Bootstrap in High Dimension with Low Computation

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Henry Lam 1Zhenyuan Liu 1

Abstract

The bootstrap is a popular data-driven method to

quantify statistical uncertainty, but for modern

high-dimensional problems, it could suffer from

huge computational costs due to the need to re-

peatedly generate resamples and reﬁt models. We

study the use of bootstraps in high-dimensional

environments with a small number of resamples.

In particular, we show that with a recent “cheap”

bootstrap perspective, using a number of resam-

ples as small as one could attain valid coverage

even when the dimension grows closely with the

sample size, thus strongly supporting the imple-

mentability of the bootstrap for large-scale prob-

lems. We validate our theoretical results and com-

pare the performance of our approach with other

benchmarks via a range of experiments.

1. Introduction

The bootstrap is a widely used method for statistical uncer-

tainty quantiﬁcation, notably conﬁdence interval construc-

tion (Efron & Tibshirani,1994;Davison & Hinkley,1997;

Shao & Tu,2012;Hall & Martin,1988). Its main idea is

to resample data and use the distribution of resample esti-

mates to approximate a sampling distribution. Typically, this

approximation requires running many Monte Carlo repli-

cations to generate the resamples and reﬁt models. This is

affordable for classical problems, but for modern large-scale

problems, this repeated ﬁtting could impose tremendous

computation concerns. This issue motivates an array of re-

cent works to curb the computation effort, mostly through a

“subsampling” perspective that ﬁts models on smaller data

sets in the bootstrap process, e.g., Kleiner et al. (2012);

Lu et al. (2020); Giordano et al. (2019); Schulam & Saria

(2019); Alaa & Van Der Schaar (2020).

In contrast to subsampling, we consider in this paper the

Department of Industrial Engineering and Operations Re-

search, Columbia University, New York, NY, USA. Correspon-

dence to: Henry Lam <khl2114@columbia.edu>.

Proceedings of the

40 th

International Conference on Machine

Learning, Honolulu, Hawaii, USA. PMLR 202, 2023. Copyright

2023 by the author(s).

reduction in bootstrap computation cost by using a fewer

number of Monte Carlo replications or resamples. In par-

ticular, we target the following question: Is it possible to

run a valid bootstrap for high-dimensional problems with

very little Monte Carlo computation? While conventional

bootstraps rely heavily on adequate resamples, recent work

(Lam,2022a;b) shows that it is possible to reduce the resam-

pling effort dramatically, even down to one Monte Carlo

replication. The rough idea of this “cheap” bootstrap is to ex-

ploit the approximate independence among the original and

resample estimates, instead of their distributional closeness

utilized in the conventional bootstraps. We will leverage

this recent idea in this paper. However, since Lam (2022a;b)

is based purely on asymptotic derivation, giving an afﬁrma-

tive answer to the above question requires the study of new

ﬁnite-sample bounds to draw understanding on bootstrap

behaviors jointly in terms of problem dimension

, sample

size nand number of resamples B.

To this end, our main theoretical contribution in this paper

is three-fold:

General Finite-Sample Bootstrap Bounds: We derive

general ﬁnite-sample bounds on the coverage error of con-

ﬁdence intervals aggregated from

resample estimates,

where

is small using the “cheap” bootstrap idea, and

B=∞

for traditional quantile-based bootstrap methods

including the basic and percentile bootstraps (e.g., Davison

& Hinkley (1997) Section 5.2-5.3). Our bounds reveal that,

given the same primitives on the approximate normality of

the original and each resample estimate, the cheap bootstrap

with ﬁxed small

achieves similar coverage error bounds

as conventional bootstraps using inﬁnite resamples. This

also simultaneously recovers the main result in Lam (2022a),

but stronger in terms of the ﬁnite-sample guarantee.

Bootstrap Bounds on Function-of-Mean Models Explicit

and

:We specialize our general bounds above to

the function-of-mean model that is customary in the high-

dimensional Berry-Esseen and central limit theorem (CLT)

literature (Pinelis & Molzon,2016;Zhilova,2020). In partic-

ular, our bounds explicit on

and

conclude vanishing

coverage errors for the cheap bootstrap when

p=o(n)

, for

any given

B≥1

. Note that the function-of-mean model

does not capture all interesting problems, but it has been

commonly used – and in fact, appears the only model used

arXiv:2210.10974v4 [stat.ME] 19 Jun 2023

Bootstrap in High Dimension with Low Computation

in deriving ﬁnite-sample CLT errors for technicality reasons.

Our bounds shed light that, at least for this wide class of

models, using a small number of resamples can achieve a

good coverage even in a dimension pgrowing closely with

Bootstrap Bounds on Linear Models Independent of

We further specialize our bounds to linear functions with

weaker tail conditions, which have orders independent of

under certain conditions on the

norm or Orlicz norm of

the linearly scaled random variable.

In addition to theoretical bounds, we investigate the empiri-

cal performances of bootstraps using few resamples on large-

scale problems, including high-dimensional linear regres-

sion, high-dimensional logistic regression, computational

simulation modeling, and a real-world data set RCV1-v2

(Lewis et al.,2004). To give a sense of our comparisons

that support using the cheap bootstrap in high dimension,

here is a general conclusion observed in our experiments:

Figure 1(a) shows the coverage probabilities of

95%

-level

conﬁdence intervals for three regression coefﬁcients with

corresponding true values

0,2,−1

in a 9000-dimensional

linear regression (in Section 4). The cheap bootstrap cov-

erage probabilities are close to the nominal level

95%

even

with one resample, but the basic and percentile bootstraps

only attain around

80%

coverage with ten resamples. In

this example, one Monte Carlo replication to obtain each

resample estimate takes around 4 minutes in the virtual ma-

chine e2-highmem-2 in Google Cloud Platform. Therefore,

the cheap bootstrap only requires 4 minutes to obtain a sta-

tistically valid interval, but the standard bootstrap methods

are still far from the nominal coverage even after more than

a 40-minute run. Figure 1(b) shows the average interval

widths. This reveals the price of a wider interval for the

cheap bootstrap when the Monte Carlo budget is very small,

but considering the low coverages in the other two methods

and the fast decay of the cheap bootstrap width for the ﬁrst

few number of resamples, such a price appears secondary.

Notation. For a random vector

, we write

as the ten-

sor power

X⊗k

. The vector norm is taken as the usual

Euclidean norm. The matrix and tensor norms are taken

as the operator norm. For a square matrix

tr(M)

de-

notes the trace of

Ip×p

denotes the identity matrix in

Rp×p

and

denotes the vector in

whose components

are all

denotes the cumulative distribution function of

the standard normal.

C2(Rp)

denotes the set of twice con-

tinuously differentiable functions on

. Throughout the

whole paper, we use

C > 0

(without subscripts) to denote

a universal constant which may vary each time it appears.

We use

C1, C2, . . .

to denote constants that could depend on

other parameters and we will clarify their dependence when

using them.

2. Background on Bootstrap Methods

We brieﬂy review standard bootstrap methods and from

there the recent cheap bootstrap. Suppose we are interested

in estimating a target statistical quantity

ψ:= ψ(PX)

where

ψ(·) : P 7→ R

is a functional deﬁned on the probability mea-

sure space

. Given i.i.d. data

X1, . . . , Xn∈Rp

following

the unknown distribution

, we denote the empirical dis-

tribution as

PX,n(·) := (1/n)Pn

i=1 I(Xi∈ ·)

. A natural

point estimator is ˆ

ψn:= ψ(ˆ

PX,n).

To construct a conﬁdence interval from

ψn

, a typical be-

ginning point is the distribution of

ψn−ψ

from which we

can pivotize. As this distribution is unknown in general,

the bootstrap idea is to approximate it using the resample

counterpart, as if the empirical distribution was the true

distribution. More concretely, conditional on

X1, . . . , Xn

we repeatedly, say for

times, resample (i.e., sample

with replacement) the data

times to obtain resamples

{X∗b

1, . . . , X∗b

n}, b = 1, . . . , B

. Denoting

P∗b

X,n

as the re-

sample empirical distributions, we construct

resample

estimates

ψ∗b

n:= ψ(ˆ

P∗b

X,n)

. Then we use the

α/2

and

(1 −α/2)

-th quantiles of

ψ∗b

n−ˆ

ψn

, called

qα/2

and

q1−α/2

to construct

[ˆ

ψn−q1−α/2,ˆ

ψn−qα/2]

as a

(1 −α)

-level

conﬁdence interval, which is known as the basic bootstrap

(Davison & Hinkley (1997) Section 5.2). Alternatively, we

could also use the

α/2

and

(1−α/2)

-th quantiles of

ψ∗b

, say

q′

α/2

and

q′

1−α/2

, to form

[q′

α/2, q′

1−α/2]

, which is known as

the percentile bootstrap (Davison & Hinkley (1997) Section

5.3). There are numerous other variants in the literature,

such as studentization (Hall,1988), calibration or iterated

bootstrap (Hall,1986a;Beran,1987), and bias correction

and acceleration (Efron,1987;DiCiccio et al.,1996;DiCic-

cio & Tibshirani,1987), with the general goal of obtaining

more accurate coverage.

All the above methods rely on the principle that

ψn−ψ

and

ψ∗

n−ˆ

ψn

(conditional on

X1, . . . , Xn

) are close in distri-

bution. Typically, this means that, with a

√n

-scaling, they

both converge to the same normal distribution. In contrast,

the cheap bootstrap proposed in Lam (2022a;b) constructs a

(1 −α)-level conﬁdence interval via

hˆ

ψn−tB,1−α/2Sn,B ,ˆ

ψn+tB,1−α/2Sn,B i,(1)

where

n,B = (1/B)PB

b=1(ˆ

ψ∗b

n−ˆ

ψn)2

, and

tB,1−α/2

the

(1 −α/2)

-th quantile of

, the

-distribution with de-

gree of freedom

. The quantity

n,B

resembles the sample

variance of the resample estimates

ψ∗b

’s, in the sense that as

B→ ∞

n,B

approaches the bootstrap variance

V ar∗(ˆ

ψ∗

(where

V ar∗(·)

denotes the variance of a resample condi-

tional on the data). In this way,

(1)

reduces to the normality

interval with a “plug-in” estimator of the standard error term

when

and

are both large. However, intriguingly,

does

not need to be large, and

n,B

is not necessarily close to

Bootstrap in High Dimension with Low Computation

2 4 6 8 10

number of resamples

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

estimated coverage probability

Cheap Boostrap βi= 0

Basic Boostrap βi= 0

Percentile Boostrap βi= 0

Cheap Boostrap βi= 2

Basic Boostrap βi= 2

Percentile Boostrap βi= 2

Cheap Boostrap βi=−1

Basic Boostrap βi=−1

Percentile Boostrap βi=−1

(a) Coverage probability

2 4 6 8 10

number of resamples

0.1

0.2

0.3

0.4

0.5

0.6

0.7

estimated conﬁdence interval width

Cheap Boostrap βi= 0

Basic/Percentile Boostrap βi= 0

Cheap Boostrap βi= 2

Basic/Percentile Boostrap βi= 2

Cheap Boostrap βi=−1

Basic/Percentile Boostrap βi=−1

(b) Conﬁdence interval width

Figure 1. Empirical coverage probabilities and conﬁdence interval widths for different numbers of resamples in a linear regression.

the bootstrap variance. Instead, the idea is to consider the

joint distribution of

ψn−ψ, ˆ

ψ∗1

n−ˆ

ψn,..., ˆ

ψ∗B

n−ˆ

ψn

that

is argued to be asymptotically independent as

n→ ∞

and

ﬁxed, which subsequently allows the construction

of a pivotal

-statistic and gives rise to

(1)

for ﬁxed

. A

more detailed explanation on the cheap bootstrap in the

low-dimensional case (i.e.,

is ﬁxed) is given in Appendix

3. High-Dimensional Bootstrap Bounds

As explained in the introduction, we aim to study the cov-

erage performance of bootstraps in high-dimensional prob-

lems, focusing on the cheap bootstrap approach that allows

the use of small computation effort. We describe our re-

sults at three levels, ﬁrst under general assumptions (Sec-

tion 3.1), then more explicit bounds under the function-of-

mean model and sub-Gaussianity of

(Section 3.2), ﬁnally

bounds for a linear function under weaker tail assumptions

on X(Section 3.3).

3.1. General Finite-Sample Bounds

We have the following ﬁnite-sample bound for the cheap

bootstrap:

Theorem 3.1. Suppose we have the ﬁnite-sample accuracy

for the estimator ˆ

ψn

sup

x∈RP(√n(ˆ

ψn−ψ)≤x)−Φ(x/σ)≤ E1,(2)

and with probability at least

1−β

we have the ﬁnite-sample

accuracy for the bootstrap estimator ˆ

ψ∗

sup

x∈RP∗(√n(ˆ

ψ∗

n−ˆ

ψn)≤x)−Φ(x/σ)≤ E2,(3)

where

σ > 0

and

are deterministic quantities, and

P∗

denotes the probability on a resample conditional on

the data. Then the coverage error of

(1)

satisﬁes, for any

B≥1,

P(|ψ−ˆ

ψn| ≤ tB,1−α/2Sn,B )−(1 −α)

≤2E1+ 2BE2+β. (4)

Condition (2) is a Berry-Esseen bound (Bentkus,2003;

Pinelis & Molzon,2016) that gauges the normal approxima-

tion for the original estimate

ψn

. Condition (3) manifests a

similar normal approximation for the resample estimate

ψ∗

and has been a focus in the high-dimensional CLT literature

(Zhilova,2020;Lopes,2022;Chernozhukov et al.,2020).

Both conditions (in their asymptotic form) are commonly

used to establish the validity of standard bootstrap methods.

Theorem 3.1 shows that, under these conditions, the cover-

age error of the cheap bootstrap interval

(1)

with any

B≥1

can be controlled. Note that Theorem 3.1 is very general in

the sense that there is no direct assumption applied on the

form of

ψ(·)

– All we assume is approximate normality in

the sense of (2) and (3). Due to technical delicacies, in the

bootstrap literature, ﬁnite-sample or higher-order coverage

errors are typically obtainable only with speciﬁc models

(Hall,2013;Zhilova,2020;Lopes,2022;Chernozhukov

et al.,2020), the most popular being the function-of-mean

model (see Section 3.2) or even simply the sample mean.

In contrast, the bound in Theorem 3.1 that concludes the

sufﬁciency in using a very small

is a general statement

that does not depend on the delicacies of

ψ(·)

. Moreover,

by plugging in suitable bounds for

E1,E2, β

under regularity

conditions, Theorem 3.1 also recovers the main result (The-

orem 1) in Lam (2022a). The detailed proof of Theorem

3.1 is in Appendix D(and so are the proofs of all other

theorems). Below we give a sketch of the main argument.

Bootstrap in High Dimension with Low Computation

Proof sketch of Theorem 3.1.Step 1: We write the cover-

age probability as the expected value (with respect to data)

of a multiple integral with respect to the distributions of

√n(ˆ

ψ∗

n−ˆ

ψn)(denoted by Q∗, conditional on data), i.e.,

P(|ψ−ˆ

ψn| ≤ tB,1−α/2Sn,B )

=P



√n(ˆ

ψn−ψ)

BPB

b=1(√n(ˆ

ψ∗b

n−ˆ

ψn))2≤tB,1−α/2



=E



Z|√n(ˆ

ψn−ψ)|

√PB

b=1 z2

b/B ≤tB,1−α/2

dQ∗(zB)···dQ∗(z1)



.

(5)

Step 2: Suppose (3) happens and denote this event by

which satisﬁes

P(Ac)≤β

. For each

b= 1, . . . , B

, given

all other

zb′, b′̸=b

, the integration region is of the form

zb∈(−∞,−q]∪[q, ∞)

for some

q≥0

. Then we can

replace the distribution

Q∗

by the distribution of

N(0, σ2)

(denoted by

) with controlled error given in (3) and obtain

E



Z|√n(ˆ

ψn−ψ)|

√PB

b=1 z2

b/B ≤tB,1−α/2

dQ∗(zB)···dQ∗(z1)





=E



Z|√n(ˆ

ψn−ψ)|

√PB

b=1 z2

b/B ≤tB,1−1−α/2

dP0(zB)···dP0(z1)





+R1,(6)

where

|R1| ≤ 2BE2+β

accounts for the error from (3) and

the small probability event Ac.

Step 3: Following the same logic in Step 2 and noticing that

the integration region for

√n(ˆ

ψn−ψ)

[−q, q]

for some

q≥0

, we can also replace the distribution of

√n(ˆ

ψn−ψ)

by the distribution

with controlled error

|R2| ≤ 2E1

according to (2):

E



Z|√n(ˆ

ψn−ψ)|

√PB

b=1 z2

b/B ≤tB,1−α/2

dP0(zB)···dP0(z1)





=Z|z0|

√PB

b=1 z2

b/B ≤tB,1−α/2

dP0(zB)···dP0(z1)dP0(z0)

+R2

= 1 −α+R2.(7)

Step 4: Plugging (6) and (7) back into (5), we can express

the coverage probability as a sum of the nominal level and

the remainder term:

P(|ψ−ˆ

ψn| ≤ tB,1−α/2Sn,B ) = 1 −α+R1+R2

with error

|R1+R2| ≤ 2E1+ 2BE2+β

. This gives our

conclusion.

Theorem 3.1 is designed to work well for small

(our target

scenario), but deteriorates when

grows. However, in the

latter case, we can strengthen the bound to cover the large-

regime with additional conditions on the variance estimator

(see Appendix B.1).

We compare with standard basic and percentile bootstraps

using B=∞. Below is a generalization of Zhilova (2020)

which focuses only on the basic bootstrap under the function-

of-mean model.

Theorem 3.2. Suppose the conditions in Theorem 3.1 hold.

qα/2

q1−α/2

are the

α/2

-th and

(1 −α/2)

-th quantiles

ψ∗

n−ˆ

ψn

respectively given

X1, . . . , Xn

, then a ﬁnite-

sample bound on the basic bootstrap coverage error is

|P(ˆ

ψn−q1−α/2≤ψ≤ˆ

ψn−qα/2)−(1 −α)|

≤2E1+ 2E2+ 2β. (8)

q′

α/2

q′

1−α/2

are the

α/2

-th and

(1 −α/2)

-th quantiles

ψ∗

respectively given

X1, . . . , Xn

, then a ﬁnite-sample

bound on the percentile bootstrap coverage error is

|P(q′

α/2≤ψ≤q′

1−α/2)−(1−α)| ≤ 2E1+2E2+2β. (9)

In view of Theorems 3.1 and 3.2, the cheap bootstrap with

any ﬁxed

can achieve the same order of coverage error

bound as the basic and percentile bootstraps with

B=∞

in the sense that

(1/2) EBQuantile ≤EBCheap ≤BEBQuantile,(10)

where

EBCheap

is the RHS error bound of

(4)

and

EBQuantile

that of

(8)

(9)

. This shows that, to attain a good coverage

that is on par with standard basic/percentile bootstraps, it

sufﬁces to use the cheap bootstrap with a small

which

could save computation dramatically.

Besides coverage, another important quality of conﬁdence

interval is its width. To this end, note that for any ﬁxed

(3) ensures that

√nSn,B ⇒σpχ2

B/B

(unconditionally as

n→ ∞

with proper model assumptions) . Therefore, the

half-width of (1) is approximately

tB,1−α/2σpχ2

B/(nB)

with expected value

E"tB,1−α/2σrχ2

nB #=tB,1−α/2σr2

Γ((B+ 1)/2)

Γ(B/2) ,

(11)

where

Γ(·)

is the gamma function. Since the dimensional

impact is hidden in

which is a common factor in the

expected width as

varies, we can see

does not affect the

relative width behavior as

changes. In particular, from

(11)

the inﬂation of the expected width relative to the case

Bootstrap in High Dimension with Low Computation

B=∞

is 417.3% for

B= 1

, and dramatically reduces to

94.6%, 24.8% and 10.9% for

B= 2,5,10

, thus giving an

interval with both correct coverage and short width using a

small computation budget B.

In the next sections, we will apply Theorem 3.1 to obtain

explicit bounds for speciﬁc high-dimensional models. Here,

in relation to

(10)

, we brieﬂy comment that the order of the

coverage error bounds for these models is of order

1/√n

both for the cheap bootstrap (which we will derive) and

state-of-the-art high-dimensional bootstrap CLT. This is

in contrast to the typical

1/n

coverage error in two-sided

bootstrap conﬁdence intervals in low dimension (see Hall

(2013) Section 3.5 for quantile-based bootstraps and Lam

(2022a) Section 3.2 for the cheap bootstrap).

3.2. Function-of-Mean Models

We now specialize to the function-of-mean model

ψ=g(µ)

for a mean vector

µ=E[X]∈Rp

and smooth function

g:Rp7→ R

, which allows us to construct more explicit

bounds. The original estimate

ψn

and resample estimate

ψ∗b

are now given by

g(¯

Xn)

and

g(¯

X∗b

respectively, where

denotes the sample mean of data and

X∗b

denotes the

resample mean of X∗b

1, . . . , X∗b

n. We assume:

Assumption 3.3. The function

g(x)∈C2(Rp)

has Hes-

sian matrix

Hg(x)

with uniformly bounded eigenvalues,

that is,

∃

a constant

CHg>0

s.t.

supx∈Rp|a⊤Hg(x)a| ≤

CHg||a||2,∀a∈Rp.

Assumption 3.4.

is sub-Gaussian, i.e., there is a constant

τ2>0

s.t.

E[exp(a⊤(X−µ))] ≤exp(||a||2τ2/2),∀a∈

. Furthermore,

has a density bounded by a constant

and its covariance matrix

is positive deﬁnite with the

smallest eigenvalue λΣ>0.

Based on Theorem 3.1, we derive the following explicit

bound:

Theorem 3.5. Suppose the function

satisﬁes Assump-

tion 3.3 and random vector

satisﬁes Assumption 3.4.

Moreover, assume

||∇g(µ)|| > C∇g√p

for some constant

C∇g>0. Then we have

P(|g(µ)−g(¯

Xn)| ≤ tB,1−α/2Sn,B )−(1 −α)

≤6

n+BC m31

√nσ3+CHgm1/3

31 tr(Σ)

√nσ2+CHgm2/3

n5/6σ

+CHgm1/3

31 m2/3

nσ2+CHgτ2

C∇g√λΣ1 + log n

prp

+||E[(X−µ)3]||

λ3/2

√n+τ3

λ3/2

Σ1 + log n

p3/21

√n

+τ4√p

λ2

Σn1 + log n

p1/2

+τ2√p

λΣn1 + log n

p1/2

+τ3√p

λ3/2

Σn1 + log n

p!

+BC1τ4(log n)3/2

λ2

Σ√n+τ2(log n)3/2

λΣ√n

+τ3

λ3/2

Σ√n1 + log n

p1/2

(log n+ log p)plog n!,

where

m31 =E[|∇g(µ)⊤(X−µ)|3]

m32 := E[||X−µ||3]

σ2=∇g(µ)⊤Σ∇g(µ)

is a universal constant and

is a constant only depending on CX.

Theorem 3.5 is obtained by tracing the implicit quantities in

Theorem 3.1 for the function-of-mean model, via extracting

the dependence on problem parameters in the Berry-Esseen

theorems for the multivariate delta method (Pinelis & Mol-

zon,2016) and the standard bootstrap (Zhilova,2020). In

particular, the sub-Gaussian assumption is required to de-

rive ﬁnite-sample concentration inequalities, in a similar

spirit as the state-of-the-art high-dimensional CLTs (e.g.,

Chernozhukov et al. (2017); Lopes (2022)). On the other

hand, the third moments such as

||E[(X−µ)3]||

(operation

norm of the third order tensor

E[(X−µ)3]

m31

and

m32

are due to the use of the Berry-Esseen theorem and a mul-

tivariate higher-order Berry-Esseen inequality in Zhilova

(2020), which generally requires this order of moments. The

bound in Theorem 3.5 can be simpliﬁed with reasonable

assumptions on the involved quantities:

Corollary 3.6. Suppose the conditions in Theorem 3.5 hold.

Moreover, suppose that

τ, CHg, C1=O(1)

C∇g, λΣ=

Θ(1)

||∇g(µ)||2=O(p)

σ2= Θ(p)

and

||E[(X−

µ)3]|| =O(1). Then as p, n → ∞,

P(|g(µ)−g(¯

Xn)| ≤ tB,1−α/2Sn,B )−(1 −α)

=B×O1 + log n

prp

√n1 + log n

p1/2

(log n+ log p)plog n!.

Consequently, for any ﬁxed

B≥1

, the cheap bootstrap

conﬁdence interval is asymptotically exact provided

o(n), i.e.,

lim

p,n→∞

p=o(n)

P(|g(µ)−g(¯

Xn)| ≤ tB,1−α/2Sn,B )=1−α.

In Corollary 3.6, the cheap bootstrap coverage error shrinks

n→ ∞

p=o(n)

, i.e., the problem dimen-

sion grows slower than

in any arbitrary fashion. Al-

though there is no theoretical guarantee that the choice of

p=o(n)

is tight, we offer numerical evidence in Section

4where the cheap bootstrap works with a small

when

p/n = 0.09

but it fails (i.e., over-covers the target with a

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BootstrapinHighDimensionwithLowComputationHenryLam1ZhenyuanLiu1AbstractThebootstrapisapopulardata-drivenmethodtoquantifystatisticaluncertainty,butformodernhigh-dimensionalproblems,itcouldsufferfromhugecomputationalcostsduetotheneedtore-peatedlygenerateresamplesandrefitmodels.Westudytheuseofbootstrap...

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Bootstrap in High Dimension with Low Computation

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