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we have 0≤FA,B(eρ)−FA,B(bρ)≤eOP(1). (145) SinceeOP(1)/FA,B(eρ) =eOP(1)/√ A2+B2≤eOP(d−b/2) =oP(1), it implies 1−oP(1) =FA,B(bρ) FA,B(eρ)=bρA√ A2+B2+p 1−bρ2B√ A2+B2=bρeρ+p 1−bρ2p 1−eρ2. Therefore, eρ= 1−oP(1) =⇒bρ= 1−oP(1) eρ=oP(1) =⇒ bρ=oP(1)(146) (a): Ifa−c<b , thendb/2≫d(a−c)/2and by Eq. (144) and (146) we have eρ,bρ=... | https://arxiv.org/abs/2502.11323v1 |
2−c∨1),p 1−bρ2⟨zsv−(bθ),bθ⟩=eOP(1). 98 (b) Ifa>b +c, then ⟨zsv+(bθ),bθ⟩=r d πn 1 +oP(1) ,⟨zsv−(bθ),bθ⟩=eOP(1). Proof.SV±(θ) may not be tractable, since it involves a nuisance term bρgias defined in Eq. (147). Therefore, we introduce a proxy of support vectors, which is easier to work with. Formally, let V+=V+(θ) := a... | https://arxiv.org/abs/2502.11323v1 |
i∈I+⟨zi−z+,eθ⟩+⟨z+,eθ−eθ+⟩, −⟨zv−(eθ),eθ⟩−∥P⊥ µz−∥2= min i∈I−⟨−zi,eθ⟩+⟨z−,eθ−⟩= min i∈I−⟨z−−zi,eθ⟩−⟨z−,eθ−eθ−⟩.(157) Now we study the two terms on the R.H.S. of Eq. (157). For the first term, based on Eq. (142), min i∈I+⟨zi−z+,eθ⟩≥− max i∈I+ ⟨zi−z+,eθ⟩ =eOP(1), min i∈I−⟨z−−zi,eθ⟩≥− max i∈I− ⟨zi−z−,eθ⟩ =eOP(1).(158) For... | https://arxiv.org/abs/2502.11323v1 |
i∈[n]|gi|=eOP(1), and Lemma F.4: τ τ+ 1p 1−bρ2⟨zsv−(bθ),bθ⟩ =eOP(1). For(a), plugging bρ= 1−oP(1) by Lemma F.3(a) and asymptotics of ⟨zsv+(bθ),bθ⟩by Lemma F.4(a). For(b), plugging bρ= 2d(b−a+c)/2 1 +oP(1) by Lemma F.3(b) from i., while bρ∥µ∥2=oP(1) from ii., and asymptotics of ⟨zsv+(bθ),bθ⟩by Lemma F.4(b). This compl... | https://arxiv.org/abs/2502.11323v1 |
b≥1 2−o(1). This concludes the proof for low signal regime. Finally, we complete the proof of Theorem 5.5. G Confidence estimation and calibration: Proofs for Section 6 G.1 Proof of Proposition 6.1 The following preliminary result summarizes the precise asymptotics of three quantities: bp(x) (max- margin confidence), p... | https://arxiv.org/abs/2502.11323v1 |
in δas MSE∗, which concludes the proof of part (b). G.2 Verification of Claim 6.2 The analytical dependence of CalErr∗and ConfErr∗on model parameters is more complicated. We provide a numerical verification of Claim 6.2. Verification of Claim 6.2 .For CalErr∗, denote h1(t) :=E σ 2t(G+t) +c −σ(G+t)2 h2(t) :=E σ ... | https://arxiv.org/abs/2502.11323v1 |
, wherecis an absolute constant. In particular, when B=Id, P ∥x∥2√ d−1 >t ≤2 exp −ct2d K4 . (c) (Hoeffding’s inequality) Let a∈Rdbe a vector. Then, for every t≥0, P|⟨x,a⟩| ∥a∥2>t ≤2 exp −ct2 K2 , wherecis an absolute constant. (d) (Bernstein’s inequality) Let B∈Rd×dbe a matrix. Then, for every t≥0, P|xTBy| ∥B∥... | https://arxiv.org/abs/2502.11323v1 |
z=proxℓ(x;λ). Moreover, eℓ(x;λ)is non-increasing in λandeℓ(x;λ)→ℓ(x)whenλ→0+. (b)proxℓ(x;λ)is continuous in (x,λ). Ifℓis twice differentiable, then proxℓ(x;λ)is also differ- 112 entiable in its domain, with partial derivatives ∂proxℓ(x;λ) ∂x=1 1 +λℓ′′(z) z=proxℓ(x;λ)∂proxℓ(x;λ) ∂λ=−ℓ′(z) 1 +λℓ′′(z) z=proxℓ(x;λ). Moreov... | https://arxiv.org/abs/2502.11323v1 |
2019. 3, 16 [12] Emmanuel J. Cand` es and Pragya Sur. The phase transition for the existence of the maximum likelihood estimate in high-dimensional logistic regression. The Annals of Statistics , 48(1):27 – 42, 2020. 3, 16 [13] Andrea Montanari, Feng Ruan, Youngtak Sohn, and Jun Yan. The generalization error of max-mar... | https://arxiv.org/abs/2502.11323v1 |
Nacson, Suriya Gunasekar, and Nathan Srebro. The implicit bias of gradient descent on separable data. Journal of Machine Learning Research , 19(70):1–57, 2018. 5, 17, 36, 37 115 [29] Ziwei Ji and Matus Telgarsky. Risk and parameter convergence of logistic regression, 2019. 5, 17 [30] Behnam Neyshabur, Ryota Tomioka, an... | https://arxiv.org/abs/2502.11323v1 |
[44] Grigoris Karakoulas and John Shawe-Taylor. Optimizing classifers for imbalanced training sets. Advances in neural information processing systems , 11, 1998. 10 [45] Gang Wu and Edward Y Chang. Class-boundary alignment for imbalanced dataset learning. InICML 2003 workshop on learning from imbalanced data sets II, W... | https://arxiv.org/abs/2502.11323v1 |
Proceedings of the National Academy of Sciences , 110(36):14557–14562, 2013. 16 [64] David Donoho and Andrea Montanari. High dimensional robust m-estimation: Asymptotic variance via approximate message passing. Probability Theory and Related Fields , 166:935–969, 2016. 16, 113 [65] Ganesh Ramachandra Kini, Orestis Para... | https://arxiv.org/abs/2502.11323v1 |
MOMENT MONOTONICITY OF WEIBULL , GAMMA AND LOG-NORMAL DISTRIBUTIONS Kang Liu˚ Independent Researcher liukangk11@gmail.com ABSTRACT This paper investigates the moment monotonicity property of Weibull, Gamma, and Log-normal distributions. We provide the first complete mathematical proofs for the monotonicity of the funct... | https://arxiv.org/abs/2502.11366v1 |
which can significantly benefit related areas of research, such as parameter estimation for these distributions. 3 Weibull, Gamma and Log-normal Distribution and their Moments The probability density function of a Weibull random variable is given by the following formula: fpx;k, λq“# k λ`x λ˘k´1e´px λqk forxě0 0 forxă0... | https://arxiv.org/abs/2502.11366v1 |
Γ1´ 1`m k¯ ´Γ´ 1`m k¯ Γ1´ 1`n k¯ ă0.(14) Therefore,dR dkă0, i.e., Ris decreasing with respect to k. Since kPp0,`8q , and lim kÑ8R“Γmp1q Γnp1q“1. Thus, we haveRě1, in other words, EpXnqměEpXmqnùñEpXnq1 něEpXmq1 m. (15) 4.2 Moment Monotonicity of Gamma Distribution The following theorem states the moment monotonicity pro... | https://arxiv.org/abs/2502.11366v1 |
on these distributions to model complex systems and processes. References Enrico Zio. Reliability engineering: Old problems and new challenges. Reliability engineering & system safety , 94(2): 125–141, 2009. Kailash C Kapur and Michael G Pecht. Reliability engineering . John Wiley & Sons, 2014. Zhaoyi Xu and Joseph Hom... | https://arxiv.org/abs/2502.11366v1 |
embeddings. arXiv preprint arXiv:2404.17606 , 2024. J Li. Production Systems Engineering . Springer, 2008. Bin Yu, William HK Lam, and Mei Lam Tam. Bus arrival time prediction at bus stop with multiple routes. Transportation Research Part C: Emerging Technologies , 19(6):1157–1170, 2011. Alfréd Rényi. Probability theor... | https://arxiv.org/abs/2502.11366v1 |
arXiv:2502.11432v2 [econ.EM] 11 Mar 2025MAXIMAL INEQUALITIES FOR SEPARATELY EXCHANGEABLE EMPIRICAL PROCESSES HAROLD D. CHIANG Abstract. This paper derives new maximal inequalities for empirical p rocesses asso- ciated with separately exchangeable random arrays. For fixe d index dimension K≥1, we establish a global maxim... | https://arxiv.org/abs/2502.11432v2 |
ar- rays exists in the literature. Establishing a local maximal inequality for SE arrays is especially challenging because their intricate multiway d ependence structure induces com- plex interactions among observations. Unlike in the i.i.d. orU-statistics settings, the multidimensional dependencies inherent to SE arra... | https://arxiv.org/abs/2502.11432v2 |
the set of positive integers and Rfor the real line. For a,b∈R, leta∨b= max{a,b}anda∧b= min{a,b}. Denote for m∈Nthat [m] = {1,2,...,n}.For two real vectors a= (a1,...,aK) andb= (b1,...,bK), we denote a≤b foraj≤bjfor all 1≤j≤K. Let supp( a) ={j:aj/\e}atio\slash= 0}. We denote by ⊙the Hadamard product, i.e., for i= (i1,.... | https://arxiv.org/abs/2502.11432v2 |
q,K, andkby E /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1/radicalbig |IN,e|/summationdisplay i∈IN,eε1,i1...εk,ik·(Pef)({Ui⊙e′}e′≤e)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleq F 1/q . 6 H. D. CHIANG DenotePIN,e=|IN,e|−1/summati... | https://arxiv.org/abs/2502.11432v2 |
zero elsewhere. By applying the symmetrisatio n of Lemma B.1 in Chiang et al. (2023), one has, for independent Rademacher r. v.’s (ε1,i1),...,(εk,ik) that are independent of ( Xi)i∈NK, that |IN,e|1/2E[/bardblHe N(f)/bardblF]/lessorsimilarE /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextend... | https://arxiv.org/abs/2502.11432v2 |
it then follows that /tildewidez2/lessorsimilar∆2+/bardblMe/bardblP,2√n/bardblPeF/bardblP,2Je(/tildewidez). By applying Lemma 4 and Lemma 2.1 of van der Vaart and Wellner ( 2011) with J=Je, A= ∆,B=/radicalbig /bardblMe/bardblP,2/√n/bardblPeF/bardblP,2andr= 1, it yields that Je(z)≤Je(/tildewidez)/lessorsimilarJe(∆)/brac... | https://arxiv.org/abs/2502.11432v2 |
called Vapnik–Chervonenkis-type (VC-type) with charac teristics (A,v) if sup QN(F,/bardbl·/bardblQ,2,ε/bardblF/bardblQ,2)≤/parenleftbiggA ε/parenrightbiggv for all 0<ε≤1, where the supremum is taken over all finite discrete distribu tions. By adapting the ar- guments used in the proofs of Corollaries 5.3 and 5.5 and Lem... | https://arxiv.org/abs/2502.11432v2 |
multiplier bootstrap: finite sample app roxi- mations to the U-process supremum with applications,” Probability Theory and Related Fields, 1–67. Chernozhukov, V., D. Chetverikov, M. Demirer, E. Duflo, C. Ha nsen, W. Newey, and J. Robins (2018): “Double/debiased machine learning for treatment and structural parameters,” ... | https://arxiv.org/abs/2502.11432v2 |
T-calibration in semi-parametric models Anja M¨ uhlemann∗and Johanna Ziegel† February 18, 2025 Abstract This note relates the calibration of models to the consistent loss functions for the target functional of the model. We demonstrate that a model is calibrated if and only if there is a parameter value that is optimal... | https://arxiv.org/abs/2502.11727v1 |
specified as at (1) with a unique correct parameter β∗, we obtain that β∗= arg min β∈ΘELϕ(m(X;β), Y). (3) The Bregman loss functions at (2) are all consistent loss functions for the mean in the sense of Gneiting (2011, Definition 1). More precisely, Savage (1971) has shown that, under mild regularity conditions, any lo... | https://arxiv.org/abs/2502.11727v1 |
moment, V(z, y) =z−y2is an identification function, and for the α-quantiles, one can take the identification function V(z, y) = 1{y < z} −α. Further examples including some robust location functionals are given in Jordan et al. (2022, Table 1). For a given identification function V, we define the elementary loss functi... | https://arxiv.org/abs/2502.11727v1 |
X= (X1, . . . , X k) and T(Y|X=x) depends on xkbut m(x;β∗) = T(Y|(X1, . . . , X k−1) = ( x1, . . . , x k−1)), or, in other words: As soon as all m(X;β) are measurable with respect to a strict sub- σ-algebra Aofσ(X) and the optimal predictor with respect to Ais in our model (and has parameter β∗), then we will have β(H)... | https://arxiv.org/abs/2502.11727v1 |
h↓0V(m(X;β∗) +h, Y) almost surely as n→ ∞ . L´ evy’s Zero-One-Law yields that 0≤E(V(m(X;β∗), Y)−V(Zn, Y)| An)→E(V(m(X;β∗), Y)|A∞)−¯Z∞ almost surely as n→ ∞ , where A∞=S nAn=σ(m(X;β∗)). Using the integrability assumption, we obtain that 0≤E(V(m(X;β∗), Y)−V(Zn, Y))→0, hence E(V(m(X;β∗), Y)|m(X;β∗)) = ¯Z∞≤0. Analogous arg... | https://arxiv.org/abs/2502.11727v1 |
η/b(η−x2)φ(x)dx=(b2−η)η2 b4φ(η/b), which is increasing in bforη <0 and has a local minimum at b=√ηforη >0. For b <0, we have d dbZη/b −∞(η−x2)φ(x)dx=(η−b2)η2 b4φ(η/b), which is decreasing in bforη <0 and has a local minimum at b=−√ηforη >0. At the local minimum the function takes the value Z−√η −∞(η−x2)φ(x)dx=Z∞ √η(η−x... | https://arxiv.org/abs/2502.11727v1 |
Fissler, and J. Ziegel. Characterizing M-estimators. Biometrika , 111: 339–346, 2024. W. Ehm, T. Gneiting, A. Jordan, and F. Kr¨ uger. Of quantiles and expectiles: Consistent scoring functions, Choquet representations, and forecast rankings (with discussion). Journal of the Royal Statistical Society: Series B , 78:505–... | https://arxiv.org/abs/2502.11727v1 |
Low-Rank Thinning Annabelle Michael Carrell1Albert Gong2Abhishek Shetty3Raaz Dwivedi2Lester Mackey4 Abstract The goal in thinning is to summarize a dataset using a small set of representative points. Re- markably, sub-Gaussian thinning algorithms like Kernel Halving and Compress can match the quality of uniform subsamp... | https://arxiv.org/abs/2502.12063v5 |
. ,xn]⊤∈Rn×dto denote the input point matrix so that Ex∼Pin[x] =X⊤pinandEx∼Pout[x] =X⊤pout. We will make use of two common measures of summa- rization quality. 1arXiv:2502.12063v5 [stat.ML] 25 Apr 2025 Low-Rank Thinning Table 1: Examples of (K, ν, δ)-sub-Gaussian thinning algorithms. For input size nin, output size nou... | https://arxiv.org/abs/2502.12063v5 |
a linear- kernel variant (LKH (δ)) with nindruntime in App. B.3. To round out our set of examples, we show in App. B.6.1 that two new thinning algorithms based on the Gram-Schmidt walk of Bansal et al. (2018) yield even smaller νat the cost of increased runtime. We call these algorithms Gram- Schmidt Thinning (GS-T HIN... | https://arxiv.org/abs/2502.12063v5 |
fact immediately yields an MMD guarantee for each algorithm in Tab. 1. We present a representative guarantee for KH (δ). Corollary 1 (Gaussian MMD of KH ).IfXin⊂Bd(R) forR > 0, then KH(δ)withk=GAUSS (η), and n=nin delivers MMD2 K(pin,pout)≤ O log(nout/δ) n2 out log(nout)∨(R2η) dd+ log(1 δ′) with probability at leas... | https://arxiv.org/abs/2502.12063v5 |
kattthat mim- ics the special structure of the softmax matrix T. Alg. 1 uses the attention kernel and a high-quality thinning algo- rithm, KH-C OMPRESS (0.5), to subselect key-value pairs and then computes exact attention (8) for the key-value subset. In total, this requires only O(d n2 out)time to run KH-C OMPRESS (0.... | https://arxiv.org/abs/2502.12063v5 |
2+2γ) time. In contrast, the bTkdeandbThypbounds require quadratic runtime to guarantee O(1√n)error in the best case (∥V∥op=O(1)) and cannot guarantee consistent sub- quadratic estimation in the worst case ( ∥V∥op= Ω(√n)). 4.2. Thinning attention in practice To gauge the practical effectiveness of Alg. 1, we recre- ate... | https://arxiv.org/abs/2502.12063v5 |
permutation-based SGD algorithm that gives a dimension-free upper bound while maintaining the same dependency on other factors.” 5 Low-Rank Thinning 0 10 20 30 40 50 Epochs0.3350.3360.3370.3380.339Full Train Loss 0 10 20 30 40 50 Epochs82.082.182.282.382.4Test Accuracy 0 200 400 600 800 1000 1200 Seconds0.3350.3360.337... | https://arxiv.org/abs/2502.12063v5 |
state-of-the-art test accu- racy of CD-GraB: Greedy and lags only slightly in terms of training convergence. See https://github.com/ microsoft/khsgd for PyTorch code replicating this experiment and App. L.2 for supplementary experiment de- tails. 6. Cheap Two-Sample Testing A core task in statistics and machine learnin... | https://arxiv.org/abs/2502.12063v5 |
size nout= 2gq m+n sfor each bin fori= 1, . . . , s mdoP(i) out←KT-C OMPRESS (δ)(X(i),g,k) fori= 1, . . . , s ndoQ(i) out←KT-C OMPRESS (δ)(Y(i),g,k) // Compute C ORESET MMD test statistic MB+1←MMD k(1 smPsm i=1P(i) out,1 snPsn i=1Q(i) out) (11) // Simulate null by randomly permuting the scoresets Btimes forb= 1, . . . ... | https://arxiv.org/abs/2502.12063v5 |
4 (Power of deep manifold kernel CTT ).Un- der the assumptions of Cor. 3, if x1,y1,(x1, ϕ(x1)), and(y1, ϕ(y1))belong to smooth compact manifolds (As- sump. E.1) with dimension d⋆< d′then CTT satisfies the conclusions of Thm. 4 with Rkdeep=O(log5d⋆ 4+3 2(n eβ)). Cors. 3 and 4 follow from explicitly bounding the eigenval... | https://arxiv.org/abs/2502.12063v5 |
and datasets. References Alman, J. and Song, Z. Fast attention requires bounded entries. Advances in Neural Information Processing Systems , 36, 2024. (Cited on page 4.) Altschuler, J., Bach, F., Rudi, A., and Niles-Weed, J. Massively scalable sinkhorn distances via the nystr ¨om method. Advances in Neural Information ... | https://arxiv.org/abs/2502.12063v5 |
page 4.) Dosovitskiy, A., Beyer, L., Kolesnikov, A., Weissenborn, D., Zhai, X., Unterthiner, T., Dehghani, M., Minderer, M., Heigold, G., Gelly, S., Uszkoreit, J., and Houlsby, N. An im- age is worth 16x16 words: Transformers for image recognition at scale. In International Conference on Learning Representa- tions , 20... | https://arxiv.org/abs/2502.12063v5 |
Information Processing Systems , 32, 2019. (Cited on page 35.) Phillips, J. M. and Tai, W. M. Near-optimal coresets of kernel density estimates. Discrete & Computational Geometry , 63 (4):867–887, 2020. (Cited on page 1.) Rahimi, A. and Recht, B. Random features for large-scale kernel machines. Advances in Neural Infor... | https://arxiv.org/abs/2502.12063v5 |
computer vision and pattern recognition , pp. 2743–2751, 2018. (Cited on page 8.) 10 Low-Rank Thinning Appendix Contents A Appendix Notation and Definitions 12 B Proof of Tab. 1: Sub-Gaussian Thinning Examples 13 B.1 S UBSAMPLING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... | https://arxiv.org/abs/2502.12063v5 |
Proof of Lipschitz kernel max seminorm bound (4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 C.4 Proof of Lem. C.4: Bounded-H ¨older sub-Gaussian process . . . . . . . . . . . . . . . . . . . . . . . . . 28 D Proof of Cor. 1: Gaussian MMD of KH 28 E Proof of Cor. 2: Intrinsic Gaussian MMD of KH 29 F Pr... | https://arxiv.org/abs/2502.12063v5 |
to several useful notions of sub-Gaussianity. Definition A.1 (Sub-Gaussian vector ).We say that a random vector w∈Rnis(K, ν)-sub-Gaussian on an event EifK is SPSD and ν >0satisfies EE exp(u⊤Kw) ≤exp(ν2 2·u⊤Ku)for all u∈Rn. (14) If, in addition, the event has probability 1, we say that wis(K, ν)-sub-Gaussian . Notably... | https://arxiv.org/abs/2502.12063v5 |
proof is complete. B. Proof of Tab. 1: Sub-Gaussian Thinning Examples This section provides supplementary details for each of the sub-Gaussian thinning algorithms of Tab. 1. B.1. S UBSAMPLING B.1.1. P ROOF OF PROP. 1:QUALITY OF UNIFORM SUBSAMPLING We begin by computing the first and second moments of pout:E[pout] =pina... | https://arxiv.org/abs/2502.12063v5 |
RKHS Hk with the same fiandaisequences employed in KH (δ), the output ψiof each round is (k, σi)-sub-Gaussian for σ2 0≜0and σ2 i≜σ2 i−1+∥fi∥2 k 1 +σ2 i−1 a2 i(∥fi∥2 k−2ai) +∀i≥1. (19) The following lemma bounds the growth of the sub-Gaussian constants σiin terms of the swapping thresholds ai. Lemma B.1 (Growth of SBH... | https://arxiv.org/abs/2502.12063v5 |
params( σ,b, δ): a←max(bσ√ 2 log(2 /δ),b2) σ2←σ2+b2(1+(b2−2a)σ2/a2)+ return (a, σ); By Lem. A.1, we thus have that the LKH (δ)output pin−poutis(K, ν)-sub-Gaussian on EforKgenerated by kand that LKH (δ)∈ Gν,δ(K). B.3.1. P ROOF OF PROP. B.3: SUB-GAUSSIANITY OF LKH (δ) We begin by studying the sub-Gaussian properties of a... | https://arxiv.org/abs/2502.12063v5 |
KH-C OMPRESS (δ)).Ifnout∈√nin2Nthen KH-C OMPRESS (δ)(Alg. B.4) is (k, ν)-sub-Gaussian with ν=1 noutq log2(nout) log(4noutlog2(nin/nout) δ) max x∈X inp k(x,x) on an event Eof probability at least 1−δ/2. Proof. Since the original Kernel Halving algorithm of Dwivedi & Mackey (2024, Alg. 2) is equal to the KT-S PLIT al- go... | https://arxiv.org/abs/2502.12063v5 |
gswalk( (xi)nin i=1)in Alg. B.6 and z′ 1,z′ 2, . . . be the fractional assignment sequence generated by kernel gswalk cubic( (xi)nin i=1)in Alg. B.7 with an identical source of randomness. If the pairwise difference matrix Q≜(k(x2i−1, x2j−1) +k(x2i, x2j)−k(x2i−1, x2j)−k(x2i, x2j−1))i,j∈[nin/2] is positive definite, the... | https://arxiv.org/abs/2502.12063v5 |
to update fractional assignment vector Compute utas(ut)A2=−CQA2×{p′},utp′= 1, anduti= 0fori /∈ A′ δ+← |max ∆ |andδ−← |min ∆ |, where ∆ =n δ∈R:zt+δut∈[−1,+1]nin/2o // Select candidate step sizes δt←δ+with probability δ−/(δ++δ−); otherwise δt← −δ−// Choose step size and sign at random zt+1←zt+δtut// Update fractional ass... | https://arxiv.org/abs/2502.12063v5 |
after the update respectively. The main difference between Algs. B.6 and B.7 is in the computation of the step direction ut, which is the solution of the program ut←argminu∈Rnu⊤Qusubject to up′= 1 andui= 0 for all i /∈ A′. uthas a closed form with entries (ut)A2=−(QA2×A2)−1·QA2×{p′}. Note that the invertibility of QA2×... | https://arxiv.org/abs/2502.12063v5 |
generated by k, then GS-C OMPRESS is (K, ν,0)-sub-Gaussian with ν≜1 noutp log2(nout)∥K∥max. Moreover, GS-C OMPRESS has an O(n3 out)runtime. Proof. By Lem. B.2 and Prop. B.8, GS-H ALVE -CUBIC is(K, νH(ℓ))-sub-Gaussian for an input point sequence of size ℓ andνH(ℓ) = 2p ∥K∥max/ℓ. Hence, by Lem. A.2, GS-H ALVE -CUBIC is a... | https://arxiv.org/abs/2502.12063v5 |
Thinning for a subset Cε,rwith|Cε,r| ≤(1 +2 ε)rand∥u∥2≤1for each u∈ Cε,r. Now fix any u∈ Cε,rand let Λr= diag( λ1, . . . , λ r). Using (23) and (24), we have Vr=Φ⊤VrΛ−1/2 r and therefore ⟨Vru,Φ⊤w⟩=⟨Φ⊤VrΛ−1/2 ru,Φ⊤w⟩=⟨VrΛ−1/2 ru,Kw⟩. In addition, we have (VrΛ−1/2 ru)⊤K(VrΛ−1/2 ru) =u⊤Λ−1/2 rV⊤ rVΛV⊤VrΛ−1/2 ru=u⊤u. Next,... | https://arxiv.org/abs/2502.12063v5 |
upper bound ∥(Pin−Pout)k∥Z,ZonEwith probability at least 1−δ′/2. To this end, we first establish that ((Pin−Pout)k(z))z∈Zis a sub-Gaussian process on Ewith respect to a particular bounded-H ¨older metric ρ. Definition C.1 (Sub-Gaussian process on an event) .We say an indexed collection of random variables (Xθ)θ∈Θis a s... | https://arxiv.org/abs/2502.12063v5 |
arbitrary, we have established (34). Finally, we bound the entropy integral using the inequality 1≤c2/u2foru∈[0, D], the concavity of the square-root function, and Jensen’s inequality: J(Z, ρ)≤RD 0p log(1 + (1 + c2/u2)r)du≤RD 0p log((3 c2/u2)r)du=RD 0q 2rlog(√ 3c/u)du ≤Dq 1 DRD 02rlog(√ 3c/u)du=Dq 2rlog(√ 3ec/D ). Toge... | https://arxiv.org/abs/2502.12063v5 |
of the approximation error for AV andA1n. Lemma F.1 (Decomposing attention approximation error) .In the notation of Alg. 1 and (37), ∥bD−1bAV−D−1AV∥max≤min ∥(1 nD)−1∥max,∥(1 nbD)−1∥max (1 n∥bAV−AV∥max+1 n∥A1n−bA1n∥∞∥V∥max). The second, proved in App. F.2, bounds the approximation error for AV andA1nin terms of the KM... | https://arxiv.org/abs/2502.12063v5 |
m∥2|˜vlj−˜vlm|√ 2+ exp(R2 √ d)∥qi−qk∥2R√ d|˜vlm| ≤exp(R2 √ d)∥ed+1 j−ed+1 m∥2√ 2∥V∥max+ exp(R2 √ d)∥qi−qk∥2R√ d∥V∥max ≤exp(R2 √ d)q R2√ d+ 2∥V∥max∥(˜qi, ed+1 j)−(˜qk,ed+1 m)∥2 by the triangle inequality, multiple applications of Cauchy-Schwarz, and the mean-value theorem applied to x7→ex. G. Proof of Thm. 3: LKH-SGD co... | https://arxiv.org/abs/2502.12063v5 |
out) where pj inandpj outare the empirical distributions over Xk in,j= (xk i)j i=1andXk out,j={xk i∈ Xk out:i∈[j]}. Since LKH (δ)is an online algorithm that assigns signs (ϵk i, ϵk i+1= 1−ϵk i)to the points (xk i,xk i+1)sequentially, we can viewXk out,jas the output of LKH (δ)applied to Xk in,jwithnout=j 2and the linea... | https://arxiv.org/abs/2502.12063v5 |
size ℓ/2isk-sub-Gaussian (Def. I.1) with shift aℓ,n in,eKand parameter vℓ,n in,eKsatisfying aℓ,n in,eK=CfK(δ,ℓ) ℓ/2and vℓ,n in,eK=MfK(δ,ℓ) ℓ/2q log(12nin4g(log4nin−g) ℓδ). 33 Low-Rank Thinning Substituting MeK(δ,2g+1√nin) = (2g√nin)v2g+1√nin,nin,eKh log(12nin4g(log4nin−g) 2g+1√ninδ)i−1 2andCeK(δ,2g+1√nin) = (2g√nin)a2g... | https://arxiv.org/abs/2502.12063v5 |
manifold kernel CTT Our reasoning is identical to that in App. J with the manifold Gaussian kernel matrix eigenvalue bound (7) now substituted for the Euclidean ball bound (6) and the approximate rank setting r⋆= (log( nnout)/c)5d⋆/2substituted for (44). L. Supplementary Experiment Details L.1. Approximating attention ... | https://arxiv.org/abs/2502.12063v5 |
Hallucinations ar e ine vit able but can be made st atisti call y negligible. The “innate” ine vit ability of hallucina tions canno t e xplain practical LLM issues. A tsushi Suzuki1Y ulan He2 3F eng T ian4Zhongyuan W ang5 1The U niv ersity of Hong K ong 2King's Colleg e London, U nited Kingdom3The Alan T ur ing Ins t... | https://arxiv.org/abs/2502.12187v2 |
Y [2] , ALICE [3] , etc. or s tatis tical languag e models based on Mark o v theor ies, e.g., [4] , [5] , [6] . Ho w e v er , the introduction of ar tificial neural netw orks in LMs, pioneered b y , e.g., [7] , [8] , [9] , [10] , has led to a paradigm shift o v er the pas t tw o decades, as adv ances in tec hniq ues an... | https://arxiv.org/abs/2502.12187v2 |
gument. This theoretical result ma y seem f atall y pessimis tic f or practitioners since hallucinations on infinite in put ins tances sound lik e an insur mountable obs tacle in practice. Indeed, those results ha v e been ref er red to as fundamental limitations of LLMs not onl y in academia but also in the g eneral p... | https://arxiv.org/abs/2502.12187v2 |
obability of hallucinations arbitrar il y close to zero b y impro ving training data and training and inf erence algor ithms. The practical significance of hallucinations occur r ing onl y on infinite in put sets with arbitrar il y small probability can ultimatel y depend on the application domain. S till, using Shanno... | https://arxiv.org/abs/2502.12187v2 |
has f ocused on specific neural netw ork arc hitectures based on T ransf or mer [15] in the continuous function appro ximation conte xt. F or e x ample, [50] and [51] ha v e pro v ed that T rans- f or mers are univ ersal appro ximators of continuous seq uence-to-seq uence functions with compact suppor t, though the y s... | https://arxiv.org/abs/2502.12187v2 |
e denote the set of real numbers, the set of integ ers, and the set of nonneg ativ e integ ers b y ℝ , ℤ , and ℤ≥ 0 , respectiv el y . W e denote the floor function and ceiling function b y ⌊ ⋅ ⌋ and ⌈ ⋅ ⌉ , respectiv el y , i.e., f or 𝑎 ∈ ℝ , ⌊ 𝑎 ⌋ ≔ max { 𝑎′∈ ℤ | 𝑎′≤ 𝑎 } and ⌈ 𝑎 ⌉ ≔ min { 𝑎′∈ ℤ | 𝑎′≥ 𝑎 } . F... | https://arxiv.org/abs/2502.12187v2 |
map ℎ is computable if there e xis ts a T ur ing mac hine halts with jus t ℎ ( 𝑠 ) on its tape f or e v er y in put 𝑠 . W e denote the set of all LMs b y ℋ . Specificall y , ℋ ≔ { ℎ : Σ∗→ Σ∗| ℎ is computable } . R ef er to, e.g. [58] , f or r igorous definitions of, e.g., T ur ing mac hines. R emar k 3 . (All LLMs ar... | https://arxiv.org/abs/2502.12187v2 |
h are special cases of s toc has tic LMs. The reasons wh y this simple discussion suffices in this paper are the f ollo wing: • T o sho w the e xis tence of a LM satisfying desirable conditions, whic h is the main goal of this paper , it suffices to raise a special case. • If w e aim to a v oid hallucinations, it is a ... | https://arxiv.org/abs/2502.12187v2 |
𝐹0 , f or a s tr ing 𝑠 ∈ Σ∗ w e call 𝐹0 ( 𝑠 ) the acceptable output set f or the in put s tr ing 𝑠 . W e sa y that the acceptable output set map 𝐹0 is non-v acuous if 𝐹0 ( 𝑠 ) ≠ { } f or all 𝑠 ∈ Σ∗. W e can reg ard 𝐹0 as a f or mulation of the gr ound truth , and w e sa y that an LM ℎ ∈ ℋ hallucinates on the ... | https://arxiv.org/abs/2502.12187v2 |
use, it hallucinates on infinitel y man y in put s tr ings. N ote that this neg ativ e result holds reg ardless of our c hoice of neural netw ork arc hitecture, algor ithms, and training data. R emar k 9 . Theorem 8 is similar to Theorems 2 and 3 in [45] both in its s tatement and its proof s trategy but tec hnicall y ... | https://arxiv.org/abs/2502.12187v2 |
f or a c hat-bot. In practice, it is sufficient to consider com put able LMT s. N e v er theless, w e do not assume the computability of a LMT to clar ify that the computability does not essentiall y matter in the f ollo wing s tatis tical results. Ob viousl y , the theoretical results holding on g eneral LMT s also ap... | https://arxiv.org/abs/2502.12187v2 |
set map 𝐹0 belongs to, lear ning is possible from Mark Gold’ s classical discussion [59] . Ho w e v er , w e ha v e no guarantee, f or e x ample, that 𝐹0 is computable. Indeed, the core of the diagonal ar guments [45] , [46] is the uncountability of the set of h ypotheses (in this paper ’ s notation, the set that 𝐹0... | https://arxiv.org/abs/2502.12187v2 |
but our vie wpoint is from hallucinations and dis tr i- butions, rather than from the h ypothesis set. This is to mak e easier its compar ison to the result in Section 3 . Definition 17 . (S tatis tical negligiblity of hallucinations) (1) W e sa y that hallucinations of a LMT 𝔄 with a q ualified random training data s... | https://arxiv.org/abs/2502.12187v2 |
long a training data seq uence w e need, but e v entuall y w e can ac hie v e the aimed hallucination probability (with high probability o v er training data dis tr ibution) if w e increase the data size. By definition, if hallucinations are unif or ml y s tatis ticall y negligible, then non-unif or ml y s tatis ticall... | https://arxiv.org/abs/2502.12187v2 |
t he pr obability measur es on Σ∗ in t he sense of Definition 17 . Pr oof. Since f or an y 𝜇 ∈ Δ ( Σ∗) , 𝜇 ∈ 𝒫CDFlen ♯ 𝜇, w e ha v e that Δ ( Σ∗) = ⋃𝜇 ∈ Δ ( Σ∗ )𝒫CDFlen ♯ 𝜇. Hence, (2) f ollo w s from (1). (1) f ollo w s from Proposition 26 immediatel y , whic h is s tated in Appendix. □ R emar k 22 . (Summar y ... | https://arxiv.org/abs/2502.12187v2 |
in R emark 23 . W e also sho w that the specific theorem is near l y optimal in that its assumptions reg arding the training data size and the a v ailability of an in put length CDF lo w er bound cannot be remo v ed. W e also point out that the optimality of our theorem enlightens future w ork directions. N o w , w e h... | https://arxiv.org/abs/2502.12187v2 |
obability . Whic h matters in practice? N o w , what should be discussed is whether the infinite but arbitrar il y small probability er rors are accepted in practice. This is no long er a mathematical discussion and can ultimatel y depend on the domain. N e v er theless, w e s till claim that it has practicall y been n... | https://arxiv.org/abs/2502.12187v2 |
il y small, the e v ent is considered to be practicall y negligible in inf or mation theor y e v en if the number of elements in the e v ent is lar g e. Theref ore, w e can conclude that, although infinite hallucinations ar e ine vit able in the sense of Theor em 8 , the y can be practicall y negligible in the applicat... | https://arxiv.org/abs/2502.12187v2 |
sis and mac hine int ellig ence , v ol. 12, no. 6, pp. 570–583, 1990. [5] D. Hiems tra, “ A linguis ticall y motiv ated probabilis tic model of inf or mation retr ie v al, ” in R esear c h and Adv anced T ec hnology f or Digital Libr aries: Second Eur opean Conf er ence, ECDL ’98 Her aklion, Cr e t e, Gr eece Sep t emb... | https://arxiv.org/abs/2502.12187v2 |
deep bidirectional transf or mers f or languag e unders tanding, ” in Pr oceedings of 2019 Annual Conf er ence of t he N or t h American Chap t er of t he Association f or Computational Linguis tics , 2019. [17] L. Ouy ang e t al. , “ T raining languag e models to f ollo w ins tr uctions with human f eedbac k, ” Adv an... | https://arxiv.org/abs/2502.12187v2 |
[Online]. A v ailable: https://blogs.microsoft. com/wp-content/uploads/prod/sites/5/2023/04/RAI-f or -the-ne w -Bing- Apr il-2023.pdf [34] J. Li, W . Zhang, T . W ang, G. Xiong, A. Lu, and G. Medioni, “GPT4R ec: A g enerativ e frame w ork f or personalized recommendation and user interes ts inter pretation, ” arXiv pr ... | https://arxiv.org/abs/2502.12187v2 |
their damag e, ” N atur e , v ol. 637, no. 8047, pp. 778–780, 2025. 11 [49] A. T . Kalai and S. S. V empala, “Calibrated languag e models mus t hallucinate, ” in Pr oceedings of t he 56t h Annual A CM Symposium on Theor y of Computing , 2024, pp. 160–171. [50] C. Y un, S. Bhojanapalli, A. S. Ra w at, S. R eddi, and S. ... | https://arxiv.org/abs/2502.12187v2 |
Else vier , 2014. [62] H. Her r lic h, Axiom of c hoice , v ol. 1876. Spr ing er , 2006. 12 R oteMemorizer ( ( ( 𝑠1 , 𝑦1 ) , ( 𝑠2 , 𝑦2 ) , … , ( 𝑠𝑚 , 𝑦𝑚 ) ) ∈ ( Σ∗, Σ∗)∗) : 1 𝑚 ← len ( ( 𝑠1 , 𝑦1 ) , ( 𝑠2 , 𝑦2 ) , … , ( 𝑠𝑚 , 𝑦𝑚 ) ) 2 Initialize an empty dictionar y 𝑑 3 f or 𝑖 ← 1 , 2 , … , 𝑚 : 4 𝑑 [... | https://arxiv.org/abs/2502.12187v2 |
is small. The pseudocode of the s traightf or w ard algor ithm is giv en in Algor ithm 1 . N ote that w e do N OT insis t that Algor ithm 1 should be used in practice. It is rather a tool f or the proof. W ith the help of Algor ithm 1 , w e can sho w the f ollo wing, whic h immediatel y giv es us Theorem 21 . Pr oposit... | https://arxiv.org/abs/2502.12187v2 |
ers the s tr ings whose length is shor ter than the threshold. W ithout the lo w er bound of the in put length CDF , w e cannot do this. Also, since it tr ies to rote memor ize all the s tr ings shor ter than the s tr ing length threshold 𝑛 , it is natural that it req uires the training data size e xponential to 𝑛 . ... | https://arxiv.org/abs/2502.12187v2 |
the c hoice of the training data 𝑇 in the w ors t case on the c hoice of 𝐹0 ( 𝑠 ) and 𝜇 . Despite its neg ativ e s tatement, w e do not consider Theorem 27 to be impl ying issues from a practical perspectiv e, since the lo w er bound could be easil y obtained as it is nothing but a probability dis tr i- bution of a... | https://arxiv.org/abs/2502.12187v2 |
, it enlightens future w ork directions, sugg es ting the necessity of s trong er assumptions reflecting the beha vior of natural languag es. Specificall y , from a theoretical perspectiv e, Theorem 29 sugg es ts that suc h assumptions are req uired to pro v e the success of LMs with practical training data size, as th... | https://arxiv.org/abs/2502.12187v2 |
1 , 𝑠𝑘 + 2 , … , whic h completes the proof. □ B.2 Mo tiv ation of a v oiding depending on the axiom of c hoice (A C) W e remark that our proof of Theorem 8 does not use the axiom of c hoice (A C), while the pre vious w ork’ s proof [45] depends on the A C. In this subsection, w e discuss its significance. As an axio... | https://arxiv.org/abs/2502.12187v2 |
the A C and ho w w e a v oided it N o w , let us see ho w the pre vious w ork’ s proof uses the A C and ho w w e ha v e a v oided it. Specificall y , when cons tr ucting 𝜓 , the pre vious w ork’ s proof [45] arbitr arily c hose an element from eac h of ̃ ℐ𝑖 from 𝑖 = 1 , 2 , … . Since 𝑖 is in the infinite set ℤ> 0 ,... | https://arxiv.org/abs/2502.12187v2 |
− 𝜀′ T o v er c hoice of tr aining data seq uence ( 𝑆1 , 𝑆2 , … , 𝑆𝑚 ) , wher e 𝜀′ H = 𝜀′ T = 2 ( 1 − CDF ( 𝑛 ) ) . Once this lemma is pro v ed, then f or an y 𝜀H , 𝜀T ∈ ( 0 , 1 ) , w e obtain that HP𝜇 ( 𝔄 ( 𝑇 ) ) < 𝜀H holds in probability at leas t 1 − 𝜀T o v er c hoice of training data seq uence if 𝑚 ... | https://arxiv.org/abs/2502.12187v2 |
Σ𝑛| =| Σ |𝑛− 1 | Σ | − 1≤ | Σ |𝑛 holds. W e inde x all the elements in Σ≤ 𝑛 in the descending order with respect to its probability . In other w ords, 𝑠1 , 𝑠2 , … , 𝑠𝑘 ∈ Σ≤ 𝑛 satisfy 𝑠𝑗 ≠ 𝑠𝑗′ and Pr ( 𝑆 = 𝑠𝑗 ) ≥ Pr ( 𝑆 = 𝑠𝑗′ ) f or an y 𝑗 , 𝑗′ satisfying 1 ≤ 𝑗 < 𝑗′≤ 𝑘 and an y random v ar iable ... | https://arxiv.org/abs/2502.12187v2 |
H 2. Theref ore, w e obtain 𝑝𝑗∗ = max { 𝑝𝑗∗ , 𝑝𝑗∗ + 1 , … , 𝑝𝑘 } ≥∑𝑘 𝑗 = 𝑗∗ 𝑝𝑗 𝑘 − 𝑗∗ + 1≥𝜀′ H 2 ( 𝑘 − 𝑗∗ + 1 )≥𝜀′ H 2 𝑘. (8) Here, the second ineq uality comes from the f act that the maximum v alue is alw a y s lar g er than or eq ual to the mean. Thus, f or 𝑗 = 1 , 2 , … , 𝑗∗, w e ha v e that �... | https://arxiv.org/abs/2502.12187v2 |
and 𝑓0 by HP𝜇 , 𝑓0( ℎ ) , whic h is defined by HP𝜇 , 𝑓0( ℎ ) = Pr ( ℎ ( 𝑋 ) ≠ 𝑓0 ( 𝑋 ) ) . Then, f or any map (lear ning algorit hm) 𝔄 : ( 𝒳 × 𝒴 )∗→ ( 𝒳 → 𝒴 ) , any nonneg ativ e int eg er (tr aining data size) 𝑚 t hat satisfies 𝑚 ≤1 2| 𝒳 | , any finit e positiv e int eg er 𝑝 satisfying 1 ≤ 𝑝 ≤ | 𝒴 |... | https://arxiv.org/abs/2502.12187v2 |
1 ) , t her e exis t a computable map 𝑓0 : 𝒳 → 𝒴 and a finit e subse t 𝒳 ⊂ 𝒳 suc h t hat bo t h t he f ollo wing ineq ualities hold: 𝔼𝑇 HPUni ( 𝒳 ) , 𝑓0( 𝔄 ( 𝑇 ) ) ≥1 2, Pr𝑇 ( HPUni ( 𝒳 ) , 𝑓0( 𝔄 ( 𝑇 ) ) ≥ 𝜆H ) ≥ 𝜆T ≔1 − 2 𝜆H 2 − 2 𝜆H. (11) Pr oof. F or an y 𝜀 ∈ ℝ> 0 , w e can pro v e that 𝔼𝑇 HPU... | https://arxiv.org/abs/2502.12187v2 |
, whic h completes the proof. □ The proof of Lemma 38 is giv en as f ollo w s. Pr oof of Lemma 38 . R ecall that 𝑛 ≔ argmin𝑛 ∈ ℤ≥ 0| Σ |𝑛 + 1− 1 ( | Σ | − 1 ) CDF ( 𝑛 ). F or 𝑛 ∈ ℤ≥ 0 , cons tr uct 𝒳𝑛 as f ollo w s. • If | Σ |𝑛 + 1− 1 | Σ | − 1≤| Σ |𝑛 + 1− 1 ( | Σ | − 1 ) CDF ( 𝑛 ), then 𝒳𝑛 ≔ Σ𝑛, • If | Σ ... | https://arxiv.org/abs/2502.12187v2 |
of 𝑛 . • If | Σ |𝑛 + 1− 1 ( | Σ | − 1 ) CDF ( 𝑛 )<| Σ |𝑛 + 1− 1 | Σ | − 1, then since 𝒳≤ 𝑛 = 𝒳 , the ineq uality | 𝒳≤ 𝑛 | | 𝒳 |= 1 ≥ CDF ( 𝑛 ) is tr ivial since CDF ( 𝑛 ) ∈ [ 0 , 1 ] b y definition. These complete the proof. □ W e conclude this section with a complete proof of Theorem 33 . Pr oof of Theor e... | https://arxiv.org/abs/2502.12187v2 |
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