ReTaSA: A Nonparametric Functional Estimation Approach for Addressing Continuous Target Shift
Authors: Hwanwoo Kim
Xin Zhang
Jiwei Zhao
Qinglong Tian
Venue: ICLR 2024 poster (2024)
Keywords: label shift
target shift
distributional shift
domain adaptation
transfer learning
importance weight
Primary area: transfer learning, meta learning, and lifelong learning
TL;DR: Addressing continuous target shift (i.e., label shift) by nonparametrically estimating the importance weight function.

Abstract: The presence of distribution shifts poses a significant challenge for deploying modern machine learning models in real-world applications. This work focuses on the target shift problem in a regression setting  (Zhang et al., 2013; Nguyen et al., 2016). More specifically, the target variable $y$ (also known as the response variable), which is continuous, has different marginal distributions in the training source and testing domain, while the conditional distribution of features $\boldsymbol{x}$ given $y$ remains the same. While most literature focuses on classification tasks with finite target space, the regression problem has an *infinite dimensional* target space, which makes many of the existing methods inapplicable. In this work, we show that the continuous target shift problem can be addressed by estimating the importance weight function from an ill-posed integral equation. We propose a nonparametric regularized approach named *ReTaSA* to solve the ill-posed integral equation and provide theoretical justification for the estimated importance weight function. The effectiveness of the proposed method has been demonstrated with extensive numerical studies on synthetic and real-world datasets.

Published as a conference paper at ICLR 2024
RETASA: A NONPARAMETRIC FUNCTIONAL ESTIMA- TION APPROACH FOR ADDRESSING CONTINUOUS TAR- GET SHIFT
Hwanwoo Kim' Xin Z Jiwei Zhao", Qinglong Tian**
University of Chicago', Meta, Inc.?, University of Visconsin-Madisoy University of Waterloo
ABSTRACT
The presence of distribution shifts poses significant challenge for deploying modern machine learning models in real-world applications. This work focuses the target shift problem in a regression setting (Zhang al., 2013 Nguyen) et al. 2016). More specifically, the target variable (also known as the response variable), which is continuous, has different marginal distributions in the training source and testing domain, while the conditional distribution of features x given y remains the same. While most literature focuses on classification tasks with finite target space, the regression problem has an infinite dimensional target space, which makes many the existing methods inapplicable. In this work, we show that the continuous target shift problem can be addressed by estimating the importance weight function from an ill- ill-posed integral equation. We propose a nonparametric regularized approach named ReTaSA to solve the ill-posed integral equation and provide theoretical justification for the estimated importance weight function. The effectiveness of the proposed method has been demonstrated with extensive numerical studies on synthetic and real-world datasets.
INTRODUCTION
Let R. be the feature vector, y E R be the target variable, and p(x, y) be their joint probability distribution. Classic supervised learning assumes the training and testing data are drawn from the same distribution (i.e., the joint distribution p(x; y) does not change across domains). However, in practice, distributional shifts between the training and testing data are pervasive due to various reasons, such as changes in the experimental objects, the data collection process, and the labeling process (Quinonero- Canerlla et al. 2008 Storkey 2009). When the distributional shift occurs, the knowledge learned from the training source data may no Tonger be appropriate to be directly generalized to the testing data.
As a common type of distributional shift, the target shift assumes that the conditional distribution of features given the target variable p(x y) remains the same in the training source and testing data domains, but the marginal distribution of the target variable p(y) differs. Existing works (Tasche)
2017 Guo al., 2020 Lipton et al.) 2018: Du Plessis & Sugiyama 2014 Iyer et al., 2014 Tian et al. 2023) on target shift primarily focused on the case where y is categorical (also known as label shift or class prior change), while the cases with continuous y (the regression task) have received less attention. However, the target shift problem in regression tasks is ubiquitous in real-world machine learning applications. For example, the Sequential Organ Failure Assessment (SOFA) score (Jones) al. 2009) is the continuous indicator associated with patients' wellness condition. Considering the scenario where the SOFA score is employed to determine the transfer destination patients: those with higher SOFA scores, indicative of severe illness, more likely to be transferred to large medical facilities with advanced medical equipment and highly skilled staff. Consequently, the distribution of SOFA scores among patients differs between large medical facilities and smaller clinics, while the distribution of features given the SOFA score remains the same across different hospitals. Nevertheless, the SOFA score predictive model trained on large medical facilities' datasets might lead to a bad performance if directly applied to smaller clinics' datasets. Another scenario for
"Correppnndence qinglong tian @ @ewaterloo.ca
Equal Contribution

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continuous target shift is the anti-causal setting (Schölkopf et al. 2012) where y is the cause of x. For example, it is reasonable to assume that height (2) is the cause of the bodyweight (x), and we can assume that p(zly) is domain invariant.
This work aims fill this research gap and focus on the target shift problem in the regression tasks. We consider the unsupervised domain adaptation setting (Kouw & Loog 202I), in which the training source dataset provides the feature-target (i.e., (x, 2)) data pairs while the testing dataset only has feature data (i.e., x) available. The goal is to improve the predictive model's performance on the testing dataset by leveraging the knowledge gained from the training source data.
Related Work The inportance- adaptation one of the most popular approaches in addressing the target shift problem. However, most of the existing importance-weighted adaptation methods are designed specifically for dealing with the shift in the classification tasks (Saerens
et al. 2002 Lipton et al., 2018 Azizzadenesheli et al., 2019; Alexandari et al. 2020 Garg et al. 2020 In the classification tasks, suppose the target variable has k classes (c.g.. h = 2 for binary classification). Then, the importance weight estimation with & parameters. However, when the target variable y is continuous (e.g., in a regression problem), these approaches are not feasible, it remains unclear how to represent the continuous target space with finite number of parameters. Existing work on the continuous target shift is scarce as opposed its categorical counterpart. Except for some heuristic approaches, we found two works (Zhang et al. 2013; N Nguyen et al., 2016) that addressed target shift using the idea of distribution matching (i.e., minimizing the distance between two distributions). However, neither of the two works provided sufficient theoretical justifications for their proposed methods: neither consistency nor error rate bounds have been established. Furthermore, both KMM (Zhang et al. (2033) and 1,21WE (Nguyen al. 2016) need to optimize quadratically constrained quadratic program.
be simplified to optimization problem
Our Contributions In this paper, we propose the Regularized Continuous Target Shift Adapatation (ReTaSA) method, novel importance- weighted adaptation method for continuous target shift problem. The key component of the ReTaaSA method is the estimation of the continuous importance weight function, which is crucial for shift adaptation. Using the estimated importance weight function, one can adjust the predictive model for the target domain via weighted empirical risk minimization (ERM). Our key results and their significance are summarized as follows:
We formulate the problem of estimating the continuous importance weight function as finding a solution to an integral equation. However, such an integral equation becomes ill-posed when the target variable is continuous. We introduce a nonparametric regularized method to overcome ill- poseddessssss and obtain stable solutions.
2. We substantiate our method by theoretically examining the proposed importance weight function estimator. The theoretical findings, outlined in Theorem [ demonstrate the statisti- cal consistency of the estimated weight function. In addition to establishing consistency, we also quantify the convergence rate of our novel importance-wedt weight function estimator, thereby enhancing the distinctive contributions of our research. To the best of our knowledge. this is the first work to offer theoretical guarantee under the continuous case.
3. We develop an easy-to-use and optimization-free importance-weight estimation formula that only involves solving linear system. Through comprehensive numerical experiments, we demonstrate that the ReTaSA method surpasses competitive approaches in estimation accuracy and computational efficiency.
2 THE TARGET SHIFT INTEGRAL EQUATION AND REGULARIZATION
We use subscripts and to represent the training source- and testing data domains, respectively. The target shift assumption says that P. (x Pe (xly) while Ps (y) + Pt (v), and the goal train a predictive model Eg (y/ x) on the target domain. In terms of data samples, we have labeled data {(x;, }:) ~ (x y) from the source domain and unlabeled data {x; nn+m {x ~ Px(x) from the testing data domain. Note that data from the testing data domain have no observation on the target variable y; thus, we cannot train R., (y) x) directly and must leverage the feature-targe target data pair (x, y) from the training source domain.

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We define the importance weight function as w(v) = pr(u)/po the target shift assumption implies that
(x(x) J (x, y)dy J Ps( y) g)w -f p=(e)x))o(ydd.
(I)
(x)
(x(
(x)
Equation ① has three terms: (x))/pp (x), Pa (x(xx), and w(y). Suppose (x(x)/p (x) and (x(y)x) are known, then (I) can be viewed
integral equation to solve the unknown function wyy). The importance weight function ((y) is the key to target shift adaptation: letting ((:,)) -) be the loss function, the risk of any predictive model f(.) on the testing data domain can be converted to weighted risk on the training source domain: E, [e {x). y = E, [w) {f(x),, y/l. where E, and E, represent the expectation with respect to the training source and testing data distribution, respectively. Such equivalence implies that we can learn predictive model for the target domain via weighted ERM using data from the source domain with the importance weight function w(y). Most existing work deals with the classification tasks with finite k classes, in which y comes from a finite discrete space (1,... k). Thus, the right-hand-sin of (T) can be represented in summation form Σ (y x (y) and {w(y), y = 1, 1.. k} can be estimated by solving the k-dimensional linear equations (Lipton et al. 2018). However, when y is continuous, the importance weight function cannot be reduced to finite number of parameters as y take an infinite number of values. Such difference pinpoints the difficulties that arise when is continuous.
Operator Notations We first introduce a few operator notations for our problem formulation. Considering the training source distribution Pa (x, y), we use (2 y) to denote the Hilbert space for mean-zero square-intes inteerable functions of (x. y). The inner product for any h2, h2 L (x, y) is defined by (h1, h2) = ha (x, ()h2(x, v)p=(x, (x, dxxyy = E, (ha (x, y))2(x, y)), and the norm is induced as ||h]| (h, h) 1/2, Similarly,
use (x) or (y) to respectively denote the subspace that contains functions of x or only. In this case, the inner product h1, h2 E (x or (v) is given by (h1,h)) define two conditional expectation operators
ha (xxx)) (x) (x)px(x)ds or (h1, h2) = hu (y)ha2y)) (y))))((dd))))) dy. Additionally,
T (3) + (2))x,, s.t. Tpp E, (x(v)))),
T (x) (2) s.l. T
E, (x(x))))-
T and T are adjoint operators because
(Te, = / ((x)E (f(U)(x) pa (x)xxxx== = /ʃ (())))())) dddydx
= f (()),E (x())l+) (v) = ()
for any y E (y) and vo e (xx.. Using the operator notations, we rewrite as
Po 1):
(4)
where vx() (x())/px(x) - and po(y) = wwy) - is the unknown function to be solved. We define 7xx) in such way satisfy the mean zero condition E, {17(²) so that n7(x) (xx)..
Identifiability With the above reformulation, natural question is whether Po is identifiable, i.e., the solution of the equation (4) is unique. To further ensure uniqueness of the solution to (4), let the null set of T be (T) = { T'p = 0, ip E (²())) }. Then po is unique solution to (4) if and only if (TT) {0}. Because T' is a conditional expectation operator, we can equivalently express the identifiability condition with the following completeness condition. We refer the readers Newee Powell] (2003) for a detailed discussion on the completeness condition.
Condition (Identifiability Condition). The importance weight function w(y) is identifiable in 7 if and only if {x(())x} almost surely implies that :ply) almost surely for any up E T.2 T.²
lol-posedness & Regularization Even though the integral equation in (4) has unique solution under Condition 1, we are still not readily able to solve for Pa. This is because belongs to the class of Fredholm integral equations of the first kind (Kress) 1999). which is known to be ill-posed. More specifically, it can be shown that even a small perturbation in 1) can result in a large change in the solution Po in This phenomenon is referred to an "explosive" behavior (Darolles et al. 2011
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for which we provide details in Section A the supplementary materials. Such sensitive dependence upon input (i.e., ?)) is problematic because, in practice, the true function 1j is unknown, and an estimated version 1j would be plugged in. No matter how accurate the estimation is, the estimated î) will inevitably differ from 77, thus leading to large error on Po-
To address the ill-posedness of Equation 4), we adopt the Tikhonov regularization technique, which has been broadly applied in various tasks, for example, the ridge regression (Hoerl & Kennard. 1970)
and the instrumental regression (Carrasco et al. 2007 Fêve & Florens 2010 Darolles et al. 2011
The Tikhonov regularization yields a regularized estimator Pa: by minimizing a penalized criterion
Pa argmin|[ 7/1² + a|lpl]? (E)
where a a regularization parameter. This minimization leads to the first-order condition for unique solution to (al + T p= T 17, where I is the identity operator. Furthermore, by the definition of and T in 2
we can rewrite the first-order condition as
aply) + E. [E, { lul = E, {n(x)lu).
(5)
3 ESTIMATION OF THE IMPORTANCE WEIGHT FUNCTION
This section focuses on how to estimate the solution function Pa using (5). Estimating Pa requires two steps: the first step is to replace (5) with a sample-based version; the second step is to solve for the estimate Pe. from the sample-basee based equation. We employ the kernel method to find the sample-based version of (5). To this end, without loss of generality, we assume x and y take values in [0, 1I2 and [0, 1] respectively. In particular, we use a generalized kernel function, whose definition is given below.
Definition 1. Let h be a bandwidth, bivariate function K₁ [0. 1] [0, Rzo satisfying:
1) (x, 2) = 0, i > or < 5 - 1, for all ã E [0, 1];
{ for q 0,
2) h (0t1) Jy 2 (z,
0 for 1 S q - 1,
is referred as a univariate generalized kernel function of order l.
Based on the above definition, the density estimates are given by
vx) = (nhp+1)
x,, M1s P.(1) = (nh)- Σ
K Ky, ys. v),
1=1
(x) (nhP)-
Kx,n(x x;, x), (x) (nhP) -1 Σ 1=++1
Kx,(x-x,,1 x),
where p is the dimension of x, h is the bandwidth parameter, Ky,h is univariate generalized kernel function, and Kx,,. is a product of univariate generalized kernel functions defined for each component of X. In addition, we estimate 77(x) with 7x() = pr (x) p. (x) - 1. Instead of simply calculating the ratio of two density estimates, there are other methods for computing the density ratio (e.g., Sugiyamal et al. 2012). Due to the space limit, we focus on the simple approach f)(x = (x)) (x) and leave more details in Section C.2.2 of the supplementary materials.
Let p be an arbitrary function in L (y)) Using the kernel density estimates, the Nadaraya- estimates of E, {p(x)(x}, E, (x(x (.), and E, [E, (p(v)) x yl are, respectively, given by
E. {(())(}} = Ei= p(y.) K (x x)
E. [x()) = Σi= (xx,) Ky V.
Σラ (x,1(x x)
E Kv,n 1).w)
and
n E. E. (p())(x) [e] = Σ
{ k=1
p(yx) Kx (x X4, xxe
Kr.ly -xx
Kr - ye.2)
2 Kx,n(x; - X,, X1)
n2
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Using the estimates above, we can obtain the sample-based version of (5). and the next step is to solve for the function p. By letting y = y1..... yn. we have the following linear system:
a (((()))...PPuu)) + Cx y yt (p(1n) ... p...)] = Cxly [7xx)) ... ((xx))..((x..))
(6)
where Cxly and Cy are both n x n matrices and their (i. j entries are given by
Ky, (vh Uz. vi)
Kx x,,X,)
(Cxiy) Di=1 KY, ye, 9:) and (Cuix) = ΣE=1 Kx,(Xx - Xx,
We can obtain the estimate of Pa on {yr...., No by solving the linear system 6 as
(
11² (al Cxly Cylx) Cxly (7(x)...... xx,))
Note that the estimates on {y 3n in (7) are all we need because the importance weights are (1x) = (y.) + 1, I,.., n; thus, for family regression models, denoted by the trained regression model for the target domain can be obtained from the weighted ERM
E =
argmin JEF 7
((x) (S(x;) y=)
4 THEORETICAL ANALYSIS
In this section, we provide a theoretical understanding of the proposed kernel-based importance weight function-basee based adaptation. In particular, we establish the consistency of P... solution of the sample-based approximation, to the solution Po of the equation 4 under suitable assumptions. We define two estimated conditional expectation operators
s.t. T4 -E, (p(W))x) = f ((u) f: :2))) + (x)x s.t.
(8)
E. (b(x)lu} = / (xl)dx
: + 13()), s.t. 1 s.t.
(xly)dx
(9)
Furthermore, by plugging 7(x) = Tor (x) (x()) - 1, we have the kernel-based approximation of equation (5)
(aT+ p= or aply) + E. E. (p(U)))) [b] = E. (7())
(10)
Now, we provide a theoretical justification for our proposed kernel-based regularization framework. Specifically, we first prove the consistency of the solution obtained from (00,, denoted by Pa. =
[al TTTTT T* Toward this end, we state several necessary assumptions.
Assumption The joint source density ps (x. u) and marginal source densities px(x) and Ps(y)
satisfy S
(x(xx)) px(xx) dxxy < 00.
px(x)p((3)
Remark The implication of Assumption is that two operators T and T defined in 2 and ( are Hilbert-Schmidt operators and therefore compact, which admits the singular value decomposition.
To establish consistency, we restrict the function space in which the solution Po resides. Consequently, we introduce the 8-regularity space of the operator T for some 3 > 0.
Definition 2. For 3 > O, the 3-regularity space of the compact operator T is defined as & = {p E (y) such that Eie (p, /223 < co) where (Ai are the decreasing sequence of nonzero singular values T and {4p: fiz is set of orthonormal sequence in T?² In other words, 100 C L2(x), To, = Aspi holds for all E N. for some orthonormal sequence (
Assumption 2. There exists 3 > 0 such that the true solution Po $2
Remark 2. In general, for any B1 2 82, & C 23 (Carrasco et al. 2007). Therefore, the regularity level 3 governs the complexity of the space. A detailed discussion of the B-regularity space given in Section A.I of the supplementary materials.
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Assumption 3. We assume X E [0.1]² E [0,1]; and the k-times continuously differen- tiable densities ps x, y), Ps(x) and pe px(x) are all bounded away from zero on their supports, i.e, Px(x, V). Ps () E [e, 00), for some >> 0.
Remark 3. The equation ① problematic when ps goes to zero. Therefore we need the assumption that Px(x) is bounded from below by some constant >> 0. Also, we need the support of (x() to be a subset of the support of ps px(x). When Pa(x) goes to zere, while Pe pe(x) is not zero, corresponds to the out -ffiiiiiiibttion (OOD) setting (Zhou al. 2023
As we adopt the class of multivariate generalized kernel functions formed by product of univariate generalized kernel functions, we impose the following common assumptions which can be found in (Darolles et al. 2011 Hall & Horowitz 2005)
Assumption 4. With a bandwidth parameter h, generalized kernel function Kr. : [0, 1] x [0.1] R20 satisfies the following conditions:
1. Kn (n v) = if x & (y 1,2] nc, where a compact interval independent of y.
2. sup [Kr(I, < 00. =EC.NE(O.1)
Remark 4. Note that the domain restriction is primarily for the purpose of theoretical analysis. In fact, any compact domain would suffice. The continuous differentiability and boundedness assumptions on the density are to obtain uniform convergence of KDE estimators, which serves as an important tool to establish consistency. Such uniform convergence results based on the ordinary kernel functions need additional restrictions on the domain, while the generalized kernel functions can obtain uniform convergence over the entire compact Darolles et al. (2071).
To guarantee convergence in probability, make assumptions the decaying rate of the bandwidth parameter h and regularization parameter a with respect to the sample size n.
Assumption 5. The kernel bandwidth h. regularization parameter a and the source sample size satisfy lim k-00 log (n)/(nh++1) 0. Furthermore, the target sample size satisfies --050
lim h-1 log (m) (m)/(mh²) =0.
m m4000
Remark 5. The assumption implies that the growth rate of n. is faster than the decaying rate of h so that nh=-1 grows faster than the log(n) rate. Similar to n, we require the growth rate of m to be faster than the decaying rate of h², As the assumption indicates, the price one must pay to establish a convergence rate in the high-dimensional setting non-negligible.
We formally state the consistency result, of which the proof is provided in Section B of the supple- mentary materials.
Theorem 1. For B 2 2, under Assumptions one can show that ||p Pall2 is of order
Op 0
a² +
1 nhetI 112))
22 +
(8-2)
nhu+t 1:22 +122++1) +1)
1
112
log m mhv
h7 +
log mhv
(11)
a²
m
In particular, if max
1127
V
1og(m)) mhh
1.,,
log(mm) mhh
= =p( and lim a-su na² = xc, Pa converges
in probability to po, i.e., l|pa poll
as a;, h + 0, n + xc, and m + xc.
Remark 6. In the proof, we conduct the theoretical analysis by decomposing the error between Po and Pa into three parts. Each term, inside the O,(-) expression in (T]. indicates the errors manifested by each of the three parts. The first term inside the Op(.) represents the error between the unregularized solution Po and the regularized solution Pa- The second term appears due to bernel-busele approximation of conditional expectation operators. The last factor reflects the cost of estimating conditional expectation operators and target/source densities. Naturally, the target sample size only gets involved in the last term.

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To establish consistency, must grow faster than a² to make no? divergent. In the meantime, the decay rate of regularization parameter a² should be slower than all three terms h?², // / and m)) (m)))) Therefore, the assumption indicates that the regular- ization parameter should decay slower as the dimension of the feature becomes larger, i.e., a larger regularization effect needed for a higher dimension. Furthermore, the assumption indicates that one use smaller regularization parameter for smoother densities with a larger order of the generalized kernel function. functtoon
Remark 7.
Remark 8. Theorem 7 provides bound for the estimation error's weighted L norm (i.e., li 2 (Pa(() po(u)}² Ps(y)dy). Compared with function's 's ordinary T,2 norm (i.e. I [(a(()) po(y))² dy). the weighted L² norm directly bounds the target domain generalization error. More details are given in Section B.1 of the supplementary materials.
EXPERIMENTS
In this section, we present the results of numerical experiments that illustrate the performance of proposed method in addressing the continuous target shift problem. We will first study with the synthetic data and then apply the methods to two real-world regression problems. Our experiments conducted Mac-Book Pro equipped with 2.9 GHz Dual-Core Intel Core 15 processor and 8GB of memory. We relegated the experimental details (e.g.. hyperparameter tuning) and extensive experiments with high- high-dimensional datasets to Section C in the supplementary material.
5.1 NONLINEAR REGRESSION WITH SYNTHETIC DATA
We first conduct synthetic data analysis with linear regression problem. Inspired by the setting in Zhang al. ()11)), we generate the dataset with v+3 3 tanh (z) te, where ~ Normal (0, 1). The target ys for source data from Normal(O, 2), while the testing data has the target y, from Normal 0.5). We use 14 to adjust the target shift between the training source and testing data. We set the size of the target test- ing data as 0.8 of this study's ' training source data size, i... = 0.8 x n unless otherwise specified. Figure shows an example with f = 0.5 and n = 200. As for adopt the polynomial regres-
Source Data
Target Data
20
-10
0
10
the target prediction, sion model with degrees as 5. In this simulation study, we consider three existing adaptation strategies as base- line methods: 1). Non-Adaptation, in which the source data trained model is directly applied for the prediction. of which the importance weight function is the ground truth. 3). KMM- AMM-Adaptation pro-
X
Figure An illustration of the nonlinear regression synthetic data with 200 and Pt = 0.5. The blue dots are the source data, and the red plus marks the target data.
2). Gracle-Adaptatinn
posed by Zhang et al. (2013). 4). L2IWE-Adaptatin proposed by Nguyen et al. (016)) It is worth noting that Tracle-Adaptatie is infeasible practice as the shift mechanism is unknown.
Evaluation Metries Two metries are used evaluate the adaptation performance in our experi- ments: 1). Weight MSE: the mean square error of the estimated adaptation weights v.s. the weights from Oracle-Adappe Aacele-ddptttioon, i.e., Σ" [w:) (a)--wll wl) /n. Non-Adaptation calculates the weight MSE by assuming no shift as ()y) = 1. 2). Delta Accuracy (A Accuracy): the percentage of the improved prediction MSE compared with that of Aon-Addptation, and the larger value represents the better improvement. We conducted all experiments with 50 replications.
Experimental Results Our first study focuses on the adaptation methods' performances under different data sizes. We fix i 0.5 and change the data size n from 100 1000. The results are shown in Figure (a) Our proposed method performs significantly better than L21WE- Adaptation and KMM-Adaptation method terms of both weight estimation and prediction accuracy. Compared with Non Non-Adaptation, KMM and L2IWE methods reduce the weight MSE by about 25%
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Nome KM L2/AE
Nove KBS LEWEE
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(a) Weight MSE vs Sample Size
(b) Weight MSE vs Target Shift
(c) Weight MSE vs Size Ratio
Oracie
KMM
L2IWE
Data Data/cccurry
Dota Accurayy
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0.7
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(d) AAccrrcc Sample Size
(e) AAccuracy Target Shift
(1) AAccuracy Size Ratio
Figure 2: Performances comparison over difference adaptation methods. The first row uses weight MSE as a metric, and the second one shows the performance measuring by A Accuracy. (d) are the performance over different sample sizes. The sample sizes range from {100, 200, 500, 1000). (b)&(c) are the performances over different target shifts μ4. The values of fe range from 0.1 to (f) are the performances over different target-source data size ratio m/ n and the values range from 0.2 to 0.8. The solid curves are the mean values and the shadow regions are 95% CI error bands.
and 60%, respectively, while our method reduces it by more than 75%. For prediction accuracy, increasing data size from 100 to 1000, our method improves accuracy from about 15% to 22% while K.MM-Adaptation improves less than 10%. The L2IWE method is better than the KMM method but still not as good as the proposed method.
Furthermore, we study the performances under different target shifts with the results shown in Figure (b)) Consistently, our proposed adaptation method still outperforms KMM and L2IWE. It can be seen from [ that as fs increases, the shift becomes severe, and the weight MSE gets larger for all the methods. However, our proposed method keeps the weight MSE smaller than 0.8 for all the jr values, while KMM and L2IWE have the weight MSE more than 1.5. Also, for the prediction performance, our proposed method is close to the Oracle- Adaptation when be increases with improved accuracy by more than 20%. But KMM and L2IWE generally get worse performance as 14 increases.
We also evaluate the impact of target data size m on the performance. We fix the source data size = 500, and the target shift μe 0.5. We tune the size ratio m/n in (0.2, 0.4, 0.6, 0.8). The experimental results are shown in Figure (c) It can be seen from Figure 2 (c) that the target data size has a relatively small impact the adaptation performances. The weight MSE and delta accuracy curves are flat over different values m/n. But when comparing the performance of methods with KMM and L2IWE, prediction accuracy and smaller weight MSE. Thus, we conclude that our proposed method performs better than the two existing methods under different sample sizes and target shift settings in this nonlinear regression study.
that method can consistently achieve higher
5.2 WORLDD DATA EXPERIMENTS
In this section, we will conduct the experiments with two real-world datasets: the SOCR datase and the Szeged weather record datase The SOCR dataset contains 1035 records of heights, weights, and position information for some current and recent Major League Baseball (MLB) players. this dataset, there is a strong correlation between the player's weights and heights. Also, it is natural to consider the causal relation: height + weight, which justifies the continuous target shift assumption.
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Published as a conference paper at ICLR 2024
Table 1: The experimental results for SOCR and Szeged weather datasets. The numbers reported
before and after + symbolize the mean and standard deviation, respectively.
SOCR Dataset
Szeged Weather Dataset
Adaption
Weight MSE
APred. MSE (%)
Weight MSE
APred. MSE (%)
Oracle
12. 12.726+1.022
35.572-26.233
KMM
0.477 10.269
3.0761..177
2.34311.488
0.5 0.516
L2IWE
0.178+0.00 025
-5.933+1.12
1.047+0.600
-8.400427.66
Ours
0.025+0.009
6.705+0.977
0.756+0.878 878
19.567=15.079
Thus, we consider players' weights the covariate I and their heights y. In our study, conducted 20 random trials. In each trial, we treat all outfielder players as the testing data and randomly select 80% of the players with the other positions as the training source data.
The second real-world task under investigation is the Szeged temperature prediction problem. This dataset comprises weather-related data recorded in Szeged, Hungary, from 2006 to 2016. In this study, we take the temperature as the target, and humidity as covariate. This is because there is a causal relationship between temperature and humidity: it tends to decrease relative humidity when temperature rises. We treat each year's 's data as an individual trial. In each trial, we utilize data from January to October as the training source dataset, while data from November and December constitute the testing dataset. In this data-splitting scheme, the testing data tends to have lower temperature values than the training source data.
The experimental results are summarized in Table As we do not know the true distribution of target variables in real-world datasets, we calculate the oracle importance weight an empirical way: we use the real target values across training and testing datasets to estimate the corresponding density functions, then leverage the ratio of the estimated density functions for the oracle importance weight. We use weight MSE and Aprediction MSE (%) the performance metrics. It can be seen that our proposed adaptation method performs better than KMM and L,2WE with smaller weight MSE and larger Aprediction MSF: Our method improves the weight estimation by reducing about 95% and 77% weight MSE for SOCR and Szeged weather datasets, respectively. Also, for Aprediction MSE, our method improves about 4% for the SOCR dataset and 19% for the Szeged weather dataset. Also, note that our method's ' estimation variation (i.e., the standard deviation) is smaller than KMM and IL2IWE. Thus, we claim that our method performs well in both estimation accuracy and stability.
DISCUSSIONS
We address the continuous target shift problem by nonparametrically estimating the importance weight function, by solving an poosdd integral equation. The proposed method does not require distribution matching and only involves solving a linear system and has outperformed existing methods in experiments. Furthermore, we offer theoretical justification for the proposed methodology. Our approach has effectively extended to high-dimensional data by integrating the black-box-shine estimation method introduced in Lipton et al. (2018). The essence of this method is to pre-trained predictive model to map the features x to scalar (y]y)), x);; thus avoiding the curse of dimensionality. Due to space constraints, we present additional experiments on high-dimensional data in the supplementary materials.
This study is primarily centered on training predictive models in the target domain. An associated question naturally arises regarding the quantification of uncertainty in predictions. While existing research has addressed this concern in the context of classification (e.g., Podkopaev & Ramdas 2021 a noticeable gap exists when considering the continuous case.To our best knowledge, this aspect remains unexplored in the current literature and could be prospective avenue for future research. Finally, it is noteworthy that we consider the case where the support of y on the target domain a subset of that in the source domain. Consequently, another potential future research question will be to address continuous target shifts under the -iitori-bttiiuiiinn setting. Lastly, we adopt a simple density ratio estimation method (for 77(x) in this paper and establish theoretical results upon it. We compare this simple density ratio method with the RuLSIF method in Section C.2.3 of the supplementary, but more extensive comparisons with other existing methods could be of interest.

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Published as a conference paper at ICLR 2024
ACKNOWLEDGEMENT
Qinglong Tian is supported by the Natural Sciences and Engineering Research Council Canada (NSERC) Discovery Grant RGPIN-2023-03479.
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Reviews:
[{"review_id": "Oe3M4mZiyZ", "reviewer": "Reviewer_D6fP", "summary": "This work addresses the challenge of distribution shifts in deploying modern machine learning models, specifically focusing on the target shift problem within a regression context. It tackles the situation where the continuous target variable y exhibits different marginal distributions between the training and testing domains, while the conditional distribution of features x given y remains constant. Notably, the regression problem's infinite-dimensional target space necessitates unique solutions. The authors propose ReTaSA, a nonparametric regularized approach to estimate the importance weight function and provide theoretical justification for it, thereby effectively addressing the continuous target shift problem. Extensive numerical studies on both synthetic and real-world datasets confirm the effectiveness of the proposed method.", "questions": "See above \"weaknesses\" section.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- The work focuses on addressing the challenge of classification tasks with an infinite-dimensional target space, while previous research on target shift primarily concentrated on scenarios where $y$ is categorical. This problem holds fundamental significance across various domains.\n\n- By precisely estimating the importance weight function, the model can effectively adapt to variations in the target variable's distribution between the training and testing domains. A continuous importance weight function offers greater flexibility compared to discrete methods.\n\n- The authors provide thorough theoretical justifications for consistency and error rate bounds, enhancing the paper's rigor and reliability.\n\n- The paper's organization is well-structured, maintaining a clear and easily-followed flow from start to finish.\n\n- Extensive references to related work offer a comprehensive overview of prior research, providing valuable context for the study and highlighting the authors' deep understanding of the field.", "weaknesses": "- Given the importance weight function, is the estimation process sensitive to noise or outliers in the data and thus potentially leading to inaccurate estimates of the importance weight function? Or this limitation is conquered by applying the regularization technique in your paper? \n\n - Estimating a continuous importance weight function is a probabilistic process, together with the regularization process, this may inherent uncertainty in the estimates, affecting the model's reliability in certain cases. How do you solve this issue? \n\n - Regularization can help stabilize the estimation process, but accurate estimation of a continuous importance weight function may still require a substantial amount of data from both source and target domains. What is the sample complexity of your algorithm?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "HXLUcu0pOX", "reviewer": "Reviewer_2aLy", "summary": "This paper studies the problem of label shift within a continuous label space, such as in regression with label shift. To tackle this issue, the authors propose a method based on importance weighting, which transforms the learning task into an estimation problem concerning the density ratio of the continuous variable $y$. The method has statistical consistency in estimating the weight function under certain identifiability assumptions. A practical algorithm is derived from the theoretical analysis.", "questions": "See the comments in weakness part. As also, it is suggested to test the performance of the proposed algorithm on large-scale and high-dimensional datasets.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This paper considers an interesting and important real-world problem with a compelling motivation.\n2. This work provides the statistical consistency analysis for the proposed estimator in the continuous label space. This technical analysis has the potential insights to benefit studies in this area.", "weaknesses": "1. In the experiments, only low-dimensional and small-scale datasets are used for comparison. Thus, the empirical results are not sufficient to support the success of the proposed algorithm. Considering the increasing need of learning with high-dimensional data such as images and the poor performance of kernel density approximation in high-dimensional data, the comparison in high-dimensional data is important. \n\n2. Determining the hyperparameter $\\alpha$ for different data sets is difficult. \n\n3. In classical ratio estimation approaches, the estimation variance is sometimes large, e.g. $p_s(y)$ goes to zero, making the algorithm unstable. In the continuous label space, it is unclear whether this problem becomes worse. It is suggested to discuss this issue or to introduce an additional variance reduction mechanism.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "cQU40eEzri", "reviewer": "Reviewer_4YDN", "summary": "This manuscript studies the continuous target shift problem. To adapt to the target shift, the authors reweighed the training samples with the density ratio of the responses: $\\omega(y)=p_t(y)/p_s(y)$. To estimate the weight $\\omega(y)$, a two-step procedure is proposed. First, the authors employed a kernel density estimator to estimate the conditional density and then used the outcomes to solve a regularized least-squares problem to obtain the weights. Additionally, the authors discussed conditions for identifiability of the weight $\\omega(y)$ and the consistency of the estimation results.", "questions": "how to select $\\alpha$ in terms of the sample size?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "Overall, the manuscript is easy to follow and is technically sound. The empirical results also show improvement compared to the baselines.", "weaknesses": "I found the major issue is insufficient comparison with other prior works.\nFor example, Nguyen et al., (2016) studied the same problem with a slightly different estimation procedure. It would be nice to discuss and compare the method. \n\nWhile the estimation procedure is straightforward, it would be nice to clarify what is the technical challenge and the novelty of this method. \n\n\n\n\nNguyen, T. D., Christoffel, M., & Sugiyama, M. (2016, February). Continuous target shift adaptation in supervised learning. In Asian Conference on Machine Learning (pp. 285-300). PMLR.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "3jPImbfmYK", "reviewer": "Reviewer_SE3s", "summary": "This paper studies unsupervised domain adaptation in the context of regression ($\\mathbb{R}$-valued labels), under the *label shift* assumption (i.e., that the label distribution $P_Y$ changes while the conditional distribution $P_{X|Y}$ of the features given the labels is invariant. It is first argued that this problem can be solved by training a model under a modified training loss, which is reweighted by the ratio of the label distributions in the test and training domains (the *importance weight function* $\\omega$). The remainder of the paper thus focuses on estimating $\\omega$. It is shown that $\\omega$ satisfies a particular intergral equation, whose other components can be estimated directly from observed data using kernel density estimation. Since this integral equation is typically undercomplete, Tikhonov regularization is added to identify a unique solution. Assuming the relevant data densities are sufficiently regular and bounded, the kernel and bandwidth are carefully selected, etc., the paper provides bounds on the rate at which the estimate of $\\omega$ converges to the true $\\omega$. The paper then presents experiments on both synthetic and real-world datasets, demonstrating that the proposed approach outperforms a prior kernel-mean-matching approach, both in terms of estimating $\\omega$ and out-of-distribution adaptation performance.", "questions": "**Major**\n\n1. I found Definition 2 quite confusing, for a few reasons:\n    1. I don't see where Definition 2 is used anywhere in the paper.\n    2. In contrast to Section 3, where the (presummably translation invariant?) kernels are written as a univariate function, the kernel here is written as a bivariate function. Is there a reason for this?\n    3. I don't understand the condition $k_h(x, y) = 0$ if $x \\notin [y - 1, y] \\cap \\mathcal{C}$. For example, usually, bivaraite kernels are symmetric in their arguments, but this appears not to be the case here.\n    4. The use of $x$ and $y$ is a bit confusing here, as it suggests the kernel is applied to the covariate $x$ and the label $y$ discussed earlier in the paper (but I don't think this is the intent, since, e.g., the condition $x \\notin [y - 1, y]$ really would not make any sense in this case). Perhaps different variables (e.g., $z_1$ and $z_2$) should be used here?\n2. Page 3, Eqs. (2)-(3): I don't understand why $T$ and $T^*$ map $L^2$ to $L^2$. Is this an additional implicit assumption? Or does it follow from the forms (conditional expectations) of $T$ and $T^*$? Relatedly, just after Eq. (4), \"$\\mathbb{E}_s\\\\{\\eta(x)\\\\} = 0$ so that $\\eta(x) \\in L^2(x)$\": I didn't understand this implication; why is $\\mathbb{E}_s\\\\{\\eta^2(x)\\\\} < \\infty$?\n3. Assumption 2: Although I'm familiar with math behind this assumption, I found it hard to understand its ramifications in this context. I understand that the $\\beta$-regularity space is the set of $\\rho$ such that $T^* T \\rho$ is approximately invertible, but is there a more concrete intuition, or maybe some examples, of what $\\beta$-regularity looks like when $T$ is the conditional expectation operator? For example, if if $\\\\{\\phi_i\\\\}_{i = 1}^\\infty$ is the Fourier basis, then $\\beta$ is related to the smoothness/differentiability of $\\rho$.\n4. Assumption 5: Similar to the previous point, can you provide some intuition or examples for when Assumption 5 is satisfied?\n\n\n**Minor**\n\n1. Assumption 4: \"Letting $\\gamma = \\min\\\\{k, \\ell\\\\}$...\": $\\gamma$ is never used again in this assumption. Is this an error? Perhaps $\\gamma$ should be defined in Assumption 5 instead?\n2. Remark 6: Does \"consistent\" here mean \"in operator norm\"? If so, please make this more explicit.\n3. I didn't understand the definition of \"Delta Accuracy\" in the experiments. Could you please provide a mathematical formula? Would $100\\%$ correspond to perfect prediction?\n4. Page 8, Last Sentence, \"In each trial, we... randomly select 80% of the players with the other positions as the training source data.\": If I understand correctly, the idea here is to use the bootstrap to obtain confidence intervals on performance. If so, this resampling should be done *with replacement*. Or is there a different reasoning here?\n5. Page 9, First Paragraph, \"For the temperature value shift between the training source and testing dataset, we use the humidity for the importance weight estimation.\": I didn't understand this sentence. What does it mean to \"use the humidity for the importance weight estimation\" (as opposed to using all of the available features)?\n6. Page 9, Second Paragraph, \"our method improves about 4% for the SOCR dataset\": I don't see this 4% in Table 1. What exactly is this refering to?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The motivation and justification for the proposed approach is quite convincing; almost every step seems natural, and so it seems to me like the \"right\" solution to this problem, under the given assumptions. The high-level writing and flow of the paper are also quite clear. The method is supported both theoretically (with some caveats; see below) and empirically.", "weaknesses": "**Major**\n\n1. The paper should discuss the theoretical computational complexity and practical scalability of ReTaSA, in terms of the source and target sample sizes $n$ and $m$, data dimension $p$, etc. Relatedly, under \"Related Work\", the paper claims \"empirical evidence from... our experimental studies confirms that KMM is computationally inefficient in categorical and continuous cases,\" but I couldn't find evidence of this in the paper.\n\n2. I found several parts of the paper a bit vague or missing some details that would help the reader:\n    1. Page 2, last sentence, \"the target shift assumption implies that... (1)\": I think it would be helpful to include a few more details on the steps by which the target shift assumption implies Eq. (1). I was eventually able to figure this out (using Bayes' rule), but it interrupted my reading of the paper and took a few minutes. This could easily be avoided by adding another intermediate equality in Eq. (1), without increasing the length of the paper.\n    2. Page 3, just after Eq. (3), \"$T$ and $T^∗$ are adjoint operators because $\\langle T\\phi, \\psi \\rangle = \\langle \\phi, T^* \\psi \\rangle$\": It's not immediately obvious why this is the case. Since this observation is important for the remainder of the paper, please include a more detailed explanation or proof to the main text, or indicate where this could be found (e.g., in an appendix).\n    3. Page 3, just after Eq. (4), \"where... $\\rho(y)$ is a unknown function to be solved\": I found this quite confusing because it sounded like $\\rho = \\omega$. Only later is it explained that $\\rho = \\omega - 1$. So perhaps this latter fact can be explained a few sentences earlier.\n\n3. First Paragraph of Section 3: $\\eta = p_t/p_s$ is estimated as the ratio of two density estimates. There is a significant body of work on density ratio estimation showing that estimating the ratio by the ratio of two density estimates is often suboptimal, both in theory and in practice (see, e.g., [K17, SSK10]). Given this, perhaps the paper should consider using direct density estimation methods for this step, both to improve practical performance and to relax the assumptions (specifically, Assumption 3).\n\n4. There are some gaps between the theoretical results Section 4 and the real-world OOD generalization problem the paper seeks to solve:\n    1. Theorem 1 bounds the $L^2$ error of the estimated $\\rho$. However, $\\rho$ is only a means to re-weighting the risk function to adapt to the test domain (as explained on Page 3), and it's not clear to me whether estimating $\\rho$ well in $L^2$ distance is necessary or sufficient to adapt to the test domain. I think the paper should provide some more concrete connection between estimation of $\\rho$ in $L^2$ distance and test-domain performance of the new risk minimizer.\n    2. It's unclear (to me) how some of the main assumptions in Section 4 relate to the real-world problem being solved; see Major Questions 3. and 4. below.\n\n**Minor**\n\n1. Page 1, Paragraph 2: It's unclear to me why the Sequential Organ Failure Assessment (SOFA) example described here satisfies the label shift assumption (in particular, why $P_{X|Y}$ is invariant between domains). I think a more convincing example here would strengthen the motivation of the paper. Perhaps it is also worth pointing up that label-shift assumptions appear naturally under anti-causal structural assumptions (see, e.g., Section 5 of [S22]).\n2. The paper would benefit from some discussion of ReTaSA's limitations or further open questions. Some examples:\n    1. The paper focuses on the non-parametric setting. While this makes weak assumptions on the relationship between $x$ and $y$, Theorem 1 suggests that its performance scales poorly with the dimension $p$ of the feature $x$. Perhaps it is worth briefly commenting on whether a parametric variant of ReTaSA (e.g., assuming a linear relationship between $x$ and $y$) could be useful, e.g., for high-dimensional data?\n    2. Do the authors believe the rate in Theorem 1 is minimax optimal under Assumptions 1-5?\n    3. How robust is ReTaSA to small violations of the label-shift assumption (e.g., small changes in $P_{X|Y}$)?\n3. Beginning of Page 4: Tikhonov regularization is added to address non-identifiability (i.e., $T^* T$ might not be invertible). Given this, it might be worth adding a sentence to point out that the regularized criterion has a unique solution (i.e., $\\alpha I + T^* T$ is always invertible). This isn't completely obvious, especially in the infinite-dimensional setting.\n4. Remark 1: If I am understanding correctly, perhaps it is worth noting that this estimate/approximation is simply the standard Nadaraya-Watson regression estimate of $\\mathbb{E}[\\rho(y)|x]$.\n5. Remark 8, \"Therefore, the assumption fits into the regime where the dimension of the feature is smaller than the smoothness level of densities and the order ofthe generalized kernel function.\": I didn't understand this sentence. It sounds like it is saying that dim$(x) = p \\leq \\min\\\\{k, \\ell\\\\} = \\gamma$, but I don't see how this follows from the previous sentence (which is about $\\alpha$).\n6. Page 7, under \"Evaluation Metrics\", \"We conducted all experiments with 50 replications on a Mac-Book Pro equipped with a 2.9 GHz Dual-Core Intel Core I5 processor and 8GB of memory.\": This seems like the wrong place to include this information. Perhaps it should be in the first paragraph of Section 5?\n7. Page 7, under \"Experimental Results\", Typo: \"performs significantly better KMM-Adaptation\" should be \"performs significantly better *than* KMM-Adaptation\"\n8. Figure 2: The lines plotted here are essentially all flat, so the plot does not illustrate much. Perhaps it would be useful to show a larger range of (smaller) sample sizes?\n9. Figure 3: I think the sub-captions are incorrect (they both say \"vs Sample Size\" but the $x$-axis here is $\\mu_t$).\n10. All Figures: Please increase the font size of the text in the plots (axis labels, legends, etc.).\n\n**References**\n\n[K17] Kpotufe, S. (2017, April). Lipschitz density-ratios, structured data, and data-driven tuning. In Artificial Intelligence and Statistics (pp. 1320-1328). PMLR.\n\n[S22] Schölkopf, B. (2022). Causality for machine learning. In Probabilistic and Causal Inference: The Works of Judea Pearl (pp. 765-804).\n\n[SSK10] Sugiyama, M., Suzuki, T., & Kanamori, T. (2010). Density ratio estimation: A comprehensive review (statistical experiment and its related topics). 数理解析研究所講究録, 1703, 10-31.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)