Taking Advantage of Sparsity in Multi-Task Learning

Introduction

We study the problem of estimating multiple regression equations under sparsity assumptions on the underlying regression coefficients. More precisely, we consider multiple Gaussian regression models,

where, for each $t=1,\dots,T$, we let $X_{t}$ be a prescribed $n\times M$ design matrix, $\beta_{t}^{*}$ the unknown vector of regression coefficients and $y_{t}$ an $n$-dimensional vector of observations. We assume that $W_{1},\dots,W_{T}$ are i.i.d. zero mean random vectors.

We are interested in estimation methods which work well even when the number of parameters in each equation is much larger than the number of observations, that is, $M\gg n$. This situation may arise in many practical applications in which the predictor variables are inherently high dimensional, or it may be “costly” to observe response variables, due to difficult experimental procedures, see, for example for a discussion.

Examples in which this estimation problem is relevant range from multi-task learning and conjoint analysis (see, for example, and references therein) to longitudinal data analysis as well as the analysis of panel data , among others. In particular, multi-task learning provides a main motivation for our study. In that setting each regression equation corresponds to a different learning task (the classification case can be treated similarly); in addition to the requirement that $M\gg n$, we are also interested in the case that the number of tasks $T$ is much larger than $n$. Following we assume that there are only few common important variables which are shared by the tasks. A general goal of this paper is to study the implications of this assumption from a statistical learning view point, in particular, to quantify the advantage provided by the large number of tasks to learn both the underlying vectors $\beta^{*}_{1},\dots,\beta^{*}_{T}$ as well as to select common variables shared by the tasks.

In this paper, we assume that the vectors $\beta_{1}^{*},\dots,\beta^{*}_{T}$ are not only sparse but also have the same sparsity pattern. This means that the set of indices which correspond to non zero components of $\beta_{t}^{*}$ is the same for every $t=1,\dots,T$. In other words, the response variable associated with each equation in (1.1) depends only on a small subset (of size $s\ll M$) of the corresponding predictor variables and the set of relevant predictors is preserved across the different equations. This assumption, that we further refer to as structured sparsity assumption, is motivated by some recent work on multi-task learning . It naturally leads to an extension of the Lasso method, the so-called group Lasso , in which the error term is the average residual error across the different equations and the penalty term is a mixed $(2,1)$-norm. The structured sparsity assumption induces a relation between the responses and, as we shall see, can be used to improve estimation.

The paper is organized as follows. In Section 2 we define the estimation method and comment on previous related work. In Section 3 we study the oracle properties of this estimator when the errors $W_{t}$ are Gaussian. Our main results concern upper bounds on the prediction error and the distance between the estimator and the true regression vector $\beta^{*}$. Specifically, Theorem 3.1 establishes that under the above structured sparsity assumption on $\beta^{*}$, the prediction error is essentially of the order of $s/n$. In particular, in the multi-task learning scenario, in which $T$ can grow, we are able to remove completely the effect of the number of predictor variables in the bounds. Next, in Section 4, under a stronger condition on the design matrices, we describe a simple modification of our method and show that it selects the correct sparsity pattern with an overwhelming probability (Theorem 4.1). We also find the rates of convergence of the estimators for mixed $(2,1)$-norms with $1\leq p\leq\infty$ (Theorem 4.2). The techniques of proofs build upon and extend those of and . Finally, in Section 5 we discuss how our results can be extended to more general noise distributions, of which we only require the variance to be finite.

Method and related work

where $\|\cdot\|$ is the standard Euclidean norm.

We have now accumulated the sufficient information to introduce the estimation method. We define the empirical residual error

and, for every $\lambda>0$, we let our estimator ${\hat{\beta}}$ be a solution of the optimization problem

In order to study the statistical properties of this estimator, it is useful to derive the optimality condition for a solution of the problem (2.2). Since the objective function in (2.2) is convex, ${\hat{\beta}}$ is a solution of (2.2) if and only if (the $MT$-dimensional zero vector) belongs to the subdifferential of the objective function. In turn, this condition is equivalent to the requirement that

where $\partial$ denotes the subdifferential (see, for example, for more information on convex analysis). Note that

Thus, ${\hat{\beta}}$ is a solution of (2.2) if and only if

Finally, let us comment on previous related work. Our estimator is a special case of the group Lasso estimator . Several papers analyzing statistical properties of the group Lasso appeared quite recently . Most of them are focused on the group Lasso for additive models or generalized linear models . Special choice of groups is studied in . Discussion of the group Lasso in a relatively general setting is given by Bach and Nardi and Rinaldo . Bach assumes that the predictors $x_{ti}$ are random with a positive definite covariance matrix and proves results on consistent selection of sparsity pattern $J(\beta^{*})$ when the dimension of the model ($p=MT$ in our case) is fixed and $n\to\infty$. Nardi and Rinaldo consider a setting that covers ours and address the issue of sparsity oracle inequalities in the spirit of . However, their bounds are too coarse (see comments in Section 3 below). Obozinski et al. replace in (2.2) the $(2,1)$-norms by $(q,1)$-norms with $q>1$ and show that the resulting estimator achieves consistent selection of the sparsity pattern under the assumption that all the rows of matrices $X_{t}$ are independent Gaussian random vectors with the same covariance matrix.

This literature does not demonstrate theoretical advantages of the group Lasso as compared to the usual Lasso. One of the aims of this paper is to show that such advantages do exist in the multi-task learning setup. In particular, our Theorem 3.1 implies that the prediction bound for the group Lasso estimator that we use here is by at least a factor of $T$ better than for the standard Lasso under the same assumptions. Furthermore, we demonstrate that as the number of tasks $T$ increases the dependence of the bound on $M$ disappears, provided that $M$ grows at the rate slower than $\exp({\sqrt{T}})$.

Sparsity oracle inequality

Let $1\leq s\leq M$ be an integer that gives an upper bound on the structured sparsity $M(\beta^{*})$ of the true regression vector $\beta^{*}$. We make the following assumption.

There exists a positive number $\kappa=\kappa(s)$ such that

where $J^{c}$ denotes the complement of the set of indices $J$.

Several simple sufficient conditions for Assumption 3.1 with $T=1$ are given in . Similar sufficient conditions can be stated in our more general setting. For example, it is enough to suppose that each of the matrices $X_{t}^{\scriptscriptstyle\top}X_{t}/n$ is positive definite or satisfies a Restricted Isometry condition as in or the coherence condition (cf. Lemma 4.1 below).

Consider the model (1.1) for $M\geq 2$ and $T,n\geq 1$. Assume that the random vectors $W_{1},\dots,W_{T}$ are i.i.d. Gaussian with zero mean and covariance matrix $\sigma^{2}I_{n\times n}$, all diagonal elements of the matrix $X^{\scriptscriptstyle\top}X/n$ are equal to $1$ and $M(\beta^{*})\leq s$. Let

where $\phi_{\rm max}$ is the maximum eigenvalue of the matrix $X^{\scriptscriptstyle\top}X/n$.

which, using $y=X\beta^{*}+W$, is equivalent to

Since we assume all diagonal elements of the matrix $X^{\scriptscriptstyle\top}X/n$ to be equal to $1$, the random variables

$t=1,\dots,T$, are i.i.d. standard Gaussian. Using this fact we can write, for any $j=1,\dots,M$,

where $\chi_{T}^{2}$ is a chi-square random variable with $T$ degrees of freedom. We now apply Lemma A.1, the union bound and the fact that $A>8$ to get

It follows from (3.4) that, on the event ${\cal A}$.

which coincides with (3.1). To prove (3.2), we use the inequality

which follows from (2.3) and (2.4). Then,

where we have used $y=X\beta^{*}+W$ and the triangle inequality. The result then follows by combining the last inequality with inequality (3.5) and using the definition of the event ${\cal A}$.

Finally, we prove (3.3). First, observe that, on the event ${\cal A}$,

This fact follows from (2.3), (2.1) and the definition of the event ${\cal A}$. The following chain yields the result:

We are now ready to state the main result of this section.

Consider the model (1.1) for $M\geq 2$ and $T,n\geq 1$. Assume that the random vectors $W_{1},\dots,W_{T}$ are i.i.d. Gaussian with zero mean and covariance matrix $\sigma^{2}I_{n\times n}$, all diagonal elements of the matrix $X^{\scriptscriptstyle\top}X/n$ are equal to $1$ and $M(\beta^{*})\leq s$. Furthermore let Assumption 3.1 hold with $\kappa=\kappa(s)$ and let $\phi_{\rm max}$ be the largest eigenvalue of the matrix $X^{\scriptscriptstyle\top}X/n$. Let

where $A>8$ and let $q=\min(8\log M,A\sqrt{T}/8)$. Then with probability at least $1-M^{1-q}$, for any solution ${\hat{\beta}}$ of problem (2.2) we have

If, in addition, Assumption RE(2$s$) holds, then with the same probability for any solution ${\hat{\beta}}$ of problem (2.2) we have

We act similarly to the proof of Theorem 6.2 in . Let $J=J(\beta^{*})=\{j:(\beta^{*})^{j}\neq 0\}$. By inequality (3.1) with $\beta=\beta^{*}$ we have, on the even ${\cal A}$, that

Moreover by the same inequality, on the event ${\cal A}$, we have $\sum_{j=1}^{M}\|{\hat{\beta}}^{j}-\beta^{*j}\|\leq 4\sum_{j\in J}\|{\hat{\beta}}^{j}-\beta^{*j}\|$, which implies that $\sum_{j\in J^{c}}\|{\hat{\beta}}^{j}-\beta^{*j}\|\leq 3\sum_{j\in J}\|{\hat{\beta}}^{j}-\beta^{*j}\|$. Thus, by Assumption 3.1

Now, (3.6) follows from (3.10) and (3.11). Inequality (3.7) follows again by noting that

and then using (3.6). Inequality (3.8) follows from (3.3) and (3.6).

Finally, we prove (3.9). Let $\Delta={\hat{\beta}}-\beta^{*}$ and let $J^{\prime}$ be the set of indices in $J^{c}$ corresponding to $s$ maximal in absolute value norms $\|\Delta^{j}\|$. Consider the set $J_{2s}=J\cup J^{\prime}$. Note that $|J_{2s}|=2s$. Let $\|\Delta_{J^{c}}^{(k)}\|$ denote the $k$-th largest norm in the set $\{\|\Delta^{j}\|:\,j\in J^{c}\}$. Then, clearly,

This and the fact that $\|\Delta_{J^{c}}\|_{2,1}\leq 3\|\Delta_{J}\|_{2,1}$ on the event ${\cal A}$ implies

In addition, $\|\Delta_{J^{c}}\|_{2,1}\leq 3\|\Delta_{J}\|_{2,1}$ easily implies that

Combining these facts and (3.13) with Assumption RE(2$s$) we find that on the event ${\cal A}$ the following holds:

This inequality and (3.12) yield (3.9). ∎

Theorem 3.1 is valid for any fixed $n,M,T$; the approach is non-asymptotic. Some relations between these parameters are relevant in the particular applications and various asymptotics can be derived as corollaries. For example, in multi-task learning it is natural to assume that $T\geq n$, and the motivation for our approach is the strongest if also $M\gg n$. The bounds of Theorem 3.1 are meaningful if the sparsity index $s$ is small as compared to the sample size $n$ and the logarithm of the dimension $\log M$ is not too large as compared to $\sqrt{T}$.

Several important conclusions can be drawn from Theorem 3.1.

The dependence on the dimension $M$ is negligible for large $T$. Indeed, the bounds of Theorem 3.1 become independent of $M$ if we choose the number of tasks $T$ larger than $\log^{2}M$. A striking fact is that no relation between the sample size $n$ and the dimension $M$ is required. This is quite in contrast to the previous results on sparse recovery where the assumption $\log M=o(n)$ was considered as sine qua non constraint. For example, Theorem 3.1 gives meaningful bounds if $M=\exp({n^{\gamma}})$ for arbitrarily large $\gamma>0$, provided that $T>n^{2\gamma}$. This is due to the structured sparsity assumption that we naturally exploit in the multi-task scenario.

Our estimator is better than the standard Lasso in the multi-task setup. Theorem 3.1 witnesses that our group Lasso estimator admits substantially better error bounds than the usual Lasso. Let us explain this considering the example of the prediction error bound (3.9). Indeed, for the same multi-task setup, we can apply a usual Lasso estimator $\hat{\beta}^{L}$, that is a solution of the following optimization problem

Assume, for instance, that we are in the most favorable situation where $M<n$, each of the matrices $\frac{1}{n}X^{T}_{t}X_{t}$ is positive definite and has minimal eigenvalue greater than $\kappa^{2}$ (this, of course, implies Assumption 3.1). We can then apply inequality (7.8) from with

where $A>2\sqrt{2}$, to obtain that, with probability at least $1-(MT)^{1-\frac{A^{2}}{8}}$, it holds

Indeed, when applying (7.8) of we account for the fact that the parameters $n$, $M$, $s$ therein correspond to $nT$, $MT$, $sT$ in our setup, and the minimal eigenvalue of the matrix $\frac{1}{nT}X^{T}X$ is greater than $\kappa^{2}/T$. Comparison with (3.9) leads to the conclusion that the prediction bound for our estimator is by at least a factor of $T$ better than for the standard Lasso under the same assumptions. Let us emphasize that the improvement is due to the property that $\beta^{*}$ is structured sparse. The second inherent property of our setting, that is, the fact that the matrix ${X^{\scriptscriptstyle\top}}X$ is block-diagonal, can be characterized as important but not indispensable. We discuss this in the next remark.

Theorem 3.1 applies to the general group Lasso setting. Indeed, the proofs in this section do not use the fact that the matrix ${X^{\scriptscriptstyle\top}}X$ is block-diagonal. The only restriction on ${X^{\scriptscriptstyle\top}}X$ is given in Assumption 3.1. For example, Assumption 3.1 is obviously satisfied if ${X^{\scriptscriptstyle\top}}X/(nT)$ (the correctly normalized Gram matrix of the regression model (2.1)) has a positive minimal eigenvalue. However, the price for having this property (or Assumption 3.1 in general), as well as the resulting error bounds, can be different for the block-diagonal (multi-task) setting and the full matrix $X$ setting.

Finally, we note that follow the scheme of the proof of to derive similar in spirit to ours but coarse oracle inequalities. Their results do not explain the advantages discussed in the points 1–3 above. Indeed, the tuning parameter $\lambda$ chosen in , pp. 614–615, is larger than our $\lambda$ by at least a factor of $\sqrt{T}$. As a consequence, the corresponding bounds in the oracle inequalities of are larger than ours by positive powers of $T$.

Coordinate-wise estimation and selection of sparsity pattern

In this section, we show how from any solution of the problem (2.2) we can reliably estimate the correct sparsity pattern with high probability.

We first introduce some more notation. We define the Gram matrix of the design $\Psi=\frac{1}{n}X^{\scriptscriptstyle\top}X$. Note that $\Psi$ is a $MT\times MT$ block-diagonal matrix with $T$ blocks of dimension $M\times M$ each. We denote these blocks by $\Psi_{t}=\frac{1}{n}X_{t}^{\scriptscriptstyle\top}X_{t}\equiv(\Psi_{tj,tk})_{j,k=1,\dots,M}$.

In this section we assume that the following condition holds true.

The elements $\Psi_{tj,tk}$ of the Gram matrix $\Psi$ satisfy

for some integer $s\geqslant 1$ and some constant $\alpha>1$.

Note that the above assumption on $\Psi$ implies Assumption 3.1 as we prove in the following lemma.

Let Assumption 4.1 be satisfied. Then Assumption 3.1 is satisfied with $\kappa=\sqrt{1-\frac{1}{\alpha}}$.

where we have used Assumption 4.1 and the Cauchy-Schwarz inequality. Next, using consecutively Assumption 4.1, the Cauchy-Schwarz inequality and the inequality $\|\Delta_{J^{c}}\|_{2,1}\leqslant 3\|\Delta_{J}\|_{2,1}$ we obtain

Note also that, by an argument as in , it is not hard to show that under Assumption 4.1 the vector $\beta^{*}$ satisfying (2.1) is unique.

Theorem 3.1 provides bounds for compound measures of risk, that is, depending simultaneously on all the vectors $\beta^{j}$. An important question is to evaluate the performance of estimators for each of the components $\beta^{j}$ separately. The next theorem provides a bound of this type and, as a consequence, a result on the selection of sparsity pattern.

Consider the model (1.1) for $M\geqslant 2$ and $T,n\geqslant 1$. Let the assumptions of Lemma 3.1 be satisfied and let Assumption 4.1 hold with the same $s$. Set

Let $\lambda$, $A$ and $W_{1},\dots,W_{T}$ be as in Lemma 3.1. Then with probability at least $1-M^{1-q}$, where $q=\min(8\log M,A\sqrt{T}/8)$, for any solution $\hat{\beta}$ of problem (2.2) we have

then with the same probability for any solution $\hat{\beta}$ of problem (2.2) the set of indices

estimates correctly the sparsity pattern $J(\beta^{*})$, that is,

Set $\Delta=\hat{\beta}-\beta^{*}$. Using Assumption 4.1 we obtain

Thus, by Lemma 3.1 and Theorem 3.1, with probability at least $1-M^{1-q}$,

By Lemma 4.1, $\alpha\kappa^{2}=\alpha-1$, which yields the first result of the theorem. The second result follows from the first one in an obvious way. ∎

Assumption of type (4.2) is inevitable in the context of selection of sparsity pattern. It says that the vectors $(\beta^{*})^{j}$ cannot be arbitrarily close to 0 for $j$ in the pattern. Their norms should be at least somewhat larger than the noise level.

The second result of Theorem 4.1 (selection of sparsity pattern) can be compared with who considered the Group Lasso. There are several differences. First, our estimator $\hat{J}$ is based on thresholding of the norms $\|\hat{\beta}^{j}\|$, while take instead the set where these norms do not vanish. In practice, the latter is known to be a poor selector; it typically overestimates the true sparsity pattern. Second, consider specific asymptotic settings, while our result holds for any fixed $n,M,T$. Different kinds of asymptotics can be therefore obtained as simple corollaries. Finally, note that the estimator $\hat{\beta}$ is not necessarily unique. Though does not discuss this fact, the proof there only shows that there exists a subsequence of solutions $\hat{\beta}$ of (2.2) such that the set $\{j:\|\hat{\beta}^{j}\|\neq 0\}$ coincides with the sparsity pattern $J(\beta^{*})$ in some specified asymptotics (we note here the “if and only if” claim before formula (23) in is not proved). In contrast, the argument in Theorem 4.1 does not require any analysis of the uniqueness issues, though it is not excluded that the solution is indeed unique. It guarantees that simultaneously for all solutions $\hat{\beta}$ of (2.2) and any fixed $n,M,T$ the correct selection is done with high probability.

Theorems 3.3 and 4.1 imply the following corollary.

Consider the model (1.1) for $M\geqslant 2$ and $T,n\geqslant 1$. Let the assumptions of Lemma 3.1 be satisfied and let Assumption 4.1 holds with the same $s$. Let $\lambda$, $A$ and $W_{1},\dots,W_{T}$ be as in Lemma 3.1. Then with probability at least $1-M^{1-q}$, where $q=\min(8\log M,A\sqrt{T}/8)$, for any solution $\hat{\beta}$ of problem (2.2) and any $1\leq p<\infty$ we have

If, in addition, (4.2) holds, then with the same probability for any solution $\hat{\beta}$ of problem (2.2) and any $1\leq p<\infty$ we have

Set $\Delta=\hat{\beta}-\beta$. For any $p\geqslant 1$ we have

Combining (3.7), (4.1) with $\kappa=\sqrt{1-\frac{1}{\alpha}}$ and the above display yields the first result. ∎

Inequalities (4.1) and (4.5) provide confidence intervals for the unknown parameter $\beta^{*}$ in mixed (2,$p$)-norms.

Consider the threshold $\tau=\frac{c}{\sqrt{n}}\sqrt{1+\frac{A\log M}{\sqrt{T}}}$ and define a thresholded estimator

This assumption says that we cannot recover arbitrarily small components. Similar assumptions are standard in the literature on sign consistency (see, for example, for more details and references).

Consider the model (1.1) for $M\geqslant 2$ and $T,n\geqslant 1$. Let the assumptions of Lemma 3.1 be satisfied and let Assumption 4.1 hold with the same $s$. Let $\lambda$ and $A$ be defined as in Lemma 3.1 and $c$ as in Theorem 4.1. Then with probability at least $1-M^{1-q}$, where $q=\min(8\log M,A\sqrt{T}/8)$, for any solution $\hat{\beta}$ of problem (2.2) we have

The proof is then similar to that of Theorem 4.1. ∎

Non-Gaussian noise

This assumption is quite mild. It is satisfied for example, if all $(x_{ti})_{j}$ are bounded in absolute value by a constant uniformly in $i,t,j$. We have the two following theorems.

Then with probability at least $1-\frac{(2e\log M-e)c^{\prime}}{(\log M)^{1+\delta}}$, for any solution ${\hat{\beta}}$ of problem (2.2) we have

If, in addition, Assumption RE(2$s$) holds, then with the same probability for any solution ${\hat{\beta}}$ of problem (2.2) we have

Consider the model (1.1) for $M\geqslant 3$ and $T,n\geqslant 1$. Let the assumptions of Theorem 5.1 be satisfied and let Assumption 4.1 hold with the same $s$. Set

Let $\lambda$ be as in Theorem as in 5.1. Then with probability at least $1-\frac{(2e\log M-e)c^{\prime}}{(\log(MT))^{1+\delta}}$, for any solution $\hat{\beta}$ of problem (2.2) we have

then with the same probability for any solution $\hat{\beta}$ of problem (2.2) the set of indices

estimates correctly the sparsity pattern $J(\beta^{*})$:

The proofs of these theorems are similar to the ones of Theorems 3.1 and 4.1 up to a modification of the bound on $P(\mathcal{A}^{c})$ in Lemma 3.1. We consider now the event

Then we use Lemma A.2 given below with the random vectors

By the definition of $\lambda$ in Theorem 5.2 and Assumption 5.1 we obtain

Thus, we see that under the finite variance assumption on the noise, the dependence on the dimension $M$ cannot be made negligible for large $T$.

Appendix A Auxiliary results

Here we collect two auxiliary results which are used in the above analysis. The first result is a useful bound on the tail of the chi-square distribution.

Let $\chi_{T}^{2}$ be a chi-square random variable with $T$ degrees of freedom. Then

where ${\mathcal{N}}$ is the standard normal random variable and $z(x)=\sqrt{x-T\log(1+x/T)}$. The result now follows from inequalities ${\rm Pr}({\mathcal{N}}>z(x))\leq\exp(-z^{2}(x)/2)$ and

The next result is a version of Nemirovski’s inequality (see , Corollary 2.4 page 5).

References