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overdispersion). LetXi= (1,Xi2,...,X ip)′andZi= (1,Zi2,...,Z iq)′be random vectors of predictors or covariates (both categorical and continuous predictors are allow ed). We shall assume in 4 the following that the Xi’s are related to the status of interest, while the Zi’s are related to the zero-inflation. XiandZiare al... | https://arxiv.org/abs/2502.16574v1 |
Conversely, sm aller values of lambda reduce the effect of the penalty, retaining more features within th e model. The log-likelihood function penalized by the L1norm is defined by lplasso n(β,γ) =ln(β,γ)+λβp/summationdisplay i=1|βi|+λγq/summationdisplay i=1|γi|. (9) 2.2.2 Ridge regularization ( L2Regularization) Ridge r... | https://arxiv.org/abs/2502.16574v1 |
next section we establish the existency, consistency and as ymptotic normality of the MLE of (7). 4 Asymptotic results We first state some regularity conditions that will be needed to ensu re identifiability. C1The covariates are bounded that is, there exist compact sets F⊂RpandG⊂Rqsuch thatXi∈FandZi∈Gfor every i= 1,2,..... | https://arxiv.org/abs/2502.16574v1 |
are calculated as N−1/summationtextN k=1ˆθ(k) j,n, whereˆθ(k) j,nis the estimate obtained from the k-th simulated sample. The quality of estimates is evaluated by using using the empirical coverag e probability and av- erage length of 95%-level confidence interval (CI), standard de viation (SD) and error (SE), Bias and ... | https://arxiv.org/abs/2502.16574v1 |
probability o f success. Empirical coverage probabilities get closer and closer to 1 as sample size increa ses. The results obtained in these tables confirm those in Table 8, where t he log-likelihood function and AIC are calculated. In particular, the Elastic-net met hod remains better and more accurate than the LASSO ... | https://arxiv.org/abs/2502.16574v1 |
0.000943 0.000914 0.000915 RMSE 0.000098 0.000156 0.000182 0.000135 0.000146 0.000199 0.000272 0.000287 0.000237 0.000211 l(CI) ¡ 2e-16 0.000169 7.05e-12 NaN 2.34e-09 0.585457 0.504791 0.144783 0.656151 0.135220 CP 0.987377 0.988389 0.985701 NaN 0.987873 0.980546 0.971664 0.975144 0.961953 0.959254 Elastic net bias -0.... | https://arxiv.org/abs/2502.16574v1 |
-0.000271 -0.000289 SD 0.000727 0.000664 0.000569 0.000316 0.000505 0.001739 0.001834 0.001992 0.002561 0.002656 SE 0.000023 0.000021 0.000018 0.000010 0.000016 0.000055 0.000058 0.000063 0.000081 0.000084 RMSE 0.000004 0.000005 0.000003 0.000008 0.000005 0.000015 0.0000192 0.000028 0.000037 0.000052 l(CI) 0.000014 0.0... | https://arxiv.org/abs/2502.16574v1 |
rel.bias 0.103528 -0.103894 -0.122717 0.213425 0.241089 0.210104 -0.18539 -0.093798 -0.992846 -0.994157 SD 2.275087 0.284495 0.238767 0.426799 0.247645 2.080304 2.19318 2.216368 2.200134 1.993499 SE 0.101745 0.012723 0.010678 0.019087 0.011075 0.093034 0.098082 0.099119 0.098393 0.089152 RMSE 0.100145 0.127845 0.129883... | https://arxiv.org/abs/2502.16574v1 |
method s for zero-inflated count data with applications to the substance abuse field. Statistic s in Medicine, No 30: 2326-2340, 2011. Chatterjee S., Chowdhury S., Mallick H. , Banerjee P. and Garai B. G roup regularization for zero-inflated negative binomial regression models with an applica tion to healthcare demand in ... | https://arxiv.org/abs/2502.16574v1 |
G.W. , Long D.L. andKincade, M. E. Review and recommendations for zero-inflated count regression modeling of dental caries indice s in epidemiological studies. Caries Research, 54(4) :413–423, 2012. Jiang J. Large sample techniques for statistics. Springer. 2022 Mullahy, J. Heterogeneity, excess zeros, and the structure... | https://arxiv.org/abs/2502.16574v1 |
arXiv:2502.16673v1 [math.ST] 23 Feb 2025On the asymptotic validity of confidence sets for linear functionals of solutions to integral equations Ezequiel Smucler1, James M. Robins2, and Andrea Rotnitzky3 1Glovo 2Harvard T.H. Chan School of Public Health 3University of Washington, Department of Biostatistics February 25, ... | https://arxiv.org/abs/2502.16673v1 |
identifies the local average treatment effect, under the model of Imbens and Angrist (1994) . Asubstantial bodyofworkproposes methodsforconstructing√n-consistent andasymptotically nor- mal estimators for general estimands of the form (1) (Chernozh ukov et al., 2023; Bennett et al., 2022, 2023), as well as for specific cas... | https://arxiv.org/abs/2502.16673v1 |
s for the local average treatment effect estimand with a binary treatment a nd instrument by inverting the score test. However, this strategy relies on the specific structure of t he local average treatment effect estimand and does not extend to the broader class of estimands in (1). In Se ction 5, we propose a method for... | https://arxiv.org/abs/2502.16673v1 |
thatαPis in range(T′ P), a strengthening typically assumed in the lireature (Severini and T ripathi, 2012; Zhang et al., 2023; Bennett et al., 2022). Our model will also accommodate further restrictions on the law P. These include two type of restrictions. The first are constraints on the possible values of ϕ(P), throug... | https://arxiv.org/abs/2502.16673v1 |
non-parametric instrumental variable f unction).SupposeYis an outcome of interest, Zis an instrumental variable, Wis a treatment variable and Xdoes not exist. Newey and Powell (2003) show that, under certain identifyin g assumptions,/integraltext E{Y(w)}ω(w)dµW(w), whereY(w)is the potential outcome under treatment W=wa... | https://arxiv.org/abs/2502.16673v1 |
LetCn=Cn(O1,...,O n)be an asymptotically uniformly valid confidence set of level at least1−αforϕ(P)overM. Then, for every δ >0there exists P∈ Msuch that liminf nP{diam(Cn)≥diam(S)}>1−2α−δ. Corollary 2 establishes that for any asymptotically uniformly valid con fidence set, there exist laws inMat which the confidence set h... | https://arxiv.org/abs/2502.16673v1 |
θ=ϕ(P) becausem(O,gP) =gP(1)−gP(0) =ϕ(P). 2. It is easy to check that gP(W) =EP(Y|Z= 1) +ϕ(P){W−EP(W|Z= 1)}and that qP(Z) =2Z−1 fP,Z(Z)1 covP(W,Z). Given the success of the strategy of inverting the score test for the local average treatment effect, a natural attempt for extending it to the general case would be to c on... | https://arxiv.org/abs/2502.16673v1 |
P:=EP(Y|Z= 1,X=x)−EP(Y|Z= 0,X=x), gde,x P:=EP(W|Z= 1,X=x)−EP(W|Z= 0,X=x). We illustrate a strategy for constructing uniformly valid confidence sets forϕ(P) whenm(O,g) = g(W,1). In such case, the estimand ϕ(P), which coincides with thecounterfactual meanunder treatment X= 1 without baseline covariates in the proximal cau... | https://arxiv.org/abs/2502.16673v1 |
binary. This strategy, illustrated for m(O,g) =g(W,1), extends to general estimands satisfying (1), for variables Z,W,Xtaking values on finite sets. However, the previously men- tioned drawbacks remain. Moreover, we were unable to come up with a general strategy for constructing uniformly valid confidence sets for ϕ(P) i... | https://arxiv.org/abs/2502.16673v1 |
level, the idea o f the proof is to approximate P∗, a law that satisfies Condition 6, by a law /tildewidePthat has a density that is a rectangular non-negative simple function of the form (7), where kZ=kW. This essentially reduces the problem to the discrete case where ZandWhave the same number of levels. We will then a... | https://arxiv.org/abs/2502.16673v1 |
X(x)=/summationdisplay h/summationtext lπi (Y,Z,X),h,l,mµZ(SZ,i l) /summationtext j,lπi (W,Z,X),j,l,mµZ(SZ,i l)µW(SW,i j)I{y∈SY,i h}. Moreover, /tildewidefi (Z,X)(z,x) =/integraldisplay /tildewidefi (W,Z,X)(w,z,x)dµW =/summationdisplay l,m/summationdisplay jπi (W,Z,X),j,l,mµW(SW,i j)I{z∈SZ,i l}I{x∈SX,i m} Now, (9) impl... | https://arxiv.org/abs/2502.16673v1 |
function of the form /summationdisplay j/summationdisplay m/tildewideαi j,mI/braceleftig W∈SW,i j/bracerightig I/braceleftbig X∈SX,i m/bracerightbig . Let/tildewideαi mbe the vector with coordinates /tildewideαi j,mforj= 1,...,ki,m= 1,...,ki X. Letιi ybe the vector with coordinates/integraltext SY,i hydµYforh= 1,...,... | https://arxiv.org/abs/2502.16673v1 |
, for some constants gi j,m. Now, when ( Z,X)∈SZ,i l×SX,i m, we have that EPi{Y|Z,X}=/summationdisplay hπY|Z,i,m l,h/integraldisplay SY,i hydµY 20 and EPi{g(W,X)|Z,X}=/summationdisplay jπW|Z,i,m l,j 1+ηi WµW(SW,i l)/integraldisplay SW,i jg(w,X)dµW =/summationdisplay jπW|Z,i,m l,j 1+ηi WµW(SW,i l)gi j,m where we used th... | https://arxiv.org/abs/2502.16673v1 |
allt. Recalling next that (11) and (23) hold, choosing a ti such thatηW,tiis small enough so that (14), (15), (16), (18) hold and moreover /bardblfi Y|Z,Xfi W|Z,X/tildewidefi (Z,X)−/tildewidefi Y|X/tildewidefi W|X/tildewidefi (Z,X)/bardblL1(µ)<1/i, we would then arrive at the conclusion that Pisatisfies all the requirem... | https://arxiv.org/abs/2502.16673v1 |
W|m/bracketleftigg diag(/tildewideαm)−/tildewideαmπ⊤ W|m π⊤ W|m/tildewide1/bracketrightigg Mmιy= γ/tildewide1/bracketleftigg π⊤ W|mdiag(/tildewideαm)−π⊤ W|m/tildewideαmπ⊤ W|m π⊤ W|m/tildewide1/bracketrightigg Mmιy= γ/tildewide1/bracketleftigg π⊤ W|mdiag(/tildewideαm)Mmιy−π⊤ W|m/tildewideαmπ⊤ W|m π⊤ W|m/tildewide1M... | https://arxiv.org/abs/2502.16673v1 |
fh,l,j,m 0,t fh,l,j,m 1,t/parenrightigg fh,l,j,m 0,tµY(SY h)µZ(SZ l)µW(SW j)µX(SX m) t→∞→/summationdisplay h,l,j,mlog/parenleftbiggfh,l,j,m fh,l,j,m/parenrightbigg fh,l,j,mµY(SY h)µZ(SZ l)µW(SW j)µX(SX m) = 0. In particular, /bardblPn 0,t−Pn 1,t/bardblTV→0, and hence by (37) inf /hatwideϕnsup P∈MEP[min{|/hatwideϕn−ϕ(P... | https://arxiv.org/abs/2502.16673v1 |
/tildewidefi)i≥1satisfies all the requirements in Lemma 1, except possibly for the fact that Lemma 1 requires that ki Z, the number of terms in the summation corresponding to ( l1,l2) is equal toki W, the number of terms in the summation corresponding to ( j1,j2). Note that ki Z=ki Z1×ki Z2 andki W=ki W1×ki W2. Next, we... | https://arxiv.org/abs/2502.16673v1 |
or more elements, the statement holds trivially. Assume then that µYis an atomless measure. Suppose that there exists c∈Rsuch that for all hit holds that/integraltext SY,i hydµY=c×µY(SY,i h). Take any hand define SY,i,+ h=SY,i h∩{y>c} SY,i,− h=SY,i h∩{y≤c}. SinceµYis atomless, it does not have an atom at c. Then,SY,i h=... | https://arxiv.org/abs/2502.16673v1 |
such that fP,A|W,L(a|W,L)> cfor alla∈ {0,1}, almost surely underP. This implies that Psatisfies (3). It follows that with probability one under P, αP(W,A,L) =/summationdisplay j,a,mαj,a,mI{W∈SW j}I{A=a}I{L∈SL m}, (39) where αj,a,m= (2a−1)/summationtext h,g,tπh,g,j,t,mµZ(SZ g)µY(SY h)/summationtext h,gπh,g,j,a,mµZ(SZg)µY... | https://arxiv.org/abs/2502.16673v1 |
fPia.e.µ→fPand the support sets of fPiandfPhave positive measure under µ. Next, we turn to Example 3. In this case αP(W,X) = (2W−1)/fP,W|X(W|X). Note that for all Pthat satisfies (3) it holds that fP,W|X(w|X)>0 forw∈ {0,1}, almost surely under P. We will first show that Condition 3 holds. Suppose Pis such that fPcoincide... | https://arxiv.org/abs/2502.16673v1 |
m} andfPia.e.µ→fP. We know already that with probability one under P, αP(W,X) =/summationdisplay j,mαj,mI{w=j}I{X∈SX m}, where αj,m= (2w−1)/summationtext h,l,jπh,l,j,mµY(SY h)/summationtext h,lπh,l,w,mµY(SY h) and that with probability one under Pi, αPi(W,X) =/summationdisplay j,mαi j,mI{w=j}I{X∈SX m}, where αi j,m= (2... | https://arxiv.org/abs/2502.16673v1 |
, 105(4):987–993. Newey, W. K. and Powell, J. L. (2003). Instrumental variable estim ation of nonparametric models. Econometrica , 71(5):1565–1578. 42 Romano, J. P. (2004). On non-parametric testing, the uniform be haviour of the t-test, and related problems. Scandinavian Journal of Statistics , 31(4):567–584. Severini... | https://arxiv.org/abs/2502.16673v1 |
Minimax Decision Trees via Martingale Approximations Zhenyuan Zhang∗Hengrui Luo† Abstract We develop a martingale-based approach to constructing decision trees that efficiently approximate a target variable through recursive conditioning. We introduce MinimaxSplit, a novel splitting criterion that minimizes the worst-c... | https://arxiv.org/abs/2502.16758v1 |
is Theorem 9 below, where the construction is given by the cyclic MinimaxSplit algorithm detailed in Section 3.1. In the rest of the Introduction, we illustrate the motivations of (P-general) and summarize our contri- bution via two concrete applications: partition-based martingale approximations and regression trees. ... | https://arxiv.org/abs/2502.16758v1 |
(Gaussian) errors with zero mean. In other words, the data set is sampled from some coupling (X∗, Y∗) where Y∗|X∗=g∗(X∗) +εwithX∗andεindependent. The function g∗can be estimated by minimizing the empirical L2risk (i.e., sum of squares; also called L2loss or mean-squared error (MSE) in this paper): ˆmN= arg min m∈M1 NNX... | https://arxiv.org/abs/2502.16758v1 |
the regression setting, researchers often assume a lower bound on the variance decay when the algorithm splits each node in the decision tree (Chi et al., 2022; Syrgkanis and Zampetakis, 2020; Mazumder and Wang, 2024; Cattaneo et al., 2024). For example, a prevalent variance decay assumption is known as the sufficient ... | https://arxiv.org/abs/2502.16758v1 |
law of U, we also establish exponential rates of Crkwhere rcan be made arbitrarily close to 1 /4 (a threshold that cannot be surpassed with binary partitions) while allowing Cto depend on the law of U(Theorem 5). Regression trees. We introduce new splitting criteria, namely the MinimaxSplit algorithm and its mul- tivar... | https://arxiv.org/abs/2502.16758v1 |
.2The sequence of partitions Π in the response space will be constructed recursively, where π0={R}and for every k⩾0, each interval A∈πksplits into two intervals, by following the same construction, forming the elements in πk+1. In the following, we introduce four distinct splitting rules that define partition-based mar... | https://arxiv.org/abs/2502.16758v1 |
pe:=P(U∈Ak+1)∈[0,1]. With each vertex v=Ak, we may associate a location ℓv:=E[U|U∈Ak]. It follows that Mkis supported on the discrete points {ℓv}v∈Vkand furthermore, E[(Mk−Mk+1)2] =X vk∈Vk, vk+1∈Vk+1vk∼vk+1p(vk,vk+1)(ℓvk−ℓvk+1)2. (10) This binary tree representation (10) of the risk will be frequently used in this pape... | https://arxiv.org/abs/2502.16758v1 |
the uniform ones (in Theorem 3). Recall from Section 2.2 that the rates obtained in Theorem 3 are asymptotically optimal for the Simons and median martingales (see Examples 11 and 12). For the variance and minimax martingales, we show in the next result that the rate r= 1/4 is optimal under certain regularity condition... | https://arxiv.org/abs/2502.16758v1 |
0 ; •P(X∈A)>0andY|X∈Ais a constant; •P(X∈A)>0andX|X∈Ais a constant. Otherwise, we say that Ais(X, Y)-splittable, or splittable. Revisiting the VarianceSplit algorithm. The greedy VarianceSplit construction (Section 8.4 of Breiman et al. (1984) and Luo and Li (2024)) proceeds by introducing nested partitions {πk}k⩾0ofRd... | https://arxiv.org/abs/2502.16758v1 |
arg min in (11) may not always be attained for general couplings ( X, Y), unless we assume that X is marginally either atomless or has a finite support, which will always be the case in our analysis. However, in the general setting, if we allow splits in which atoms can be duplicated and assigned to both nodes (instead... | https://arxiv.org/abs/2502.16758v1 |
rule. Assuming marginal non-atomicity, we show that up to universal constants and model mis-specification, the approximation error decays exponentially with rate rkfor some explicit r∈(0,1) that depends only on dimension d. 3.2 Cyclic MinimaxSplit risk bound We first claim that if Xis marginally atomless and j∈[d], the... | https://arxiv.org/abs/2502.16758v1 |
. Consider the additive function class G:={g(x) :=g1(x1) +···+gd(xd)}, 11 where x= (x1, . . . , x d). For g∈ G, define ∥g∥TVas the infimum ofPd i=1∥gi∥TVover all such additive representations of g. (A continuous version of) Theorem 4.2 of Klusowski and Tian (2024) states that if (X, Y,{Mk}k⩾0) is constructed from the V... | https://arxiv.org/abs/2502.16758v1 |
+ε:=E[Y∗|X∗] +ε, where the error ε=Y∗−g∗(X∗) is sub-Gaussian, i.e., for some σ >0, P∗(|ε|⩾u)⩽2 exp −u2 2σ2 , u⩾0. Also, we consider the empirical law of ( X, Y) ofNsamples from P∗. A limitation of Theorem 6 is that it applies only if the covariate Xis marginally atomless (for (20) to hold). Therefore, we need the fol... | https://arxiv.org/abs/2502.16758v1 |
= 7 split points, optimized simultaneously using sequential least squares programming to minimize the overall MSE on the training data. The topology of the tree is predetermined and enforced through constraints in the optimization process. The optimal tree approach contrasts with traditional variance-based, top-down de... | https://arxiv.org/abs/2502.16758v1 |
higher-order polynomial terms, creating a complex surface with multiple peaks and valleys. The visualizations of the prediction surfaces, as shown in Figure 2, further illustrate the improved accuracy and smoother transitions in the predicted values when using the proposed methods. In this example (as well as in Sectio... | https://arxiv.org/abs/2502.16758v1 |
evaluating the models. The next four subplots depict the predictions of the VarianceSplit, Weighted VarianceSplit, MinimaxSplit, cyclic MinimaxSplit for L1(top row) and L2(bottom row) decision trees. All trees are fitted with a max depth of 10. smoother reconstruction with better preservation of large-scale features an... | https://arxiv.org/abs/2502.16758v1 |
values across the input space, serving as the benchmark for evaluating the models. The next three subplots depict the predictions of the variance-based decision tree, minimax decision tree, and cyclic minimax decision tree (depth k= 10 with L1(top) and L2(bottom) norm), respectively. Each plot is annotated with the cor... | https://arxiv.org/abs/2502.16758v1 |
Universal guarantees for decision tree induction via a higher-order splitting criterion. Advances in Neural Information Processing Systems , 33:9475–9484, 2020. Yu V Borovskikh and VS Korolyuk. Martingale Approximation . Walter de Gruyter GmbH & Co KG, 1997. Leo Breiman, Jerome Friedman, Richard Olshen, and Charles Sto... | https://arxiv.org/abs/2502.16758v1 |
, pages 3453–3454. PMLR, 2020. Yan Shuo Tan, Abhineet Agarwal, and Bin Yu. A cautionary tale on fitting decision trees to data from additive models: generalization lower bounds. In International Conference on Artificial Intelligence and Statistics , pages 9663–9685. PMLR, 2022. 20 Aad van der Vaart and Jon Wellner. Wea... | https://arxiv.org/abs/2502.16758v1 |
the above expression) and the upper bounds for the length of the edges (which is d2j−1, d2j). The path leading to a mass Mat level k−1 in the binary tree representation (10) consists of knodes: ∅=N0,N1, . . . ,Nk−1=M, where Nj∈Vj. We denote the lengths of the edges connecting the node Nj−1 to its two descendants on lev... | https://arxiv.org/abs/2502.16758v1 |
disjoint intervals). By H¨ older’s inequality, E[(Mk−1−Mk)2] =2kX j=1pjd2 j⩽2kX j=1pjd3 j2/32kX j=1pj1/3 ⩽ max j∈[2k](pjd2 j)2kX j=1dj2/3 ⩽max j∈[2k](pjd2 j)2/3⩽2−2(k−1)/3.(37) By the martingale property, we then have E[(U−Mk)2] =∞X j=kE[(Mj−Mj+1)2]⩽∞X j=k2−2j/3⩽2−2k/3 1−2−2/3⩽2.71·2−2k/3, as desired. Simons mart... | https://arxiv.org/abs/2502.16758v1 |
martingale. At level k, there are 2kvertices (denoted by j∈[2k]) in the binary tree representation of the partition-based martingale approximation. Denote the probabilities of the vertices by pj(which correspond to P(U∈Aj), Aj∈πk) and the locations by ℓj. It follows that E[(U−Mk)2] =2kX j=1pjE[(U−ℓj)2|U∈Aj] =:2kX j=1pj... | https://arxiv.org/abs/2502.16758v1 |
< y . Take M > sup{x, y}. Let τ+ Mbe the first hitting time to {x:x⩾M},τ− Mbe the first hitting time to {x:x⩽−M}, and τM= min {τ+ M, τ− M}. It is straightforward to prove (see Zhang et al. (2024)) that for some r∈(0,1) and C >0, E[(U−Mk)2]⩽E[ 1{τM<∞}U2] +CM2rk. 26 Since the supports of {Mk}k⩾0correspond to the means of... | https://arxiv.org/abs/2502.16758v1 |
of generality, we assume that E[Y] = 0. Equivalent to (38) is (1 +δ)E[X2]−2(1 + δ)E[XY] +δE[Y2] +C(sup supp X−inf supp X)2⩾0. After completing the square, it remains to show Eh (√ δ Y−1 +δ√ δX)2i + C(sup supp X−inf supp X)2−1 +δ δE[X2] ⩾0. (39) If sup supp X/inf supp X⩾(1−p (1 +δ)/(Cδ))−1, the second term is always n... | https://arxiv.org/abs/2502.16758v1 |
right-hand side of (46), we recall from (21) that as a consequence of the marginally atomless property, max j∈JkE[(Y−Mk)21{X∈Ij}]⩽2−kVar(Y). (47) Next, we claim that s Vj pj⩽(1 +δ−1)3/2∆gj. (48) This is immediate for case (ii) above (when yk,j̸∈Ran(gj)). By definition and using yk,j∈Ran(gj), we have for case (i) above ... | https://arxiv.org/abs/2502.16758v1 |
(B.28) therein (where we used δ⩾2−2⌊k/d⌋/3). This extra term is carried until (B.34) therein, where it can be absorbed by the term U42klog(Nd)/Ntherein. The rest of the proof remains unchanged. B Further numerics B.1 Convergence rates of partition-based martingale approximations The experiments in Figure 5 are designed... | https://arxiv.org/abs/2502.16758v1 |
size (100, 1000 or 10000) exceeds the dimensionality (4, 16 or 64), we observe relatively consistent performance across all three methods. This suggests that when data is abundant re- lative to the input space, the choice of splitting criterion (be it the standard impurity measure used by Scikit-learn, MinimaxSplit, or... | https://arxiv.org/abs/2502.16758v1 |
in child nodes but can be implemented with different loss norms ( L1orL2). Cyclic minimax, a variant of minimax, alternates through features (i.e., the first and second coordinates) in a predetermined order for splitting, which can be beneficial in high-dimensional spaces when the depth is larger than the input dimensi... | https://arxiv.org/abs/2502.16758v1 |
as different tree levels might require different considerations. In fact, the minimax/variance alternating strategy with L2norm (RMSE= 0.112334 ) further improves the lowest RMSE compared to a homogeneous splitting criteria. Con- sequently, a heterogeneous strategy can lead to more accurate and robust models by optimal... | https://arxiv.org/abs/2502.16758v1 |
arXiv:2502.16916v1 [math.PR] 24 Feb 2025Sharp concentration of simple random tensors OMAR AL-GHATTAS Department of Statistics, University of Chicago, IL 60637, USA JIAHENG CHEN* Committee on Computational and Applied Mathematics, University o f Chicago, IL 60637, USA AND DANIEL SANZ-ALONSO Department of Statistics, Uni... | https://arxiv.org/abs/2502.16916v1 |
is to establish concentration inequalities for thep-th order sample moment tensor1 N∑N i=1X⊗p i,which is a sum of i.i.d. simple (rank-one) random tensors. Specifically, we seek to obtain high-probability and expectation bounds on the deviation /vextenddouble/vextenddouble/vextenddouble/vextenddouble1 NN ∑ i=1X⊗p i−EX⊗p/... | https://arxiv.org/abs/2502.16916v1 |
where the supremum in (2.3)is taken over a symmetric convex body K ⊂Rd. Here, we focus on the specific case where K is the unit ball 4 O. AL-GHATTAS, J. CHEN, D. SANZ-ALONSO {v:/bardblv/bardbl ≤1}. In the sub-Gaussian setting, applying [ 84, Lemma 2.7] together with a union bound gives/parenleftbigg Emax 1≤i≤N/bardblXi/... | https://arxiv.org/abs/2502.16916v1 |
empirical process framework. In particular, [ 75] proved (2.5)using decoupling and Slepian-Fernique comparison inequalities. More recently, [ 25] obtained a sharp version of (2.5)with optimal constants via the Gaussian min-max theorem. Beyond the operator norm, [ 65] derived dimension-free bounds on the Frobenius dista... | https://arxiv.org/abs/2502.16916v1 |
hold for the empirical process supf∈F/vextendsingle/vextendsingle1 N∑N i=1|f|p(Xi)−E|f|p(X)/vextendsingle/vextendsingle, where fp(Xi)is replaced by its absolute value. Remark 2.4 For p≥3/2, a minor modification of our proof yields Esup f∈F/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 NN ∑ i=1fp(Xi)−Efp(X)/ve... | https://arxiv.org/abs/2502.16916v1 |
Esup /bardblv/bardbl=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 NN ∑ i=1/an}bracketle{tXi,v/an}bracketri}htp−E/an}bracketle{tX,v/an}bracketri}htp/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥/bardblΣ/bardblp/2/parenleftbigg t1 N/parenleftBig r(Σ)p/2−C(p)/parenrightBig +(1−t)c(p) N/parenrightb... | https://arxiv.org/abs/2502.16916v1 |
Combining ( 3.1), (3.2) and ( 3.5) completes the proof. /square 4. Multi-product empirical processes This section studies multi-product empirical processes f/ma√sto→1 NN ∑ i=1fp(Xi)−Efp(X), f∈F. (4.1) We leverage generic chaining techniques to derive sharp high-probabilit y upper bounds on their suprema in terms of qua... | https://arxiv.org/abs/2502.16916v1 |
it holds with probability at least 1−2exp(−t2/2)that /vextendsingle/vextendsingle/vextendsingle/vextendsingleN ∑ i=1εizi/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤∑ i∈I|zi|+t/parenleftbigg ∑ i∈Icz2 i/parenrightbigg1/2 . In particular, for every 1≤k≤N,it holds with probability at least 1−2exp(−t2/2)that /v... | https://arxiv.org/abs/2502.16916v1 |
sup f∈F/parenleftbiggN ∑ i=1|f|m(Xi)/parenrightbigg1/m /lessorsimilarmγ(F,ψ2)+N1/mdψ2(F)+N1/2md(m−1)/m ψ2(F)γ1/m(F,ψ2) +N1/2mdψ2(F)(ϕ−1(Z))1/m, where ϕ(x)=2min{(√ Nx)2/m,x2}−1,x≥0, andϕ−1(x)denotes its inverse function. Here, Z is a nonnegative random variabl e satisfying EZ≤C for some absolute constant C >0. Moreover,... | https://arxiv.org/abs/2502.16916v1 |
i/parenleftbig |πs+1f|p−1+|πsf|p−1/parenrightbig2 i/parenrightbigg1/2 /lessorsimilarp∑ i∈Is|(∆sf)i||(πs+1f)i|p−1+u2s/2/parenleftbigg ∑ i∈Ics(∆sf)2 i(πs+1f)2(p−1) i/parenrightbigg1/2 +∑ i∈Is|(∆sf)i||(πsf)i|p−1+u2s/2/parenleftbigg ∑ i∈Ics(∆sf)2 i(πsf)2(p−1) i/parenrightbigg1/2 =:T(s) 1+T(s) 2, where in the inequality ( ⋆... | https://arxiv.org/abs/2502.16916v1 |
/bardblπsf/bardblp L8(p−1)/lessorsimilarp/bardblπsf/bardblp ψ2≤dp ψ2(F),/bardblπs0f/bardblp L8(p−1)/lessorsimilarp/bardblπs0f/bardblp ψ2≤dp ψ2(F); (⋆⋆) follows by ∑ s≥s02s/2/bardbl∆sf/bardblψ2≤∑ s≥s02s/2(/bardblf−πsf/bardblψ2+/bardblf−πs+1f/bardblψ2)/lessorsimilarγ(F,ψ2). 20 O. AL-GHATTAS, J. CHEN, D. SANZ-ALONSO Now w... | https://arxiv.org/abs/2502.16916v1 |
anym≥1. Proof of Theorem 4.5Let(Fs)s≥0⊂ F be an increasing sequence which satisfies Fs⊂ F s+1,|F0|= 1,|Fs|≤22s−1 for s≥1,and/uniontext∞ s=0Fsis dense in F. Note that here we slightly change the requirement of the size of Fs(see Definition 2.1) for technical reasons. For any f∈F ands≥0,πsfis the nearest 22 O. AL-GHATTAS, ... | https://arxiv.org/abs/2502.16916v1 |
≤∞ ∑ n=s∗2n/2/bardblf−πnf/bardblψ2+2√ 2∞ ∑ k=02(sk+1−1)/2/bardblf−πsk+1−1f/bardblψ2 ≤4∞ ∑ n=02n/2/bardblf−πnf/bardblψ2/lessorsimilarγ(F,ψ2), where (⋆) follows by the definition of {sk}∞ k=0:/bardblπskf−f/bardblψ2≤2/bardblπsk+1−1f−f/bardblψ2. To prove the inequality ( 4.7), we observe that by the definition of {sk}∞ k=0, ... | https://arxiv.org/abs/2502.16916v1 |
NN ∑ i=1Xi⊗Xi−EX⊗X/vextenddouble/vextenddouble/vextenddouble/vextenddouble≤/bardblΣ/bardbl/parenleftbigg 1+C/radicalbig r(Σ)/parenrightbigg/parenleftbigg 2/radicalbigg r(Σ) N+r(Σ) N/parenrightbigg , where C>0 is some absolute constant. A natural question is whether similarly ti ght constants, which should depend on p, ... | https://arxiv.org/abs/2502.16916v1 |
Xim), which is a natural higher-order generalization of the multiplier empirical processes. The framework developed in this paper provides a natural way for studying an additional generalization: (f1,..., fp)/ma√sto→N ∑ i=1ξif1(Xi)···fp(Xi), where fk∈F(k)(1≤k≤p)for some function classes (F(k))p k=1. We leave a detailed... | https://arxiv.org/abs/2502.16916v1 |
opportunities and theoretical challenges. Annual Review of Statistics and Its Application , 12, 2024. 4. A. Auddy and M. Yuan. Large dimensional independent compo nent analysis: Statistical optimality and computational tractability. arXiv preprint arXiv:2303.18156 , 2023. 5. S. Bamberger, F. Krahmer, and R. Ward. The H... | https://arxiv.org/abs/2502.16916v1 |
arXiv:2207.13594v3 , 2024. 26. Q. Han and J. A. Wellner. Convergence rates of least squar es regression estimators with heavy-tailed errors. The Annals of Statistics , 47(4):2286–2319, 2019. 27. R. Han, R. Willett, and A. R. Zhang. An optimal statistica l and computational framework for generalized tensor estimation. T... | https://arxiv.org/abs/2502.16916v1 |
52(6):2583–2612, 2024. 46. P. McCullagh. Tensor methods in statistics: Monographs on statistics and applied probability . Chapman and Hall/CRC, 2018. 47. S. Mendelson. On weakly bounded empirical processes. Mathematische Annalen , 340(2):293–314, 2008. 48. S. Mendelson. Empirical processes with a bounded ψ1diameter. Ge... | https://arxiv.org/abs/2502.16916v1 |
M. Talagrand. Upper and Lower Bounds for Stochastic Processes: Decomposi tion Theorems , volume 60. Springer Nature, 2022. 71. K. Tikhomirov. Sample covariance matrices of heavy-tai led distributions. International Mathematics Research Notices , 2018(20):6254–6289, 2018. 72. J. A. Tropp. An introduction to matrix conce... | https://arxiv.org/abs/2502.16916v1 |
with probability at least 1 −exp(−v2ω)that (∗)/lessorsimilarpuA+upB/lessorsimilarpvA+vpB. IfC4(p,ω)≤1, the proof is complete. Otherwise, if C4(p,ω)>1, we further observe that C4(p,ω)≤/radicalBig c3(p)C2 1(p)sincelogC2(p) ω≥0. In this case, for 1 ≤v<C4(p,ω), it holds with probability at least 1−exp(−C2 4(p,ω)ω)≥1−exp(−v... | https://arxiv.org/abs/2502.16916v1 |
Convergence of Shallow ReLU Networks on Weakly Interacting Data Léo Dana1, Loucas Pillaud-Vivien2, and Francis Bach1 1Sierra, Inria. 2CERMICS, ENPC. Abstract Weanalysetheconvergenceofone-hidden-layerReLUnetworkstrainedbygradi- ent flow on ndata points. Our main contribution leverages the high dimensionality of the ambi... | https://arxiv.org/abs/2502.16977v1 |
convergence speed to a global minimizer of the loss. We summarize our contributions in the analysis of the learning dynamics of a one- hidden-layer ReLU network on a finite number of data nvia gradient flow. •Our main contribution is that shallow neural networks in high dimension d≥Cn2 interpolates exactly random white... | https://arxiv.org/abs/2502.16977v1 |
symmetry in the function θ7→hθand hence the loss2. This feature is known to lead automati- cally to invariants in the gradient flow as explained generally by Marcotte et al. (2024). The following lemma is not new (Wojtowytsch, 2020, p.11), and shows that, from this invariance, we deduce that the two layers have balance... | https://arxiv.org/abs/2502.16977v1 |
feature learning because of the initialization scale. Another field of study is themean-field regime, where feature learning can happen but where the optimization has been shown to converge only in the infinite width case (Mei et al., 2018; Chizat and Bach, 2018; Rotskoff and Vanden-Eijnden, 2018). Note that it is also... | https://arxiv.org/abs/2502.16977v1 |
the system, and define for all t≥0, µ(t) :=p∥∇L(θt)∥2 L(θt)=−d dtL(θt) L(θt)(5) with the second equality being a property of the gradient flow. Intuitively, this coeffi- cientdescribesthecurvatureinparameterspacethat θt“sees” attime t≥0. Thefollowing lemma is classical and shows how it can be used to prove the global c... | https://arxiv.org/abs/2502.16977v1 |
Lemma 3. Let(xi, yi)1≤i≤nbe generated i.i.d. from a probability distribution PX,Ysuch that the marginal PXis sub-Gaussian, has zero-mean, and satisfy Ex∼PX[xxT] =λ2 dIdfor some λindependent of d, n, while, on R∗, the marginal law PYhas compact support.5 There exists C > 0depending only on the constants C+,− x,yand the ... | https://arxiv.org/abs/2502.16977v1 |
exists jsuch that |aj|21j,iis strictly positive. Thanks to the initialization of the weights, |aj|2≥ |aj(0)|2− ||wj(0)||2>0, and to Assumption 2, 1√ 2(C− x)2>||XTX−DX||. Thus, we have convergence if at any time, for any data input, one neuron remains active, i.e., formally, for all t≥0, and all i∈J1, nK, there exists j... | https://arxiv.org/abs/2502.16977v1 |
the inputs: the interaction between inputs should accelerate the dynamics. However, although all quantities have well defined limits as n→+∞, the limits cannot be understood as a gradient descent in an infinite dimensional space7. Proposition 1 is in fact valid for pfixed, and an initialization of the weights for which... | https://arxiv.org/abs/2502.16977v1 |
can see two phase transitions in the very high-dimensional regime. The constant β0is strictly positive as soon as the limit network does not directly equal the labels, which is natural to assume since they are unknown a priori. Thus the exponential rate of decrease of the loss in the early times of the dynamics is of o... | https://arxiv.org/abs/2502.16977v1 |
1 holds when the number of data points increase. Intuitively, as the number of examples ngrows, the neural network becomes less and less overparametrized, and hence is expected to fail to globally converge. Knowing if and when this occurs with high probability is important for us to understand how much our current thre... | https://arxiv.org/abs/2502.16977v1 |
a special initialization of the network, a phase transition in this rate occurs during the dynamics. Future Directions. We are most enthusiastic about proving the convergence of the networks for linear threshold d≥Cn, which should require new proof techniques, as well as quantifying the impact of large amounts of neuro... | https://arxiv.org/abs/2502.16977v1 |
Libin Zhu, and Mikhail Belkin. Loss landscapes and optimization in over- parameterized non-linear systems and neural networks. Applied andComputational Harmonic Analysis, 59:85–116, 2022. Kaifeng Lyu and Jian Li. Gradient descent maximizes the margin of homogeneous neural networks. International Conference onLearning R... | https://arxiv.org/abs/2502.16977v1 |
j=1|aj|21j,i(15) Proof.of Lemma 6 Let us start by writing the derivatives of the two variables of the system. We define Pj∈Rn×nthe diagonal matrix with elements 1j,i. d dtwj=aj nnX i=1rixi1j,i=aj nXPjR d dtaj=1 nnX i=1ri⟨wj|xi⟩+=1 nwT jXPjR(16) We now compute the derivatives for the residuals ri. d dtri=−1 ppX j=1d dt... | https://arxiv.org/abs/2502.16977v1 |
ifri yi≥1, then ⟨wj∗ i|xi⟩ strictly increases, and otherwise we have 0<yi−ri yi=1 ppX j=1aj yi⟨wj|xi⟩+≤ ⟨wj∗ i(t)|xi⟩1 ppX j=1|aj| |yi|(27) Which implies that ⟨wj∗ i|xi⟩stays strictly positive throughout the dynamics. (iii)As shown in Lemma 5, for p≥4 log(n ε), we have the strict positivity with probability 1−ε. P(∀i,∃... | https://arxiv.org/abs/2502.16977v1 |
at initialization since other- wise nothing happens. We observe that, as long as all ri̸= 0, then the neurons vary monotonously. This implies that if ajyi<0, then the corresponding neuron will reach 0 in finite time. Let t∗ nthe first time any |ri−yi|=yi 2, which is finite with high probability. Forj, isuch that ajyi<0... | https://arxiv.org/abs/2502.16977v1 |
andsjdoes not change by Lemma 1. This implies that the dynamics is decoupled: wj andwqcan be studied separately. Let us compute the dynamics for the neuron j. We let Dn j=1 knPkn i=1yj ixj i,||wj||2 +=Pn i=1⟨wj|xi⟩2 +, and ¯x+=x ||x||+. We first consider the alignment between wjandDn j: d dt⟨¯Dn j|sj¯w+ j⟩= ¯Dn j Id−¯w... | https://arxiv.org/abs/2502.16977v1 |
+ p V(ξ)E[(wTXP)T(wTXP)]˜Y 2 + p V(ξ)E[|a|2(XP)T(XP)]˜Y 2 .(66) where the first term is the deviation of the mean, and the two other come from the deviation of the residuals. A.6 Theorem 2 Proof.of Theorem 2 We consider the setting of Proposition 2 but with a fixed number of neuron p, and as in its proof, we focus on o... | https://arxiv.org/abs/2502.16977v1 |
Section 5.1, andxi ||xi||be an orthogonal family in Section 5.2. Experiment 1. For the experiment in Figure 2, we trained 500 networks in dimension 100, with nbetween 2500 and 3500, with 25 runs for each value of n. We used p= ⌊log(n ε) log(4 3)⌋+1neurons for each experiment with ε= 0.05, since this is the optimal thre... | https://arxiv.org/abs/2502.16977v1 |
the neurons are correctly initialized. Proposition 3 gives an example of such collapse in low dimension. Proposition 3. Suppose that d=n=p= 2. Let (x1, x2)be the canonical basis of R2, with the outputs satisfying y1y2<0, λ=|y2 y1|. Let |a1(0)|,|a2(0)| ≤ δ, and let min j,i⟨wj(0)|xi⟩>0. Then, for δsmall enough, y1large e... | https://arxiv.org/abs/2502.16977v1 |
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