RMTL: an R library for multi-task learning

Abstract

Motivation

Multi-task learning (MTL) is a machine learning technique for simultaneous learning of multiple related classification or regression tasks. Despite its increasing popularity, MTL algorithms are currently not available in the widely used software environment R, creating a bottleneck for their application in biomedical research.

Results

We developed an efficient, easy-to-use R library for MTL (www.r-project.org) comprising 10 algorithms applicable for regression, classification, joint predictor selection, task clustering, low-rank learning and incorporation of biological networks. We demonstrate the utility of the algorithms using simulated data.

Availability and implementation

The RMTL package is an open source R package and is freely available at https://github.com/transbioZI/RMTL. RMTL will also be available on cran.r-project.org.

Supplementary information

Supplementary data are available at Bioinformatics online.

1 Introduction

Multi-task learning (MTL) is a machine learning technique that explores and exploits the relatedness across a set of different learning tasks. Since its inception (Caruana, 1998), MTL has been used in numerous data-intensive research areas, including biomedical informatics (Feriante, 2015; Li et al., 2016; Widmer and Ratsch, 2012; Xu et al., 2011; Yuan et al., 2016; Zhou et al., 2013), speech and natural language processing [i.e. (Wu et al., 2015)], image processing and computer vision [i.e. (Wang et al., 2009)], as well as web based applications [i.e. (Chapelle et al., 2010)].

A strong motivation to develop biomedical MTL applications stems from the necessity to integrate diverse data sources to explore the biological underpinning of complex illnesses, such as schizophrenia. Previous research has already shown that for such illnesses, integrative multi-omics open a new avenue for identification of etiological mechanisms, for example by taking into account genetic, expression and methylation data simultaneously [i.e. (Lin et al., 2014)]. For such applications, multi-task learning offers the possibility to directly explore illness-related biological profiles that are linked across data modalities and therefore a new route toward the identification of biomarker signatures.

Previous implementations of MTL have focused on knowledge transfer via regularization (Zhou et al., 2011), Bayesian methods (Greenlaw et al., 2017) or deep architectures (Yang and Hospedales, 2016). Here, we developed the first R library for MTL, offering a comprehensive machine learning pipeline that covers several types of MLT algorithms and can be easily applied to high-dimensional data.

In the following section, we briefly describe the RMTL package, including the implemented MTL methods (for detailed information see Supplementary Methods). The results section describes the application of the algorithms on a simulation study, to demonstrate the performance and interpretability of the respective models.

2 Materials and methods

This package provides an automated, simple-to-use implementation of MTL, comprising five classification and five regression algorithms, which share knowledge across tasks according to different priors via regularization. All algorithms aim to minimize the same objective:

$$\underset{W}{\min }\sum _i^t\frac{1}{n_i}\mathcal{L}\left(W_i\right|X_i, Y_i)+\Omega \left(W\right)$$

where $L\left(\circ \right)$ is the loss function (logistic loss for classification or least square loss for regression). $X=\left\{X_i\right\}, Y=\left\{Y_i\right\}$are sets of predictor matrices and the corresponding responses for all $t$ tasks where $X_i\in R^{n_i\times p}$ and $Y_i\in R^{n_i\times 1}$ is the predictor matrix and the response vector of task $i\in \{1, 2,\dots , t\}$⁠. Accordingly, $n_i$ and $p$ refer to the number of subjects and predictors (all tasks share the same predictor space) of task $i$⁠, respectively. Moreover, $W=R^{p\hspace{0.5em}\times \hspace{0.5em}t}$ is the coefficient matrix for all tasks, where $W_i$⁠, the $i$th column of$W$⁠, is the coefficient vector for task $i$⁠.

Knowledge transfer among tasks is achieved via a convex term $\Omega \left(W\right)$that jointly modulates models according to specific functionalities. In this package, five common regularization techniques are implemented to suit different applications, i.e. sparse structure, joint predictor selection, low-rank structure, network constraint for task relatedness and task clustering. Here, we refer to the above regularization strategies as MTL_Lasso, MTL_L21, MTL_Trace, MTL_Graph and MTL_CMTL, in the same sequence. These strategies can be broadly categorized into two classes: strategies for predictor selection (MTL_Lasso and MTL_L21) and strategies for task relatedness exploration (MTL_Graph, MTL_Trace and MTL_CMTL). While the former class explores sparse patterns are explored over the predictor space, the latter class exploits task relatedness based on additional assumptions. For all algorithms, we implemented a solver based on the accelerated gradient descent method (Nesterov, 2013). To solve the non-smooth and convex regularization, the proximal operator (Parikh and Boyd, 2014) was applied. Overall, the solver achieves a complexity of O(1/k²), which is optimal among first-order gradient methods.$$Further methodological details are shown in the Supplementary Methods.

3 Results

Predictive performance and model interpretability of the implemented algorithms were explored using simulated data. The simulated datasets were constructed by the ground truth model$W$⁠, which is specified for a given prior (Supplementary Fig. S1). We compared the ground truth and the learnt model as an indicator of model interpretability. For predictive comparison, the primary baseline method was the conventional lasso, which reflects single task learning performance. We further applied MTL with lasso (MTL_Lasso), to explore the effect of inappropriate prior choice as a second baseline method.

3.1 Model interpretability

Supplementary Figure S1a shows the coefficient matrix of MTL_Lasso and MTL_L21 and demonstrates that the number of predictors identified by MTL_Lasso was approximately half the number of ground truth predictors. This may be due to the fact that highly correlated predictors exist in the high-dimensional space (Zou and Hastie, 2005). As a consequence and similar to conventional Lasso, MTL_Lasso tended to select one among several correlated predictors. Despite this, 75% (precision) of selected predictors were ground truth predictors. For MTC_L21, the ground truth was highly sparse: only 40 out of 400 predictors were active predictors for all tasks. The simulation demonstrates that 39 of the predictors were successfully identified (sensitivity: 97.5%), with a precision of 72%. These results indicate that MTL algorithms could successfully identify ground truth predictors.

The relatedness of tasks was represented by pairwise correlation between models. Supplementary Figure S1b shows that all methods were able to capture correctly the pairwise relatedness compared to the ground truths. Particularly, MTL_Graph incorporated a strong network prior such that the “in-group” differences became zero. This may be because the network prior provided the most complete information about task relatedness among all priors.

3.2 Predictive performance

Supplementary Figure S2 indicates that conventional Lasso failed to yield accurate predictions on all simulated datasets except when using the $l_{21}$prior. Compared to this baseline, the MTL models improved the accuracy by 18.7% on average. The MTL_Lasso incorporating an inappropriate prior achieved an average accuracy of 67% and was substantially inferior to MTL models with appropriate priors (average accuracy: 79.2%).

4 Conclusion

In this study, we developed an R library for multi-task learning comprising 10 algorithms incorporating five different priors. MTL models outperformed two baseline methods when applied on simulated data. High model-interpretability was observed in terms of predictor selection and task-relatedness compared to the respective ground truths.

Funding

This study was supported by the Deutsche Forschungsgemeinschaft (DFG), SCHW 1768/1-1. In addition, this work is supported in part by the National Science Foundation under grants IIS-1615597 (to JZ) and IIS-1749940 (to JZ).

Conflict of Interest: none declared.