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model (H), it holds that: Dz,θ (J)φ(ξ) =X γ∈IH(J)Wz γ(ξ, θ)∂γφ(ξ), (ξ, z, θ )∈RN×Zκ×Θ, (C.8) where the coefficient Wz γ:RN×Θ→Ris explicitly given and takes the form of (C.7) with the function hγ satisfying: hγ(ξ, z, θ )≡0,∀(ξ, z, θ )∈RN×Zκ×Θ for any γ∈IH(J), J= 1,2, satisfying ∥γ∥H ≡3 2|γS|+1 2|γR| =1 2⌊3J 2⌋+3 2. Be... | https://arxiv.org/abs/2502.07047v1 |
W[2],z γ:RN×Θ→Rspecified as (C.7) and W[1],z γis defined as (C.11). Due to (C.12), for any γ∈ZN ≥0 satisfying |γS|=|γR|= 1, it holds that X 1≤k1≤N ek1≤γX 1≤k2≤NW[1],z γ(ξ, θ) [Az,θ]k2k1=X 1≤k1≤N ek1≤γX 1≤k2≤NX 1≤i≤Ngi γ(ξ, z, θ )(ξi−zi)[Az,θ]k2k1 (C.15) with gi γ∈Sκdefined in (C.12). Then the term (C.14) is expressed a... | https://arxiv.org/abs/2502.07047v1 |
Y. Iguchi and A. Beskos •When α= 0, F(α)≤1 2 =α+1 2 ; (C.25) •When α= 1orα= 2, F(α) = 3 2 =α+1 2 α= 1; 5 2 =α+1 2 α= 2,(C.26) •When α≥3, F(α) =1 2⌊3α 2⌋+3 2=α 2+1 2⌊α 2⌋+3 2≤ |α|+1 2, (C.27) since⌊α 2⌋ ≤α−2forα≥3. Thus, (C.24) holds for α∈Z. We now assume that (C.24) holds for α∈Zj ≥0withj≤kand consider the c... | https://arxiv.org/abs/2502.07047v1 |
functions hj1j2 i, hj i1i2∈Sκunder Assumptions 1, 3 and 4. Note that we have performed Taylor expansion around ξ=zto obtain the last line. Then, (C.32) and the assumption of mathematical induction yield that: Dz,θ αφ(x)|z=x=Dz,θ α(k)fLz θφ(x)|z=x=U(i) 1+U(i) 2, where we have set: U(i) 1≡X γ∈JH(α(k))X ν≤γX 1≤i≤NX 1≤j1,j... | https://arxiv.org/abs/2502.07047v1 |
the adjoint operators ( fLz θ)∗and (Dz,θ (J))∗. We divide the proof into the following two cases: (i) J= 0 and (ii) J≥1. Case (i). J= 0. We have from (C.32) that: ∂αfLz θ ψ(H)(t,·) (ξ) ≤C1X ν≤αX 1≤j1,j2≤NX 1≤i≤N ∂ν ξ (ξj1−zj1)(ξj2−zj2) ∂α−ν+eiψ(H)(t, ξ) +C2X ν≤αX 1≤j≤NX NS+1≤i1,i2≤N ∂ν ξ (ξj−zj) ∂α−ν+ei1+ei2ψ(H)(t... | https://arxiv.org/abs/2502.07047v1 |
s +b(I) z,θ ds+ Σ(θ)Bt, t ≥0, (D.2) with Bt= (B1,t, B2,t), t≥0 and A(I) z,θ="1 ε 1−3(z1)2 −1 ε γ −1# , b(I) z,θ=b(z;θ)−A(I) z,θz. The transition density of the scheme (D.2) writes: ¯p(I),z ∆(x, y;θ)≡1 √ (2π)2deta(I)(∆,z)exp −1 2 y−µ(I),z(∆, x, θ)⊤a(I)(∆, z)−1 y−µ(I),z(∆, x, θ) where µ(I),z(∆, x, θ) =z+e∆A(I) z... | https://arxiv.org/abs/2502.07047v1 |
and A. Beskos where the residual Ris characterised as (D.4) and the correction terms are given as: e(II) k(∆, x, y;θ) = 0 , k = 1,2; e(II) 3(∆, x, y;θ) =−∆1/2 2·(s−x1+x3 1+x2)γ ε·H(II) (2)(∆, x, y;θ) −∆3/2 6·6x1(s−x1+x3 1+x2)2+2(s−x1+x3 1+x2)γϵ ε3 ·H(II) (1)(∆, x, y;θ); e(II) 4(∆, x, y;θ) =−∆ 6·γσ2 ε·H(II) (2,2)(∆, x, ... | https://arxiv.org/abs/2502.07047v1 |
2425 3147 1 0.118γ 2300 3032 1 β 4335 5493 1 σ 2482 3018 1 P2’ (CF-expansion, J= 4)ε 2715 2876 1 0.548γ 2926 2809 1 β 5216 4928 1 σ 2799 2908 1 510152025 0.150.300.450.60 246810 0.81.62.43.24.0 510152025 0.15 0.30 0.45 0.60 0.81.62.43.24.0 (a)P1’ (Local Gaussian scheme) 510152025 0.150.300.450.60 246810 0.81.62.43.24.0... | https://arxiv.org/abs/2502.07047v1 |
Schneider. Bayesian inference for discretely sampled Markov processes with closed-form likelihood expansions. J. Financ. Econom. , 8(4):450–480, 2010. A. Vehtari, A. Gelman, D. Simpson, B. Carpenter, and P.-C. B¨ urkner. Rank-normalization, folding, and localization: An improved ˆR for assessing convergence of MCMC (wi... | https://arxiv.org/abs/2502.07047v1 |
General-Purpose f-DP Estimation and Auditing in a Black-Box Setting Önder Askin1,Holger Dette1,Martin Dunsche1,∗,Tim Kutta2,Yun Lu3,Yu Wei4,Vassilis Zikas4† 1Ruhr-University Bochum 2Aarhus University 3University of Victoria 4Georgia Institute of Technology Abstract In this paper we propose new methods to statistically ... | https://arxiv.org/abs/2502.07066v1 |
black-box access. We formulate our two objectives: •Estimation: Given black-box access to a mecha- nism M, estimate its true privacy parameter (i.e., the function finf-DP). •Auditing: Given black-box access to a mechanism Mand a target privacy f, check whether Mviolates the targeted privacy level (i.e., given f, does M... | https://arxiv.org/abs/2502.07066v1 |
the decision to reject/fail to reject can be erroneous and the error rates of these decisions are called α, the "type-I error", and β, the "type-II error". Their formal definitions are α(g):=Pr X∼P[g(X) =1],β(g):=Pr X∼Q[g(X) =0]. One test gis better than another g′, if simultaneously α(g)≤α(g′)and β(g)≤β(g′). This comp... | https://arxiv.org/abs/2502.07066v1 |
2.3 Kernel Density Estimation Kernel density estimation (KDE) is a well-studied tool from non-parametric statistics to approximate an un- known density pby an estimator ˆp. More concretely, in the presence of sample data X1,..., Xn∼pwith Xi∈Rd, the KDE for pis given by ˆp(t):=1 nbdn ∑ i=1Kt−Xi b . One can think of th... | https://arxiv.org/abs/2502.07066v1 |
for the optimal privacy curve fof a mechanism M. This task can be broken down into two parts: (1) Se- lecting datasets D,D′that cause the largest difference inM’s output distributions and (2) Developing an esti- mator/auditor for the trade-off curve given that choice ofD,D′. In line with previous works on black-box est... | https://arxiv.org/abs/2502.07066v1 |
error αis associated with the smallest possible type-II error β(see our introduction for details). Understood as a function in αwe denote the type-II error byT:[0,1]→[0,1]and call it a trade-off curve. We note that any trade-off curve is continuous, non-increasing and convex (see [19]). 4.1 Estimation of the f-DP curve... | https://arxiv.org/abs/2502.07066v1 |
the level sets {q/p=η}for all η. Indeed, suppose we have two estimators ˆp,ˆq, we can run a perturbed LR test with them, just as in equation (2). A short theoretical derivation (found in the appendix) then shows that running the perturbed LR test for ˆp,ˆqand some threshold η, yields the following type-I and type-II er... | https://arxiv.org/abs/2502.07066v1 |
of theT(0)-privacy claim and it is therefore ideally suited as a starting point for our auditing approach in Section 5.2. 5 Goal 2: Auditing f-DP In this section, we develop methods for uncertainty quan- tification in our assessment of T. We begin, with Section 5.1, where we derive (two dimensional) confidence re- gion... | https://arxiv.org/abs/2502.07066v1 |
5.1 states that, assuming the Bayes optimal classifier can be constructed, one can establish simulta- neous confidence intervals for the parameters α(η)and β(η)with a user-specified failure probability γ, which 4Refer to Section 3.5 for the notation and problem setup for Bayesian classification problem. 6 can be made a... | https://arxiv.org/abs/2502.07066v1 |
is below T(0), there seems no plausible way that T(0)-DP is satisfied and the auditor will detect a privacy violation. If, on the other hand, some or all of the values in □γare on or above T(0), our auditor does not detect a violation. Algorithm 2 summarizes the procedure we have just described. It uses a small geometr... | https://arxiv.org/abs/2502.07066v1 |
difference in the output distributions of M(D) andM(D′). We can achieve this by simply choosing D andD′to be as far apart as possible (while still remaining neighbors) and we settle on the choice D= (0,..., 0)and D′= (1,0,..., 0) (10) for all our experiments.6.1 Mechanisms In this section, we test our methods on two fr... | https://arxiv.org/abs/2502.07066v1 |
are depicted in Figure 1 for the DP algorithms described in this section and the appendix. On top of that, we also construct figures that upper and lower bound the worst case errors for the Gaussian mechanism and DP- SGD over the 1000 simulation runs. These plots visually show how the error of the estimator ˆThshrinks ... | https://arxiv.org/abs/2502.07066v1 |
For DP-SGD, we have used the trade-off curve cor- responding to τ=5instead of the true τ=10iterations (Figure 5). Implementation Details The implementation is done using python and R.6. For the simulations, we have used a local device and a server. All runtimes were collected on a local device with an Intel Core i5-113... | https://arxiv.org/abs/2502.07066v1 |
Decision: "No Violation" ✓ (e)n2=107,Ground truth: No Violation; Decision: "No Violation" ✓ (f)n2=108,Ground truth: No Violation; Decision: "No Violation" ✓ Figure 4: Auditing a correct Mechanism: Claimed curve T(0)=TGauss (a,b,c) and T(0)=TSGD(d,e,f). Obtain critical vertical line with step 3 in Algorithm 2 with inter... | https://arxiv.org/abs/2502.07066v1 |
the exact achievable trade-off between type 1 and 2 errors for a given mechanism M. For instance, consider the Gaussian mechanism that adds random noise N(0,σ2)withσ=1 to a statistic Swith sensitivity ∆=1. Given fixed ε>0, δ=Φ∆ 2−ε ∆ −eεΦ −∆ 2−ε ∆ is the optimal achievable δfor this algorithm [7]. Figure 6 shows th... | https://arxiv.org/abs/2502.07066v1 |
DP-SGD. In [4], the same procedure is deployed and combined with specially crafted worst-case initial param- etersθ0to obtain tighter audits for DP-SGD in the black- box setting of [35]. The same method is also used to study various implementations of DP-SGD [5] or the im- pact of shuffling on its privacy [3]. The appr... | https://arxiv.org/abs/2502.07066v1 |
B., AND WANG, Y.Improving the gaussian mechanism for differential privacy: Analytical calibration and optimal de- noising. In Proceedings of the 35th International Conference on Machine Learning (ICML) (2018). [8]BARTHE , G., F ONG, N., G ABOARDI , M., G RÉGOIRE , B., H SU, J.,AND STRUB , P.Advanced probabilistic coupl... | https://arxiv.org/abs/2502.07066v1 |
P., C OURNAPEAU , D., W IESER , E., T AY- LOR, J., B ERG, S., S MITH , N. J., K ERN, R., P ICUS , M., H OYER , S.,VAN KERKWIJK , M. H., B RETT , M., H ALDANE , A., F ER- NÁNDEZ DEL RÍO, J., W IEBE , M., P ETERSON , P., G ÉRARD - MARCHANT , P., S HEPPARD , K., R EDDY , T., W ECKESSER , W., ABBASI , H., G OHLKE , C., AND... | https://arxiv.org/abs/2502.07066v1 |
HIRION , B., G RISEL , O., B LONDEL , M., P RETTENHOFER , P., W EISS, R., D UBOURG , V., ET AL .Scikit-learn: Machine learning in python. Journal of machine learning research 12 , Oct (2011), 2825–2830. [38] SCOTT , D. W. Multivariate Density Estimation: Theory, Prac- tice, and Visualization , 2nd ed. Wiley Series in P... | https://arxiv.org/abs/2502.07066v1 |
distributions of M(D),M(D′) [P]η Mixture distribution with parameter η p,q Probability densities of P,Q α,β type-I & type-II errors (typically of the Neyman-Pearson test) ˆαh,ˆβh Estimated errors using KDE ˜α,˜β Estimated errors using k-NN (typically of the Neyman-Pearson test) T optimal trade-off curve for p,q T(0)tra... | https://arxiv.org/abs/2502.07066v1 |
want to demonstrate the conver- gence|ˆT(α)−T(α)|=o(1)pointwise. More precisely, we will demonstrate that for the pair (α,T(α)), there ex- ist values of ηsuch that ˆαh(η)→αandˆβh(η)→T(α). Since the proofs of both convergence results work ex- actly in the same way, we restrict ourselves in this proof to show that ˆαh(η)... | https://arxiv.org/abs/2502.07066v1 |
choose ζ′′small enough such that Z p·I{η<q/p≤η+ζ′′}<ζ. (16) This explains the second term on the right of equation (15). The third term corresponds to the probability of rejecting whenever q/p=η(this probability is L) times the probability that the coin shows heads (reject) with 7We omit the simpler case where η=0 and ... | https://arxiv.org/abs/2502.07066v1 |
φ∗(x) =argmax {0,1}{Pr[Y=0|X=x],Pr[Y=1|X=x]} (by Bayes classifier φ∗’s construction) =argmax {0,1}{Pr[Y=0,X=x],Pr[Y=1,X=x]} (by Bayes Theorem) =argmax {0,1}{1 ηp(x),q(x)} =ISη(x). (byISη’s definition) For an observation x=⊥, it is easy to check φ∗(x) = ISη(x) =0,asq(x) =0. Then, we also observe that α(η) = Pr X∼P[X∈Sη]... | https://arxiv.org/abs/2502.07066v1 |
also define the maximum violation v∗=sup α∈[0,1] T(0)(α)−T(α) and the set of thresholds Ψ:= η≥0 :T(0)(α(η))−T(α(η))≥v∗/2 . It holds by the proof of Theorem 4.2 case 1) that sup η|ˆαh(η)−α(η)|P→0,as n 1→∞. In particular, it follows that Pr[ˆη∗∈Ψ] =1−rn1, where rn1→0asn1→∞. If the above statement were false, it would ... | https://arxiv.org/abs/2502.07066v1 |
noise scale σ, closed and convex space Θ. Ensure: Final parameter θτ. 1:fort=1,...,τdo 2: Subsampling: Take a uniformly random subsample It⊆ {1,..., r}with batch size m. 3: fori∈Itdo 4: Compute gradient: v(i) t←∇θℓ(θt,xi) 5: end for 6: Average, perturb, and descend: θt+1←θt−ρ ΠΘ 1 m∑ i∈Itv(i) t+Zt! 7:end for 8:Output: ... | https://arxiv.org/abs/2502.07066v1 |
that r=10). Similar to the experiments section, we construct figures that upper and lower bound the worst case errors for the Laplace mechanism and the Subsampling algorithm over 1000 simulation runs. We can see again that the error of the estimator ˆThshrinks significantly, as n1grows. B.3 Additional simulations We pr... | https://arxiv.org/abs/2502.07066v1 |
arXiv:2502.07135v2 [cs.DS] 27 Apr 2025One-Shot Learning for k-SAT Andreas Galanis, Leslie Ann Goldberg, Xusheng Zhang University of Oxford, Oxford OX1 3QD, UK April 29, 2025 Abstract Consider a k-SAT formula Φ where every variable appears at most dtimes, and let σbe a satisfying assignment of Φ sampled proportionally t... | https://arxiv.org/abs/2502.07135v2 |
by [ BM18] and subsequently extended to tensor or weighted variants of the Ising model in [ GM20,MSB22,DDDK21 ]. Beyond the Ising model, [DDP19,DDP20] examined one-shot learning in more general settings, nota bly including logistic regression and higher-order spin systems, obtain ing various algorithmic results in “sof... | https://arxiv.org/abs/2502.07135v2 |
β. The goal is to estimate βusing these inputs. To quantify the accuracy of our estimate, we say that ˆβis anǫ-estimate if|β−ˆβ|≤ǫ. Typically we want ǫto decrease as nincreases so that ǫ→0 whenn→∞. In this case we call ˆβa consistent estimator. On the other hand, if there exists a constant ǫ0>0 such that limsupn|ˆβ−β|≥... | https://arxiv.org/abs/2502.07135v2 |
that, at least for the k-SAT model, the sampling threshold is more relevant to one-shot learning than the satisfiability thres hold. Inaddition, wefindthatifweallow β∗tobeproportionalto k, thenlearningbecomesimpossible for a significantly larger range of d. Specifically, unlike condition ( 3), which requires dto be exponen... | https://arxiv.org/abs/2502.07135v2 |
Ψ 0to show the existence of a stronger gadget Ψ 2, parametrized by b >1 in Lemma 3.4, which guarantees that the all-true assign- mentσ+satisfies Ψ 2and any other assignment with fewer than n/bvariables set to FALSEfails to satisfy Ψ 2. Then we choose bappropriately in terms of β∗to make sure that σ+carries nearly all of... | https://arxiv.org/abs/2502.07135v2 |
the k-SAT problem. A formula Φ = ( V,C) denotes a CNF (Conjunc- tive Normal Form) formula with variables V={x1,...,x n}and clauses C. We use σ(xi) andσi to denote the truth value of the variable xiunder an assignment σ:V→{TRUE,FALSE}. For any clausec∈C,var(c) denotes the set of variables appearing in c(negated or not).... | https://arxiv.org/abs/2502.07135v2 |
/BD[σ−i∧(σi←FALSE)] can take only three values {0,1,eβ/(1+eβ)}. Hence, by grouping the summands according to their values, we obtain m(σ)−/summationdisplay i∈VSi(β) =m(σ)−|{i∈V:Si(β) = 1}|−eβ 1+eβ·/vextendsingle/vextendsingle/vextendsingle/vextendsingle/braceleftbigg i∈V:Si(β) =eβ 1+eβ/bracerightbigg/vextendsingle/vext... | https://arxiv.org/abs/2502.07135v2 |
cis unsatisfied. If there exists a sequence {xc}c∈Csuch that for each c∈C, Pr µ[Ac]≤xc·/productdisplay j∈Γ(c)(1−xj), (14) whereΓ(c)⊆Cis the set of clauses that contain a variable in var(c), thenΦhas a satisfying assignment. Moreover, the distributions PrΦ,βandPrµcan be related as follows: for any event E that can be com... | https://arxiv.org/abs/2502.07135v2 |
m/parenrightbig ≤(ne/m)m) that Pr Φ,β[E]≤Pr µ[E]·/parenleftbigg 1−1 d2k+1/parenrightbigg−|Γ(E)| ≤/parenleftbiggR 2R/3/parenrightbigg/bracketleftBigg d·/parenleftbiggeβ 1+eβ/parenrightbiggk−1/bracketrightBigg2R/3/parenleftbigg 1−1 d2k+1/parenrightbigg−Rd2k ≤(3e/2)2R/3/bracketleftBigg d·/parenleftbiggeβ 1+eβ/parenrightbi... | https://arxiv.org/abs/2502.07135v2 |
of the Claim: Suppose we are not in the first case, so we will show that σassigns FALSEtoxc∈C(ℓ+2)k/2andxd∈Cℓk/2\{xa}. Sinceσsatisfies Ψ 0, it satisfies at least one of Wℓk/2,aand Π (ℓ+1)k/2. Since variables in C(ℓ+1)k/2are all assigned to TRUEby the assumption that we are not in the first case, σdoes not satisfy Π (ℓ+1)k/... | https://arxiv.org/abs/2502.07135v2 |
Corollary 3.3. Letk≥4be an even integer and let n≥kbe a multiple of k/2. Letπbe a permutation of N. Ifσ/\e}atio\slash=σ+satisfies Ψπ 0then there exists a subset Mπ(σ)⊆{0,1,...,2n k−1}of size at leastn ksuch that for all ℓ∈Mπ(σ),σassigns at least one variable in Cπ ℓk/2toFALSE. Remark. While Corollary 3.3does not guarant... | https://arxiv.org/abs/2502.07135v2 |
1−1 b/parenrightbigk/2,n b/bracerightbigg ≥min/braceleftbigg 2·3n 4k·k 2b,4k 3b/parenleftbig 1−1 b/parenrightbig−k/2·3n 4k/parenleftbig 1−1 b/parenrightbigk/2,n b/bracerightbigg ≥n b. The second to last inequality needs some explanation. Let x= (2k/b)(1−1/b)−k/2. Ifx≥2 then ⌊x⌋≥x−x/3 = 2x/3, and this is applied in the ... | https://arxiv.org/abs/2502.07135v2 |
function from (0 ,1) to (0,∞). By (21), we have β≥f(α). We will show that for any β∗≥β, samples from both Pr Ψ2,β∗will beσ+with probability 1 −e−C1nfor some 14 C1>0. Hence, not only does one-shot learning fail with high prob ability, but even eC1n/2many independent samples provide no additional information wit h high p... | https://arxiv.org/abs/2502.07135v2 |
Paramet er estimation for undirected graphical models with hard constraints. IEEE Transactions on Information Theory , 67(10):6790–6809, 2021. [Bre15] Guy Bresler. Efficiently learning Ising models on arb itrary graphs. In Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing , STOC ’15, page 771–7... | https://arxiv.org/abs/2502.07135v2 |
. InAdvances in Neural Information Processing Systems , volume 30, 2017. [HWY23] Kun He, Chunyang Wang, and Yitong Yin. Deterministi c counting lov´ asz local lemma beyondlinearprogramming. In Proceedings of the 2023 Annual ACM-SIAMSymposium on Discrete Algorithms (SODA) , pages 3388–3425, 2023. 17 [JPV22] Vishesh Jain... | https://arxiv.org/abs/2502.07135v2 |
arXiv:2502.07199v1 [cs.LG] 11 Feb 2025Fixed-Confidence Best Arm Identification with Decreasing Variance Tamojeet Roychowdhury1, Kota Srinivas Reddy2, Krishna P Jagannathan2, and Sharayu Moharir1 1Department of Electrical Engineering, IIT Bombay,2Department of Electrical Engineering, IIT Madras Email: tamojeet@iitb.ac.in,... | https://arxiv.org/abs/2502.07199v1 |
time- invariant. The cost incurred by the learner is the number of samples needed by the learner to identify the best arm [1], [2]. In our problem, since the variance of the arms decreases over time and the cost incurred by the learner increases with the number of arms sampled, unlike the classical version of the probl... | https://arxiv.org/abs/2502.07199v1 |
without knowing the number of changes. In [10], the focus is on a non-stationary multiarm bandit setti ng, and the authors show the connection between the extent of allowable reward “variation” and the minimal achievable regret. In [11], the focus is on abruptly changing and slowly varying environments. The authors pro... | https://arxiv.org/abs/2502.07199v1 |
guarantees on their performance. We consider two settings: the first where the difference between the means of the best and the second best arms is known and the second setting where the learner has no side information about the means of theKarms. A. Known Sub-optimality Gap We first focus on the setting where the sub-op... | https://arxiv.org/abs/2502.07199v1 |
the variance of the arm reward decreases with time and therefore, samples collected later are more representative of the mean rewards as compared to earlier samples. Let Xj,tdenote the reward for armjin roundt. Then, the empirical mean of the reward of armjat the end of sampling round r=t/λ, denoted by ˆµj(r)is given b... | https://arxiv.org/abs/2502.07199v1 |
stopping time, and the to tal cost incurred by all four candidate policies. 1. The number of arms is fixed to 5, and the arm means are in an arithmetic progression with the common difference being a varying parameter, from 0.3 to 1 in steps of 0.1. The cost of sampling an arm, c, is fixed to 1 and σ= 10 . 2. The arm mean... | https://arxiv.org/abs/2502.07199v1 |
samples is ηπ=K/parenleftbigtW Kc/parenrightbig =tW c, and the time of the last sample is τπ=tW+tW Kc. Therefore, the cost is given by τπ+cηπ=tW+tW Kc+c/parenleftbiggtW c/parenrightbigg =2σ ∆/parenleftbigg 2+1 Kc/parenrightbigg/radicalBigg Kc·ln/parenleftbiggK δ/parenrightbigg .Now, we need to prove that the fixed-confid... | https://arxiv.org/abs/2502.07199v1 |
Following this, the policy samples all arms for a fixed duration and outputs the arm with the highest empirical mean at the end of this period as the best arm. The second policy samples arms periodically at a frequency that decreases with the number of arms and the sampling cost. It computes an appropriately weighted av... | https://arxiv.org/abs/2502.07199v1 |
and Analytics , vol. 3, pp. 267–283, 2017. [14] G. Ghatak, “Best arm identification based beam acquisit ion in sta- tionary and abruptly changing environments,” IEEE Transactions on Signal Processing , 2024. [15] S. Kalyanakrishnan, A. Tewari, P. Auer, and P. Stone, “P ac subset selection in stochastic multi-armed bandi... | https://arxiv.org/abs/2502.07199v1 |
Riemannian Proximal Sampler for High-accuracy Sampling on Manifolds Yunrui Guan1, Krishnakumar Balasubramanian2, and Shiqian Ma1 1Department of Computational Applied Mathematics and Operations Research, Rice University. 2Department of Statistics, University of California, Davis. 1{yg83,sqma }@rice.edu 2{kbala }@ucdavis... | https://arxiv.org/abs/2502.07265v1 |
of manifold Brownian motion increments–feasible on the sphere. Gatmiry and Vempala (2022) extended this approach to general Hessian manifolds, proving convergence results under the same assumption of exact Brownian motion implementation, which is generally infeasible. Both works require the target density to satisfy a ... | https://arxiv.org/abs/2502.07265v1 |
the following natural question arises: 2 Can one develop high-accuracy algorithms for sampling on Riemannian manifolds? To the best of our knowledge, no prior work exists on providing an affirmative answer to this question. In this work, we develop the Riemannian Proximal Sampler which generalizes the Euclidean Proxima... | https://arxiv.org/abs/2502.07265v1 |
M∥grad logρ1 ρ2∥2dρ1is the relative Fisher information. For more details on LSI, see Appendix G.2. In Euclidean space, such a condition is a relaxation of strongly convex assumption, and is used to establish convergence of sampling algorithms in KL divergence. See, for example, Vempala and Wibisono (2019) (for the Lang... | https://arxiv.org/abs/2502.07265v1 |
x, y )is then defined as the minimal solution of the following heat equation: ∂ ∂tν(t, x, y ) =1 2∆yν(t, x, y ) with ν(0, x, y) =δx(y). 4 Algorithm 1 Riemannian Proximal Sampler fork= 0,1,2, ...do Step 1 (MBI): From xk, sample yk∼πY|X(·, xk)which is a manifold Brownian increment. Step 2 (RHK): From yk, sample xk+1∼πX|Y... | https://arxiv.org/abs/2502.07265v1 |
Recall that ν(η, x, y )denotes the density of manifold Brownian motion with time η. Define a joint distribution πη(x, y)∝e−f(x)ν(η, x, y ). Then, step 2 consists of sampling from the aforementioned distribution. When there is no ambiguity, we omit the step size ηand simply write π(x, y)∝e−f(x)ν(η, x, y ). Algorithm 1 i... | https://arxiv.org/abs/2502.07265v1 |
(2022), assuming that the target distribution satisfies the LSI. In Section 4.1, we consider the case where both steps of Algorithm 1 are implemented exactly, and in Section 4.2, we consider the case when MBI and RHK oracles are inexact. Regarding notation, we let ρX k(x), ρY k(y)denote the law of xandygenerated by Alg... | https://arxiv.org/abs/2502.07265v1 |
section, we derive rates of convergence in the setting where both the MBI and RHK oracles are implemented inexactly. Specifically, we assume we are able to approximately implement the MBI oracle by generating y∼ˆπY|X η(·|x), and approximately implement the RKH oracle by generating x∼ˆπX|Y η(·|y), see Assumption 1 below... | https://arxiv.org/abs/2502.07265v1 |
Assume we want to generate samples from ρthrough rejection sampling. We choose a suitable proposal distribution denoted as µ, and a suitable scaling constant Ksuch that the acceptance rate Kρ(x) µ(x)≤1,∀x. We generate a random proposal x∼µandu∈[0,1]being a uniform random number. Then we compute Kρ(x) µ(x), and accept x... | https://arxiv.org/abs/2502.07265v1 |
generates inexact Brownian motion starting from xwith time η. 5.2 Verification of Assumption 1 We now show that Assumption 1 is satisfied for the aforementioned inexact implementation of the Riemannian Proximal Sampler. To do so, we specifically consider the case when the manifold Mis compact and is a homogeneous space... | https://arxiv.org/abs/2502.07265v1 |
provided in Appendix A.3. Verifying Assumption 1 for this implementation is open. 1IfMis a smooth compact Riemannian manifold then the Wasserstein space P2(M)is the space of Borel probability measures onM, equipped with the Wasserstein metric W2. We refer the reader to Villani (2021) for background on Wasserstein space... | https://arxiv.org/abs/2502.07265v1 |
(iii) extending these techniques to broader classes of manifolds (and other metric-measure spaces). References J. M. Altschuler and S. Chewi. Faster high-accuracy log-concave sampling via algorithmic warm starts. Journal of the ACM , 71(3):1–55, 2024. D. Andersson. Estimates of the spherical and ultraspherical heat ker... | https://arxiv.org/abs/2502.07265v1 |
Riemannian score-based generative modelling. Advances in Neural Information Processing Systems , 35:2406–2422, 2022. P. Dubey and H.-G. M ¨uller. Fr ´echet analysis of variance for random objects. Biometrika , 106(4):803–821, 2019. R. Dwivedi, Y . Chen, M. J. Wainwright, and B. Yu. Log-concave sampling: Metropolis-Hast... | https://arxiv.org/abs/2502.07265v1 |
Bodies. arXiv preprint arXiv:2405.01425 , 2024. J. M. Lee. Introduction to Riemannian manifolds , volume 2. Springer, 2018. Y . T. Lee, R. Shen, and K. Tian. Logsmooth gradient concentration and tighter runtimes for Metropolized Hamiltonian Monte Carlo. In Conference on learning theory , pages 2565–2597. PMLR, 2020. Y ... | https://arxiv.org/abs/2502.07265v1 |
32, 2019. C. Villani. Topics in optimal transportation , volume 58. American Mathematical Soc., 2021. A. Wibisono. Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem. In Conference on Learning Theory , pages 2093–3027. PMLR, 2018. K. Wu, S. Schmidler, and Y . Ch... | https://arxiv.org/abs/2502.07265v1 |
samples independently via different runs) and compute1 10000P10000 i=1d(xi, x∗)2. We use rejection sampling to generate unbiased samples and get an estimation of the true value. Due to the biased nature of the Riemannian LMC method, to achieve a high accuracy we need a small step size. Contrary to the Riemanian LMC met... | https://arxiv.org/abs/2502.07265v1 |
can compute the interior multiplication as iY(dVg) =q det(gij)nX j=1((−1)j+1dxj(Y))dx1∧...∧dˆxj∧...∧dxn =q det(gij)nX j=1((−1)j+1Yj)dx1∧...∧dˆxj∧...∧dxn. We can then compute its exterior derivative as d(iY(dVg)) =nX j=1((−1)j+1∂(Yjp det(gij)) ∂xjdxj)dx1∧...∧dˆxj∧...∧dxn 17 =1p det(gij)nX j=1∂(Yjp det(gij)) ∂xjq det(gij... | https://arxiv.org/abs/2502.07265v1 |
We can now compute the time derivative of the ϕ-divergence along certain flow. LetµX tbe the law of the continuous-time Langevin diffusion with target distribution πX∝e−f(x). That is, we have the following SDE, dXt=−grad f(Xt)dt+√ 2dBt. Then, µX tsatisfies the following Fokker-Planck equation (see Lemma 28 for a proof)... | https://arxiv.org/abs/2502.07265v1 |
t πX t), ρX tgrad log ρX tE 20 +D ϕ′(ρX t πX t) gradρX t πX t, πX tgrad log πX tE +DρX t πX tgrad ϕ′(ρX t πX t), πX tgrad log πX tE dVg(x). Now, notice that D grad ϕ(ρX t πX t), πX tgrad log πX tE =D ϕ′(ρX t πX t) gradρX t πX t, πX tgrad log πX tE . So we get 2∂ ∂tΦπX t(ρX t) =Z MDρX t πX tgrad ϕ′(ρX t πX t), πX tgrad ... | https://arxiv.org/abs/2502.07265v1 |
to obtain Z M−ϕ(ρ− t π− t)∆π− t−ϕ′(ρ− t π− t)∆ρ− t+ϕ′(ρ− t π− t)ρ− t π− t∆π− tdVg(x) =Dπ− t(ρ− t), and used integration by parts, to obtain 2Z Mϕ′(ρ− t π− t) div ( ρ− tgrad logρ− t π− t)dVg(x) =−2Z M⟨grad ϕ′(ρ− t π− t), ρ− tgrad logρ− t π− t⟩dVg(x) =−2Eρ− t[⟨grad ϕ′(ρ− t π− t),grad logρ− t π− t⟩] =−2Dπ− t(ρ− t). C.3 Co... | https://arxiv.org/abs/2502.07265v1 |
Here, the last inequality follows from Lemma 36. Together, we have ∥ρX k(x)−˜ρX k(x)∥TV≤ζRHK+ζMBI+∥˜ρX k−1(x)−ρX k−1(x)∥TV. Iteratively applying this inequality and noting that ∥˜ρX 0(x)−ρX 0(x)∥TV= 0, we obtain ∥ρX k(x)− ˜ρX k(x)∥TV≤k(ζRHK+ζMBI). Recall that Pinsker’s inequality states ∥µ−ν∥TV≤q 1 2Hν(µ). Proof. [Proo... | https://arxiv.org/abs/2502.07265v1 |
)dVg(x)|dVg(x) ≤1 2Z M|Z2e−f(x)νl(η, x, y )−Z1e−f(x)ν(η, x, y )| Z1Z2dVg(x) ≤Z Mmin{Z1, Z2} · |e−f(x)νl(η, x, y )−e−f(x)ν(η, x, y )| 2Z1Z2dVg(x) 25 +Z M|Z2−Z1| ·max{e−f(x)ν(η, x, y ), e−f(x)νl(η, x, y )} 2Z1Z2dVg(x) ≤˜O(ζ) +˜O(Z M|Z2−Z1| ·max{ν(η, x, y ), νl(η, x, y )}dVg(x)) = ˜O(ζ), where by Lemma 20, we obtainmin{Z1... | https://arxiv.org/abs/2502.07265v1 |
forR M|ν(η, x, y )−νl(η, x, y )|dVg(y). Hence we get the desired bound, i.e.,R M|ν(η, x, y )−νl(η, x, y )|dVg(x) =˜O(ζ)andR M|ν(η, x, y )−νl(η, x, y )|dVg(y) =˜O(ζ). 27 E.2 Truncation method on hypersphere LetMbe a hypersphere. In the last subsection, we discussed some existing results which provided a bound on the L2n... | https://arxiv.org/abs/2502.07265v1 |
≥ −logC5+d 2logη+d(x, y)2 2sη−log(1 +1 C5). For all1 Cη≤d(x,y)2 2≤2−1 s Cη, we have δ(x, η) =e−1 ηCη C5 ηd 2exp(−d(x,y)2 2η)≤ηd 2exp(1−1 s ηCη) C5≤1 C5ε1−1 s, so that when εis small, for some C6we have log(1 + δ(x, η))≤C6+ log δ(x, η)≤C6+ log1 C5+ (1−1 s) log1 ε =C6+ log1 C5+1−1 s ηCη≤C6+ log1 C5+1−1 s ηd(x, y)2 2,∀1 C... | https://arxiv.org/abs/2502.07265v1 |
1: We follow exactly the same proof as in Proposition 23, with parameters chosen as s=d d−1, 1 η=L2 1dlog1 ε,Cη=L2 1d,r2/2 =2−d−1 d L2 1dand1 t=L2 1(d−2) log1 ε. Note that t=d d−2η. We know, for all x∈Br(y), we have (for some constant C) −logνl(η, x, y ) + log νl(η, y, y )≥d(x, y)2 2(sη)−C. We want to find tbeing the v... | https://arxiv.org/abs/2502.07265v1 |
truncation: hypersphere In this subsection, we show that on hyperspheres Sd, the truncation error bound ∥ν−νL∥∞=˜O(ζ)can be achieved with truncation level L=˜O(Poly (log1 ε)). As proved in Zhao and Song (2018), the heat kernel on Sdcan be written as the following uniformly convergent series (with φ:=⟨x, y⟩Rd+1) ν(η, x,... | https://arxiv.org/abs/2502.07265v1 |
variation of Lagrangian is given by Z M×Mφ·1 2ηd(x, y)2+ log( γ) + 1−β(x) dVg(x)dVg(y). We want the above to be zero for all φ. Thus we need1 2ηd(x, y)2+log( γ)+1−β(x) = 0 which is equivalent to γ(x, y) =eβ(x)−1 2ηd(x,y)2−1. This implies γ(x, y)∝eβ(x)−1 2ηd(x,y)2Integrating with respect to the yvariable, we get ρX(x)... | https://arxiv.org/abs/2502.07265v1 |
div ( bh) +1 2∆h. By Kolmogorov forward equation (Bakry et al., 2014, Equation 1.5.2), we get ∂tρt=L∗ρt= div ( b(Xt)ρt) +1 2∆ρt= div ( b(Xt)ρt+1 2grad ρt). We briefly mention some properties of Markov semigroup. The following results are from Bakry et al. (2014, Section 1.2). Definition 29. 1.Given a markov process, th... | https://arxiv.org/abs/2502.07265v1 |
Proposition 34. LetMbe a Riemannian manifold with Ricci curvature bounded below by κ. Letρ0be any initial distribution. Assume ρ0satisfies LSI with constant1 d0: Z Mg2logg2dρ0−Z Mg2dρ0logZ Mg2dρ0≤2d0Z M∥grad g∥2dρ0,∀g. Then the propagation of ρ0along heat flow, denoted as ρt, satisfies LSI with constant 1 2c(t) +d0e−κt... | https://arxiv.org/abs/2502.07265v1 |
the situation that we cannot compute the minimizer of g(x) =f(x) +1 2ηd(x, y)2, and instead use yas the approximation of it. Notice that when ηis small, since f(x)is uniformly bounded, the function g(x)is dominated by1 2ηd(x, y)2, thus the minimizer of gwill be close to y. Therefore it is reasonable to use yas an appro... | https://arxiv.org/abs/2502.07265v1 |
follows: R Mexp(−1 2td(x, y)2)dVg(x)R Mexp(−f(x) +f(y)−1 2ηd(x, y)2−Cε 2)dVg(x)≤R Mexp(−1 2td(x, y)2)dVg(x)R Mexp(−1 2Td(x, y)2−Cε)dVg(x). Using Li and Erdogdu (2023, Lemma 8.2) and Li and Erdogdu (2023, Lemma C.5), when β≥d R2, using Riemannian normal coordinates we have the following lower bound on the integral: Z Me... | https://arxiv.org/abs/2502.07265v1 |
Uniform Kernel Prober Soumya Mukherjee and Bharath K. Sriperumbudur Department of Statistics Pennsylvania State University, University Park, PA 16802, USA. {szm6510,bks18}@psu.edu February 12, 2025 Abstract The ability to identify useful features or representations of the input data based on training data that achieves... | https://arxiv.org/abs/2502.07369v1 |
et al. (2019); Boix-Adsera et al. (2022). An ideal pseudometric must be interpretable and efficiently computable based on a reasonably small amount of data samples. It must also be sensitive only to differences in features that will lead to differences in predictive performance, but be fairly insensitive to any other d... | https://arxiv.org/abs/2502.07369v1 |
i.e., pairs of inputs and outputs to the model. 2 5.It is possible to design a statistically efficient estimator for the UKP distance based on a finite number ( n) of samples from the input domain, that enjoys an estimation error rate of n−1/2. 6.The UKP distance enables us to even compare representations that differ i... | https://arxiv.org/abs/2502.07369v1 |
as, dUKP λ,K(ϕ, ψ):= sup ∥η∥L2(PX)≤1 E[αλ(X)−βλ(X)]21 2, where αλandβλare defined in Equations (1)and(2), respectively. 3 Properties of dUKP λ,K LetIϕ:Hϕ→L2(PX), f→fbe the inclusion operator, which maps any f∈ H ϕto its representation f∈L2(PX). Then the adjoint of the inclusion operator is given by I∗ ϕ:L2(PX)→ Hϕ, f... | https://arxiv.org/abs/2502.07369v1 |
For any function ϕ:Rd→Rkfor some k∈N,dUKP λ,K(ϕ, ϕ) = 0, 5 2.(Non-negativity) For any two functions ϕ:Rd→Rkandψ:Rd→Rlfor some k, l∈N, dUKP λ,K(ϕ, ψ)≥0, 3.(Symmetric) For any two functions ϕ:Rd→Rkandψ:Rd→Rlfor some k, l∈N, dUKP λ,K(ϕ, ψ) =dUKP λ,K(ψ, ϕ), 4.(Triangle inequality) For any three functions ϕ:Rd→Rk,ψ:Rd→Rland... | https://arxiv.org/abs/2502.07369v1 |
Proposition 1, we arrive at the following V-statistic type estimator of dUKP λ,K(ϕ, ψ): ˆdUKP λ,K(ϕ, ψ) =h Tr ˆΣ−λ ϕˆΣϕˆΣ−λ ϕˆΣϕ +Tr ˆΣ−λ ψˆΣψˆΣ−λ ψˆΣψ −2Tr ˆΣ−λ ϕˆΣϕψˆΣ−λ ψˆΣψϕi 1 2,(3) where ˆΣϕ=1 nnX i=1Kϕ(·, Xi)⊗HϕKϕ(·, Xi) =1 nnX i=1K(ϕ(·), ϕ(Xi))⊗HϕK(ϕ(·), ϕ(Xi)), ˆΣψ=1 nnX i=1Kψ(·, Xi)⊗HψKψ(·, Xi) =1 nnX i... | https://arxiv.org/abs/2502.07369v1 |
ˆdRCCA λ,K (ϕ, ψ) =Tr ˆΣ−λ ϕˆΣϕψˆΣ−λ ψˆΣψϕ =nX i=1nX j=1µ(i) ϕµ(j) ψ µ(i) ϕ+nλ µ(j) ψ+nλ c(i),(j) ϕ,ψ2 . However, the machine learning literature has largely focused on the original Ridge-CCA formulation with a linear kernel, as discussed in Kornblith et al. (2019). The classical CCA distance ˆdCCAcan be derive... | https://arxiv.org/abs/2502.07369v1 |
data and computational requirements. 4.2 Finite sample convergence rate of ˆdUKP λ,K From a statistical estimation viewpoint, it is possible that the estimator ˆdUKP λ,Kconverges to dUKP λ,K as the number of data samples X1, . . . , X nfrom the input domain grows to infinity. In addition, we also provide a rate of conv... | https://arxiv.org/abs/2502.07369v1 |
are estimated for each pair of representations using 5,000 test images from the same dataset. We create synthetic kernel ridge regression tasks where we randomly sample 5000 images and randomly assign a standard Gaussian label to each image to create the synthetic label/target vector. We obtain the kernel ridge regress... | https://arxiv.org/abs/2502.07369v1 |
pseudometric. Concurrently, we perform an agglomerative (bottom-up) hierarchical clustering of the representations based on the pairwise UKP distances and obtain the corresponding dendrogram. We observe in Fig. 2 that similar architectures which share important properties, such as the Regnets and Resnets are clustered ... | https://arxiv.org/abs/2502.07369v1 |
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