roam/thesismethod.org

thesis/method thesis/method

• Guidelines
• Section Structure
• Gotchas

Chapter 3 of the BGU MSc thesis on ROLL. Core contribution chapter. Currently ~30 pages. Parent: thesis

Guidelines

• sec:problem-formulation is now notation-only — motivation deferred to Ch. 2 via \Cref{sec:imbalanced-tpr-fpr,sec:neyman-pearson}
• sec:imbalanced-tpr-fpr and sec:neyman-pearson were removed from this chapter and live in Ch. 2
• Both objectives (TPR@FPR and FPR@TPR) are equivalent by negating scores and swapping labels; all derivations given for TPR@FPR only — FPR@TPR follows by transformation

Section Structure

Section	Label	Status
Problem Formulation (notation only)	sec:problem-formulation	Done
The ROLL Framework	sec:roll-framework	Written
Differentiability Problem	sec:differentiability-problem	Written
Score Distribution Fitting	sec:score-distribution-fitting	Written
General ROLL Formulation and Derivation	sec:roll-formulation	Written — core theoretical contribution
Gaussian ROLL (forward + gradient)	sec:roll-gaussian	Written
Beta ROLL (forward + gradient)	sec:roll-beta	Stub — TBD
KDE ROLL (forward + gradient)	sec:roll-kde	Written
Properties	sec:roll-properties	Written — gradient balance + locality
Custom Backward Pass	sec:roll-backward	Stub — TBD
Numerical Stability	sec:roll-numerical-stability	Stub — TBD
Bandwidth Estimation (ISJ + scheduling)	sec:kde-bandwidth	Written

Gotchas

• KDE backward pass uses inverse-function-theorem approach deliberately; implicit differentiation was considered and rejected — user needs to explain the derivation in their own words
• Correct gradient of threshold w.r.t. negative-class score: σ'(τ − x_i) / Σ_j σ'(τ − x_j) where sum is over ALL of B_0 — sympy-verified
• Beta ROLL: translation-invariance does NOT hold when scores pass through sigmoid before fitting; gradient-balance property breaks for Beta — documented in sec:roll-properties
• Gradient-balance proof uses a uniform-shift argument; requires translation-invariant CDF estimator