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the last step holds due to αK/2+1≤1/2,(c1logK)/K≤1/2, and(1−1/x)x≤exp(−1)for x>1. B.2 Proof of Lemma 2 Let us first control the KL divergence between pXKandpYK. Recall that YK∼ N(0,Id)andXK∼ N(√αKx0,(1−αK)Id)givenX0=x0. We can derive KL/parenleftbig pXK/⌊a∇d⌊lpYK/parenrightbig(i) ≤EX0∼p⋆ 0/bracketleftBig KL/parenleftbig... | https://arxiv.org/abs/2503.09583v1 |
Guntuboyina (2020). For the sake of completeness and to ensure our analysis is s elf- contained, we present a derivation of this bound below. In light of the expression of the score function in ( 25), one can derive /⌊a∇d⌊lst(x)/⌊a∇d⌊l2 2≤1 t2E/bracketleftbig /⌊a∇d⌊lZ0−x/⌊a∇d⌊l2 2|Zt=x/bracketrightbig (i) ≤2 tlogE/brac... | https://arxiv.org/abs/2503.09583v1 |
ϕ(1) t(x):= (2πt)−1/2exp/parenleftbig −x2/(2t)/parenrightbig . We can derive /vextendsingle/vextendsingle/vextendsinglep(1) t(y1)−p(1) t(z1)/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay x1∈R/parenleftBig ϕ(1) t(y1−x1)−ϕ(1) t(z1−x1)/parenrightBig p(1)... | https://arxiv.org/abs/2503.09583v1 |
can derive E/bracketleftBig/vextenddouble/vextenddoubleJt(Zt)/vextenddouble/vextenddouble1/braceleftbig pt(Zt)<cηηt/bracerightbig 1/braceleftbig Zt∈B/bracerightbig/bracketrightBig =/integraldisplay B1/braceleftbig pt(x)<cηηt/bracerightbig/vextenddouble/vextenddoubleJt(x)/vextenddouble/vextenddoublept(x)dx /lessorsimila... | https://arxiv.org/abs/2503.09583v1 |
bounds gives the desired bound. 50 Lemma 11. Given a random vector Z0inRd, considerZt=Z0+√ tWwhereW∼N(0,Id)is a standard Gaussian random vector in Rdthat is independent of Z0. Letst(x):=∇logpZt(x)denote the score function ofZtand letJt(x)denote the Jacobian of st(x). Then for any integer p≥1, one has E/bracketleftBig/v... | https://arxiv.org/abs/2503.09583v1 |
Batch List-Decodable Linear Regression via Higher Moments Ilias Diakonikolas∗ University of Wisconsin-Madison ilias@cs.wisc.eduDaniel M. Kane† UC San Diego dakane@ucsd.edu Sushrut Karmalkar‡ University of Wisconsin-Madison s.sushrut@gmail.comSihan Liu UC San Diego sil046@ucsd.edu Thanasis Pittas§ University of Wisconsi... | https://arxiv.org/abs/2503.09802v1 |
model through small batches of samples collected from different sources. Importantly, as motivated by our running example, we consider the setting where mostbatches might not be collected from reliable sources. Our formal setup is encapsulated in the following definition. Definition 1.1 (List-Decodable Linear Regressio... | https://arxiv.org/abs/2503.09802v1 |
rarely large enough in real-world applications with high-dimensional data. This leaves the regime of 1< n≪das the most meaningful. [ DJKS23] showed that using m=poly(d, n,1/α)batches of sizen≥˜Ω(1/α), it is possible to efficiently recover a list of size O(1/α2)containing an element ˆβwith∥ˆβ−β∥2=O(σ/√nα). Their algorit... | https://arxiv.org/abs/2503.09802v1 |
can cover the entire regime of Clog2(1/α)≤n≤C/α, if we do not necessarily restrict kto be an absolute constant. A limitation is that reaching the lower end of the regime would require kto be super-constant, namely k∼log2(1/α), which would result in quasi-polynomial runtime. Interestingly, even for that lower regime of ... | https://arxiv.org/abs/2503.09802v1 |
2 I(assuming ∥β∗∥2≫σ). They then use an algorithm for robust mean estimation for bounded-covariance distributions to derive an initial estimate ˆβwith a bounded error relative to β∗. They then improve the error by bootstrapping this approach. To do this, they adjust the labels via the transformation y′=y−ˆβ⊤X. This re... | https://arxiv.org/abs/2503.09802v1 |
one step, we bootstrap this to design an iterative algorithm such that the final list will contain an element that is sufficiently close. A significant challenge arises during the iterative phase of our list decoding algorithm. Initially, we generate a list of O(1/α)hypotheses, with the guarantee that at least one of t... | https://arxiv.org/abs/2503.09802v1 |
[ KKK19,RY20]. Unfortunately the runtime and sample complexity had an exponential dependence on 1/α, this was later shown to be necessary for SQ algorithms [DKP+21]. Robust Learning from Batches The problem of learning discrete distributions from untrusted batches was introduced in [ QV18], which gave exponential-time ... | https://arxiv.org/abs/2503.09802v1 |
every iteration, at least one candidate from the list is close to the target regressor. It does so by iteratively applying two subroutines. In Subsection 3.1, we discuss a list-decoding subroutine that, given batch sample queries, generates a list of candidates containing some near-optimal regressor. In Subsection 3.2,... | https://arxiv.org/abs/2503.09802v1 |
bound. The SoS moment bound on the empirical distribution over samples then follows by a careful analysis on the concentration properties of the empirical moments of ZB. See Appendix D for the detailed argument. Lemma 3.4 (SoS Moment Bound) .Letα∈(0,1/2),σ>0,k∈ Z+,β∗∈Rd. Let Tbe a set of mbatches drawn according to the... | https://arxiv.org/abs/2503.09802v1 |
most O(1/α)candidate regressors L′⊆L such that there is at least one regressor β∈L′satisfying ∥β−β∗∥2 2≤O R+kα−1/kσQ1/k/√n with probability at least 1−δover the randomness of the batches drawn. 8 The Pruningalgorithm involves two phases. Initially, it filters regressors β∈Lby keeping those matching a certain set of s... | https://arxiv.org/abs/2503.09802v1 |
show that y−XTβ′2must be weakly anti-concentrated. On the other hand, due to the bounds on the higher-order moments of X, we can show that y−XTβ2must be sufficiently concentrated around its mean. Combining the two observations then we show that Equation (6) must hold with high probability over the inlier distributi... | https://arxiv.org/abs/2503.09802v1 |
This ensures that for at least one transformed instance, the norm of the optimal regressor decreases significantly. Applying Corollary 3.1 to these instances and merging the resulting lists yields a list containing a candidate regressor that is closer to β∗. We iterate this process until we get a list with an element t... | https://arxiv.org/abs/2503.09802v1 |
time via fourier moments. In Proc. 52nd Annual ACM Symposium on Theory of Computing (STOC) , 2020. [CSV17] M. Charikar, J. Steinhardt, and G. Valiant. Learning from untrusted data. In Proc. 49th Annual ACM Symposium on Theory of Computing (STOC) , pages 47–60, 2017. [DeV89] R. D. DeVeaux. Mixtures of linear regressions... | https://arxiv.org/abs/2503.09802v1 |
ArXiv Preprint arxiv:2309.01973 , 2023. [KKK19] S. Karmalkar, A. Klivans, and P. K. Kothari. List-decodable Linear Regression. In Advances in Neural Information Processing Systems 32 (NeurIPS) , 2019. [KS17] P. K. Kothari and J. Steinhardt. Better agnostic clustering via relaxed tensor norms. ArXiv preprint arXiv:1711.... | https://arxiv.org/abs/2503.09802v1 |
setting, this problem suffers from an exponential dependence on t. This is inherent in moment-based approaches, as shown in [ CLS20]. The most efficient algorithm for the problem is due to [ DK20] which runs in time and needs samples quasi-polynomial in t. More recently, [ DK24] gave a fully polynomial-time learner, fo... | https://arxiv.org/abs/2503.09802v1 |
the number of variables appeared is at most k/2since otherwise some yimust have degree- 1by the pigeonhole principle. By a simple counting argument, we have that the number of monomials with non-zero expectations is then at most n (k/2) kk/2.LetQn i=1ywi ibe one of such monomial with non-zero expectation, wherePn i=1w... | https://arxiv.org/abs/2503.09802v1 |
the maximum of a set of non-negative numbers by their sum, and in the last inequality we use the fact that there are at most tnon-zero si’s, and that Dhas its t-th central moments bounded from above by M. This concludes the proof of Claim D.2. We can therefore bound from above the variance of Yby Var[Y]≤E[Y2] =E[T2(X−µ... | https://arxiv.org/abs/2503.09802v1 |
X i−µ⟩t + 2tE i∼[m] ⟨v, µ−µ⟩t ≤2t+1E i∼[m] ⟨v, X i−µ⟩t + 2t∥v∥t 2∥µ−µ∥t 2 ≤2t+2∥v∥t 2M , where in the first line we use the SoS triangle inequality (Fact B.2), in the second line we use SoS Cauchy’s inequality (Fact B.1), and the last inequality follows from Equations (9) and (11). Lemma 3.4 (SoS Moment Bound) .Le... | https://arxiv.org/abs/2503.09802v1 |
then follows by an application of Lemma D.4 with t= 2k,κ:=C1 n σ2+∥β∗∥2 2 ,M:=(2k)2k nkQ σ2k+∥β∗∥2k 2 , F:=(4k)4k n2kQ σ4k+∥β∗∥4k 2 ≤24kM2Q−1, and m≫(2kd)8k24kQ−1+d n σ2+∥β∗∥2 2 M−1/k+ 1. It is not hard to see that (2kd)8k24kQ−1+d n σ2+∥β∗∥2 2 M−1/k+1≤O(1) (4kd)8kQ−1+dQ−1/k+ 1 ≤O (4kd)8kQ−1 , where the la... | https://arxiv.org/abs/2503.09802v1 |
following two subsets of batches: E1:= B∈T:X (X,y)∈B y−X⊤β12 ≤X (X,y)∈B y−X⊤β22 , andE2:= B∈T:X (X,y)∈B y−X⊤β22 ≤X (X,y)∈B y−X⊤β12 . Since each batch Bbelongs to either E1orE2, we have either (W1∩ W 2)(E1)≥(W1∩ W 2)(T)/2or (W1∩ W 2)(E2)≥(W1∩ W 2)(T)/2. Without loss of generality, assume that we are in the f... | https://arxiv.org/abs/2503.09802v1 |
sufficiently large by the Chernoff bound. 23 Next we show Condition 6 is satisfied with high probability over the randomness of T. Fix some β′satisfying ∥β′−β∥2≫R+kα−1/kσQ1/k/√n. We will analyze the random variable Zβ′(B) :=X (X,y)∼B y−X⊤β′2 −X (X,y)∼B y−X⊤β2 , where B∼Dβ∗. Recall that we have y=X⊤β∗+ξ, where ξ∼ N(... | https://arxiv.org/abs/2503.09802v1 |
claim that ∥β′−β∗∥2≫ ∥β−β∗∥2 (23) ∥β′−β∗∥2≥(1−o(1))∥β−β′∥2. (24) To prove Equation (24), we note that ∥β′−β∗∥2≥ ∥β′−β∥2− ∥β−β∗∥2≥(1−o(1))∥β−β′∥2. 25 where the first inequality is the triangle inequality, and the second inequality is true by our assumption that∥β−β∗∥2< R≪ ∥β′−β∥2. Equation (23) then follows immediately ... | https://arxiv.org/abs/2503.09802v1 |
our ourselves to the case α <1/2, which corresponds to more than half of the data being corrupted. We are interested only in this since this is the truly “list-decodable setting”. For this reason, we will use n≪log(1/αB)in the claim below (because we have already mentioned that α=α1/n B, thus in order to have α <1/2we ... | https://arxiv.org/abs/2503.09802v1 |
Generalized network autoregressive modelling of longitudinal networks with application to presidential elections in the USA Guy Nason∗†, Daniel Salnikov∗† Imperial College London and Mario Cortina-Borja†∗ Great Ormond Street Institute of Child Health, University College London March 14, 2025 Abstract Longitudinal netwo... | https://arxiv.org/abs/2503.10433v1 |
NCND OH OKOR PARI SCSD TNTXUTVT VAWA WVWIWY DCRed States Blue States Swing States Figure 1: USA state-wise network, blue nodes are Blue states (Democrat nominee won at least 75% of elections), orange nodes are Redstates (Republican nominee won at least 75% of elections), and grey nodes are Swing states (neither party w... | https://arxiv.org/abs/2503.10433v1 |
to community K2={1,5}. 3 Section 3 studies the parsimonious GNAR family, proposes a conditional least-squares estimator based on a linear model representation, presents idealised results for our estimator and shows attractive finite sample properties of GNAR models. These form the basis of our asymptotic results and al... | https://arxiv.org/abs/2503.10433v1 |
are constant with respect to time, then we drop the tsubscript and write W. In the absence of prior weights, GNAR assigns equal importance to each r-stage neighbour in a neighbourhood regression at all times t, i.e., wij={|N r(i)|}−1, where |Nr(i)| ≤d−1 is the number of r-stage neighbours of node iandNr(i)⊂ K is the se... | https://arxiv.org/abs/2503.10433v1 |
either urban, rural or a hub-town. Each cluster is a collection of nodes that defines a community inG. We select the c-community defined by Kcusing the vectors ξc∈Rd. These are given byξc:= (ξ1,c, . . . , ξ d,c), where ξi,c:=I(i∈Kc). Each entry in ξcis non-zero if and only if i∈Kc, thus, we can select the community lin... | https://arxiv.org/abs/2503.10433v1 |
correspond to nodes in community Kc), and by the parameter γk,r,c:˜c∈R, which is the interaction coefficient. The interaction term for the effect community K˜c has on community Kcat the kth lag and rthr-stage is γk,r,c:˜cZr,c:˜c t−k. The community- α GNAR model is composed of the following additive terms: autoregressiv... | https://arxiv.org/abs/2503.10433v1 |
and incorporate clustering algorithms into model estimation. However, the CNAR model estimates its parameters by plugging in an estimated cluster membership matrix result- ing from the eigendecompostion of the Laplacian for the observed adjacency matrix, i.e., diag{(Pd j=1[S1]ij)} −S1. Thus, said clustering algorithm i... | https://arxiv.org/abs/2503.10433v1 |
(5), CNAR in (6) and BNAR in (7). Each column indicates if the model can incorporate said specification, as well as prior assumptions imposed on the model and/or underlying network. See text for a precise description of each column. Model tv weights missing data diff. or. int. spec. r-stage net. const. latent GNAR ✓ ✓ ... | https://arxiv.org/abs/2503.10433v1 |
b) in Remark 1, results in the more parsimonious local- α GNAR( p,[sk]) model. We interpret it as local autoregressive effects that interact with sets of r-stage neighbours, thus, we use knowledge of the network to study the effect that r-stage neighbours have, rather than the node-wise effects on the local processes, ... | https://arxiv.org/abs/2503.10433v1 |
utkepohl (2005). Corollary 2. LetXtbe acommunity- αGNAR ([ pc],{[sk(c)]},[C],I[C])process such that the autoregressive matrices in (8)satisfy det( Id−pX k=1zpΦk(t−k)) ̸= 0, (11) for all |z| ≤1. Then, Xtgiven by the VAR representation in (9)is stationary. We remark that condition (10) implies condition (11) and is easie... | https://arxiv.org/abs/2503.10433v1 |
note thatPC c=1yc=y, where y:= (Xp+1, . . . ,XT)∈Rndis the ‘response vector’ for the whole model, and we set n= min( nc). Let Rbe the matrix that results from concatenating by columns each Rcforc= 1, . . . , C and further expand the linear model to obtain y=Rθ+u, (13) where u=PC c=1ucis a vector of independent and iden... | https://arxiv.org/abs/2503.10433v1 |
cRc)≥ {τc|Kc|(T−pc)}1/2for all c∈[C]. 13 2. There exists a positive constant Γsuch that the ℓ2-norm of all columns in Rare upper bounded by n−1/2Γ, i.e., there exists Γ>0s.t.max j∈[q]{||[R]·j||2} ≤n−1/2Γ, where n:= min c∈[C]{|Kc|(T−pc)}(i.e., the number of rows in R). 3. The residuals uare independent sub-Gaussian whit... | https://arxiv.org/abs/2503.10433v1 |
it is necessary that no community is empty, and that the linear model obtained by conditioning the realisation is not underdetermined (i.e., K[C]>0 and τ > 0). Also, as the number of communities and/or interactions increases the bound becomes looser, i.e., largerPC c=1qcresult in a looser bound for a given realisation ... | https://arxiv.org/abs/2503.10433v1 |
autoregressive processes. Remark 5. LetXt∈Rbe a stationary zero-mean AR(p)process, which we analyse as a one-node network GNAR( p)process. Then, by Corollary 3 we see that ˆθgiven by (14) is consistent and asymptotically normal, i.e., p (T−p) ˆθ−θ0 d−→Np 0, σ2 uΓ−1 p , where Γp∈Rp×pis the autocovariance matrix of X... | https://arxiv.org/abs/2503.10433v1 |
both communities. Further, it shows that the PNACF cuts-off after the first lag andr-stage one for K1, i.e., it suggests the order (1 ,[1]) for K1, and after the second lag at the first r-stage for lags one and two for K2, i.e., it suggests the pair (2 ,[1,1]) for K2. In this case, the R-Corbit in Figure 3 reflects the... | https://arxiv.org/abs/2503.10433v1 |
Following Shin & Webber (2014), Ishise & Matsuo (2015), we classify states into communities based on the percentage of elections won by either party, i.e., each node (state) is classified as either Red,Blue orSwing , which are identified by the covariates c∈ {1,2,3}and communities: i∈K1if the Republican nominee won at ... | https://arxiv.org/abs/2503.10433v1 |
for the standardised network time series of vote percentages for the Republican nominee in presidential elections in the USA from 1976 to 2020. The fit is a community- αGNAR(2 ,{[1,0]},3). [Recall from (2) that the coefficients are αkcand βkrc, where kis lag, ris network r-stage and cis community.] ˆα11ˆβ111 ˆα21 ˆα12ˆ... | https://arxiv.org/abs/2503.10433v1 |
is, we compare the model’s forecasting performance us- ing the standardised time series Yi,t:=√ 11{P11 t=1(Xi,t−Xi)2}−1/2(Xi,t−Xi), where Xi= (1 /11)P11 t=1Xi,t. Each model produces a one-step ahead standardised forecast ˆYi,12, which we use to compute the one-step ahead forecast as ˆXyi,12={P11 t=1(Xi,t− Xi)2}1/2ˆYi,1... | https://arxiv.org/abs/2503.10433v1 |
alternating between a Democrat and Republican president every eight-years might not be the only governing dynamic. Following the 2016 election, the new pattern could be that parties take turns at the presidency every four rather than eight years. Interestingly, our results corroborate the analysis of political polarisa... | https://arxiv.org/abs/2503.10433v1 |
GNAR. References Anton, C. & Smith, I. (2022), Model based clustering of functional data with mild outliers, in‘Classification and Data Science in the Digital Age’, Springer. Armillotta, M. & Fokianos, K. (2024), ‘Count network autoregression’, J. Time Ser. Anal. 45(4), 584–612. URL: https://onlinelibrary.wiley.com/doi... | https://arxiv.org/abs/2503.10433v1 |
URL: https://CRAN.R-project.org/package=sparsevar Wainwright, M. J. (2019), High-Dimensional Statistics: A Non-Asymptotic Viewpoint , Cambridge University Press, Cambridge. Xiaojing, Z., Cantay, C., P, C. D., Konstantinos, S., Dylan, W. & Eric, K. (2023), ‘Disen- tangling positive and negative partisanship in social me... | https://arxiv.org/abs/2503.10433v1 |
T= 1, . . . , 1000, and “true” parameter vector θ0; see Algorithm 1. Each curve is ℓ2-error-norm, orange is for K1, set of nodes identified as Red states , blue is for K2, (Blue states ), green is for K3, (Swing states ), and black is for the whole model. Algorithm 1 Simulation study 1:set.seed(2024) 2:forT∈ {1, . . . ... | https://arxiv.org/abs/2503.10433v1 |
andRednetwork effects remain after one-lag differencing, whereas, Swing state network effects are not approx statistically significant at the 5% level. This suggests that state-wise communities might have different second-order effects. 29 123 4 5 6 78abs(pnacf) 0.000 0.025 0.050 0.075 0.100 0.125 −0.10−0.050.00pnacf p... | https://arxiv.org/abs/2503.10433v1 |
proof. ■ 32 D Nonasymptotic bounds We present a proof of Theorem 2. We begin by recalling the fixed design we obtain by conditioning the observed realisation, recall from Section 3.2 that a realisation of length T∈Z+coming from a stationary community- αGNAR process can be expressed as the linear model (13), i.e., y=Rθ0... | https://arxiv.org/abs/2503.10433v1 |
the error we note that each community-wise vector norm is the same if we augment the vector with zeros in the entries that correspond to members that do not belong to community Kc, i.e.,||θc||2 2=||ξc⊙θ||2 2, hence, by orthogonality of the ξc, we have that ||θ||2 2=PC c=1||θc||2 2. The community-wise decomposition allo... | https://arxiv.org/abs/2503.10433v1 |
we include it as an assumption for completeness and leave the proper derivation for future work. We assume that A.1, A.2, A.3 in Assumptions 1 hold. Further, assume that as T→+∞, all three conditions hold, i.e., for all T > 0 there exists a τ >0 such that λmin(RT cRc)≥ {τ|Kc|(T−pc)}1/2for all c∈[C], so we have that lim... | https://arxiv.org/abs/2503.10433v1 |
matrix, i.e., the second order ap- proximation is well defined as θ→θ0. See Condition (b) in Chapter 10.2 of Van de Geer (2020). 3. The ℓ2-norm of gradient differences is continuous, i.e., lim θ→θ0 .gθ−.gθ0 2= 0. See Condition (c.2) in Chapter 10.2 of Van de Geer (2020). 4. There exists an ϵ >0 such that the function c... | https://arxiv.org/abs/2503.10433v1 |
.|Zsk(c),c:˜c t−ki , Rk,t:= [Rk,t,c|···|Rk,t,C], Zti:= [R1,t|···|Rp,t], (E.5) where each predictor column for ˜ c∈ I cis concatenated in ascending order with respect to ˜c, i.e., if ˜ c2>˜c1, then the column for ˜ c1precedes the one for ˜ c2,utiis a sub-Gaussian multivariate white noise process with a diagonal covarian... | https://arxiv.org/abs/2503.10433v1 |
jares.t. k is cross-within , E Zr,c:˜c k,t+hZr,c:˜c k,t ,ifi, jares.t. k is cross-interaction , E Zr,c k,t+hZr,c:˜c k,t ,ifi, jares.t. k is within-interaction .(E.12) Since all ki-th processes are stationary, by the spectral representation of the autocovari- ance function; see Brockwell & Davis (2006), we have that... | https://arxiv.org/abs/2503.10433v1 |
the sum of two non-negative numbers, thus, it is minimised if θ=θ0, i.e.,θ0is an idealised minimiser of gθ. By Proposition 1 the risk is finite, and notice that 0≤E∥Zti(θ−θ0)∥2 2≤τ(q)∥(θ−θ0)∥2 2, thus, by iterated expectation we have that Fyt,Zt(gθ0)≤Fyt,Zt(gθ)≤σ2 u+τ(q)∥(θ−θ0)∥2 2<+∞, (E.18) where τ(q):=λmax E{(ZT tZ... | https://arxiv.org/abs/2503.10433v1 |
integral is bounded for all θ, thus, we can differentiate both sides of (E.24) to get Fyt,Zt(.gθ−.gθ0) =E{(ZT tZt)}(θ−θ0) +o(1)∥θ−θ0∥2, i.e., we have shown (E.22). Alternatively, we can look at (E.20) and notice that the error is accounted for by the smoothness and nice properties of squared error loss. ■ Proposition 4... | https://arxiv.org/abs/2503.10433v1 |
Gϵis asymptotically continuous at.gθ0. ■ E.3.2 Completion of asymptotic normality We are ready to finish our proof of asymptotic normality. Proof. By Propositions 1–5 we see that ˆθsatisfies the assumptions in Lemma 2. Define the following Γ(p,[C]) := E{(ZT tZt)}. Then, by (E.22) we see that the influence function isd−... | https://arxiv.org/abs/2503.10433v1 |
arXiv:2503.10580v1 [math.PR] 13 Mar 2025On the Injective Norm of Sums of Random Tensors and the Moment s of Gaussian Chaoses Ishaq Aden-Ali∗ March 14, 2025 Abstract We prove an upper bound on the expected ℓpinjective norm of sums of subgaussian random tensors. Our proof is simple and does not rely on any explicit geome... | https://arxiv.org/abs/2503.10580v1 |
the permutation that maps the elements t1<···< t|I|∈Ito{1,...,|I|}in the same relative order and maps the elements t|I|+1<···< tr∈Icto{|I|+1,...,r}in the same relative order.2We write πI(i) to denote the ith element of the permutation. For a tensor A∈Rd1⊗···⊗Rdrand a subset of indices I⊆[r], we define the permuted tenso... | https://arxiv.org/abs/2503.10580v1 |
which required understanding the underlying (complicated) geometry of the Gaussian process. In this work, we completely avoid using any explicit geometric or chainin g arguments. Instead, we control the relevant (sub)gaussian process by appealing to the so-called PAC-Bayesian lemma . At a high level, the PAC-Bayesian l... | https://arxiv.org/abs/2503.10580v1 |
moment of a random variable X∈Ris defined as /bardblX/bardblp= (E|X|p)1 p. A fundamental result in probability theory due to Lata/suppress la [ Lat06 ] is a sharp upper bound on the moments of decoupled Gaussian chaoses of all orders.4This generalizes a well known result due to Hanson and Wright 4When the tensor Acontai... | https://arxiv.org/abs/2503.10580v1 |
slightly abuse notation by also using subscripts to index a collect ion of vectors x1∈Rd1...xr∈Rdr. In this case we will write ( xk)ito denote the ith entry of the kth vector. We also generalize this notation to tensors, e.g. ( Ak)i1,...,irdenotes the ( i1,...,ir)th entry of the kth tensor. We further overload subscrip... | https://arxiv.org/abs/2503.10580v1 |
a proof in Appendix A for completeness. Lemma 2.3. LetZ1,...,Z nbe independent random variables on a common measurable spac eZ. Let Θ(the parameter space) be asubset of Rdand letπbe aprobability measure (the prior) on Θ. Letf1,...,f n:Z×Θ→R be measurable functions such that EZi[exp(fi(Zi,θ))]<∞π-almost surely for all i... | https://arxiv.org/abs/2503.10580v1 |
arbitrary. Every parameter θ= (θ1,...,θ r)∈Θ has a corresponding posterior ρθ=N(θ1,β−1Id1)⊗···⊗N (θr,β−1Idr). Define the set Mp={ρθ:θ∈Bd1p×···×Bdrp}. As we will see below, these probability measures play a crucial role in how we a pply the PAC-Bayesian lemma. For k∈[n] define the function fk(ξk,θ) =λξkd/summationdisplay ... | https://arxiv.org/abs/2503.10580v1 |
2/vextenddouble/vextenddoubleA({Tk}n k=1)Q/vextenddouble/vextenddouble I2. 8 Proof. Letm/lessorequalslantr−1 be the cardinality of the partition P={I1,...,I m} ∈S(r−1). For i∈[m] define di=d|Ii|and notice that /vextenddouble/vextenddoubleTP/vextenddouble/vextenddouble I2= sup x1∈Bd1 2,...,xm∈Bdm 2d/summationdisplay ir=1... | https://arxiv.org/abs/2503.10580v1 |
Giuseppe Durisi, Benjamin Gu edj, and Maxim Raginsky. Generalization bounds: Perspecti ves from information theory and pac-bayes. Foundations andTrends ®inMachine Learning, 18(1):1–223, 2025. 6 [HW71] D. L. Hanson and F. T. Wright. A Bound on Tail Probabili ties for Quadratic Forms in Independent Random Variables. TheA... | https://arxiv.org/abs/2503.10580v1 |
divergence yields E θ∼ρg(θ)−KL(ρ/bardblπ)/lessorequalslantlog/parenleftbigg E θ∼πexp(g(θ))/parenrightbigg . (12) Noticing that ρ≪πwas chosen arbitrarily together with the fact that Eq. (12) is an equality when ρ=πg yields the claim. We are now ready to prove the PAC-Bayesian lemma. Define the func tiong:Zn×θ→Ras g(Z1,..... | https://arxiv.org/abs/2503.10580v1 |
arXiv:2503.10794v1 [math.ST] 13 Mar 2025Nonparametric Exponential Family Regression Under Star-S haped Constraints Guanghong Yi and Matey Neykov Department of Statistics and Data Science, Northwestern Un iversity guanghongyi2025@u.northwestern.edu mneykov@northwest ern.edu Abstract In this paper, we study the minimax r... | https://arxiv.org/abs/2503.10794v1 |
exponential families with star-shaped constraints. We not e that our approach is inspired by Neykov [2023], where the author constructed an algorithm which iterativ ely selects a vector that minimizes theℓ2norm within designed packing sets. In contrast, our algorit hm introduces a new criterion based on the likelihood ... | https://arxiv.org/abs/2503.10794v1 |
Section3introduces the Bernoulli example we introduced earlier. We derive the minimax rate in this example assuming that the probabilities piare evaluations of a multivariate monotonic function on a lattice. We analyze this model by dividing it in to one-dimensional, two-dimensional, and higher-dimensional cases (three... | https://arxiv.org/abs/2503.10794v1 |
components of θ andθ′. Furthermore, in Lemma 2.1, we constrain the exponential family distribution to be non singular, i.e., Varθi[T(Yi)]>0, for allθi∈[−M,M]. This ensures that the sufficient statistics, T(Yi), are not constant. We also require that the cumulant function is twice continuously di fferentiable and its secon... | https://arxiv.org/abs/2503.10794v1 |
bound. The algorithm constructs a sequence that converges to a desired point. In a nutshell we first build a directed tree of points in Kwhoseconstruction is describedin subsubsection 2.2.2. Next, we usethe data to iteratively traverse this tree and take the limiting point (or any point which is sufficiently far ahead in ... | https://arxiv.org/abs/2503.10794v1 |
take the first element of Uk, sayuk 1. Then construct the set Tk(uk 1) ={uk j∈Uk:/bardbluk j−uk 1/bardbl≤d 2k−1c,j/ne}ationslash= 1} ensuring that uk 1∪Tk(uk 1) does not contain any nodes that have already been pruned (i. e., removed fromUk). For each uk j∈Tk(uk 1), remove the directed edge from P(uk j) touk jand instea... | https://arxiv.org/abs/2503.10794v1 |
parameter estimation process within a specific neighborhoo d of the true parameter θ. This lemma demonstrates that when a candidate is selected by Algorithm 1, it is highly improbable to be far from the true parameter, which bounds the error of our algori thm. Lemma 2.9. Suppose now, we are given NpointsNk:={θ1,...,θN}∈... | https://arxiv.org/abs/2503.10794v1 |
sufficiently large constant allowed to depend onM. Then the minimax rate for estimatingθis given by ε∗2∧d2up to absolute constant factors. Note that the proof of Theorem 2.14heavily utilizes the results of Theorems 2.3and2.12. Theorem 2.14establishes that for all sufficiently smooth nonsingular exp onential families with s... | https://arxiv.org/abs/2503.10794v1 |
order of εso that ε2/greaterorsimilarlogNloc Q−M,M(ε,c). Substituting the covering number bound for the right side, w e have to find the smallest εsuch that: ε4/greaterorsimilar8nM·/parenleftBigg log2√ 2nM ε/parenrightBigg2 . It is simple to see that ε2≍√nlognworks, and thus an upper bound on the minimax rate is given b... | https://arxiv.org/abs/2503.10794v1 |
Q. Han, T. Wang, S. Chatterjee, and R. J. Samworth. Isotonic r egression in general dimensions. The Annals of Statistics , 47(5):2440 – 2471, 2019. doi: 10.1214/18-AOS1753. M. I. Jordan. The exponential family: Basics, 2009. URL https://people.eecs.berkeley.edu/ ~jordan/courses/260-spring10/other-readings/chapter8 .pdf... | https://arxiv.org/abs/2503.10794v1 |
constants c(M),C(M). Taking into account that KL( θ/bardblθ′) =/summationtext i∈[n]KL(θi/bardblθ′ i), we have: c(M)/bardblθ′−θ/bardbl2≤KL(θ/bardblθ′)≤C(M)/bardblθ′−θ/bardbl2(A.2) This completes the proof. Proof of Lemma 2.3.For exponential families under star-shaped constraint spa ce, we have inf /hatwideνsup θE/bardbl... | https://arxiv.org/abs/2503.10794v1 |
from Lemma 2.8, we know that P(H(δ,θα,S)>0) =P ∃β:/bardblθα−θβ/bardbl≥Cδand/summationdisplay i∈[n](θα i−θβ i)T(Yi)+A(θβ i)−A(θα i)≤0 ≤Ne−κ(M)δ2 Thus, we have shown that P/parenleftbig /bardblθ−θ/bardbl≥(C+1)δ/parenrightbig ≤Ne−κ(M)δ2 for any fixed C >2. Before proceeding, we introducefourlemmas that play a sign ific... | https://arxiv.org/abs/2503.10794v1 |
holds. By Lemma A.5, for any 1≤J≤/tildewideJ(which implies J≤J∗) we have 20 /bardblΥJ−θ∗/bardbl≤d(2+4c) c2J=/parenleftbigg1 C+1+4/parenrightbigg ·d 2J usingc= 2(C+1). Note that P(Ac J)≤P(Bc J), for if/bardblΥJ−θ/bardbl≤d/2J−1, then by the triangle inequality and the definition of ωandεJ, we have /bardblθ∗−θ/bardbl≤/bard... | https://arxiv.org/abs/2503.10794v1 |
ε∗2κ(M)<8log2. Case 1:ε∗2κ(M)>8log2 For the case that ε∗satisfiesε∗2κ(M)>8log2, where we also have ε∗2κ(M)≤logNloc(ε∗,c). Whenε∗>0, and define δ∗:=ε∗[/radicalBig κ(M) 8C(M)∧1/2], logNloc(δ∗,c)≥lim γ→0logNloc(ε∗−γ,c)≥lim γ→0[(ε∗−γ)2κ(M)] =ε∗2κ(M) 2+ε∗2κ(M) 2 ≥4δ∗2C(M)+4log2 Thus, we may see that in this case δ∗satisfies th... | https://arxiv.org/abs/2503.10794v1 |
arXiv:2503.10851v3 [econ.EM] 6 May 2025A New Design-Based Variance Estimator for Finely Stratified Experiments∗ Yuehao Bai Department of Economics University of Southern California yuehao.bai@usc.eduXun Huang Department of Economics University of Chicago xhuang520@uchicago.edu Joseph P. Romano Departments of Economics &... | https://arxiv.org/abs/2503.10851v3 |
of a fixed size kaccording to baseline covariates and then, within each group, a fixed number ℓ < kare assigned uniformly at random to treatment and the remainder to control. A prominent special case of this fram ework is a matched pairs design, in which k= 2 and ℓ= 1. While our primary focus is on experiments where trea... | https://arxiv.org/abs/2503.10851v3 |
unless these excep tional restrictions hold. The literature on stratified block randomization dates back to at lea stFisher(1935). For the case in whichℓ= 1 ork−ℓ= 1, the variance estimator in Imai(2008) has been extended to different settings byImai et al. (2009),Pashley and Miratrix (2021), andZhu et al. (2024).Fogarty... | https://arxiv.org/abs/2503.10851v3 |
it should be understood that m→ ∞. Using this notation, the (joint) distribution of treatment assignment is characterized by the following assumption : 2 Assumption 2.1. Treatment status D(n)is assigned independently for each 1 ≤j≤mso that (Di:i∈λj)∼Unif/parenleftbigg/braceleftbigg (d1,...,d k)∈ {0,1}k:/summationdispla... | https://arxiv.org/abs/2503.10851v3 |
the assumption that D(n)satisfiesAssumption 2.1. In contrastto these other estimators, however,we further argue that the magnitude of the bias of our estimator depends exp licitly on the quality of the groupings of the experimental units into strata, in a sense made precise by Theo rem3.2below. 3.1 Motivating Result The... | https://arxiv.org/abs/2503.10851v3 |
k/summationdisplay i∈λj(Yi(1)−Yi(0)). In settings where min {ℓ,k−ℓ}>1, the construction of an upward-biased variance estimator is str aight- forward: an unbiased estimator of S2 j(d) ford∈ {0,1}is given by the sample variance of the outcomes for units within stratum jassigned to treatment d, which we denote by ˆS2 j(d)... | https://arxiv.org/abs/2503.10851v3 |
Estimator of Var[ˆ∆n] Buildingonourearlierworkonsuper-populationapproachestoinfer enceforfinelystratifieddesigns( Bai et al. , 2022,2024a,b,d,2025), we now propose a novel estimator of Var[ ˆ∆n], primarily for settings where min {ℓ,k− ℓ}= 1. Our estimator is constructed by pairing together strata; for simplicity, we pair... | https://arxiv.org/abs/2503.10851v3 |
varianceestimators of Var[ˆ∆n] discussed in Section 3. This framework is formalized by restrictions on the sequence of fin ite populations and experimental designs ( W(n),Λn) specified in the assumptions below. These restrictions are motivated by a limiting thought experiment in which W(n)is itself a realization of an i.... | https://arxiv.org/abs/2503.10851v3 |
Q. Using Assumption 4.1, we have the following expression for the limit of Var[ ˆ∆n] after normalizing appro- priately: Theorem 4.1. Under Assumption 4.1, n·Var[ˆ∆n]→V , where V:=EQ/bracketleftigg VarQ[˜Y(1)|˜X] η+VarQ[˜Y(0)|˜X] 1−η/bracketrightigg −EQ/bracketleftig VarQ[˜Y(1)−˜Y(0)|˜X]/bracketrightig . We can now ... | https://arxiv.org/abs/2503.10851v3 |
n andˆVF ndo not. Specifically, ˆVIM nonly attains Vobswhen treatment effects are sufficiently homogeneous, in the sense that EQ[˜Y(1)−˜Y(0)|˜X] is constant; ˆVF nonly attains Vobswhen the conditional average treatment effectEQ[˜Y(1)−˜Y(0)|˜X] is linear and therefore equal to the best linear predictor of ˜Y(1)−˜Y(0) given 1... | https://arxiv.org/abs/2503.10851v3 |
design (i.e., a finely stratified design with ℓ= 1 and k= 2), under two alternative matching methods, which we call “good match” and “bad match.” For “good match,” we sort units in increasin g values of Xiand then pair adjacent units. For “bad match,” we sort units in increasing values of Xiand then pair them such that t... | https://arxiv.org/abs/2503.10851v3 |
n 1.000 5.288 0.944 8.185 ˆVF n 1.000 1.389 0.939 8.173 ˆVn 0.980 0.848 0.930 8.102 ˆValt n 0.990 0.910 0.880 6.813 250ˆVIM n 1.000 3.274 0.947 5.174 ˆVF n 1.000 0.909 0.944 5.153 ˆVn 1.000 0.812 0.940 5.133 ˆValt n 1.000 0.992 0.892 4.292 500ˆVIM n 1.000 2.256 0.952 3.587 ˆVF n 1.000 0.675 0.952 3.580 ˆVn 0.992 0.370 ... | https://arxiv.org/abs/2503.10851v3 |
To complete the proof, we apply Lemma A.1withXn=√n(ˆ∆n−∆n),σ2 n=nVar[ˆ∆n], and ˆσ2 n=nˆVn. A.2 Proof of Theorem 3.2 First we establish (a). By direct calculation using Assumption 2.1and Theorem 2.1 in Cochran (1977), we have E[ˆτ2 n−ˆκn] =1 nℓ/summationdisplay 1≤i≤nYi(1)2+2(ℓ−1) n(k−1)ℓ/summationdisplay 1≤j≤m/summation... | https://arxiv.org/abs/2503.10851v3 |
The result in ( 22) then follows upon noting 1 m/summationdisplay 1≤j≤m1/parenleftbigk 2/parenrightbig/summationdisplay i<i′∈λjRiRi′=1/parenleftbigk 2/parenrightbig/summationdisplay 1≤γ1<γ2≤k1 m/summationdisplay 1≤j≤mRij,γ1Rij,γ2. We thus obtain that 1 m/summationdisplay 1≤j≤m/summationdisplay i∈λj(Ri−¯Rj)2=1 m/summati... | https://arxiv.org/abs/2503.10851v3 |
follows from the definition of the operator norm, the second inequality follows from repeated application of the inequality that ( a−b)2≤2(a2+b2), and the convergence follows from Assumption 4.2(a) and that Σ X,n→VarQ[˜X], which further follows from Assumption 4.2(b)–(c) by similar arguments to those used to prove ( 23)... | https://arxiv.org/abs/2503.10851v3 |
asymptotic normality, we verify the Lindeberg condition : /summationdisplay 1≤j≤mE/bracketleftigg A2 j s2nI/braceleftigg A2 j s2n> ǫ2/bracerightigg/bracketrightigg →0, wheres2 n=/summationtext 1≤j≤mVar[Aj]. Note that by our previous calculations s2 n∝1 m/summationdisplay 1≤j≤m/summationdisplay i∈λj(Ri−¯Rj)2, 25 hen... | https://arxiv.org/abs/2503.10851v3 |
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