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the ith stage to maximize the probability of {√n(¯Zn−µ0)< c}when µ=µ0(i.e., maximize the actual type I error rate). Until the final stage n, the experimenter follows a sequential strategy θ= (σ1, . . . , σ n), where σi=σindicates Zi=W1iand σi=σindicates Zi=W2i. That is, Zi=( W1i,ifσi=σ, W2i,ifσi=σ, where the {σi}n i=1a... | https://arxiv.org/abs/2502.11038v1 |
when the data is actually generated under the null hypothesis. Theorem B.2. (Optimal manipulation strategy θ∗) For a given constant c, we can construct the asymptotically optimal strategy θ∗to maximize the probability of {√n|¯Zn−µ0|> c}when µ=µ0(i.e., attain the supremum in (B.7) ) as follows: for i≥2, σi=σ, if|ξi−1|√n... | https://arxiv.org/abs/2502.11038v1 |
Consistency of heritability estimation from summary statistics in high-dimensional linear models David Azriel, Samuel Davenport, Armin Schwartzman February 27, 2025 Abstract In Genome-Wide Association Studies (GWAS), heritability is defined as the fraction of variance of an outcome explained by a large number of geneti... | https://arxiv.org/abs/2502.11144v2 |
the choice of regularization may affect the results. Second, more importantly from a practical point of view, public datasets from genetic studies do not reveal the original values of the outcome and the genetic predictors due to privacy regulations. Instead, they contain so-called “summary statistics”, measuring the u... | https://arxiv.org/abs/2502.11144v2 |
but neither with any pre- scribed distribution. Under this model, the heritability can be defined in two different ways depending on whether the expectations are conditional on the coefficients or not. We show that both definitions coincide in the limit of increasing number of predictors if two important conditions are... | https://arxiv.org/abs/2502.11144v2 |
that the LDSC regression estimator with free intercept does not account for population stratification as originally claimed but it remains with a large bias. Table 1 summarizes the various forms of the estimators considered in this study. Both the GWASH and LDSC regression estimators are studied theoretically and via s... | https://arxiv.org/abs/2502.11144v2 |
x im) for n independent subjects i= 1, . . . , n . In GWAS, the predictors are SNP allele counts taking values 0, 1, 2, but the theory is more general. The poly-additive linear model, called polygenic linear model in genetics (Fisher, 1918; Lynch and Walsh, 1998), in row or vector form respectively, is yi=⃗xiβ+εi, i = ... | https://arxiv.org/abs/2502.11144v2 |
i)/σ4 ε, respectively. Their excess kurtosis with respect to the Gaussian distribution are then Kurt( βi)−3 and Kurt( εi)−3. 2.2 The essential conditions In general, h2in (3) is not equal to the expectation of h2 β(2). However, the two quantities are close in high dimensions. Our first result shows that the two FVE def... | https://arxiv.org/abs/2502.11144v2 |
basic estimators 3.1 Motivation for the estimators To motivate the estimators, recall that genetics data is often not publicly available. Instead, the data is reduced to so-called “summary statistics”, specifically correlation scores (or χ2statistics) and LD (Linkage Disequilibrium) scores, defined as follows. Letuj:=1... | https://arxiv.org/abs/2502.11144v2 |
addition, the summary statistics are typically computed from standardized predictors. Thus d2 jis replaced by 1, corresponding to the practice of standardizing the columns of X(andXR) in the sample. With these two modifications, let ˆℓj,Rdenote the bias corrected LD scores from the reference dataset: ˆℓj,R:=1 n2xT j,RX... | https://arxiv.org/abs/2502.11144v2 |
One difficulty in this analysis is that the denominators of the estimators are random variables that could get arbitrarily close to 0, which could produce unbounded moments. In fact, because the entries of Xcontain zeros, it is possible that a denominator could be exactly equal to 0 in real data, even if extremely unli... | https://arxiv.org/abs/2502.11144v2 |
dependence, this term is of order 1 /m; otherwise, it may be large. In order to present the consistency results more formally, we consider first the case where the predictors ⃗xiare Gaussian and then extend the results to the non-Gaussian case. 11 4.1.1 Gaussian predictors Suppose that ⃗xi∼N(0,Σ). We consider the follo... | https://arxiv.org/abs/2502.11144v2 |
the second moments are ‘weak’ in the sense that the entries Σj,kare mostly small, but the absolute fourth moment |Cov(Xi,jXi,p1,Xi,kXi,p2)|may be large. It is easy to verify that condition M 1(b) is satisfied when ⃗xiis a linear combination of iid variables with finite fourth moment. Condition M 1(b) also holds under t... | https://arxiv.org/abs/2502.11144v2 |
is bounded. 14 Compared to M 1, M2(a) requires moments of order 16 to be bounded. This is because in DLDSC there are higher moments of Xthan in DGWASH . The main requirement, namely, the fourth moment condition M2(b) is the same. The parallel result for Theorem 3 is now given. Theorem 5. Consider Model (1)and assume th... | https://arxiv.org/abs/2502.11144v2 |
) =1 m[Kurt( βi)−3]h m [r(m)]2Pm j=1ℓ2 j+O(1)i +O(1)tr(Σ4) [r(m)]2. Theorem 6 indicates that the denominator of ¯h2 GWASH converges in L2. The rate of convergence of the numerator depends on the convergence rate r(m) of the second spectral moment of the covariance Σ. Corollary 1. For¯h2 GWASH to be consistent (in L2) i... | https://arxiv.org/abs/2502.11144v2 |
in the numerator and denominator of (10) by weighted averages, using as weights the inverse of the variances of the terms being averaged. That is, ˆh2 GWASH −W:=1 mPm j=1wj u2 j−1 1 mPm j=1wjn m¯ℓj,R, (20) with weights wj= 1 +ˆh2n m¯ℓj,R−2 . (21) The justification for these weights is the same as in Bulik-Sullivan ... | https://arxiv.org/abs/2502.11144v2 |
are ˜ℓj,R:=1 n2˜xT j,R˜XR˜XT R˜xj,R−m n=1 n2·1 d2 j,RxT j,RXRD−2 RXT Rxj,R−m n. In order to analyze the bias of the estimators we consider E(˜ u2 j|X) and compare it to E( u2 j|X) as given in (6). If the difference is not negligible, then the resulting estimators are biased. Write ˜u2 j=Nuj Duj, N uj:=1 n(xT jy)2, D uj... | https://arxiv.org/abs/2502.11144v2 |
1are satisfied, then the estimators are asymptotically unbiased. 7 Sources of bias: Population stratification The LDSC regression estimator considered above, both in its unweighted and weighted versions, estimated the slope of the regression of the squared correlation scores on the LD scores without including an interc... | https://arxiv.org/abs/2502.11144v2 |
is strong dependence under population stratification and hence we do not expect the above estimators (basic, weighted, standardized) to have asymptotically vanishing variance. Here we show that population stratification also produces bias. To see this, consider the correlation scores uj=xT jy/√n. A parallel expression ... | https://arxiv.org/abs/2502.11144v2 |
σ2 ξ/Cfdiverges to ∞when Cf→0, there is a discontinuity point at Cf= 0. Importantly, Theorem 9 implies that if the |fj|’s are small, the bias can be very significant. In particular, the claim by Bulik-Sullivan et al. (2015) that the free intercept offers a correction to issues of population stratification in the data, ... | https://arxiv.org/abs/2502.11144v2 |
the distribution of βto be t-distributed with degrees of freedom taking values 2 ,2.3,2.5,2.8,3 and 5. For the t-distribution, the kurtosis is infinite 24 for degrees of freedom ≤4, which violates the BKE condition. The SE and bias of ˆh2 GWASH andˆh2 LDSC are shown in Figures A2 and A8, respectively. The results show ... | https://arxiv.org/abs/2502.11144v2 |
a bias-variance trade-off. However ˆh2 GWASH andˆh2 GWASH −Ware slightly less biased than ˆh2 LDSC andˆh2 LDSC −Win this setting. As in previous sections, standardizing the data leads to a decrease in the SE of the estimator. 8.3 Bias caused by population stratification In Section 7 it was claimed that when population ... | https://arxiv.org/abs/2502.11144v2 |
and GWASH are biased. To deal with population stratification, Bulik-Sullivan et al. (2015) suggested using the LDSC regression estimator with free intercept. However, as shown in theory in Theorem 9 and in practice in Section 8.3, the free intercept estimator does not eliminate the bias in this scenario. Hence, we reco... | https://arxiv.org/abs/2502.11144v2 |
al. (2015). Partitioning heritability by functional annotation using genome-wide association summary statistics. Nature genetics 47 (11), 1228–1235. Fisher, R. A. (1918). The correlation between relatives on the supposition of Mendelian inheritance. Transactions of the Royal Society of Edinburgh 52 , 399–433. Gupta, A.... | https://arxiv.org/abs/2502.11144v2 |
the estimators under population stratifica- tion. Figure A5 illustrates the performance of ˆh2 LDSC for Gaussian predictors under weak and strong correlation. Figure A6 illustrates the performance of ˆh2 GWASH for binomial predictors under weak and strong correlation. Figure A7 illustrates the performance of ˆh2 LDSC f... | https://arxiv.org/abs/2502.11144v2 |
for a range of distributions for β. The results can be interpreted similarly to those in Figure A2. Simulation SE was at most 0.005 for the standardized data. 38 B Further theoretical results B.1 Sufficient conditions for the weighted estimators To analyze the effect of weighting and allow for different weighting schem... | https://arxiv.org/abs/2502.11144v2 |
and let ξ= (ξ1, . . . , ξ d)Tbe a vector of iid random variables with mean zero and finite fourth moment. Let M2andM4denote the second and fourth moment of ξi. Then, Cov(ξTAξ,ξTBξ) = (diag( A))Tdiag(B)(M4−3M2 2) + 2tr( AB)M2 2. Proof of Lemma 1: We have that Cov(ξTAξ,ξTBξ) =X i,i′,h,h′Cov(Aii′ξiξi′, Bhh′ξhξh′) = ... | https://arxiv.org/abs/2502.11144v2 |
nd2 j +1 mmX j=1E(d2 j−1) = h2n m1 mmX j=1E ˆℓj−m n because E(d2 j) = 1. And by the computation in Part (i) we have that E( NGWASH )→h2µ2/λ. Consider now Var( NGWASH ). We have the variance decomposition Var(NGWASH ) = Var( E(NGWASH |X)) + E(Var( NGWASH |X)). 42 By similar arguments as in Part (i), Var( E(NGWASH |X)... | https://arxiv.org/abs/2502.11144v2 |
are 4 n(n−1) such cases. 4. For the case i1=i2=i3=i4we just bound the covariance by a constant. It follows that Cov((Sj,p1)2,(Sk,p2)2) =4 +O(1/n) n(Σj,kΣp1,p2+Σj,p2Σk,p1)Σj,p1Σk,p2 +2 +O(1/n) n2(Σj,kΣp1,p2+Σj,p2Σk,p1)2 +8 +O(1/n) n2(Σk,p1Σp1,p2Σk,p2+Σj,kΣj,p2Σk,p2) +O(1/n3). Therefore, Cov( ˆℓj,ˆℓk) = Cov X p(Sj,p)2,X ... | https://arxiv.org/abs/2502.11144v2 |
that Cov((Sj,p1−Σj,p1)2,(Sk,p2−Σk,p2)2)≤1 n2C(¯Σj,k¯Σp1,p2+¯Σj,p2¯Σk,p1)2≤2 n2C[(¯Σj,k)2+¯Σ2 j,p2],(39) where the last inequality is true because for every a, b, we have that ( a+b)2≤2(a2+b2) and ¯Σp1,p2≤1. Thus, (37), (38) and (39) imply that Var 1 mmX j=1ˆℓj =1 m2X j,kCov( ˆℓj,ˆℓk) =4 m2X j,k,p 1,p2Σj,p1Σk,p2Cov(... | https://arxiv.org/abs/2502.11144v2 |
1,p2 (¯Σp1,p2+¯Σj,p1¯Σj,p2)(¯Σp1,p2+¯Σk,p1¯Σk,p2) + (¯Σj,k¯Σp1,p2+¯Σj,p2¯Σk,p1)2 (44) +O(1/m) +C m2n2X j,k,p 1,p2(¯Σj,p2+¯Σk,p2+¯Σj,p1+¯Σk,p1) (45) +C m2nX j,k,p 1,p2¯Σj,p1¯Σj,p2(¯Σp1,p2+¯Σk,p1¯Σk,p2) +¯Σj,p1¯Σk,p2(¯Σj,k¯Σp1,p2+¯Σj,p2¯Σk,p1) (46) +C m2nX j,k,p 1,p2¯Σk,p1¯Σj,p2(¯Σj,k¯Σp1,p2+¯Σj,p1¯Σk,p2) +¯Σk,p1¯Σk... | https://arxiv.org/abs/2502.11144v2 |
imply that NLDSC→h2µ∗ 2/λinL2asn, m→ ∞ . Indeed, (50) is true because E 1 mmX j=1ℓj u2 j−1 =1 mmX j=1ℓj E[E(u2 j|X)]−1 =1 mmX j=1ℓj Eh h2n mˆℓj−d2 j +d2 ji −1 =1 mmX j=1ℓjh2En mˆℓj−1 =1 mmX j=1ℓjh2n mE ˆℓj−m n =h2n m1 mmX j=1ℓ2 j→h2µ∗ 2/λ, where the first and second equality in the second line are due t... | https://arxiv.org/abs/2502.11144v2 |
j,p1, where we used the inequality in M 2(b). Notice that the set of indices j, p1, p2where the inequality does not hold (which is of order O(m3)) is negligible because we divide by n4. Therefore, we assume without loss of generality that the inequality holds for all j, p1, p2. Case (a) occurs for 4 n(n−1)(n−2) indices... | https://arxiv.org/abs/2502.11144v2 |
i1,jX2 i1,p1 X2 i2,jX2 i2,p2−E X2 i2,jX2 i2,p2 X2 i3,jX2 i3,p3−E X2 i3,jX2 i3,p3 X2 i4,jX2 i4,p4−E X2 i4,jX2 i4,p4 i =O(1/n6), where the last equality is true because the expectation is different from zero when i1=i2,i3=i4(or symmetric cases), and there are order of n2such cases. Therefore, E mX p=1Wj,p... | https://arxiv.org/abs/2502.11144v2 |
m21 r(m)mX j=1E(ˆℓ2 j) + [Kurt( εi)−3](1−h2)2 nm2 [r(m)]21 nnX i=1E"∥⃗xi∥2 m2# +2 [r(m)]2tr E h2n mS2+ (1−h2)S2 .(55) Consider the first term in (55). We have 1 r(m)mX j=1E(ˆℓ2 j) =1 r(m)mX j=1n [E(ˆℓj)]2+ Var( ˆℓj)o . By (5), E( ˆℓj) =ℓj(1 + 1 /n) +m n. Therefore, 1 r(m)mX j=1[E(ˆℓj)]2=1 r(m)mX j=1ℓ2 j+O(m) r(m)... | https://arxiv.org/abs/2502.11144v2 |
term (II). We have by Lemma 7 below, (II) =1 r(m)mX j=1Cov Nuj, Duj|X E2 Duj|X =1 (1−h2+h2tr(S)/m)21 r(m)mX j=1n(s2 j)Td2[E(β4 i)−3h4/m2] d2 j+2nsT jSsjh4/m2 d2 j +1 n2(x2 j)T1[E(ε4 i)−3(1−h2)2] d2 j+2 n2d2 j(1−h2)2 d2 j+4(1−h2)h2 mˆℓj d2 j =1 (1−h2+h2tr(S)/m)2 (IIa) + (IIb) + (IIc) + (IId) + (IIe) . (59) 60 Be... | https://arxiv.org/abs/2502.11144v2 |
n[E(ε4 i)−3(1−h2)2] +2 n(1−h2)2+ 4(1−h2)h2 nmtr(S) 62 By similar arguments as before we have that 1 (1−h2+h2tr(S)/m)3= 1 + op(1) and the term 1 n[E(ε4 i)−3(1−h2)2] +2 n(1−h2)2+ 4(1−h2)h2 nmtr(S) is negligible. It follows that (III) =1 r(m)mX j=1Var Duj|X E Nuj|X E3 Duj|X ={1 +op(1)}1 r(m)mX j=1h2 n mˆℓj−d2 j +... | https://arxiv.org/abs/2502.11144v2 |
∪ {W2> ε}··· ∪ { Wm> ε})→0asm→ ∞ . Then,Pm j=1VjWjconverges in probability to zero as m→ ∞ . Proof of Lemma 8: Define an event E:=n {W1≤ε} ∩ {W2≤ε}··· ∩ { Wm≤ε} ∩ {mX j=1Vj≤C}o ; by the assumptions, P(E)→1. Write mX j=1VjWj=mX j=1VjWjI(E) +mX j=1VjWjI(Ec), where I(·) is the indicator function. When EoccursPm j=1VjWj≤Cε... | https://arxiv.org/abs/2502.11144v2 |
lemma. C.10 Proof of Corollary 2 The bias term is asymptotically equal to 1 m[Kurt( βi)−3]h4(h2−1) +2h4 m2µ2/λ h2µ2/λ+m r(m) tr(S2)−n r(m)tr(S3) . The first term is negative when h2<1 and it converges to zero iff the BKE condition holds. For the second term, notice that sincePm j=1ˆℓj= tr(S2), (60) implies that n m... | https://arxiv.org/abs/2502.11144v2 |
˜r(m)mX j=1[¯Σ3 j,j+ℓ2 j+1 ntr(Σ2)] 1/2 ,(69) where the last inequality is due to Cauchy-Schwarz. We assumed that1 ˜r(m)Pm j=1¯Σ3 j,j=tr(¯Σ3) ˜r(m)is bounded. The assumption that1 ˜r(m)Pm j=1ℓ2 j→µ∗ 2implies that1 ˜r(m)Pm j=1ℓ2 jis bounded because the difference between ℓjandℓjis1 nPm p=1[Var(Xi,jXi,p)−1] (see (14)),... | https://arxiv.org/abs/2502.11144v2 |
LD scores. We have that ℓj=mX p=1Σ2 j,p=1 1 +f2 jmX p=1[(Σ0)j,p+fjfp]2 1 +f2p=1 1 +f2 jmX p=1[(Σ0)j,p]2+ 2(Σ0)j,pfjfp+f2 jf2 p 1 +f2p. Now, mX p=1[(Σ0)j,p]2 1 +f2p≤mX p=1[(Σ0)j,p]2=ℓj,0≤C, where in the last inequality we used the assumption that ℓj,0is bounded. The mixed term is 2fjmX p=1(Σ0)j,pfp 1 +f2p≤2fjvuutmX p=1[... | https://arxiv.org/abs/2502.11144v2 |
j) = 1 as we assumed that the data is standardized. We conclude that term IIconverges to zero in probability. Consider now term (III) in (74). Again we rewrite it (III) =1 mPm j=1ˆℓj,R m−¯ˆℓR m 1 mn(xT jξ)2 1 mPm j=1n mˆℓj,R m−¯ˆℓR m2. (77) We now consider approximating the term1 mn(xT jξ)2by its expectation, which... | https://arxiv.org/abs/2502.11144v2 |
that 1 mmX j=1ˆℓj,R m1 mn(xT jξ)2L1−→σ2 ξCf λlim m→∞1 mmX j=1f4 j (1 +f2 j)2 and that 1 mmX j=1 ˆℓj,R m!2 L1−→C2 f λlim m→∞1 mmX j=1f4 j (1 +f2 j)2. Therefore, term Cconverges in probability toσ2 ξ Cf. It follows that ˆh2 LDSC has the same asymptotic bias as ˆh2 LDSC −free. The proof that ˆh2 GWASH has the same asympto... | https://arxiv.org/abs/2502.11144v2 |
A statistical theory of overfitting for imbalanced classification Jingyang Lyu∗Kangjie Zhou†Yiqiao Zhong∗ February 18, 2025 Abstract Classification with imbalanced data is a common challenge in data analysis, where certain classes (minority classes) account for a small fraction of the training data compared with other ... | https://arxiv.org/abs/2502.11323v1 |
kz2326@columbia.edu 1arXiv:2502.11323v1 [math.ST] 17 Feb 2025 4 Precise asymptotics of empirical logit distribution 18 4.1 Separable data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Non-separable data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5... | https://arxiv.org/abs/2502.11323v1 |
well understood for standard classification methods. Generally, we expect estimation consistency, where the estimated parameter vector bβis close to the target β, and a small generalization gap where the training error is close to the test error. However, for high-dimensional problems where dis typically comparable to ... | https://arxiv.org/abs/2502.11323v1 |
in empirical studies, such as adjusting decision boundaries, reweighting loss functions, subsampling the majority classes, and oversampling the minority classes [3, 22], among others. Overfitting in high dimensions further compounds the challenges of imbalanced classification. Empirical studies have shown that deep lea... | https://arxiv.org/abs/2502.11323v1 |
distribution Understanding why test accuracy drops more for the minority class requires a more refined char- acterization of overfitting. To this end, we study the empirical distribution of the logits bf(xi) = ⟨xi,bβ⟩+bβ0,i∈[n] on the training set. Let (x1,y1),..., (xn,yn)i.i.d.∼Px,ybe the training data, and ( xtest,yt... | https://arxiv.org/abs/2502.11323v1 |
+1 and Law bf(xtest)|ytest=−1 . Note that the randomness in TLDs is taken over both the classifier bfand the test point (xtest,ytest), so they are deterministic probability measures. When bκn>0, the training set is linearly separable, and the training accuracy of bfis 100%. While linear separability is common in high... | https://arxiv.org/abs/2502.11323v1 |
data . We use IMDb movie review dataset [40] to perform binary sentiment classification. The BERT base model (110M) [41] is applied to extract the features of dimension d= 768. We subsample an imbalanced training set. Negative reviews belong to the minority class, with imbalance ratio π= 0.02 and total sample size n= 6... | https://arxiv.org/abs/2502.11323v1 |
p 1−ρ2ξ= max{κ,ρ∥µ∥+G+Yβ0}−(ρ∥µ∥+G+Yβ0) = (κ−ρ∥µ∥−G−Yβ0)+, wherea+:= max{0,a}. (7) Thus, we can view ξas a map that pushes the overlapping probability masses in the TLD to the margin boundaries in the ELD. It is the cause for the discrepancy between the ELD and the TLD. •More truncation for minority class. Due to imbal... | https://arxiv.org/abs/2502.11323v1 |
Errb=1 2Err++1 2Err−. (10) Empirical phenomenon. For the proportional regime, we generate imbalanced 2-GMM based on Eq. (9) with sample size n= 100 and dimension d= 200, under different settings of ∥µ∥2and 10 τ/hatwideκ /hatwideκ /angbracketleftx,/hatwideβ/angbracketright+/hatwideβ0=τ/hatwideκ /angbracketleftx,/hatwide... | https://arxiv.org/abs/2502.11323v1 |
SVM Eq. (8) for each configuration. Figure 6 shows that there are three phases in terms of the majority/minority errors. In particular, the margin rebalancing is crucial for one phase with moderate signal strength. Theoretical foundation. For the proportional regime, denote Err∗ +, Err∗ −, Err∗ bas the limits of Err+, ... | https://arxiv.org/abs/2502.11323v1 |
high dimensions, which is difficult. 13 0.0 0.1 0.2 0.3 0.4 0.5 π0.000.050.100.150.20CalErr Calibration Error 0.0 0.1 0.2 0.3 0.4 0.5 π0.100.120.140.160.180.200.22MSE Mean Squared Error 0.0 0.1 0.2 0.3 0.4 0.5 π0.0750.1000.1250.1500.1750.200ConfErr Confidence Estimation Error /bardblµ/bardbl2= 1.25 /bardblµ/bardbl2= 1.5... | https://arxiv.org/abs/2502.11323v1 |
δ. Table 2 summarizes the monotone behavior of test errors and miscalibration metrics that we establish in this paper. Err∗ +,Err∗ −,Err∗ b CalErr∗MSE∗ConfErr∗ imbalance ratio π↑↓(Prop. 5.4) ↓(Prop. 6.1)↓(Claim 6.2) signal strength∥µ∥2↑↓(Prop. 5.4)↓(Claim 6.2)↓(Prop. 6.1) aspect ratio n/d→δ↑↓(Prop. 5.4)↓(Claim 6.2)↓(Pr... | https://arxiv.org/abs/2502.11323v1 |
(2b). Proposition 3.1. (a) When data is linearly separable, Eq. (8)has a unique solution. (b) Let (bβ(τ),bβ0(τ),bκ(τ))be an optimal solution to Eq. (8)under hyperparameter τ. Then bβ(τ) =bβ(1),bβ0(τ) =bβ0(1) +τ−1 τ+ 1bκ(1),bκ(τ) =2 τ+ 1bκ(1). (17) 16 Remark 3.1. As shown in Figure 3, there is a clear geometric interpre... | https://arxiv.org/abs/2502.11323v1 |
normal distribution. Let Law(X) denote the distribution of random variable (or vector)X. We write X⊥ ⊥YifXandYare independent random variables. We useO(·) ando(·) for the standard big- Oand small-onotations. For real sequences ( an)n≥1, (bn)n≥1, we writean≲bnorbn≳anifan=O(bn), andan≍bnifan≲bnandan≳bn. We also writean≪b... | https://arxiv.org/abs/2502.11323v1 |
Y,Ymax{s(Y)κ∗,ρ∗∥µ∥+G+Yβ∗ 0} , νtest ∗:=Law Y,Y(ρ∗∥µ∥+G+Yβ∗ 0) . which we will prove to be the limiting ELD and TLD respectively. Theorem 4.1 (Separable data) .Assumen,d→∞ withn/d→δ∈(0,∞). Fixτ∈(0,∞). (a)(Phase transition) With probability tending to one, the data is linearly separable if δ<δ∗(0) and is not linearly... | https://arxiv.org/abs/2502.11323v1 |
(20). Let ρ∗,R∗,β∗ 0,ξ∗be a solution to the variational problem minimize ρ∈[−1,1],R≥0,β0∈R,ξ∈L2E" ℓρ∥µ∥2R+RG+β0Y+Rp 1−ρ2ξ s(Y)# , subject to E ξ2 ≤1/δ.(26) where (Y,G)∼Py×N(0,1). We define ν∗:=Law Y,Ys (Y)proxλ∗ℓ s(Y)ρ∗∥µ∥2R∗+R∗G+β∗ 0Y s(Y) . νtest ∗:=Law Y,Y(R∗ρ∗∥µ∥+R∗G+Yβ∗ 0) , aiming to show they are the l... | https://arxiv.org/abs/2502.11323v1 |
summarized in the following result. Proposition 5.2. Err∗ +is decreasing in τ∈(0,∞), and Err∗ −is increasing in τ∈(0,∞). Choosing the optimal τ.A natural idea for margin rebalancing is to choose τsuch that the balanced error Err∗ bis minimized. Proposition 5.3 (Optimalτ).Letτoptbe the optimal margin ratio τdefined in P... | https://arxiv.org/abs/2502.11323v1 |
2-subGMM. Suppose that a−c <1. A margin-rebalanced SVM is trained, with test errors calculated according to Eq. (6). Then asd→∞ , the conclusions of the three phases in Theorem 1.3 still hold. 6 Consequences for confidence estimation and calibration Recall the definition of confidence of the max-margin classifier bp(x)... | https://arxiv.org/abs/2502.11323v1 |
25 (a) Let (ρ∗,β∗ 0)be defined as per Theorem 4.1, and (Y,G)∼Py×N(0,1). Denote MSE∗:=Eh σ −ρ∗∥µ∥2−β∗ 0Y+G2i , mMSE∗= MSE∗−π(1−π), CalErr∗:=E" σ 2ρ∗∥µ∥2(ρ∗∥µ∥2Y+G) + logπ 1−π −σ ρ∗∥µ∥2Y+G+β∗ 02# , V∗ y|x:=E σ −2∥µ∥2(∥µ∥2+G)−logπ 1−πY2 , ConfErr∗= MSE∗−V∗ y|x. then lim n→∞MSE(bp) = MSE∗, lim n→∞CalErr( bp) = ... | https://arxiv.org/abs/2502.11323v1 |
logit for label 215161718logit for label 1 −12.5−10.0−7.5 logit for label 30.00.10.20.30.40.5 Density Logit distribution of class 1 Figure 9: Joint empirical logit distributions of multinomial logistic regression. The heatmaps display empirical joint logits bf1(xi),bfk(xi) for featuresxifrom class 1, where k= 2,3. Ove... | https://arxiv.org/abs/2502.11323v1 |
. . . . . . . . . . . . . 32 A.3 Function plot for the proximal operator proxλℓ(x) . . . . . . . . . . . . . . . . . . . . 32 B Preliminaries: Proofs for Section 3 33 B.1 Proof of Proposition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 B.2 Proof of Proposition 3.2 . . . . . . . . . . . . ... | https://arxiv.org/abs/2502.11323v1 |
Proofs of Lemma D.5, D.9 . . . . . . 69 D.1.5 Parameter convergence and optimality analysis: Proofs of Lemma D.10—D.7 73 D.1.6 ELD convergence: Proof of Lemma D.8 . . . . . . . . . . . . . . . . . . . . . 77 D.1.7 Completing the proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . 80 E Margin rebalancing in... | https://arxiv.org/abs/2502.11323v1 |
H.3 Properties of the Moreau envelope and proximal operator . . . . . . . . . . . . . . . 112 I Miscellaneous 113 A Experiment details A.1 Experiment setup and details We present the details of our experiments, including the computational configurations, information about the datasets, and the pretrained neural network... | https://arxiv.org/abs/2502.11323v1 |
stanfordnlp/imdb . The maximum length in number of tokens for inputs was set as 512. Pretrained models. We downloaded and used pretrained models from Huggingface. •ResNet-18 [38]: 18-layer, 512-dim, 11.2M parameters, convolutional neural network (CNN), pretrained on CIFAR-10 training set (50,000 images). The pretrained... | https://arxiv.org/abs/2502.11323v1 |
m=1|Bm| n cacc(Bm)−dconf(Bm)2 . This is a variant of the prominent estimator called expected calibration error (ECE) [49]. The confidence reliability diagrams for additional 2-GMM simulations and IMDb movie review dataset are shown in Figures 10—12. These plots confirm a similar trend: miscalibration is getting worse... | https://arxiv.org/abs/2502.11323v1 |
solid curve represents the functiony=proxλℓ(x) and the dashed line represents the identity map y=x. orn(this happens with nonzero probability for any fixed n), we haveκ(bβ,bβ0) =∞. It motivates us to redefine the maximum margin properly. Definition B.1. The well-defined maximum margin is bκ:= κ(bβ,bβ0) = min i∈[n]e... | https://arxiv.org/abs/2502.11323v1 |
It holds even if the data is not linearly separable. 35 (a): Next, we show that bβdoes not depend on τ. According to Eq. (41) and Eq. (39), bβ= arg max β∈Sd−1κ(β,ˇβ0(β)) = arg max β∈Sd−1τκ+(β,ˇβ0(β)) +κ−(β,ˇβ0(β)) τ+ 1 = arg max β∈Sd−1⟨xsv+(β),β⟩−⟨xsv−(β),β⟩ τ+ 1= arg max β∈Sd−1⟨xsv+(β)−xsv+(β),β⟩, where⟨xsv+(β)−xsv+(β... | https://arxiv.org/abs/2502.11323v1 |
the event En,κ= κ(bβn,bβ0,n)≥κ =n ∃β∈Rd,∥β∥2≤1,β0∈R, such that eyi ⟨xi,β⟩+β0 ≥κfor alli∈[n]o =n ∃β∈Rd,∥β∥2≤1,β0∈R, such that (κsy−y⊙Xβ−β0y)+ 2= 0o , where sy= (s(y1),...,s (yn))Tandsis the function defined in Eq. (21). Therefore, the data ( X,y) is linearly separable if and only if En,κholds for some κ>0. We would l... | https://arxiv.org/abs/2502.11323v1 |
ξ′(1) n,κ,B−v ≥t . See Appendix C.1.2 for the proof. Step 3: Dimension reduction (from ξ′(1) n,κ,Btoξ′(2) κ,B)It turns out that ξ′(1) n,κ,Bcan be further simplified for analytical purposes. We define a new (deterministic) quantity ξ′(2) κ,B:= min ρ2+r2≤1,r≥0 |β0|≤B−r+√ δ Eh s(Y)κ−ρ∥µ∥2+ρG1+rG2−β0Y2 +i1/2 , which i... | https://arxiv.org/abs/2502.11323v1 |
op+1 n−∥Z−∥2 op =:eB0,n, where in (i) we denote Z+∈Rn+×das a Gaussian random matrix with rows zisuch thatyi= +1, Z−∈Rn−×dwith rowszjsuch thatyj= +1, while in (ii) we use Cauchy–Schwarz inequality, the definition of operator norm, and ∥eβn∥2≤1. Next, we show that eB0,nis asymptotically bounded. Notice Z+,Z−have i.i.d. ... | https://arxiv.org/abs/2502.11323v1 |
ξ′(1) n,κ,B≥v−t , which proves Lemma C.2. C.1.3 Step 3 — Dimension reduction: Proof of Lemma C.3 Proof of Lemma C.3 .The expression of ξ′(1) n,κ,Bcan be further simplified to ξ′(1) n,κ,B= min ρ2+∥θ∥2 2≤1 |β0|≤Bmax ∥λ∥2≤1 λ⊙y≥01√ d ∥λ∥2gTθ+λT(κsy⊙y−ρ∥µ∥2y+ρu+∥θ∥2h−β01) (i)= min ρ2+∥θ∥2 2≤1 |β0|≤B1√ d gTθ+ (κsy−ρ∥µ∥2... | https://arxiv.org/abs/2502.11323v1 |
the proof of Lemma C.1, we can choose Blarge enough such that ξ′(2) κ,B=ξ(2) κ. We can rewrite ξ(2) κas follows by introducing an auxiliary parameter c: ξ(2) κ= min ρ2+r2≤1,r≥0,β0∈R, ρ2+r2+β2 0=c2,c≥0−r+√ δ Eh s(Y)κ−ρ∥µ∥2+ρG1+rG2−β0Y2 +i1/2 , and we also define the following quantity eξ(2) κ:= min ρ2+r2≤1,r≥0,β0∈R,... | https://arxiv.org/abs/2502.11323v1 |
κ∈R:ξ(3) κ= 0o =n κ∈R:eξ(3) κ= 0o .(49) Now we can consider the following two regimes, each with a chain of equivalence: δ≤δ∗(κ)(i)⇐⇒κ≤κ∗(i)⇐⇒ξ(3) κ,eξ(3) κ≤0(ii)⇐⇒ξ(2) κ≤0 (iii)⇐⇒ξ′ n,κ,Bp− → ξ(2) κ += 0(iv)⇐⇒ P(ξn,κ= 0)→1, δ>δ∗(κ)(i)⇐⇒κ>κ∗(i)⇐⇒ξ(3) κ,eξ(3) κ>0(ii)⇐⇒ξ(2) κ>0 (iii)⇐⇒ξ′ n,κ,Bp− → ξ(2) κ +>0(iv)⇐⇒ P(... | https://arxiv.org/abs/2502.11323v1 |
of [20, Theorem 4.6]. We show the convergence of logit margins W2(bLn,L∗)p− →0 first, where bLn:=1 nnX i=1δyi(⟨xi,bβ⟩+bβ0),L∗:=Law max κ∗,ρ∗∥µ∥2+G+β∗ 0Y . (52) Throughout this subsection, all the expectations (including the one in Hκ) are conditional on {(yi,zi)}n i=1, which will be denoted as E·|n[·]. Now, let 1 n... | https://arxiv.org/abs/2502.11323v1 |
we also have (for ε>0 small enough) II≤|κ∗−ε|+ E·|nbV21/2≤κ∗+ E·|n (G+bρ∥µ∥2+bβ0Y)21/2 ≤κ∗+ E[G2]1/2+|bρ|∥µ∥2+|bβ0| ≤κ∗+ 1 +∥µ∥2+B, by using Minkowski inequality and |bβ0|≤B(with high probability) from Lemma C.1. Based on these results and Eq. (59), with high probability, we have E·|nh V−max{κ∗−ε,bV}2i ≤I2−... | https://arxiv.org/abs/2502.11323v1 |
R). Takeρ∈(−1,1). According to the Karush–Kuhn–Tucker (KKT) and Slater’s conditions for variational problems [80, Theorem 2.9.2], ( κ,ξ) is the solution to Eq. (62) if and only if it satisfies the following for some Λ ∈L1,Λ≥0 (a.s.) and ν≥0: −1 +E[s(Y)Λ] = 0,−p 1−ρ2Λ + 2νξ= 0 (a.s.), ν E[ξ2]−δ−1 = 0, Λ s(Y)κ−X−p 1−ρ... | https://arxiv.org/abs/2502.11323v1 |
(63b) imply that ∥ezn∥2 2 11≤n+≤n−1converges inL1, and thus is u.i.. So bκ2 nis also u.i.. By Vitali convergence theorem, convergence in probability of bκncan be strengthen toL2convergence. This concludes the proof of part (c) for δ<δ∗(0). (c),δ>δ∗(0):For non-separable regime, we cannot work with ξn,κin Eq. (43) to sho... | https://arxiv.org/abs/2502.11323v1 |
(68), it follows that bβ0,n(τ)p− →β∗ 0(τ) for anyτ >0. In Lemma C.7, we have shown that W2 bνn,ν∗ =oε(1) forτ= 1 with high probability, i.e., W2 Law Y′1Dcn, ⟨x′,bβn⟩+bβ0,n(1) 1Dcn| {z } =:Un ,Law Y,Ymax κ∗(1),G+ρ∗∥µ∥2+β∗ 0(1)Y | {z } =:U∗ =oε(1). (70) Then there exists a coupling ( Y′,Y,Un,U∗) such that, wit... | https://arxiv.org/abs/2502.11323v1 |
combining Eq. (75) and (76), we get Eq. (72a). Eq. (72b) and (72c) directly come from Eq. (75). Note that function g:R>0→R>0satisfiesg(0+) = 0. Asρvaries from 0 to 1, the L.H.S. of Eq. (72a) increases from 0 to a positive number while the R.H.S. decays to 0, which guarantees the existence and uniqueness of ρ∗>0. Since ... | https://arxiv.org/abs/2502.11323v1 |
from the case of arbitraryτ >0. In Appendix D.1.7, we will discuss how to extend our proof to general τ >0. Recall the original unconstrained empirical risk minimization (ERM) problem Eq. (2a): Mn:= min β∈Rd,β0∈RbRn(β,β0) := min β∈Rd,β0∈R1 nnX i=1ℓ yi(⟨xi,β⟩+β0) . (78) We first provide an outline for the proof of The... | https://arxiv.org/abs/2502.11323v1 |
Θβ=n (β,β0)∈Rd×R: cos(µ,β),∥β∥2,β0 ∈Θco . Then we can simplify M(2) n(Θβ,Ξu) as follows: M(2) n(Θc,Ξu) (i)= min (ρ,R,β 0)∈Θc u∈Ξumin ∥β∥2=R cos(µ,β)=ρmax γ≥0max ∥v0∥2=1( 1 nnX i=1ℓ(ui) +γ nvT 0(ρ∥µ∥2R1+Rg+β0y−u) +γ nhTβ) (ii)= min (ρ,R,β 0)∈Θc u∈Ξumin ∥β∥2=R cos(µ,β)=ρmax γ≥0( 1 nnX i=1ℓ(ui) +γ n ρ∥µ∥2R1+Rg+β0y−u 2+γ... | https://arxiv.org/abs/2502.11323v1 |
Q∞:=Pthe population measure of ( G,Y) (so that ( G,Y)∼N(0,1)×PyunderQ=Q∞, and we have EQ∞:=E,∥U∥Q∞:= (E[U2])1/2). Then we also define the asymptotic counterpart of M(3) n(Θc) by replacing QnwithQ∞: M∗(Θc) := min (ρ,R,β 0)∈Θc ξ∈L2(Q∞),∥ξ∥Q∞≤1/√ δEh ℓ ρ∥µ∥2R+RG+β0Y+Rp 1−ρ2ξi . The following lemma shows that M(3) n(Θc) ... | https://arxiv.org/abs/2502.11323v1 |
yj= +1,yk=−1. Then as bβ0→±∞ , we have ℓ(0)≥1 nnX i=1ℓ yi(⟨xi,bβ⟩+bβ0) ≥1 nℓ ⟨xj,bβ⟩+bβ0 +1 nℓ −⟨xk,bβ⟩−bβ0 →+∞, which leads to a contradiction. So |bβ0|is also bounded with high probability. Finally, in the minimax representation of Mn, the optimal umust satisfy ui=yi(⟨xi,bβ⟩+bβ0) for alli∈[n]. Therefore, accord... | https://arxiv.org/abs/2502.11323v1 |
1−ρ2 √ δ ≤ρ∥µ∥2Rmax+Rmax∥G∥Qn+B0,max∥Y∥Qn+Rmax√ δ (∗)=ρ∥µ∥2Rmax+Rmax 1 +oP(1) +B0,max+Rmax√ δ, by denoting Rmax:= max (ρ,R,β 0)∈ΘcR,B0,max:= max (ρ,R,β 0)∈Θc|β0|, andC > 0 is some constant. Here, (∗) is from the law of large numbers: ∥G∥Qnp− →∥G∥Q∞= (E[G2])1/2= 1. Combining these estimates, we finally deduce that M(2... | https://arxiv.org/abs/2502.11323v1 |
For any α∈(0,1) andξ1,ξ2∈L2(Q), with a shorthand V:=ρ∥µ∥2R+RG+β0Y, we notice that RQ(αξ1+ (1−α)ξ2) =EQh ℓ α V+Rp 1−ρ2ξ1 + (1−α) V+Rp 1−ρ2ξ2i ≤EQh αℓ V+Rp 1−ρ2ξ1 + (1−α)ℓ V+Rp 1−ρ2ξ2i =αRQ(ξ1) + (1−α)RQ(ξ2), where the inequality follows from strong convexity of ℓ, and it becomes equality if and only if Q(ξ1̸=ξ... | https://arxiv.org/abs/2502.11323v1 |
is convex in ( A,B,β 0). It comes from Lemma H.5(a) that ( x,λ)∝⇕⊣√∫⊔≀→eℓ(x;λ) is convex, and the fact that integration EQpreserves convexity. •Rν,Q(A,B,β 0) is concave in ν. This comes from Eq. (170) that eℓ A∥µ∥2+AG1+BG2+β0Y;B ν = min t∈Rn ℓ(t) +ν 2B(A∥µ∥2+AG1+BG2−t)2o , with the fact that pointwise minimum and int... | https://arxiv.org/abs/2502.11323v1 |
any nonnegative Z∈L2(Q∞), we know that aρ∥µ∥2+aG+bY+ap 1−ρ2ξ≥0,almost surely , or equivalently, there exists ( ρ,R,β 0)∈[−1,1]×R>0×Randξ∈L2(Q∞),E[ξ2]≤1/δsatisfying Rρ∥µ∥2+RG+β0Y+Rp 1−ρ2ξ≥0,almost surely . It implies the constraint of the variational problem for the separable regime (SVM) Eq. (22), i.e., ρ∥µ∥2+G+β′ 0Y+p... | https://arxiv.org/abs/2502.11323v1 |
Stein’s identity, we also have relation E ℓ′(U)G1 =AE ℓ′′(U) ,E ℓ′(U)G2 =BE ℓ′′(U) . Combine the above with Eq. (103), we obtain E ℓ′(U) =−νBA δ∥µ∥2,E ℓ′(U)G1 =νBA δ,E ℓ′(U)G2 =νBB δ, which is equivalent to (recall that AG1+BG2d=RG) E ℓ′(U) =−νBA δ∥µ∥2,E ℓ′(U)G =νBR δ. (105) The above implies A> 0 sin... | https://arxiv.org/abs/2502.11323v1 |
0)−(ρ∗,R∗,β∗ 0)∥2<η is aη-L2open ball around the global minimizer ( ρ∗,R∗,β∗ 0). For the first term, with Θclarge enough such that ( ρ∗,R∗,β∗ 0)∈Θc, by Lemma D.5 we have I≥ min (ρ,R,β 0)∈Θc\B2,c∗(η)min u∈Nδn(ρ,R,β 0)1 nnX i=1ℓ(ui) = min (ρ,R,β 0)∈Θc\B2,c∗(η) ξ∈L2(Qn),∥ξ∥2 Qn≤1/δEQnh ℓ ρ∥µ∥2R+RG+β0Y+Rp 1−ρ2ξi =M(3) n ... | https://arxiv.org/abs/2502.11323v1 |
the one at the end of the proof of Lemma C.7, we can show the convergence of empirical logit distribution W2(bνn,ν∗)p− →0 fromW2(bLn,L∗)p− →0 given by Lemma D.8. D.1.7 Completing the proof of Theorem 4.3 Proof of Theorem 4.3 .Consider the ERM problem Eq. (25) with arbitrary τ >0. Recall that eyi=yi/s(yi) wheres:{±1}→{ ... | https://arxiv.org/abs/2502.11323v1 |
the corresponding convergence of ELD still hold. The convergence of TLD directly comes from the proof of part (c). This concludes the proof of part (d). Finally, we complete the proof of Theorem 4.3. E Margin rebalancing in proportional regime: Proofs for Sec- tion 5.1 E.1 Proofs of Proposition 5.1 and 5.2 We show the ... | https://arxiv.org/abs/2502.11323v1 |
an increasing function of π∈(0,1 2),∥µ∥2, andδ, when τ= 1(without margin rebalancing). Moreover, β∗ 0<0. Proof. Whenτ= 1, the above equations reduce to β0=1 2 g−1 1ρ 2(1−π)∥µ∥2δ −g−1 1ρ 2π∥µ∥2δ , (121) κ=1 2 g−1 1ρ 2(1−π)∥µ∥2δ +g−1 1ρ 2π∥µ∥2δ + 2ρ∥µ∥2 . Clearlyβ∗ 0<0, sinceg−1 1is an increasing function and... | https://arxiv.org/abs/2502.11323v1 |
observing f′(x) =φ(−a+x)−φ(−a−x)<0 for all x<0, andf′(x)>0 for allx>0. Hence we conclude β∗ 0= 0 and Err∗ += Err∗ −= Err∗ b. Settingβ0= 0 in Eq. (120a) and solving for τ, we get Eq. (29). This completes the proof. As stated in Remark 5.1, when ∥µ∥2,δare fixed and πis small, the numerator of τoptscales asp 1/π. We forma... | https://arxiv.org/abs/2502.11323v1 |
the function r(y) is defined in the proof of Lemma E.6, and we know that r(y)>0 for all y∈R. Therefore, g1(y)/g′ 1(y) is increasing. This completes the proof. F Margin rebalancing in high imbalance regime: Proof of Theo- rem 5.5 Without loss of generality, we may consider the following case as a substitute of Eq. (9): ... | https://arxiv.org/abs/2502.11323v1 |
bρ,bθ,bβ0) well. 2. In Appendix F.2, we derive the asymptotic orders of ( bρ,bθ,bβ0) by using ( eρ,eθ,eβ0). 3. In Appendix F.3, we use these asymptotics to analyze test errors and conclude Theorem 5.5. F.1 A tight upper bound on maximum margin: Proof of Lemma F.1 The following Lemma provides a data-dependent upper boun... | https://arxiv.org/abs/2502.11323v1 |
Clearly, ( eρ,eθ,eβ0) satisfies the constraints in Eq. (125). This candidate solution is motivated by the optimal ( ρ,θ) that makes (i) and (ii) equal in Eq. (132), i.e., κ=eρ(∥µ∥2+eg) +p 1−eρ2⟨ez,eθ⟩, andβ0that balances the magnitude of average logit margins from the two classes, i.e., we choose β0such thatκ+=κ−in Eq.... | https://arxiv.org/abs/2502.11323v1 |
max i∈I− j∈I− ⟨zi−zj,P⊥ µez⟩p d/πn 94 =OP(logn2 −) =OP(logd). (142) Finally, incorporating Eq. (140) and Eq. (142) into Eq. (139), we have max i∈[n] κ−κi(eρ,eθ,eβ0) ≤ max i∈I+ gi−g+ + max i∈I+ ⟨zi−z+,eθ⟩ ∨ max i∈I− gi−g− + max i∈I− ⟨zi−z−,eθ⟩ ≤n OP(p logd) +OP(logd)o ∨n OP(p logd) +OP(logd)o =OP(logd). Therefore,... | https://arxiv.org/abs/2502.11323v1 |
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