Learning What Not to Model: Gaussian Process Regression with Negative Constraints
Authors: Gaurav Shrivastava
Harsh Shrivastava
Abhinav Shrivastava
Venue:  (2021)
Keywords: Gaussian Process
Gaussian Process Regression

Abstract: Gaussian Process (GP) regression fits a curve on a set of datapairs, with each pair consisting of an input point '$\mathbf{x}$' and its corresponding target regression value '$y(\mathbf{x})$' (a positive datapair). But, what if for an input point '$\bar{\mathbf{x}}$', we want to constrain the GP to avoid a target regression value '$\bar{y}(\bar{\mathbf{x}})$' (a negative datapair)? This requirement can often appear in real-world navigation tasks, where an agent would want to avoid obstacles, like furniture items in a room when planning a trajectory to navigate. In this work, we propose to incorporate such negative constraints in a GP regression framework. Our approach, 'GP-NC' or Gaussian Process with Negative Constraints, fits over the positive datapairs while avoiding the negative datapairs. Specifically, our key idea is to model the negative datapairs using small blobs of Gaussian distribution and maximize its KL divergence from the GP. We jointly optimize the GP-NC for both the positive and negative datapairs. We empirically demonstrate that our GP-NC framework performs better than the traditional GP learning and that our framework does not affect the scalability of Gaussian Process regression and helps the model converge faster as the size of the data increases.

Under review as a conference paper at ICLR 2021
LEARNING WHAT NOT TO MODEL: GAUSSIAN PRO- CESS REGRESSION WITH NEGATIVE CONSTRAINTS
Anonymous authors Paper under double-blind review
ABSTRACT
We empirically demonstrate that our GP-NC framework performs better than the traditional GP learning and that framework does not affect the scalability of Gaussian Process regression and helps the model converge faster as the size of the data increases. Gaussian Process (GP) regression fits a curve a of datapairs, with each pair consisting of an input point 'x' and its corresponding target regression value 'y(x)" (a positive datapair). But, what if for an input point 'x', we want to constrain the GP to avoid a target regression value 'у(x)' (a negative datapair)? This requirement can often appear in real-world navigation tasks, where an agent would want to avoid obstacles, like furniture items in room when planning a trajectory to navigate. In this work, we propose to incorporate such negative constraints in a GP regression framework. Our approach, 'GP-NC' or Gaussian Process with Negative Constraints, fits over the positive datapairs while avoiding the negative datapairs. Specifically, our key idea is model the negative datapairs using small blobs of Gaussian distribution and maximize its KL divergence from the GP. We jointly optimize the GP-NC for both the positive and negative datapairs. We empirically demonstrate that our GP-NC framework performs better than the traditional GP learning and that our framework does not affect the scalability of Gaussian Process regression and helps the model converge faster as the size of the data increases.
INTRODUCTION
Gaussian process are one of the most studied model class for data-driven learning as these are nonparametric, flexible function class that requires little prior knowledge of the process. Traditionally, GPs have found their applications in various fields of research, including Navigation systems (e.g., in Wiener and Kalman filters) (Jazwinski) 2007), Geostatistics, Meteorology (Kriging (Handcock) & Stein 1993)) and Machine learning (Rasmussen] 2006). The wide range of applications can be attributed to the property of GPs to model the target uncertainty by providing the predictive variance over the target variable.
Gaussian process regression in its current construct fits only on a set of positive datapairs, with each pair consisting of an input point and its desired target regression value, learn the distribution on functional space. However, in some cases, more information is available in the form of datapairs, where particular input point, we want avoid range of regression values during the curve fitting of GP. We designate such data as negative datapairs.
An illustration where modeling such negative datapairs would be extremely beneficial is given in [ig[[ In [ig an agent wants to model a trajectory such that it covers all the positive datapairs marked by 'x". However, its trajectory based only on the positive datapairs. We can handle this problem of navigating in the presence of obstacles in two ways, one way is to get high density of positive datapairs near the negative
essential note that the agent would run into an obstacle if it models
obstacle, and the other more straightforward approach is to just mark the obstacle datapair. The former approach would unnecessarily increase the number of positive datapairs for gP to regress. Hence, it may run into scalability issues. However, in the latter approach, if the point is denoted as a negative datapair with a sphere of negative influence around it as illustrated by Fig [T. the new trajectory can be modeled with less number of datapairs that accounts for all obstacles on the

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Figure I: An illustration our problem setup. (a) top view of the room where the agent wants to travel to particular location while avoiding obstacles; (b) the agent has been given the location of the positive datapairs that are needed to be covered in its trajectory. Since the number of these observed points is low, the agent is not able to avoid the obstruction (coffee table) while forecasting course: (c) the agent given both the positive datapairs which it needs to reach along with negative datapairs (area of influence is given by shaded red region) that should be avnided during the modeling of future trajectory.
way. Various GP methods in their current framework lack the ability to incorporate these negative datapairs for the regression paradigm.
Contributions:In this paper, explore the concept of negative datapairs. We provide simple yet effective GP regression framework, called GP-NC which can on the positive datapairs while avoiding the negative datapairs. Specifically, our key idea is to model the negative datapairs using a small Gaussian blob and maximize its KI. divergence from the GP Our framework can be easily incorporated for various types of GP models (e.g., exact, SVGP (Hensman et al., 2013). PPGPR (Jankowiak et al. 2019)) and works well in the scalable settings too. We empirically show in that the inclusion of negative datapairs in training helps with both the increase in accuracy and the convergence rate of the algorithm.
2 REVIEW OF GAUSSIAN PROCESS REGRESSION
We briefly review the basics of Gaussian Process regression, following the notations in Willon et al. 2015). For more comprehensive discussion of GPs, refer to (Rasmussen 2006).
Gaussian process collection of random variables, any finite number which have a joint Gaussian distribution (Rasmussen, 2006). We consider dataset D with n D-dimensional input vectors, X = {x1,- X,, } and corresponding n X 1 vector of targets = (v(x1), ... (xx,) The goal of GP regression is learn function f that maps elements from input space target space, i.e., y(x) = [x) + where is i.i.d. noise. If f(x) GP( ko), then any collection of function values has a joint multivariate normal distribution given by.
f(X) (f(x),, [(xx)) ~ (ux,KxxX)
with the mean vector and covariance matrix defined by the functions of the Gaussian Process, as (ux); = p(x;) and (Kx,x)s = ko (x,,x;).. The kernel function ka of the GP is parameterized by Assuming additive Gaussian noise, fxx)]f(xx ~N(x(x); f(x), a²), then the predictive distribution of the GP evaluated at the n,, test points indexed by X., is given by
f. f, X,y,0,0 (EEEE))
Eff. px. Kx.,x [xxx +001] y.
Cov (f.) Kx.xx Kx..x [Kxxx o² - Kx.x.
Kx..x represents the n. x n covariance matrix between the GP evaluated at X, and X. Other covariance matrices follow similar conventions. μx, is the mean vector of size test points and Kx,x the n x n covariance matrix calculated using the training inputs X. The underlying hyperparameter 0 implicitly affects all the covariance matrices under consideration.
for the
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2.1
GPs: LEARNING AND MODEL SELECTION
We can view the GP in terms of fitting a joint probability distribution as,
p (y, f[x) = p (ylf, (f(X)
and we derive the marginal likelihood of the targets for the GP by integrating out the functions the joint distribution of Eq. (3) nice property of the GP is that this marginal likelihood has analytical form given by,
function of kernel parameters alone
()) = log p(y x) = - (y" (Ko + +²1) y + log ( + + + N Nlgg (2t)
(4)
where we have used Kę as a shorthand for Kx,x given 0. The process kernel learning is that of optimizing Eq. (4) w.r.t. 0.
The first term on the right hand side in Eq. is used for model fitting, while the second term is complexity penalty term that maintains the Occam's 's razor for realizable functions shown by (Rasmussen & Ghahramani, (001). The marginal likelihood involves matrix inversion and evaluating determinant for n matrix, which the naive implementation would require cubic order of computations (nn²) and (n22) ) of storage. Approaches like Scalable Variational GP (SVGP) (Hensman et al., (2013) and parametric GPR (PPGPR) (Jankowiak al. 2019) have proposed approximations that lead to much better scalability. Please refer to Appexdix for details.
3 GP REGRESSION WITH NEGATIVE DATAPAIRS
As shown in Fig. [ we want the model to avoid certain negative datapairs in its trajectory. In other words, we want the trajectory of the Gaussian Process to have a very low probability of passing through these negative datapairs. In this section, we will first formalize the functional form of the negative datapairs and then subsequently describe our framework called GP-NC regression.
DEFINITION OF POSITIVE NEGATIVE DATAPAIRS
Positive datapairs: The set of datapairs through which the GP should pass defined pos- itive datapairs. We assume a set n datapairs (input, positive target) with D-dimensional in- put vectors, X (x1,... Xn) and corresponding n vector of target regression values y = (y(x), ... ,,(xx)))
Negative datapairs: The set of datapairs which the GP should avoid (obstacles) defined as negative datapairs. We assume a set of m datapairs (input, negative target) with D-dimensional input vectors X {_,,..., Xm3 and corresponding set of negative targets y {x) . ((mm) The sample value of GP at input 8,, given by f(x:), should be far from the negative target regression value gxx)).
Note that it is possible that a particular input x can be in both the positive and negative data pair set. This will happen, when a particular input we want the GP regression value be close its positive target regression value yxx and far from its negative target regression value 7xx).
3.2 FUNCTIONAL REPRESENTATION OF NEGATIVE DATAPAIRS
For our framework, we first get functional representation of the negative datapairs. We define a Gaussian distribution around each of the negative datapair, q x) N (x((), oay), with mean equal to the negative target value (xx) and neg is the variance which is a hyperparameter, The Gaussian blob can also be thought of as the area influence for the negative datapair with the variance dneg indicating the spread of its influence.
3.3 GP-NC REGRESSION FRAMEWORK
The aim of GP-NC regression framework is simultaneously fit the GP regression the positive datapairs (X) and avoid the negative datapairs (X) (i.e., using them as negative constraints (NC)). The former is achieved by maximizing the marginal likelihood given in the Eq. 44. To avoid
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Algorithm 1: Training of GP-NC
Algorithm describing the GP-NC regression. We alternatively up- date between the negative log- likelihood and KL divergence term with respect to the kernel parame- ters 6. For different GP methods we can appropriately plug-in the log-likelihood term (NLL).
Input
Datapairs {x, ytt, {X.8} y)
Parameters
GP Kernel Parameters '0'
Hyperparameters: Oneg. X while until convergence do
NLL= ply |0, X);
<minimize (NLL);
ai = À log DEL (p (510,, X) IW (5, (egg)
0 t maximize (KLaiv)
end
the negative datapairs, we want our GP model adjust its distribution curve so that while drawing samples from the predictive GP distribution, its values do not lic in the influence region of the negative datapairs. To this end, we propose to fit the GP regression model on the positive datapairs along with maximizing the Kullback-Leible (KL) divergence between the distributions of the GP regression model and the Gaussian distributions defined over the negative datapairs.
Thus, mathematicllly, we want maximize the following KL divergence given by
= arg max DxL. (p 8, ll X))
(5)
We chose to maximize the DKL term in the p - q direction, as this fixes the negative datapairs distribution q() (X) the reference probability distribution. Now, since the KL divergence is an unbounded distance metric, the following section describes a practical workaround to maximize it.
3.3.1
MAXIMIZING KL DIVERGENCE USING THE LOGARITHM TRICK
Eq. (5) is increasing the distance between the GP distribution and the negative datapairs distribution by maximizing the KI. divergence. However, KI. divergence is an unbounded function, i.e., DKL. E [0, oo). Implementing the DKL divergence directly in the form of Eq. (5) can create problems for the gradient updates and convergence.
We also want maximize the marginal log-likelihood Eq. (4) and the A terms simultaneously. This raises a problem of mismatch in the magnitude of scale for the marginal log- likeliied term and DKL divergence term as the values of DKL divergence will be significantly higher. Thus, the gradient update would be dominated by A term. In essence, the model would fixate more on avoiding the negative datapairs than fitting the curve on the positive datapairs. We also observed this empirically. Hence, to suppress the gradient update from the A term, we encapsulate Eq. (5) in logarithmic function. This turns out to be beneficial in multiple ways. Firstly, maximizing the DRL term is equivalent to maximizing the log( log(Drt).. as log is an monotonically increasing function. Secondly, and more importantly, the scale of magnitude A term becomes equivalent to the scale of magnitude for the marginal ligeeeildoood term which makes the convergence stable.
3.3.2 GP-NC: LEARNING AND MODEL SELECTION
We apply a log function to the DEL given in Eq. 5 and write the combined objective function for our GP-NC regression,
[()) arg min loggp(y 0, X) Alog DKL (P
(6)
where p(y X) is the marginal log-likelidood term that represents the model to be fitted on the observed datapoints. The parameter À is the tradcoff hyperparameter between curve fitting and avoidance of the negative datapoints, or how relaxed can the negative constraints be.
We already know the analytical form of the log-likelihood term from Eq. (4). We now focus on the log( DKL) term. Since, both the likelihood p(y 10, X) and negative datapair distributions g (X) are modeled using Gaussians, we can simply use the analytical form of KL divergence between any two Gaussian distributions given by.
of + (w 222 1
(p.(p. 9) log
22 +
σ1
20
2
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here p. are Gaussian distributions defined as [[1 01) and (1 02) 02) respectively. The DKL term
is adjusting the mean and variance of the likelihood pY 0, X) with respect to the fixed blohs of Gaussian distributions around the negative datapairs. Specifically, we can consider the distribution = NW 01) = ply|0, X)" and "q = NN22, x2) g ) in Eq. 7 Now, if we refer Eq. 2 Pi = [ff.] and σ = cfv(f.) which contain the parameters of the GP that are optimized. μ22, 02 correspond to the hyperparameters of the Gaussian distribution representing the negative datapairs and are constant. Algorithm ① gives an overview of the training of GP-NC regression.
A note the difference between the GP-NC regression and general classification settings: In the case of GP-NC regression, the boundary for every negative datapair is optimized independent cach other. In classification settings, all the negative points belong to class and they jointly affect the decision boundary of the GP for their class.
3.4 SPARSE GAUSSIAN PROCESSES WITH NEGATIVE DATAPOINTS
In Appendix A we show that is straightforward modify the class of scalable and Sparse GP regression models to account for the negative datapairs in their formulation. In particular we review the SVGP model by (Hensman et al. 2013). which is popular scalable implementation of GPs. We also investigate recent parametric Gaussian Process regressors (PPGPR) method by (Jankowiak jet al. 2019). We evaluate the performance of these methods with our GP-NC framework in the experiments section.
4 RELATED WORKS
To the best of our knowledge, the classical GP regression introduced in (Rasmussen 2006) and many subsequent works primarily focus on positive datapairs for curve fitting. Even with the absence of the concept of negative datapairs, GP regression methods have been widely used for obstacle-aware navigation task which is one of the relevant applications to evaluate our GP-NC framework.
Classical GP:
S navigation: GPs are extensively used the field of navigation and often are a component of path planning algorithms. (Ellis al., 2009) used GP regression in modeling the pedestrian trajectories by using positive datapairs. (Aoude et al. 2013) used heuristic based approach over GP regression to incorporate dynamic changes and environmental constraint in the surroundings. Their solution, named RR-GP, builds learned motion pattern model by combining the flexibility of GP with the efficiency of RRTReach, a sampling- predictions are conditioned on dynamically feasible paths identified from the reachability analysis, yielding more accurate predictions of future behavior. (Goli et al.) 2018) introduced the use of GP regression for long-term location prediction for collision avoidance in Connected Vehicle (CV) environment. The GPs are used to model the trajectory of the vehicles using the historical data. The collected data from vehicles together with GPR models received from infrastructure are then used to predict the future trajectories of vehicles in the scene. (Meera et al. 2019) designed an Obstacle-aware Adaptive Informative Path Planning (OA-IPP) algorithm for target search in cluttered environments using UAVs. This method uses GP to detect the obstacles/target, which the UAV gets by marking dense number points (positive datapairs) around the obstacles. (Hewing al. 2020)] Yuan & Kitani 2219)) are some of the works using sampling based techniques for trajectory prediction. (Choi et al. 2015) one of the work which tries to incorporate the concept of negative datapairs in classical GP construct by introducing a leveraged parameter in kernel function. The authors demonstrate that having the ability to incorporate negative targets increases the efficiency of trajectory predictions. However, this approach fundamentally differs from ours in terms of incorporation of negative datapairs ours try to maximize the KI.-Divergence between j(x) and y(x) while theirs utilizes additional leveraged parameter in the kernel function. Besides, our approach is more scalable as the size of the covariance matrix docsn't 't increase to incorporate the negative datapairs.
reachability computation. Obstacle trajectory GP
Scalable GP: Naive implementation of GP regression is not scalable for large datasets the model selection and inference of GP requires a cubic order of computations ((na) and O(n²) of storage. Since the GP-NC framework is quite generic and work for various scale GP methods, highlight few of these methods. (Hensman et al., 2013 Dai et al. 20044) Gal et al. 2014) are some of the well known scalable methods suitable for framework they use stochastic gradient descent
want to
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SVGP
SVGP-NC
5.0
5.0
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0.0
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-2.
-1.
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SVGP-NC
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* Positive Targets
Negotive Targets
Inducing Points
Curve Fitted
Confidence
Figure 2: Visualizing GP-NC regression framework: The figures compare how the SVGP regression fits using the classical GP framework (left) vs the GP-NC framework (right). The aim is fit the regression targets marked in 'black' (positive datapairs) and avoid the targets marked 'red' (negative datapairs). The classical GP framework only uses the positive datapairs whereas our proposed GP-NC framework uses both the positive and negative datapairs for fitting the regression curve. The points in "yellow" are the inducing points used to fit the GP. We used two inducing points setting. Top figures: locations of inducing points curve. Bottom figures: we randomly sampled the inducing points from the whole range of training inputs. GP-NC framework (right), param--rraeeterss were selected as À = 0.1 and dng 1.2
taken the start of
for optimization. Furthermore, recent works (Wilson & Nickisch 2015 Wilson et al. 2015 2016) have improved scalability by reducing the learning to O(n) and test prediction to O(I) under some assumptions.
Negative datapairs in other domains: The concept of negative datapairs has been extensively utilized in the self-supervised learning. Applications include learning word embeddings (Mikolov et al. 2013
Mnih & Kavukcuoglu 2013), image representations (He et al. [2000] Misra & Maaten, 2020
Feng
al. 2019)) video representations (Sermanet et al.) 2018] Fernando et al. 2017 Misra al.
2016
Harley et al. )020), etc. In these works negative and positive samples are created as pseudo labels to
train neural network to learn the deep representations of the inputs.
EXPERIMENTS
We compared various GP regression models in their classical form (using only positive datapairs) with their corresponding GP-NC regression models where we used our negative constraints framework. We used Negative Log-likelihode (NLL) and Root Mean Squared Error (RMSE) our evaluation metrics. We compared our framework on a synthetic dataset and six real world datasets. Throughout experiments, found that for every GP model, the GP-NC regression framework outperforms its corresponding classical GP regression setting. We used GPytorch (Gardner et al.) (2018) to implement all the GP (ours + baselines) models. We use zero mean value and the RBF kernel for GP prior for all of the models unless mentioned otherwise.
5.1 SYNTHETIC DATASET: VISUALIZING THE GP-NC REGRESSION FRAMEWORK
We aim to visualize the GP-NC regression framework using a toy dataset. We sampled 400 positve datapairs from sinusoidal function and randomly sampled 15 negative datapairs as represented by Fig. We trained a sparse SVGP model to regress curve on the positive datapair using the classical GP framework and the one with negative constraints GP-NC. For the top figures of Fig 2 we trained SVGP with 80 inducing points, all at the starting location training input range. For the bottom figures of Fig. 2 randomly sampled 10 inducing points from the range of training inputs. SVGP with a constant mean and RBF kernel for the GP prior was used. After training the SVGP model in both settings for 100 epochs we obtain the curves as depicted by the figures on the

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Wine Quality
Elevators (N=16599)
Protein
Prud. (N=59381)
3DRoad (N=434874)
(e5(N)1599)
white(N=4898)
(N=45730)
PPGPR
PPGPR-NC
SVGP
SVGP-NC
Exact
Exact-NC
1.68
1.52
0.41
0.52
0.47
1.01
2.11
3.15
0.19
0.73
NLL
Comparison real world data: We plot test negative log-likelihoods (NLL) for 6 univariate regression datasets (lower better). Results averaged over 10 random train/test/valid splits.
Figure 3:
Table I: Runtime comparison the classical GP and GP-NC frameworks which includes negative datapairs on different datascts. A. is the runtime difference of the GP model in GP-NC framework vs the classical GP framework. We used GPU accelerated GP implementation of Torch library.
Datasets
Size of data
Type of Target Variable
A Exact GP
Ar sparse SVGP
Wine quality red
1.599
Discrete
2 mss
3.1s
Wine quality white
4898
Discrete
40ms
3.7s
Elevators
16599
Continuous
5s
34s
Protein
45730
Continuous
50s
37s
Prudential
59381
Discrete
1.2
34s
3DRoad
434874
Continuous
2208s
292s
left-side in Fig. 2 The inability to incorporate the information provided by the negative datapairs in the classical GTP construct hinders its ability to fit the data well as patently visible in the left figure. Mean and predictive variance not only losing
some positive datapairs but are also engulfing the negative datapairs in the confidence region which is undesirable.
Our GP-NC framework re-calibrates the curve by integrating the information provided by the negative datapairs as seen in the right-hand side figures of Fig 2 As evident from the figure, the additional information from a few negative datapairs helps the model fit better to the positive datapairs in addition to avoiding most of the negative datapairs. We can tune the values of À given in Eq. (6) to balance between the weightage given by the GP to positive and negative datapairs. Decreasing the value of À results in reduction of influence of the negative datapairs. Notice that curved learned by our approach even with sub-optimal inducing points.
5.2 TRAJECTORY PREDICTION USING GP-NC REGRESSION FRAMEWORK
We want to model an agent's trajectory using a GP regression model such that it takes the agent's s present location (z. y) input and predicts agents next location (й, 9).. For this set of experiments, we synthesize a 2d-virtual traffic scene given by Fig. Furthermore, the road contain pitfalls, roadblocks, accidents, etc. that need to be avoided are represented as red diamonds in the figure. We designate these targets as negative targets that are to be avoided for ensuring the safety of traffic. There are a total number 10 negative datapairs present in the scene. Next, We sample 250 observed co-ordinates on the 2d virtual path for modeling the future trajectory of the agent.
We trained classical SVGP model and ours SVGP-NC model predict the trajectory of agent. For both the models we utilize constant mean and an RBF kernel for the GP prior. Both the models were trained for 100 epochs. It can be observed from the Fig. 4 that the classical GP framework lacks the ability incorporate negative datapairs, which results in loosely fitted GP model. On the other hand, when trained with an additional constraints given by negative datapairs GP-NC fits tighter curve on the observed datapairs and avoiding all the negative datapairs shown in Fig 4b
Morcover, these set of experiments it is easy to demonstrate the impact of the values from Eq. (6) the GP regression. Decreasing the value of À results in reduction of influence of the negative
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Positive Targets
Negative Targets
Curve Fitted
Confidence
The figures compare trajectory prediction in 2D-virtual environment using the classical GP framework (a) vs. the GP-NC framework (b,c). The navigating through the forest and our aim is to avoid the roadblocks marked in 'red while maintaining car's proximity to the 'black' trajectory markers. The classical GP framework only uses the positive datapairs whereas our proposed GP-NC framework uses both the positive & negative datapairs for prediction of agent's trajectory. (h) depicts the GP-NC framework with the hyper parameter À 1 (c) depicts the NPPNCC framework with the hyper parameter À 0.1
Figure 4: Trajectory prediction with GP-NC regression framework:
datapairs can be observed from Fig b Fig However, it can be observed by juxtaposing both the figures (Fig and Fig Fig cc) that predicted mean trajectory by classical GP and GP-NC with A 0.1, the latter fits better to the black trajectory markers. Hence, for saftey critical applications like navigation GP-NC is superior than the classical GP in incorporating negative constraints.
5.3 REALWORLD DATASETS
We evaluated our GP-NC framework six real world datasets with the number of datapoints ranging from N ~ 1500, 5000, 15000. 50000, 450000 and the number of input dimension d E [3, 127]. Among the six datasets five of them are from UCI repository (Dua & Graff 2017) (Wine quality red, white, Elevators, Protein, and 3DRoad), while the sixth one is from Kaggle Competition (Prudential life insurance risk assessment). These datasets consists of two different kind of prediction/regression variables namely discrete variable and continuous variable. For discreet variable the value of elements lies between certain range, i.e., integer values lie between [0, 10). For continuous variable, the value of target regression can be any real number. Datasets (Prudential risk assessment, Wine quality red, white) are all discrete target variable datasets while datasets (Elevators, Protein and 3DRoad) have continuous target variables.
Random shuffling technique for creating negative datapairs for GP-NC: As we are only given positive regression target values in these datasets, we create eseudo-negative regression targets by randomly shuffling the labels and pairing them with the inputs to create negative datapairs. This generates a of valid datapairs as given the input only yxx can be associated as true regression value/label, we can assume whatever label we get by random shuffling as a negative target.
We compared model against the standard baselines of Exact GP (Gardner al. 2019 Wang et al. 2019). and Sparse GP methods like SVGP (Hensman et al. 2013) and PPGPR (Jankowiak al. 2019 For training the sparse methods, we used 1000 inducing points. We trained the models using Adam optimizer with a learning rate of 0.1 for 400 epochs on each dataset. For GP-NC framework, we used 200 negative datapairs. We maintained consistency all the models in all terms of maintaining a constant mean and RBF kernel for a GP prior.
Fig. 3 compares negative log likelihood values of various GP regression methods, in both classical and PNC frameworks, all six real datasets. The orange dots represents our methods while the blue dots in the plots depict the baselines. It can be observed that GP-NC framework outperforms the classical GP framework. Methods like SVGP- NC and Exact GP NC performs 0.2 nats better than bascline SVGP and Exact GP respectively. It is interesting to note that including negative datapairs by the 'random shuffling technique' is quite effective and we observe gains in terms of model performance.
average
Table compares runtime difference between the classical GP and the GP-NC to how the additional KL divergence term that accounts for negative datapairs affects the runtime of GP-NC framework. We train all the GP models for 50 epochs and measure the average excess time (A,)
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Figure 5: RMSE plots real world data (top- Exact GP; middle- SVGP; bottom PPGPR): Plots show the test RMSE for six univariate regression datasets (lower is better). Models are fitted by using cross validation on training data. Convergence GP-NC framework consistently faster than its classical GP counterpart for all the models.
value over 10 runs. can be observed from Table [ that the A, depends on the size of the dataset and also on the type of target regression variable. For the Exact GP and discrete target variables setting the A does not increase much with size of the dataset, however for continuous target variable there a considerable amount of increase. For SVGP model the increase in A, can be attributed to the size of the dataset. Overall, the average increase in training is not very significant. Thus, our experiments indicate that the added penalty term to likelihood term in Eq. (6) does not significantly affect the scalability of current scalable GP architectures.
Figure 5 shows the RMSE plots for Exact GP, SVGP, and PPGPR models with the classical GP juxtaposed on to the models with the negative constraint GP-NC framework. It can be observed from the plots that our GP-NC models converges faster, and better, than the classical GP models for the six univariate real world datasets. In addition, it is interesting to note that as the size of the data increases, the convergence curve of the gP-NC model becomes steeper.
CONCLUSION
We presented a novel and generic Gaussian Process regression framework GP-NC that incorporates negative constraints along with fitting the curve for the positive datapairs. Our key idea was to assume small blobs of Gaussian distribution on the negative datapairs. Then, while fitting the GP regression on the positive datapairs, our GP-NC framework simultaneously maximizes the KL divergence from the negative datapairs. Our work highlights the benefits of modeling the negative datapairs for GPs and our experiments support the effectiveness of our approach. We hope that this successful realization of the concept of negative datapairs for GP regression will be useful in variety of applications.
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Under review as a conference paper at ICLR 2021
A GP-NC FOR SCALABLE GP METHODS
We can replace the NLL term in Algorithm T by the log likelihood of the different scalable GP methods. We have a scalable implementation of the DKL update, so the entire Algorithm scales well with the input data size. It is straightforward to plug-in the class of scalable and Sparse gP regression models in the likelihood term of Algorithm (I) account for the negative datapairs in their formulation. In particular we review the SVGP model by (Hensman et al. 2013), which is popular scalable implementation of GPs. We also investigate a recent parametric Gaussian Process regressors (PPGPR) method by (Jankowiak et al. (219).. In this section. we follow the notations given in their respective research works and give their derivations of the log likelihood function here for the sake of completeness.
A.1 SVGP REGRESSION MODEL
(Hensman et al. 2013) proposed the Scalable Variational GP (SVGP) method. The key technical innovation the development of inducing point methods which review. By introducing inducing variables that depend on variational parameters {zm m==1 where M = dim(u) < N
LM
and with each 2m Rd. augment the GP prior follows:
X) + p(flu, X, Z)p(u(z)
We then appeal to Jensen's inequality and lower bound the log joint density over the targets and inducing variables:
logply, u|x, 2) log dfply (f)((())p(())
2 (((E()) [logply[() + logp(u)]
Σ
log.1 N(v k. K MM robe) 20 Tr-KKK logp(u)
1=1
where k; = k(x,, Z), KMM k(2, Z) and KINN is given by KINN KNN KNMK MM KMN
with KNM = K = k(X, 2). The essential characteristics of Eqn. 8 are that: i) it replaces expensive computations involving K KNN with cheaper computations like K MM that scale as ((M)); and ii) is amenable data subsampling, since the log likelihood and trace terms factorize sums over datapoints (yi, X:).
A.1.1 SVGP LIKELIHOOD FUNCTION
SVGP proceeds by introducing a multivariate Normal variational distribution q(u) = (mm, S). The parameters and are optimized using the ELBO (evidence lower bound), which the expectation of Eqn. w.r.t. q(u) plus an entropy term term [[(u]]:
Lavgp Eq(u) [lygply x/X, 2)] + H[(())
or(x,)
(10)
Σ
log Nog.((w/rrxx)) a
DKL (a(()))))(
20
where KL denotes the Kullback-Leibe divergence, (x(x)) is the predictive mean function given by (x,) = k.¹ KrrKKin and of Var[fi[x]] = Kta + k. K KKykk denotes the latent function variance.
Laugp which depends on m, S, Z, oobs and the various kernel hyperparameters 0, can then be maximized with gradient methods. We refer the resulting GP regression method as SVGP.
A.2 PPGPR-NC REGRESSION MODEL: LIKELIHIOD FUNCTION
Jankowiak et al. 2019) recently proposed parametric Gaussian Process regressors (PPGPR) method,
defer the reader Section (3.2) their paper for details about their likelihood function.
12


Reviews:
[{"review_id": "zb8mYTUEKvr", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 5, "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Summary\n--------\nThe paper addresses Gaussian process regression (GPR) over 'positive' and 'negative' data points, such that the regression model fit the former and avoid the latter. The 'negative' data points in this work are each represented by individually distributed Gaussian random variables, with known parameters (mean vector and variance).\nThe authors propose a novel GPR loss (6) for optimizing GP model parameters $\\theta$, striking a compromise between the regular GP loss and the KL divergence between the predictive GP distribution and the set of 'negative' data points. The approach is novel and an interesting take on the problem, and it has several beneficial properties which are investigated empirically.\n\nStrong points\n-------------\n1. Compared to previous work on GPR with 'positive' and 'negative' data points, the proposed solution is elegant in its simplicity, and scalable for predictions (since the 'negative' data is only used during kernel parameter optimization). I especially like the scalability aspect, since it would allow the use of a otherwise prohibitively large amount of 'negative' examples. This is useful in navigation, for example in contextes where the boundaries of static obstacles are commonly represented by potential fields.\n2. The experiments are fairly extensive and demonstrate several beneficial properties of the proposed method, such as fast learning convergence and compatibility with sparse GP methods.\n\nWeak points\n-------------\n1. The problem of GPR with both 'positive' and 'negative' data points (and in the context of robot navigation) has been investigated before [1], where a non-stationary kernel function is proposed. The approach in [1] should be addressed, and compared with, at least qualitatively.\n2. The experiments are not representing any situation where the distinction of 'positive' and 'negative' data points naturally occur. This makes it hard to assess how well the proposed method works for real problems of the kind the paper intends to address.\n3. The technical clarity is varied:\n     1. Equation (5) is incorrect? The two multivariate Gaussian distributions have different dimension (n vs m), and they do not share the same support ($X$ and $\\bar X$ can differ). Maybe it should be \"$p(y|\\theta, \\bar X)$\". But is this then entirely compatible with the prior sentence: $\\theta$ is in essence the \"GP regression model learned from the positive data pairs\" and yet it is stated that you maximize KL divergence between the GP regression model and the positive data pairs with respect to $\\theta$? If $\\theta$ is changed, then the first distribution in the KL divergence is no longer the GP model learned from the positive data points. I suggest that this section is made clearer and more precise. Equation (6) on the other hand corresponds to the stated proposal.\n   2. It is not obvious to me how you make use of the re-parameterization trick proposed by (Kingma & Welling, 2013).\n   3. Why do you not minimize the loss (6) directly, instead of first minimizing NLL and then maximizing KL at each iteration (Algorithm 1)?\n   4. What value does $\\lambda$ and $\\sigma_{neg}$ have in the experiments?\n4. The 'Random shuffling technique' is not at all clear to me. \n   1. How can you get a set of *valid* data pairs (i.e. $x_i$ and $y_i$) by shuffling the labels (\"$y_i$\")? \n   2. If randomly create pseudo-negative data points, especially using existing data points (that are either designated 'negative' or shuffled in some way to produce new input-output pairs as 'negative'), then isn't it likely given real data sets that these data points will be in conflict (e.g. overlap) with other 'positive' data points? If so, what do you mean with that these are *valid* and what does this mean for the experiment (more details on the limitations of the experiment and interpretations of the results should be present in the paper)?\n\nReason for score\n----------------\nAlthough an interesting take on the problem, the paper in its current form is lacking technical clarity, comparisons to directly related works, and the evaluation is weak in that it does not consider actual instances of the addressed problem in the empirical evaluation.\n\nMinor comments\n--------------\n\"to model the uncertainty by providing the predictive variance\" -> \"to model [target] uncertainty by providing predictive variance\"\n\n\"in both the positive and negative data pair.\" -> \"in both the positive and negative data pair set.\"\n\nPage 4: \"$P(y|\\theta, X)$\" -> \"$p(y|\\theta, X)$\"\n\n\"loglikelihood\" -> \"log likelihood\"\n\n--------------\n\n[1] S. Choi, E. Kim, K. Lee and S. Oh, \"Leveraged non-stationary Gaussian process regression for autonomous robot navigation,\" 2015 IEEE International Conference on Robotics and Automation (ICRA), Seattle, WA, 2015, pp. 473-478, doi: 10.1109/ICRA.2015.7139222.\n\nEdit: Markdown problem with nested lists\n\nEdit: Upgraded rating from 4 to 5\n\n-----\nThank you for addressing most of my concerns.\n\nI have increased my rating, but the paper still needs some work in my opinion. More specifically,\n* a more thorough comparison with [1], e.g. by including the quantitative comparison you were working on.\n* an improved representative experiment from the problem domain. The newly added one is a bit simplistic and not motivated enough. Although stated to be 2D, it is (as far as I can tell) indistinguishable from a 1D regression task? How do you handle the 2D output in GPR? How are the data points generated that are used as training data, and what could motivate such a generation of data in a plausibly real setting?", "extra_scores": {}, "extra_text": {}}, {"review_id": "jSWh4WAxmQZ", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Summary\n--\n\nThis paper in concerned with Gaussian process regression under constraints that aim to discourage the model from learning certain values (negative constraints). These are called negative data pairs, and the authors propose an extension to the standard GP methodology to incorporate these constraints in the model. This is done by iteratively training a standard GP and maximising the KL between the GP and blobs of the negative data pairs.\n\nThe problem tackled in the paper is of relevance in fields such as spatial statistics and robotics, with possible applications also in more general ML tasks. The paper could be of interest to a narrow audience at ICLR. The presentation is rather clear and the main idea is easy to follow.\n\n\nConcerns\n--\n\n1. My main concern is originality and novelty of the approach presented in the paper. The authors tackle an important practical problem (or limitation), but the presented contribution alone is small. The proposed approach for including negative data pairs is straightforward, and something that one might find in an application paper (where the main interest is in solving a task related to an application). \n\n2. Limitations. The limitations of the approach should have been discussed in detail. Questions related to uni- vs. multimodality (splitting), bias influenced by the parameter \\lambda, behaviour in multi-output (2d, like Fig. 1), would perhaps best have been illustrated with suitable simulated test cases and included as figures (including random samples from the GP posterior). Some of the issues are visible in Fig. 2, but these are not discussed in detail.\n\n3. Experiments. The experiments do not appear convincing. Rather than artificially changing standard test data sets to fit your task setting, it would have been more interesting to see actual problems, where the model class would have been beneficial.\n\n4. Presentation. Even if the method itself is easy to understand from how it is presented, the paper would deserve improvements in the presentation. As a practical suggestion: Background material could be presented more concisely, the methods section refined to be more concise, and the additional space used for improving experiments and discussion.\n\n5. From how Alg. 1 now reads, I would interpret \\lambda not to have any effect on the training. This should not be the case. Alg. 1 does not agree with the objective in Eq. (6).\n\n6. Minor: Typo in Eq. (4).", "extra_scores": {}, "extra_text": {}}, {"review_id": "S1zuJcxOC_M", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Summary:\nThis paper incorporates information of obstacles to avoid (e.g robot navigation trajectory in the room where the robot has to avoid items such as furniture) into Gaussian process regression fit. They call the obstacles, negative datapairs and the rest of data, positive datapairs. The aim is to have a GP where the probability of passing through the negative datapairs is low. The proposed method is called the Gaussian process with negative constraints (GP-NC). \n\nTo be able to fit the GP regression to positive datapairs and avoid negative datapairs, they maximize the KL-divergence between the distributions of GP learned from positive datpairs $(p(y|\\theta. X))$ and negative datapiars $(q(\\hat{y}|\\hat{X}))$ which will have a bound between $[0, \\inf]$.\nFor being able to maximize the KL-divergence with the marginal log-likelihood they change the scale of KL-divergence to the log scale and a parameter $\\lambda$. $\\lambda$ is a tradeoff between curve fitting and avoidance of the negative datapoints. KL term has an analytical solution since both distributions are Gaussians.\nThey compare their method against SVGP (Hensman et al., 2013) and PPGPR (Jankowiak et al., 2019) and the exact GP.\n\nComments and questions: \nThe paper is well-written and easy to follow. This paper is also technically sound and to the best of my knowledge is novel and relevant to the community.\n\nIn figure two you have mentioned you sampled the inducing inputs randomly from the whole range of training inputs. Did you choose the same inducing inputs for the SVGP and SVGP-NC? \n\nIn the error calculation, standardized mean squared error (SMSE) could be a better choice than RMSR since the former incorporates variance information as well.\n\nIn the experiment section, all negative datapairs are synthetically made. I was wondering if you could apply your method to a dataset with real negative datapiars (like a robot in the room avoiding obstacles)? \n\nIn figure 3 (3DRoad) the approximate results are better than the exact GP. Is this because of the scale of the data and not being able to converge?\n\nMiscellaneous comments:\nIn Table 1 row 2 should be wine quality - white\n\n################### After the rebuttal ################\n\nI thank the authors for addressing the issues that were raised. The paper is indeed addressing a very practical issue in the ML community.\nAfter reading other reviewers' comments and concerns I decreased my score. \n", "extra_scores": {}, "extra_text": {}}, {"review_id": "FqQrALQKeJG", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "SUMMARY \n\nThis paper introduces GP-NC, a new methodology that allows Gaussian processes to stay far from certain data points (referred to as negative datapairs). These negative datapairs complement the standard training points to be fitted. The authors show enhanced performance when using this technique with standard GPs, SVGP and PPGPR.\n\n###################################################\n\nPROS\n\n1) I am not aware of previous works in the GP literature dealing with this scenario, i.e. data pairs to be avoided during training.\n\n####################################################\n\nCONS\n\n1) I think this work is not sufficiently motivated. The authors claim that the brand-new negative datapairs can be useful for modelling navigation problems with GPs. However, this type of problem is not addressed in the experimental section. I would expect to see how the proposed method compares to the previous approaches that have modelled the navigation problem with GPs. \n\n2) In fact, due to the lack of real problems that require the presented technique, in the experimental section they just simulate some negative datapairs (following the so-called random shuffling technique). Moreover, this simulation technique is pretty poor in my view, since different inputs could have very similar outputs, which may lead to a simulated datapair almost identical to a standard training datapair. \n\n3) Some arguments are too vague/ambiguous. For instance, the equivalence between the scale of magnitude of both terms in eq. (5). It is stated that, by using the so-called logarith trick, both terms have the same \"scale of magnitude\". How is that defined? and how can the equivalence be proved? Or the sentence that says \"A Gaussian process is a Bayesian non-parametric approach that fits Gaussian distributions to functions\". \n\n4) Low quality writing in general. I have come across many typos and sentences that are not well written. See e.g.:\n* caption of Fig. 1 \"is been given\".\n* after eq. (5) \"minimize\",\n* \"Hewing et al. (2020) are some of the works using sampling based techniques for trajectory prediction\"\n* \"We trained a sparse SVGP model to regress a curve on the using the classical GP framework and the one with negative constraints GP-NC\"\n* Caption of Figure 4: it says five datasets, but there are six\n* Many references are missing the journal/conference/arxiv reference. Format not homogeneous across references.\n\n\n############################################\n\nAFTER THE REBUTTAL\n\nI have decided to keep my score unchanged. The authors include a new \"road navigation experiment\" in Section 5.2. However, I have several concerns about it:\n* This is a synthetic experiment that the authors have created themselves. Therefore, it does not address my concern on seeing actual real-world problems that can be solved with this approach.\n* The results of GP-NC (the proposed approach) and GP are very similar. I don't really see that the predictive mean of GP-NC avoids the \"negative samples\" more than the standard GP.\n* In fact, the design of this experiment is not clear to me. They say at the beginning that they use the present location ($x,y$) to predict the next location ($\\hat x, \\hat y)$. How do they model these two outputs? How do they obtain the plotted error bars for this model? This is not sufficiently explained. My impression is that they are just fitting a standard 1-dimensional GP being the input the x-dimension and the output the y-dimension. ", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper is very pleasant to read. The reviewers also like the key idea discussed and find the targeted application interesting and practical. However, after reading the indeed interesting motivation, all four reviewers expected to see more from the evaluation section, including more challenging and realistic set-ups and clearer gains over standard methods. The reviewers also discuss how both the navigation problem as well as the GP constraint problem have been tackled in the past, often in combination (e.g. reference [1] by R1). Therefore, it would be needed to see additional experimental evaluation in line with those previous works.
Decision: Reject