ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Deterministic zero-reward arms have zero regret

Example

Fix any horizon $n \geq 1$ and take two arms with deterministic reward stacks $X_{s, 1} = X_{s, 2} = 0$ for every $s \in [n]$ . Then $μ_{1} = μ_{2} = μ^{*} = 0$ and $Δ_{1} = Δ_{2} = 0$ . Every admissible policy has identically zero pseudo-regret and realized reward regret, so both expected regrets are zero.

Example references

Finite stochastic bandit model, gaps, and regret: the finite-arm means, gaps, pseudo-regret, and realized reward regret used in the example.

Facts & Assumptions

Given: The two-arm deterministic reward table in the example and the definitions of gaps and the two regret quantities.

[F1]

The arm means, gaps, pseudo-regret, and realized reward regret are those in the finite stochastic bandit model (Finite stochastic bandit model, gaps, and regret).

Verification

1.1givenF1algebra

Each arm's law is the point mass at $0$ , so both finite means equal $0$ . Hence $μ^{*} = 0$ and $Δ_{1} = Δ_{2} = 0$ .

2.1givenF1step 1.1algebra

For every policy and every round, $μ^{*} - μ_{A_{t}} = 0$ and the observed reward $Y_{t} = 0$ . The defining sums therefore give ${\overset{ˉ}{R}}_{n} = 0$ and $R_{n}^{r e w} = 0$ pointwise.

3.1givenstep 2.1algebra∎

Taking expectations of the two identically zero random variables shows that expected pseudo-regret and expected reward regret both equal zero.

Depends on

Finite stochastic bandit model, gaps, and regret

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.