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the definition of rΛ(k) :=Tr(Λ≥k) ∥Λ≥k∥opas the intrinsic dimension of Λat level k, we sequentially define the following quantities that can be found in Misiakiewicz and Saeed (2024); Defilippis et al. (2024). MΛ(k) = 1 +rΛ(⌊η∗·k⌋)∨k klog (rΛ(⌊η∗·k⌋)∨k), (B.27) ρκ(p) = 1 +p·ξ2 ⌊η∗·p⌋ κMΛ(p), (B.28) eρκ(n, p) = 1 + 1{n≤... | https://arxiv.org/abs/2502.01585v2 |
i.e. Φ2(X;A)−Ψ2(µ∗;A) ≤eO(n−1 2)·Ψ2(µ∗;A), that is required to derive our non-asymptotic deterministic equivalence for the bias term of the ℓ2 norm. By introducing a change of variable µ∗:=µ∗(λ) =λ/λ∗, we find that µ∗satisfies the following fixed-point equation: µ∗=n 1 + Tr( Σ(µ∗Σ+λ)−1). (C.1) We define tandTas follows... | https://arxiv.org/abs/2502.01585v2 |
+ nEκ−S (1 +κ)(1 + S)t⊤M−At .(C.6) For the first term, recall that eµ∗is the solution of the equation (C.1) where we replaced nby n−1, and eµ−:=n/(1 +κ). By Misiakiewicz and Saeed (2024, Proposition 2), we have |E[Tr(AM−)]−Ψ1(eµ∗;A)| ≤ E(D) 1,n−1·Ψ1(eµ∗;A), where E(D) 1,n−1=C∗,Kρλ(n)5/2 √n−1. For n≥C, we have E(D) 1,... | https://arxiv.org/abs/2502.01585v2 |
iTiMi) , where Miis the rescaled resolvent removes xi, and we used that Ei AT⊤ iTiMi =Ei−1 AT⊤ iTiMi , and we’ll write (recall that Si=t⊤ iMiti) Tr(AT⊤TM)−Tr(AT⊤ iTiMi) = Tr( A(tit⊤ i+T⊤ iTi)M)−Tr(AT⊤ iTiMi) =t⊤ iMAt i+ Tr(AT⊤ iTiM)−Tr(AT⊤ iTiMi) =1 (1 +Si)n t⊤ iMiAti−Tr(AT⊤ iTiMitit⊤ iMi)o =1 (1 +Si)Tr(tit⊤ iMiA(... | https://arxiv.org/abs/2502.01585v2 |
We give the asymptotic deterministic equivalents for BLS N,λandVLS N,λ, respectively. For the bias term BLS N,λ, we use Eq. (B.3) by taking A=β∗β⊤ ∗andB=Iand thus obtain BLS N,λ=⟨β∗,(X⊤X)2(X⊤X+λI)−2β∗⟩ = Tr( β∗β⊤ ∗(X⊤X)2(X⊤X+λI)−2) ∼Tr(β∗β⊤ ∗Σ2(Σ+λ∗I)−2) +λ2 ∗Tr(β∗β⊤ ∗Σ(Σ+λ∗I)−2)·Tr(Σ(Σ+λ∗I)−2)·1 n−Tr(Σ2(Σ+λ∗I)−2) =⟨β∗... | https://arxiv.org/abs/2502.01585v2 |
also face with the issue on the difference between two deterministic equivalents. We generalize Theorem 3.4 as below. Theorem D.6 (Deterministic equivalents of the ℓ2-norm of the estimator. Full version of Theo- rem 3.4) .Assume well-behaved data {xi}n i=1satisfy Assumption 3.2 and Assumption D.5. Then for anyD, K > 0,... | https://arxiv.org/abs/2502.01585v2 |
for minimum ℓ2-norm 4Due to the complexity of the calculations, we use Mathematica Wolfram to eliminate n. The same approach is applied later whenever norpelimination is required. 39 estimator and Σ=Id, for the under-parameterized regime ( d < n ), we have BLS R,0= 0,VLS R,0=σ2d n−d; BLS N,0=∥β∗∥2 2,VLS N,0=σ2d n−d. Fr... | https://arxiv.org/abs/2502.01585v2 |
1 andβ= 0, when n→d, we have5 VLS R,0≈2(VLS N,0)2 dVLS N,0−d2σ2,BLS R,0≈2BLS N,0(d−BLS N,0) d2. Remark: The relationship between RLS 0andNLS 0is still linear in the under-parameterized regime, but is quite complex in the over-parameterized regime. We characterize more special cases (e.g., β=±1) in Appendix D.3 and find... | https://arxiv.org/abs/2502.01585v2 |
N,0)4−324d2(BLS N,0)3+126 d4(BLS N,0)2+d6BLS N,0−5d8 2d5(6BLS N,0−d2). Here we present some experimental results to check the relationship between BLS R,0andBLS N,0, as well as VLS R,0andVLS N,0, see Fig. 5. We can see that our approximate relationship on variance (see the red line in Fig. 5(d)) provides the precise es... | https://arxiv.org/abs/2502.01585v2 |
2=Eε∥(Z⊤Z+λI)−1Z⊤(Gθ∗+ε)∥2 2 =∥(Z⊤Z+λI)−1Z⊤Gθ∗∥2 2+Eε∥(Z⊤Z+λI)−1Z⊤ε∥2 2 =⟨θ∗,G⊤Z(Z⊤Z+λI)−2Z⊤Gθ∗⟩+σ2Tr Z⊤Z(Z⊤Z+λI)−2 =:BRFM N,λ+VRFM N,λ. Accordingly, we conclude the proof. Now we are ready to present the proof of Theorem 4.2 as below. 45 Proof of Theorem 4.2. We give the asymptotic deterministic equivalents for the ... | https://arxiv.org/abs/2502.01585v2 |
regime ( p < n ), when λgoes to zero, we have eλ→0 and edf2(eλ)→p(Bach, 2024). For the bias term, we use Eq. (B.7) with T=G,Σ=F⊤F,A=θ∗θ⊤ ∗,B=F⊤Fand then obtain BRFM N,0= Tr( θ⊤ ∗G⊤Z(Z⊤Z+λI)−2Z⊤Gθ∗) =pTr(θ⊤ ∗G⊤GF⊤(FG⊤GF⊤+pλI)−2FG⊤Gθ∗) =pTr(θ∗θ⊤ ∗G⊤(GF⊤FG⊤+pλI)−1GF⊤FG⊤(GF⊤FG⊤+pλI)−1G) ∼pTr(θ∗θ⊤ ∗(F⊤F+eλI)−1F⊤F(F⊤F+eλI)−1... | https://arxiv.org/abs/2502.01585v2 |
For the bias term, it can be decomposed into BRFM R,0=∥θ∗−p−1/2F⊤(Z⊤Z+λI)−1Z⊤Gθ∗∥2 2 =θ⊤ ∗θ∗−2p−1/2θ⊤ ∗F⊤(Z⊤Z+λI)−1Z⊤Gθ∗ +θ⊤ ∗G⊤Z(Z⊤Z+λI)−1bΛF(Z⊤Z+λI)−1Z⊤Gθ∗. For the second term: p−1/2θ⊤ ∗F⊤(Z⊤Z+λI)−1Z⊤Gθ∗, we can use Eq. (B.6) with T=G,Σ=F⊤F, A=θ∗θ⊤ ∗F⊤Fand obtain p−1/2θ⊤ ∗F⊤(Z⊤Z+λI)−1Z⊤Gθ∗= Tr( θ∗θ⊤ ∗F⊤FG⊤(GF⊤FG⊤+pλ... | https://arxiv.org/abs/2502.01585v2 |
that AFis the event defined in Defilippis et al. (2024, Eq. (79)). Under the assumptions, we have P(AF)≥1−p−D. Hence, applying Proposition B.11 for F∈ AFand via union bound, we obtain that with probability at least 1 −p−D−n−D, nΦ4(Z;bΛ−1 F, λ)−neΦ5(F;bΛ−1 F, pν1) ≤C∗,D,K· E1(p, n)·neΦ5(F;bΛ−1 F, pν1), (E.6) and we reca... | https://arxiv.org/abs/2502.01585v2 |
p−n. RRFM 0=nλ2 n⟨θ∗,(Λ+λnI)−2θ∗⟩ n−Tr(Λ2(Λ+λnI)−2)+nλn⟨θ∗,(Λ+λnI)−1θ∗⟩ p−n+σ2Tr(Λ2(Λ+λnI)−2) n−Tr(Λ2(Λ+λnI)−2)+σ2n p−n =nλ2 n⟨θ∗,(Λ+λnI)−2θ∗⟩+σ2Tr(Λ2(Λ+λnI)−2) n−Tr(Λ2(Λ+λnI)−2)+ nλn⟨θ∗,(Λ+λnI)−1θ∗⟩+σ2n1 p−n. Then we eliminate pand obtain that the deterministic equivalents of the estimator’s test risk and norm, RRFM... | https://arxiv.org/abs/2502.01585v2 |
show that when p→n, which also implies that BRFM N,0→ ∞ andBRFM R,0→ ∞ , this relationship is approximately linear. Recall that the relationship between BRFM R,0andBRFM N,0is given by BRFM R,0=(m−n) nBRFM N,0, and is equivalent toBRFM N,0=n (m−n)BRFM R,0:=f(BRFM R,0). We then do a difference and get BRFM N,0−f(BRFM R,0... | https://arxiv.org/abs/2502.01585v2 |
((C1C4−C2C3)n+C1(C3−C4)p) (C1−C2)(p−n), BRFM N,0=ν2 ν1⟨θ∗,(Λ+ν2)−1θ∗⟩ −λ nν2 2 ν2 1⟨θ∗,(Λ+ν2)−2θ∗⟩+χ(ν2)⟨θ∗,Λ(Λ+ν2)−2θ∗⟩ 1−Υ(ν1, ν2) ≈ν2 ν1⟨θ∗,(Λ+ν2)−1θ∗⟩ ≈ n C1−α(2r−1) C3p p−n. Then we eliminate pand obtain BRFM R,0≈n C1−α BRFM N,0+n C1−2αrC2C3−C1C4 C1−C2. (E.11) 60 Condition 2: r∈[1 2,1) BRFM R,0=ν2 2 1−Υ(ν1, ... | https://arxiv.org/abs/2502.01585v2 |
≈C1C4 C1−C2+p n−pC3 ν(2r−1)∧0 2 . Then we use the approximation ν2≈(p C1)−αand obtain BRFM R,0≈n n−pC3ν2r∧1 2≈n n−pC3p C1−α(2r∧1) , 63 BRFM N,0≈C1C4 C1−C2+p n−pC3 ν(2r−1)∧0 2 ≈C1C4 C1−C2+p n−pC3p C1−α[(2r−1)∧0] . Similarly to the bias term, we derive the relationship in the under-parameterized regime ( p < n ... | https://arxiv.org/abs/2502.01585v2 |
to (E.15), we know that the relationship between RRFM 0and NRFM 0in the under-parameterized regime when p→ncan be written as RRFM 0≈(n/Cα)−αNRFM 0+Cn,α,r, 2. F Scaling laws To derive the scaling laws based on norm-based capacity, we first give the decay rate of the ℓ2 norm w.r.t. n. The rate of the deterministic equiva... | https://arxiv.org/abs/2502.01585v2 |
q >ℓ αIn this region, according to Defilippis et al. (2024, Corollary 4.1), we have RRFM λ= Θ n−2ℓr , and according to Fig. 6, we have NRFM λ= Θ n−ℓ(2r−1) , combing the above rate, we can obtain that RRFM λ= Θ n−1·NRFM λ . 69 Region 5: q <1 2αr+1and q <ℓ αIn this region, according to Defilippis et al. (2024, Coro... | https://arxiv.org/abs/2502.01585v2 |
defined as φ(x,w) =φ(⟨w,x⟩) with random Gaussian initialization w∼ N(0,Id), where the activation function φ(·) is chosen from ReLU,erf,tanh, orsigmoid . ReLU erf tanh sigmoidWith ridge 0 3 6 9 12 2 norm 0.07000.18250.29500.40750.5200T est lossTest loss vs. 2 norm 2004006008001000 p λ= 0.1 0.0 57.5 115.0 172.5 230.0 2 n... | https://arxiv.org/abs/2502.01585v2 |
norm 0.71000.76750.82500.88250.9400T est errorTest error vs. 2 norm 2004006008001000 pλ= 10−4 3.00 21.25 39.50 57.75 76.00 2 norm 0.71000.76750.82500.88250.9400T est errorTest error vs. 2 norm 2004006008001000 pλ= 10−4 0.0 52.5 105.0 157.5 210.0 2 norm 0.7100.7650.8200.8750.930T est errorTest error vs. 2 norm 200400600... | https://arxiv.org/abs/2502.01585v2 |
( = 1.5) Theoretical Empirical (e) Norm vs. λ 0 510 15 20 25 30 35 40 2 2 0.040.060.080.100.120.14T est RiskTest Risk vs. Norm ( = 1.5) Theoretical Empirical (f) Test risk vs. Norm Figure 7: Relationship between test risk, ℓ2norm, and λfor different γ=p nfor random feature ridge regression. Points in these figures are ... | https://arxiv.org/abs/2502.01585v2 |
the regularization strength λincreases, model complexity decreases. From Definition B.9, we have n−λ λ∗=Tr(Σ(Σ+λ∗)−1), implying that an increase in λraises λ∗, leading to a reduction in df1(λ∗) anddf2(λ∗). This suggests that degrees of freedom can, to some extent, represent model complexity. However, it is worth noting... | https://arxiv.org/abs/2502.01585v2 |
data {(xi, yi)}i∈[n],d= 1000, sampled from a linear model yi=x⊤ iβ∗+εi,σ2= 0.0004, xi∼ N(0,Σ), with σk(Σ) =k−1,β∗,k=k−3/2. 0 1 2 3 (p n) 0.00.20.4test riskTest Risk vs. ridgeless =0.001 =0.005 =0.01 =0.02 =0.05 =0.1 =1 =5 (a)Test Risk vs. γ:=p/n 0 1 2 3 (p n) 0601201802 norm 2 Norm vs. ridgeless theory empirical (b)ℓ2n... | https://arxiv.org/abs/2502.01585v2 |
The relationship between test risk, ℓ2norm and the number of features p. Solid lines are obtained from the deterministic equivalent, and points are numerical simulations, with the different curves denoting different regularization strengths. Training data {(xi, yi)}i∈[n],n= 300, sub-sampled from the MNIST data set (LeC... | https://arxiv.org/abs/2502.01585v2 |
MNIST data set (LeCun et al., 1998), consisting of 4,000 training samples from all the 10 classes. To simulate real-world noisy data, a noise level ηis introduced, meaning η·100% of the training labels are randomly corrupted. The model is chosen as a two-layer fully connected neural network with parameters including a ... | https://arxiv.org/abs/2502.01585v2 |
500 Frobenius Distance0.0650.0700.0750.0800.0850.090T est Loss T est Loss vs Frobenius Distance T est Loss vs Frobenius Distance (mean) (c) Test Loss vs. µfro-dis Figure 17: Experiments on two-layer fully connected neural networks with noise level η= 0.2. Theleftfigure is the same as Fig. 16(a). The middle figure shows... | https://arxiv.org/abs/2502.01585v2 |
the noise level η= 0.1 and η= 0.3 in Figs. 20 and 21, respectively. We can see that, when the noise level increases, we observe stronger peaks in the test loss for double descent. However, the trend of test loss is similar at different noise levels with Path norm µpath-norm as the model capacity, i.e., it shows a U-sha... | https://arxiv.org/abs/2502.01585v2 |
Learning with Differentially Private (Sliced) Wasserstein Gradients David Rodríguez-Vítores Universidad de Valladolid and IMUV A david.rodriguez.vitores@uva.esClément Lalanne Institut de Mathématiques de Toulouse clement.lalanne@math.univ-toulouse.fr Jean-Michel Loubes Université de Toulouse ANITI & Regalia INRIA jean-... | https://arxiv.org/abs/2502.01701v2 |
(2010); Barber & Duchi (2014); Diakonikolas et al. (2015); Karwa & Vadhan (2018); Bun et al. (2019, 2021); Kamath et al. (2019); Biswas et al. (2020); Kamath et al. (2020); Acharya et al. (2021); Lalanne (2023); Aden-Ali et al. (2021); Cai et al. (2019); Brown et al. (2021); Cai et al. (2019); Kamath et al. (2022a); La... | https://arxiv.org/abs/2502.01701v2 |
work, we address the challenge of optimizing objectives which depend on sliced Wasserstein distances between empirical measures. Assume that we have samples X= (x1, . . . , x n)∈ Xn, Z= (z1, . . . , z m)∈ Zm, and denote by PXandPZthe empirical distributions associated with XandZ. Assuming that we have a vector of train... | https://arxiv.org/abs/2502.01701v2 |
conditional distributions based on sensitive attributes. This offers a novel private in-processing approach to mitigate bias of Machine Learning algorithms. This privacy-preserving fairness regularization strategy constitutes, to our knowledge, a new contribution to the literature. 1.2 Related Work Our work directly co... | https://arxiv.org/abs/2502.01701v2 |
“The difference between Dand˜Donly comes from one individual’s data”. In our paper, due to the splitting of the data into separate categories in the Wasserstein distance, and because of potential asymmetry that may arise in their treatments, we will occasionally employ modified definitions of neighboring relations, whi... | https://arxiv.org/abs/2502.01701v2 |
given sample of observations xi∈Rfori∈[n], we denote its order statistics by x(1)≤x(2)≤ ··· ≤ x(n).Given two discrete probabilities on the real line PU=1 nPn i=1δUiand PV=1 mPm j=1δVj, using the characterization of W2 2(PU, PV)in terms of quantile functions, it follows that if we define the weights Ri,j=λ i−1 n,i n ∩... | https://arxiv.org/abs/2502.01701v2 |
in Remark 4.2. 5 Proposition 3.2 and the chain rule under suitable assumptions give the following expression, ∇θW2 2= 2nX i=1mX j=1Rσ(i),τ(j)(gθ(xi)−hθ(zj))∇θgθ(xi) + 2nX i=1mX j=1Rσ(i),τ(j)(hθ(zj)−gθ(xi))∇θhθ(zj). This decomposition enables the following sensitivity analysis : Theorem 4.1. With all the previous notati... | https://arxiv.org/abs/2502.01701v2 |
training procedure of Rakotomamonjy & Ralaivola (2021) is considerably more limited in scope and primarily suited for simple tasks such as data generation. A numerical comparison of both methods when applicable is provided in Appendix F. Additionally, Section 5 presents a comparison in the context of data generation, l... | https://arxiv.org/abs/2502.01701v2 |
the ADAM rule (Kingma & Ba (2015)). The clipping values used were M= 1.5for the decoder output, L=√ 6for clipping the Jacobian matrix of the encoder’s last layer (following the naive approach of Remark C.2), and C= 1 for clipping the individual gradients of the reconstruction loss. The number of training iterations was... | https://arxiv.org/abs/2502.01701v2 |
α(θ) = (1 −α)L(θ) +α SW2 2(φθ#PX0, φθ#PX1), (3) where α∈[0,1]controls the trade-off between prediction accuracy and fairness, and φθ=gθ(if the distribution loss is applied to the output) or gθ=ψθ◦φθ(if the distribution loss is applied to intermediate representation, e.g., an intermediate layer in a NN). To ensure priva... | https://arxiv.org/abs/2502.01701v2 |
other research directions. Our results do not directly extend to other Wasser- stein losses, such as Wp. As shown in Appendix H, it is not possible to bound the sensitivity of the gradient of Wp, for general p≥1, by a factor that decreases approximately at a rate of 1/n. Nevertheless, we believe that the generalization... | https://arxiv.org/abs/2502.01701v2 |
thou r3579x?: anonymized social networks, hidden patterns, and structural steganography. In Williamson, C. L., Zurko, M. E., Patel-Schneider, P. F., and Shenoy, P. J. (eds.), Proceedings of the 16th International Conference on World Wide Web, WWW 2007, Banff, Alberta, Canada, May 8-12, 2007 , pp. 181–190. ACM, 2007. do... | https://arxiv.org/abs/2502.01701v2 |
Y ., and Zhang, L. The cost of privacy: Optimal rates of convergence for parameter estimation with differential privacy. CoRR , abs/1902.04495, 2019. URL http://arxiv.org/ abs/1902.04495 . Chiappa, S., Jiang, R., Stepleton, T., Pacchiano, A., Jiang, H., and Aslanides, J. A general approach to fairness with optimal tran... | https://arxiv.org/abs/2502.01701v2 |
Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, USA, pp. 3571–3580, 2017. URL https://proceedings.neurips.cc/paper/2017/hash/ 253614bbac999b38b5b60cae531c4969-Abstract.html . Ding, J., Zhang, X., Li, X., Wang, J., Yu, R., and Pan, M. Differentially private and fair classifica- tion via calibrat... | https://arxiv.org/abs/2502.01701v2 |
fairness in differential privacy. In Zhang, B., Popa, R. A., Zaharia, M., Gu, G., and Ji, S. (eds.), PPMLP’20: Proceedings of the 2020 Workshop on Privacy-Preserving Machine Learning in Practice, Virtual Event, USA, November, 2020 , pp. 15–19. ACM, 2020. doi: 10.1145/3411501.3419419. URL https://doi.org/10.1145/3411501... | https://arxiv.org/abs/2502.01701v2 |
Li, J., Singhal, V ., and Ullman, J. R. Privately learning high-dimensional distributions. In Beygelzimer, A. and Hsu, D. (eds.), Conference on Learning Theory, COLT 2019, 25-28 June 2019, Phoenix, AZ, USA , volume 99 of Proceedings of Machine Learning Research , pp. 1853–1902. PMLR, 2019. URL http://proceedings.mlr.pr... | https://arxiv.org/abs/2502.01701v2 |
Garivier, A., and Gribonval, R. Private Statistical Estimation of Many Quantiles. In ICML 2023 - 40th International Conference on Machine Learning , Honolulu, United States, July 2023a. URL https://hal.science/hal-03986170 . Lalanne, C., Gastaud, C., Grislain, N., Garivier, A., and Gribonval, R. Private Quantiles Estim... | https://arxiv.org/abs/2502.01701v2 |
July 21-27, 2024 . OpenReview.net, 2024. URL https://openreview.net/forum?id=VZsxhPpu9T . Rabin, J., Peyré, G., Delon, J., and Bernot, M. Wasserstein barycenter and its application to texture mixing. In Bruckstein, A. M., ter Haar Romeny, B. M., Bronstein, A. M., and Bronstein, M. M. (eds.), Scale Space and Variational... | https://arxiv.org/abs/2502.01701v2 |
systematic survey. ACM Comput. Surv. , 51 (3):57:1–57:38, 2018. doi: 10.1145/3168389. URL https://doi.org/10.1145/3168389 . Wang, X., Zhang, Y ., and Zhu, R. A brief review on algorithmic fairness. Management System Engineering , 1(1):7, 2022. ISSN 2731-5843. doi: 10.1007/s44176-022-00006-z. URL https: //doi.org/10.100... | https://arxiv.org/abs/2502.01701v2 |
This field of research has received a growing attention over the last few years as pointed out in the following papers and references therein Chouldechova & Roth (2020); Dwork et al. (2012); Oneto & Chiappa (2020); Wang et al. (2022); Barocas et al. (2018); Besse et al. (2022). The Wasserstein distance offers a compell... | https://arxiv.org/abs/2502.01701v2 |
i=1mX j=1Rσ(i),τ(j)(Ui−Vj) ProjL1(∇θgθ(xi)) + 2nX i=1mX j=1Rσ(i),τ(j)(Vj−Ui) ProjL2(∇θhθ(zj))(4) where for all i and j, we have Ui= ProjM(gθ(xi)),Vj= ProjM(hθ(zj)), andσ, τare defined as in Section 3. This technique is known as “clipping" and was historically introduced as a preprocessing of the gradients for problems ... | https://arxiv.org/abs/2502.01701v2 |
the privacy accounting framework in the asymptotic regime described in Section 5.2 of Dong et al. (2019). In our experiments, we implement this privacy accountant using the Yousefpour et al. (2021) library. Note that this accountant can be replaced by any accountant, tailored for fixed size batch sampling without repla... | https://arxiv.org/abs/2502.01701v2 |
2,k(gθ#PX, hθ#PZ) then ∆2Φθ≤4M3L1+L2 n. (b)Under neighboring relation ∼2inD=Xn× Zm, if we define Ψθ(X,Z)as ∇θSW2 2(gθ#PX, hθ#PZ)or its Monte Carlo approximation ∇θSW2 2,k(gθ#PX, hθ#PZ), then ∆2Ψθ≤4Mmaxn3L1+L2 n,L1+ 3L2 mo . Remark C.2.From a computational point of view, if we want to define a clipped approximation JL1 ... | https://arxiv.org/abs/2502.01701v2 |
6, we include it again for completeness. We present this section for the general case of the sliced Wasserstein distance, note that the case d= 1coincides with the one-dimensional Wasserstein distance. As in Section 6, let φθdenote the representation to which the fairness penalty is applied. That is, φθ=gθif the penalt... | https://arxiv.org/abs/2502.01701v2 |
an illustrative synthetic model to simulate bias in algorithmic decision-making. Note that we do not provide comparisons with other application-specific methodologies, as our approach is highly general and does not include any of the statistical, convergence, or fairness guarantees described by other methods, see Xu et... | https://arxiv.org/abs/2502.01701v2 |
penalized loss functions (6) and (7). Following the above notation, Xj= (xi:ai=j),nj=length (Xj)forj= 0,1, and Xj,k= (xi:ai=j, yi=k),nj,k=length (Xj,k)forj, k∈ {0,1}. Given our data generation procedure, we know that E(nj) =n/2,E(nj,j) =pn/2andE(nj,1−j) = (1 −p)n/2forj∈ {0,1}. In all the subsequent experiments, the bat... | https://arxiv.org/abs/2502.01701v2 |
at the expense of accuracy, as expected. The second important conclusion is that adding privacy does not significantly alter the results of the optimization. For different privacy budgets ε, both the histogram and the computed measures do not change much across the rows. Moreover, Figure 9 shows the training loss curve... | https://arxiv.org/abs/2502.01701v2 |
distribution of YCconditioned on the sensitive attribute. If T0denotes the triangle with vertices (0,0),(0,1),(1,0)andT1the triangles with vertices (0,1),(1,1),(1,0), then we know that YC|A= jfollows a mixture of the uniform distributions on T0andT1, with weight pinT0and(1−p)inT1if A= 0, and vice versa if A= 1. The aim... | https://arxiv.org/abs/2502.01701v2 |
0.1/n. Above each graph we indicate the noise added at each step of DP-SGD to obtain the desired privacy level, the value of the loss (6)in the training procedure, together with the individual value of the regression loss (RL) and the sliced Wasserstein loss (SW). Last line includes the indexes OD0andOD1. 𝛼𝛼=0 𝛼𝛼=0... | https://arxiv.org/abs/2502.01701v2 |
sliced Wasserstein gradient, weighted by α. •Noise multiplier: The standard Gaussian noise is scaled by a factor that depends on both α (via sensitivity) and the required privacy budget ε. To illustrate the effectiveness of this isolation strategy, we repeated the Statistical Parity classification experiment—chosen for... | https://arxiv.org/abs/2502.01701v2 |
dataset is sub-sampled at first, and we then use this sub-sampled dataset until the end of the training procedure. Note that by design, and contrary to our method, this approach is unable to provide privacy guarantees w.r.t. the source data. As described in Algorithm 1 in Greenewald et al. (2024), the same amount of du... | https://arxiv.org/abs/2502.01701v2 |
The four initial subproblems of fairness experiment required 1.18 hours in total for the 4 subproblems, whereas the randomness strategy required another 1.72 hours. The synthetic circles experiment took 1.64 hours. Thus, the total GPU time for the reported experiments was approximately 9.5 hours. The overall computatio... | https://arxiv.org/abs/2502.01701v2 |
(13), replacing W2withSW2(or its Monte Carlo approximation). In this setting, the dimension of the hidden dimension in the vector–matrix product becomes dminstead of than m, further increasing the computational burden. A further study of potential improvements to the naive strategy presented here is left for future res... | https://arxiv.org/abs/2502.01701v2 |
define Ui:=gθ(xi)and˜Ui:=gθ(˜xi)for each i∈[n], and Vj:=hθ(zj) forj∈[m]. Again, Ui=˜Uifor every i̸= 1. Define now the rank permutations σ,˜σandτsuch that Ui=U(σ(i)), i ∈[n], ˜Ui=˜U(˜σ(i)), i ∈[n], Vj=V(τ(j)), j ∈[m]. Denote U= (U1, . . . , U n)andV= (V1, . . . , V m). Corollary 3.2 ensures if we define PU= gθ#PX=1 nPn ... | https://arxiv.org/abs/2502.01701v2 |
. . , V m, then by definition of Ri,j, mX j=1V(j)(Rσ(i),j)−R˜σ(i),j) = =mX j=1V(j) Zσ(i) n σ(i)−1 nIj−1 m< t≤j m dt−Z˜σ(i) n ˜σ(i)−1 nIj−1 m< t≤j m dt! =Zσ(i) n σ(i)−1 nmX j=1V(j)Ij−1 m< t≤j m dt−Z˜σ(i) n ˜σ(i)−1 nmX j=1V(j)Ij−1 m< t≤j m dt =Zσ(i) n σ(i)−1 nG−1(t)dt−Z˜σ(i) n ˜σ(i)−1 nG−1(t)dt =Zσ(i) n σ(i)−1 nG... | https://arxiv.org/abs/2502.01701v2 |
arXiv:2502.01919v1 [stat.ML] 4 Feb 2025Poisson Hierarchical Indian Buffet Processes-With Indications for Microbiome Species Sampling Models Lancelot F. James∗ Department of ISOM, HKUST Juho Lee The Graduate School of AI, KAIST Abhinav Pandey Department of ISOM, HKUST February 5, 2025 Abstract In this work, we present a ... | https://arxiv.org/abs/2502.01919v1 |
examine how our model can be interpreted in terms of its capabilities for complex c ount models arising in recent microbiome and ecological species sampling studies. see fo r instance [1, 7, 31, 33, 37], Relevant to that context, our analysis addresses a potentia lly infinite number of species and unknown parameters, fr... | https://arxiv.org/abs/2502.01919v1 |
cies sampling models, without going into all the formal technical details. This discussio n will also highlight forthcoming results, serving as an outline. In Section 2, we will describe the specifics of the Poisson HIB P, Throughout we will use the notation [ n] ={1,2,...,n}for an integer n. 1.1 Poisson HIBP as Sparse ... | https://arxiv.org/abs/2502.01919v1 |
diversity across and within groups of species, our constructions induce priors and posterior distributions o ver sequences of parameters that indicate mean rates and relative rates of species abundance across and within groups. In particular, we introduce pairs ( λl,Yl)l≥1, which are points of a Poisson Process, where ... | https://arxiv.org/abs/2502.01919v1 |
[37]. However, our work also allows for a potential myriad of implementable measure s either modelling parameters of interest, including indicators of mean rates or relative rates of abundance, or utilizing latent or partially observed data. For example, we may propo se (new) random measures of alpha-diversity acrossth... | https://arxiv.org/abs/2502.01919v1 |
{sj,k:wj,k∈Qj,l,k≥1}, are independent across each j,l,and specified by L´ evy densities B0(Qj,l)τj, with atoms selected iid proportional to I{ω∈Qj,l}B0(dω). Remark 2.1. Throughout we use P(λ)to denote a Poisson(λ)random variable which may otherwise be indexed by i,j,k,letc. Now, following [12, 36, 38], we define Poisson ... | https://arxiv.org/abs/2502.01919v1 |
scale 1.Additionally, a variable, say Tα(y), is a (simple) generalized gamma random variable if its Laplace transform is of the form E[e−sTα(y)] =e−y((1+sα)−1) for 0<α<1, and otherwise has the density of an exponentially tilted s table variable. This is a commonly used variable and corresponding process that y ields po... | https://arxiv.org/abs/2502.01919v1 |
has the same distribution as in (2.2). Use (2.3). Note importantly that there is also the representation (Z(i) j,i∈[Mj])d= (∞/summationdisplay l=1P(i) j,l(σj,l(λl))δYl,i∈[Mj]), where P(i) j,l(σj,l(λl))d=/summationtext∞ k=1P(i) j,k,l(sj,k,l), which reveals the random intensities σj,l(λl) that we can interpret as the mea... | https://arxiv.org/abs/2502.01919v1 |
represents the species ( Y′ l)l≥1that do not appear in the ( Mj,j∈[J]) samples. Here, (λ′ l)l≥1are the corresponding random rates determined by the L´ evy d ensityτ0,J(λ) = e−λ/summationtextJ j=1ψj(Mj)τ0(λ). Forgeneralbutrelatedinterpretationsof( Hl,˜Xl)l≥1,see[27][Section3]and [12][Proposition 3.3]. Note that in many ... | https://arxiv.org/abs/2502.01919v1 |
2.1, it suffices to work with the info rmation in the sum process (Nj,l,˜Yl,l∈[r],ϕ=r,j∈J),where again Nj,l=/summationtextXj,l k=1˜Cj,k,l,for descriptions of the posterior distribution. The task then becomes to descri be distributions of ( ˜Cj,k,l,k∈ [Xj,l],Xj,l) given this information, which in general is a difficult prob... | https://arxiv.org/abs/2502.01919v1 |
a bundance rates, denoted as (˜σj,l(Hl),j∈[J]), of˜Ylselected in the sample for each group j∈[J]. This result follows from the randomization of the posterior distribution of the total mass in [15][Theorem 1], or equivalently in [11][Corollary 5.1]. See also [28]. Proposition 3.2. Consider the Poisson HIBP setting in Th... | https://arxiv.org/abs/2502.01919v1 |
(Nj,l,j∈[J]) We now provide descriptions of the general marginal process . First note that ( Nj,l,j∈ [J])|(˜σj,l(Hl) =tj,j∈[J]),Hlis tP(t1M1,...,tJMJ),and hence there is a joint distribution of (Nj,l,˜σj,l(Hl),j∈[J]),Hl,withnl:=/summationtextJ j=1nj,l= 1,2,..., /producttextJ j=1Mnj,l jtnj,l je−tjMj (1−e−/summationtextJ... | https://arxiv.org/abs/2502.01919v1 |
nges [17]. Indeed, our setting is quite different; samples correspond to (Z(i) j,i∈Mj,j∈[J]): -Z(i) jis a single sample unit from group j, producing a sparse matrix of counts with many components (some latent), including counts, distinct species, and samples of existing species. - The counterpart of mwould be a random v... | https://arxiv.org/abs/2502.01919v1 |
that appears for the first time in a new sampleZ(Mj+1) jfrom group j. Assuming all terms given ( Nj,l,j∈[J],l∈[r]) are random, as indicated in Proposition 3.4, we can provide a posterior o r predictive distribution for a measure of previously unseen alpha-diversity. This gives a parameter-based answer to Q2 in this sett... | https://arxiv.org/abs/2502.01919v1 |
as topic models. For practi cal illustration, we will soon include, in an updated version, both detailed synthetic and real data studies based broadly on information from thepapers[32, 33, 37], showcasing some of thenew schemes we propose here. We will also provide relevant computational informat ion that can be access... | https://arxiv.org/abs/2502.01919v1 |
arXiv:1908.07186 . [14] James, L. F., Lee, J. and Pandey, A. [2023], ‘Bayesian an alysis of generalized hierar- chical indian buffet processes for within and across group sha ring of latent features’, arXiv preprint arXiv:2304.05244 . 20 [15] James, L. F., Lijoi, A. and Pr¨ unster, I. [2009], ‘Poste rior analysis for no... | https://arxiv.org/abs/2502.01919v1 |
[35] Thibaux, R. andJordan, M. I. [2007], Hierarchical beta processes andtheIndianbuffet process, in‘International conference on artificial intelligence and s tatistics’, pp. 564– 571. [36] Titsias, M. K. [2008], ‘The infinite gamma-Poisson feat ure model’, Advances in Neural Information Processing Systems. 2008. [37] Wil... | https://arxiv.org/abs/2502.01919v1 |
Local minima of the empirical risk in high dimension: General theorems and convex examples Kiana Asgari∗Andrea Montanari†Basil Saeed‡ February 5, 2025 Abstract We consider a general model for high-dimensional empirical risk minimization whereby the data xi ared-dimensional isotropic Gaussian vectors, the model is param... | https://arxiv.org/abs/2502.01953v1 |
trivialization under convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 ∗Department of Management Science and Engineering, Stanford University †Department of Statistics and Department of Mathematics, Stanford University ‡Department of Electrical Engineering, Stanford University 1arXiv:2502.01953... | https://arxiv.org/abs/2502.01953v1 |
. . 28 7.1.2 The critical point optimality condition: Proof of point 2. . . . . . . . . . . . . . . 29 7.2 Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 A Random matrix theory: the asymptotics of the Hessian 33 A.1 Preliminary results . . . . . . . . . . . . . . .... | https://arxiv.org/abs/2502.01953v1 |
47 A.6.1 Proof of Lemma 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 A.6.2 Proof of Lemma 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 A.6.3 Proof of Lemma 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 A.6.4 Proof of Lemma 11 ... | https://arxiv.org/abs/2502.01953v1 |
the gradient process: Proof of Lemma 4 . . . . . . . . . . . . . . 63 B.4 Analysis of the determinant: Proof of Lemma 5 . . . . . . . . . . . . . . . . . . . . . . . . 65 B.4.1 Relating the determinant of the differential to the determinant of the Hessian: Proof of Lemma 3 . . . . . . . . . . . . . . . . . . . . . . . ... | https://arxiv.org/abs/2502.01953v1 |
per sample becomes of order one [Mon18]. Among the most interesting of such phenomena: exact or weak recovery phase transitions [DMM09, BM12, LM19, BKM+19, MKL+20]; information-computation gaps [CM22, SW22]; benign overfitting and double descent [HMRT22]. The proportional regime is increasingly of interest because of t... | https://arxiv.org/abs/2502.01953v1 |
binary classification problem whereby yi∈ {0,1}, withP(yi= 1|xi) =φ(ΘT 0xi). It makes sense to fit a two-layer neural network, with k≥k0(for Θ·,ithei-th column of Θ): f(x;Θ) =kX i=1aiσ(ΘT ·,ix). For simplicity, we can think of the second layer weights ( ai) as fixed. The ERM problem can be recast in the form (1) by set... | https://arxiv.org/abs/2502.01953v1 |
of application. In our companion paper we will characterize regimes in which (4) holds for non-convex losses hence determining the asymptotics in those problems as well. 5 1.3 Related work A substantial line of work studies the existence and properties of local minima of the empirical risk for a variety of statistics a... | https://arxiv.org/abs/2502.01953v1 |
see e.g. [DM16, SC19, LSG+21, C ¸LO24]. While our main motivation is to move beyond convexity, our approach recovers and unifies, with some distinctive advantages. Notations We denote by P(Ω) the set of (Borel) probability measure on Ω (which will always be Polish equipped with its Borel σ-field). Additionally, we deno... | https://arxiv.org/abs/2502.01953v1 |
n)∈Rn,ℓ:Rk+k∗+1→R, (u,v, w)7→ℓ(u,v, w), and ρ:R→R. Recall that ˆ µ√ d[Θ,Θ0]denotes the empirical distribution of rows of√ d Θ,Θ0 ∈Rd×(k+k0). We further define R(ˆµ√ d[Θ,Θ0]) :=ΘTΘ ΘTΘ0 ΘT 0Θ ΘT 0Θ0 =Z ttTˆµ√ d[Θ,Θ0](dt). (6) 7 Given block matrix R∈Sk+k0we define the Schur complement of the k0×k0block as: R=R11R10 ... | https://arxiv.org/abs/2502.01953v1 |
formula that upper bounds the number of critical points of the risk (5). To state it, let V:=V(A,B,Π,Pw,µ0) be defined by V(A,B,Π,Pw,µ0) :=n (µ, ν)∈A×B:AR≻R(µ)≻σR,AV≻Eν[vvT]≻σV,Eν[∇ℓ∇ℓT]≻σL, Eν[∇ℓ(v,v0, w)(v,v0)T+Eµ[ρ′(θ)(θ,θ0)T] =0k×(k+k0), µ(θ0)=µ0, ν(w)=Pw, µ⋆(µ, ν)((−∞,0)) = 0o . (14) Further, define Φ gen:P(Rk×Rk0... | https://arxiv.org/abs/2502.01953v1 |
O(1) iterations. For the case k >1, this approach was initiated in [LSG+21], with important steps completed in [C ¸LO24], albeit under the assumption of strong convexity. Even in the convex case, the Kac-Rice approach has some important advantages. Most notably, it can be used to characterizes allpotential minimizers, ... | https://arxiv.org/abs/2502.01953v1 |
F(K,M) := inf u∈S(K)E[ℓ(u+Kz1+Mz 0,R1/2 00z0, w)] +λ 2Tr(K2+MMT). (28) Then 1.Fis convex on Sk ⪰×Rk×k0. 2. The minimizers of Fare in one-to-one correspondence with the solutions Roptof the critical point optimality condition of Definition 1, via (Ropt/R00,Ropt 10R−1/2 00) = ((Kopt)2,Mopt) where Kopt,Moptare the minimiz... | https://arxiv.org/abs/2502.01953v1 |
1 +kX j=1evj . (38) For multinomial regression, Eqs. (32), (33) take the even more explicit form αE[(p(v)−y)(p(v)−y)T] =S−1(R/R00)S−1, (39) E[(p(v)−y)(vT,gT 0)] =0, where p(v) := pj(v) j∈[k]forpjdefined in Eq. (2), and the random variables v,yhave joint distribution defined by v= Prox a(·)(g+Sy;S),P(y=ej) =pj(g0), j... | https://arxiv.org/abs/2502.01953v1 |
(/bardblΘ−Θ0/bardblF) R00=R(1) 00 R00=R(2) 00 R00=R(3) 00Figure 2: Train/test error (log loss), estimation error, and classification error of multinomial regression, for (k+ 1) = 3 symmetric classes, as a function of αfor different values of R00specified in the text. Empirical results are averaged over 100 independent ... | https://arxiv.org/abs/2502.01953v1 |
α= 20. See Fig. 3. 5.2 Fashion-MNIST data 250 features 350 features 500 features 5.0 7.5 10.0 12.5 15.0 17.5 20.0 α0.240.260.280.300.320.340.360.38Test error (classification) 5.0 7.5 10.0 12.5 15.0 17.5 20.0 α0.240.260.280.300.320.340.360.38Test error (classification) 5.0 7.5 10.0 12.5 15.0 17.5 20.0 α0.240.260.280.300.3... | https://arxiv.org/abs/2502.01953v1 |
[HL22, MS22, PKLS23] points in this direction, there is still much unexplained in this agreement. •We constructed isotropic feature vectors xithrough the ‘whitening’ step xi=bΣ−1/2xi. We expect it to be possible to generalize our results to non-isotropic feature vectors, but leave it for future work. 20 6 General empir... | https://arxiv.org/abs/2502.01953v1 |
its covariance has rank mn−rkforrk:=k(k+k0) while Theorem 5 requires the dimension of the index set to be the same as the rank of the covariance of the process. On the other hand, by the KKT conditions, the points ( Θ,¯V) corresponding to critical points of ˆRnbelong to the mn−rkdimensional manifold M0:={G(Θ,¯V) = 0}wh... | https://arxiv.org/abs/2502.01953v1 |
determinant term appearing in Eq. (58) is that of the differential d z(Θ,¯V) defined on the tangent space of M. We relate this to the Euclidean Jacobian of ζ(Θ,¯V) which will be defined on M(1)⊆Rmn. LetBT(Θ,¯V)andBΣ(Θ,¯V)be a basis for the tangent space of Mand the column space of Σ(Θ,¯V) at (Θ,¯V), respectively. Furth... | https://arxiv.org/abs/2502.01953v1 |
det( LTL⊗Id)·det(R(Θ)⊗In) for large nand fixed k, k0. If we use this heuristics in Eq. (72) rewrite various quantities in terms of the empirical measures ˆ µand ˆν, we reach the conclusion of the following lemma. (We refer to Appendix B.3 for its proof.) Lemma 4 (Bounding the density) .Define the Gaussian densities p1(... | https://arxiv.org/abs/2502.01953v1 |
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