Alphabeta MathPreference learning and direct preference optimization
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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
- 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
Information Theory
Linear Algebra
Mathematical Statistics
Measure Theory
Numerical Optimization Inverse Problems and Uncertainty Quantification
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
- Convex and Semicontinuous Functions on Rⁿ
- Convexity
- Countability and Uncountability
- Darboux, L'Hôpital, and Taylor's Theorem
- Foundations of the Real Numbers for Analysis
- 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
- 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: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
Statistical Learning and Classical Models
2 · Logical flowchart
3 · Items
Definition
Finite KL-regularized policy improvement
Definition
Finite Bradley–Terry preference model
Theorem
The finite-action Gibbs policy uniquely maximizes reward minus KL
Theorem
Bradley–Terry reward-policy reparameterization and finite DPO gradient
Remark
Preference coverage and Bradley–Terry model limits
Algorithm
Finite-budget direct preference optimization by mini-batch gradient descent
Lemma
Finite-budget DPO preserves its valid pair and terminates with finite oracle work
Sources
- Rafael Rafailov et al., Direct Preference Optimization: Your Language Model is Secretly a Reward Model, arXiv:2305.18290v3 (2024)
- Purdue course lecture notes, Preference Optimization (PrefOpt)
- Rafailov et al., Direct Preference Optimization, §4 and Appendix B (procedure background; the finite-budget and oracle-work properties are proved locally here)