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proof. Define the following sequences of events: F(1) k:∥Xk∆t−xk∆t∥ ≤rk∆t F(2) k:∥Xt−¯xt,k∥ ≤r∆,∀t∈(k∆t,(k+ 1)∆ t).(78) By union bound inequality, the probability that both F(1) kandF(2) k hold for the whole time period can be bounded by P N\ k=1F(1) k!\ N\ k=1F(2) k!! ≥1−(NX k=1δ 2N+NX k=1δ 2N) = 1−δ.(79) When the joi... | https://arxiv.org/abs/2503.03328v1 |
arXiv:2503.03347v1 [math.ST] 5 Mar 2025Drift estimation for rough processes under small noise asymptotic : trajectory fitting method Arnaud Gloter1and Nakahiro Yoshida2,3 1Laboratoire de Math´ ematiques et Mod´ elisation d’Evry, Universit´ e d’Evry∗ 2Graduate School of Mathematical Sciences, University of Tokyo† 3Japan ... | https://arxiv.org/abs/2503.03347v1 |
observation of Xε, it is possible to consider the MLE estimator of the model. In the context of a Volterra Ornstein-Uhle nbeck model, this method is used in [29] in order to estimate the drift parameter, w henT→ ∞. The consistency of the estimator is proved in [29]. In practice, the MLE methods necessitates to approxim... | https://arxiv.org/abs/2503.03347v1 |
implies that a version of the p ro- cess is such that the sample paths t/ma√sto→Xε tare a.s.α′-H¨ older for any α′∈(0,α). We assume that the set Θ is such that the following Sobolev embedding holds true. For f∈ C1(◦ Θ),p > dΘ, we have sup θ∈Θ|f(θ)| ≤c[/ba∇dblf/ba∇dblLp(Θ)+/ba∇dbl∇f/ba∇dbl Lp(◦ Θ)]. The true value of th... | https://arxiv.org/abs/2503.03347v1 |
=/integraldisplayT 0|X0 t(θ)−X0 t(θ⋆)|2dt=/integraldisplayT 0|K ⋆b(t)|2dt. We denote B(t) =/integraltextt 0b(s)ds. AsL⋆K(t) =1{t>0}, we have B(t) =/parenleftbig (L⋆K)⋆ b/parenrightbig (t). Hence, /ba∇dblB/ba∇dbl2 L2([0,T])=/ba∇dbl(L⋆K)⋆b/ba∇dbl2 L2([0,T])=/ba∇dblL⋆(K ⋆b)/ba∇dbl2 L2([0,T]) ≤ /ba∇dblL/ba∇dbl2 L1([0,T])×/... | https://arxiv.org/abs/2503.03347v1 |
0K(t−s)a(Xε s)dBs +/integraldisplayt 0K(t−s)(b(Xε s,θ⋆)−b(X0 s(θ⋆),θ⋆))ds. 7 We use the Burkholder-Davies-Gundy inequality to get, E[|Eε t|p]≤c(p)εpE/bracketleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt 0K(t−s)2|a(Xε s)|2ds/vextendsingle/vextendsingle/vextendsingle/vextendsinglep/2/b... | https://arxiv.org/abs/2503.03347v1 |
ε, we remark that by Cauchy-Schwarz’s inequality /vextendsingle/vextendsingle/vextendsingleQ(3) ε(θ)/vextendsingle/vextendsingle/vextendsingle≤2/parenleftBig Q(1) ε(θ)/parenrightBig1/2/parenleftBig Q(2) ε(θ)/parenrightBig1/2 ≤2/parenleftBig/vextendsingle/vextendsingle/vextendsingleQ(1) ε(θ)−Q0(θ)/vextendsingle/vextends... | https://arxiv.org/abs/2503.03347v1 |
Proof.We start with the second part. Les us write for M >0, P(|ˆθε−θ⋆| ≥M)≤P( inf |θ−θ⋆|≥M,θ∈ΘQε(θ)≤Qε(θ⋆)) ≤P( inf |θ−θ⋆|≥M,θ∈ΘQ0(θ)−sup θ∈Θ|Qε(θ)−Q0(θ)| ≤ Q0(θ⋆)+sup θ|Qε(θ)−Q0(θ)|) ≤P(2sup θ∈Θ|Qε(θ)−Q0(θ)| ≥ inf |θ−θ⋆|≥M,θ∈ΘQ0(θ)−Q0(θ⋆)). (27) AsQ0(θ⋆) = 0, by Lemma 2.1 2), we know that inf |θ−θ⋆|≥M,θ∈ΘQ0(θ)− Q0(θ⋆)... | https://arxiv.org/abs/2503.03347v1 |
ds/parenrightbigg∗ ×/parenleftbigg/integraldisplayt 0K(t−s)/parenleftbig ∇xb(X0 s(θ⋆),θ⋆)Y0 s(θ⋆)+∇θb(X0 s(θ⋆),θ⋆)/parenrightbig ds/parenrightbigg dt, ˙Q2 ε(θ⋆) =−2/integraldisplayT 0/parenleftbigg/integraldisplayt 0K(t−s)a(Xε s)dBs/parenrightbigg∗ ×/parenleftbigg/integraldisplayt 0K(t−s)/parenleftbig ∇xb(X0 s(θ⋆),θ⋆)Y... | https://arxiv.org/abs/2503.03347v1 |
b(X0 s(θ⋆),θ⋆)−b(X0 s(θ),θ)/parenrightbig ds +ε/integraldisplayt 0K(t−s)a(Xε s)dBs/parenrightBig∗ ×/parenleftBig/integraldisplayt 0K(t−s)∂2 ∂θu∂θv/parenleftbig b(X0 s(θ),θ)/parenrightbig ds/parenrightBig dt.(34) This expression gives∂Qε ∂θu∂θv(θ) =Ju,v(θ)+Eu,v(θ). Now, using that ˆθεis a minimizer of the contrast funct... | https://arxiv.org/abs/2503.03347v1 |
that the R.H.S. in the equation above is bounded independently of ε∈(0,1]. As a result (38) is proved. 21 5 Appendix Letb: [0,T]×Rq×Θ→Rqsuch that for all t∈[0,T], (x,θ)/ma√sto→b(t,x,θ) is C1(Rq×◦ Θ) and the functions ∇xb(t,x,θ) and∇θb(t,x,θ) areC0([0,T]×Rq×◦ Θ). Moreover, we assume that for some c >0. sup t∈[0,T];θ∈◦ Θ... | https://arxiv.org/abs/2503.03347v1 |
of Integral Equations 2.3 (1980), 187–245. issn: 0163-5549. JSTOR: 26164035 . [4] Marc A. Berger and Victor J. Mizel. “Volterra Equations with Itˆ o Inte- grals—II”. In: Journal of Integral Equations 2.4 (1980), 319–337. issn: 0163-5549. JSTOR: 26164044 . [5] CarstenH.Chongetal.“StatisticalInferenceforRoughVolat ility:... | https://arxiv.org/abs/2503.03347v1 |
(Mar. 7, 2022), pp. 109–138. issn: 0304-4149. doi: 10.1016/j.spa.2021.07.003 .arXiv:2004.00340 [math] .Pre-published. 25 [24] MichaelSørensenand MasayukiUchida.“Small-Diffusion Asympt otics for Discretely Sampled Stochastic Differential Equations”. In: Bernoulli 9.6 (Dec. 2003), 1051–1069. issn: 1350-7265. doi:10.3150/bj... | https://arxiv.org/abs/2503.03347v1 |
STATISTICAL LIMITS IN RANDOM TENSORS WITH MULTIPLE CORRELATED SPIKES YANG QI AND ALEXIS DECURNINGE Abstract. We use tools from random matrix theory to study the multi-spiked tensor model, i.e., a rank- rdeformation of a symmetric random Gaussian tensor. In particular, thanks to the nature of local optimization methods ... | https://arxiv.org/abs/2503.03356v1 |
in [SGG23] using random matrix tools. However, these results cannot be applied directly to the rank- rtensor decomposition and it is necessary to study Model (1.1) as a whole, which motivates this paper. In addition to the low-rank approximation problems arising from data science, another source of motivation comes fro... | https://arxiv.org/abs/2503.03356v1 |
that the algorithmic threshold for tensor unfolding is N(d−2)/4. Moreover, the algorithmic threshold for first-order opti- mization methods has been shown to be N(d−2)/2in [MR14, AGJ20], and the algorithmic thresholds for sum-of-squares and spectral methods have been given in [HSS15, PWBM18, PWB20]. When r >1, the dete... | https://arxiv.org/abs/2503.03356v1 |
to distinguish random quantities that depend on the random tensor Xfrom deterministic quantities, we use hat notation for the former, e.g., ˆvvv1denotes a random vector depending on X. The tensor contraction operation is denoted by ⟨,⟩. For instance, the (id−1, id)th entry of ⟨Y,vvv⊗(d−2) 1⟩is given by ⟨Y,vvv⊗(d−2) 1⟩i... | https://arxiv.org/abs/2503.03356v1 |
f(X) ≤E∥∇f(X)∥2 2. 2.4.Symmetric Gaussian distribution. Let us explicitly describe the distribution of the sym- metric Gaussian noise tensor X. It is characterized by the following equality: Xi1...id=1 d!X π∈SdWiπ(1)...iπ(d), (2.2) where Wis a Gaussian noise tensor with i.i.d. entries Wi1...id∼ N(0,1)andSdis the symme... | https://arxiv.org/abs/2503.03356v1 |
ˆRˆRˆR⊙(d−1) vv,r = 1 ⟨ˆvvv1,ˆvvv2⟩d−1⟨ˆvvv1,ˆvvv3⟩d−1. . .⟨ˆvvv1,ˆvvvr⟩d−1 ⟨ˆvvv1,ˆvvv2⟩d−1⟨ˆvvv2,ˆvvv3⟩d−1. . .⟨ˆvvv2,ˆvvvr⟩d−1 ............... ⟨ˆvvv1,ˆvvvr⟩d−1⟨ˆvvv2,ˆvvvr⟩d−1⟨ˆvvv3,ˆvvvr⟩d−1. . . 1 . In addition, let ˆWˆWˆWr= ˆRˆRˆR⊙(d−1) vv,r−1 ˆγr ˆγ10. . . 0 0ˆγr ˆγ2. . . 0 ............ 0 0 . . ... | https://arxiv.org/abs/2503.03356v1 |
spectrum of ♭(T) 4.1.Sequence of critical points with fully convergent summary statistics. Consider the following assumption where we assume that the summary statistics of the restimated signals are converging. 8 Y. QI AND A. DECURNINGE Assumption 4.1. The sequence (ˆγ1, . . . , ˆγr,ˆvvv1, . . . , ˆvvvr)satisfies ˆγia.... | https://arxiv.org/abs/2503.03356v1 |
in particular Subsection D.2, it is straightforward to verify that the result still hold for sequence of vectors (ˆγ1ˆvvv1, . . . , γ rˆvvvr)drawn independently from T. Therefore, in order to illustrate this result for r= 2, we generate 100random symmetric pure noise tensors Tand independent vectors ˆγ1ˆvvv1,ˆγ2ˆvvv2sa... | https://arxiv.org/abs/2503.03356v1 |
Assumption 4.6 may not be satisfied in practice, we conjecture that Theorem 4.7 remains true when only assuming that⟨ˆvvvi,ˆvvvj⟩converges to 0when Ngoes to infinity. Remark 4.9. Let us define UUUi= uuui0 0 0...0 0 0 uuui and AAAi=βiˆWˆWˆWr ⟨uuui,ˆvvv1⟩d−20 0 0... 0 0 0 ⟨uuui,ˆvvvr⟩d−2 . Then ♭(T) =♭(rX j=1... | https://arxiv.org/abs/2503.03356v1 |
αd−1 12 αd−1 21 αd−1 22 = 1τ τ1 MMM+1 d(d−1)GGGγ2 d−1 ν0 0 1,(5.7) LOW-RANK RANDOM TENSORS 13 with GGG(z) = g11(z)g12(z) g21(z)g22(z) with gij(z)defined in Corollary 4.5, and MMM= γ1γ1λ γ2λ γ 2 +1 d(1−λ2) νg11 γ2 d−1 −λg12 γ2 d−1 τd−2 νg21 γ2 d−1 −λg22 γ2 d−1 τd−2 g12 γ2 d−1 −λνg 11 γ2 d−1... | https://arxiv.org/abs/2503.03356v1 |
γ1(2-signal solution) (a)Theoretical γ1and empirical ˆγ10 1 2 3 4 5 β212345Empirical ˆγ2 Asymptotic γ2(2-signal solution) (b)Theoretical γ2and empirical ˆγ2 0 1 2 3 4 5 β20.00.20.40.60.81.0 Empirical ⟨u1,ˆ v1⟩ Asymptotic lim⟨u1,ˆ v1⟩(1 signal solution #1) Asymptotic lim⟨u1,ˆ v1⟩(1-signal solution #2) Asymptotic lim⟨u1,... | https://arxiv.org/abs/2503.03356v1 |
visible in Figure 2 and Figure 3. 6.Summary statistics inference from estimated signals For a given received tensor Twe can compute a critical point of the rank- rapproximation loss, i.e., the squared error in (1.2). Then, if (ˆγiˆvvvi)icorresponds to an optimal point of the likelihood, it is the maximum likelihood est... | https://arxiv.org/abs/2503.03356v1 |
j=1ˆγjˆvvv⊗d j,ˆvvv⊗(d−1) i⟩= 0. Appendix B.Proof of Proposition 3.2 We reformulate the first equation of (3.4) as follows ⟨T,ˆvvv⊗(d−1) 1⟩= ˆγ1ˆvvv1+ ˆγ2⟨ˆvvv1,ˆvvv2⟩d−1ˆvvv2+···+ ˆγr⟨ˆvvv1,ˆvvvr⟩d−1ˆvvvr . . . ⟨T,ˆvvv⊗(d−1) r⟩= ˆγ1⟨ˆvvv1,ˆvvvr⟩d−1ˆvvv1+···+ ˆγr−1⟨ˆvvvr−1,ˆvvvr⟩d−1ˆvvvr−1+ ˆγrˆvvvr, (B.1) 18 ... | https://arxiv.org/abs/2503.03356v1 |
− →0ifs̸=t. (D.2) Therefore, we need to control the term1 NP k̸=iETr[QQQsu(z)AAAtt]for any 1≤s, t≤r. For the sake of simplicity of exposition, we will consider in the sequel the case s=u= 1since the other terms can be controlled in a similar way. Recall that A11 ji=1√ NNX ℓ1,...,ℓd−2=1Xℓ1...ℓd−2jiˆv1 ℓ1···ˆv1 ℓd−2, 20 ... | https://arxiv.org/abs/2503.03356v1 |
of notation, we use Q11 ijto denote the (i, j)th entry of QQQ11(z). In particular, for A2, we focus on the following term appearing in A2as an example of the argument 1 N√ NNX i,j,ℓ 1,...,ℓd−2=1σ2 ℓ1...ℓd−2jiE∂ˆv1 ℓ1 ∂Xℓ1...ℓd−2jiˆv1 ℓ2···ˆv1 ℓd−2Q11 ij . (D.11) Assume CCC−1+CCC−1RRRSSS−1LLLCCC−1has the form CCC−1+CC... | https://arxiv.org/abs/2503.03356v1 |
0. The proof of this claim splits into two cases, i.e., only one of kandlis contained in I, or both of them are in I. Since the spirits of the proofs are the same, here we only present the argument for the case k∈Iandl /∈I. In this case, since ΩI,k,l−1√ N1 d(d−1)ˆv1 I ≤1√ N|ˆv1 I|, we have 1 N√ NX IX {k,l}∩I̸=∅|ˆv1 I| ... | https://arxiv.org/abs/2503.03356v1 |
to compute the eigenvalues ofGGG(z)which we denote by ζ1(z), . . . , ζ 2(z). Then (D.22) becomes 1 4 (κ(r) 1)2ζ2 1(z). . . 0 0... 0 0 0 ( κ(r) 1)2ζ2 r(z) + zζ1(z) 0 0 0... 0 0 0 zζr(z) + IdIdId = 0 00.(D.23) Because the Stieltjes transform behaves as O(z−1)at infinity, Equation (D.23) has a unique solution ... | https://arxiv.org/abs/2503.03356v1 |
=2 πκ2 is κ2 i−x2 +dx. (E.2) We compute similarly the inverse Stieltjes transform in the case where κi= 0. Appendix F.Proof of Theorem 4.7 We will follow the same lines of proof as in Section D. For the sake of clarity of the exposition, we provide details in the case s= 1, focusing only on ˆvvv1andˆγ1. Therefore, we... | https://arxiv.org/abs/2503.03356v1 |
notations of (D.12). For the purpose of computing (G.3), we start from the term d−1√ NX i1,...,id,jσ2 i1...idE D11 i1j∂⟨T,ˆvvv⊗(d−1) 1⟩j ∂Xαˆv1 i2···ˆv1 id−1 u1 id. (G.4) To this end, thanks to (D.6), let us decompose DDD11as DDD11=1 d−1 QQQ(ˆγs d−1)ˆWˆWˆW 11+MMM, (G.5) 30 Y. QI AND A. DECURNINGE where QQQ(ˆγs d−1... | https://arxiv.org/abs/2503.03356v1 |
σ2 i1...idE D11 ℓjqϕ1 q ˆv1 ℓ1m1··· ˆv1 ℓj−1mj−1 ˆv1 ℓjmj−1 ˆv1 ℓj+1mj+1··· ˆv1 ℓkmk u1 id.(G.9) As before, we focus on the term 1√ NX ℓ2,...,ℓk,id,qm1d−1 m1, . . . , m k σ2 i1...idE D11 ℓ1qϕ1 q ˆv1 ℓ1m1−1 ˆv1 ℓ2m2··· ˆv1 ℓkmk u1 id =1 NX ℓ2,...,ℓk,idm1 dEd−1 m1−1, m2, . . . , m k,1 D11 ℓ1ℓ1 ˆ... | https://arxiv.org/abs/2503.03356v1 |
βr − γ1 ... γr ⊤ RRR⊙d vv,r γ1 ... γr +o(1). (H.4) Appendix I.Proof of Theorem 5.5 Let us consider a sequence of critical points satisfying Assumption 4.6 and Assumption 5.1, where ˆγris the largest eigenvalue. Recall from (4.15) that ♭(T) =♭(rX j=1βjuuu⊗d j) +1√ N♭(X) =rX i=1UUUiAAAiUUU⊤ i+1√ N♭(X), (I.... | https://arxiv.org/abs/2503.03356v1 |
(J.5) Then System (6.2) becomes( eKKK=eNNNˆDDDeNNN−1 d−1 eNNN⊤eNNNd−1=ˆCCC. (J.6) SinceeKKKis a function of eNNNfrom the first equation of (J.6), it suffices to solve eNNN⊤eNNNd−1=ˆCCC, (J.7) i.e., the second equation of (J.6). LOW-RANK RANDOM TENSORS 35 Let us parameterize eNNNbyeNNN= ξ10 0ξ2 1η θ1 . Then eNNNd−1=... | https://arxiv.org/abs/2503.03356v1 |
, 47(5):2734–2756, 2019. [CHL21] Wei-Kuo Chen, Madeline Handschy, and Gilad Lerman. Phase transition in random tensors with multiple independent spikes. The Annals of Applied Probability , 31(4):1868–1913, 2021. [CMDL+15] Andrzej Cichocki, Danilo Mandic, Lieven De Lathauwer, Guoxu Zhou, Qibin Zhao, Cesar Caiafa, and Hu... | https://arxiv.org/abs/2503.03356v1 |
Fromthe gaussian hidden clique problem to rank-one perturbations of gaussian tensors. In C. Cortes, N. Lawrence, D. Lee, M. Sugiyama, and R. Garnett, editors, Advances in Neural Information Processing Systems , volume 28. Curran Associates, Inc., 2015. [OMH13] Alexei Onatski, Marcelo J Moreira, and Marc Hallin. Asympto... | https://arxiv.org/abs/2503.03356v1 |
Early-Stopped Mirror Descent for Linear Regression over Convex Bodies Tobias Wegel∗1, Gil Kur1, and Patrick Rebeschini2 1Department of Computer Science, ETH Zürich, Switzerland 2Department of Statistics, University of Oxford, UK March 6, 2025 Abstract Early-stopped iterative optimization methods are widely used as alte... | https://arxiv.org/abs/2503.03426v1 |
in explicit regularization. However, most works on early-stopping investigate existing optimization algorithms or geometries, and a (principled) development of early-stopped iterative algorithms for many settings is lacking in the literature. In this work, we aim to show that such a correspondence exists between Early-... | https://arxiv.org/abs/2503.03426v1 |
defined as pαLSEParg min αPKτRnpαq. 2 The predictions of the LSE on the sample Xare given by the orthogonal projection of yontoXKτ“ tXα|αPKτu, which is convex, and thus the projection is unique. Hence, while the minimizer in Defini- tion 1 is not necessarily unique (especially in the high-dimensional setting where dąn)... | https://arxiv.org/abs/2503.03426v1 |
we provide sufficient conditions on the optimization potential for our bound to apply (Section 3.1). We use these conditions for developing new (and analyzing existing) potentials in several examples (Section 4). •We apply our risk bounds to ℓp-norm balls with pPr1,2qas well as general M-convex hulls and derive sharp s... | https://arxiv.org/abs/2503.03426v1 |
Definition 4. For a convex body KĂRd, the function φK:RdÑR, defined as φKpαq:“inftτą0|αPτKu is called the Minkowski functional ofK(also referred to as distance or gauge function). 4 The Minkowski functional has the following properties (Schneider, 2013, p.53): It is positive on Rdzt0u, non-negative homogeneous, sub-add... | https://arxiv.org/abs/2503.03426v1 |
or not strongly convex (for example, the ℓ1-norm), and we need to approximate it with a different function. This is possible for all K, as we now show in Lemma 1. Specifically, we can a smoothen and a convexify φ2 K: To that end, we denote the Moreau envelope (Moreau, 1965) of a closed and proper convex function fwithλ... | https://arxiv.org/abs/2503.03426v1 |
of discrete time, the strong convexity parameter does not influence the bound in (6), however, it impacts the bound on the stopping time. Finally, we would like to highlight that Theorem 1 can easily be used to derive bounds on the estimation risk whenever the design matrix has a vanishing kernel width, cf. Raskutti et... | https://arxiv.org/abs/2503.03426v1 |
discussed the general case of arbitrary convex bodies and design matrices, which yields the full generality of our main results Theorem 1 and Corollary 1. These results allow us to view Assumption A as a blueprint. For a given convex body, one can construct a potential that satisfies (I)-(IV). Using this potential, ESM... | https://arxiv.org/abs/2503.03426v1 |
Rpαt‹qď˜ 16 rkpXq n^48caτc logd nd1{q¸ `8 logp1{δq n with probability at least 1´expp´0.1nq´δover the noise. If Xis Gaussian, then for all α‹PτBd p, ESMD achieves Rpαt‹qÀ1^τ?n#?logdif1ăpď1`1 logd, ?qd1{qif1`1 logdăpă2, with probability at least 0.99´2 expp´0.1nqjointly over draws of the design matrix Xand the noise ξ. ... | https://arxiv.org/abs/2503.03426v1 |
same as in Lemma 1, which would also be a valid potential for Theorem 4. Here we show pMλ}¨}1`dλ 2q2`ρ 2}¨}2 2, because it has a closed-form solution. See Figure 3 in Appendix A.2 for example optimization paths. 4.2ℓ1-norm When the convex body is an ℓ1-norm ball, the corresponding LSE is the LASSO estimator in its cons... | https://arxiv.org/abs/2503.03426v1 |
Therefore, they cannot be improved upon beyond the constants and ESMD is minimax optimal, as is the LSE. Notice how the third potential in Table 1 is an adjusted version of the hypentropy potential from Ghai et al. (2020). With only a few changes to the potential, we closed the logarithmic gap from (15)to(14); In parti... | https://arxiv.org/abs/2503.03426v1 |
draws of the noise. Using the bound on localized Gaussian width of M-convex hulls from Bellec (2017), we prove Proposition 5 in Appendix B.11, where we also specify the constants. Noticably, when Kis anℓ1-norm ball, we recover the bound from Theorem 4 with M“2d,m“2dandϕ“1. 5 Discussion The main contribution of this wor... | https://arxiv.org/abs/2503.03426v1 |
Probability Theory and Related Fields , 97:113–150, 1993. T. Bonnesen and W. Fenchel. Theorie der konvexen Körper . Springer Berlin Heidelberg, 1934. Stéphane Boucheron, Gábor Lugosi, and Pascal Massart. Concentration Inequalities: A Nonasymptotic Theory of Independence . Oxford University Press, 2013. Peter Bühlmann a... | https://arxiv.org/abs/2503.03426v1 |
Co KG, 2014. Trevor Hastie, Robert Tibshirani, and Jerome Friedman. The Elements of Statistical Learning . Springer New York Inc., 2001. Anatoli Juditsky and Arkadii S Nemirovski. Large deviations of vector-valued martingales in 2-smooth normed spaces. arXiv preprint arXiv:0809.0813 , 2008. Varun Kanade, Patrick Rebesc... | https://arxiv.org/abs/2503.03426v1 |
, pages 89–130, 2019. Akshay Prasadan and Matey Neykov. Some facts about the optimality of the LSE in the Gaussian sequence model with convex constraint. arXiv preprint arXiv:2406.05911 , 2024. Garvesh Raskutti, Martin J. Wainwright, and Bin Yu. Minimax Rates of Estimation for High-Dimensional Linear Regression Over ℓq... | https://arxiv.org/abs/2503.03426v1 |
Journal of the IMA , 12(2):633–713, 2022. Yuhong Yang and Andrew Barron. Information-theoretic determination of minimax rates of convergence. The Annals of Statistics , 27(5):1564 – 1599, 1999. Yuan Yao, Lorenzo Rosasco, and Andrea Caponnetto. On Early Stopping in Gradient Descent Learning. Constructive Approximation ,... | https://arxiv.org/abs/2503.03426v1 |
problem, we plot paths on one data instance. Note that the paths can (somewhat) deviate from the LASSO path, but early-stopping still achieves minimax rates (Section 4.2). (b): For n“d“100, we take Xto be Gaussian and α‹to be 1-sparse. We repeat the experiment 50 times and plot mean, 10th and 90th quantiles. B Proofs B... | https://arxiv.org/abs/2503.03426v1 |
convergence implies uniform convergence, i.e., for the function ζpλq“minφKpαq“1b Mλφ2 Kpαqď1thatlimλÑ0ζpλq“1. Therefore, there exists aλ0such that for all λďλ0it holds ζpλqě1{2and hence alsob Mλφ2 KpαqěφKpαq{2for all αwith φKpαq“1. Becauseb Mλφ2 Kis another Minkowski functional (Planiden and Wang, 2019, Theorem 5.6) an... | https://arxiv.org/abs/2503.03426v1 |
conditioned on A2we have Zr‹ďr2 ‹ 2`r‹a 2 logp1{δq. We now make a case distinction between Rpαt‹qďr2 ‹{nandRpαt‹qąr2 ‹{nconditioned on A2. In the first case, we get trivially Rpαt‹qďr2 ‹ nď1 n´ r‹`a 2 logp1{δq¯2 `ε. In the second case when Rpαt‹q“1 n}Xpαt‹´α‹q}2 2ąr2 ‹{n, we define λ“r‹{}Xpαt‹´α‹q}2P p0,1qandv“λXαt‹`p1... | https://arxiv.org/abs/2503.03426v1 |
n. where we denoted r‹“r‹pXKτq. This concludes the corollary. 24 B.6 Proof of Remark 1 Proof of Remark 1. An argument akin to the proof of Theorem 7 in Bellec (2017) shows that if we let r“2a rkpXqand we denote the orthogonal projection onto the column space of XasΠX, we have Er}ΠXξ}2sďa rkpXq, and hence for every α‹PK... | https://arxiv.org/abs/2503.03426v1 |
draws of Xit holds r2 ‹p3τXBd pqÀτ?n#?logdif1ăpď1`1 logd, ?qd1{qif1`1 logdăpă2. 26 Combining this with Theorem 1 and Remark 1 (noting that rkpXq“nwith probability 1) we get from a union bound that with probability 1´2 expp´0.1nq´δit holds sup α‹PτBdpRpαt‹qď4r2 ‹p3caτXBd pq n`8 logp1{δq n À˜ 1^τ?n#?logdif1ăpď1`1 logd?qd... | https://arxiv.org/abs/2503.03426v1 |
the entropy of St. Define the i-th dyadic entropy number of any set Sas eipSq“inftεą0 : log2Npε, Sqďi´1u. By definition, we have that eipSqďε, if and only if logNpε, Sqďi. We know by Schütt’s Theorem Schütt (1984) that the i-th dyadic entropy number of Bk pis given by eipBk pq—$ ’’& ’’%1 if1ďiďlogk,´ logpek{iq i¯1{p´1{... | https://arxiv.org/abs/2503.03426v1 |
least 1´Cexpp´cnq. We outline the proof of (28)after finishing the main proof. Using (28), for εÁd1{qwe can solve the inequality logMpε{?n, Bd pq—ˆε?n˙´2p 2´p log˜ dˆε?n˙2p 2´p¸ Áε2, where we used (25). Analogous to the fixed-design setting (where thad the role of?n), we can see that choosing ε2“np{2plogdq1´p{2satisfie... | https://arxiv.org/abs/2503.03426v1 |
1´Cexpp´cnq, which is (28). B.10 Proof of Theorem 4 Proof of Theorem 4. We split the proof of Theorem 4 into several auxiliary lemmas for each potential, which we prove in Appendix B.12. Specifically, we show for each potential from Table 1 that it satisfies Assumption A with the parameters stated in Table 1. Lemma B.1... | https://arxiv.org/abs/2503.03426v1 |
concentration (Wainwright, 2019, Exercise 2.12), which yields the second upper bound. Plugging the second bound into (6) 33 from Theorem 1 we get that with probability at least 1´expp´0.1nq´δit holds sup α‹PτBd 1Rpαt‹qď4r2 ‹pXK3caτq n`8 logp1{δq n ď4˜ 4rkpXq n^12caτc logd n¸ `8 logp1{δq n ď˜ 16rkpXq n^48caτc logd n¸ `8... | https://arxiv.org/abs/2503.03426v1 |
(II). The Moreau envelope Mλ}¨}1`dλ 2is convex (Beck and Teboulle, 2012, §4.2) and non-negative, so that a ψpαq“dˆ pMλ}¨}1qpαq`dλ 2˙2 `ρ 2}α}2 2 is convex, as it is the Euclidean norm of two non-negative convex functions. (III). E.g., by Nikodem and Pales (2011), the ρ-strong convexity of ψwith respect to ℓ2-norm holds... | https://arxiv.org/abs/2503.03426v1 |
řd i“1b x2 i`γ2 ěinf x}x}1_darcsinhpγ´1q´1 }x}1`dγ “darcsinhpγ´1q´1 darcsinhpγ´1q´1`dγ“1 1`γarcsinhpγ´1qě1 2 where we used Fact 2 that γarcsinhp1{γq ď 1for all γą0. Therefore, infx,}y}1“1yJ∇2ψpxqyě arcsinhpγ´1q´2which implies ργ-strong convexity with respect to the ℓ1-norm (Yu, 2015, Theorem 3). With a similar argument... | https://arxiv.org/abs/2503.03426v1 |
1–23 Visual tests using several safe confidence intervals Timoth ´ee Mathieu TIMOTHEE .MATHIEU @INRIA .FR Inria, Universit ´e de Lille, CNRS, Centrale Lille, UMR 9189 – CRIStAL Abstract We propose a new statistical hypothesis testing framework which decides visually, using confidence intervals, whether the means of two... | https://arxiv.org/abs/2503.03567v1 |
overlap provided that the variances of PandQare known. Our main contribution is to provide confidence intervals and associated visual tests of overlap. We bound all the probabilities of error of our test giving theoretical guarantees in a nonparametric setting in which we suppose that the distributions are bounded. We ... | https://arxiv.org/abs/2503.03567v1 |
Moreover, e-values are often anytime-valid , in other words they can deal with data that are collected sequentially. Of particular interest for our article, we will look at the technique leveraging e-values to construct confidence intervals introduced in Waudby-Smith and Ramdas (2024) in the case of bounded random vari... | https://arxiv.org/abs/2503.03567v1 |
we denote L(I) = max x∈Ix−miny∈Iyfor the length of the interval I. We denote I≥aifminx∈Ix≥aandI≥J+aifminx∈Ix≥max y∈Jy+a. 3 MATHIEU 2.2. Tests of overlap LetP,Qbe two distributions with bounded supports. Inspired by works on directional hypotheses tests (Leventhal, 1996), we define the following three hypotheses: H− 1:E... | https://arxiv.org/abs/2503.03567v1 |
to bound the associated type I and type II error. 3. Length of confidence intervals and probability of non-intersection The first step of our analysis is to study some basic properties of our confidence intervals. In this paper we have two concurrent objectives: having tight confidence intervals (which we will also lin... | https://arxiv.org/abs/2503.03567v1 |
c= 1to recover a Hoeffding-type confidence interval. If instead we think that the data will be well concentrated we can use a larger c. Whatever the value ofc,Cn(α;X, w)remain a valid confidence interval, cimpacts on the theoretical error bounds of the test and the width of the confidence intervals. In the case of anyt... | https://arxiv.org/abs/2503.03567v1 |
Theorem 8 (Type I errors) Suppose EP[X] =EQ[Y]. Let t0≥0and suppose Wt(X)and Wt(Y)are deterministic for t≤t0. Let Ct0= max Pt0 t=1Wt(X) 1−Wt(X)(bP−aP)Pt0 t=1Wt(X) 1+Wt(X)(bP−aP),Pt0 t=1Wt(Y) 1−Wt(Y)(bQ−aQ)Pt0 t=1Wt(Y) 1+Wt(Y)(bQ−aQ) . then, P(∃n, m≥t0|Cn(α;X, W )> C m(α;Y, W ))≤ααCt0 4 1 1+Ct0+1 4α24 αCt0Ct0 1+... | https://arxiv.org/abs/2503.03567v1 |
we bound the type III error. This type of error does not happen in a bilateral test and is often assumed too small to care in practice. Theorem 11 (Bound on type III errors) Suppose EP[X]>EQ[Y] + ∆ and for t≤t0, suppose thatWt(X)andWt(Y)are deterministic. Then P(∃n, m≥t0|Cn(α;X)< C m(α;Y))≤α2+α e−∆Pt0 t=1Wt(Y) 1+Wt(Y)... | https://arxiv.org/abs/2503.03567v1 |
≤0.014 Ber equal Ber lower Unif vs Ber equal Unif vs Ber lower Beta equal Beta lower lower 0.0002 0.8512 0.0002 0.8770 0.0000 1.0000 equal 0.9990 0.1488 0.9998 0.1230 1.0000 0.0000 larger 0.0008 0.0000 0.0000 0.0000 0.0000 0.0000 Figure 3: Probabilities of obtaining each decision. Fixed time, with 1000 samples each. Th... | https://arxiv.org/abs/2503.03567v1 |
reward of several al- gorithms on a bandit problem with Bernoulli distribution of arms. The algorithms’ the goal is to maximize the cumulative reward. The means of the arms are 0.6,0.85,0.9and the horizon is 1000 (which can be considered a small sample regime for bandit problems). We use the anytime test de- fined in A... | https://arxiv.org/abs/2503.03567v1 |
of a collection of means. Journal of the Royal Statistical Society: Series A (Statistics in Society) , 158(1):175–177, 1995. Peter Gr ¨unwald, Rianne de Heide, and Wouter Koolen. Safe testing. Journal of the Royal Statistical Society Series B: Statistical Methodology , 86(5):1091–1128, 03 2024. ISSN 1369-7412. doi: 10.... | https://arxiv.org/abs/2503.03567v1 |
Y, wf)) ;Y ) ≥log(Em(B(Y) + ∆ −(LH(P) +LH(Q)) ;Y)) ≥log(Em(B(Y);Y)) + (∆ −(LH(P) +LH(Q))) inf z d dzlog(Em(z;Y)) ≥log1 α + (∆−(LH(P) +LH(Q)))nX t=1wf t(Q) 1 +wf t(Q)(bQ−aQ). • IfEQ[Y]∈Cn(α;X, wf), similarly, we have log(En(EQ[X];X))≥log(En(A(X)−(∆−(LH(P) +LH(Q)));X)) ≥log1 α + (∆−(LH(P) +LH(Q)))nX t=1wf t(P) 1 +w... | https://arxiv.org/abs/2503.03567v1 |
result, −nX t=1Wt(X) 1−Wt(X)(bP−aP)≤d dzlog(En(z;X))≤ −nX t=1Wt(X) 1 +Wt(X)(bP−aP). B.2. Proof of Lemma 4 The following inequality on the logarithm will be used to bound the length of the confidence interval. Lemma 14 (Inequality on the logarithm) For any xsuch that |x| ≤c≤1, we have log(1+ x)≥ x(2−c) 2−c+x. Proof Letf... | https://arxiv.org/abs/2503.03567v1 |
LX, LYsuch that LX−ℓn,m≤ηlog(1/α) Pn t=1Wt(X) 1−Wt(X)(bP−aP),andLY−ℓn,m≤ηlog(1/α) Pn t=1Wt(Y) 1−Wt(Y)(bQ−aQ). (9) Then, we have on Event 1, log(1/α)≤log 1 2mY t=1(1 +Wt(Y)(Yt−µ))! +ηlog(1/α). and on Event 2, log(1/α)≤log 1 2nY t=1(1−Wt(X)(Xt−µ))! +ηlog(1/α). We get a similar result on Event 2 which implies that the fol... | https://arxiv.org/abs/2503.03567v1 |
arXiv:2503.04100v1 [math.ST] 6 Mar 2025IMPROVING DISCREPANCY BY MOVING A FEW POINTS GLEB SMIRNOV AND ROMAN VERSHYNIN Abstract. We show how to improve the discrepancy of an iid sample by movi ng only a few points. Specifically, modifying O(m)sample points on average reduces the Kolmogorov–Smirnov distance to the populati... | https://arxiv.org/abs/2503.04100v1 |
distributions. Proof. LetXbe a random variable with cumulative distribution function F(x), which may not be continuous or strictly increasing. Consider the general ized inverse of F: F−1(u)=inf{x:F(x)/greaterorequalslantu}. For allx∈Randu∈[0,1], we have {F−1(u)/lessorequalslantx}iff{u/lessorequalslantF(x)}. (2.2) IfXisF... | https://arxiv.org/abs/2503.04100v1 |
the total number of points moved by Algorithm 1. We sha ll show that M1isO(m) on average. To bound the expected total number of replaced po ints, we compare Algorithm 1 to the following alternative algorithm: Algorithm 2: •Start with I=[0,1]. •DivideIinto two equal halves, I1andI2. Then randomly and independently selec... | https://arxiv.org/abs/2503.04100v1 |
there exists a dyadic (i.e., of the form 2−j) integer k0such that /radicalbig nk0=m. It then suffices to show that EM2=O/parenleftbig/radicalbig nk0/parenrightbig , 6 GLEB SMIRNOV AND ROMAN VERSHYNIN since, by Lemma 3, the bound on EM1then follows. For each interval Iencountered in Algorithm 2, let m(I)=#{points moved be... | https://arxiv.org/abs/2503.04100v1 |
via the Berry–Esseen theorem or dir ect lower bounds for the binomial tail) imply that P{Ijis dense}>1 100. Thus, the expected number of dense intervals is at least k/100. To achieve the discrepancy Dn<1/(4m), we must make each interval regular, and this requires movin g at least 0.5/radicalbig n/k 8 GLEB SMIRNOV AND R... | https://arxiv.org/abs/2503.04100v1 |
Published as a conference paper at ICLR 2025 GENERALIZABILITY OF NEURAL NETWORKS MINIMIZ - INGEMPIRICAL RISKBASED ON EXPRESSIVE ABILITY Lijia Yu1, Yibo Miao2, 3, Yifan Zhu2, 3, Xiao-Shan Gao2, 3, 4∗, Lijun Zhang1, 3, 5 1Key Laboratory of System Software of Chinese Academy of Sciences Institute of Software, Chinese Acad... | https://arxiv.org/abs/2503.04111v1 |
VC-dimension is equal to the product of the number of parameters and the depth for ReLU networks (Bartlett et al., 2021), which renders the bound in equation 1 useless for over-parameterized models. Most of the algorithmic-dependent generalization bounds make strong and unrealistic assumptions. For example, the NTK con... | https://arxiv.org/abs/2503.04111v1 |
difficult than memorization of Dtr, then the robustness accuracy of FoverDhas an upper bound which may be low, or Fhas no robustness generalization over D. Importance of over-parameterization. (Section 6.2) It is recognized that over-parameterized networks have nice generalizaility (Belkin et al., 2019; Bartlett et al.... | https://arxiv.org/abs/2503.04111v1 |
Sun et al. (2023) gave stability generalization bounds under asynchronous SGD. However, these algorithmic-dependent generalization bounds always impose strong assumptions on the training process or dataset. Generalization bounds for memorization networks were given in Yu et al. (2024b). However, minimizing empirical ri... | https://arxiv.org/abs/2503.04111v1 |
AD(F) =P(x,y)∼D(Sgn(F(x)) =y), where Sgnis the sign function. We use Dtr∼ DNto mean that Dtris a dataset of Nsamples drawn i.i.d. according to D. 3.3 M INIMUM EMPIRICAL RISK Consider the loss function L(F(x), y) = ln(1 + e−F(x)y), which is the cross-entropy loss for binary classification problems. For a dataset Dtr⊂[0,... | https://arxiv.org/abs/2503.04111v1 |
(n)be expressed byHσ W0with confidence c. Then for any W≥W0+ 1,N∈N+,δ∈(0,1), with probability at least 1−δofDtr∼ DN, the following bound stands for any F ∈Mσ W(Dtr, n): AD(F)≥1−O(W0 cW+nLp(W0+c)p log(4n) c√ N+r ln(2/δ) N). (3) Proof Idea. There are two main steps in the proof. The first step tries to estimate the minim... | https://arxiv.org/abs/2503.04111v1 |
expressed byHσ W0with confidence c. Then for any W≥W0+ 1,N∈N+,q≥1andδ∈(0,1), with probability at least 1−δofDtr∼ DN, we have AD(F)≥1−O(qW0 cW+nLp(W0+c)p log(4n) c√ N+r ln(2/δ) N), for any q-approximation F ∈Hσ W(n)to minimize the empirical risk. The conditions of the above theorem can be achieved much easier than those... | https://arxiv.org/abs/2503.04111v1 |
network interpolates the positive separation distribution, and it stands for any distribution D ∈ D (n) as mentioned in Proposition 4.3. Stability bounds represent another algorithm-dependent approach to generalization bound, as shown below: Theorem 4.11 (Theorem 3.7 in Hardt et al. (2016)) .Assume that for every sampl... | https://arxiv.org/abs/2503.04111v1 |
on ReLU networks, by the result in (Bartlett et al., 2019), we have 7 Published as a conference paper at ICLR 2025 Corollary 5.3. For any given n, W, W 0∈N+, there is a D ∈ D (n)that satisfies the following properties. (1) There is an F ∈HW0(n)such that AD(F) = 1 . (2) For any given ϵ, δ∈(0,1), ifN≤O(nW0(1−4ϵ−δ)), then... | https://arxiv.org/abs/2503.04111v1 |
and O(1)width to ensure generalization, but ReLU networks require at least Ω(n)samples and width to ensure generalization. This demonstrates the crucial role of selecting the appropriate network models. Remark 5.8.It is worth mentioning that for some very simple distributions like the Bernoulli distribution, the perfor... | https://arxiv.org/abs/2503.04111v1 |
c0≫c1, such as the example given in the proof of Theorem 4.3 in (Li et al., 2022). 6.2 I MPORTANCE OF OVER -PARAMETERIZED NETWORKS In the above section, we mainly consider F ∈ MW(Dtr, n). But what we really need is F ∈ arg maxG∈HW(n)AD(G). By Theorem 4.4, it is easy to show that when the number of data and the size of ... | https://arxiv.org/abs/2503.04111v1 |
any N≥0, there is a W0≥0, such that if W≥W0, then with probability 0.99ofDtr∼ DN, we have AD(F)≤0.5for some F ∈arg min G∈HW(n)P (x,y)∈DtrLb(G(x), y). This theorem means that to ensure generalizability, it is important to choose the appropriate loss function. The proof is given in the Appendix J. 7 C ONCLUSION In this p... | https://arxiv.org/abs/2503.04111v1 |
Raef Bassily, Vitaly Feldman, Cristóbal Guzmán, and Kunal Talwar. Stability of stochastic gradient descent on nonsmooth convex losses. Advances in Neural Information Processing Systems , 33: 4381–4391, 2020. Alexander Bastounis, Anders C Hansen, and Verner Vla ˇci´c. The mathematics of adversarial attacks in ai–why dee... | https://arxiv.org/abs/2503.04111v1 |
bound for deep neural networks: Cnns, resnets, and beyond. arXiv preprint arXiv:1806.05159 , 2018. Yingcong Li, Muhammed Emrullah Ildiz, Dimitris Papailiopoulos, and Samet Oymak. Transformers as algorithms: Generalization and stability in in-context learning. In International Conference on Machine Learning , pp. 19565–... | https://arxiv.org/abs/2503.04111v1 |
networks. Advances in Neural Information Processing Systems , 35:9139–9150, 2022. Martin J Wainwright. High-dimensional statistics: A non-asymptotic viewpoint , volume 48. Cam- bridge university press, 2019. Mingze Wang and Chao Ma. Generalization error bounds for deep neural networks trained by sgd. arXiv preprint arX... | https://arxiv.org/abs/2503.04111v1 |
Theorem A.1, there exist a Wand aF ∈Zσ W(n)such that |F(x)−f(x)| ≤0.1for all x∈[0,1]n. Thus, F ∈Zσ W(n)⊂Z2W(n)andP(x,y)∼D(yF(x)≥0.9) = 1 . Let the maximum of the absolute value of the parameters of FbeA. IfA≤1, thenFis what we want. IfA >1, then we write F=aReLU( Wx+b)+c, letFA= (a/A)ReLU(( W/A )x+b/A)+c/A2, then there... | https://arxiv.org/abs/2503.04111v1 |
of the three transition matrices plus bias vectors are n+ 1,W+1+a [W/2]and2[W/2], as shown in the below. The first transition matrices: this layer is the same as the first layer of f. Consider the bound of values of parameters of fis not more than 1, and the first transition matrix of has nweights in each row, and with... | https://arxiv.org/abs/2503.04111v1 |
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