Multi-task regression using minimal penalties

By Matthieu Solnon, Sylvain Arlot, Francis R. Bach. 2011

60 citations recorded by the source corpus.

In this paper we study the kernel multiple ridge regression framework, which we refer to as multitask regression, using penalization techniques. The theoretical analysis of this problem shows that the key element appearing for an optimal calibration is the covariance matrix of the noise between the different tasks. We present a new algorithm to estimate this covariance matrix, based on the concept of minimal penalty, which was previously used in the single-task regression framework to estimate the variance of the noise. We show, in a non-asymptotic setting and under mild assumptions on the target function, that this estimator converges towards the covariance matrix. Then plugging this estimator into the corresponding ideal penalty leads to an oracle inequality. We illustrate the behavior of our algorithm on synthetic examples.

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1. Yu Zhang
2. Shaogang Gong
3. Tao A. Xiang
4. Yu Zhang
5. Qiang Yang
6. Chen Change Loy
7. Xiaogang Wang
8. Ke Chen

Citations in the corpus

• A Survey on Multi-Task Learning (2017, methodology): “For example, the methods in [73], [74], [75], [76], [77], which utilize a priori information on task relations, directly learn W and b by defining g(Σ) = 0, some task clustering methods [65], [69] identify task clusters by assuming that Σ has a block structure, several task relation learning methods”
• Multi-Task Learning and Algorithmic Stability (2015, methodology): “…works, such as (Evgeniou and Pontil 2004; Evgeniou, Micchelli, and Pontil 2005; Kato et al. 2007) which assume that Ω is given as a priori information and (Jacob, Bach, and Vert 2008; Zhang and Yeung 2010a; Solnon, Arlot, and Bach 2012; Solnon 2013) which aim to learn Ω from data in different ways.”
• A Bias and Variance Analysis for Multistep-Ahead Time Series Forecasting (2016, unknown): “This is a special case of the broader concept of multitask regression [41] or multiresponse regression [42], developed in the machine learning and the statistics literature.”
• Parallel Multi-task Learning (2015, methodology): “We classify regularized multi-task algorithms into two main categories: learning with feature covariance [5], [7], [6], [8], [9], [10], [11], [19], [20], [21], [22] and learning with task relations [2], [3], [4], [13], [15], [14], [17], [18], [16].”
• Flexible Clustered Multi-Task Learning by Learning Representative Tasks (2016, methodology): “Several previous works also studied the non-linear extension of MTL methods [3], [16], [20], [30], [45], [55], [57].”
• Cumulative Attribute Space for Age and Crowd Density Estimation (2013, methodology): “Following established design principle of multi-task learning [2, 3, 18, 30], we formulate the following multioutput attribute learning problem.”