Robust Unsupervised Multi-task and Transfer Learning on Gaussian Mixture Models

Unsupervised learning that learns patterns from unlabeled data is a prevalent problem in statistics and machine learning. Clustering is one of the most important problems in unsupervised learning, where the goal is to group the observations based on some metrics of similarity. Researchers have developed numerous clustering methods including $k$-means (Forgy, , 1965), $k$-medians (Jain and Dubes, , 1988), spectral clustering (Ng et al., , 2001), and hierarchical clustering (Murtagh and Contreras, , 2012), among others. On the other hand, clustering problems have been analyzed from the perspective of the mixture of several probability distributions (Scott and Symons, , 1971). The mixture of Gaussian distributions is one of the simplest models in this category and has been widely applied in many real applications (Yang and Ahuja, , 1998; Lee et al., , 2012).

In the binary Gaussian mixture models (GMMs) with common covariances, each observation $Z\in\mathbb{R}^{p}$ comes from the following mixture of two Gaussian distributions:

where $w\in(0,1)$, $\bm{\mu}_{1}\in\mathbb{R}^{p}$, $\bm{\mu}_{2}\in\mathbb{R}^{p}$ and $\bm{\Sigma}\succ 0$ are parameters. This is the same setting as the linear discriminant analysis (LDA) problem in classification (Hastie et al., , 2009), except that the label $Y$ is unknown in the clustering problem, while it is observed in the classification case. It has been shown that the Bayes classifier for the LDA problem is

where $\bm{\beta}=\bm{\Sigma}^{-1}(\bm{\mu}_{2}-\bm{\mu}_{1})\in\mathbb{R}^{p}$ and $\delta=\bm{\beta}^{\top}(\bm{\mu}_{1}+\bm{\mu}_{2})/2$. Note that $\bm{\beta}$ is usually referred to as the discriminant coefficient (Anderson, , 1958; Efron, , 1975). Naturally, this classifier is useful in clustering too. In clustering, after learning $w$, $\bm{\mu}_{1}$, $\bm{\mu}_{2}$ and $\bm{\beta}$, we can plug their estimators into (3) to group any new observation $Z^{\textup{new}}$. Generally, we define the mis-clustering error rate of any given clustering method $\mathcal{C}$ as

where $\pi$ is a permutation function, $Y^{\textup{new}}$ is the label of a future observation $Z^{\textup{new}}$, and the probability is taken w.r.t. the joint distribution of $(Z^{\textup{new}},Y^{\textup{new}})$ based on parameters $w$, $\bm{\mu}_{1}$, $\bm{\mu}_{2}$ and $\bm{\Sigma}$. Here the error is calculated up to a permutation due to the lack of label information. It is clear that in the ideal case where the parameters are known, $\mathcal{C}(\cdot)$ in (3) achieves the optimal mis-clustering error. Multi-cluster Gaussian mixture models with $R\geq 3$ components can be described in a similar way. We provide the details in Section S.1 of the supplementary materials.

There is a large volume of published studies on learning a GMM. The vast majority of approaches can be roughly divided into three categories. The first category is the method of moments, where the parameters are estimated through several moment equations (Pearson, , 1894; Kalai et al., , 2010; Hsu and Kakade, , 2013; Ge et al., , 2015). The second category is the spectral method, where the estimation is based on the spectral decomposition (Vempala and Wang, , 2004; Hsu and Kakade, , 2013; Jin et al., , 2017). The last category is the likelihood-based method including the popular expectation-maximization (EM) algorithm as a canonical example. The general form of the EM algorithm was formalized by Dempster et al., (1977) in the context of incomplete data, though earlier works (Hartley, , 1958; Hasselblad, , 1966; Baum et al., , 1970; Sundberg, , 1974) have studied EM-style algorithms in various concrete settings. Classical convergence results on the EM algorithm (Wu, , 1983; Redner and Walker, , 1984; Meng and Rubin, , 1994; McLachlan and Krishnan, , 2007) guarantee local convergence of the algorithm to fixed points of the sample likelihood. Recent advances in the analysis of EM algorithm and its variants provide stronger guarantees by establishing geometric convergence rates of the algorithm to the underlying true parameters under mild initialization conditions. See, for example, Dasgupta and Schulman, (2013); Wang et al., (2014); Xu et al., (2016); Balakrishnan et al., (2017); Yan et al., (2017); Cai et al., (2019); Kwon and Caramanis, (2020); Zhao et al., (2020) for GMM related works. In this paper, we propose modified versions of the EM algorithm with similarly strong guarantees to learn GMMs, under the new context of multi-task and transfer learning.

Multi-tasking is an ability that helps people pay attention to more than one task simultaneously. Moreover, we often find that the knowledge learned from one task can also be useful in other tasks. Multi-task learning (MTL) is a learning paradigm inspired by the human learning ability, which aims to learn multiple tasks and improve performance by utilizing the similarity between these tasks (Zhang and Yang, , 2021). There has been numerous research on MTL, which can be classified into five categories (Zhang and Yang, , 2021): feature learning approach (Argyriou et al., , 2008; Obozinski et al., , 2006), low-rank approach (Ando et al., , 2005), task clustering approach (Thrun and O’Sullivan, , 1996), task relation learning approach (Evgeniou and Pontil, , 2004) and decomposition approach (Jalali et al., , 2010). The majority of existing works focus on the use of MTL in supervised learning problems, while the application of MTL in unsupervised learning, such as clustering, has received less attention. Zhang and Zhang, (2011) developed an MTL clustering method based on a penalization framework, where the objective function consists of a local loss function and a pairwise task regularization term, both of which are related to the Bregman divergence. In Gu et al., (2011), a reproducing kernel Hilbert space (RKHS) was first established, and then a multi-task kernel k-means clustering was applied based on that RKHS. Yang et al., (2014) proposed a spectral MTL clustering method with a novel $\ell_{2,p}$-norm, which can also produce a linear regression function to predict labels for out-of-sample data. Zhang et al., (2018) suggested a new method based on the similarity matrix of samples in each task, which can learn the within-task clustering structure as well as the task relatedness simultaneously. Marfoq et al., (2021) established a new federated multi-task EM algorithm to learn the mixture of distributions and provided some theory on the convergence guarantee, but the statistical properties of the estimators were not fully understood. Zhang and Chen, (2022) proposed a distributed learning algorithm for GMMs based on transportation divergence when all GMMs are identical. In general, there are very few theoretical results about unsupervised MTL.

Transfer learning (TL) is another learning paradigm similar to multi-task learning but has different objectives. While MTL aims to learn all the tasks well with no priority for any specific task, the goal of TL is to improve the performance on the target task using the information from the source tasks (Zhang and Yang, , 2021). According to Pan and Yang, (2009), most of TL approaches can be classified into four categories: instance-based transfer (Dai et al., , 2007), feature representation transfer (Dai et al., , 2008), parameter transfer (Lawrence and Platt, , 2004) and relational-knowledge transfer (Mihalkova et al., , 2007). Similar to MTL, most of the TL methods focus on supervised learning. Some TL approaches are also developed for the semi-supervised learning setting (Chattopadhyay et al., , 2012; Li et al., , 2013), where only part of target or source data is labeled. There are much fewer discussions on the unsupervised TL approaches ¹¹1There are different definitions for unsupervised TL. Sometimes people call the semi-supervised TL an unsupervised TL as well. We follow the definition in Pan and Yang, (2009) here., which focuses on the cases where both target and source data are unlabeled. Dai et al., (2008) developed a co-clustering approach to transfer information from a single source to the target, which relies on the loss in mutual information and requires the features to be discrete. Wang et al., (2008) proposed a TL discriminant analysis method, where the target data is allowed to be unlabeled, but some labeled source data is necessary. In Wang et al., (2021), a TL approach was developed to learn Gaussian mixture models with only one source by weighting the target and source likelihood functions. Zuo et al., (2018) proposed a TL method based on infinite Gaussian mixture models and active learning, but their approach needs sufficient labeled source data and a few labeled target samples.

There are some recent studies on TL and MTL under various statistical settings, including high-dimensional linear regression (Xu and Bastani, , 2021; Li et al., 2022b, ; Zhang et al., , 2022; Li et al., 2022a, ), high-dimensional generalized linear models (Bastani, , 2021; Li et al., , 2023; Tian and Feng, , 2023), functional linear regression (Lin and Reimherr, , 2022), high-dimensional graphical models (Li et al., 2022b, ), reinforcement learning (Chen et al., , 2022), among others. The recent work Duan and Wang, (2023) developed an adaptive and robust MTL framework with sharp statistical guarantees for a broad class of models. We discuss its connection to our work in Section 2.

Our main contributions in this work can be summarized in the following:

1. (i)

   We develop efficient polynomial-time iterative procedures to learn GMMs in both MTL and TL settings. These procedures can be viewed as adaptations of the standard EM algorithm for MTL and TL problems.

2. (ii)

   The developed procedures come with provable statistical guarantees. Specifically, we derive the upper bounds of their estimation and excess mis-clustering error rates under mild conditions. For MTL, it is shown that when the tasks are close to each other, our method can achieve better upper bounds than those from the single-task learning; when the tasks are substantially different from each other, our method can still obtain competitive convergence rates compared to single-task learning. Similarly for TL, our method can achieve better upper bounds than those from fitting GMM only to target data when the target and sources are similar, and remains competitive otherwise. In addition, the derived upper bounds reveal the robustness of our methods against a fraction of outlier tasks (for MTL) or outlier sources (for TL) from arbitrary distributions. These guarantees certify our procedures as adaptive (to the unknown task relatedness) and robust (to contaminated data) learning approaches.

3. (iii)

   We derive the minimax lower bounds for parameter estimation and excess mis-clustering errors. In various regimes, the upper bounds from our methods match the lower bounds (up to small order terms), showing that the proposed methods are (nearly) minimax rate optimal.

4. (iv)

   Our MTL and TL approaches require the initial estimates for different tasks to be “well-aligned”, due to the non-identifiability of GMM. We propose two pre-processing alignment algorithms to provably resolve the alignment problem. Similar problems arise in many unsupervised MTL settings. However, to our knowledge, there is no formal discussion of the alignment issue in the existing literature on unsupervised MTL (Gu et al., , 2011; Zhang and Zhang, , 2011; Yang et al., , 2014; Zhang et al., , 2018; Dieuleveut et al., , 2021; Marfoq et al., , 2021). Therefore, our rigorous treatment of the alignment problem is an important step forward in this field.

The rest of the paper is organized as follows. In Section 2, we first discuss the multi-task learning problem for binary GMMs, by introducing the problem setting, our method, and the associated theory. The above-mentioned alignment problem is discussed in Section 2.4. We present a simulation study in Section 3 to validate our theory. Finally, in Section 4, we point out some interesting future research directions. Due to the space limit, the extension to multi-cluster GMMs, additional numerical results, a full treatment of the transfer learning problem, and all the proofs are delegated to the supplementary materials.

We summarize the notations used throughout the paper here for convenience. We use bold capital letters (e.g., $\bm{\Sigma}$) to denote matrices and use bold small letters (e.g., $\bm{x}$, $\bm{y}$) to denote vectors. For a matrix $\bm{A}=[a_{ij}]_{p\times q}\in\mathbb{R}^{p\times q}$, its 2-norm or spectral norm is defined as $\|\bm{A}\|_{2}=\max_{\bm{x}:\|\bm{x}\|_{2}=1}\|\bm{A}\bm{x}\|_{2}$. If $q=1$, $\bm{A}$ becomes a $p$-dimensional vector and $\|\bm{A}\|_{2}$ equals its Euclidean norm. For symmetric $\bm{A}$, we define $\lambda_{\max}(\bm{A})$ and $\lambda_{\min}(\bm{A})$ as the maximum and minimum eigenvalues of $\bm{A}$, respectively. For two non-zero real sequences $\{a_{n}\}_{n=1}^{\infty}$ and $\{b_{n}\}_{n=1}^{\infty}$, we use $a_{n}\ll b_{n}$, $b_{n}\gg a_{n}$ or $a_{n}=\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.7}{$\scriptscriptstyle\mathcal{O}$}}(b_{n})$ to represent $|a_{n}/b_{n}|\rightarrow 0$ as $n\rightarrow\infty$. And $a_{n}\lesssim b_{n}$, $b_{n}\gtrsim a_{n}$ or $a_{n}=\mathcal{O}(b_{n})$ means $\sup_{n}|a_{n}/b_{n}|<\infty$. For two random variable sequences $\{x_{n}\}_{n=1}^{\infty}$ and $\{y_{n}\}_{n=1}^{\infty}$, the notation $x_{n}=\mathcal{O}_{\mathbb{P}}(y_{n})$ means that for any $\epsilon>0$, there exists a positive constant $M$ such that $\sup_{n}\mathbb{P}(|x_{n}/y_{n}|>M)\leq\epsilon$. For two real numbers $a$ and $b$, $a\vee b$ and $a\wedge b$ represent $\max(a,b)$ and $\min(a,b)$, respectively. For any positive integer $K$, both $1:K$ and $[K]$ stand for the set $\{1,2,\ldots,K\}$. For any set $S\subseteq[K]$, $|S|$ denotes its cardinality, and $S^{c}$ denotes its complement. Without further notice, $c$, $C$, $C_{1}$, $C_{2}$, $\ldots$ represent some positive constants and can change from line to line.

Suppose there are $K$ tasks, for which we have $n_{k}$ observations $\{\bm{z}_{i}^{(k)}\}_{i=1}^{n_{k}}$ from the $k$-th task. Suppose there exists an unknown subset $S\subseteq 1:K$, such that observations from each task in $S$ independently follow a GMM, while samples from tasks outside $S$ can be arbitrarily distributed. This means,

for all $k\in S$, $i=1:n_{k}$, and

where $\mathbb{Q}_{S}$ is some probability measure on $(\mathbb{R}^{p})^{\otimes n_{S^{c}}}$ and $n_{S^{c}}=\sum_{k\in S^{c}}n_{k}$. In unsupervised learning, we have no access to the true labels $\{y^{(k)}_{i}\}_{i,k}$. To formalize the multi-task learning problem, we first introduce the parameter space for a single GMM:

where $M$, $c_{w}\in(0,1/2]$ and $c_{\bm{\Sigma}}$ are some fixed positive constants. For simplicity, throughout the main text, we assume these constants are fixed. Hence, we have suppressed the dependency on them in the notation $\overline{\Theta}$. The parameter space $\overline{\Theta}$ is a standard formulation. Similar parameter spaces have been considered, for example, in Cai et al., (2019).

Our goal of multi-task learning is to leverage the potential similarity shared by different tasks in $S$ to collectively learn them all. The tasks outside $S$ can be arbitrarily distributed and they can be potentially outlier tasks. This motivates us to define a joint parameter space for GMMs in $S$:

where $\bm{\beta}^{(k)}=(\bm{\Sigma}^{(k)})^{-1}(\bm{\mu}^{(k)}_{2}-\bm{\mu}^{(k)}_{1})$ is called the discriminant coefficient in the $k$-th task (recall Section 1.1). For convenience, we define $\delta^{(k)}=(\bm{\beta}^{(k)})^{\top}(\bm{\mu}^{(k)}_{1}+\bm{\mu}^{(k)}_{2})/2$, which together with $\log((1-w^{(k)})/w^{(k)})$ is part of the decision boundary. Note that this parameter space is defined only for GMMs of tasks in $S$. To model potentially corrupted or contaminated data, we do not impose any distributional constraints for tasks in $S^{c}$. Such a modeling framework is reminiscent of Huber’s $\epsilon$-contamination model (Huber, , 1964). Similar formulations have been adopted in recent multi-task learning research such as Konstantinov et al., (2020) and Duan and Wang, (2023).

For GMMs in $\overline{\Theta}_{S}(h)$, we assume that they share similar discriminant coefficients. The similarity is formalized by assuming that all the discriminant coefficients in $S$ are within Euclidean distance $h$ from a “center”. Given that the discriminant coefficient has a major impact on the clustering performance (see the discriminant rule in (3)), the parameter space $\overline{\Theta}_{S}(h)$ is tailored to characterize the task relatedness from the clustering perspective. A similar viewpoint that focuses on modeling the discriminant coefficient has appeared in the study of high-dimensional GMM clustering (Cai et al., , 2019) and sparse linear discriminant analysis (Cai and Liu, , 2011; Mai et al., , 2012). With both $S$ and $h$ being unknown in practice, we aim to develop a multi-task learning procedure that is robust to outlier tasks in $S^{c}$, and achieves improved performance for tasks in $S$ (compared to the single-task learning), in terms of discriminant coefficient estimation and clustering, whenever $h$ is small.

The parameter space does not require the mean vectors $\{\bm{\mu}^{(k)}_{1},\bm{\mu}^{(k)}_{2}\}_{k\in S}$ or the covariance matrices $\{\bm{\Sigma}^{(k)}\}_{k\in S}$ to be similar, although they are not free parameters due to the constraint on $\{\bm{\beta}^{(k)}\}_{k\in S}$. And the mixture proportions $\{w^{(k)}\}_{k\in S}$ do not need to be similar either. We thus avoid imposing restrictive conditions on those parameters. On the other hand, it implies that estimation of the mixture proportions, mean vectors, and covariance matrices in multi-task learning may not be generally improvable over that in the single-task learning. This is verified by the theoretical results in Section 2.3. While the current treatment in the paper does not consider similarity structure among $\{\bm{\mu}^{(k)}_{1},\bm{\mu}^{(k)}_{2}\}_{k\in S},\{\bm{\Sigma}^{(k)}\}_{k\in S}$ or $\{w^{(k)}\}_{k\in S}$, our methods and theory can be readily adapted to handle such scenarios, if desired.

There are two main reasons why this MTL problem can be challenging. First, commonly used strategies like data pooling are fragile with respect to outlier tasks and can lead to arbitrarily inaccurate outcomes in the presence of even a small number of outliers. Also, since the distribution of data from outlier tasks can be adversarial to the learner, the idea of outlier task detection in the recent literature (Li et al., , 2021; Tian and Feng, , 2023) may not be applicable. Second, to address the nonconvexity of the likelihood, we propose to explore the similarity among tasks via a generalization of the EM algorithm. However, a clear theoretical understanding of such an iterative procedure requires a delicate analysis of the whole iterative process. In particular, as in the analysis of EM algorithms (Cai et al., , 2019; Kwon and Caramanis, , 2020), the estimates of similar discriminant vectors $\{\bm{\beta}^{(k)*}\}_{k\in S}$ and other potentially dissimilar parameters are entangled in the iterations. It is highly non-trivial to separate the impact of estimating $\{\bm{\beta}^{(k)*}\}_{k\in S}$ and other parameters to derive the desired statistical error rates. We manage to address this challenge through a localization technique by carefully shrinking the analysis radius of estimators as the iteration proceeds.

We aim to tackle the problem of GMM estimation under the context of multi-task learning. The EM algorithm is commonly used to address the non-convexity of the log-likelihood function arising from the latent labels. In the standard EM algorithm, we “classify” the observations (update the posterior) in E-step and update the parameter estimations in M-step (Redner and Walker, , 1984). For multi-task and transfer learning problems, the penalization framework is very popular, where we solve an optimization problem based on a new objective function. This objective function consists of a local loss function and a penalty term, forcing the estimators of similar tasks to be close to each other. For examples, see Zhang and Zhang, (2011); Zhang et al., (2015); Bastani, (2021); Xu and Bastani, (2021); Li et al., (2021); Duan and Wang, (2023); Lin and Reimherr, (2022); Li et al., (2023); Tian and Feng, (2023). Thus motivated, our method seeks a combination of the EM algorithm and the penalization framework.

In particular, we adapt the penalization framework of Duan and Wang, (2023) and modify the updating formulas in the M-step accordingly. The proposed procedure is summarized in Algorithm 1. For simplicity, in Algorithm 1 we have used the notation

Note that $\gamma_{\bm{\theta}}(\bm{z})$ is the posterior probability $\mathbb{P}(Y=2|Z=\bm{z})$ given the observation $\bm{z}$. The estimated posterior probability is calculated in every E-step given the updated parameter estimates.

Recall that the parameter space $\overline{\Theta}_{S}(h)$ introduced in (LABEL:eq:_parameter_space_mtl) does not encode similarity for the mixture proportions $\{w^{(k)*}\}_{k\in S}$, mean vectors $\{\bm{\mu}^{(k)*}_{1},\bm{\mu}^{(k)*}_{2}\}_{k\in S}$, or covariance matrices $\{\bm{\Sigma}^{(k)*}\}_{k\in S}$. Hence, the updates of them in Steps 5-7 are kept the same as in the standard EM algorithm. Regarding the update for discriminant coefficients in Step 9, the quadratic loss function is motivated by the direct estimation of the discriminant coefficient in high-dimensional GMM (Cai et al., , 2019) and high-dimensional LDA literature (Cai and Liu, , 2011; Witten and Tibshirani, , 2011; Fan et al., , 2012; Mai et al., , 2012, 2019). The penalty term in Step 9 penalizes the contrasts of $\bm{\beta}^{(k)}$’s to exploit the similarity structure among tasks. Having the “center” parameter $\overline{\bm{\beta}}$ in the penalization induces robustness against outlier tasks. We refer to Duan and Wang, (2023) for a systematic treatment of this penalization framework. It is straightforward to verify that when the tuning parameters $\{\lambda^{[t]}\}_{t=1}^{T}$ are set to zero, Algorithm 1 reduces to the standard EM algorithm performed separately on the $K$ tasks. That is, for each $k=1:K$, given the parameter estimate from the previous step $\widehat{\bm{\theta}}^{(k)[t-1]}=(\widehat{w}^{(k)[t-1]},\widehat{\bm{\beta}}^{(k)[t-1]},\widehat{\delta}^{(k)[t-1]})$, we update $\widehat{w}^{(k)[t]}$, $\widehat{\bm{\mu}}^{(k)[t]}_{1}$, $\widehat{\bm{\mu}}^{(k)[t]}_{2}$, $\widehat{\bm{\Sigma}}^{(k)[t]}$, and $\widehat{\delta}^{(k)[t]}$ as in Algorithm 1, and update $\widehat{\bm{\beta}}^{(k)[t]}$ via

For the maximum number of iteration rounds, $T$, our theory will show that $T\gtrsim\log(\max_{k=1:K}n_{k})$ is sufficient to reach the desired statistical error rates. In practice, we can terminate the iteration when the change of estimates within two successive rounds falls below some pre-set small tolerance level. We discuss the initialization in detail in Sections 2.3 and 2.4.

In this section, we develop statistical theories for our proposed procedure MTL-GMM (see Algorithm 1). As mentioned in Section 2.1, we are interested in the performance of both parameter estimation and clustering, although the latter is the main focus and motivation. First, we impose conditions in the following assumption set.

We first establish the rate of convergence for the estimation. Recalling the parameter space $\overline{\Theta}_{S}(h)$ in (LABEL:eq:_parameter_space_mtl), let us denote the true parameter by

To better present the results for parameters related to the optimal discriminant rule (3), we further denote

where $\bm{\beta}^{(k)*}=(\bm{\Sigma}^{(k)*})^{-1}(\bm{\mu}^{(k)*}_{2}-\bm{\mu}^{(k)*}_{1}),\delta^{(k)*}=\frac{1}{2}(\bm{\beta}^{(k)*})^{\top}(\bm{\mu}^{(k)*}_{1}+\bm{\mu}^{(k)*}_{2})$. Note that $\bm{\theta}^{(k)*}$ is a function of $\overline{\bm{\theta}}^{(k)*}$. For the estimators returned by MTL-GMM (see Algorithm 1), we are particularly interested in the following two error metrics:

where $\pi:[2]\rightarrow[2]$ is a permutation on $\{1,2\}$. Again, we take the minimum above because binary GMM is identifiable up to label permutation. The first error metric $d(\widehat{\bm{\theta}}^{(k)[T]},\bm{\theta}^{(k)*})$ involves the error for discriminant coefficients and is closely related to the clustering performance. It reveals how well our method utilizes similarity structure in multi-task learning. The second error metric is about the mean vectors and covariance matrix. As discussed in Section 2.1, we shall not expect it to be improved compared to that in single-task learning, as these parameters are not necessarily similar.

We are ready to present upper bounds for the estimation error of MTL-GMM. We recall that $\overline{\Theta}_{S}(h)$ and $\mathbb{Q}_{S}$ are the parameter space and probability measure that we use in Section 2.1 to describe the data distributions for tasks in $S$ and $S^{c}$, respectively.

The upper bound of $\big{(}\min_{\pi:[2]\rightarrow[2]}\max_{r=1:2}\|\widehat{\bm{\mu}}^{(k)[T]}_{r}-\bm{\mu}^{(k)*}_{\pi(r)}\|_{2}\big{)}\vee\|\widehat{\bm{\Sigma}}^{(k)[T]}-\bm{\Sigma}^{(k)*}\|_{2}$ contains two parts. The first part is comparable to the single-task learning rate (Cai et al., , 2019) (up to a $\sqrt{\log K}$ term due to the simultaneous control over all tasks in $S$), and the second part characterizes the geometric convergence of iterates. As expected, since $\bm{\mu}^{(k)*}_{1}$, $\bm{\mu}^{(k)*}_{2}$, $\bm{\Sigma}^{(k)*}$ in $S$ are not necessarily similar, an improved error rate over single-task learning is generally impossible. The upper bound for $d(\widehat{\bm{\theta}}^{(k)[T]},\bm{\theta}^{(k)*})$ is directly related to the clustering performance of our method. Thus we will provide a detailed discussion about it after presenting the clustering result in the next theorem.

As introduced in Section 1.1, using the estimate $\widehat{\bm{\theta}}^{(k)[T]}=(\widehat{w}^{(k)[T]},\widehat{\bm{\beta}}^{(k)[T]},\widehat{\delta}^{(k)[T]})$ from Algorithm 1, we can construct a classifier for task $k$ as

Recall that for a clustering method $\mathcal{C}:\mathbb{R}^{p}\rightarrow\{1,2\}$, its mis-clustering error rate under GMM with parameter $\overline{\bm{\theta}}=(w,\bm{\mu}_{1},\bm{\mu}_{2},\bm{\Sigma})$ is

where $Z^{\textup{new}}\sim(1-w)\mathcal{N}(\bm{\mu}_{1},\bm{\Sigma})+w\mathcal{N}(\bm{\mu}_{2},\bm{\Sigma})$ is a future observation associated with the label $Y^{\textup{new}}$, independent from $\mathcal{C}$; the probability $\mathbb{P}_{\overline{\bm{\theta}}}$ is w.r.t. $(Z^{\textup{new}},Y^{\textup{new}})$, and the minimum is taken over two permutation functions on $\{1,2\}$. Denote $\mathcal{C}_{\overline{\bm{\theta}}}$ as the Bayes classifier that minimizes $R_{\overline{\bm{\theta}}}(\mathcal{C})$. In the following theorem, we obtain the upper bound of the excess mis-clustering error of $\widehat{\mathcal{C}}^{(k)[T]}$ for $k\in S$.

The upper bounds of $d(\widehat{\bm{\theta}}^{(k)[T]},\bm{\theta}^{(k)*})$ in Theorem 1 and $R_{\overline{\bm{\theta}}^{(k)*}}(\widehat{\mathcal{C}}^{(k)[T]})-R_{\overline{\bm{\theta}}^{(k)*}}(\mathcal{C}_{\overline{\bm{\theta}}^{(k)*}})$ in Theorem 2 consist of five parts with one-to-one correspondence. It is sufficient to discuss the bound of $R_{\overline{\bm{\theta}}^{(k)*}}(\widehat{\mathcal{C}}^{(k)[T]})-R_{\overline{\bm{\theta}}^{(k)*}}(\mathcal{C}_{\overline{\bm{\theta}}^{(k)*}})$. Part (\@slowromancapi@) represents the “oracle rate”, which can be achieved when all tasks in $S$ are the same. This is the best rate to possibly achieve. Part (\@slowromancapii@) is a dimension-free error caused by estimating scalar parameters $\delta^{(k)*}$ and $w^{(k)*}$ that appears in the optimal discriminant rule. Part (\@slowromancapiii@) includes $h$ that measures the degree of similarity among the tasks in $S$. When these tasks are very similar, $h$ will be small, contributing a small term to the upper bound. Nicely, even when $h$ is large, the term becomes $\frac{p+\log K}{n_{k}}$ and it is still comparable to the minimax error rate of single-task learning $\mathcal{O}_{\mathbb{P}}(p/n_{k})$ (e.g., Theorems 4.1 and 4.2 in Cai et al., (2019)). We have the extra $\log K$ term here due to the simultaneous control over all tasks in $S$. Part (\@slowromancapiv@) quantifies the influence from the outlier tasks in $S^{c}$. When there are more outlier tasks, $\epsilon$ increases, and the bound becomes worse. On the other hand, as long as $\epsilon$ is small enough to make this term dominated by any other part, the error rate induced by outlier tasks becomes negligible. Given that data from outlier tasks can be arbitrarily contaminated, we can conclude that our method is robust against a fraction of outlier tasks from arbitrary sources. The term in Part (\@slowromancapv@) decreases geometrically in the iteration number $T$, implying that the iterates in Algorithm 1 converge geometrically to a ball of radius determined by the errors from Parts (\@slowromancapi@)-(\@slowromancapiv@).

After explaining each part of the upper bound, we now compare it with the convergence rate $\mathcal{O}_{\mathbb{P}}(\frac{p+\log K}{n_{k}})$ (including $\log K$ here since we consider all the tasks simultaneously) in the single-task learning and reveal how our method performs. With a quick inspection, we can conclude the following: