Coherence-based Label Propagation over Time Series for Accelerated Active Learning

1 INTRODUCTION . A time series is a sequence of data points at successive timestamps . Supervised learning ( e.g. , classification ) with a time series requires the label of every data point , but unfortunately labels are often missing and hard to obtain due to lack of domain-specific knowledge ( Shen et al. , 2018 ; Malhotra et al. , 2019 ; Li et al. , 2020 ) . It is worse for a time series collected for an extended length of time , as manually labeling so many data points is labor-intensive and time-consuming ( Perslev et al. , 2019 ; Tonekaboni et al. , 2021 ) . Active learning ( Settles , 2009 ) , a method that iteratively selects the most informative data point and queries a user for its label , can mitigate the high labeling cost . However , most active learning methods are not geared for time-series data , as they assume that data points are independent of one other ( Sener & Savarese , 2018 ; Yoo & Kweon , 2019 ; Ash et al. , 2020 ) , which is obviously not true in time-series data . Time-series data typically has the characteristic of temporal coherence ; that is , temporally consecutive data points tend to have the same label ( Wang et al. , 2020 ; Barrow et al. , 2020 ; Ishikawa et al. , 2021 ) . Let us refer to a sub-sequence of temporally coherent data points as a segment . For example , in motion-sensing time-series data , a segment consists of data points with the same motion status ( e.g. , walking , running ) . This temporal coherence of a segment can be exploited in time-series active learning . Specifically , when the label of a certain data point is obtained from a user , the same label can be propagated to other data points in the same segment . One challenge here is that the segment length is not known but needs to be estimated . If it is too short , unnecessarily frequent queries are issued ; if too long , data points on the fringe of the segment are labeled incorrectly , consequently damaging the learning performance ( e.g. , classification accuracy ) . Thus , accurate estimation of the segments is important to enable the label propagation to achieve the maximum learning performance with the minimum number of queries . This paper addresses the label propagation segment estimation problem in time-series active learning through a novel framework called Temporal Coherence-based Label Propagation ( TCLP ) . Figure 1 illustrates the overall workflow in the time-series active learning centered on TCLP . TCLP receives the class probabilities ( i.e. , softmax output ) for the label of each data point from a classifier model and estimates the extent of the segment to propagate the label . This estimation is challenging in a real time series , as the classifier model output is uncertain and the time-series segments are unknown . TCLP meets this challenge by taking advantage of the temporal coherence via a quadratic plateau model ( Moltisanti et al. , 2019 ) , by fitting it to the classifier model output to smooth out the fluctuations of class probabilities across consecutive data points . To the best of our knowledge , TCLP is the first that performs label propagation for time-series active learning . The previous work closest to ours is pseudo-labeling in single-timestamp supervised learning , where labels are known for at least one data point in each segment ( Moltisanti et al. , 2019 ; Ma et al. , 2020 ; Li et al. , 2021 ) . The approximate location and true class of a segment must be known in their work , which is often impractical in the real world . Moreover , the known labels are relatively dense in single-timestamp supervised learning , but they are very sparse in active learning—typically , no more than 5 % of segments in our experiments . Thus , finding the boundaries between segments is more challenging in active learning than in single-timestamp supervised learning . To cope with the sparsity of labeled data points , TCLP performs sparsity-aware label propagation by exploiting temperature scaling ( Guo et al. , 2017 ) and plateau regularization . Contributions of this paper are summarized as follows : • It proposes a novel time-series active learning framework equipped with a sparsity-aware label propagation within an accurately estimated segment . • It verifies the merit of TCLP through extensive experiments . The classification accuracy is improved by up to 7.1 times with TCLP compared to without label propagation . Moreover , TCLP works with any query selection strategy including core-set sampling ( Sener & Savarese , 2018 ) and BADGE ( Ash et al. , 2020 ) , boosting the effect of individual labeling . 2 RELATED WORK . 2.1 ACTIVE LEARNING . Active learning is a special case of machine learning that ‘ actively ’ queries a user for the labels of data points , to the effect of using fewer labels to achieve the same learning performance . Recent studies have focused on developing such query strategies for machine learning based on deep neural networks ( Settles , 2009 ; Ren et al. , 2020 ) . These approaches exploit prediction probabilities ( Beluch et al. , 2018 ) , embeddings ( Sener & Savarese , 2018 ) , gradients ( Ash et al. , 2020 ) , and losses ( Yoo & Kweon , 2019 ) from deep neural networks to estimate the impact of each unlabeled data point if it were to be labeled . However , these methods are not suitable for time-series data , because they assume that data points are independent . Several methods have been developed for time-series or sequence data , but most of them are applicable to only segmented time-series data under the assumption that a time series is already divided into labeled and unlabeled segments . Treating these segments as independent and identically distributed , these methods simply apply existing active learning frameworks to the segments . For example , He et al . ( 2015 ) select unlabeled segments that are far from labeled segments to maximize diversity ; Peng et al . ( 2017 ) select unlabeled segments with distinctive patterns to maximize diversity ; and Zhang et al . ( 2017 ) select unlabeled segments with high gradients to consider uncertainty for sentence classification . In addition , new neural network architectures or measures have been developed for sequence-data applications such as named entity recognition ( Shen et al. , 2018 ) , video action recognition ( Wang et al. , 2018 ) , and speech recognition ( Malhotra et al. , 2019 ) . None of these methods is applicable to our problem , which handles unsegmented time-series data . 2.2 PSEUDO-LABELING . Pseudo-labeling has been actively studied for label-deficient learning environments , such as semisupervised learning , to exploit unlabeled data points in training a classifier ( Lee et al. , 2013 ; Liu et al. , 2019 ) . In general , a pseudo-label is given to an unlabeled data point based on the predictions from a classifier trained with labeled data points . Confidence-based methods create a pseudo-label if it is confidently predicted by a classifier ( Lee et al. , 2013 ) . Consistency-based methods create a pseudo-label if it is consistently predicted for the original and augmented data points ( Sajjadi et al. , 2016 ; Rizve et al. , 2021 ) . Graph-based methods propagate pseudo-labels from labeled data points ( nodes ) to unlabeled data points based on a similarity graph constructed from the features of all data points ( Shi et al. , 2018 ; Liu et al. , 2019 ; Wagner et al. , 2018 ) . However , these methods are not designed for time-series data , and therefore are not directly applicable to our problem . Coherence-based methods are developed for single-timestamp supervised learning for unsegmented time-series data ; they assume that at least one data point in each segment is given a true class label through weak-supervision . Ma et al . ( 2020 ) propose probability thresholding propagation ( PTP ) , which propagates known labels bidirectionally unless the predicted class probability for each data point is decreased by more than a threshold . Deldari et al . ( 2021 ) propose embedding similarity propagation ( ESP ) , which propagates known labels bidirectionally unless the embedding of each data point changes rapidly . Recently , Moltisanti et al . ( 2019 ) adopt a plateau model that represents class probabilities across consecutive data points , where a plateau model is constructed for each labeled data point and fitted to the classifier output ; a known label is propagated as long as the value of a plateau model is higher than a threshold . While this work shares the idea of using a plateau model with our work , using the plateau model as it is for active learning results in performance degradation owing to the difference in the density of known labels , as will be shown in Section 4 . 3 TCLP : TEMPORAL COHERENCE-BASED LABEL PROPAGATION . 3.1 PRELIMINARIES AND PROBLEM SETTING . Active learning : LetD = { ( xt , yt ) , t ∈ T } be a time series where T is the index set of timestamps ; xt is a multi-dimensional data point at timestamp t , and yt is one of the class labels if xt is labeled or null otherwise . Let DL ⊆ D be a labeled set , i.e. , a set of labeled data points , and DU ⊆ D be an unlabeled set , i.e. , a set of unlabeled data points , where DU ∪ DL = D. At each round of active learning , b data points are selected from DU by a query selection strategy , such as entropy sampling ( Wang & Shang , 2014 ) , core-set selection ( Sener & Savarese , 2018 ) , and BADGE ( Ash et al. , 2020 ) , and their ground-truth labels are obtained from a user ; these newly-labeled b data points are then removed from DU and added to DL . After DL is updated , a classifier model fθ is re-trained using the updated labeled set . Label propagation : Given a data point at timestamp tq and its label , ( xtq , ytq ) , obtained from a user in response to a query , TCLP assigns the label ytq to nearby data points in the timestamp range [ ts : te ] ( ts ≤ tq ≤ te ) estimated according to its temporal coherence property criteria . We call the sub-sequence of data points in [ ts : te ] an estimated segment at tq . There are two properties : ( i ) accuracy , which indicates that as many data points in the segment as possible should have the same ground-truth label ytq ; and ( ii ) coverage , which indicates that the length of the segment ( te − ts ) should be as long as possible . More formally , we estimate the segment for tq by ts , te = arg min t′s , t ′ e 1 t′e − t′s t′e∑ t=t′s ( 1− fθ ( xt ) [ ytq ] ) , ( 1 ) where t′s ≤ tq ≤ t′e holds , fθ ( xt ) is the softmax output vector of the classifier model at timestamp t , and fθ ( xt ) [ ytq ] is the estimated probability of the label ytq . In Equation ( 1 ) , the accuracy is achieved by minimizing the sum of errors in the numerator , and the coverage is achieved by maximizing the candidate segment length in the denominator . Note that estimated probabilities are used to calculate the errors , since the true probabilities are not known . Once segment estimation is done , all data points in the estimated segment ( i.e. , in [ ts : te ] ) are removed fromDU and added toDL with the label ytq , thus doing coherence-based label propagation . At each round of active learning , the segment estimation repeats for each of the b queried data points ; as a result , the size of DL is increased by b times the average length of estimated segments . Besides , we allow the data points in [ ts : te ] , except tq , to be queried again in subsequent rounds so that the propagated labels can be refined subsequently .

Summary: The goal of the paper is to improve active learning for time-series.  The goal is to learn a function from x to y, with data coming from a time series (x_t) and (y_t). It is assumed that (y_t) is piecewise constant. At the beginning no (y_t) are known, but query can be made in a active learning framework.  This paper tries to find the pieces where (y_t) is constant based on the features (x_t). It is done using the labeled timestamps (y_t_q) at (t_q) and the current learnt model f from X to Y with a thresholding rule based on f(x_t) around the (t_q). This gives segments on which (y_t) is supposed constant egal to the (y_t_q). Those labels are added as if they were true labels. Then the model for f is updated, and some active query are made based on any model. The loop is repeated for a given number of rounds.