Alphabeta MathHow 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.
Simulation objectives and unbiased payoff oracles
Definition
Fix a finite dimension , an open set , and a probability space . A simulation payoff is a jointly measurable map such that is integrable for each . Its objective is A payoff-oracle call at returns an integrable real random variable with . Two calls may use the same random input or different random inputs; their joint law is part of the chosen coupling. A single oracle call is one function evaluation, regardless of the cost of its internal simulation.
Depends on
Used by
- Two-evaluation simultaneous-perturbation gradient estimator Algorithm
- Coupled finite-difference simulation-gradient estimator Definition
- Pathwise and reparameterized simulation-gradient estimators Definition
- A moving uniform support contributes a missing score boundary term Example
- A dominated pathwise simulation gradient is unbiased Theorem
- A finite stochastic adjoint gives the discrete gradient under domination Theorem
- Summably biased SPSA has convergent objective values and zero gradient liminf Theorem
- The likelihood-ratio score gradient is unbiased on fixed support Theorem
Dependency tree · two levels
7 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.
Sources
- Michael C. Fu, Stochastic Gradient Estimation, preprint of Handbook of Simulation Optimization, §3 (standard reference, not scraped)
- Paul Glasserman, Monte Carlo Methods in Financial Engineering, §7 (standard reference, not scraped)