an optimization failure and from an evaluation-design failure; each requires a different repair. A numerical answer within range is not automatically meaningful, and a rising training metric is not proof that the intended signal is being learned. The strongest debugging move is usually to shrink the input until the complete computation fits on paper, then compare the paper trace with the program line by line.

Productionizing Max Q Value changes the question from “does it work once?” to “does it remain trustworthy under load and change?” Measure environment interactions, exploration budget, replay reuse, and evaluation games. Define observability for inputs, outputs, latency, failures, drift, and resource saturation. Decide what happens on malformed data, cancellation, partial worker failure, unavailable accelerators, or a distribution outside the training envelope. Version configuration and schemas with the code, preserve reproducible seeds where appropriate, and expose a safe fallback. Optimization is accepted only when the reference tests, numerical comparisons, and task-level metrics remain within an explicitly documented budget.

• Part: Q-Table Setup. Initialize the tabular value store and basic lookups over states and actions.
• Normal case: choose the smallest input that exercises the intended transformation.
• Boundary case: use an empty, singleton, saturated, masked, terminal, or maximum-size input as appropriate.
• Invariant: verify shape, range, conservation, normalization, symmetry, immutability, or monotonicity.
• Production evidence: record correctness, latency, memory or cost, and the exact configuration.

Greedy Action — from contract to production evidence

Greedy Action is the decision at milestone 3 of Q-Learning on FrozenLake. Its purpose is not merely to make the next function run. It establishes a contract between q-table setup and every downstream stage. Begin by naming the accepted inputs, their axes, units, legal ranges, ownership rules, and whether mutation is permitted. Then name the output with the same precision. In this project the surrounding ideas—Bellman update, Epsilon greedy, Q-table—only compose correctly when this boundary preserves those invariants. A useful implementation note records one representative shape, one smallest valid example, one boundary example, and one invalid example before any optimization is attempted.

From first principles, treat Greedy Action as a mapping from available information to a new state transition. Ask which information is genuinely known at this point and which information would leak from the future, evaluation set, opposing player, held-out client, or later pipeline stage. Write the transformation symbolically before translating it into array operations. Every reduction must state its axis; every probability must state its normalization set; every random choice must state its distribution and seed; every learned quantity must state the objective that changes it. This discipline turns an appealing formula into an executable specification that can be challenged with small counterexamples.

The reference implementation should favor clarity over cleverness. Separate validation, the mathematical core, and state updates so each can be tested independently. Use explicit intermediate names that correspond to the derivation rather than compressing the work into one expression. Confirm dtype promotion, broadcasting, device placement, and empty-input behavior. If Greedy Action depends on randomness, pass a generator instead of reading hidden global state. If it owns mutable state, return or document the updated state explicitly. The optimized implementation may later fuse operations or reuse buffers, but it must remain numerically comparable with this small version on deterministic fixtures.

Verification for Greedy Action needs more than a happy-path assertion. Prove a hand-computable normal case, a boundary case, an invalid case, and at least one invariant. Compare against seeded rollouts, learning curves, confidence intervals, and baseline matches. Add metamorphic tests when an exact answer is awkward: permutation, scaling, symmetry, conservation, monotonicity, or equivalence under a harmless representation change. Run the test repeatedly under fixed seeds to distinguish deterministic defects from statistical variation. When floating-point arithmetic is involved, justify tolerances from expected rounding error instead of choosing a loose threshold simply because the test passes.

Failure analysis asks how Greedy Action can look plausible while being wrong. Inspect reward leakage, unstable bootstrapping, invalid actions, and optimistic evaluation. Trace one example through every intermediate value and preserve enough logging to reproduce it. Distinguish a contract violation from an optimization failure and from an evaluation-design failure; each requires a different repair. A numerical answer within range is not automatically meaningful, and a rising training metric is not proof that the intended signal is being learned. The strongest debugging move is usually to shrink the input until the complete computation fits on paper, then compare the paper trace with the program line by line.

Productionizing Greedy Action changes the question from “does it work once?” to “does it remain trustworthy under load and change?” Measure environment interactions, exploration budget, replay reuse, and evaluation games. Define observability for inputs, outputs, latency, failures, drift, and resource saturation. Decide what happens on malformed data, cancellation, partial worker failure, unavailable accelerators, or a distribution outside the training envelope. Version configuration and schemas with the code, preserve reproducible seeds where appropriate, and expose a safe fallback. Optimization is accepted only when the reference tests, numerical comparisons, and task-level metrics remain within an explicitly documented budget.

• Part: Q-Table Setup. Initialize the tabular value store and basic lookups over states and actions.
• Normal case: choose the smallest input that exercises the intended transformation.
• Boundary case: use an empty, singleton, saturated, masked, terminal, or maximum-size input as appropriate.
• Invariant: verify shape, range, conservation, normalization, symmetry, immutability, or monotonicity.
• Production evidence: record correctness, latency, memory or cost, and the exact configuration.

Sample Random Action — from contract to production evidence

Sample Random Action is the decision at milestone 4 of Q-Learning on FrozenLake. Its purpose is not merely to make the next function run. It establishes a contract between epsilon-greedy action selection and every downstream stage. Begin by naming the accepted inputs, their axes, units, legal ranges, ownership rules, and whether mutation is permitted. Then name the output with the same precision. In this project the surrounding ideas—Bellman update, Epsilon greedy, Q-table—only compose correctly when this boundary preserves those invariants. A useful implementation note records one representative shape, one smallest valid example, one boundary example, and one invalid example before any optimization is attempted.

From first principles, treat Sample Random Action as a mapping from available information to a new state transition. Ask which information is genuinely known at this point and which information would leak from the future, evaluation set, opposing player, held-out client, or later pipeline stage. Write the transformation symbolically before translating it into array operations. Every reduction must state its axis; every probability must state its normalization set; every random choice must state its distribution and seed; every learned quantity must state the objective that changes it. This discipline turns an appealing formula into an executable specification that can be challenged with small counterexamples.

The reference implementation should favor clarity over cleverness. Separate validation, the mathematical core, and state updates so each can be tested independently. Use explicit intermediate names that correspond to the derivation rather than compressing the work into one expression. Confirm dtype promotion, broadcasting, device placement, and empty-input behavior. If Sample Random Action depends on randomness, pass a generator instead of reading hidden global state. If it owns mutable state, return or document the updated state explicitly. The optimized implementation may later fuse operations or reuse buffers, but it must remain numerically comparable with this small version on deterministic fixtures.

Verification for Sample Random Action needs more than a happy-path assertion. Prove a hand-computable normal case, a boundary case, an invalid case, and at least one invariant. Compare against seeded rollouts, learning curves, confidence intervals, and baseline matches. Add metamorphic tests when an exact answer is awkward: permutation, scaling, symmetry, conservation, monotonicity, or equivalence under a harmless representation change. Run the test repeatedly under fixed seeds to distinguish deterministic defects from statistical variation. When floating-point arithmetic is involved, justify tolerances from expected rounding error instead of choosing a loose threshold simply because the test passes.

Failure analysis asks how Sample Random Action can look plausible while being wrong. Inspect reward leakage, unstable bootstrapping, invalid actions, and optimistic evaluation. Trace one example through every intermediate value and preserve enough logging to reproduce it. Distinguish a contract violation from an optimization failure and from an evaluation-design failure; each requires a different repair. A numerical answer within range is not automatically meaningful, and a rising training metric is not proof that the intended signal is being learned. The strongest debugging move is usually to shrink the input until the complete computation fits on paper, then compare the paper trace with the program line by line.

Productionizing Sample Random Action changes the question from “does it work once?” to “does it remain trustworthy under load and change?” Measure environment interactions, exploration budget, replay reuse, and evaluation games. Define observability for inputs, outputs, latency, failures, drift, and resource saturation. Decide what happens on malformed data, cancellation, partial worker failure, unavailable accelerators, or a distribution outside the training envelope. Version configuration and schemas with the code, preserve reproducible seeds where appropriate, and expose a safe fallback. Optimization is accepted only when the reference tests, numerical comparisons, and task-level metrics remain within an explicitly documented budget.

• Part: Epsilon-Greedy Action Selection. Implement exploration vs exploitation using random sampling and epsilon decay.
• Normal case: choose the smallest input that exercises the intended transformation.
• Boundary case: use an empty, singleton, saturated, masked, terminal, or maximum-size input as appropriate.
• Invariant: verify shape, range, conservation, normalization, symmetry, immutability, or monotonicity.
• Production evidence: record correctness, latency, memory or cost, and the exact configuration.

Should Explore — from contract to production evidence

Should Explore is the pipeline boundary at milestone 5 of Q-Learning on FrozenLake. Its purpose is not merely to make the next function run. It establishes a contract between epsilon-greedy action selection and every downstream stage. Begin by naming the accepted inputs, their axes, units, legal ranges, ownership rules, and whether mutation is permitted. Then name the output with the same precision. In this project the surrounding ideas—Bellman update, Epsilon greedy, Q-table—only compose correctly when this boundary preserves those invariants. A useful implementation note records one representative shape, one smallest valid example, one boundary example, and one invalid example before any optimization is attempted.

From first principles, treat Should Explore as a mapping from available information to a new state transition. Ask which information is genuinely known at this point and which information would leak from the future, evaluation set, opposing player, held-out client, or later pipeline stage. Write the transformation symbolically before translating it into array operations. Every reduction must state its axis; every probability must state its normalization set; every random choice must state its distribution and seed; every learned quantity must state the objective that changes it. This discipline turns an appealing formula into an executable specification that can be challenged with small counterexamples.

The reference implementation should favor clarity over cleverness. Separate validation, the mathematical core, and state updates so each can be tested independently. Use explicit intermediate names that correspond to the derivation rather than compressing the work into one expression. Confirm dtype promotion, broadcasting, device placement, and empty-input behavior. If Should Explore depends on randomness, pass a generator instead of reading hidden global state. If it owns mutable state, return or document the updated state explicitly. The optimized implementation may later fuse operations or reuse buffers, but it must remain numerically comparable with this small version on deterministic fixtures.

Verification for Should Explore needs more than a happy-path assertion. Prove a hand-computable normal case, a boundary case, an invalid case, and at least one invariant. Compare against seeded rollouts, learning curves, confidence intervals, and baseline matches. Add metamorphic tests when an exact answer is awkward: permutation, scaling, symmetry, conservation, monotonicity, or equivalence under a harmless representation change. Run the test repeatedly under fixed seeds to distinguish deterministic defects from statistical variation. When floating-point arithmetic is involved, justify tolerances from expected rounding error instead of choosing a loose threshold simply because the test passes.

Failure analysis asks how Should Explore can look plausible while being wrong. Inspect reward leakage, unstable bootstrapping, invalid actions, and optimistic evaluation. Trace one example through every intermediate value and preserve enough logging to reproduce it. Distinguish a contract violation from an optimization failure and from an evaluation-design failure; each requires a different repair. A numerical answer within range is not automatically meaningful, and a rising training metric is not proof that the intended signal is being learned. The strongest debugging move is usually to shrink the input until the complete computation fits on paper, then compare the paper trace with the program line by line.

Productionizing Should Explore changes the question from “does it work once?” to “does it remain trustworthy under load and change?” Measure environment interactions, exploration budget, replay reuse, and evaluation games. Define observability for inputs, outputs, latency, failures, drift, and resource saturation. Decide what happens on malformed data, cancellation, partial worker failure, unavailable accelerators, or a distribution outside the training envelope. Version configuration and schemas with the code, preserve reproducible seeds where appropriate, and expose a safe fallback. Optimization is accepted only when the reference tests, numerical comparisons, and task-level metrics remain within an explicitly documented budget.

• Part: Epsilon-Greedy Action Selection. Implement exploration vs exploitation using random sampling and epsilon decay.
• Normal case: choose the smallest input that exercises the intended transformation.
• Boundary case: use an empty, singleton, saturated, masked, terminal, or maximum-size input as appropriate.
• Invariant: verify shape, range, conservation, normalization, symmetry, immutability, or monotonicity.
• Production evidence: record correctness, latency, memory or cost, and the exact configuration.

Epsilon Greedy Action — from contract to production evidence

Epsilon Greedy Action is the decision at milestone 6 of Q-Learning on FrozenLake. Its purpose is not merely to make the next function run. It establishes a contract between epsilon-greedy action selection and every downstream stage. Begin by naming the accepted inputs, their axes, units, legal ranges, ownership rules, and whether mutation is permitted. Then name the output with the same precision. In this project the surrounding ideas—Bellman update, Epsilon greedy, Q-table—only compose correctly when this boundary preserves those invariants. A useful implementation note records one representative shape, one smallest valid example, one boundary example, and one invalid example before any optimization is attempted.

From first principles, treat Epsilon Greedy Action as a mapping from available information to a new state transition. Ask which information is genuinely known at this point and which information would leak from the future, evaluation set, opposing player, held-out client, or later pipeline stage. Write the transformation symbolically before translating it into array operations. Every reduction must state its axis; every probability must state its normalization set; every random choice must state its distribution and seed; every learned quantity must state the objective that changes it. This discipline turns an appealing formula into an executable specification that can be challenged with small counterexamples.

The reference implementation should favor clarity over cleverness. Separate validation, the mathematical core, and state updates so each can be tested independently. Use explicit intermediate names that correspond to the derivation rather than compressing the work into one expression. Confirm dtype promotion, broadcasting, device placement, and empty-input behavior. If Epsilon Greedy Action depends on randomness, pass a generator instead of reading hidden global state. If it owns mutable state, return or document the updated state explicitly. The optimized implementation may later fuse operations or reuse buffers, but it must remain numerically comparable with this small version on deterministic fixtures.

Verification for Epsilon Greedy Action needs more than a happy-path assertion. Prove a hand-computable normal case, a boundary case, an invalid case, and at least one invariant. Compare against seeded rollouts, learning curves, confidence intervals, and baseline matches. Add metamorphic tests when an exact answer is awkward: permutation, scaling, symmetry, conservation, monotonicity, or equivalence under a harmless representation change. Run the test repeatedly under fixed seeds to distinguish deterministic defects from statistical variation. When floating-point arithmetic is involved, justify tolerances from expected rounding error instead of choosing a loose threshold simply because the test passes.

Failure analysis asks how Epsilon Greedy Action can look plausible while being wrong. Inspect reward leakage, unstable bootstrapping, invalid actions, and optimistic evaluation. Trace one example through every intermediate value and preserve enough logging to reproduce it. Distinguish a contract violation from an optimization failure and from an evaluation-design failure; each requires a different repair. A numerical answer within range is not automatically meaningful, and a rising training metric is not proof that the intended signal is being learned. The strongest debugging move is usually to shrink the input until the complete computation fits on paper, then compare the paper trace with the program line by line.

Productionizing Epsilon Greedy Action changes the question from “does it work once?” to “does it remain trustworthy under load and change?” Measure environment interactions, exploration budget, replay reuse, and evaluation games. Define observability for inputs, outputs, latency, failures, drift, and resource saturation. Decide what happens on malformed data, cancellation, partial worker failure, unavailable accelerators, or a distribution outside the training envelope. Version configuration and schemas with the code, preserve reproducible seeds where appropriate, and expose a safe fallback. Optimization is accepted only when the reference tests, numerical comparisons, and task-level metrics remain within an explicitly documented budget.

• Part: Epsilon-Greedy Action Selection. Implement exploration vs exploitation using random sampling and epsilon decay.
• Normal case: choose the smallest input that exercises the intended transformation.
• Boundary case: use an empty, singleton, saturated, masked, terminal, or maximum-size input as appropriate.
• Invariant: verify shape, range, conservation, normalization, symmetry, immutability, or monotonicity.
• Production evidence: record correctness, latency, memory or cost, and the exact configuration.

Decay Epsilon — from contract to production evidence

Decay Epsilon is the pipeline boundary at milestone 7 of Q-Learning on FrozenLake. Its purpose is not merely to make the next function run. It establishes a contract between epsilon-greedy action selection and every downstream stage. Begin by naming the accepted inputs, their axes, units, legal ranges, ownership rules, and whether mutation is permitted. Then name the output with the same precision. In this project the surrounding ideas—Bellman update, Epsilon greedy, Q-table—only compose correctly when this boundary preserves those invariants. A useful implementation note records one representative shape, one smallest valid example, one boundary example, and one invalid example before any optimization is attempted.

From first principles, treat Decay Epsilon as a mapping from available information to a new state transition. Ask which information is genuinely known at this point and which information would leak from the future, evaluation set, opposing player, held-out client, or later pipeline stage. Write the transformation symbolically before translating it into array operations. Every reduction must state its axis; every probability must state its normalization set; every random choice must state its distribution and seed; every learned quantity must state the objective that changes it. This discipline turns an appealing formula into an executable specification that can be challenged with small counterexamples.

The reference implementation should favor clarity over cleverness. Separate validation, the mathematical core, and state updates so each can be tested independently. Use explicit intermediate names that correspond to the derivation rather than compressing the work into one expression. Confirm dtype promotion, broadcasting, device placement, and empty-input behavior. If Decay Epsilon depends on randomness, pass a generator instead of reading hidden global state. If it owns mutable state, return or document the updated state explicitly. The optimized implementation may later fuse operations or reuse buffers, but it must remain numerically comparable with this small version on deterministic fixtures.

Verification for Decay Epsilon needs more than a happy-path assertion. Prove a hand-computable normal case, a boundary case, an invalid case, and at least one invariant. Compare against seeded rollouts, learning curves, confidence intervals, and baseline matches. Add metamorphic tests when an exact answer is awkward: permutation, scaling, symmetry, conservation, monotonicity, or equivalence under a harmless representation change. Run the test repeatedly under fixed seeds to distinguish deterministic defects from statistical variation. When floating-point arithmetic is involved, justify tolerances from expected rounding error instead of choosing a loose threshold simply because the test passes.

Failure analysis asks how Decay Epsilon can look plausible while being wrong. Inspect reward leakage, unstable bootstrapping, invalid actions, and optimistic evaluation. Trace one example through every intermediate value and preserve enough logging to reproduce it. Distinguish a contract violation from an optimization failure and from an evaluation-design failure; each requires a different repair. A numerical answer within range is not automatically meaningful, and a rising training metric is not proof that the intended signal is being learned. The strongest debugging move is usually to shrink the input until the complete computation fits on paper, then compare the paper trace with the program line by line.

Productionizing Decay Epsilon changes the question from “does it work once?” to “does it remain trustworthy under load and change?” Measure environment interactions, exploration budget, replay reuse, and evaluation games. Define observability for inputs, outputs, latency, failures, drift, and resource saturation. Decide what happens on malformed data, cancellation, partial worker failure, unavailable accelerators, or a distribution outside the training envelope. Version configuration and schemas with the code, preserve reproducible seeds where appropriate, and expose a safe fallback. Optimization is accepted only when the reference tests, numerical comparisons, and task-level metrics remain within an explicitly documented budget.

• Part: Epsilon-Greedy Action Selection. Implement exploration vs exploitation using random sampling and epsilon decay.
• Normal case: choose the smallest input that exercises the intended transformation.
• Boundary case: use an empty, singleton, saturated, masked, terminal, or maximum-size input as appropriate.
• Invariant: verify shape, range, conservation, normalization, symmetry, immutability, or monotonicity.
• Production evidence: record correctness, latency, memory or cost, and the exact configuration.

Td Target — from contract to production evidence

Td Target is the pipeline boundary at milestone 8 of Q-Learning on FrozenLake. Its purpose is not merely to make the next function run. It establishes a contract between q-learning update rule and every downstream stage. Begin by naming the accepted inputs, their axes, units, legal ranges, ownership rules, and whether mutation is permitted. Then name the output with the same precision. In this project the surrounding ideas—Bellman update, Epsilon greedy, Q-table—only compose correctly when this boundary preserves those invariants. A useful implementation note records one representative shape, one smallest valid example, one boundary example, and one invalid example before any optimization is attempted.

From first principles, treat Td Target as a mapping from available information to a new state transition. Ask which information is genuinely known at this point and which information would leak from the future, evaluation set, opposing player, held-out client, or later pipeline stage. Write the transformation symbolically before translating it into array operations. Every reduction must state its axis; every probability must state its normalization set; every random choice must state its distribution and seed; every learned quantity must state the objective that changes it. This discipline turns an appealing formula into an executable specification that can be challenged with small counterexamples.

The reference implementation should favor clarity over cleverness. Separate validation, the mathematical core, and state updates so each can be tested independently. Use explicit intermediate names that correspond to the derivation rather than compressing the work into one expression. Confirm dtype promotion, broadcasting, device placement, and empty-input behavior. If Td Target depends on randomness, pass a generator instead of reading hidden global state. If it owns mutable state, return or document the updated state explicitly. The optimized implementation may later fuse operations or reuse buffers, but it must remain numerically comparable with this small version on deterministic fixtures.

Verification for Td Target needs more than a happy-path assertion. Prove a hand-computable normal case, a boundary case, an invalid case, and at least one invariant. Compare against seeded rollouts, learning curves, confidence intervals, and baseline matches. Add metamorphic tests when an exact answer is awkward: permutation, scaling, symmetry, conservation, monotonicity, or equivalence under a harmless representation change. Run the test repeatedly under fixed seeds to distinguish deterministic defects from statistical variation. When floating-point arithmetic is involved, justify tolerances from expected rounding error instead of choosing a loose threshold simply because the test passes.

Failure analysis asks how Td Target can look plausible while being wrong. Inspect reward leakage, unstable bootstrapping, invalid actions, and optimistic evaluation. Trace one example through every intermediate value and preserve enough logging to reproduce it. Distinguish a contract violation from an optimization failure and from an evaluation-design failure; each requires a different repair. A numerical answer within range is not automatically meaningful, and a rising training metric is not proof that the intended signal is being learned. The strongest debugging move is usually to shrink the input until the complete computation fits on paper, then compare the paper trace with the program line by line.

Productionizing Td Target changes the question from “does it work once?” to “does it remain trustworthy under load and change?” Measure environment interactions, exploration budget, replay reuse, and evaluation games. Define observability for inputs, outputs, latency, failures, drift, and resource saturation. Decide what happens on malformed data, cancellation, partial worker failure, unavailable accelerators, or a distribution outside the training envelope. Version configuration and schemas with the code, preserve reproducible seeds where appropriate, and expose a safe fallback. Optimization is accepted only when the reference tests, numerical comparisons, and task-level metrics remain within an explicitly documented budget.

• Part: Q-Learning Update Rule. Compute TD targets, errors, and apply the Q-learning update to table entries.
• Normal case: choose the smallest input that exercises the intended transformation.
• Boundary case: use an empty, singleton, saturated, masked, terminal, or maximum-size input as appropriate.
• Invariant: verify shape, range, conservation, normalization, symmetry, immutability, or monotonicity.
• Production evidence: record correctness, latency, memory or cost, and the exact configuration.

Td Error — from contract to production evidence

Td Error is the pipeline boundary at milestone 9 of Q-Learning on FrozenLake. Its purpose is not merely to make the next function run. It establishes a contract between q-learning update rule and every downstream stage. Begin by naming the accepted inputs, their axes, units, legal ranges, ownership rules, and whether mutation is permitted. Then name the output with the same precision. In this project the surrounding ideas—Bellman update, Epsilon greedy, Q-table—only compose correctly when this boundary preserves those invariants. A useful implementation note records one representative shape, one smallest valid example, one boundary example, and one invalid example before any optimization is attempted.

From first principles, treat Td Error as a mapping from available information to a new state transition. Ask which information is genuinely known at this point and which information would leak from the future, evaluation set, opposing player, held-out client, or later pipeline stage. Write the transformation symbolically before translating it into array operations. Every reduction must state its axis; every probability must state its normalization set; every random choice must state its distribution and seed; every learned quantity must state the objective that changes it. This discipline turns an appealing formula into an executable specification that can be challenged with small counterexamples.

The reference implementation should favor clarity over cleverness. Separate validation, the mathematical core, and state updates so each can be tested independently. Use explicit intermediate names that correspond to the derivation rather than compressing the work into one expression. Confirm dtype promotion, broadcasting, device placement, and empty-input behavior. If Td Error depends on randomness, pass a generator instead of reading hidden global state. If it owns mutable state, return or document the updated state explicitly. The optimized implementation may later fuse operations or reuse buffers, but it must remain numerically comparable with this small version on deterministic fixtures.

Verification for Td Error needs more than a happy-path assertion. Prove a hand-computable normal case, a boundary case, an invalid case, and at least one invariant. Compare against seeded rollouts, learning curves, confidence intervals, and baseline matches. Add metamorphic tests when an exact answer is awkward: permutation, scaling, symmetry, conservation, monotonicity, or equivalence under a harmless representation change. Run the test repeatedly under fixed seeds to distinguish deterministic defects from statistical variation. When floating-point arithmetic is involved, justify tolerances from expected rounding error instead of choosing a loose threshold simply because the test passes.

Failure analysis asks how Td Error can look plausible while being wrong. Inspect reward leakage, unstable bootstrapping, invalid actions, and optimistic evaluation. Trace one example through every intermediate value and preserve enough logging to reproduce it. Distinguish a contract violation from an optimization failure and from an evaluation-design failure; each requires a different repair. A numerical answer within range is not automatically meaningful, and a rising training metric is not proof that the intended signal is being learned. The strongest debugging move is usually to shrink the input until the complete computation fits on paper, then compare the paper trace with the program line by line.

Productionizing Td Error changes the question from “does it work once?” to “does it remain trustworthy under load and change?” Measure environment interactions, exploration budget, replay reuse, and evaluation games. Define observability for inputs, outputs, latency, failures, drift, and resource saturation. Decide what happens on malformed data, cancellation, partial worker failure, unavailable accelerators, or a distribution outside the training envelope. Version configuration and schemas with the code, preserve reproducible seeds where appropriate, and expose a safe fallback. Optimization is accepted only when the reference tests, numerical comparisons, and task-level metrics remain within an explicitly documented budget.

• Part: Q-Learning Update Rule. Compute TD targets, errors, and apply the Q-learning update to table entries.
• Normal case: choose the smallest input that exercises the intended transformation.
• Boundary case: use an empty, singleton, saturated, masked, terminal, or maximum-size input as appropriate.
• Invariant: verify shape, range, conservation, normalization, symmetry, immutability, or monotonicity.
• Production evidence: record correctness, latency, memory or cost, and the exact configuration.

Q Learning Update — from contract to production evidence

Q Learning Update is the learning update at milestone 10 of Q-Learning on FrozenLake. Its purpose is not merely to make the next function run. It establishes a contract between q-learning update rule and every downstream stage. Begin by naming the accepted inputs, their axes, units, legal ranges, ownership rules, and whether mutation is permitted. Then name the output with the same precision. In this project the surrounding ideas—Bellman update, Epsilon greedy, Q-table—only compose correctly when this boundary preserves those invariants. A useful implementation note records one representative shape, one smallest valid example, one boundary example, and one invalid example before any optimization is attempted.

From first principles, treat Q Learning Update as a mapping from available information to a new state transition. Ask which information is genuinely known at this point and which information would leak from the future, evaluation set, opposing player, held-out client, or later pipeline stage. Write the transformation symbolically before translating it into array operations. Every reduction must state its axis; every probability must state its normalization set; every random choice must state its distribution and seed; every learned quantity must state the objective that changes it. This discipline turns an appealing formula into an executable specification that can be challenged with small counterexamples.

The reference implementation should favor clarity over cleverness. Separate validation, the mathematical core, and state updates so each can be tested independently. Use explicit intermediate names that correspond to the derivation rather than compressing the work into one expression. Confirm dtype promotion, broadcasting, device placement, and empty-input behavior. If Q Learning Update depends on randomness, pass a generator instead of reading hidden global state. If it owns mutable state, return or document the updated state explicitly. The optimized implementation may later fuse operations or reuse buffers, but it must remain numerically comparable with this small version on deterministic fixtures.

Verification for Q Learning Update needs more than a happy-path assertion. Prove a hand-computable normal case, a boundary case, an invalid case, and at least one invariant. Compare against seeded rollouts, learning curves, confidence intervals, and baseline matches. Add metamorphic tests when an exact answer is awkward: permutation, scaling, symmetry, conservation, monotonicity, or equivalence under a harmless representation change. Run the test repeatedly under fixed seeds to distinguish deterministic defects from statistical variation. When floating-point arithmetic is involved, justify tolerances from expected rounding error instead of choosing a loose threshold simply because the test passes.

Failure analysis asks how Q Learning Update can look plausible while being wrong. Inspect reward leakage, unstable bootstrapping, invalid actions, and optimistic evaluation. Trace one example through every intermediate value and preserve enough logging to reproduce it. Distinguish a contract violation from an optimization failure and from an evaluation-design failure; each requires a different repair. A numerical answer within range is not automatically meaningful, and a rising training metric is not proof that the intended signal is being learned. The strongest debugging move is usually to shrink the input until the complete computation fits on paper, then compare the paper trace with the program line by line.

Productionizing Q Learning Update changes the question from “does it work once?” to “does it remain trustworthy under load and change?” Measure environment interactions, exploration budget, replay reuse, and evaluation games. Define observability for inputs, outputs, latency, failures, drift, and resource saturation. Decide what happens on malformed data, cancellation, partial worker failure, unavailable accelerators, or a distribution outside the training envelope. Version configuration and schemas with the code, preserve reproducible seeds where appropriate, and expose a safe fallback. Optimization is accepted only when the reference tests, numerical comparisons, and task-level metrics remain within an explicitly documented budget.

• Part: Q-Learning Update Rule. Compute TD targets, errors, and apply the Q-learning update to table entries.
• Normal case: choose the smallest input that exercises the intended transformation.
• Boundary case: use an empty, singleton, saturated, masked, terminal, or maximum-size input as appropriate.
• Invariant: verify shape, range, conservation, normalization, symmetry, immutability, or monotonicity.
• Production evidence: record correctness, latency, memory or cost, and the exact configuration.

Interaction Step — from contract to production evidence

Interaction Step is the decision at milestone 11 of Q-Learning on FrozenLake. Its purpose is not merely to make the next function run. It establishes a contract between training loop and every downstream stage. Begin by naming the accepted inputs, their axes, units, legal ranges, ownership rules, and whether mutation is permitted. Then name the output with the same precision. In this project the surrounding ideas—Bellman update, Epsilon greedy, Q-table—only compose correctly when this boundary preserves those invariants. A useful implementation note records one representative shape, one smallest valid example, one boundary example, and one invalid example before any optimization is attempted.

From first principles, treat Interaction Step as a mapping from available information to a new state transition. Ask which information is genuinely known at this point and which information would leak from the future, evaluation set, opposing player, held-out client, or later pipeline stage. Write the transformation symbolically before translating it into array operations. Every reduction must state its axis; every probability must state its normalization set; every random choice must state its distribution and seed; every learned quantity must state the objective that changes it. This discipline turns an appealing formula into an executable specification that can be challenged with small counterexamples.

The reference implementation should favor clarity over cleverness. Separate validation, the mathematical core, and state updates so each can be tested independently. Use explicit intermediate names that correspond to the derivation rather than compressing the work into one expression. Confirm dtype promotion, broadcasting, device placement, and empty-input behavior. If Interaction Step depends on randomness, pass a generator instead of reading hidden global state. If it owns mutable state, return or document the updated state explicitly. The optimized implementation may later fuse operations or reuse buffers, but it must remain numerically comparable with this small version on deterministic fixtures.

Verification for Interaction Step needs more than a happy-path assertion. Prove a hand-computable normal case, a boundary case, an invalid case, and at least one invariant. Compare against seeded rollouts, learning curves, confidence intervals, and baseline matches. Add metamorphic tests when an exact answer is awkward: permutation, scaling, symmetry, conservation, monotonicity, or equivalence under a harmless representation change. Run the test repeatedly under fixed seeds to distinguish deterministic defects from statistical variation. When floating-point arithmetic is involved, justify tolerances from expected rounding error instead of choosing a loose threshold simply because the test passes.

Failure analysis asks how Interaction Step can look plausible while being wrong. Inspect reward leakage, unstable bootstrapping, invalid actions, and optimistic evaluation. Trace one example through every intermediate value and preserve enough logging to reproduce it. Distinguish a contract violation from an optimization failure and from an evaluation-design failure; each requires a different repair. A numerical answer within range is not automatically meaningful, and a rising training metric is not proof that the intended signal is being learned. The strongest debugging move is usually to shrink the input until the complete computation fits on paper, then compare the paper trace with the program line by line.

Productionizing Interaction Step changes the question from “does it work once?” to “does it remain trustworthy under load and change?” Measure environment interactions, exploration budget, replay reuse, and evaluation games. Define observability for inputs, outputs, latency, failures, drift, and resource saturation. Decide what happens on malformed data, cancellation, partial worker failure, unavailable accelerators, or a distribution outside the training envelope. Version configuration and schemas with the code, preserve reproducible seeds where appropriate, and expose a safe fallback. Optimization is accepted only when the reference tests, numerical comparisons, and task-level metrics remain within an explicitly documented budget.

• Part: Training Loop. Combine action selection, environment stepping, and updates into episodes and full training runs.
• Normal case: choose the smallest input that exercises the intended transformation.
• Boundary case: use an empty, singleton, saturated, masked, terminal, or maximum-size input as appropriate.
• Invariant: verify shape, range, conservation, normalization, symmetry, immutability, or monotonicity.
• Production evidence: record correctness, latency, memory or cost, and the exact configuration.

Run Training Episode — from contract to production evidence

Run Training Episode is the learning update at milestone 12 of Q-Learning on FrozenLake. Its purpose is not merely to make the next function run. It establishes a contract between training loop and every downstream stage. Begin by naming the accepted inputs, their axes, units, legal ranges, ownership rules, and whether mutation is permitted. Then name the output with the same precision. In this project the surrounding ideas—Bellman update, Epsilon greedy, Q-table—only compose correctly when this boundary preserves those invariants. A useful implementation note records one representative shape, one smallest valid example, one boundary example, and one invalid example before any optimization is attempted.

From first principles, treat Run Training Episode as a mapping from available information to a new state transition. Ask which information is genuinely known at this point and which information would leak from the future, evaluation set, opposing player, held-out client, or later pipeline stage. Write the transformation symbolically before translating it into array operations. Every reduction must state its axis; every probability must state its normalization set; every random choice must state its distribution and seed; every learned quantity must state the objective that changes it. This discipline turns an appealing formula into an executable specification that can be challenged with small counterexamples.

The reference implementation should favor clarity over cleverness. Separate validation, the mathematical core, and state updates so each can be tested independently. Use explicit intermediate names that correspond to the derivation rather than compressing the work into one expression. Confirm dtype promotion, broadcasting, device placement, and empty-input behavior. If Run Training Episode depends on randomness, pass a generator instead of reading hidden global state. If it owns mutable state, return or document the updated state explicitly. The optimized implementation may later fuse operations or reuse buffers, but it must remain numerically comparable with this small version on deterministic fixtures.

Verification for Run Training Episode needs more than a happy-path assertion. Prove a hand-computable normal case, a boundary case, an invalid case, and at least one invariant. Compare against seeded rollouts, learning curves, confidence intervals, and baseline matches. Add metamorphic tests when an exact answer is awkward: permutation, scaling, symmetry, conservation, monotonicity, or equivalence under a harmless representation change. Run the test repeatedly under fixed seeds to distinguish deterministic defects from statistical variation. When floating-point arithmetic is involved, justify tolerances from expected rounding error instead of choosing a loose threshold simply because the test passes.

Failure analysis asks how Run Training Episode can look plausible while being wrong. Inspect reward leakage, unstable bootstrapping, invalid actions, and optimistic evaluation. Trace one example through every intermediate value and preserve enough logging to reproduce it. Distinguish a contract violation from an optimization failure and from an evaluation-design failure; each requires a different repair. A numerical answer within range is not automatically meaningful, and a rising training metric is not proof that the intended signal is being learned. The strongest debugging move is usually to shrink the input until the complete computation fits on paper, then compare the paper trace with the program line by line.

Productionizing Run Training Episode changes the question from “does it work once?” to “does it remain trustworthy under load and change?” Measure environment interactions, exploration budget, replay reuse, and evaluation games. Define observability for inputs, outputs, latency, failures, drift, and resource saturation. Decide what happens on malformed data, cancellation, partial worker failure, unavailable accelerators, or a distribution outside the training envelope. Version configuration and schemas with the code, preserve reproducible seeds where appropriate, and expose a safe fallback. Optimization is accepted only when the reference tests, numerical comparisons, and task-level metrics remain within an explicitly documented budget.

• Part: Training Loop. Combine action selection, environment stepping, and updates into episodes and full training runs.
• Normal case: choose the smallest input that exercises the intended transformation.
• Boundary case: use an empty, singleton, saturated, masked, terminal, or maximum-size input as appropriate.
• Invariant: verify shape, range, conservation, normalization, symmetry, immutability, or monotonicity.
• Production evidence: record correctness, latency, memory or cost, and the exact configuration.

Train Q Learning — from contract to production evidence

Train Q Learning is the learning update at milestone 13 of Q-Learning on FrozenLake. Its purpose is not merely to make the next function run. It establishes a contract between training loop and every downstream stage. Begin by naming the accepted inputs, their axes, units, legal ranges, ownership rules, and whether mutation is permitted. Then name the output with the same precision. In this project the surrounding ideas—Bellman update, Epsilon greedy, Q-table—only compose correctly when this boundary preserves those invariants. A useful implementation note records one representative shape, one smallest valid example, one boundary example, and one invalid example before any optimization is attempted.

From first principles, treat Train Q Learning as a mapping from available information to a new state transition. Ask which information is genuinely known at this point and which information would leak from the future, evaluation set, opposing player, held-out client, or later pipeline stage. Write the transformation symbolically before translating it into array operations. Every reduction must state its axis; every probability must state its normalization set; every random choice must state its distribution and seed; every learned quantity must state the objective that changes it. This discipline turns an appealing formula into an executable specification that can be challenged with small counterexamples.

The reference implementation should favor clarity over cleverness. Separate validation, the mathematical core, and state updates so each can be tested independently. Use explicit intermediate names that correspond to the derivation rather than compressing the work into one expression. Confirm dtype promotion, broadcasting, device placement, and empty-input behavior. If Train Q Learning depends on randomness, pass a generator instead of reading hidden global state. If it owns mutable state, return or document the updated state explicitly. The optimized implementation may later fuse operations or reuse buffers, but it must remain numerically comparable with this small version on deterministic fixtures.

Verification for Train Q Learning needs more than a happy-path assertion. Prove a hand-computable normal case, a boundary case, an invalid case, and at least one invariant. Compare against seeded rollouts, learning curves, confidence intervals, and baseline matches. Add metamorphic tests when an exact answer is awkward: permutation, scaling, symmetry, conservation, monotonicity, or equivalence under a harmless representation change. Run the test repeatedly under fixed seeds to distinguish deterministic defects from statistical variation. When floating-point arithmetic is involved, justify tolerances from expected rounding error instead of choosing a loose threshold simply because the test passes.

Failure analysis asks how Train Q Learning can look plausible while being wrong. Inspect reward leakage, unstable bootstrapping, invalid actions, and optimistic evaluation. Trace one example through every intermediate value and preserve enough logging to reproduce it. Distinguish a contract violation from an optimization failure and from an evaluation-design failure; each requires a different repair. A numerical answer within range is not automatically meaningful, and a rising training metric is not proof that the intended signal is being learned. The strongest debugging move is usually to shrink the input until the complete computation fits on paper, then compare the paper trace with the program line by line.

Productionizing Train Q Learning changes the question from “does it work once?” to “does it remain trustworthy under load and change?” Measure environment interactions, exploration budget, replay reuse, and evaluation games. Define observability for inputs, outputs, latency, failures, drift, and resource saturation. Decide what happens on malformed data, cancellation, partial worker failure, unavailable accelerators, or a distribution outside the training envelope. Version configuration and schemas with the code, preserve reproducible seeds where appropriate, and expose a safe fallback. Optimization is accepted only when the reference tests, numerical comparisons, and task-level metrics remain within an explicitly documented budget.

• Part: Training Loop. Combine action selection, environment stepping, and updates into episodes and full training runs.
• Normal case: choose the smallest input that exercises the intended transformation.
• Boundary case: use an empty, singleton, saturated, masked, terminal, or maximum-size input as appropriate.
• Invariant: verify shape, range, conservation, normalization, symmetry, immutability, or monotonicity.
• Production evidence: record correctness, latency, memory or cost, and the exact configuration.

Extract Greedy Policy — from contract to production evidence

Extract Greedy Policy is the pipeline boundary at milestone 14 of Q-Learning on FrozenLake. Its purpose is not merely to make the next function run. It establishes a contract between policy extraction and evaluation and every downstream stage. Begin by naming the accepted inputs, their axes, units, legal ranges, ownership rules, and whether mutation is permitted. Then name the output with the same precision. In this project the surrounding ideas—Bellman update, Epsilon greedy, Q-table—only compose correctly when this boundary preserves those invariants. A useful implementation note records one representative shape, one smallest valid example, one boundary example, and one invalid example before any optimization is attempted.

From first principles, treat Extract Greedy Policy as a mapping from available information to a new state transition. Ask which information is genuinely known at this point and which information would leak from the future, evaluation set, opposing player, held-out client, or later pipeline stage. Write the transformation symbolically before translating it into array operations. Every reduction must state its axis; every probability must state its normalization set; every random choice must state its distribution and seed; every learned quantity must state the objective that changes it. This discipline turns an appealing formula into an executable specification that can be challenged with small counterexamples.

The reference implementation should favor clarity over cleverness. Separate validation, the mathematical core, and state updates so each can be tested independently. Use explicit intermediate names that correspond to the derivation rather than compressing the work into one expression. Confirm dtype promotion, broadcasting, device placement, and empty-input behavior. If Extract Greedy Policy depends on randomness, pass a generator instead of reading hidden global state. If it owns mutable state, return or document the updated state explicitly. The optimized implementation may later fuse operations or reuse buffers, but it must remain numerically comparable with this small version on deterministic fixtures.

Verification for Extract Greedy Policy needs more than a happy-path assertion. Prove a hand-computable normal case, a boundary case, an invalid case, and at least one invariant. Compare against seeded rollouts, learning curves, confidence intervals, and baseline matches. Add metamorphic tests when an exact answer is awkward: permutation, scaling, symmetry, conservation, monotonicity, or equivalence under a harmless representation change. Run the test repeatedly under fixed seeds to distinguish deterministic defects from statistical variation. When floating-point arithmetic is involved, justify tolerances from expected rounding error instead of choosing a loose threshold simply because the test passes.

Failure analysis asks how Extract Greedy Policy can look plausible while being wrong. Inspect reward leakage, unstable bootstrapping, invalid actions, and optimistic evaluation. Trace one example through every intermediate value and preserve enough logging to reproduce it. Distinguish a contract violation from an optimization failure and from an evaluation-design failure; each requires a different repair. A numerical answer within range is not automatically meaningful, and a rising training metric is not proof that the intended signal is being learned. The strongest debugging move is usually to shrink the input until the complete computation fits on paper, then compare the paper trace with the program line by line.

Productionizing Extract Greedy Policy changes the question from “does it work once?” to “does it remain trustworthy under load and change?” Measure environment interactions, exploration budget, replay reuse, and evaluation games. Define observability for inputs, outputs, latency, failures, drift, and resource saturation. Decide what happens on malformed data, cancellation, partial worker failure, unavailable accelerators, or a distribution outside the training envelope. Version configuration and schemas with the code, preserve reproducible seeds where appropriate, and expose a safe fallback. Optimization is accepted only when the reference tests, numerical comparisons, and task-level metrics remain within an explicitly documented budget.

• Part: Policy Extraction and Evaluation. Derive a greedy policy from the trained Q-table and measure its success rate on FrozenLake.
• Normal case: choose the smallest input that exercises the intended transformation.
• Boundary case: use an empty, singleton, saturated, masked, terminal, or maximum-size input as appropriate.
• Invariant: verify shape, range, conservation, normalization, symmetry, immutability, or monotonicity.
• Production evidence: record correctness, latency, memory or cost, and the exact configuration.

Run Greedy Episode — from contract to production evidence

Run Greedy Episode is the pipeline boundary at milestone 15 of Q-Learning on FrozenLake. Its purpose is not merely to make the next function run. It establishes a contract between policy extraction and evaluation and every downstream stage. Begin by naming the accepted inputs, their axes, units, legal ranges, ownership rules, and whether mutation is permitted. Then name the output with the same precision. In this project the surrounding ideas—Bellman update, Epsilon greedy, Q-table—only compose correctly when this boundary preserves those invariants. A useful implementation note records one representative shape, one smallest valid example, one boundary example, and one invalid example before any optimization is attempted.

From first principles, treat Run Greedy Episode as a mapping from available information to a new state transition. Ask which information is genuinely known at this point and which information would leak from the future, evaluation set, opposing player, held-out client, or later pipeline stage. Write the transformation symbolically before translating it into array operations. Every reduction must state its axis; every probability must state its normalization set; every random choice must state its distribution and seed; every learned quantity must state the objective that changes it. This discipline turns an appealing formula into an executable specification that can be challenged with small counterexamples.

The reference implementation should favor clarity over cleverness. Separate validation, the mathematical core, and state updates so each can be tested independently. Use explicit intermediate names that correspond to the derivation rather than compressing the work into one expression. Confirm dtype promotion, broadcasting, device placement, and empty-input behavior. If Run Greedy Episode depends on randomness, pass a generator instead of reading hidden global state. If it owns mutable state, return or document the updated state explicitly. The optimized implementation may later fuse operations or reuse buffers, but it must remain numerically comparable with this small version on deterministic fixtures.

Verification for Run Greedy Episode needs more than a happy-path assertion. Prove a hand-computable normal case, a boundary case, an invalid case, and at least one invariant. Compare against seeded rollouts, learning curves, confidence intervals, and baseline matches. Add metamorphic tests when an exact answer is awkward: permutation, scaling, symmetry, conservation, monotonicity, or equivalence under a harmless representation change. Run the test repeatedly under fixed seeds to distinguish deterministic defects from statistical variation. When floating-point arithmetic is involved, justify tolerances from expected rounding error instead of choosing a loose threshold simply because the test passes.

Failure analysis asks how Run Greedy Episode can look plausible while being wrong. Inspect reward leakage, unstable bootstrapping, invalid actions, and optimistic evaluation. Trace one example through every intermediate value and preserve enough logging to reproduce it. Distinguish a contract violation from an optimization failure and from an evaluation-design failure; each requires a different repair. A numerical answer within range is not automatically meaningful, and a rising training metric is not proof that the intended signal is being learned. The strongest debugging move is usually to shrink the input until the complete computation fits on paper, then compare the paper trace with the program line by line.

Productionizing Run Greedy Episode changes the question from “does it work once?” to “does it remain trustworthy under load and change?” Measure environment interactions, exploration budget, replay reuse, and evaluation games. Define observability for inputs, outputs, latency, failures, drift, and resource saturation. Decide what happens on malformed data, cancellation, partial worker failure, unavailable accelerators, or a distribution outside the training envelope. Version configuration and schemas with the code, preserve reproducible seeds where appropriate, and expose a safe fallback. Optimization is accepted only when the reference tests, numerical comparisons, and task-level metrics remain within an explicitly documented budget.

• Part: Policy Extraction and Evaluation. Derive a greedy policy from the trained Q-table and measure its success rate on FrozenLake.
• Normal case: choose the smallest input that exercises the intended transformation.
• Boundary case: use an empty, singleton, saturated, masked, terminal, or maximum-size input as appropriate.
• Invariant: verify shape, range, conservation, normalization, symmetry, immutability, or monotonicity.
• Production evidence: record correctness, latency, memory or cost, and the exact configuration.

Evaluate Success Rate — from contract to production evidence

Evaluate Success Rate is the measurement at milestone 16 of Q-Learning on FrozenLake. Its purpose is not merely to make the next function run. It establishes a contract between policy extraction and evaluation and every downstream stage. Begin by naming the accepted inputs, their axes, units, legal ranges, ownership rules, and whether mutation is permitted. Then name the output with the same precision. In this project the surrounding ideas—Bellman update, Epsilon greedy, Q-table—only compose correctly when this boundary preserves those invariants. A useful implementation note records one representative shape, one smallest valid example, one boundary example, and one invalid example before any optimization is attempted.

From first principles, treat Evaluate Success Rate as a mapping from available information to a new state transition. Ask which information is genuinely known at this point and which information would leak from the future, evaluation set, opposing player, held-out client, or later pipeline stage. Write the transformation symbolically before translating it into array operations. Every reduction must state its axis; every probability must state its normalization set; every random choice must state its distribution and seed; every learned quantity must state the objective that changes it. This discipline turns an appealing formula into an executable specification that can be challenged with small counterexamples.

The reference implementation should favor clarity over cleverness. Separate validation, the mathematical core, and state updates so each can be tested independently. Use explicit intermediate names that correspond to the derivation rather than compressing the work into one expression. Confirm dtype promotion, broadcasting, device placement, and empty-input behavior. If Evaluate Success Rate depends on randomness, pass a generator instead of reading hidden global state. If it owns mutable state, return or document the updated state explicitly. The optimized implementation may later fuse operations or reuse buffers, but it must remain numerically comparable with this small version on deterministic fixtures.

Verification for Evaluate Success Rate needs more than a happy-path assertion. Prove a hand-computable normal case, a boundary case, an invalid case, and at least one invariant. Compare against seeded rollouts, learning curves, confidence intervals, and baseline matches. Add metamorphic tests when an exact answer is awkward: permutation, scaling, symmetry, conservation, monotonicity, or equivalence under a harmless representation change. Run the test repeatedly under fixed seeds to distinguish deterministic defects from statistical variation. When floating-point arithmetic is involved, justify tolerances from expected rounding error instead of choosing a loose threshold simply because the test passes.

Failure analysis asks how Evaluate Success Rate can look plausible while being wrong. Inspect reward leakage, unstable bootstrapping, invalid actions, and optimistic evaluation. Trace one example through every intermediate value and preserve enough logging to reproduce it. Distinguish a contract violation from an optimization failure and from an evaluation-design failure; each requires a different repair. A numerical answer within range is not automatically meaningful, and a rising training metric is not proof that the intended signal is being learned. The strongest debugging move is usually to shrink the input until the complete computation fits on paper, then compare the paper trace with the program line by line.

Productionizing Evaluate Success Rate changes the question from “does it work once?” to “does it remain trustworthy under load and change?” Measure environment interactions, exploration budget, replay reuse, and evaluation games. Define observability for inputs, outputs, latency, failures, drift, and resource saturation. Decide what happens on malformed data, cancellation, partial worker failure, unavailable accelerators, or a distribution outside the training envelope. Version configuration and schemas with the code, preserve reproducible seeds where appropriate, and expose a safe fallback. Optimization is accepted only when the reference tests, numerical comparisons, and task-level metrics remain within an explicitly documented budget.

• Part: Policy Extraction and Evaluation. Derive a greedy policy from the trained Q-table and measure its success rate on FrozenLake.
• Normal case: choose the smallest input that exercises the intended transformation.
• Boundary case: use an empty, singleton, saturated, masked, terminal, or maximum-size input as appropriate.
• Invariant: verify shape, range, conservation, normalization, symmetry, immutability, or monotonicity.
• Production evidence: record correctness, latency, memory or cost, and the exact configuration.