roam/thesismethod.org

thesis/method

• How roll-aoc Works (CRITICAL — post-pivot hero method)
• Guidelines
• Section Structure
• Appendix Structure (after 2026-08-29 restructure, +2 appendices added 2026-09-09)
• Mentor Audit Progress (2026-09-07 audit, thesis_mathematical_audit.pdf)
• Third Mentor Review (2026-09-10, screenshots) — AOC divisor mismatch
• Second Mentor Review (2026-09-09) — 5 points, status
• Neyman-Pearson Theory: now Appendix L, pointed to from Discussion 5.1.1
  • sec:np-class-weighting — class weighting as an implicit choice of eta
  • sec:np-noise-invariance — class weighting under label noise: an implicit, distorted eta
  • Build status
• Gotchas

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

How roll-aoc Works (CRITICAL — post-pivot hero method)

roll-aoc optimizes full AUC by:

1. Sampling 100 evenly-spaced FPR points across [0, 1]
2. Computing ROLL's TPR estimate at each point
3. Averaging — this is a Monte Carlo approximation of the AUC integral

This is not partial AUC (which integrates over a sub-range). This is not the pairwise ranking formulation (LibAUC/compositional AUC). It is a direct numerical integration of the ROC curve via 100 ROLL evaluations.

This distinction must be stated clearly in Ch. 3 and Ch. 2 related work.

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
• EXCEPTION to the above (2026-09-10): sec:np-class-weighting and sec:np-noise-invariance are NP-theoretic content that DOES live in this chapter now, despite NP itself being background — see the dedicated section below for why and where.

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
Distribution Instantiations (intro only)	sec:roll-instantiations	Rewritten — KDE only, parametric → appendix
KDE ROLL (forward + gradient)	sec:roll-kde	Written
Properties	sec:roll-properties	Written — gradient balance + locality + sign correctness
Custom Backward Pass	sec:roll-backward	Stub — TBD
Numerical Stability	sec:roll-numerical-stability	Written — normalization invariance proposition + frozen-c convention (2026-09-09)
Bandwidth Estimation (ISJ + scheduling)	sec:kde-bandwidth	Written

sec:np-class-weighting and sec:np-noise-invariance are NOT in this chapter as of the final 2026-09-10 move – they live in Appendix L now (appendix:np-theory), pointed to from Discussion 5.1.1. See the dedicated section below for the full story; don't re-add them here if seen in stale context.

Gaussian and Beta subsections removed from Ch. 3 body; full math lives in appendix:

• Gaussian (forward + analytic gradient): appendix:gauss-grad (appendix_gauss_grad.tex, expanded)
• Beta (forward + autograd note): appendix:beta-roll (appendix_beta_roll.tex, new)

Appendix Structure (after 2026-08-29 restructure, +2 appendices added 2026-09-09)

Label	Chapter title	File	Hebrew letter
appendix:gauss-grad	Gaussian ROLL: Forward Pass and Gradient Derivation	appendix_gauss_grad.tex	נספח ב
appendix:beta-roll	Beta ROLL: Forward Pass and Gradient	appendix_beta_roll.tex	נספח ג
appendix:kde-grad	KDE ROLL: Gradient Derivation	appendix_kde_grad.tex	נספח ד (now also has sec:kde-grad-stability, 2026-09-09)
appendix:kde-sigmoid	KDE ROLL: Sigmoid Kernel — Explicit Equations	appendix_kde_sigmoid.tex	נספח ה
appendix:kde-bisection	KDE ROLL: Bisection for CDF Inversion	appendix_kde_bisection.tex	נספח ו
appendix:kde-nr	KDE ROLL: Newton–Raphson Inversion (Superseded)	appendix_kde_nr.tex	נספח ז
appendix:roll-code	ROLL Loss: Implementation Listing	appendix_roll_code.tex	נספח ח
appendix:keel-noise	KEEL Benchmark: Noise Tables at 5% and 10%	appendix_keel_noise.tex	נספח ח (verify)
appendix:grad-balance	Proof of the Gradient Balance Proposition	appendix_grad_balance.tex	נספח ט
appendix:bandwidth-freeze	KDE ROLL: Adaptive-Bandwidth Gradients and the Freezing Convention	appendix_bandwidth_freeze.tex	נספח י (appended at end, 2026-09-09)
appendix:score-normalization	Score Normalisation: Invariance, ISJ, and the Frozen Chain Rule	appendix_score_normalization.tex	נספח יא (appended at end, 2026-09-09)
appendix:np-theory	Theoretical Justification for AUC-Optimizing Losses via the Neyman-Pearson Lemma	appendix_np_theory.tex	נספח יב (appended at end, 2026-09-10; NOT ROLL-derived math, holds sec:np-class-weighting/sec:np-noise-invariance, pointed to from Discussion 5.1.1)

Note: not every audit fix gets its own appendix — small, self-contained formula-level fixes (like the sigmoid-derivative stability fix) go directly into the existing relevant appendix as a new paragraph/section; only fixes needing real counterexamples/propositions (bandwidth freezing, score normalization) got whole new appendix chapters. Keep it that way — the user pushed back once already on the thesis getting too long.

Mentor Audit Progress (2026-09-07 audit, thesis_mathematical_audit.pdf)

Full narrative for each item (derivations, counterexamples, code changes, test results) lives in thesis/todos under "Mentor mathematical audit fixes" — not duplicated here. This audit list is now FULLY CLOSED (all items DONE as of 2026-09-09).

DONE (2026-09-09), in priority order: bandwidth backprop convention (frozen v_0,v_1 + appendix:bandwidth-freeze) → bisection stopping criterion (certified-loss bound, appendix:kde-bisection + roll.py fix) → bisection bracket range (exact target-dependent bracket, same files) → score-normalization contradiction (normalization-invariance proposition + frozen-c convention, appendix:score-normalization) → cached-sigmoid derivative underflow (log-space/softmax fix, sec:kde-grad-stability in appendix_kde_grad.tex

• roll.py `_log_sigma_prime`) → indicator-derivative-at-jump wording (undefined, not

infinite, + finite-sample-vs-population-rate distinction) → score/label reversal scope (footnote: negation trick needs reflection-equivariance; Beta uses s→1−s instead, verified exact via the â↔b̂ swap identity) → softened "structural property" overclaim in Ch. 5's noise-robustness discussion.

CORRECTED framing of the common thread (2026-09-09 — user caught my oversimplification, do NOT put the old version in the thesis): it is NOT "freeze anything numerically estimated" — tau is also numerically estimated (via bisection) and we deliberately DO differentiate through it, since that's the entire mechanism by which negative-class scores get any gradient at all. The real distinction: differentiate through quantities that are part of the TRUE objective's own definition (tau — TPR@FPR is defined via "the threshold achieving this FPR," so tau's score-dependence is structural, not an artifact); freeze quantities that exist ONLY because of the smoothing/approximation scheme used to make the true (indicator-based) objective differentiable in the first place, and have no counterpart in that true objective at all (v_0, v_1 bandwidth; c normalization scale). User's framing: bandwidth is "an estimation of our approximation of a theoretical distribution… could be wrong and flawed, and produces artifacts" when differentiated through — that's the right way to say it. This principle is NOT yet written into the thesis (user wants it kept out of the discussion chapter for now, maybe later) — this memory entry is for continuity across sessions only.

Third Mentor Review (2026-09-10, screenshots) — AOC divisor mismatch

Separate from the "second mentor review" 5-point list below (different reviewer/round, delivered as screenshots so no direct file/page cite – mentor referred to it loosely as "equation 3.13" but the actual affected equation turned out to be eq:roll-aoc, now (3.15)).

Mentor's claim: thesis defines L_ROLL-AOC with normalization 1/99 (eq:roll-aoc, method.tex), but the implementation's _AOC_DIVISOR is 100. Mentor's argument: grid spacing is 0.01 (i.e. 1/100), so a Riemann-sum approximation should divide by 100, assuming the omitted alpha=1 endpoint is zero.

VERIFIED: real mismatch, confirmed directly in impl/src/roll.py (sibling repo, NOT under thesis/ – lives at /home/aner/projects/research-projects/roll/impl/src/roll.py): `_AOC_DIVISOR = 100` (line 114), used in BOTH the forward pass (`total_loss = losses.sum() / divisor`, line 296) and the backward-pass gradients (lines 320, 333) – code's own comment even states "Sum divided by 100 approximates integral [sec:roll-aoc]", i.e. the code was ALWAYS intended to match the thesis and the thesis just had the wrong constant written down. Since the code is what actually produced every reported result, the FIX belongs in the thesis text, not the code.

IMPORTANT NUANCE (checked numerically before agreeing with the mentor's full framing): g(1) = 1-TPR(FPR^-1(1)) = 1-TPR(-infinity) = 0 EXACTLY (verified: thresholding at -infinity classifies everything positive) – so (1/100)*sum_i=1⁹⁹ IS a complete, valid right-Riemann sum over 100 equal subintervals of [0,1] (the "missing" 100th term is known to be exactly zero, not actually omitted). This gives a clean, PROVABLE error bound of exactly 1/100=0.01 (classical monotone-function Riemann-sum bound: error <= range*step = (g(0)-g(1))*(1/100) <= 1*(1/100)), tighter and cleaner than the old "1/99≈0.0101" claim. BUT: numerically stress-tested both /99 and /100 against the true integral on 200 random Gaussian-type AOC curves, and /99 had SMALLER actual error than /100 in ALL 200 trials (though both stayed well under the 0.01 worst-case bound, ~0.005 typical). So it is NOT mathematically true that "100 is more accurate than 99" for realistic AOC integrands – it's simply that 100 is what the code computes and is a cleanly-justifiable, provably- bounded choice, not a superior one. Told the user this nuance directly before fixing, rather than just conceding "99 was a bug, 100 is better" at face value.

FIX applied (method.tex, sec:roll-aoc), all three affected spots:

1. eq:roll-aoc (now eq 3.15): 1/99 -> 1/100.

2. Darboux-bound paragraph right after it: rewritten around the g(1)=0-exact fact and the right-Riemann-sum framing, replacing the old approximate "1/99≈0.0101 by a standard Darboux-sum argument" with the new EXACT derivation and clean 0.01 bound (shown as

   L_ROLL-AOC - L_AOC

   <= (g(0)-g(1))*(1/100) <= 1/100 = 0.01).

3. eq:roll-aoc-gradient (now eq 3.16): 1/99 -> 1/100 (gradient of a linear combination scales the same way as the combination itself).

Checked for other stale "1/99" or "99-point average" mentions elsewhere in the thesis (appendix_gauss_grad.tex, discussion_and_conclusions.tex, method.tex intro) – all other occurrences just say "99-point" or "99 evenly-spaced" descriptively (correctly describing the NUMBER of evaluated points, unaffected by the divisor value), none needed changes. Rebuilt thesis clean (nix build exit 0), confirmed eq 3.15/3.16 both show 1/100 and the new bound derivation renders correctly in the compiled PDF.

Second Mentor Review (2026-09-09) — 5 points, status

New round of feedback, distinct from the first audit. ALL 5 POINTS NOW RESOLVED as of 2026-09-10 (episode-count justification, the last open one, closed same day as this note). Full detail in thesis/todos; summary here:

1. Justify varying episode counts — RESOLVED 2026-09-10. User's corrections/actions: CIFAR-10N was actually already n=5 (a stale "n=3" caption, not a real 3-episode run) — just a caption fix, no data change. ChestXray14 and Food-101N WERE genuinely at n=3 — user ran 2 more real episodes each and updated the mean/std numbers in tab:results:large accordingly (git diff showed real numeric changes to those two rows only, e.g. ChestXray14 ROLL-AOC 0.839->0.838, Food-101N 0.867->0.867 with updated std devs) before asking me to verify entry + rewrite the counts. I caught one formatting nit while verifying: Food-101N/Focal cell had "$\pm$0.01" (2 decimals) vs 3 decimals everywhere else in that table – fixed to "$\pm$0.010". Final state: n=5 EVERYWHERE except KEEL noise experiments (5%/10%/20%), which stay at n=10, with justification added inline in sec:experiments:configurations (experiments.tex ~line 76): "KEEL's datasets are small enough that each episode trains quickly despite the number of datasets involved, making it practical to run more episodes there for a more reliable noise-robustness comparison" – user's own justification, given verbatim in the request ("keel dataset benchmarks ran very fast, despite their number, so they ran for more episodes"). Also fixed the two stale table captions (tab:results:cifar10n "n=3"->"n=5"; tab:results:large dropped the "Food-101N and ChestXray14: n=3" exception entirely, now just states "n=5 episodes" for the whole table). Rebuilt thesis clean (nix build exit 0), confirmed every episode-count mention in the compiled PDF is now n=5 except the three KEEL-noise tables (4.3, H.1, H.2), which correctly stay n=10.

2. AUC integral definition: same variable (alpha) on page 5 (Ch.2) vs Ch.3? — RESOLVED 2026-09-10. Was a real inconsistency: TPR is defined everywhere else (eq 3.1, eq 3.6) as a function of a THRESHOLD tau; FPR^-1(alpha) is the separate function mapping an FPR level to a threshold; Ch.2's page-5 definition and eq:objective-auc (eq 3.5) correctly write the composition TPR(FPR^-1(alpha)), but sec:roll-aoc (method.tex, starting ~line

   1. silently dropped the FPR^-1 wrapper and wrote bare "TPR(alpha)" six separate times

   (lines 300, 303, 306, 324, 325, and the fig:roll-aoc-roc caption) – overloading TPR to take an FPR-level directly, contradicting its own prior definition. Not a computational bug (the AUC value computed is identical either way) but a real notation-consistency flaw – exactly the kind of thing a careful reader/mentor would catch. Fix: global replace_all of the literal string "\operatorname{TPR}(α)" -> "\operatorname{TPR}(\operatorname{FPR}^-1(α))" across method.tex (safe as a blanket replace since grep confirmed all 6 occurrences in the whole file needed the same fix, and no correctly-already-composed instance used that exact bare substring). Note: $\mathcal{L}_{\text{ROLL-TPR@FPR}}(\alpha)$ notation right next to these was NOT part of the problem – that loss was already, by its own original definition (eq:roll-tpr-at-fpr), a function of alpha directly, so no inconsistency there; only the TPR symbol specifically was being overloaded. Rebuilt thesis clean (nix build exit 0), confirmed all instances now render as "TPR(FPR⁻¹(α))" consistently throughout the PDF (eq 3.5, eq 3.13, sec:roll-aoc prose, and the figure caption all match).

3. Theoretical justification for noise robustness — RESOLVED 2026-09-09, see below.
4. Exact gradient or stop-gradient approximation for the backward pass? — ALREADY FIXED, pre-existing text. method.tex:470-476 explicitly: "We adopt this as a deliberate stop-gradient convention"; appendix:bandwidth-freeze derives the fully-adaptive (exact) gradient alongside specifically to show what's traded away. No ambiguity.
5. "Chapters A and B" should say "Appendix" — WAS NOT FIXED, now IS. Root cause: this wasn't literal hardcoded text (source already used \Cref{appendix:…}) — it was a cleveref config bug. Appendices are typeset via \chapter, so cleveref classified every appendix label under the generic "chapter" counter and printed "Chapter X" everywhere (~10+ instances across the whole PDF: "Chapter A", "Chapters A and B", "Chapter I", "Chapter H", etc.), even though main.tex:170-171 already correctly defined \Crefname{appendix}{Appendix}{Appendices} — that declaration was just never triggered. Fix: added \crefalias{chapter}{appendix} right after \appendix in main.tex. Rebuilt and confirmed every appendix ref now says "Appendix", the 5 real chapters still say "Chapter N" (crefalias only affects labels defined after it in source order, and no appendix file refers back to a main chapter{:…} label, so this was safe).

Neyman-Pearson Theory: now Appendix L, pointed to from Discussion 5.1.1

Answers second-mentor point 3 ("is there a theoretical justification for noise robustness?"). Has moved TWICE now (2026-09-10, same day, both moves user-initiated): Ch.2 (related_work.tex, original) -> Ch.3 (method.tex, inside sec:roll-formulation) -> final location: Appendix L (content/appendixs/appendix_np_theory.tex, new chapter added 2026-09-10, label appendix:np-theory, \crefalias makes it render as "Appendix L" not "Chapter L" per the cleveref fix). Do not be surprised by the two intermediate locations if they show up in old context/diffs – Appendix L + Discussion 5.1.1 is the FINAL, current state; Ch.2 and Ch.3 no longer contain this material at all.

User's reasoning for the final move: read the Ch.3-embedded version and felt it was still in the wrong place – genuinely new theoretical content shouldn't be buried inline in the Method chapter either. Final structure: (1) Appendix L holds the full two-section analysis verbatim (sec:np-class-weighting = L.1, sec:np-noise-invariance = L.2, both §ion now, not ⊂section, matching appendix-chapter convention where each appendix's top-level divisions are §ion); (2) Discussion sec:discussion:np-justification (new subsection, FIRST subsection of Section 5.1 "Discussion", before "Noise Robustness of ROLL-AOC") is a short pointer paragraph, wording lifted close to verbatim from what the user dictated, ending "The full analysis is given in \Cref{appendix:np-theory}"; (3) Ch.3 (method.tex) has ZERO remaining trace – fully reverted to how it read before any of this was added; (4) Ch.2 sec:neyman-pearson also has zero trace, ends at "The practical gap" as it always did.

Generalization the user explicitly asked to be stated: this is NOT ROLL-specific. It justifies any order-dependent/ranking-based loss – primarily ROLL-AOC, but also LibAUC and similar AUC-optimizing methods – since none of them ever form a class-weight ratio or compare to a fixed eta in the first place. Added this framing to: the appendix's own lead-in paragraph, the end of L.1 (explicit LibAUC mention), and the Discussion pointer paragraph.

Appendix L.1's "ROLL avoids this" sentence now says "The TPR@FPR objective of \Cref{sec:roll-formulation}, \eqref{eq:roll-tpr-at-fpr}" (explicit chapter-crossing reference, since the appendix is no longer inside the section that defines it) – still resolves correctly cross-chapter via \eqref/\Cref, no different from how other appendices already reference Ch.3 equations (e.g. appendix:kde-bisection referencing method.tex equations).

Follow-up (2026-09-10, same day): user asked to add a per-sample "outcome" framing to L.1, calling it "the main driver" showing the practical difference between sample-wise and order-dependent losses. Not a new derivation – it's the same q*/eta result from the existing footnote, unpacked concretely: since q*(x)>1/2 iff L(x)>eta, for ANY given x the classifier ends up calling it positive or negative purely by which side of eta its own L(x) falls on, independent of how separable x actually is. Two samples with nearly identical L(x) straddling eta can land on opposite sides purely because of where eta sits – and since eta is fixed entirely by the chosen weight ratio, the weight choice (not the data) decides borderline samples' fate. Added as a new paragraph in appendix_np_theory.tex sec:np-class-weighting (L.1), right after the eta-derivation paragraph, before "The standard heuristic…" paragraph. Closing sentence makes the contrast explicit: "This is the central difference from an order-dependent loss: a per-sample loss must commit to some eta before training even begins, and every sample's outcome is filtered through that fixed, weight-determined cutoff." Rebuilt thesis clean (nix build exit 0), confirmed the new paragraph renders correctly in Appendix L.1.

Follow-up 2 (2026-09-10, same day): user asked why eta = (w0/w1)*(pi0/pi1) specifically had the pi0/pi1 factor – explained it's NOT part of NP itself (NP just says "some eta exists, thresholds the likelihood ratio"); the pi0/pi1 comes from Bayes' rule converting between POSTERIOR odds (what a weighted loss naturally compares: w1*P(y=1|x) vs w0*P(y=0|x)) and LIKELIHOOD ratio (NP's native currency) – posterior odds = likelihood ratio * prior odds, so translating the weighted-loss condition into likelihood-ratio form introduces pi1/pi0.

Follow-up 3 (2026-09-10, same day): user then asked to SIMPLIFY the whole section to avoid needing eta's exact formula at all – "it exists, proportional to the class-weight ratio, same for all samples; changing weights changes target eta; noise changes interaction with eta, regardless of how eta is defined; don't need the eta=1 part, keep it simple." Realized the cleanest fix wasn't to hedge/soften the existing likelihood-ratio version, but to REWRITE L.1 entirely in POSTERIOR-ODDS terms (matching L.2, which already worked this way) instead of likelihood-ratio terms – this eliminates the Bayes'-rule/prior-odds conversion entirely, since a weighted loss compares posteriors NATIVELY, with no translation needed: q*>1/2 iff O(x)=P(y=1|x)/P(y=0|x) > w0/w1, exactly, no pi0/pi1 anywhere. Kept this as an EXACT stated value (not hedged as merely "proportional") since the footnote genuinely proves the exact identity for weighted CE – hedging it as merely proportional right next to a footnote proving exact equality would have been inconsistent, so instead added one sentence generalizing the QUALITATIVE takeaway (one constant, same for every sample, moves with the weight ratio) as what carries over to other per-sample losses, without claiming the exact w0/w1 formula does. REMOVED entirely per explicit request: the "eta=1 exactly" class-imbalance-cancellation paragraph (was the whole reason pi0/pi1 existed in the old version). Also removed L.2's opening footnote bridging its posterior-odds eta to L.1's old likelihood-ratio eta – no longer needed since both sections now use the SAME O(x) object throughout, L.2 just directly continues from L.1's result. L.2's OWN "shrinks to w=1" result (the noise-validity-window boundary case) was NOT removed – that's a different, still-wanted finding (about noise, not about class-imbalance cancellation), user's "don't need the eta=1 part" referred specifically to L.1's paragraph. Rebuilt thesis clean (nix build exit 0), confirmed the removed content is gone and no new undefined refs.

CAVEAT ADDED BY ME (not yet confirmed by user): translated the new appendix chapter title and the Discussion subsection's Hebrew TOC lines myself (I am not a native speaker) – "הצדקה תיאורטית לפונקציות הפסד ממטבות AUC לפי למת ניימן-פירסון" (appendix) and "הצדקה תיאורטית לפונקציות הפסד ממטבות AUC" (discussion subsection). Flagged to the user directly that these should be reviewed/corrected before treating this as final, unlike the rest of this thesis's Hebrew content which presumably had more careful authorship.

sec:np-class-weighting — class weighting as an implicit choice of eta

Core claim, sympy-verified before writing: for a weighted per-sample loss (weighted CE), minimizing pointwise risk and thresholding at q>1/2 is exactly equivalent to thresholding the likelihood ratio p(x|y=1)/p(x|y=0) > eta with eta = (w0/w1)*(pi0/pi1). Standard inverse-class-frequency weighting (w1/w0 = pi0/pi1) makes eta collapse to EXACTLY 1, regardless of noise/separability/distribution shape — i.e. it cancels the prior imbalance rather than targeting any actual desired FPR; this is a pure algebraic cancellation baked into the recipe (w0/w1 is DEFINED as the reciprocal of pi0/pi1), not a data-dependent coincidence — shown explicitly via a displayed substitution, added after the user asked why eta=1 held in general, not just in one example.

Derivation placed as a footnote (short, ~4 lines), not a full appendix — user's call ("footnote if it's a few lines, appendicise if more"). No new bib entry added; self-contained derivation, not attributed to a specific citation (Elkan-style cost-sensitive learning literature is the likely place this already exists, but wasn't verified/added — flag for the user's own citation pass if wanted later).

Scope note (user corrected an earlier, weaker framing before this was even written into the thesis): an early draft claimed ROLL-AOC "doesn't commit to a single eta" by sweeping 99 alpha values — user rejected this: at any actual decision point a classifier commits to one threshold, hence one implied eta; sweeping alphas during AOC training isn't the same as avoiding commitment to one eta at inference. Dropped; only the TPR@FPR "discovers eta as it goes" point was kept.

sec:np-noise-invariance — class weighting under label noise: an implicit, distorted eta

User's own idea (not mine), refined through two full passes:

PASS 1 (superseded): user proposed noise gives each sample a different "target eta." Checked with sympy — NOT what happens under class-conditional (x-independent) noise: the noisy likelihood ratio L~(x) depends on x ONLY through the true ratio L(x), via a fixed Mobius function g(L), strictly monotone for rho<1/2 — i.e. ONE global bijection, not per-sample. Written up using this density-ratio g(L) form directly.

PASS 2 (current thesis text): while explaining the g(L) derivation to the user, caught a real bug — p(x|ytilde=k) needs dividing by the normalizing constant D_k=P(ytilde=k) to be a proper conditional density, which the written g(L) omitted; true L~(x) = (D0/D1)*g(L(x)), not g(L(x)) exactly (sympy-verified). Didn't affect the monotonicity conclusion (D0/D1 is a positive x-independent constant) but was imprecise as written. Fixing it surfaced the real goal more clearly: not just "ranking is preserved" abstractly, but "noise implies an eta implicitly chosen by score/class weighting, one ROLL never has to choose." Reframed entirely in POSTERIOR-ODDS space instead of density-ratio space, which sidesteps the bug rather than patching it (no priors/normalizing constants needed at all): O(x)=P(y=1|x)/P(y=0|x), and P(ytilde=1|x) = (1-rho)P(y=1|x)+rho*P(y=0|x) via elementary total probability (no Bayes'-rule density manipulation, no joint-vs-conditional distinction to get wrong). Gives P(ytilde=1|x)/P(ytilde=0|x) = h(O(x)), h(O)=[rho+O(1-rho)]/[(1-rho)+O*rho], h'(O)= (1-2*rho)/[…]^2 > 0 for rho<1/2 (sympy-verified; h(O)->O at rho=0 sanity check passes).

Core claim: training with weight ratio w=w0/w1 on NOISY labels doesn't threshold O(x) at w – it thresholds h(O(x)) at w, equivalent to thresholding the TRUE posterior odds at eta_true = h^-1(w) = [w(1-rho)-rho]/[(1-rho)-w*rho]. Whatever w you pick, you actually get eta_true, which depends on the (usually unknown) rho — choosing w no longer means choosing eta, it means choosing eta up to an unknown noise-dependent distortion. Extra result found and verified (2000-trial sympy/numeric check, 0 mismatches): eta_true is only a valid (positive) posterior odds when rho/(1-rho) < w < (1-rho)/rho; outside that window no classifier could produce the implied threshold at all. As rho->1/2 this window shrinks to just {w=1} — heavy noise leaves ONLY the no-tilt threshold achievable by weighting, regardless of intent. ROLL sidesteps the whole mechanism: no weight, no eta, no need to know rho — it targets a rate directly, and ranking by O(x) is preserved under this noise model (same h monotonicity fact) so the target still means what it's supposed to.

Caveat (user's explicit request, carried unchanged across both passes and the Ch.2->Ch.3 move — "show it's problematic under the specific condition, doesn't mean it's untrue in general"): the whole argument needs rho to depend on y only (class-conditional), not x. Instance-dependent noise rho(x) breaks THIS ARGUMENT (h would vary with x directly) — did NOT claim ranking invariance actually fails there, only that this proof doesn't cover it.

Build status

Rebuilt thesis clean (nix build exit 0) at every step: first Ch.2 writeup, the pass-2 rewrite, the eta=1 follow-up, and the final Ch.2->Ch.3 move. Confirmed after the move: sections now number 3.2.4/3.2.5 (Method chapter), \Cref{eq:roll-tpr-at-fpr} resolves to "(3.8)" instead of a forward-pointer, and the Ch.2 back-references (Section 2.4, Section 2.5 for sec:imbalanced-tpr-fpr/sec:neyman-pearson) and Ch.4 forward-reference (\Cref{chapter:experiments} -> "Chapter 4") all resolve correctly.

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
• eq:roll-tpr-at-fpr-gaussian and eq:gauss-grad-combined now defined in appendix_gauss_grad.tex (moved from method.tex) — appendix_gauss_grad.tex references both, so they must stay there
• New .tex files must be git-added before nix build (untracked files are invisible to the flake's source enumeration) — bit both appendix:bandwidth-freeze and appendix:score-normalization on this, now routine to check
• Never verify a derivation mentally — use sympy in the nix devShell (`nix develop –command python3`), and cross-check with an independent finite-difference derivative when possible
• impl/src/roll.py's KernelizedROLLoss is a custom torch.autograd.Function with a hand-written backward() — it never auto-differentiates through forward()'s intermediate computations, so it was already implementing "frozen bandwidth" and "frozen normalization scale" by construction (omission), not by explicit design, until the thesis text caught up to state this as a deliberate convention (2026-09-09)
• When a KEEL regression test doesn't need bit-exact old-vs-new comparison (user will say so explicitly, e.g. "don't need numerical exactness"), a single clean run (no NaN, sane AUC in the range prior runs on the same dataset/config produced) is sufficient — no need for the git-stash old-vs-new dance every time
• cleveref config gotcha: \Crefname{appendix}{Appendix}{Appendices} alone does NOT make \Cref{appendix:…} say "Appendix" if appendices are typeset with \chapter (shares the "chapter" counter) — need \crefalias{chapter}{appendix} after \appendix too, otherwise every appendix cross-reference silently renders as "Chapter X" with no error/warning
• Chapter boundaries aren't fixed by topic (prior work vs. original contribution) alone — NP the LEMMA is prior work (Ch.2), but original NP-flavored ANALYSIS the thesis contributes can live in Ch.3 even though it's built on Ch.2 background; don't assume "cites prior work" means "belongs in related work" for a whole section wholesale