Batch Multi-Fidelity Active Learning with Budget Constraints Shibo Li Jeff M. Phillips Xin Yu Robert M. Kirby and Shandian Zhe

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Batch Multi-Fidelity Active Learning with Budget

Constraints

Shibo Li∗∗, Jeff M. Phillips∗, Xin Yu, Robert M. Kirby, and Shandian Zhe

School of Computing, University of Utah

Salt Lake City, UT 84112

{shibo, jeffp, xiny, kirby, zhe}@cs.utah.edu

Abstract

Learning functions with high-dimensional outputs is critical in many applications,

such as physical simulation and engineering design. However, collecting training

examples for these applications is often costly, e.g., by running numerical solvers.

The recent work (Li et al., 2022) proposes the ﬁrst multi-ﬁdelity active learning

approach for high-dimensional outputs, which can acquire examples at different

ﬁdelities to reduce the cost while improving the learning performance. However,

this method only queries at one pair of ﬁdelity and input at a time, and hence has a

risk to bring in strongly correlated examples to reduce the learning efﬁciency. In this

paper, we propose Batch Multi-Fidelity Active Learning with Budget Constraints

(BMFAL-BC), which can promote the diversity of training examples to improve

the beneﬁt-cost ratio, while respecting a given budget constraint for batch queries.

Hence, our method can be more practically useful. Speciﬁcally, we propose a novel

batch acquisition function that measures the mutual information between a batch

of multi-ﬁdelity queries and the target function, so as to penalize highly correlated

queries and encourages diversity. The optimization of the batch acquisition function

is challenging in that it involves a combinatorial search over many ﬁdelities while

subject to the budget constraint. To address this challenge, we develop a weighted

greedy algorithm that can sequentially identify each (ﬁdelity, input) pair, while

achieving a near

(1 −1/e)

-approximation of the optimum. We show the advantage

of our method in several computational physics and engineering applications.

1 Introduction

Applications, such as in computational physics and engineering design, often demand we calculate

a complex mapping from low-dimensional inputs to high-dimensional outputs, such as ﬁnding the

optimal material layout (output) given the design parameters (input), and solving the solution ﬁeld on

a mesh (output) given the PDE parameters (input). Computing these mappings is often very expensive,

e.g., iteratively running numerical solvers. Hence, learning a surrogate model to outright predict the

mapping, which is much faster and cheaper, is of great practical interest and importance (Kennedy

and O’Hagan, 2000; Conti and O’Hagan, 2010)

However, collecting training examples for the surrogate model becomes another bottleneck, since

each example still requires a costly computation. To alleviate this issue, Li et al. (2022) developed

DMFAL, a ﬁrst deep multi-ﬁdelity active learning algorithm, which can acquire examples at different

ﬁdelities to reduce the cost of data collection. Low-ﬁdelity examples are cheap to compute (e.g.,

with coarse meshes) yet inaccurate; high-ﬁdelity examples are accurate but much more expensive

(e.g., calculated with dense grids). See Fig. 1 for an illustration. DMFAL uses an optimization-based

acquisition method to dynamically identify the input and ﬁdelity at which to query new examples, so

as to improve the learning performance, lower the sample complexity, and reduce the cost.

∗Equal contribution Correspondence to: Jeff M. Phillips, Shandian Zhe.

36th Conference on Neural Information Processing Systems (NeurIPS 2022).

arXiv:2210.12704v1 [cs.LG] 23 Oct 2022

Despite its effectiveness, DMFAL can only optimize and query at one pair of input and ﬁdelity each

time and hence ignores the correlation between consecutive queries. As a result, it has a risk of

bringing in strongly correlated examples, which can restrict the learning efﬁciency and lead to a

suboptimal beneﬁt-cost ratio. In addition, the sequential querying and training strategy is difﬁcult to

utilize parallel computing resources that are common nowadays (e.g., multi-core CPUs/GPUs and

computer clusters) to query concurrently and to further speed up.

In this paper, we propose BMFAL-BC, a batch multi-ﬁdelity active learning method with budget

constraints. Our method can acquire a batch of multi-ﬁdelity examples at a time to inhibit the

example correlations, promote diversity so as to improve the learning efﬁciency and beneﬁt-cost

ratio. Our method can respect a given budget in issuing batch queries, hence are more widely

applicable and practically useful. Speciﬁcally, we ﬁrst propose a novel acquisition function, which

measures the mutual information between a batch of multi-ﬁdelity queries and the target function.

The acquisition function not only can penalize highly correlated queries to encourage diversity,

but also can be efﬁciently computed by an Monte-Carlo approximation. However, optimizing

the acquisition function is challenging because it incurs a combinatorial search over ﬁdelities and

meanwhile needs to obey the constraint. To address this challenge, we develop a weighted greedy

algorithm. We sequentially ﬁnd one pair of ﬁdelity and input each step, by maximizing the increment

of the mutual information weighted by the cost. In this way, we avoid enumerating the ﬁdelity

combinations and greatly improve the efﬁciency. We prove that our greedy algorithm nearly achieves

a(1 −1

e)-approximation of the optimum, with a few minor caveats.

For evaluation, we examined BMFAL-BC in ﬁve real-world applications, including three benchmark

tasks in physical simulation (solving Poisson’s, Heat and viscous Burger’s equations), a topology

structure design problem, and a computational ﬂuid dynamics (CFD) task to predict the velocity

ﬁeld of boundary-driven ﬂows. We compared with the budget-aware version of DMFAL, single

multi-ﬁdeity querying with our acquisition function, and several random querying strategies. Under

the same budget constraint, our method consistently outperforms the competing methods throughout

the learning process, often by a large margin.

2 Background

2.1 Problem Setting

Suppose we aim to learn a mapping

f: Ω ⊆Rr→Rd

where

is small but

is large, e.g., hundreds

of thousands. To economically learn this mapping, we collect training examples at

ﬁdelities. Each

ﬁdelity

corresponds to mapping

fm: Ω →Rdm

. The target mapping is computed at the highest

ﬁdelity, i.e.,

f(x) = fM(x)

. The other

can be viewed as a (rough) approximation of

. Note

that

is unnecessarily the same as

for

m < M

. For example, solving PDEs on a coarse mesh

will give a lower-dimensional output (on the mesh points). However, we can interpolate it to the

-dimensional space to match

f(·)

(this is standard in physical simulation (Zienkiewicz et al., 1977)).

Denote by λmthe cost of computing fm(·)at ﬁdelity m. We have λ1≤. . . ≤λM.

2.2 Deep Multi-Fidelity Active Learning (DMFAL)

To effectively estimate

while reducing the cost, Li et al. (2022) proposed DMFAL, a multi-ﬁdelity

deep active learning approach. Speciﬁcally, a neural network (NN) is introduced for each ﬁdelity

, where a low-dimensional hidden output

hm(x)

is ﬁrst generated, and then projected to the

high-dimensional observation space. Each NN is parameterized by

(Am,Wm,θm)

, where

the projection matrix,

is the weight matrix of the last layer, and

θm

consists of the remaining

NN parameters. The model is deﬁned as follows,

xm= [x;hm−1(x)],hm(x) = Wmφθm(xm),ym(x) = Amhm(x) + ξm,(1)

where

is the input to the NN at ﬁdelity

ym(x)

is the observed

dimensional output,

ξm∼

N(·|0, τmI)

is a random noise,

φθm(xm)

is the output of the second last layer and can be viewed

as a nonlinear feature transformation of

. Since

includes not only the original input

, but

also the hidden output from the previous ﬁdelity, i.e.,

hm−1(x)

, the model can propagate information

throughout ﬁdelities and capture the complex relationships (e.g., nonlinear and nonstationary) between

different ﬁdelities. The whole model is visualized in Fig. 5 of Appendix. To estimate the posterior

of the model, DMFAL uses a structural variational inference algorithm. A multi-variate Gaussian

high-ﬁdelity, costly to solvelow-ﬁdelity, cheap to solve

multi-ﬁdelity surrogate to predict solution ﬁelds

PDE parameters:

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ut+uux⌫·uxx =0

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⌫

PDE solvers

Figure 1: Illustration of the motivation and goal with physical simulation as an example. It is costly to solve

every PDE from scratch. We aim to train a multi-ﬁdelity surrogate model to directly predict high-ﬁdelity solution

ﬁelds given the PDE parameters. To further reduce the cost of collecting the training data with numerical solvers,

we seek to develop multi-ﬁdelity active learning algorithms.

posterior is introduced for each weight matrix,

q(Wm) = Nvec(Wm)|µm,Σm

. A variational

evidence lower bound (ELBO) is maximized via stochastic optimization and the reparameterization

trick (Kingma and Welling, 2013).

To conduct active learning, DMFAL views the most valuable example at each ﬁdelity

as the one

that can best help its prediction at the highest ﬁdelity

(i.e., the target function). Accordingly, the

acquisition function is deﬁned as

a(m, x) = 1

λm

Iym(x),yM(x)|D=1

λm

(H(ym|D) + H(yM|D)−H(ym,yM|D)) ,(2)

where

I(·,·)

is the mutual information,

H(·)

is the entropy, and

is the current training dataset.

The computation of the acquisition function is quite challenging because

and

are both high

dimensional. To address this issue, DMFAL takes advantage of the fact that each low-dimensional

output

hm(x)

is a nonlinear function of the random weight matrices

{W1,...,Wm}

. Based on the

variational posterior

{q(Wj)}

, DMFAL uses the multi-variate delta method (Oehlert, 1992; Bickel

and Doksum, 2015) to estimate the mean and covariance of

h= [hm;hM]

, with which to construct

a joint Gaussian posterior approximation for

by moment matching. Then, from the projection

(1)

, we can derive a Gaussian posterior for

[ym;yM]

. By further applying Weinstein-Aronszajn

identify (Kato, 2013), we can compute the entropy terms in (2) conveniently and efﬁciently.

Each time, DMFAL maximizes the acquisition function

(2)

to identify a pair of input and ﬁdelity at

which to query. The new example is added into

, and the deep multi-ﬁdelity model is retrained and

updated. The process repeats until the maximum number of training examples have been queried or

other stopping criteria are met.

3 Batch Multi-Fidelity Active Learning with Budget Constraints

While effective, DMFAL can only identify and query at one input and ﬁdelity each time, hence

ignoring the correlations between the successive queries. As a result, highly correlated examples are

easier to be acquired and incorporated into the training set. Consider we have found

(x∗, m∗)

that

maximizes the acquisition function

(2)

. It is often the case that a single example will not cause a

signiﬁcant change of the model posterior

{q(Wm)}

(especially when the current dataset

is not very

small). When we optimize the acquisition function again, we are likely to obtain a new pair

(ˆ

x∗,ˆm∗)

that is close to

(x∗, m∗)

(e.g.,

ˆm∗=m∗

and

x∗

close to

x∗

). Accordingly, the queried example

will be highly correlated to the previous one. The redundant information within these examples can

restrict the learning efﬁciency, and demand for more queries (and cost) to achieve the satisfactory

performance. Note that the similar issue have been raised in other single-ﬁdelity, pool-based active

learning works, e.g., (Geifman and El-Yaniv, 2017; Kirsch et al., 2019); see Sec 4.

To overcome this problem, we propose BMFAL-BC, a batch multi-ﬁdelity active learning approach

to reduce the correlations and to promote the diversity of training examples, so that we can improve

the learning performance, lower the sample complexity, and better save the cost. In addition, we take

into account the budget constraint in querying the batch examples, which is common in practice (like

cloud or high-performance computing) (Mendes et al., 2020). Under the given budget, we aim to ﬁnd

a batch of multi-ﬁdelity queries that improves the beneﬁt-cost ratio as much as possible.

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BatchMulti-FidelityActiveLearningwithBudgetConstraintsShiboLi,JeffM.Phillips,XinYu,RobertM.Kirby,andShandianZheSchoolofComputing,UniversityofUtahSaltLakeCity,UT84112{shibo,jeffp,xiny,kirby,zhe}@cs.utah.eduAbstractLearningfunctionswithhigh-dimensionaloutputsiscriticalinmanyapplications,suchasphysi...

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Batch Multi-Fidelity Active Learning with Budget Constraints Shibo Li Jeff M. Phillips Xin Yu Robert M. Kirby and Shandian Zhe

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