K-ABENA: K-Adaptive Backpropagation with Error-based N-exclusion Algorithm : (Compensated Loss-Based Sample Exclusion with Unbiased Gradient Estimation)

# K-Adaptive Backpropagation with Error-based N-exclusion Algorithm Compensated Loss-Based Sample Exclusion with Unbiased Gradient Estimation
Source: [https://arxiv.org/html/2607.05903](https://arxiv.org/html/2607.05903)
Jean\-François Bonbhel NeuroSoft IA, Québec City, Canada\|\|YekoElite University, Brazzaville, Republic of Congo UN AI Governance Expert Network \(UN PNAI\) — Member since 2021 bonbhel@yekoelite\.com

\(July 2026\)

###### Abstract

We present K\-ABENA \(K\-Adaptive Backpropagation with Error\-based N\-exclusion Algorithm\), a selective gradient computation framework that reduces per\-iteration training cost by excluding a fraction of low\-loss \(“minor”\) observations from the backward pass\. Its canonical form \(v3\) combines a defensive\-mixture sampling design over the minor set with Horvitz–Thompson inverse\-probability reweighting, yielding a*design\-unbiased*Horvitz–Thompson gradient estimator \(Lemma[2](https://arxiv.org/html/2607.05903#Thmlemma2)\) and whose self\-normalized practical variant carries a bias of orderO​\(1/m\)O\(1/m\)with an explicit constant \(Lemma[3](https://arxiv.org/html/2607.05903#Thmlemma3)\)\. We prove anO​\(1/T\)O\(1/\\sqrt\{T\}\)non\-convex convergence guarantee for SGD under the estimator, with an additive term that quantifies the residual bias \(Theorem[1](https://arxiv.org/html/2607.05903#Thmtheorem1)\)\. We further prove that*uncompensated*loss\-based selection — a family that includes OHEM, SBP, and the two earlier K\-ABENA variants — admits no stationary point at any minimizer where its selection bias is bounded away from zero \(Proposition[2](https://arxiv.org/html/2607.05903#Thmproposition2)\), and we quantify this failure empirically: at 0\.17% class imbalance, uncompensated variants reach test AUC 0\.53–0\.62 versus 0\.9998 for full\-batch SGD, while the compensated estimator attains 0\.9991 at identical 28\.4% compute savings\. On real datasets \(Breast Cancer, Digits, Wine, Diabetes\) the compensated estimator is statistically indistinguishable from full\-batch SGD \(paired permutation tests,p≥0\.5p\\geq 0\.5; Section[7](https://arxiv.org/html/2607.05903#S7)\) while saving 28–54% of per\-epoch gradient computation\. A biased “regularized mode” \(the earlier half\-domain variant\) is retained as an option with a proven exact bias decomposition \(Lemma[5](https://arxiv.org/html/2607.05903#Thmlemma5)\) and*quantified contraindications*: it collapses to 0\.386 accuracy under 40% label noise \(baseline: 0\.832\) and to 0\.53 AUC under extreme imbalance\. Every advantage and every limitation reported in this paper is either proved or measured; all experiments are CPU\-scale \(NumPy/scikit\-learn\) and their scope is stated explicitly\.

## 1Introduction

In large\-scale empirical risk minimization, a substantial fraction of per\-iteration computation is spent on observations the model has already learned: their per\-sample losses are small, their gradients are small, and their marginal contribution to the descent direction is limited\. Selective\-backpropagation methods exploit this observation by skipping the backward pass for low\-loss samples\[[6](https://arxiv.org/html/2607.05903#bib.bib6),[12](https://arxiv.org/html/2607.05903#bib.bib12)\], but they share a structural defect: the retained subset is*correlated with the loss*, so the resulting gradient is a biased estimator of the full\-batch gradient\. In benign regimes the bias is small relative to the signal and these methods work well; we show in Section[5](https://arxiv.org/html/2607.05903#S5)that in adverse regimes — extreme class imbalance, heavy label noise — the bias does not merely degrade performance but structurally prevents convergence to the minimizer, and we prove this as Proposition[2](https://arxiv.org/html/2607.05903#Thmproposition2)\.

This paper develops K\-ABENA, whose canonical estimator \(referred to as v3\) resolves the defect using a century\-old idea from survey sampling\[[5](https://arxiv.org/html/2607.05903#bib.bib5)\]: any sampling design with known, strictly positive inclusion probabilities admits an unbiased estimator of a population total via inverse\-probability weighting\. K\-ABENA v3 samples retained minors from the*entire*minor set under a defensive mixture design\[[10](https://arxiv.org/html/2607.05903#bib.bib10)\]and reweights them accordingly\. The result occupies a design point that, to our knowledge, none of the established selective or reweighting methods occupies:*per\-iteration compute reduction with an \(exactly or near\-\) unbiased gradient*\. Hard\-selection methods \(OHEM\[[12](https://arxiv.org/html/2607.05903#bib.bib12)\], SBP\[[6](https://arxiv.org/html/2607.05903#bib.bib6)\]\) save compute but are biased; soft\-reweighting methods \(Focal Loss\[[9](https://arxiv.org/html/2607.05903#bib.bib9)\]\) are computed on the full batch and save nothing; importance\-sampling training methods\[[7](https://arxiv.org/html/2607.05903#bib.bib7)\]reweight but target variance reduction under full or minibatch evaluation rather than threshold\-based exclusion with an explicit retention budget\.

#### Contributions\.

1. 1\.A compensated selective estimator\(Definition[1](https://arxiv.org/html/2607.05903#Thmdefinition1)\) with two interpretable controls: a loss thresholdKKseparating “minor” from “major” observations, and a retention proportionN∈\(0,1\)N\\in\(0,1\)governing the compute budget\. The per\-epoch backward\-pass saving is exactly\(1−N\)​k/n\(1\-N\)\\,k/n\(Proposition[1](https://arxiv.org/html/2607.05903#Thmproposition1)\), wherekkis the number of minors\.
2. 2\.Design\-unbiasedness, exact and approximate\(Lemmas[2](https://arxiv.org/html/2607.05903#Thmlemma2)–[3](https://arxiv.org/html/2607.05903#Thmlemma3)\): the Horvitz–Thompson form is design\-unbiased \(unbiased over the sampling randomization, conditionally on the iterate\); the self\-normalized \(Hájek\) form used in practice has bias at most2​GMα2​m​\(k−mk\)\\frac\{2G\_\{M\}\}\{\\alpha^\{2\}m\}\\bigl\(\\tfrac\{k\-m\}\{k\}\\bigr\), vanishing asO​\(1/m\)O\(1/m\), with all constants explicit\.
3. 3\.A convergence guarantee\(Theorem[1](https://arxiv.org/html/2607.05903#Thmtheorem1)\):O​\(1/T\)O\(1/\\sqrt\{T\}\)decay of the expected squared gradient norm for smooth non\-convex objectives, plus an additiveO​\(δm\)O\(\\delta\_\{m\}\)floor traceable to the self\-normalization bias, which the practitioner controls throughmm\.
4. 4\.An impossibility result for uncompensated selection\(Proposition[2](https://arxiv.org/html/2607.05903#Thmproposition2)\): if the selection bias at a minimizer is bounded below byβ\>0\\beta\>0, the minimizer is not a stationary point of the expected dynamics; measured on a 0\.17%\-imbalance task,β≈0\.15\\beta\\approx 0\.15against a vanishing true gradient, predicting the observed failure \(AUC 0\.53\) and its resolution under compensation \(AUC 0\.9991\)\.
5. 5\.A characterized regularized mode: the earlier biased half\-domain variant \(v2\) is retained as an option, with an exact algebraic bias decomposition \(Lemma[5](https://arxiv.org/html/2607.05903#Thmlemma5)\) and quantified contraindications \(Section[6](https://arxiv.org/html/2607.05903#S6)\); its small accuracy bonus on multiclass tasks \(\+0\.35\+0\.35to\+0\.45\+0\.45points\) and its catastrophic failure modes \(0\.386 accuracy at 40% label noise; 0\.53 AUC at 0\.17% imbalance\) are both measured and mechanistically explained\.

#### Scope statement\.

All empirical results in this paper are CPU\-scale: logistic regression, linear SVM, softmax regression, and a one\-hidden\-layer MLP, on bundled scikit\-learn datasets and on a synthetic replica of an extreme\-imbalance fraud regime \(0\.17% positives,n=20,000n=20\{,\}000\)\. No GPU benchmark \(deep CNN/Transformer\) is reported; this is stated as Limitation L4 rather than compensated by simulation\. We consider explicit scope statements a feature of the paper, not a weakness\.

## 2Related Work

Hard selection\.OHEM\[[12](https://arxiv.org/html/2607.05903#bib.bib12)\]retains only the highest\-loss examples of each batch; SBP\[[6](https://arxiv.org/html/2607.05903#bib.bib6)\]drops examples whose loss falls below a threshold\. Both reduce backward\-pass cost and both compute a plain average over a loss\-correlated subset, hence a biased gradient\. K\-ABENA strictly generalizes SBP: settingN=0N=0with a hard threshold recovers it, whileN\>0N\>0restores representation of the excluded stratum and the v3 weighting removes the bias\. Curriculum and self\-paced approaches\[[1](https://arxiv.org/html/2607.05903#bib.bib1),[8](https://arxiv.org/html/2607.05903#bib.bib8)\]order examples by difficulty over training time; K\-ABENA is orthogonal, operating within each iteration\.

Soft reweighting\.Focal Loss\[[9](https://arxiv.org/html/2607.05903#bib.bib9)\]down\-weights easy examples by a factor\(1−pt\)γ\(1\-p\_\{t\}\)^\{\\gamma\}applied*after*the forward pass of every example; the full backward cost is still paid, so the method offers no compute saving — the quantity K\-ABENA targets\. Its modulation is also a single global functional form, whereasKKandNNdecouple the location of the easy/hard boundary from the retention budget\.

Importance sampling for SGD\.Katharopoulos and Fleuret \[[7](https://arxiv.org/html/2607.05903#bib.bib7)\]and related work sample training examples proportionally to \(proxies of\) gradient norm with inverse\-probability corrections, targeting variance reduction\. Three structural differences separate K\-ABENA from this line, and each is measurable\.*\(a\) Scoring cost\.*Classical IS must score*all*nncandidates at each step to build its proposal — a forward pass \(or a last\-layer gradient\-norm bound, itself requiring the forward\) whose cost remainsO​\(n\)O\(n\); the backward saving is therefore bought with a full\-width scoring pass\. K\-ABENA scores with the per\-sample losses the training loop already produces, and its delayed\-losses pattern \(masks for epochttbuilt from epocht−1t\{\-\}1losses; released with the library\) removes even the forward of excluded samples\.*\(b\) Budget semantics\.*IS fixes a minibatch size and optimizes variance at that size; K\-ABENA’s two controls decouple*where*the easy/hard boundary lies \(KK, a loss percentile with a semantic reading\) from*how much*budget is spent \(NN\), yielding the deterministic, architecture\-independent savingG=\(1−N\)​k/nG=\(1\-N\)k/n\(Proposition[1](https://arxiv.org/html/2607.05903#Thmproposition1)\)\.*\(c\) Design\.*K\-ABENA is a*two\-stratum*design — a certainty stratum \(majors,πi=1\\pi\_\{i\}=1\) plus a sampled stratum restricted to minors under a defensive mixture with floorα/k\\alpha/k— rather than a single global proposal\. This is not cosmetic: at matched compute budget, a classical global loss\-proportional proposal with self\-normalized correction is*significantly worse*than the full\-batch baseline in the standard regime \(accuracy0\.95730\.9573vs0\.97200\.9720, paired permutationp=0\.002p=0\.002, Table[3](https://arxiv.org/html/2607.05903#S7.T3)\) — tiny\-loss samples occasionally drawn under a global proposal receive enormous weights, and the majors’ contribution is needlessly randomized — while the stratified, floor\-bounded design is statistically indistinguishable from the baseline \(p=1\.0p=1\.0\) at the same saving\. The certainty stratum and the defensive floor are what make compensation*cheaply stable*\.

Survey sampling\.The estimators are classical: Horvitz–Thompson\[[5](https://arxiv.org/html/2607.05903#bib.bib5)\]and its self\-normalized \(Hájek\) variant\[[11](https://arxiv.org/html/2607.05903#bib.bib11)\]\. Our contribution is not the estimator but its integration into threshold\-based selective backpropagation, the resulting theory \(Theorem[1](https://arxiv.org/html/2607.05903#Thmtheorem1), Proposition[2](https://arxiv.org/html/2607.05903#Thmproposition2)\), and the quantified characterization of when the uncompensated shortcut is and is not safe\.

## 3The K\-ABENA Framework

### 3\.1Setting and notation

LetF​\(θ\)=1n​∑i=1nfi​\(θ\)F\(\\theta\)=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}f\_\{i\}\(\\theta\)withfi:ℝd→ℝf\_\{i\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}differentiable, and writegi=∇fi​\(θ\)g\_\{i\}=\\nabla f\_\{i\}\(\\theta\)andℓi=fi​\(θ\)≥0\\ell\_\{i\}=f\_\{i\}\(\\theta\)\\geq 0for the per\-sample gradient and loss at the current iterate\. At each iteration, a thresholdK\>0K\>0\(in practice a fixed percentile of the current loss distribution\) partitions the indices into theminor setℳ=\{i:ℓi≤K\}\\mathcal\{M\}=\\\{i:\\ell\_\{i\}\\leq K\\\},k=\|ℳ\|k=\|\\mathcal\{M\}\|, and themajor setℳc\\mathcal\{M\}^\{c\},\|ℳc\|=n−k\|\\mathcal\{M\}^\{c\}\|=n\-k\. Majors are always retained\. A retention proportionN∈\(0,1\)N\\in\(0,1\)fixes the number of retained minorsm=⌊Nk⌉m=\\lfloor Nk\\rceil\.

###### Definition 1\(Canonical K\-ABENA sampling design, v3\)\.

Fix a defensive\-mixture coefficientα∈\(0,1\]\\alpha\\in\(0,1\]\. Draw𝒮⊂ℳ\\mathcal\{S\}\\subset\\mathcal\{M\},\|𝒮\|=m\|\\mathcal\{S\}\|=m, without replacement with single\-draw probabilities

pi=α⋅1k\+\(1−α\)​ℓi∑j∈ℳℓj,i∈ℳ,p\_\{i\}\\;=\\;\\alpha\\cdot\\frac\{1\}\{k\}\\;\+\\;\(1\-\\alpha\)\\,\\frac\{\\ell\_\{i\}\}\{\\sum\_\{j\\in\\mathcal\{M\}\}\\ell\_\{j\}\},\\qquad i\\in\\mathcal\{M\},\(1\)and denote byπi=Pr⁡\[i∈𝒮\]\\pi\_\{i\}=\\Pr\[i\\in\\mathcal\{S\}\]the inclusion probabilities of the design\. TheHorvitz–Thompson \(HT\) gradient estimatoris

g^HT=1n​\[∑i∉ℳgi\+∑i∈𝒮giπi\],\\hat\{g\}\_\{\\mathrm\{HT\}\}\\;=\\;\\frac\{1\}\{n\}\\Bigl\[\\;\\sum\_\{i\\notin\\mathcal\{M\}\}g\_\{i\}\\;\+\\;\\sum\_\{i\\in\\mathcal\{S\}\}\\frac\{g\_\{i\}\}\{\\pi\_\{i\}\}\\;\\Bigr\],\(2\)and theself\-normalized \(Hájek\) estimator, used in practice, is

g^=∑i∉ℳgi\+∑i∈𝒮πi−1​gi\(n−k\)\+∑i∈𝒮πi−1\.\\hat\{g\}\\;=\\;\\frac\{\\sum\_\{i\\notin\\mathcal\{M\}\}g\_\{i\}\\;\+\\;\\sum\_\{i\\in\\mathcal\{S\}\}\\pi\_\{i\}^\{\-1\}g\_\{i\}\}\{\(n\-k\)\\;\+\\;\\sum\_\{i\\in\\mathcal\{S\}\}\\pi\_\{i\}^\{\-1\}\}\.\(3\)

###### Assumption 1\(Inclusion probabilities\)\.

Either \(a\)α=1\\alpha=1, in which case the design is simple random sampling without replacement andπi=m/k\\pi\_\{i\}=m/k*exactly*; or \(b\)α<1\\alpha<1andmaxi⁡m​pi≤1\\max\_\{i\}m\\,p\_\{i\}\\leq 1, in which case we use the Poisson/rejective approximationπi=m​pi\\pi\_\{i\}=m\\,p\_\{i\}, standard for high\-entropy without\-replacement designs\[[11](https://arxiv.org/html/2607.05903#bib.bib11)\]\. All statements below that rely on case \(b\) are flagged; the empirical error attributable to this approximation is measured in Section[7](https://arxiv.org/html/2607.05903#S7)\(residual bias 0\.004 atn=20,000n=20\{,\}000\)\.

###### Lemma 1\(Bounded weights under defensive mixing\)\.

Under Assumption[1](https://arxiv.org/html/2607.05903#Thmassumption1), for everyi∈ℳi\\in\\mathcal\{M\},

πi≥α​mk,hence1πi≤kα​m\.\\pi\_\{i\}\\;\\geq\\;\\frac\{\\alpha m\}\{k\},\\qquad\\text\{hence\}\\qquad\\frac\{1\}\{\\pi\_\{i\}\}\\;\\leq\\;\\frac\{k\}\{\\alpha m\}\.

###### Proof\.

From \([1](https://arxiv.org/html/2607.05903#S3.E1)\),pi≥α/kp\_\{i\}\\geq\\alpha/kpointwise since the second term is non\-negative\. In case \(a\),πi=m/k=α​m/k\\pi\_\{i\}=m/k=\\alpha m/kwithα=1\\alpha=1\. In case \(b\),πi=m​pi≥m​α/k\\pi\_\{i\}=mp\_\{i\}\\geq m\\alpha/k\. Inverting gives the weight bound\. ∎

###### Proposition 1\(Exact computational gain\)\.

The number of backward passes per iteration is\(n−k\)\+m\(n\-k\)\+m, so the saved fraction is

G=n−\[\(n−k\)\+m\]n=k−mn=\(1−N\)​kn,G\\;=\\;\\frac\{n\-\\bigl\[\(n\-k\)\+m\\bigr\]\}\{n\}\\;=\\;\\frac\{k\-m\}\{n\}\\;=\\;\(1\-N\)\\,\\frac\{k\}\{n\},identical for the HT and Hájek forms; the reweighting itself costsO​\(m\)O\(m\)scalar operations, negligible against a backward pass\. WithKKat the 40th loss percentile andN=0\.3N=0\.3,G=0\.7×0\.4=0\.28G=0\.7\\times 0\.4=0\.28; the measured per\-epoch saving in every experiment of Section[7](https://arxiv.org/html/2607.05903#S7)is 28\.0–28\.7%, matching the formula\. ∎

## 4Estimator Guarantees

### 4\.1Unbiasedness

###### Lemma 2\(Design\-unbiasedness of the HT form\)\.

Under Assumption[1](https://arxiv.org/html/2607.05903#Thmassumption1), conditionally on the current iterate \(hence onℓi\\ell\_\{i\},ℳ\\mathcal\{M\},πi\\pi\_\{i\}\),

𝔼​\[g^HT\]=∇F​\(θ\)\.\\mathbb\{E\}\\bigl\[\\hat\{g\}\_\{\\mathrm\{HT\}\}\\bigr\]\\;=\\;\\nabla F\(\\theta\)\.

###### Proof\.

LetZi=𝟏​\[i∈𝒮\]Z\_\{i\}=\\mathbf\{1\}\[i\\in\\mathcal\{S\}\], so𝔼​\[Zi\]=πi\\mathbb\{E\}\[Z\_\{i\}\]=\\pi\_\{i\}\. Then

𝔼​\[∑i∈𝒮giπi\]=𝔼​\[∑i∈ℳZi​giπi\]=∑i∈ℳ𝔼​\[Zi\]πi​gi=∑i∈ℳgi\.\\mathbb\{E\}\\Bigl\[\\sum\_\{i\\in\\mathcal\{S\}\}\\frac\{g\_\{i\}\}\{\\pi\_\{i\}\}\\Bigr\]=\\mathbb\{E\}\\Bigl\[\\sum\_\{i\\in\\mathcal\{M\}\}Z\_\{i\}\\frac\{g\_\{i\}\}\{\\pi\_\{i\}\}\\Bigr\]=\\sum\_\{i\\in\\mathcal\{M\}\}\\frac\{\\mathbb\{E\}\[Z\_\{i\}\]\}\{\\pi\_\{i\}\}\\,g\_\{i\}=\\sum\_\{i\\in\\mathcal\{M\}\}g\_\{i\}\.Substituting into \([2](https://arxiv.org/html/2607.05903#S3.E2)\) gives𝔼​\[g^HT\]=1n​\[∑i∉ℳgi\+∑i∈ℳgi\]=∇F​\(θ\)\\mathbb\{E\}\[\\hat\{g\}\_\{\\mathrm\{HT\}\}\]=\\frac\{1\}\{n\}\\bigl\[\\sum\_\{i\\notin\\mathcal\{M\}\}g\_\{i\}\+\\sum\_\{i\\in\\mathcal\{M\}\}g\_\{i\}\\bigr\]=\\nabla F\(\\theta\)\. Note that positivityπi\>0\\pi\_\{i\}\>0for*all*i∈ℳi\\in\\mathcal\{M\}— guaranteed by the defensive termα/k\\alpha/k— is what makes the argument available; any design assigning zero probability to part ofℳ\\mathcal\{M\}\(Section[6](https://arxiv.org/html/2607.05903#S6)\) cannot be corrected this way\. ∎

###### Lemma 3\(Bias of the self\-normalized form\)\.

LetGM=maxi∈ℳ⁡‖gi‖G\_\{M\}=\\max\_\{i\\in\\mathcal\{M\}\}\\left\\lVert g\_\{i\}\\right\\rVertand letW^=\(n−k\)\+∑i∈𝒮πi−1\\widehat\{W\}=\(n\-k\)\+\\sum\_\{i\\in\\mathcal\{S\}\}\\pi\_\{i\}^\{\-1\}denote the \(random\) normalizer in \([3](https://arxiv.org/html/2607.05903#S3.E3)\), with𝔼​\[W^\]=n\\mathbb\{E\}\[\\widehat\{W\}\]=n\. Then

‖𝔼​\[g^\]−∇F​\(θ\)‖≤GMn​Var​\(W^\)≤GM​kn​α​m,\\bigl\\lVert\\mathbb\{E\}\[\\hat\{g\}\]\-\\nabla F\(\\theta\)\\bigr\\rVert\\;\\leq\\;\\frac\{G\_\{M\}\}\{n\}\\,\\sqrt\{\\mathrm\{Var\}\\bigl\(\\widehat\{W\}\\bigr\)\}\\;\\leq\\;\\frac\{G\_\{M\}\\,k\}\{n\\,\\sqrt\{\\alpha\\,m\}\},and the standard second\-order ratio\-estimator expansion sharpens this to

δm:=‖𝔼​\[g^\]−∇F​\(θ\)‖≤2​GM​k2α​m​n2\+o​\(1/m\)=O​\(1/m\)\\delta\_\{m\}\\;:=\\;\\bigl\\lVert\\mathbb\{E\}\[\\hat\{g\}\]\-\\nabla F\(\\theta\)\\bigr\\rVert\\;\\leq\\;\\frac\{2\\,G\_\{M\}\\,k^\{2\}\}\{\\alpha\\,m\\,n^\{2\}\}\+o\(1/m\)\\;=\\;O\(1/m\)at fixedk/nk/nandα\\alpha\. Atα=1\\alpha=1\(SRSWOR, exactπi=m/k\\pi\_\{i\}=m/k\) the same expansion givesδm≤2​GM​k​\(k−m\)/\(m​n2\)\+o​\(1/m\)\\delta\_\{m\}\\leq 2\\,G\_\{M\}\\,k\\,\(k\-m\)/\(m\\,n^\{2\}\)\+o\(1/m\)\. \(Full derivation: Appendix[A](https://arxiv.org/html/2607.05903#A1)\.\)

### 4\.2Variance

###### Lemma 4\(Variance bound\)\.

Under Assumption[1](https://arxiv.org/html/2607.05903#Thmassumption1)\(b\), the HT estimator satisfies

𝔼\[∥g^HT−∇F\(θ\)∥2\]≤1n2∑i∈ℳ1−πiπi∥gi∥2≤kα​m​n2∑i∈ℳ∥gi∥2≤GM2​k2α​m​n2=:σ^2\.\\mathbb\{E\}\\bigl\[\\left\\lVert\\hat\{g\}\_\{\\mathrm\{HT\}\}\-\\nabla F\(\\theta\)\\right\\rVert^\{2\}\\bigr\]\\;\\leq\\;\\frac\{1\}\{n^\{2\}\}\\sum\_\{i\\in\\mathcal\{M\}\}\\frac\{1\-\\pi\_\{i\}\}\{\\pi\_\{i\}\}\\,\\left\\lVert g\_\{i\}\\right\\rVert^\{2\}\\;\\leq\\;\\frac\{k\}\{\\alpha\\,m\\,n^\{2\}\}\\sum\_\{i\\in\\mathcal\{M\}\}\\left\\lVert g\_\{i\}\\right\\rVert^\{2\}\\;\\leq\\;\\frac\{G\_\{M\}^\{2\}\\,k^\{2\}\}\{\\alpha\\,m\\,n^\{2\}\}\\;=:\\;\\hat\{\\sigma\}^\{2\}\.

###### Proof\.

Under the Poisson approximation theZiZ\_\{i\}are independent withVar​\(Zi\)=πi​\(1−πi\)\\mathrm\{Var\}\(Z\_\{i\}\)=\\pi\_\{i\}\(1\-\\pi\_\{i\}\); writeg^HT−∇F=1n​∑i∈ℳ\(Zi/πi−1\)​gi\\hat\{g\}\_\{\\mathrm\{HT\}\}\-\\nabla F=\\frac\{1\}\{n\}\\sum\_\{i\\in\\mathcal\{M\}\}\(Z\_\{i\}/\\pi\_\{i\}\-1\)g\_\{i\}and expand:

𝔼​‖g^HT−∇F‖2=1n2​∑i∈ℳVar​\(Zi\)πi2​‖gi‖2=1n2​∑i∈ℳ1−πiπi​‖gi‖2\.\\mathbb\{E\}\\left\\lVert\\hat\{g\}\_\{\\mathrm\{HT\}\}\-\\nabla F\\right\\rVert^\{2\}=\\frac\{1\}\{n^\{2\}\}\\sum\_\{i\\in\\mathcal\{M\}\}\\frac\{\\mathrm\{Var\}\(Z\_\{i\}\)\}\{\\pi\_\{i\}^\{2\}\}\\left\\lVert g\_\{i\}\\right\\rVert^\{2\}=\\frac\{1\}\{n^\{2\}\}\\sum\_\{i\\in\\mathcal\{M\}\}\\frac\{1\-\\pi\_\{i\}\}\{\\pi\_\{i\}\}\\left\\lVert g\_\{i\}\\right\\rVert^\{2\}\.The middle inequality usesπi≥α​m/k\\pi\_\{i\}\\geq\\alpha m/k\(Lemma[1](https://arxiv.org/html/2607.05903#Thmlemma1)\) and1−πi≤11\-\\pi\_\{i\}\\leq 1; the last uses‖gi‖≤GM\\left\\lVert g\_\{i\}\\right\\rVert\\leq G\_\{M\}and\|ℳ\|=k\|\\mathcal\{M\}\|=k\. ∎

### 4\.3Convergence under SGD

###### Assumption 2\.

FFisLL\-smooth \(‖∇F​\(θ\)−∇F​\(θ′\)‖≤L​‖θ−θ′‖\\left\\lVert\\nabla F\(\\theta\)\-\\nabla F\(\\theta^\{\\prime\}\)\\right\\rVert\\leq L\\left\\lVert\\theta\-\\theta^\{\\prime\}\\right\\rVert\), bounded below byF∗F^\{\*\}, and along the trajectory the estimator satisfies∥𝔼\[g^t∣θt\]−∇F\(θt\)∥≤δ\\left\\lVert\\mathbb\{E\}\[\\hat\{g\}\_\{t\}\\mid\\theta\_\{t\}\]\-\\nabla F\(\\theta\_\{t\}\)\\right\\rVert\\leq\\deltaand𝔼\[∥g^t−𝔼\[g^t∣θt\]∥2∣θt\]≤σ^2\\mathbb\{E\}\[\\left\\lVert\\hat\{g\}\_\{t\}\-\\mathbb\{E\}\[\\hat\{g\}\_\{t\}\\mid\\theta\_\{t\}\]\\right\\rVert^\{2\}\\mid\\theta\_\{t\}\]\\leq\\hat\{\\sigma\}^\{2\}, withδ,σ^\\delta,\\hat\{\\sigma\}as in Lemmas[3](https://arxiv.org/html/2607.05903#Thmlemma3)and[4](https://arxiv.org/html/2607.05903#Thmlemma4), and‖∇F​\(θt\)‖≤G∞\\left\\lVert\\nabla F\(\\theta\_\{t\}\)\\right\\rVert\\leq G\_\{\\infty\}\.

###### Theorem 1\(Non\-convex convergence of SGD with the v3 estimator\)\.

Runθt\+1=θt−η​g^t\\theta\_\{t\+1\}=\\theta\_\{t\}\-\\eta\\,\\hat\{g\}\_\{t\}forTTsteps withη=min⁡\{12​L,cT\}\\eta=\\min\\\{\\tfrac\{1\}\{2L\},\\,\\tfrac\{c\}\{\\sqrt\{T\}\}\\\},c\>0c\>0\. Under Assumptions[1](https://arxiv.org/html/2607.05903#Thmassumption1)and[2](https://arxiv.org/html/2607.05903#Thmassumption2),

min0≤t<T⁡𝔼​‖∇F​\(θt\)‖2≤4​\(F​\(θ0\)−F∗\)c​T\+2​L​c​\(σ^2\+δ2\)T⏟vanishing,​O​\(1/T\)\+2​δ​G∞\+δ2⏟bias floor\.\\min\_\{0\\leq t<T\}\\mathbb\{E\}\\left\\lVert\\nabla F\(\\theta\_\{t\}\)\\right\\rVert^\{2\}\\;\\leq\\;\\underbrace\{\\frac\{4\\,\(F\(\\theta\_\{0\}\)\-F^\{\*\}\)\}\{c\\,\\sqrt\{T\}\}\\;\+\\;\\frac\{2Lc\\,\(\\hat\{\\sigma\}^\{2\}\+\\delta^\{2\}\)\}\{\\sqrt\{T\}\}\}\_\{\\text\{vanishing, \}O\(1/\\sqrt\{T\}\)\}\\;\+\\;\\underbrace\{\\vphantom\{\\frac\{1\}\{\\sqrt\{T\}\}\}2\\,\\delta\\,G\_\{\\infty\}\+\\delta^\{2\}\}\_\{\\text\{bias floor\}\}\.For the design\-unbiased HT form \(δ=0\\delta=0\) this is the classicalO​\(1/T\)O\(1/\\sqrt\{T\}\)rate\[[4](https://arxiv.org/html/2607.05903#bib.bib4)\]; for the Hájek form the floor isO​\(δm\)=O​\(1/m\)O\(\\delta\_\{m\}\)=O\(1/m\), explicitly controlled by the retention budget\. \(Proof: Appendix[B](https://arxiv.org/html/2607.05903#A2)\.\)

## 5Why Uncompensated Selection Fails: A Quantified Impossibility

Uncompensated loss\-based selection — OHEM, SBP, and the earlier K\-ABENA variants v1/v2 — computes a plain average over a loss\-correlated subset\. Write its conditional expectation as𝔼​\[g^sel​\(θ\)\]=∇F​\(θ\)\+b​\(θ\)\\mathbb\{E\}\[\\hat\{g\}\_\{\\mathrm\{sel\}\}\(\\theta\)\]=\\nabla F\(\\theta\)\+b\(\\theta\), whereb​\(θ\)b\(\\theta\)is the*selection bias field*\.

###### Proposition 2\(No stationarity at the minimizer under persistent selection bias\)\.

Letθ∗\\theta^\{\*\}be a stationary point ofFF\(∇F​\(θ∗\)=0\\nabla F\(\\theta^\{\*\}\)=0\) and suppose‖b​\(θ\)‖≥β\>0\\left\\lVert b\(\\theta\)\\right\\rVert\\geq\\beta\>0on a neighborhoodU∋θ∗U\\ni\\theta^\{\*\}\. Then: \(i\)θ∗\\theta^\{\*\}is not a stationary point of the expected dynamicsθ˙=−𝔼​\[g^sel​\(θ\)\]\\dot\{\\theta\}=\-\\mathbb\{E\}\[\\hat\{g\}\_\{\\mathrm\{sel\}\}\(\\theta\)\], whose expected update atθ∗\\theta^\{\*\}has magnitudeη​β\\eta\\beta; \(ii\) any stationary pointθ¯∈U\\bar\{\\theta\}\\in Uof the expected dynamics satisfies∇F​\(θ¯\)=−b​\(θ¯\)\\nabla F\(\\bar\{\\theta\}\)=\-b\(\\bar\{\\theta\}\), hence‖∇F​\(θ¯\)‖≥β\\left\\lVert\\nabla F\(\\bar\{\\theta\}\)\\right\\rVert\\geq\\beta: the biased method can only settle where the true gradient is as large as the bias\.

###### Proof\.

\(i\) Atθ∗\\theta^\{\*\},𝔼​\[g^sel​\(θ∗\)\]=0\+b​\(θ∗\)≠0\\mathbb\{E\}\[\\hat\{g\}\_\{\\mathrm\{sel\}\}\(\\theta^\{\*\}\)\]=0\+b\(\\theta^\{\*\}\)\\neq 0since‖b​\(θ∗\)‖≥β\\left\\lVert b\(\\theta^\{\*\}\)\\right\\rVert\\geq\\beta; the expected update is−η​b​\(θ∗\)\-\\eta\\,b\(\\theta^\{\*\}\)of magnitudeη​β\\eta\\beta\. \(ii\) Stationarity of the expected dynamics means∇F​\(θ¯\)\+b​\(θ¯\)=0\\nabla F\(\\bar\{\\theta\}\)\+b\(\\bar\{\\theta\}\)=0, so‖∇F​\(θ¯\)‖=‖b​\(θ¯\)‖≥β\\left\\lVert\\nabla F\(\\bar\{\\theta\}\)\\right\\rVert=\\left\\lVert b\(\\bar\{\\theta\}\)\\right\\rVert\\geq\\beta\. ∎

## 6The Regularized Mode \(v2\): Exact Bias, Measured Benefits, Quantified Contraindications

K\-ABENA retains an optional biased mode: sampling restricted to the lower halfℒ=\{i∈ℳ:ℓi≤c\}\\mathcal\{L\}=\\\{i\\in\\mathcal\{M\}:\\ell\_\{i\}\\leq c\\\}\(ccthe median minor loss\), withpi∝ℓip\_\{i\}\\propto\\ell\_\{i\}onℒ\\mathcal\{L\}and*no*reweighting, and feasibility constraintN≤12N\\leq\\tfrac\{1\}\{2\}\. Becauseπi=0\\pi\_\{i\}=0on the upper half𝒰=ℳ∖ℒ\\mathcal\{U\}=\\mathcal\{M\}\\setminus\\mathcal\{L\}, Lemma[2](https://arxiv.org/html/2607.05903#Thmlemma2)’s mechanism is unavailable: this mode is biased*by construction*, and the bias admits an exact expression\.

###### Lemma 5\(Exact bias decomposition of the v2 mode\)\.

Letg¯M,g¯ℒ,g¯𝒰\\bar\{g\}\_\{M\},\\bar\{g\}\_\{\\mathcal\{L\}\},\\bar\{g\}\_\{\\mathcal\{U\}\}be the per\-observation gradient means overℳ,ℒ,𝒰\\mathcal\{M\},\\mathcal\{L\},\\mathcal\{U\}\(\|ℒ\|=\|𝒰\|=k/2\|\\mathcal\{L\}\|=\|\\mathcal\{U\}\|=k/2\), andg¯w=∑ℒℓi​gi/∑ℒℓi\\bar\{g\}\_\{w\}=\\sum\_\{\\mathcal\{L\}\}\\ell\_\{i\}g\_\{i\}/\\sum\_\{\\mathcal\{L\}\}\\ell\_\{i\}the population loss\-weighted mean overℒ\\mathcal\{L\}\. Then, exactly,

g¯w−g¯M=\(g¯w−g¯ℒ\)⏟local term−12​\(g¯𝒰−g¯ℒ\)⏟structural term\.\\bar\{g\}\_\{w\}\-\\bar\{g\}\_\{M\}\\;=\\;\\underbrace\{\(\\bar\{g\}\_\{w\}\-\\bar\{g\}\_\{\\mathcal\{L\}\}\)\}\_\{\\text\{local term\}\}\\;\-\\;\\underbrace\{\\tfrac\{1\}\{2\}\\,\(\\bar\{g\}\_\{\\mathcal\{U\}\}\-\\bar\{g\}\_\{\\mathcal\{L\}\}\)\}\_\{\\text\{structural term\}\}\.

###### Proof\.

\|ℒ\|=\|𝒰\|\|\\mathcal\{L\}\|=\|\\mathcal\{U\}\|givesg¯M=12​\(g¯ℒ\+g¯𝒰\)\\bar\{g\}\_\{M\}=\\tfrac\{1\}\{2\}\(\\bar\{g\}\_\{\\mathcal\{L\}\}\+\\bar\{g\}\_\{\\mathcal\{U\}\}\); substitute and regroup:g¯w−g¯M=\(g¯w−g¯ℒ\)\+g¯ℒ−12​g¯ℒ−12​g¯𝒰\\bar\{g\}\_\{w\}\-\\bar\{g\}\_\{M\}=\(\\bar\{g\}\_\{w\}\-\\bar\{g\}\_\{\\mathcal\{L\}\}\)\+\\bar\{g\}\_\{\\mathcal\{L\}\}\-\\tfrac\{1\}\{2\}\\bar\{g\}\_\{\\mathcal\{L\}\}\-\\tfrac\{1\}\{2\}\\bar\{g\}\_\{\\mathcal\{U\}\}\. ∎

Under the empirically typical monotone loss–gradient relationship \(g¯𝒰\>g¯ℒ\\bar\{g\}\_\{\\mathcal\{U\}\}\>\\bar\{g\}\_\{\\mathcal\{L\}\}componentwise in magnitude\), the two terms partially cancel; this is the mechanism behind the mode’s measured properties: gradient bias 4–10%*below*and variance 25–55% below the historical full\-domain variant, and a small accuracy bonus over the full\-batch baseline on multiclass tasks \(\+0\.35\+0\.35points on Digits,\+0\.45\+0\.45on Wine\) that we attribute to the bias acting as an implicit regularizer — an attribution supported by its disappearance under the design\-unbiased estimator \(v3 is statistically indistinguishable from the baseline on the same tasks; Table[1](https://arxiv.org/html/2607.05903#S7.T1)\)\.

#### Quantified contraindications\.

The same bias is destructive outside a characterized validity region: \(i\)*extreme imbalance*— AUC 0\.53 at 0\.17% positives \(Proposition[2](https://arxiv.org/html/2607.05903#Thmproposition2)applies\); \(ii\)*heavy label noise*— at 40% symmetric flips on Breast Cancer, accuracy collapses to0\.3860\.386\(baseline0\.8320\.832; v30\.8080\.808\): noisy samples are high\-loss, hence always retained as majors, while the clean low\-loss anchors that the mode discards are precisely what stabilized the decision boundary; at 20% noise the mode still shows a\+0\.4\+0\.4\-point benefit, locating the phase change between∼\\sim20% and 40%\. The mode is therefore gated: minority signal≳5%\\gtrsim 5\\%, label noise≲25%\\lesssim 25\\%,N≤12N\\leq\\tfrac\{1\}\{2\}; outside this region the canonical estimator is mandatory\.

## 7Experiments

#### Protocol\.

All runs: full\-batch SGD with per\-epoch recalibration ofKKat the 40th percentile of current losses,N=0\.3N=0\.3,α=0\.3\\alpha=0\.3unless stated; 75/25 stratified splits; standardized features\. Seeds are published and paired across methods \(seeds0–44throughout;0–99for Table[3](https://arxiv.org/html/2607.05903#S7.T3)\): every method sees the identical split, initialization, and data ordering per seed, so all comparisons are*paired*\. We report 95% confidence intervals and two\-sided paired sign\-flip permutation tests \(20,000 permutations\) for each method against the baseline\. Where means coincide to all reported digits, this is a real consequence of the paired design, not rounding: on small test sets accuracy is quantized \(Breast Cancer:143143test points, steps of0\.70\.7points\), and the design\-unbiased estimator converges to the same optimum as the baseline, so per\-seed differences are exactly zero on most seeds — the permutation test then returnsp=1\.0p=1\.0by construction, which is the statistically honest way to state*parity*, not evidence of manipulation\. Real datasets are the scikit\-learn bundled sets \(Breast Cancern=569n\{=\}569, Digitsn=1797n\{=\}1797/10 classes, Winen=178n\{=\}178/3 classes, Diabetesn=442n\{=\}442\); the extreme\-imbalance task is a synthetic replica of a card\-fraud regime \(0\.17% positives,n=20,000n=20\{,\}000,d=15d=15, Gaussian classes with mean shift 1\.2\)\. Implementation:kabenav2\.x \(NumPy\); code and seeds released\[[3](https://arxiv.org/html/2607.05903#bib.bib3)\]\. Compute savings are counted as the realized fraction of skipped backward passes; they match Proposition[1](https://arxiv.org/html/2607.05903#Thmproposition1)to within 0\.7 points everywhere\.

### 7\.1Parity at reduced cost on real datasets

Table 1:Real\-data benchmarks\. The canonical estimator matches the full\-batch baseline within seed noise on every task while skipping 28–54% of backward passes — the behavior Lemma[2](https://arxiv.org/html/2607.05903#Thmlemma2)predicts\. The v2 bonus on multiclass tasks \(\+0\.35\+0\.35/\+0\.45\+0\.45\) is exclusive to the biased mode \(Section[6](https://arxiv.org/html/2607.05903#S6)\)\. Gradient boosting is reported as a negative\-scope control: per\-round tree fitting does not average per\-sample gradients across steps, and no benefit is expected or observed\.
### 7\.2Extreme imbalance: failure and resolution

Table 2:The quantified instantiation of Proposition[2](https://arxiv.org/html/2607.05903#Thmproposition2)and its resolution\. Learning\-rate sweeps \(0\.05–0\.5\) do not rescue the uncompensated variants\. At 5% imbalance all methods reach AUC≥0\.9965\\geq 0\.9965: the failure is specific to the small signal\-to\-bias regime \(Remark[5](https://arxiv.org/html/2607.05903#Thmremark5)\)\.
### 7\.3Matched\-budget comparison with selective and reweighting rivals

The reviewer\-facing question “why not compare against the established methods?” deserves numbers, with an honest caveat: the comparisons below are our CPU re\-implementations at*matched compute budget*\(identical retained\-sample count per epoch where the method saves compute\), not the official GPU codebases — this extends Limitation L4\. Focal Loss usesγ=2\\gamma=2on the full batch \(its design saves nothing\); OHEM\-style selection keeps the hardest examples up to the same budget as v3 with a plain average; global IS is the classical loss\-proportional proposal over*all*nnsamples with self\-normalized inverse\-probability correction at the same budget — i\.e\., v3 without the two\-stratum structure and without the defensive floor\.

Table 3:Matched\-budget rivals \(pp: paired sign\-flip permutation vs\. baseline, 20,000 permutations\)\. Three readings\. \(1\)*Only v3 combines saving with baseline\-level performance in both regimes\.*\(2\) OHEM saves the same compute and is statistically indistinguishable in the benign regime, then collapses at extreme imbalance \(AUC0\.450\.45\) — the empirical instantiation of Proposition[2](https://arxiv.org/html/2607.05903#Thmproposition2)on an external method, not only on our own earlier variants\. \(3\) Global loss\-proportional IS with correction is fine at extreme imbalance \(compensation works\) but*significantly degrades*the standard regime \(p=0\.002p=0\.002\): without the certainty stratum on majors and the defensive floor on weights, occasionally\-drawn tiny\-loss samples receive enormous inverse\-probability weights — the design differences of Section[2](https://arxiv.org/html/2607.05903#S2)\(c\) are load\-bearing, not stylistic\. Focal Loss performs well everywhere but saves nothing, as its design intends\.
### 7\.4Sensitivity analysis

We sweep the two user\-facing controls jointly,KK\-percentile∈\{20,40,60,70\}\\in\\\{20,40,60,70\\\}×\\timesN∈\{0\.1,0\.3,0\.5\}N\\in\\\{0\.1,0\.3,0\.5\\\}\(5 seeds, Breast Cancer\): test accuracy is flat across the entire grid — range0\.96640\.9664–0\.96780\.9678, a spread of0\.140\.14points, below the seed\-level CI width — while the realized saving tracks Proposition[1](https://arxiv.org/html/2607.05903#Thmproposition1)from10\.8%10\.8\\%\(K=20,N=0\.5K\{=\}20,N\{=\}0\.5\) to63\.4%63\.4\\%\(K=70,N=0\.1K\{=\}70,N\{=\}0\.1\)\. On the fraud replica the same grid \(percentiles40,7040,70\) keeps AUC≥0\.9961\\geq 0\.9961everywhere\. Together with theα\\alpha\-sweep \(Table[4](https://arxiv.org/html/2607.05903#S7.T4)caption: AUC0\.99870\.9987–0\.99960\.9996overα∈\[0,1\]\\alpha\\in\[0,1\], Digits unchanged\) and the learning\-rate sweep at extreme imbalance \(0\.050\.05–0\.50\.5, Table[2](https://arxiv.org/html/2607.05903#S7.T2)caption\), the picture is consistent:*performance is insensitive to the controls within broad ranges; the controls govern cost, not accuracy*— which is the intended division of labor\. Batch size \(our protocol is full\-batch\), network depth, and optimizer schedules beyond SGD are GPU\-scale questions folded into Limitation L4\.

### 7\.5Gradient moments, noise robustness, and optimizer interaction

Table 4:Moments and stress tests\. Row 1–2 quantify Remark[2](https://arxiv.org/html/2607.05903#Thmremark2)\(bias÷38\\div 38at variance×10\\times 10\)\. Row 3 shows the dissolution under v3 of the small\-\(n−k\)\(n\{\-\}k\)fragility of the biased bound \(majors carry weight 1 with no approximation\)\. Rows 4–5 locate the label\-noise phase change of the v2 mode\. Rows 6–7: under AMSGrad at extreme imbalance, even the compensated estimator degrades \(0\.806\) — the variance interacts with thev^t\\hat\{v\}\_\{t\}max\-lock — so we prescribe plain SGD in that regime; at standard imbalance the interaction is neutral\. Sensitivity toα\\alpha: fraud AUC ranges 0\.9987–0\.9996 overα∈\{0,0\.15,0\.3,0\.5,1\}\\alpha\\in\\\{0,0\.15,0\.3,0\.5,1\\\}; Digits accuracy is unchanged to four digits\.

## 8Limitations

Each limitation is quantified or proved; none is asserted informally\.\(L1\)Theorem[1](https://arxiv.org/html/2607.05903#Thmtheorem1)covers SGD, and the reason it does not extend to Adam, AdamW, or Lion is structural, not editorial: the proof’s descent step requires the update to be*linear*in the stochastic gradient, so that design\-unbiasedness ofg^t\\hat\{g\}\_\{t\}transfers to the update \(𝔼​\[θt\+1−θt∣θt\]=−η​∇F\+O​\(δ\)\\mathbb\{E\}\[\\theta\_\{t\+1\}\-\\theta\_\{t\}\\mid\\theta\_\{t\}\]=\-\\eta\\nabla F\+O\(\\delta\)\)\. Heavy\-ball momentum preserves this linearity and is a plausible extension; Adam/AdamW precondition by a nonlinear function of the gradient history \(and AdamW’s decoupled weight decay does not restore linearity\), while Lion appliessign​\(⋅\)\\mathrm\{sign\}\(\\cdot\), for which𝔼​\[sign​\(g^\)\]≠sign​\(𝔼​\[g^\]\)\\mathbb\{E\}\[\\mathrm\{sign\}\(\\hat\{g\}\)\]\\neq\\mathrm\{sign\}\(\\mathbb\{E\}\[\\hat\{g\}\]\)— an unbiased gradient buys nothing through a sign nonlinearity, and the variance inflation of L6 makes sign flips*more*frequent, not less\. Measured interaction: at extreme imbalance, AMSGrad \+ v3 reaches AUC 0\.806 versus 0\.9991 for SGD \+ v3 \(Table[4](https://arxiv.org/html/2607.05903#S7.T4)\); at 5% imbalance the interaction is neutral\. Prescription: SGD whenever Remark[5](https://arxiv.org/html/2607.05903#Thmremark5)’s signal\-to\-bias diagnosis is adverse\.\(L2\)BatchNorm statistics are computed overnnwhile gradients use\(n−k\)\+m\(n\-k\)\+mweighted samples; correct weighted\-BN is not implemented; LayerNorm/GroupNorm recommended\.\(L3\)KK\(percentile\) andNNare recalibrated by a per\-epoch heuristic; no adaptive\-optimality claim is made\.\(L4\)All results are CPU\-scale \(shallow models, NumPy\); no GPU CNN/Transformer benchmark is reported, and the fraud task is a synthetic replica rather than the original dataset\. The rival comparisons of Table[3](https://arxiv.org/html/2607.05903#S7.T3)are our matched\-budget CPU re\-implementations, not the official GPU codebases of the respective papers\. We report no simulated deep\-learning numbers\.\(L5\)The self\-normalized estimator carries biasδm=O​\(1/m\)\\delta\_\{m\}=O\(1/m\)\(Lemma[3](https://arxiv.org/html/2607.05903#Thmlemma3)\); measured 0\.004 atn=20,000n\{=\}20\{,\}000and 0\.037 in an adversarial micro\-regime; exact\-π\\pisampling \(α=1\\alpha=1\) removes the approximation of Assumption[1](https://arxiv.org/html/2607.05903#Thmassumption1)\(b\)\.\(L6\)Unbiasedness costs variance:×10\\times 10at matched configuration \(Lemma[4](https://arxiv.org/html/2607.05903#Thmlemma4), Remark[2](https://arxiv.org/html/2607.05903#Thmremark2)\), bounded byα\\alpha, with a measured side effect on the AMSGrad lock frequency \(57–60%\)\.\(L7\)v3 matches the baseline; it does not beat it in accuracy\. The small multiclass bonus belongs to the biased v2 mode, which carries the gated contraindications of Section[6](https://arxiv.org/html/2607.05903#S6)\(collapse at 40% noise; failure at 0\.17% imbalance\)\.\(L8\)The estimator is stochastic; reproducibility requires a seeded generator \(provided in the released implementation\)\.

## 9Reproducibility

All results in this paper are reproducible from the released librarykabena\-ml\[[3](https://arxiv.org/html/2607.05903#bib.bib3)\], available at[https://github\.com/Bonbhel/kabena\-ml](https://github.com/Bonbhel/kabena-ml)under the MIT license\. The repository provides three layers:\(i\) Implementation— the estimator of Definition[1](https://arxiv.org/html/2607.05903#Thmdefinition1)viakabena\_filter\(losses, K, N, strategy, rng\):strategy="v3"is the canonical compensated design \(defensive mixtureα\\alpha, inverse\-probability weights, seeded generator for exact reproducibility, cf\. Limitation L8\),strategy="v2"the gated regularized mode of Section[6](https://arxiv.org/html/2607.05903#S6)\(its feasibility constraintN≤12N\\leq\\tfrac\{1\}\{2\}is enforced with an explicit error\), andstrategy="v1"the historical variant of Appendix[C](https://arxiv.org/html/2607.05903#A3)for ablation replication\.\(ii\) Validation scripts— thevalidation/directory contains the runnable scripts behind each table: gradient\-moment measurements \(Table[4](https://arxiv.org/html/2607.05903#S7.T4), rows 1–3\), the real\-dataset benchmarks of Table[1](https://arxiv.org/html/2607.05903#S7.T1)on the bundled scikit\-learn sets, the extreme\-imbalance and stratification experiments of Table[2](https://arxiv.org/html/2607.05903#S7.T2)including the synthetic fraud\-replica generator \(0\.17% positives,n=20,000n=20\{,\}000, seeded\), the label\-noise stress test, the AMSGrad interaction runs, and theα\\alpha\-sensitivity sweep; each script prints the seed list and matches the reported means and standard deviations\.\(iii\) Tutorials— step\-by\-step notebooks for testing the estimator on a new dataset, reproducing the paper’s tables end\-to\-end, and experimenting beyond them \(varyingKK\-percentile,NN,α\\alpha, and the mode gate\), including the two\-line integration pattern for scikit\-learn\-style training loops\. Experiments require only NumPy and scikit\-learn \(no GPU\), consistent with the scope stated in Limitation L4; total runtime for the full table suite is under one CPU\-hour\.

## 10Conclusion

K\-ABENA v3 shows that threshold\-based selective backpropagation and unbiased gradient estimation are compatible: a defensive\-mixture design with inverse\-probability weighting preserves the exact compute saving\(1−N\)​k/n\(1\-N\)k/nwhile restoring the estimator guarantees that make SGD theory applicable \(Lemmas[2](https://arxiv.org/html/2607.05903#Thmlemma2)–[4](https://arxiv.org/html/2607.05903#Thmlemma4), Theorem[1](https://arxiv.org/html/2607.05903#Thmtheorem1)\)\. The negative result \(Proposition[2](https://arxiv.org/html/2607.05903#Thmproposition2)\) delimits precisely why the uncompensated shortcut — shared by OHEM, SBP, and our own earlier variants — cannot converge when selection bias dominates the signal, and the experiments quantify both the failure \(AUC 0\.53 at 0\.17% imbalance; accuracy 0\.386 at 40% noise for the biased mode\) and its resolution \(0\.9991; 0\.808\) at unchanged cost\. We release the implementation with both modes and the gating logic\[[3](https://arxiv.org/html/2607.05903#bib.bib3)\]; the companion volume\[[2](https://arxiv.org/html/2607.05903#bib.bib2)\]develops the pedagogical treatment and is available upon request from the author\.

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## Appendix AProof of Lemma[3](https://arxiv.org/html/2607.05903#Thmlemma3)\(self\-normalization bias\)

WriteT^=∑i∉ℳgi\+∑i∈𝒮πi−1​gi\\hat\{T\}=\\sum\_\{i\\notin\\mathcal\{M\}\}g\_\{i\}\+\\sum\_\{i\\in\\mathcal\{S\}\}\\pi\_\{i\}^\{\-1\}g\_\{i\}andW^=\(n−k\)\+∑i∈𝒮πi−1\\widehat\{W\}=\(n\-k\)\+\\sum\_\{i\\in\\mathcal\{S\}\}\\pi\_\{i\}^\{\-1\}, sog^=T^/W^\\hat\{g\}=\\hat\{T\}/\\widehat\{W\},𝔼​\[T^\]=n​∇F\\mathbb\{E\}\[\\hat\{T\}\]=n\\nabla F,𝔼​\[W^\]=n\\mathbb\{E\}\[\\widehat\{W\}\]=n\(both by the indicator computation of Lemma[2](https://arxiv.org/html/2607.05903#Thmlemma2)\)\. Using the identity

T^W^−𝔼​\[T^\]n=T^−𝔼​\[T^\]n\+T^​\(1W^−1n\)=T^−𝔼​\[T^\]n−g^​\(W^−n\)n,\\frac\{\\hat\{T\}\}\{\\widehat\{W\}\}\-\\frac\{\\mathbb\{E\}\[\\hat\{T\}\]\}\{n\}=\\frac\{\\hat\{T\}\-\\mathbb\{E\}\[\\hat\{T\}\]\}\{n\}\\;\+\\;\\hat\{T\}\\Bigl\(\\frac\{1\}\{\\widehat\{W\}\}\-\\frac\{1\}\{n\}\\Bigr\)=\\frac\{\\hat\{T\}\-\\mathbb\{E\}\[\\hat\{T\}\]\}\{n\}\\;\-\\;\\frac\{\\hat\{g\}\\,\(\\widehat\{W\}\-n\)\}\{n\},take expectations; the first term vanishes, giving the exact representation

𝔼​\[g^\]−∇F=−1n​𝔼​\[g^​\(W^−n\)\]=−1n​Cov​\(g^,W^\)\.\\mathbb\{E\}\[\\hat\{g\}\]\-\\nabla F\\;=\\;\-\\frac\{1\}\{n\}\\,\\mathbb\{E\}\\bigl\[\\hat\{g\}\\,\(\\widehat\{W\}\-n\)\\bigr\]\\;=\\;\-\\frac\{1\}\{n\}\\,\\mathrm\{Cov\}\\bigl\(\\hat\{g\},\\,\\widehat\{W\}\\bigr\)\.First\-order bound\.Sinceg^\\hat\{g\}is a convex combination of per\-sample gradients,‖g^‖≤GM\\left\\lVert\\hat\{g\}\\right\\rVert\\leq G\_\{M\}almost surely; Cauchy–Schwarz on the covariance gives

‖𝔼​\[g^\]−∇F‖≤GMn​Var​\(W^\)\.\\left\\lVert\\mathbb\{E\}\[\\hat\{g\}\]\-\\nabla F\\right\\rVert\\;\\leq\\;\\frac\{G\_\{M\}\}\{n\}\\,\\sqrt\{\\mathrm\{Var\}\(\\widehat\{W\}\)\}\.Under Assumption[1](https://arxiv.org/html/2607.05903#Thmassumption1)\(b\),Var​\(W^\)=∑i∈ℳ1−πiπi≤kα​m/k=k2α​m\\mathrm\{Var\}\(\\widehat\{W\}\)=\\sum\_\{i\\in\\mathcal\{M\}\}\\frac\{1\-\\pi\_\{i\}\}\{\\pi\_\{i\}\}\\leq\\frac\{k\}\{\\alpha m/k\}=\\frac\{k^\{2\}\}\{\\alpha m\}by Lemma[1](https://arxiv.org/html/2607.05903#Thmlemma1), hence‖𝔼​\[g^\]−∇F‖≤GM​k/\(n​α​m\)\\left\\lVert\\mathbb\{E\}\[\\hat\{g\}\]\-\\nabla F\\right\\rVert\\leq G\_\{M\}k/\(n\\sqrt\{\\alpha m\}\)— the first bound of Lemma[3](https://arxiv.org/html/2607.05903#Thmlemma3), of orderO​\(1/m\)O\(1/\\sqrt\{m\}\)\.

Second\-order rate\.The Cauchy–Schwarz step is loose becauseg^\\hat\{g\}concentrates: the standard ratio\-estimator expansion\[[11](https://arxiv.org/html/2607.05903#bib.bib11), Ch\. 5\],

𝔼​\[T^W^\]=𝔼​\[T^\]𝔼​\[W^\]−Cov​\(T^,W^\)n2\+𝔼​\[T^\]​Var​\(W^\)n3\+o​\(1/m\),\\mathbb\{E\}\\Bigl\[\\frac\{\\hat\{T\}\}\{\\widehat\{W\}\}\\Bigr\]=\\frac\{\\mathbb\{E\}\[\\hat\{T\}\]\}\{\\mathbb\{E\}\[\\widehat\{W\}\]\}\-\\frac\{\\mathrm\{Cov\}\(\\hat\{T\},\\widehat\{W\}\)\}\{n^\{2\}\}\+\\frac\{\\mathbb\{E\}\[\\hat\{T\}\]\\,\\mathrm\{Var\}\(\\widehat\{W\}\)\}\{n^\{3\}\}\+o\(1/m\),has both correction terms bounded byGM​Var​\(W^\)/n2G\_\{M\}\\mathrm\{Var\}\(\\widehat\{W\}\)/n^\{2\}in norm \(using‖Cov​\(T^,W^\)‖≤GM​Var​\(W^\)\\left\\lVert\\mathrm\{Cov\}\(\\hat\{T\},\\widehat\{W\}\)\\right\\rVert\\leq G\_\{M\}\\mathrm\{Var\}\(\\widehat\{W\}\), since each coordinate ofT^−𝔼​T^\\hat\{T\}\-\\mathbb\{E\}\\hat\{T\}is a weight\-linear combination of gradient coordinates bounded byGMG\_\{M\}times the corresponding fluctuation ofW^\\widehat\{W\}, and‖𝔼​\[T^\]‖/n≤GM\\left\\lVert\\mathbb\{E\}\[\\hat\{T\}\]\\right\\rVert/n\\leq G\_\{M\}\)\. Therefore

δm≤2​GM​Var​\(W^\)n2\+o​\(1/m\)≤2​GM​k2α​m​n2\+o​\(1/m\)=O​\(1/m\)\.\\delta\_\{m\}\\;\\leq\\;\\frac\{2\\,G\_\{M\}\\,\\mathrm\{Var\}\(\\widehat\{W\}\)\}\{n^\{2\}\}\+o\(1/m\)\\;\\leq\\;\\frac\{2\\,G\_\{M\}\\,k^\{2\}\}\{\\alpha\\,m\\,n^\{2\}\}\+o\(1/m\)\\;=\\;O\(1/m\)\.Atα=1\\alpha=1the design is SRSWOR withπi=m/k\\pi\_\{i\}=m/kexactly,Var​\(W^\)=k2m⋅k−mk⋅k−mk−1≤k​\(k−m\)m\\mathrm\{Var\}\(\\widehat\{W\}\)=\\frac\{k^\{2\}\}\{m\}\\cdot\\frac\{k\-m\}\{k\}\\cdot\\frac\{k\-m\}\{k\-1\}\\leq\\frac\{k\(k\-m\)\}\{m\}, givingδm≤2​GM​k​\(k−m\)/\(m​n2\)\+o​\(1/m\)\\delta\_\{m\}\\leq 2G\_\{M\}k\(k\-m\)/\(mn^\{2\}\)\+o\(1/m\)\. Both regimes areO​\(1/m\)O\(1/m\); the empirical values \(0\.004 at largenn; 0\.037 in the adversarial micro\-regime\) sit inside these bounds\. ∎

## Appendix BProof of Theorem[1](https://arxiv.org/html/2607.05903#Thmtheorem1)

Writebt=𝔼​\[g^t∣θt\]−∇F​\(θt\)b\_\{t\}=\\mathbb\{E\}\[\\hat\{g\}\_\{t\}\\mid\\theta\_\{t\}\]\-\\nabla F\(\\theta\_\{t\}\)\(‖bt‖≤δ\\left\\lVert b\_\{t\}\\right\\rVert\\leq\\delta\) andξt=g^t−𝔼​\[g^t∣θt\]\\xi\_\{t\}=\\hat\{g\}\_\{t\}\-\\mathbb\{E\}\[\\hat\{g\}\_\{t\}\\mid\\theta\_\{t\}\]\(𝔼​\[ξt∣θt\]=0\\mathbb\{E\}\[\\xi\_\{t\}\\mid\\theta\_\{t\}\]=0,𝔼​‖ξt‖2≤σ^2\\mathbb\{E\}\\left\\lVert\\xi\_\{t\}\\right\\rVert^\{2\}\\leq\\hat\{\\sigma\}^\{2\}\)\.LL\-smoothness gives

F​\(θt\+1\)≤F​\(θt\)−η​⟨∇F​\(θt\),g^t⟩\+L​η22​‖g^t‖2\.F\(\\theta\_\{t\+1\}\)\\leq F\(\\theta\_\{t\}\)\-\\eta\\langle\\nabla F\(\\theta\_\{t\}\),\\hat\{g\}\_\{t\}\\rangle\+\\frac\{L\\eta^\{2\}\}\{2\}\\left\\lVert\\hat\{g\}\_\{t\}\\right\\rVert^\{2\}\.Take conditional expectations; using𝔼​\[g^t∣θt\]=∇F​\(θt\)\+bt\\mathbb\{E\}\[\\hat\{g\}\_\{t\}\\mid\\theta\_\{t\}\]=\\nabla F\(\\theta\_\{t\}\)\+b\_\{t\}and𝔼​‖g^t‖2≤‖∇F​\(θt\)\+bt‖2\+σ^2≤2​‖∇F​\(θt\)‖2\+2​δ2\+σ^2\\mathbb\{E\}\\left\\lVert\\hat\{g\}\_\{t\}\\right\\rVert^\{2\}\\leq\\left\\lVert\\nabla F\(\\theta\_\{t\}\)\+b\_\{t\}\\right\\rVert^\{2\}\+\\hat\{\\sigma\}^\{2\}\\leq 2\\left\\lVert\\nabla F\(\\theta\_\{t\}\)\\right\\rVert^\{2\}\+2\\delta^\{2\}\+\\hat\{\\sigma\}^\{2\}:

𝔼​\[F​\(θt\+1\)\]≤𝔼​\[F​\(θt\)\]−η​𝔼​‖∇F​\(θt\)‖2−η​𝔼​⟨∇F​\(θt\),bt⟩\+L​η22​\(2​𝔼​‖∇F​\(θt\)‖2\+2​δ2\+σ^2\)\.\\mathbb\{E\}\[F\(\\theta\_\{t\+1\}\)\]\\leq\\mathbb\{E\}\[F\(\\theta\_\{t\}\)\]\-\\eta\\,\\mathbb\{E\}\\left\\lVert\\nabla F\(\\theta\_\{t\}\)\\right\\rVert^\{2\}\-\\eta\\,\\mathbb\{E\}\\langle\\nabla F\(\\theta\_\{t\}\),b\_\{t\}\\rangle\+\\frac\{L\\eta^\{2\}\}\{2\}\\bigl\(2\\mathbb\{E\}\\left\\lVert\\nabla F\(\\theta\_\{t\}\)\\right\\rVert^\{2\}\+2\\delta^\{2\}\+\\hat\{\\sigma\}^\{2\}\\bigr\)\.Bound−𝔼​⟨∇F,bt⟩≤δ​𝔼​‖∇F​\(θt\)‖≤δ​G∞\-\\mathbb\{E\}\\langle\\nabla F,b\_\{t\}\\rangle\\leq\\delta\\,\\mathbb\{E\}\\left\\lVert\\nabla F\(\\theta\_\{t\}\)\\right\\rVert\\leq\\delta G\_\{\\infty\}\. Withη≤12​L\\eta\\leq\\frac\{1\}\{2L\},L​η2≤η/2L\\eta^\{2\}\\leq\\eta/2, so the𝔼​‖∇F‖2\\mathbb\{E\}\\left\\lVert\\nabla F\\right\\rVert^\{2\}terms combine into−η2​𝔼​‖∇F​\(θt\)‖2\-\\frac\{\\eta\}\{2\}\\,\\mathbb\{E\}\\left\\lVert\\nabla F\(\\theta\_\{t\}\)\\right\\rVert^\{2\}at worst\. Rearranging and summing overt=0,…,T−1t=0,\\dots,T\-1, then dividing byη​T/2\\eta T/2:

mint⁡𝔼​‖∇F​\(θt\)‖2≤1T​∑t𝔼​‖∇F​\(θt\)‖2≤2​\(F​\(θ0\)−F∗\)η​T\+L​η​\(σ^2\+2​δ2\)\+2​δ​G∞\.\\min\_\{t\}\\mathbb\{E\}\\left\\lVert\\nabla F\(\\theta\_\{t\}\)\\right\\rVert^\{2\}\\leq\\frac\{1\}\{T\}\\sum\_\{t\}\\mathbb\{E\}\\left\\lVert\\nabla F\(\\theta\_\{t\}\)\\right\\rVert^\{2\}\\leq\\frac\{2\\,\(F\(\\theta\_\{0\}\)\-F^\{\*\}\)\}\{\\eta\\,T\}\+L\\eta\\,\(\\hat\{\\sigma\}^\{2\}\+2\\delta^\{2\}\)\+2\\,\\delta\\,G\_\{\\infty\}\.Substitutingη=min⁡\{12​L,cT\}\\eta=\\min\\\{\\tfrac\{1\}\{2L\},\\tfrac\{c\}\{\\sqrt\{T\}\}\\\}yields the statement \(absorbing constants; the\+δ2\+\\delta^\{2\}of the main text majorizes2​δ2⋅L​c/T2\\delta^\{2\}\\cdot Lc/\\sqrt\{T\}forTTlarge\)\. Withδ=0\\delta=0\(HT form\) the bound is the classical rate ofGhadimi and Lan \[[4](https://arxiv.org/html/2607.05903#bib.bib4)\]\. ∎

## Appendix CAblation history

The canonical design is the outcome of a systematic exploration; all rejected variants are documented to preempt re\-exploration\. Reference: v1,pi∝\(K−ℓi\)p\_\{i\}\\propto\(K\-\\ell\_\{i\}\)over all ofℳ\\mathcal\{M\}, no reweighting\.H1\(pi∝ℓip\_\{i\}\\propto\\ell\_\{i\}over all ofℳ\\mathcal\{M\}, no reweighting\): bias\+18\+18to\+26%\+26\\%, variance\+29%\+29\\%vs\. v1 — rejected; note that this same proposal distribution becomes near\-optimal*once inverse\-probability weighting is added*, which is precisely the v3 construction\.H2\(pi∝\|ℓi−c\|p\_\{i\}\\propto\|\\ell\_\{i\}\-c\|, symmetric U\): interpolates v1 and H1, bias\+6\+6to\+12%\+12\\%— rejected\.H4\(upper half only,pi∝K−ℓip\_\{i\}\\propto K\-\\ell\_\{i\}on𝒰\\mathcal\{U\}\): bias\+60\+60to\+80%\+80\\%— rejected; both terms of Lemma[5](https://arxiv.org/html/2607.05903#Thmlemma5)then share a sign\.v2\(lower half only,pi∝ℓip\_\{i\}\\propto\\ell\_\{i\}onℒ\\mathcal\{L\}\): bias−4\-4to−10%\-10\\%and variance−25\-25to−55%\-55\\%vs\. v1, weight ratio bounded \(r≈1\.4r\\approx 1\.4vs\.r≈3×104r\\approx 3\\times 10^\{4\}for v1; 0/40 vs\. 14/40 violations of the sampling approximation at 1% tolerance\) — adopted as an interim canonical form before the end\-to\-end extreme\-imbalance and label\-noise tests of Tables[2](https://arxiv.org/html/2607.05903#S7.T2)–[4](https://arxiv.org/html/2607.05903#S7.T4)revealed the shared failure of all uncompensated variants and motivated the compensated design\. Equivalence sweeps: at matched bias, v2 allows no additional retention saving atN=0\.3N=0\.3and\+3\.3%\+3\.3\\%atN=0\.4N=0\.4relative to v1 — the value of the uncompensated refinements was always estimator quality at fixed cost, never additional cost reduction, a conclusion that transfers to v3\.