Learning from Asymmetrically-corrupted Data in Regression for Sensor Magnitude
Authors: Takayuki Katsuki
Takayuki Osogami
Venue: Submitted to ICLR 2023 (2023)
Keywords: regression
sensor data analytics
healthcare

Abstract: This paper addresses a regression problem in which output label values represent the results of sensing the magnitude of a phenomenon. A low value of such labels can either mean that the actual magnitude of the phenomenon has been low or that the sensor has made an incomplete observation. This leads to a bias toward lower values in labels and its resultant learning because labels for incomplete observations are recorded as lower than those for typical observations, even if both have monitored similar phenomena. Moreover, because an incomplete observation does not provide any tags indicating incompleteness, we cannot eliminate or impute them. To address this issue, we propose a learning algorithm that explicitly models the incomplete observations to be corrupted with an asymmetric noise that always has a negative value. We show that our algorithm is unbiased with a regression learned from the uncorrupted data that does not involve incomplete observations. We demonstrate the advantages of our algorithm through numerical experiments.

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LEARNING FROM
ASYMMETRICALLY-CORRUPPDD
DATA IN REGRESSION FOR SENSOR MAGNITUDE
Anonymous authors Paper under double-blind review
ABSTRACT
This paper addresses a regression problem in which output label values represent the results of sensing the magnitude of a phenomenon. A low value of such labels can either mean that the actual magnitude of the phenomenon has been low or that the sensor has made an incomplete observation. This leads a bias toward lower values in labels and its resultant learning because labels for incomplete observations are recorded lower than those for typical observations, even if both have monitored similar phenomena. Moreover, because an incomplete observation does not provide any tags indicating incompleteness, we cannot eliminate or impute them. To address this issue, we propose learning algorithm that explicitly models the incomplete observations to corrupted with an asymmetric noise that always has a negative value. We show that our algorithm is unbiased with regression learned from the uncorrupted data that does not involve incomplete observations. We demonstrate the advantages of our algorithm through numerical experiments.
INTRODUCTION
This paper addresses a regression problem for predicting the magnitude of phenomenon when an observed magnitude involves particular measurement error. The magnitude typically represents how large a phenomenon or how strong the nature of the phenomenon is. Such examples of predicting the magnitude are found in several application areas, including pressure, vibration, and temperature (Vandal et al. 2017 (Sh et al. 2017 Wilby et al.
2004: Tanaka et al. 2019). In medicine and healthcare, the magnitude may represent pulsation, respiration, or body movements (Inan et al.)
2009 Nukaya et al.
2010 Lee et al. 2016 Alaziz et al.
2016
2017 Carlson et al. 2018).
More specifically, we learn regression function predict the label representing the magnitude of a phenomenon from explanatory variables. The training data consists of pairs of the label and explanatory variables, but note that the label in the data is observed with sensor and is not necessarily in agreement with the actual magnitude of the phenomenon. We note that we use the term "label" even though we address the regression problem, and refers real-valued label in this paper.
In the example of predicting the magnitude of body movements, the label in the data is measured with an intrusive sensor attached the chest or the wrist, and the explanatory variables are the values measured with non-intrusive bed sensors (Mullaney et al.) 1980, Webster et al. 1982 Colc al. 1992. Tryon 2013). regression function for this example would make it possible to replace intrusive sensors with non- intrusive ones, which in turn will reduce the burden on patients.
Although the sensors that measure the label generally have high accuracy, they often make incomplete observations, and such incomplete observations are recorded as low values instead of missing values. This leads to the particular challenge where low value of the label can either mean that the actual magnitude of the phenomenon has been low that the sensor has made incomplete observation, and there are no clues that allow us to tell which is the case. We illustrate this challenge in Fig. (a).
Such incomplete observations are prevalent in measuring the magnitude of phenomenon. For example, the phenomenon may be outside the coverage of a sensor, or the sensing system may experience temporal mechanical failures. In the example of body movements, the sensor may be temporarily detached from the chest or wrist. In all cases, the sensor keeps recording low values, while the actual magnitude may be high, and tag indicating incompleteness can be provided.

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Olserved sensor ourput
--. Actual magnitude of phenomenon
W Sensor value
Upper-side
Lower-side data
ooer-ssie data with
Incomplete observation
weideniifiable incumplete obvervations
Actual low vulues
valuex from incouplete points
-.:
Biased regression line
Explanatory variable
Unbiased regression linc
Points without incomplete observations
Points with incomplete observations
(a) Sensing results for magnitude of phenomenon
(b) Asymmetrically-comupped data
Figure 1: (a) Low sensor value can either mean actual low magnitude or incomplete observation. (b) Labels for incomplete observations, depicted triangles, become lower than those of typical observations, depicted as circles or squares.
This incomplete observation is particularly severe for the sensor measuring the label since it is single-souree and has narrower data coverage. This stems from the fact that the sensor is usually intrusive it is costly to produce highly accurate observations for measuring the label. Examples of this can be seen in chest or wrist sensors that focus on the movements of a local body part with high accuracy and often miss movements outside their coverage, such as those of parts located far from where the sensor is attached. At most, a single intrusive sensor can be attached to a patient to avoid burdening them. In contrast, the sensors measuring the explanatory variables are usually multi-source and provide broader data coverage. For example, multiple sensors can be attached to various places of bed and globally monitor the movements of all body parts on the bed but with lower accuracy.
One cannot simply ignore the problem that the observations of labels may be incomplete because the estimated regression functions trained on such data with incomplete observations are severely biased toward lower values regardless of the amount of available training data. This bias comes from the fact that incomplete observations always have lower values than the actual magnitude of phenomenon, and they occur intensively on label sensors, while explanatory variables are usually observed completely. Moreover, incomplete observations can be much more frequent than expected.
Unfortunately, since cannot identify which observations incomplete, cannot climinate or impute them by using existing methods that require identifying incomplete observations. Such methods include thresholding, missing value detection (Pearson 2006 Qahtan et al.) 2018), imputa- tion (Enders] 2010; Smieja et al. 2018 Ma & Chen
2019 Sportisse et al. 2020). and semi-supervised
regression (Zhou Li 2005 Zhu & Goldberg 2009 Jean et al. 2018 Zhou al. 2019).
The issues of incomplete observations also cannot be solved with robust regression (Huber et al.)
1964 Narula & Wellington 1982 Draper & Smith 1998, Wilcox 1997), which takes into account the
possibility that the observed labels contain outliers. While robust regression is an established approach and state-of-the-are against corrupted labels in regression, assumes symmetric label corruption. Namely, the noise is assumed to not be biased either positively or negatively. Since incomplete observations induce the noise that severely biased toward lower values, robust regression methods still produce regression functions that are biased toward lower values than the one that would be learned from the data without incomplete observations.
In this paper, to mitigate the bias toward lower values, we explicitly assume the existence of the noise from incomplete observations, which always has negative values, addition the ordinary symmetric noise. That is, consider training data to be caammetrically-coruppee data. We then formulate a regression problem from our asymmetrically-comnptt data and design a principled learning algorithm for this regression problem.
By explicitly modeling the incomplete observation, we derive a learning algorithm that has a rather drastic feature: namely, it ignores the labels that have relatively low values (lower-side labeled data). In other words, our algorithm uses the data whose labels have relatively high values (upper-side labeled data) and the data whose labels are ignored (unlabeled data). Hence, we refer to algorithm
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upper and unlabeled regression (U2 regression). This aligns with the intuition that the labels with low values are unreliable, since those low values may be due to incomplete observations.
Our main result is that U2 regression, which learns from the a regression function that is, under some technical assumptions, unbiased and consistent with the one that is produced from the uncorrupted data that does not involve incomplete observations. This counterintuitive result is achieved by considering a specific class of loss functions and deriving their gradient, which only requires upper-side labeled data and unlabeled data in the asymmetrically- corrupted data and can still be shown to be asymptotically equivalent to the expression of the gradient that has access to the uncorrupted data, The main novelty in our approach is thus in the loss function, and we will empirically demonstrate the effectiveness of the proposed class of loss functions over existing common loss functions dealing with caanmttticallycoorpppp data synthetic and six real-world regression tasks.
wwt.mnttrccllllooorppptoe data, produces
Contributions.
The main contributions of this paper are summarized as follows.
We formulate a novel problem of learning a regression function from asymmetrically- corrupted data. This is important for applications where the magnitude of phenomenon measured with sensor that is susceptible to unidentifiable incomplete observations.
We derive unbiased and consistent learning algorithm (U2 regression) for this problem from the new class of loss functions.
Extensive experiments synthetic and six real-world regression tasks including a real use case for healthcare demonstrate the effectiveness of the proposed method.
REGRESSION FROM ASYMMETRICALLY-CORRUPDD DATA
Our goal is derive learning algorithm with training data are corrupted with negative-valued noise due to incomplete observations, in manner that unbiased and consistent with the regression that uses uncorrupted data without involving incomplete observations. We first consider the regression problem that uses the uncorrupted data in Section 2.1 and then formulate learning from the asymmetrically-commee
data, i.e., labels in the
data Section 2.2
2.1 REGRESSION PROBLEM FROM DATA WITHOUT INCOMPLETE OBSERVATIONS
Let PD E N) be a D-dimensional explanatory variable and E be a real-valued label. We assume that, without incomplete observations, is observed in accordance with
f (a) + 6x
(1)
where /' the oracle regressor and ex the symmetric noise with as the center, such as additive white Gaussian noise (AWGN).
We learn a regression function f(x: that computes the value of the estimation of a label, ú, for newly observed æ as ÿ = f(x). The optimal regression function, f. given by
arg min L(f), JEF
where F hypothesis space for f. and L(f) is the expected loss when the regression function f() is applied to data (ar, v), distributed in accordance with an underlying distribution p(ær, y):
C(f) Ep(n,)) (L((x)), v)],
(3)
where [ denotes the expectation over the distribution p. and ((f(x). y) is the loss function between f(x) and y. e.g., the squared loss, L(f(x),) 3) = llf(x) 211². The expectation Epiza) can be estimated by computing a sample average for the training data D = {x yn yn) which N pairs of explanatory variables and labels.
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2.2 REGRESSION PROBLEM FROM ASYMMETRICALLY
DATA
In this paper, we consider a scenario in which we only have access to the asymmetrically-coruuppie data D' = ((r,: (n))n=: where a label y' may be corrupted due to incomplete observations. corrupted label y is observed from the uncorrupted y with an asymmetric negative-valuee noise, eai
= y + far
where the asymmetric noise always has a random negative value, which means y' S.
Using only D'. we learn a regression function (x) as the solution for Equation 2 in an unbiased and consistent manner. Although AWGN can be handled even when we use naive regression method such least squares, the asymmetric noise ca. which always has negative value, is problematic.
Intuitively, the asymmetric noise makes lower-side labeled data particularly unreliable and inappropriate for learning, while keeping upper-side labeled duta reliable, where the upper-side labeled data refers to the data {(x, y)} whose label is above the regression line (i.e., [(x) S and data whose label below the regression line. The regression line represents the estimation of regression function. Figure (b) illustrates this
the lower-side labeled data refers
scatter plot of the value the label against the value of am explanatory variable. Here, the data with incomplete observations appear only in the lower side of the regression line because ca makes observations have lower label values than those of typical observations, where the regression line represents such typical observations. This asymmetry leads biased learning compared with the learning from the uncorrupted data without incomplete observations.
To address the asymmetric noise 6n and its resultant bias, we formalize the assumption the observation process for the asymmetrically-coruppp data D and derive lemma representing the nature of D Then, we propose a learning algorithm based on the lemma in the next section.
The observation processes of 'D and D' are formally characterized as follows.
Ep(e,]]s]] = 0; c. f. Ca S 0 almost surely (a.s.); 2/eg Eal a.s. when ea O; and {(I V, ( are i.i.d. observations in accordance with Equa- tion Aand Equation 4
Assumption 2.1. Assume 64 L (xz),
This assumption means that D' has enough information to estimate f. and the asymmetric noise ca is significant enough compared the symmetric noise cs, which are necessary assumptions that the learning problem is solvable, and ca should be handled separately from Es.
From Assumption 2.1 we then have the following lemma.
Lemma 2.2. Let F If e (f())= f=)) S c. a.s.). When f E F,, the following holds for (m) cx and y = y + 6a under Assumption 2.7
E (f (G. v')l (m(elrrm) GG(,
(5)
for any function G: RRD x x R R as long as the expectations exist.
Proof. We outline proof here and provide complete one in Appendix A.I We first show that ca does not change the distribution for upper-side labeled data (fx < y/) on the basis of the oracle regression function f* before and after adding ca. i.e., ca
when f" < y/. With the condition f E F', we can further prove that = 0 when f(x) < 3/ which is for upper-side labeled data on the basis of f. This establishes p(x, fll v') = p(z, y f S 3) and implies Lemma 2.2
The condition parts of these conditional distributions represent the relationships between labels and the estimations of the regression function f. e.g., p(x, flf(x) S y) the distribution of x and when y is higher than what is given by f. The condition f E F' represents natural expectation that the regression function f well approximates f²
Lemma 2.2 shows that ca does not change the expectation for upper-side labeled data (f(x) s y²) before and after adding e,, which makes them still reliable for regression. In the next section, we derive unbiased learning algorithm based this lemma.
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3 U2 REGRESSION
We seek to find the minimizer of the objective in Equation from the asymmerrically-coruppp data D'. To this end, propose a gradient that relies only the knowledge of the distribution of the corrupted data p(æ, y²) but is still equivalent to the gradient of Equation 3 which relies on the knowledge of the distribution of the uncorrupted data p a, y). Based Lemma 2.2 gradient based on p(z, y) into the one that only requires p( p( y²).
we rewrite the
3.1 GRADIENT FOR LEARNING FROM SSYMMETRICALLY-CORRUTI
DATA
Here, we address Equation with the gradient descent. At step + 1 in the gradient descent, the gradient of Equation 3 with respect to the parameters 0 of f represented with regression function, ft, which estimated at step i, as follows:
where VI((((x))))) @T(J(x))))) v)
V((f)) (p(().)) (L((((()) v)),
(6)
00
f=f+
Note that this holds for any step in the gradient descent. When fu the initial value of f. and when oo, we suppose Soo = f. We can decompose VL((f) as
vL(ft) =x( S (Ltp:a/wwwnalssss///LLLL//V v)]
+ pyy < ft ( VL(
We then assume that, when y < f()), the gradient of the loss function does not depend on y and only depends (ar); thus write (z)) y) g( (x))) when y < f(:) emphasize this independence. Formally,
Condition 3.1. Let g()())) (x)((r); y) for y < [(x)). (((())) a gradient function depend- ing only f(x) and not the value y.
Such common losses are the absolute loss and pinball loss, which are respectively used in least absolute regression and quantile regression and work well real data (Lee et al.) 2016 Yeung et al. 2002
Wang et al. 22005 Srinivas et al.) 2020). For example, the gradient of the absolute loss is
flf()) yl 0f(x)
when y < f((),
00
00
which does not depend the value of but only on f(x).
We now propose a gradient that does not rely on the knowledge of p(x:, y) but instead uses only p(x, y²). Namely,
vê(f) = p(f((x)
(LL((()))))
(9)
+ Ep(()) [(f((=))) p(f(()) S (() [g (f((()))
In Section 3.2 we will formally establish the equivalence between the gradient in Equation 9 and
that in Equation 6 under our assumptions. Note that in the second and third terms of Equation
apply expectations over p(x) and (()) < y²) to g g(f(x)), even though (x)) is defined to be
the gradient (L(((x). y) for y < f(x) This is tractable due to the nature of (z())) which only depends on fxx) and does not depend on the value of
Since the expectations in Equation g only depend on æ and y', they can be estimated by computing data D' as
sample average for our asymmetricallyyccruyy
Vā(f))
Tup
Σ
VL(f((z),v)
+ N
Σ BE X
g(f((z))
Tup
Σ
g(fc(z))
Tap
Thp
(x. Win
2r x
(10)
where { (Xup= }dp represents the set of coupled pairs of and y in the upper-side labeled sample set, {x, y/: fx I s/}. in D': X,m is a sample set of æ in D' ignoring labels y/: nap is the number of samples in the upper-side labeled set; Tup Tup (((()) < v)-
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Note that Tup depends the current estimation of the function ft and the label y with complete observation. Thus, it changes at each step of the gradient descent, and we cannot determine its value in general way. In this paper, we propose simple approach of choosing Tup the hyperparameter. We optimize it with the grid search based on the validation set, which enables us to flexibly handle data variation. We will show that it works practically in our experiments.
single value of
As we will show in Section 3.2 we can use Equation to design an algorithm that gives an unbiased and consistent regression function. By using the gradient in Equation [0] we can optimize Equation and learn the regression function only with upper-side labeled samples and unlabeled samples from D' independent of lower-side labels. This addresses the issue that our lower-side labeled data is particularly unreliable and leads to overcoming the bias that stems from this unreliable part of the data. We refer to our algorithm as upper and unlabeled regression (U2 regression).
See Appendix B for the specific implementation of the algorithm based on stochastic optimization. The gradient in Equation (10 can be interpreted in an intuitive manner. The first term in Equation TO has the effect of minimizing the spper-side loss. Recall that the upper-side data are not affected by the asymmetric noise under our assumptions. Thus, U2 regression seeks to learn the regression function f on the basis of this reliable upper-side data. Notice that the first term becomes zero when all of the data points are below ff (i.e.. v/ S ft()), V(x, y') E D'), since then {Xup. Fup becomes empty. The second term thus has the effect of pushing down f at all of the data points that some data points are above f. Meanwhile, the third term partially cancels this effect of the second term for the upper-side data to control the balance between the first and the second terms.
3.2 UNBIASEDNESS AND CONSISTENCY OF GRADIENT
U2 regression is the learning algorithm based on the gradient, V(((f)) in Equation 10 and uses only symmmetricallly-corppte data D'. The use of v V(((f) can be justified follows:
Proposition 3.2. Suppose that Assumption 2.7 holds and the loss function L(f(z). y) satisfies
Condition 3.1 Then, the gradient Vā(fe) in Equation and its empirical approximation (ff))
in Equation 10 are unbiased and consistent with the gradient ((f()t) in Equation 6 a.s.
Proof. We outline proof here and provide complete one in Appendix A.2 First, we rewrite Equa- tion iinto a gradient that only contains the expectation over p(x, y fe (z) S y) with Condition 3.1
Then, apply Lemma 2.2 to the gradient, and it becomes expression identical to Equation g
In other words, U2 regression asymptotically produces the same result the learning algorithm based on the gradient V (fi)) in Equation 6 which requires the uncorrupted data without incomplete observations. D. The convergence rate of U2 regression is of the order Op(% accordance with the central limit theorem (Chung) 1968), where Op denotes the order in probability.
1/VN)) in
We further justify our approach of having the specific form of Equation9bbs showing that straightfor- ward variant that uses D' as if it does not involve incomplete observations (i.e., p(a., y) as p(n, y can fail for our problem. To this end, we introduce an additional assumption on the observation process:
Assumption 3.3. Assume ca Il xC
Then, we have
Lemma 3.4. Let vū(f)) be a variant of the gradient in Equation] 7 replacing p(x, y) with p(æ, v'), 8 be the difference between the expectations of the gradients in the upper side and the lower side Bptea/lv<((ee) [V (x:) ()), 0 O<<<> be probability when 0 -<e ess and 0 < < be probability when = O. Then, VL(e)) () is not consistent with the gradient in Equation 0 a.s., and the difference (bias) between them at step + 1 in the gradient descent
= Ep(a.el
y)] (V [VL(f(x)
17(1 (1 5) (1 - 1)) + (
(11)
Proof. We outline proof here and provide a complete one in Appendix A.3 We first show that the bias ((())) ((()) can be represented by the difference between the expectation of (()(

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5 5
Proposed
MSE
Data index
Data index
Data index (c) Breathing, K=50%
(a) LowNoise, K=50%
(b) HighNoisc, K=50%
Figure 2: Errors (predicted value minus true value) by proposed method (blue) and by MSE (green) for three tasks; (a) LowNoise, (b) HighNoise, and (c) Breathing, with =500 of incomplete training samples. Error of each data-point is shown by dot (for proposed method) or by cross mark (for MSF), and average error shown by solid line (for proposed method) by dashed line (for MSE).
with the upper-side data and that with the lower-side data, which can be written by 8. The bias also has the coefficient which contains the proportions for the lower-side data and the original upper-side data mixed into the lower side due incomplete observations. These values can be written by 17 and & from their definitions.
Lemma 3.4 shows that the bias caused by incomplete observations becomes severe when there is large difference between the expectations of the gradients in the upper side and the lower side. is usually higher than zero because
implies there is difference between the expectations the gradients in the upper side and the lower side or both of the expectations are zero. Furthermore, larger < = p(a, < 0) makes the bias more significant, which agrees with the intuition that as the proportion of incomplete observations increases, the problem becomes more difficult.
4
EXPERIMENTS
We now evaluate the proposed method through numerical experiments. We first introduce the baselines be compared in the experiments. Then, present the experimental results to show the effectiveness of our unbiased learning.
Baselines. Recall that the novelty of the proposed approach lies in the unbiased gradient in Equa- tion I0 which is derived from the new class of loss functions in equation with Condition 3.] An objective of our experiments is thus to validate the effectiveness this new class of loss functions and the corresponding gradients against common loss functions in the literature. Specifically, we compare proposed method with MSE (mean squared error), MAE (mean absolute error), and Huber losses (Huber et al. 1964 Narula & Wellington| 1982. Wilcox 1997). For robust loss function in many studies and librarics. We use the same model and optimization method with all of the loss functions under considcration, and hence the only difference among the proposed method and the baselines is in the gradients. Since the loss function uniquely determines the baseline, refer to each baseline method as MSE, MAE, or Huber.
regression, MAE and Huber losses are considered the facto standard and state-of-the-an
4.1 EXPERIMENTAL PROCEDURE AND RESULTS
The experiments organized into three parts. In Section 4.1.1
visually demonstrate the effectiveness of the proposed approach in giving unbiased prediction. In Section 4.1.2] we intensively and quantitatively evaluate the predictive error of the proposed method and baselines with five real-world regression tasks. In Section 4.1.3] we demonstrate the practical benefit of our approach in real healthcare use case, which has motivated this work. See the appendix for the details of the experimental settings.
4.1.1 DEMONSTRATION OF UNBIASED LEARNING
Procedure. We start by conducting the experiments with synthetic data to show the effectiveness of our method in obtaining unbiased learning results from asymmetrically-comppte data with
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Table I: Comparison between proposed method and baselines in terms of MAE (smaller is better). Best methods are in bold. Confidence intervals are standard errors.
Specification
Throwing A
Lifting
Lowering
Throwing B
Avg.
MSE 2.38
1.54
1.42400001
1.37+0.01
1.21
1.58
MAE
2.14+0002
1.46=0.01
1.44+0.01
1.33
1.31
1.54
Euber
2.04
1.66=0.01
1.45
1.50
1.32
1.59
Proposed-1 1.55+0.02
1.18=0001
1.11#0.01
1.14#0.01
1.03
1.20
Proposed-2
1.32
0.99
0.9410.01
0.86
0.97
1.02
Table 2: Proportion of correct prediction period and rate of false prediction in real use case for healthcare. We estimate intrusive sensor output from outputs of non-intrusive sensors.
Proportion of correct prediction period Rate false prediction
0.89
0.016
different proportions of incomplete observations, K {25, 50, 75... We use three synthetic tasks, LowNoise, HighNoise, and Breathing collected from the Kaggle dataset (Sen) 2016). We compare the proposed method against MSE, which assumes that both upper- and lower-side data correctly labeled. This comparison shows whether our method can learn from esymmetrically-coruug data an unbiased manner, which MSE cannot do.
Results. In Fig. 2 plot the error in prediction (i.e., the predicted value minus the true value) given by the proposed method and MSE for each data-point of the three tasks with K = 50%. Note that, for evaluating the unbiasedness, these test sets do have incomplete observations. Since MSE regards both upper- and lower-side data as correctly labeled, it produces biased results due to the incomplete observations, where the average error (shown by the green dashed line) is negative, which means the estimation has negative bias. In contrast, the average error by the proposed method (shown by the blue solid line) is approximately zero. This clearly shows that the proposed method obtained unbiased learning results. The figures for the other settings and tables showing quantitative performance are Appendix F
4.1.2 PERFORMANCE COMPARISON AMONG DIFFERENT Loss FUNCTIONS
Procedure. We next apply the proposed method and baselines to five different real-world healthcare tasks from the UCI Machine Learning Repository (Velloso 2013 Velloso et al., 2013) show a more extensive comparison between the proposed method and the baselines (MSE, MAE, and Huber). For the proposed method, we use two implementations of L(f(x), y) for f(x) $ y in Equation I0 the absolute loss (P roposed-1) and the squared loss (Proposed-2) report the mean absolute error (MAE), and its standard error, of the predictions 2). Here, we {ύ, | against the corresponding true labels y across 5-fold cross-validation, each with a different randomly sampled sraining-testing split. MAE the common metric used in the healthcare domain (Lee et al.) 2016 Yeung et al. 2002 Wang et al. 2005 Srinivas al. 2020) and is defined as MAE(y, 3)
I/N >m=1 ly. Inl. For each fold of the vross-validation, we use randomly sampled 20% of the training set as a validation set to choose the best hyperparameters for each algorithm, in which hyperparameters providing the highest MAE in the validation set are chosen.
Results. As seen in Table Proposed- and Proposed-2 largely outperformes the baselines. The robust regression methods (MAE and Huber) did not improve in performance against MSE. E. In particular, Pr Proposed-1 and Proposed-2 respectively reduced the MAE by more than 20% and 30% on average, compared with baselines.
4.1.3 REAL USE CASE FOR HEALTHCARE
Procedure Finally, we demonstrate the practicality of our approach in real use case in healthcare. From non- ion-intrusive bed sensors installed under each the four legs a bed, we estimate the motion intensity of subject that could be measured accurately but intrusively with ActiGraph, gold
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standard sensor wrapped around the wrist (Tryon 2013 Mullaney al., 1980; Webster et al. 1982
Cole et al. 1992). If we can mimic the outputs from ActiGraph with outputs from the bed sensors, we can measure the motion with high accuracy and high coverage, while also easing the burden on the subject. We divide the dataset into three pieces and evaluate the results with 3-fold cross-validation. We here the evaluation metrics that are specifically designed for sleep-wake discrimination (Cole) ct al.
1992) i.e., proportion of correct prediction period and rate of false prediction.
Results. Table shows the proportion of correct prediction period and rate of false prediction, which indicate that the proposed method captured 89 percent of the total time period of the motions that were captured by ActiGraph, and false detection due to factors such as floor vibration was only 1.6 percent. Furthermore, the proposed method captured 15 additional motions that were not captured by ActiGraph. The baseline method MSE was severely underfitted, and most of the weights were zero; thus, we omitted these results. Overall, our findings here demonstrate that ActiGraph can be replaced with bed sensors, and we can also use the bed sensors for the inputs of certain ActiGraph functions, such aleep-wake discrimination (Cole al. 1992). See also Appendix G for further details, including the actual estimation results of the motion intensity.
DISCUSSION
Limitations. In this paper, we do not address symmetric label corruption, such as ordinary outliers, where the coverage and incompleteness are consistent between a label and explanatory variables. Other established approaches can handle such cases. Only when the corruption asymmetric does lead to the technical challenge we address here. In that sense, we can handle the opposite asymmetric corruption, in which labels for some observations may become inconsistently higher than those for typical observations. This can be handled as learning from lower-side labeled data and unlabeled data, i.e., LU regression. Since our derivation of U2 regression is straightforwardly applicable to this LU regression case, we show only its learning algorithm in Appendix
Asymmetric Label Corruption in Classification. In the classification problem setting, asymmet- ric label corruption is addressed with positive-unlabeled (PU) learning, where it is assumed that negative data cannot be obtained, but unlabeled data are available as well as positive data (Denis)
1998 De Comité al.
Letouzey al. 2000 Shi et al. 2018 Kato et al. 2019: Sakai & Zhang el al.) 2019) 9 2020) Chen el al. 2020b a Luo et al.) al. 2021 et al. 2021). An unbiased risk estimator has also been proposed (Du Plessis et al. 2014 2015). However, PU classification cannot be used for regression problem, where labels are real values and we need to handle order and gradation between labels. This is because its derivation and algorithm are based on the nature that labels must be binary, i.e., only positive or negative. We overcome this limitation with a novel approach based on an unbiased gradient.
1999
Shimizu 2019 Li et al.
2019
Future work. We showed that our approach to estimating hyperparameters based on the grid search with the validation set was effective even for the one contains the important ratio for upper-side labeled data, (x) $ y). It also provides the flexibility needed to handle data variation. Most studies on PU learning assume that a hyperparameter corresponding to Tup is given (Hammoudeh & Lowd 2020 Sonntag al., 2021 Lin et al. 2022), and some papers have addressed this hyperparameter estimation as their main contribution (Jain et al. 2016) Ramaswamy et al.] 2016 Christoffel et al.) 2016 Jain al. 2020 Yao al. (021). ). Developing a method for the hyperparameter estimation to improve performance would be a worthwhile next step of our study. Also, in Assumption 2. we assumed 6g JL (x) and 6, 1 fx),, which is a common noise assumption. Addressing the case when the noises are not independent of f (xx is another future direction of our work.
Conclusion. We formulated a regression problem from asymmetically-cooupte data in which training data are corrupted with an asymmetric noise that always has negative value. This causes labels for data with relatively lower label values to be particularly unreliable. To address this problem, we proposed learning algorithm, U2 regression. Under some technical assumptions, we showed that our algorithm is unbiased and consistent with regression that uses uncorrupted data without incomplete observations. Our analysis is based the equivalence of the gradient between them. An experimental evaluation demonstrated that the proposed method was significantly better than the methods without the assumption of the asymmetrical label corruption.

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REFERENCES
Zubin Abraham and Pang-Ning Tan. An integrated framework for simultaneous classification and regression of time-series data. In SDM. pp. 653-664, 2010.
Zubin Abraham, Pang-Ning Tan, Perdinan, Julie A Winkler, Shiyuan Zhong, and Malgorzata Liszewska. Distribution regularized regression framework for climate modeling. In SDM, pp. 333-341, 2013.
Musaab Alaziz, Zhenhua Jia, Jian Liu, Richard Howard, Yingying Chen, and Yanyong Zhang. Motion scale: A body motion monitoring system using bed-mounted wireless load cells. In CHASE, pp. 183-192, 2016.
Musaab Alaziz, Zhenhua Jia, Richard Howard, Xiaodong Lin, and Yanyong Zhang. Motiontree: tree-based in-bed body motion classification system using load-cells, In CHASE, pp. 127-136, 2017.
Charles Carlson, Ahmad Suliman, Alaleh Alivar, Punit Prakash, David Thompson. Balasubramaniam Natarajan, and Steve Warren. A pilot study of an unobtrusive bed-based sleep quality monitor for severely disabled autistic children. EMBC, pp. 4343 4346, 2018.
Hui Chen, Fangqing Liu. Yin Wang. Liyue Zhao, and Hao Wu. A variational approach for learning from positive and unlabeled data. In NeurlPS, 2020a.
Xuxi Chen, Wuyang Chen, Tianlong Chen, Ye Yuan, Chen Gong, Kewei Chen, and Zhangyang Wang. Self-PU: Self boosted and calibrated sessitveeuullbblle training. In ICML, 2020b.
Haibin Cheng and Pang-Ning Tan. Semi-supervised learning with data calibration for long-term time series forecasting. In KDD, pp. 133-141, 141, 2008.
Marthinus Christoffel, Gang Niu, and Masashi Sugiyama. Class-prior estimation for learning from positive and unlabeled data. In ACML, pp. 221-236 PMLR, 2016.
Kai Lai Chung. course in probability theory. Academic press, 1968.
Roger Cole, Daniel Kripke, William Gruen, Daniel J Mullaney, and J Christian Gillin, Automatic sleep/wake identification from wrist activity. Sleep, 15(5):461446, 1992.
Francesco De Comité, François Denis, Rémi Gilleron, and Fabien Letouzey. Positive and unlabeled examples help learning. In ALT, pp. 219-230, 1999.
François Denis, Pac learning from positive statistical queries. In ALT, pp. 112-126, 1998.
Norman R Draper and Harry Smith. Applied regression analysis, volume 326. John Wiley & Sons, 1998.
Marthinus Du Plessis, Gang Niu, and Masashi Sugiyama. Convex formulation for learning from positive and unlabeled data. In ICML, pp. 1386-1394, 2015.
Marthinus C Du Plessis, Gang Niu, and Masashi Sugiyama. Analysis of learning from positive and unlabeled data. In NIPS, pp. 703-711, 2014.
Craig Enders. Applied missing data analysis. Guilford press, 2010.
Zayd Hammoudeh and Daniel Lowd. Learning from positive and unlabeled data with arbitrary positive shift. In NeurlPS, 2020.
Wenpeng Hu, Ran Le, Bing Liu, Feng Ji, Jinwen Ma, Dongyan Zhao, and Rui Yan. Predictive adversarial learning from positive and unlabeled data. In AAAI, volume 35, pp. 7806-7814, 2021.
Peter J Huber et al. Robust estimation a location parameter. The Annals of Mathematical Statistics, 35(1):73-101, 1964.
OT Inan, M Etemadi, A Paloma, L Giovangrandi, and GTA Kovacs. Non-invasive cardiac output trending during exercise recovery on a bathroom- scale-based ballistocardiograph. Physiological measurement, 30(3):261, 2009.
10

---

Under review as a conference paper at ICLR 2023
Shantanu Jain, Martha White, and Predrag Radivojac. Estimating the class prior and posterior from noisy positives and unlabeled data. In NeurlPS, pp. 2693-2701, 2016.
Shantanu Jain, Justin Delano, Himanshu Sharma, and Predrag Radivojac. Class prior estimation with biased positives and unlabeled examples. In AAAI, volume 34, pp. 4255-4263, 2020.
Neal Jean, Sang Michael Xie, and Stefano Ermon. Semi-supervised deep kernel learning: Regression with unlabeled data by minimizing predictive variance. In NeurIPS, 2018.
Masahiro Kato, Takeshi Teshima, and Junya Honda. Learning from positive and unlabeled data with selection bias. ICI.R, 2019.
Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In ICLR, 2015.
Won Lee, Heenam Yoon, Chungmin Han, Kwang Joo, and Kwang Park. Physiological signal monitoring bed for infants based on load-cell sensors. Sensors, 16(3):409, 2016.
Fabien Letouzey, François Denis, and Rémi Gilleron. Learning from positive and unlabeled examples. In AL,, pp. 71-85, 2000.
Changchun Li, Ximing Li, Lei Feng, and Jihong Ouyang. Who is your right mixup partner in positive and unlabeled learning. In ICLR, 2021.
Tianyu Li, Chien-Chih Wang, Yukun Ma, Patricia Ortal, Qifang Zhao, Bjor Stenger, and Yu Hirate. Learning classifiers positive and unlabeled data with policy gradient. In ICDM, pp. 399-408, 2019.
Xinyang Lin, Hanting Chen, Yixing Xu, Chao Xu, Xiaolin Gui, Yiping Deng, and Yunhe Wang. Federated learning with positive and unlabeled data. In ICML, pp. 13344-13355, 2022.
Chuan Luo, Pu Zhao, Chen Chen, Bo Qiao, Chao Du, Hongyu Zhang, Wei Wu, Shaowei Cai. Bing He, Saravanakumar Rajmohan, et al. Pulns: Positive-unlabeled learning with effective negative sample selector. In AAAI, volume 35, pp. 8784-8792, 2021.
Wei Ma and George H Chen. Missing estimating missingness probabilities under a low nuclear norm assumption. In NeurlPS, 2019.
random in matrix completion: The effectiveness of
DJ Mullaney, DF Kripke, and S Messin. Wrist-actigraphic estimation of sleep time. Sleep, 3(1): 83-92, 1980.
Vinod Nair and Geoffrey E Hinton. Rectified linear units improve restricted boltzmann machines. In ICML, pp. 807- 814, 2010.
Subhash C Narula and John F Wellington. The minimum sum of absolute errors regression: A state of the art survey. International Statistical Review/Revue Internationale de Statistique, pp. 317-326, 1982.
Shoko Nukaya, Toshihiro Shino, Yosuke Kurihara, Kajiro Watanabe, and Hiroshi Tanaka. Noninvasive bed sensing of human biosignals via piezoceramic devices sandwiched between the floor and bed. IEEE Sensors journal, 12(3):431-44 -438, 2010.
Ronald K Pearson. The problem of disguised missing data. Acm Sigkdd Explorations Newsletter, (1):83-92, 2006.
Abdulhakim A Qahtan, Ahmed Elmagarmid, Raul Castro Fernandez, Mourad Ouzzani, and Nan Tang. Fahes: robust disguised missing values detector. In KDD, pp. 2100-2109, 2018.
Harish Ramaswamy, Clayton Scott, and Ambuj Tewari. Mixture proportion estimation via kernel embeddings of distributions. In ICML, pp. 2052-2060. PMLR, 2016.
Tomoya Sakai and Nobuyuki Shimizu. Covariate shift adaptation learning from positive and unlabeled data. AAAI, pp. 4838-4845, 2019.
Sagar Sen. Kaggle dataset. http://www...gggg..oooossggssseg data- data-from- from- cheschhsbbl 2016.

---

Under review as a conference paper at ICLR 2023
Hong Shi, Shaojun Pan, Jian Yang. and Chen Gong. Positive and unlabeled learning via loss decomposition and centroid estimation. In IJCAI, pp. 2689-2695, 2018.
Xingjian Shi, Zhihan Gao, Leonard Lausen, Hao Wang, Dit- Yan Yeung, Wai-kin Wong, and Wang- chun Woo. Deep learning for precipitation noweasting: A benchmark and new model. In NIPS, pp. 5617-5627, 2017.
Marek Smieja, Lukasz Struski, Jacek Tabor, Bartosz Zieliñski, and Przemystaw Spurek. Processing of missing data by neural networks. In NeurlPS, 2018.
Jonas Sonntag, Michael Engel, and Lars
Predicting parking availability from mobile payment transactions with positive unlabeled learning. In AAAI, volume 35, pp. 1.408-15415, 2021.
Aude Sportisse, Claire Boyer, and Julie Josse. Estimation and imputation in probabilistic principal component analysis with missing not at random data. In NeurlPS, 2020.
K Srinivas, Kavitha Rani, M Rao, Raj Kumar Patra, G Madhukar, and A Mahendar. Prediction of heart disease using hybrid linear regression. European Journal of Molecular & Clinical Medicine, 7(5): 159--171, 2020.
Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ily Sutskever, and Ruslan Salakhutdinov. Dropout: simple way prevent neural networks from overfitting. The journal of machine learning research, 1529::1929-195, 2014.
Yusuke Tanaka, Tomoharu Iwata, Toshiyuki Tanaka, Takeshi Kurashima, Maya Okawa, and Hiroyuki Toda. Refining coarse-grained spatial data using auxiliary spatial data sets with various granularities. In AAAI, volume 33, pp. 5091-5099, 2019.
Warren W Tryon. Activity measurement in psychology and medicine. Springer Science & Business Media, 2013.
Thomas Vandal, Evan Kodra, Sangram Ganguly, Andrew Michaelis, Ramakrishna Nemani, and Auroop R Ganguly. Deepsd: Generating high resolution climate change projections through single image super-resolution. In KDD, pp. 1663-1672, 2017.
Eduardo Velloso. UCI machine learning repository.
2013.
Eduardo Velloso, Andreas Bulling, Hans Gellersen, Wallace Ugulino, and Hugo Fuks. Qualitative activity recognition of weight lifting exercises. In AH, pp. 1 116-123, 2013.
Zhishun Wang, Zhenya He, and Jiande DZ Chen. Robust time delay estimation of bioelectric signals using least absolute deviation neural network. IEEE Transactions on biomedical engineering, 52 (3):4544462, 2005.
John B Webster, Daniel F Kripke, Sam Messin, Daniel J Mullaney, and Grant Wyborney. An activity-based sleep monitor system for ambulatory use. Sleep, 5(4):389-399, 1982.
Robert L Wilby, SP Charles, Eduardo Zorita, Bertrand Timbal, Penny Whetton, and LO Mearns. Guidelines for of climate scenarios developed from statistical downscaling methods. Supporting material of the Intergovernmental Panel on Climate Change, available from the DDC of IPCC TGCIA, 27, 2004.
Rand R Wilcox. Introduction robust estimation and hypothesis testing. Academic Press, 1997.
Yu Yao, Tongliang Liu, Bo Han, Mingming Gong, Gang Niu, Masashi Sugiyama, and Dacheng Tao. Rethinking class-prior estimation for positive-unlabelee learning. In /CLR, 2021.
MK Stephen Yeung, Jesper Tegnér, and James Collins. Reverse engineering gene networks using singular value decomposition and robust regression. Proceedings of the National Academy of Sciences, 99(9):6163-6118 2002.
Chenguang Zhang, Yuexian Hou, and Yan Zhang. Learning from positive and unlabeled data without explicit estimation of class prior. In AAAI, pp. 6762-6769, 2020.
12

---

Under review as a conference paper at ICLR 2023
Chuang Zhang, Dexin Ren, Tongliang Liu, Jian Yang, and Chen Gong. Positive and unlabeled learning with label disambiguation. In IJCAI, pp. 1-7, 2019.
Fan Zhou, Tengfei Li, Haibo Zhou, Hongtu Zhu, and Jieping Ye. Graph-based semi-supervised learning with non-ignorable non-response. In NeurlPS, 2019.
Zhi-Hua Zhou and Ming Li. Semi-supervised regression with co-training. In IJCAI, pp. 908-913, 2005.
Xiaojin Zhu and Andrew B Goldberg. Introduction to semi-supervisee learning. Synthesis lectures on artificial intelligence and machine learning, 3(1):1-130, 2009.
A PROOFS
A.1 PROOF OF LEMMA
2.2
Proof. For the proof of Lemma 2.2 we will derive two important lemmas from Assumption 2.1 Then, will prove Lemma 2.2 by using them.
We first show (*) S y
O. When (*)x) S 1/, we have from Equation and Equation
(x) I (x) + 6 +
(12)
0 S 6 ca
Ca S Cs.
Since ca S 0 by Assumption
2.1 we have
Ccal S C,.
(13)
If ca 0, Assumption
2.1 implies |c,l < kall which contradicts Equation 13 Hence, must have
0.
(14)
Since = =y' when Ea O, we have
p(x, ff) / 0) 5v)
(15)
uf 4,6 0)
p(x, (l) st),
which establishes Lemma A.1. Let p(a, v. be the underlying probability distribution for x, and y/. Then, p(æ, 1/'""(xx (a) S v') = p(æ // S y)-
(16)
The condition parts these conditional distributions represent the relationships between labels and regression functions, e.g., p(æ, (z) I is the distribution of and y when is higher than what given by the oracle regression function f*
Similar to Lemmal A.I le, a.s.), which represents our natural expectation that the regression function of well approximates f". When (x: & y', we have from Equation and Equation 4 with the condition f F.:
show f(z) S y ca Let {f e E f( <
f(x) < fx) t
(17)
f(x)) < ((x) 6x + + lex
0 < 6x + + leal
Ea < Es + leal-
Since ea < by Assumption 2.1 we have
|cal S 6, lcsl.
(18)
If < O, Assumption 2.1 implies 2|cal < kal. which contradicts Equation 18 Hence, we must have
E = O.
(19)
Since y = of when 6, 0, by replacing fr with f for the argument in the derivation of Lemma A.I
in Equation 15 we have
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Lemma A.2. Let F'
((()) k.l). When f F,, the following holds: p(a,v/1((z) S w/) p< vl) s))
(20)
Lemma A. immediately implies
Pylea.ll//(ssss (( v/) (a)s
(21)
for any function G IR. x
long as the expectations exist. When f e F², from Lemma A.2
we then have
(22)
A.2 PROOF OF PROPOSITION 3.2
Proof. From the decomposed gradients VZ(()) in Equation 7 we derive the proposed gradient only with the expectations over p(æ, v/).
From Condition 3.1 for L(f(x)), y), =T) (((())) when y < f(x)). Thus, Equation can be rewritten as
(p(()r)(((a)))) /
(23)
ply (alla
where y is marginalized out in the expectation in the second term since g(f((=)) does not depend on
g.
Here, Equation and Equation
can be rewritten by replacing VL(f( v) with g(f(())) as
(p = x p) ( [((((()))
(24)
+ ply (() (9 ((f((z)))
ply ((n)())) [9((((=))
(25)
= Ep( [(((()))) p(f(())))))))))))))))(<))) s () [(((()))) (1())))))))
Since ((f(())) does not depend on y. we can marginalize out in Equation 25 as
plu f((zz) [9(f(=)))
(26)
= p [(f(()))) - ((f(())= S B [[())))
From Equation 26
we can express Equation 23 as
vC(ff) p(f((z) S LE Wt
(27)
+ Ep(a) [((((z))) p(f((()) y) (y (() [(((
Finally, from Lemma 2.2 can rewrite Equation
27 as:
v VL()) p(f((1))))))))))))))))(())) ie
(28)
E (e)) g(f((z)))
S v) E [g(f(())).
which is identical to Equation Thus, the gradient Equation is unbiased and consistent with the gradient in Equation 6 a.s.
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A.3 PROOF OF LEMMA 3.4
Proof. The difference between the decomposed gradients Vλ(li and V (G)) at step / + in the gradient descent is
VC(A) -
(29)
p(S((w) S 5 WuDepeeee//Meesss11W
+ < f((z)) ((iaa)Baaaa///cccaaa)
-p(S((x)= S /y Pt (ineaae)
- -p < (ittt))lssseeeoccccooooLXLL....d
From Lemma 2.2
and Condition 3.1
[VL(J)) V.())
(30)
ply (awl)))))))
- -l < <ave))
plu =(a (({(())))
- ppy (
We decompose (ega) again as v(()) VC())
(31)
= ply < f(()))
p(f((m) s vlv < </laa)/eeeerrccaassssssaaalaa))l
+ < f(())/ (()) (/000//////llBBeeeeeeeeeehhhoee
ppy (a
The condition yf f((x) Ay (z) is equivalent to the condition f((x) since y' s y from Assumption 2.1 and thus p(u < f((x)ly < fx()) = 1. Then, we have
vD(f)) vC(fi)
(32)
=ply < f((x))
p(((1)) s vlv' < <iit))Blarrrddeesssssalllll
+ ply < f((x)ly < (aaaa
- p(y < [
Additionally, since ply < f((x)ly/ < f((x)) = -(-())))) f())),
vV((f)) vC(f))
(33)
= p(v < ff())))(((())) f((x))
E Estes//ciioMossss - Exelec(oonl((((())))
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This equation shows that the bias is represented by the difference between the expectation of gf with the lower-side data and that with the original upper-side data mixed into the lower side due to incomplete observations and the corresponding proportions.
From Assumption
3.3, since εд I æ.
(V(())) vC(fi)
(34)
= - < [(x))) vwl < {(x))
Since If f*1 S le a.s., ass. s and p(y fi(x)) = 1 from their definition,
p(f((a) S v)p(ca <0) p(fi(z) slu < f((x)) p(u f((x)) p(f((()))))))))) s v)p(e, 0)
n1 - E
(35)
(1 n) + 111 -)
Therefore, from the definition of 6,
n - (1-1 8.
( (1 + 1n(1 - E)
(36)
B IMPLEMENTATION OF LEARNING ALGORITHM BASED ON STOCHASTIC OPTIMIZATION
We scale up our U2 regression algorithm by stochastic approximation with M/ mini-batches and add a regularization term, R(f):
v (m)) (f))
Σ
VL(f((x).))
(37)
(m. E x(mm -up
Σ a(file))
Σ g(f((z)) + + +
aR(A))
p
00
(m
(
{m) {m) and X {m} respectively where v v((m) (ft) is the gradient for the m-th mini-batch. (X Yup
are upper-side and unlabeled sets in the m-th mini-batch based on the current ft. X regularization parameter, and the regularization term R(f) is, for example, the L1 or L2 norm of the parameter vector of f. We also convert M) N) to a hyperparameter p. ignoring constant coefficients instead of directly handling Tup The hyperparameters p and X are optimized in training based on the grid-search with the validation set.
The U2 regression algorithm based on stochastic optimization is described in Algorithm We learn the regression function with the gradient in Equation (37 by using any stochastic gradient method. Here, we used Adam with the hyperparameters recommended in Kingma & Ba (2015). and the number of samples in the mini-batches was set to 32, We set the candidates the hyperparameters, estimate ÿ f(x) for new data
and X, to (10 10 10-2, 10-1, 100}. ). By using the learned x.
C ALGORITHM FOR LU REGRESSION
We show the algorithm for the lower and unlabeled regression (LU regression), where labels for some observations may become inconsistently higher than those for typical observations. Let LLU (f) v) be a loss function for LU regression and gl.u( be a gradient V (x))) when f(x) S y. Similar to Condition 3.1 for U2 regression, we assume that the class of (fv(((()) y) satisfies the
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Algorithm 1 U2 regression based on stochastic gradient method.
Input: Training data D {In, Vn n== hyperparameters À 2 0: an external stochastic gradient
method A
Output: Model parameters 0 for f
while No stopping criterion has been met
2: Shuffle DD' into M mini-batches, (X (m) (m) M m==
3: for m to MM
4: Compute the gradient Ci (J:) in Equation 37 with {x ( (y(m)
5: Update by with vLtm) (ft)
condition that (r)(((())) is a gradient function depending only on [(x) and not on the value of y. Then, LU regression is Algorithm 1, with the following gradient, V zimm (fe), instead of V V (fr) Equation 37 as
V (fo)
Σ
VLLv(f(((),))
(38)
(a. of x(!"!
(m)
{mi)
glu(fr())) - aR(f))
Σ g(v(f((at)) #F (m))
Σ
+
00
#E mEX (m))
where {x/0 {m))
to y1o and X ( batch based on the current fa.
(mm)
(m))
respectively are lower-side and unlabeled sets in the m-th mini-
D COMPUTING INFRASTRUCTURE
All of the experiments were carried out with Python and TensorFlow implementation on workstations having 80 GB of memory, 4.0 GHz CPU, and an Nvidia Titan X GPU. In this environment, the computational time produce the results was a few hours.
E DETAILS OF EXPERIMENTS IN SECTION 4.1.1
E. SYNTHETIC DATASETS
We conducted the experiments on synthetic data to evaluate the feasibility of our method for obtaining unbiased learning results from cemmmettiically-oorppeee data containing different proportions of incomplete observations. We generated synthetic data on the basis of Assumption 2. and Equation 4 We randomly generated N = 1000 training samples, X {a,, ).Y. m= from the standard Gaussian distribution (xxi 0, I). where the number of features in æ was = 10, and I is the identity matrix. Then, using X, we generated the corresponding N sets of true labels y {vx J.N from the
distribution N (ym: u In. 3)), where w are coefficients that were also randomly generated from the standard Gaussian distribution (ww O, I), / is the noise precision, and T denotes the transpose. For simulating the situation in which a label has incomplete observations, we created corrupted labels y [y= by randomly selecting K percent of data in and subtracting the absolute value of white Gaussian noise with twice the value of the precision as that of y. 233 from their values. We repeatedly evaluated the proposed method for each of the following settings. The noise precision was B = {100 10 10-1}, which corresponded to low-noise setting task (LowNoise) and high-noise setting task (HighNoise)) and the proportion of incomplete training samples was K (25, 50, 75) %. In the case of K 75%, only 25 percent of the samples correctly corresponded to labels, and all of the other samples were attached with labels that were lower than the corresponding true values. It is quite difficult to learn regression functions using such data. In these tasks, we used linear model, 0 x, for f f=x) and an implementation for Equation 37 with the absolute loss, which satisfies Condition 3.1] for the loss function L and L Li-regularization for the regularization term. We set the candidates of the hyperparameters, P and X, to {10 3 10 10 10"). J. We standardized the data by subtracting their mean and dividing by their standard deviation in the training split. We used Adam
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Error
3. 1a
Proposed x MSE
Data index
Data index
Data index
(8) LowNoise, K-25%
(b) HigeNoise, K-25%
(c) Breathing, K-25%
d. Eroo
index (d) LowNoise, K-75%
index (1) Breathing, K-75%
HighNoise, K-75%
Figure 3: Errors in prediction (predicted value minus true value) by proposed method (blue) and by MSE (green) for tasks; (a) LowNoise, K 25%. (b) HighNoise. K 25%, (c) Breathing. 75%, and (f) Breathing, K 75%.
K = 75%, (d) LowNoise, K = 75%, (e) HighNoise, Error of each data-point is shown by dot (for proposed method) by eross mark (for MSE), and average error is shown by solid line (for proposed method) or by dashed line (for MSE).
with the hyperparameters recommended in Kingma & Ba (015)), and the number of samples in the mini-batches was set to 32.
We also used woald sensor dataset collected from the Kaggle dataset (Sen 2016) that contains breathing signals (Breathing). The dataset consisted of N = 432 samples. We used signals from a chest belt as X = {ær, J and æ in each sample had D 2 number of features, i.e., the period and height of the expansion/contraction of the chest. We used signals obtained by the Douglas bag (DB) method, which is the gold standard for measuring ventilation, as true labels = {8n J For our problem setting, created corrupted labels y {3/ J.W through the same procedure for synthetic corruption as that for Noise and HighNoise with = {25, 50, 75) %. In the experiment Breathing. for its non-linearity, function with the training set as its bases, and is a hyperparameter representing the kernel width that is also optimized by using the validation We the candidates of the hyperparameter o to {10-3, 10-2, 2, 10-1, 10"). The other implementation details were the same as those for Low Noise and HighNoise.
used 0 Φf (z.00 o) for f(x), where Φb is radial basis
E.2 DETAILED RESULTS
Figure B shows the error between the estimation results of the proposed method and their true values and those of MSE for LowNoise. HighNoise, and Breathing with 25 and 75 percent of incomplete training samples. Table 3 shows the performance on LowNoise, HighNoise, and Breathing for the proposed method and MSE. As shown in Figure 3 the proposed method obtains unbiased learning results in all cases, while MSE produces biased results. From Table we can see that the proposed method outperformes MSE overall. We found that the performance our method is not significantly affected by the increase in the proportion of incomplete training samples K even for K 75%, unlike that of MSE.
E.3 PERFORMANCE OVER DIFFERENT SIZES OF VALIDATION SET
To demonstrate the robustness of our validation iontdaatio--ee--baee set-based approach to estimating the hyperparameter Tup show the performance of the proposed method over different sizes of the validation set in Fig. 4 This analysis is conducted on the tasks in Section 4.1.I] LowNoise, HighNoise, and Breathing, with K 50%. Figure 4 shows that the proposed method does not degrade its performance much,
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Table 3: Comparison of proposed method and MSF in terms of MAE (smaller is better). Best methods are in bold. Confidence intervals are standard errors.
LowNoise
HighNoise
25%
50%
75%
25%
50%
75%
MSR
0.77+
0.02
2.30++002
1.03=00.02
1.62-+0.03 0.80
2.36
Proposed
0.50
0.04
0.08
0.79
0.80=30.02
Breathing
K 25%
K 50%
K 75%
MSE Proposed
0.55 0.43
0.91
0.32
0.46
0.59 0.001
1.8
LowNoise
HighNoise
Breathing
1.4
Average
MAE
1.0
0.6
10-1
10-2
10-3
Size of validation set (proportion within training set)
Figure 4: Performance (MAE, lower is better) of proposed method over different sizes validation set. Blue line represents results on LowNoise task, orange line represents results HighNoise task, green line represents results on Breathing task, with K = =50% of incomplete training samples, and red line is their average. Shaded areas confidence intervals. Leftmost point is results when 20% training set as validation set, which is setting we used in experiments throughout this paper, and rightmost point those of 0.1%.
even when we use only 1% of the training set as the validation set. This demonstrates that the proposed approach is robust enough also for the small size of the validation set well the high proportion incomplete validation samples. In Fig. 5 we also show a chart similar to Fig, 2 (the the validation set. We can that even in
error in prediction) when used 1% of the training this case, the proposed method achieved unbiased learning (the average error shown by the blue solid line is approximately zero.).
F DETAILS OF EXPERIMENTS IN SECTION 4
We applied the algorithm five different real-world healthcare tasks recorded in the datasets from the UCI Machine Learning Repository (Velloso) 2013) Velloso et al. 2013). which contains sensor outputs from wearable devices attached the arm while subjects exereised. From the non-intrusive sensors attached to gym equipment, we estimated the motion intensity of a subject that was measured accurately with an arm sensor that was an intrusive sensor wrapped around the arm. If we can mimic outputs from the arm sensor with outputs from the equipment sensor, it could contribute to the subjects' comfort, as they would not need to wear sensors to measure their motion intensity. We used all the features from the equipment sensor that took "None" values less than ten times as X {æ,, J.v where each sample had D = 13 number of features. The corrupted labels y' = {1// magnitude of acceleration from the arm sensor, which accurately sense motion intensity on the arm, but it had insufficient data coverage and incomplete or missing observations for the movements of other body parts. For performance evaluation, we used the magnitude of acceleration for the
N n== were the
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Under review as a conference paper at ICLR 2023
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Figure 5: Errors in prediction when we choose hyperparameters with 1% of training set validation set. Tasks are (a) LowNoise, (b) HighNoise, and (c) Breathing, with K = 50% of incomplete training samples. Error of each data-point is shown by blue dot (for proposed method) or by green cross mark (for MSE), and average error is shown by blue solid line (for proposed method) or by green dashed line (for MSE).
entire body as true labels = {yn J.N n=1- The number of samples were N = 11, 159, N 7, 593, N 844, N 6, 432, and N 7,2 respectively for the tasks, Specification, Throwing A, Lifting, Lowering, and Throwing B. For the complex nature of the tasks, we used 6-layer multilayer perceptron with ReLU (Nair & Hinton 2010) (more specifically. 1-100-100-1000100-11 as f(x). which demonstrates the usefulness of the proposed method for training deep neural networks, We also used dropout (Srivastava et al., 2014) with a rate of 50% after each fully connected layer. We used two implementations for f.([(x), loss (P rrrooseed- 1) and the squared loss (Proposed 2). For both implementations, we used the absolute loss, which satisfies Condition 3.1 for the loss function L(f(x), y) when y' < f()) and used L --rreullarizatin for the regularization term. The other implementation details were the same as those for Lowse HighN and Breathing.
when f(x) S y in Equation 37 with the absolute
G DETAILS OF EXPERIMENTS IN SECTION 4.1.3
We demonstrate the practicality of approach in real use case in healthcare. From non-intrusive bed sensors installed under each of the four legs of bed, we estimated the motion intensity of subject that was measured accurately with ActiGraph, gold standard intrusive sensor wrapped around the wrist [Troon 2013 Mullaney et al. 1980 Webster et al. 1982; Cole et al. 1992). The sensing results of ActiGraph are used for tasks such as discriminating whether subject is asleep or awake (Cole et al. 1992). While ActiGraph can accurately sense motion on the forearm, it has insufficient data coverage other areas and often causes observations of movements on other body parts to be missing. The bed sensors have broader data coverage since they can sense global motion on all body parts; however, the sensing accuracy is limited due to their non-intrusiveness. can mimic the outputs from ActiGraph with outputs from the bed sensors, we can expect to achieve sufficient accuracy and coverage while also easing the burden on the subject. The dataset we used included three pieces of data, Data (i), (ii), and (iii), which were respectively recorded over 20, 18, and 18.5 minutes. Each piece of data consisted of pairs of bed-sensor-data sequences and the corresponding motion intensity sequence obtained by ActiGraph. We used the "magnitude" attribute of ActiGraph as corrupted labels y for the motion intensity, whose sampling rate was about one sample per second. For true labels y. we manually measured the motion intensity every minute under the management of a domain expert. For X, we first computed the gravity center of the four sensor outputs that were obtained from the bed sensors under the four legs of bed. Then, we computed the time derivatives and cross terms of the raw sensor outputs and the gravity center. The sampling rate of the bed sensors was different from that of ActiGraph, about one sample per five milliseconds. Thus, X was finally generated (1 second) subsequences of the time series of the above computed variables, where second was the same as the sampling interval of ActiGraph. The statistics (0.05, 0.25, 0.5, 0.75, 0.95) quantiles. In this task, we used the linear model 0 x for f(x) due to its interpretability, which is inevitable in real-world healthcare and medical applications.
we
sliding window of statistics in 1, 1000000llisccoom
means, standard deviations, and
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Under review as a conference paper at ICLR 2023
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Figure 6: Experiment on real use case for healthcare. Blue line represents our estimation results for motion intensity. White (non-colored) area shows that both proposed method and ActiGraph correctly estimated motion intensity the subject this duration. Green area shows that our method could capture motion this duration while ActiGraph could not. Orange shows that our method could not capture motion at this duration but ActiGraph could. Gray area shows that our method mistakenly captured noise subject's motion.
G.1 ESTIMATION RESULTS FOR MOTION INTENSITY
Figure 6 compares estimation results for motion intensity with the output of ActiGraph and true labels.
G.2 IMPORTANT FEATURES ESTIMATING MOTION INTENSITY
The important features selected by L1 regularization were the statistics of the gravity center and the cross terms and time derivatives of the raw sensor outputs. The largest weight was assigned to the standard deviation of the gravity center, which represents the amplitude of the gravity center, so directly related to the motion of subjects.
H
OTHER POSSIBLE USE CASES OF REGRESSION FOR SENSOR MAGNITUDE
Examples of predicting the magnitude values of a sensor, which is a field of application of U2 regression, can be found in several areas. Besides the medical and healthcare applications discussed in the main text, another example is estimating the wind speed rainfall in specific region from observable macroscopic information (Cheng & Tan 2008 Abraham & Tan 2010; Abraham et al. 2013 Vandal et al. 2017), known as statistical downscaling (Wilby et al.] 2004). Wind speed and rainfall, which are labels in these tasks, can sensed locally in a limited number of locations and provide incomplete observations and biased labels compared with macroscopic information, which is considered to be explanatory variables.
21


Reviews:
[{"review_id": "sdQElxOFe7S", "reviewer": "Reviewer_gU2i", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "This draft studies a special weakly supervised regression problem in which the labels are generalized by sensors. Therefore, the low value displayed by the sensor could either be actual or suggest that the label is missing. The author formulates this problem as a regression from asymmetrically corrupted data and proposes a new approach based on an unbiased gradient estimator to solve it. Both synthetic and real-world experiments demonstrate the effectiveness of the proposed approach. ", "summary_of_the_review": "This draft studies a well-motivated problem that covers a wide range of real-world sensor-based applications. The proposed method constructs an unbiased gradient estimator with asymmetrically corrupted data under certain mild assumptions. There is a theoretical basis for the proposed method and empirical studies support its effectiveness.", "strength_and_weaknesses": "Strength:\n+ The work is well motivated by many sensor-based applications and covers a wide range of real-world applications. To address this task, the proposed method appears promising. Also, empirical results support the proposed method's effectiveness.\n+ The proposed method has nice theoretical properties. According to certain assumptions, the proposed approach produces an unbiased estimator with asymmetrically corrupted data and thus obtains a well-generalized model.\n\nWeakness:\n- The estimation of $\\pi_{up}$ is heuristic and changes according to the training procedure based on the validation set. Therefore, the proposed method and the theoretical results do not end-to-end match exactly. Due to the small validation data set, the estimation can be unreliable and unstable due to overfitting.\n- This method can only be applied to absolute loss and pinball loss, which limits its use.\n\nThere is a question regarding the implementation of the proposed method. The proposed method relays on an adaptive estimation of $/pi_[up]$ and an upper-side labeled sample set based on the validation set. What is the size of the validation set and how does it affect the performance of the proposed method? How to avoid the overfit to the validation data?", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "qyPjqPkEHH", "reviewer": "Reviewer_QdLV", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper considered a regression problem where incomplete observations have labels biased towards lower values. The authors proposed an algorithm which utilizes only high-valued labels and discarded the lower-valued labels. They conducted several experiments on both synthetic and real-world datasets to demonstrate their algorithm.", "summary_of_the_review": "Overall I do not buy the idea or arguments made in this paper. For detailed comments please look at the Strength and Weakness section above.", "strength_and_weaknesses": "Strength:\nThe idea is explained well. Their work is motivated by real-world use cases which makes it interesting.\n\nWeakness:\nIn general I do not really buy this idea for addressing asymmetrically corrupted dataset, and the assumptions they made seem problematic. On a high level, completely ignoring the lower-valued labels, no matter they are true low values or come from incomplete observations, could potentially lose some information. Couple of questions are in order.\n\n1. As the authors showed in the experiments, Huber regression performed badly. But have you tried using Huber regression as an outlier detection step? Namely, identify observations whose (x,y) relationship differs significantly from others, and remove those from your sample set, and then apply Least Squares regression to the rest of normal data points. The logic is that, I do not think we can completely remove lower-valued labels, but instead, we can do some pre-processing to identify abnormal data points. The high-value / low-value scheme does not seem to well supported.\n\n2. Section 2.1 is not needed in my view. It is common knowledge and should be described very briefly. \n\n3. Based on your definition of upper-side labeled data, their noise \\epsion_s >=0. But technically \\epsilon_s could also be negative. So you are missing a portion of complete observations by restricting to the upper-side labeled data.\n\n4. Lemma 2.2 also seems problematic. Ideally when there is no corruption, namely, (x,y), we want to use all the data, not just data with f(x) <= y. Eq. (5) doesn't seem to establish the consistent estimator as the authors claimed.\n\n5. The authors claimed that for some loss functions, when y<f(x), the gradient of the loss function does not depend on y, e.g., Eq. (8). But notice that even when y>f(x), the gradient of this type of loss also does not depend on y. So it is about the loss function itself, not the upper or lower sides of the labels. When you switch to squared loss (which was used in your Proposal-2 in the experimental section and thus violated your assumption here), the gradient would depend on y no matter you are in the upper or lower sided region.\n\n6. In order to identify the upper or lower sides, you need to do an iterative process where you use the f_t from previous iteration to define upper or lower region. Does this unnecessarily complicate the problem?\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "jScqu5EU8h", "reviewer": "Reviewer_Sduq", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper studies the problem of learning a decision function when the output might be corrupted by the fact that the sensor in charge of collecting it has failed to record it properly, underestimating it. Without a debiasing procedure, the decision function naturally underestimate the magnitude of the event. Under the assumption that the loss function does not depend on the prediction (when the latter is greater than the output), the authors propose an estimator of the gradient that is unbiased. Experiments complement the paper.", "summary_of_the_review": "Overall, I feel the contribution of this paper might not be enough to warrant acceptance. In particular, the restricted choice of loss functions is shortcuting the interest of the approach. The experiments need strengthening. ", "strength_and_weaknesses": "**Strengths**\n- the paper is globally clear and well written\n- the problem studied is of interest and the proposed approach is natural\n\n**Weaknesses**\n- it seems to me that the interest of the approach is completely shortcut by the choice of loss functions, which do not depend on the label. This is a very strong limitation of the approach to me\n- in the same vein, could the authors think about other examples where to apply a similar method? This could make the contribution of greater interest\n- the experiments are pretty rudimentary, especially on real data. For instance how would a model with censorship behave, when given the information if the output has been altered or not?\n- I am also pointing out the literature on Median-of-Means (MoM)-based methods for robust regression, see in particular [1], that do not assume the outliers to be symmetric and could be interesting to benchmark\n\n[1] Robust classification via MOM minimization, Lecué et al. 2020", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "X5HuxEIJaCR", "reviewer": "Reviewer_mxCJ", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "\nThe authors address an interesting problem of the asymmetrically corrupted regression problem. They motivate the problem using several real-world examples. They propose a solution by modeling the target value as corrupted with asymmetric noise. To learn the regression function, they derive a loss function based on the data model and use gradient descent to find the solution. They use real and synthetic data to demonstrate the proposed approach. \n", "summary_of_the_review": "Overall the authors present an interesting problem with a novel solution. They prove that the gradient of the proposed loss is unbiased and consistent. Then, they demonstrate that the method works on synthetic and real data. This is a valuable contribution to the community, and I believe that the paper should be accepted.", "strength_and_weaknesses": "Strengths: The paper is well written, and the English level is satisfactory. The proposed problem is interesting with a novel solution. They prove the unbiasedness and consistency of the gradient of the proposed loss function. Several examples are provided to demonstrate the advantages of the proposed approach. \n\nWeaknesses: The authors only compare their method to simple baselines, while several solutions exist to the related problem of regression with asymmetric loss functions.\n\n\nMinor comments:\nFor clarity, please mention what K is in the caption of figure 2.\n\nPage 4 results of figure 2, I assume that the data in the test set is not incomplete, but this is not explained.\n\nPlease expand on the caption of Table 2.\n\nP9 the results related to table 2, can be viewed as a classification task, so why not compare to baselines that focus on asymmetric classification?\n\nWhy is the word Rate capitalized?\n \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject