Alphabeta MathStochastic finite-armed bandits
About this page
Pipeline-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.
1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Combinatorics
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Machine Learning Algorithms
Measure Theory
Probability
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Equivalent Forms of Completeness
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Further Trigonometric Identities and Inverse Functions
- Improper and Parameter-Dependent Multiple Integrals
- Improper Integrals
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Inverse and Implicit Function Theorems
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Centered 1-sub-Gaussian reward
Lemma
The zeta sums at 2 and 3 are finite
Lemma
Expectation of a countably valued nonnegative random variable is its weighted series
Lemma
A deterministic-prefix envelope sum for the sharper UCB count bound
Lemma
Tail bounds for deterministic prefixes of independent sub-Gaussian rewards
Definition
Finite stochastic bandit model, gaps, and regret
Algorithm
Explore-then-commit for a finite stochastic bandit
Algorithm
Finite-horizon upper-confidence-bound bandit algorithm
Lemma
The independent reward-stack model has the prescribed conditional arm laws
Lemma
Expected pull count for a suboptimal UCB arm
Theorem
Expected regret of explore-then-commit
Lemma
A sharper expected suboptimal-arm pull count for the unchanged UCB rule
Theorem
Gap-dependent expected regret of UCB
Theorem
Gap-free expected regret of UCB
Sources
- Tor Lattimore and Csaba Szepesvári, Bandit Algorithms
- Sébastien Bubeck and Nicolò Cesa-Bianchi, Regret Analysis of Stochastic and Nonstochastic Multi-armed Bandit Problems
- Stephen Semmes, Elements of Analysis
- John K. Hunter, Measure Theory Notes
- Lattimore and Szepesvári, Bandit Algorithms, Theorems 7.1–7.2; independent local sharpening of the expected-pull argument