We consider high-dimensional estimation problems, where the number of variables can be much larger that the number of observations . In this regime, consistent estimation can be achieved by imposing low-dimensional structural constraints on the estimation parameters. Sparsity is a prototypical structural constraint, where at most a small set of parameters can be non-zero.
A key class of sparsity-constrained estimators is based on regularized M-estimators using convex penalties, with the penalty by far the most common. In the context of linear regression, the Lasso estimator Tibshirani (1996) solves an regularized (or constrained) least squares problem, and has strong statistical guarantees, including prediction error consistency van de Geer and Buhlmann (2009), consistency of the parameter estimates in some norm van de Geer and Buhlmann (2009); Meinshausen and Yu (2009); Candes and Tao (2007), and variable selection consistency Meinshausen and Bühlmann (2006); Wainwright (2009a); Zhao and Yu (2006). In the context of sparse Gaussian graphical model (GMRF) estimation, the graphical Lasso estimator minimizes the Gaussian negative log-likelihood regularized by the norm of the (off-diagonal) entries of the concentration Yuan and Lin (2007); Friedman et al. (2007); Bannerjee et al. (2008). Strong statistical guarantees for this estimator have been established (see Ravikumar et al. (2011) and references therein).
Recently, there has been significant interest in M-estimators with non-convex penalties, including SCAD and MCP penalties Fan and Li (2001); Breheny and Huang (2011); Zhang et al. (2010); Zhang and Zhang (2012). In particular, Zhang and Zhang (2012) establishes consistency for the global optima of least-squares problems with certain non-convex penalties. Loh and Wainwright (2015) shows that under some regularity conditions on the penalty, any stationary point of the objective function will lie within statistical precision of the underlying parameter vector and thus provide - and - error bounds for any stationary point. Compared to convex penalties, perhaps the strongest point in favor of non-convex regularization is made by authors of Loh and Wainwright (2017), who proved that for a class of amenable non-convex regularizers with vanishing derivative away from the origin (including SCAD and MCP), any stationary point is able to recover the parameter support without requiring the typical incoherence conditions needed for convex penalties.
In this paper, we study a family of M-estimators with trimmed regularization, which leaves the largest parameters unpenalized. This family includes as special cases the recently proposed Trimmed Lasso estimator (Gotoh et al. (2017); Bertsimas et al. (2017)) and its counterpart for sparse graphical model estimation, which we call Graphical Trimmed Lasso. This work complements efforts of Yang et al. (2016), who analyze statistical benefits of trimming losses. We apply the trimming mechanism to separable components of the regularizer.
We present the first statistical analysis of M-estimators with trimmed regularization111Trimmed Lasso has been studied from an optimization perspective and with respect to its connections with existing penalties, but has not been analyzed from a statistical standpoint.. These estimators are non-convex, but unlike SCAD and MCP regularizers, they are not amenable and hence the analyses of Loh and Wainwright (2015, 2017) cannot be applied. Our main theoretical result shows that if the trimming parameter is smaller than the true support size, for any local optimum of the resulting non-convex program all the zero entries of the true parameter vector are successfully estimated as zero; while if is larger than the true support size, the non-relevant parameters of the local optimum have smaller absolute values than relevant parameters and hence relevant parameters are not penalized. In addition to error bounds, we provide error bounds. These are asymptotically the same as those for amenable regularized problems such as SCAD or MCP, but have better constants and do not require the additional constraint where is a safety radius. We specialize our main results and derive corollaries for the special cases of linear regression and graphical model estimation. To optimize the trimmed regularized problem we develop and analyze a specialized algorithm, which performs better than recent methods based on difference of convex (DC) functions optimization Khamaru and Wainwright (2018). Experiments on simulated and real data demonstrate the value of trimming compared to SCAD, MCP and vanilla penalties.
Beyond regularization, the trimming strategy can be seamlessly applied to other decomposable regularizers including group-sparsity promoting regularization Tropp et al. (2006); Zhao et al. (2009); Yuan and Lin (2006); Jacob et al. (2009). Our work therefore motivates a future line of research on trimming a wide class of regularizers.
2 Problem Setup and the Trimmed Regularizer
Trimming has been typically applied to the loss function of
-estimators. We can handle outliers and heavy tailed noise by trimmingobservations with large residuals in terms of a loss function : given a collection of samples, , we solve the problem
which trims outliers (see Yang et al. (2016) and references therein).
Here, we consider a family of -estimators with trimmed regularization for general high-dimensional problems. We trim entries of that incur the largest penalty using the following program:
where denotes the parameter space (e.g., for linear regression). Defining the order statistics of the parameter , we can partially minimize over (setting to or based on the size of ), and rewrite the reduced version of problem (2) in alone:
For statistical analysis, we focus on the reduced problem (2). When optimizing, we exploit the structure of (2), treating weights as auxiliary optimization variables. This gives us both a new fast algorithm and analysis technique it that is not based on the DC structure of (2).
Prior art (e.g., Tibshirani (1996); Negahban et al. (2012); Loh and Wainwright (2015, 2017)) derives the estimation upper bounds for diverse sparsely regularized estimators motivated by many real applications. In this paper, we mainly consider two typical examples using the trimmed regularizers, but the results generalize.
Example: sparse linear models.
In high-dimensional linear regression problems, we have observation pairs of a real-valued target and its covariates in a linear relationship:
Here, , and are independent observation noises. The goal is to estimate the -sparse vector . According to the framework (2), we use the least squares loss function with trimmed regularizer (instead of the standard norm in Lasso Tibshirani (1996)):
Example: sparse graphical models.
GGMs form a powerful class of statistical models for representing distributions over a set of variables Lauritzen (1996), using undirected graphs to encode conditional independence conditions among variables.
In such high-dimensional settings, graph sparsity constraints are particularly pertinent for estimating GGMs. The most widely used estimator, the graphical Lasso minimizes the negative Gaussian log-likelihood regularized by the norm of the entries (or the off-diagonal entries) of the precision matrix (see Yuan and Lin (2007); Friedman et al. (2007); Bannerjee et al. (2008)). In our framework, we replace with the trimmed version:
where denotes the convex cone of symmetric and strictly positive definite matrices, does the smallest absolute sum of off-diagonals.
3 Theoretical Guarantees of Trimmed Regularization
Our goal is to estimate the true -sparse parameter vector (or matrix) that is the minimizer of expected loss: . We use to denote the support set of , namely the set of non-zero entries (i.e., ). In this section, we derive the upper bounds of estimation consistency (in terms of and support set recovery) under the following standard assumptions:
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The loss function is differentiable and convex.
(Restricted strong convexity on ) Let be the possible set of error vector on the parameter . Then, for all ,
where is a “curvature” parameter, and is a “tolerance” constant.
Note that the convex loss function in general cannot be strongly convex under the high dimensional setting (). 2 imposes strong curvature only in some limited directions where the ratio is small. This condition has been extensively studied and known to hold for several popular high dimensional problems (see Raskutti et al. (2010); Negahban et al. (2012); Loh and Wainwright (2015) for instance). The convexity condition of in 1 can be relaxed by introducing additional mild constraint, as shown in Loh and Wainwright (2017). However in this paper, we focus on the convex loss for clarity.
We begin with the bound. Toward this, we adopt the primal-dual witness (PDW) technique, specifically devised for the trimmed regularizer . Note that a line of works uses the PDW technique and shows the support set recovery for regularizer (Wainwright, 2009c; Yang et al., 2015) as well as amenable non-convex regularizers (Loh and Wainwright, 2017). However, even though is also symmetric and concave, it is not amenable. The key step of PDW is to build the restricted program. Let be an arbitrary subset of whose size is . Denoting and , we consider the following restricted program:
where we fix for all . We further construct the dual variable to satisfy the zero sub-gradient condition
where for (after re-ordering indices properly) and . Note that we suppress the dependency on in and for clarity. In order to derive the final statement, we will establish the strict dual feasibility of , i.e., .
The following theorem describes our main theoretical result concerning any local optimum of the non-convex program (2). The theorem guarantees under strict dual feasibility that non-relevant parameters of local optimum have smaller absolute values than relevant parameters and hence relevant parameters are not penalized (as long as is set as larger than ). Consider the problem with the trimmed regularizer (2) that satisfies 1 and 2. Let be an any local minimum of (2) with a sample size and . Suppose that:
given any selection of s.t. , the dual vector from the PDW construction (7) satisfies the strict dual feasibility with some ,
where is the union of true support and ,
letting , the minimum absolute value is lower bounded by
where denotes the maximum absolute row sum of the matrix.
Then, we have
for every pair , we have ,
if , all are successfully estimated as zero and we have
if , at least the smallest (in absolute) entries in have exactly zero but instead we have simpler (possibly tighter) bound as
where is defined as the largest absolute entries of including .
We will derive the actual bounds on terms involving , and in the corollaries for actual problems (for instance will be upper bounded by and we can choose accordingly). Though (8) and (9) seem apparently more stringent than
case (which is Lasso), we will see in corollaries that they are uniformly upper bounded for all selections under the asymptotically same probability as.
Note also that if is set as , the results will recover those of regular norm. Furthermore, by the statement in the theorem, if , only contains relevant feature indices and some relevant features are not penalized. If , includes all relevant indices (and some non-relevant indices). In this case, the second term in (10) disappears, but the term increases as gets larger. Moreover, the condition that will be violated as approaches . While we do not know the true sparsity a priori in many problems, we implicitly assume that we can set (i.e., by cross-validation).
Now we turn to the estimation bound under the same conditions: Consider the problem with a trimmed regularizer (2) where all conditions in Theorem 3 hold. Then, for any local minimum of (2), the parameter estimation error in terms of norm is upper bounded as: for some constant ,
The bound in (12) can be trivially derived since for any local optimum, the size of guaranteed to be smaller than or equal to by Theorem 3 hence we can apply the known results (e.g., Negahban et al. (2012)) to obtain bound of restricted program (6) given . From Theorem 3 and Corollary 3, we can observe that the estimation bounds are asymptotically same as those for )-amenable regularized problems such as SCAD or MCP: and . However, the constant for those regularizers might be large since it involves term (instead of for the trimmed ). In addition, those non-convex regularizers require additional constraint in their optimization problems for the theoretical guarantees, introducing additional assumptions on and tuning parameter .
Now we apply our main theorem to popular high dimensional problems introduced in Section 2: sparse linear regression and sparse graphical model, but due to the space constraint, the results for sparse graphical models are provided in the supplementary materials.
Sparse least squares.
Motivated by the information theoretic bound for arbitrary methods, all previous analyses of sparse linear regression assume for sufficiently large constant . We also assume , provided . Consider the model (3) where is sub-Gaussian. Suppose that we solve the program (4) with the selection of for some constant and satisfying
the sample covariance matrix satisfies the condition: for any selection of ,
Further suppose that is lower bounded by for some constant . Then with high probability at least , any local minimum of (4) satisfies
for every pair , we have ,
if , all are successfully estimated as zero and we have
if , at least the smallest entries in have exactly zero and we have
The conditions in Corollary 3 are also studied in the line of previous work. Especially, (1) is known as an incoherence condition for the sparse least square estimators Wainwright (2009b). All conditions may be shown to hold with high probability via the standard concentration bounds for sub-Gaussian matrices. It is important to note that in case of Lasso, the estimation will fail if the incoherence condition is violated Wainwright (2009b). Unlike Lasso, we confirm by simulations in Section 5 that the trimmed problem (4) can succeed even when this condition is not met.
We develop and analyze a block coordinate descent algorithm for solving objective (2). By leaving the weights as an explicit block rather than projecting them out, we give the weights more freedom to vary before setting down. We can also analyze the algorithm using the structure of (2) instead of relying on the DC formulation for(2). The approach is detailed in Algorithm 1, and the convergence analysis is summarized in Theorem 4.
Consider the general objective function
where is the convex indicator function. Let
Then, we assume following assumptions are satisfied, (a) is a smooth closed convex function with an -Lipchitz continuous gradient; (b) are convex and (c) is a closed convex set and is bounded below.
Moreover, define , if we choose the step size , we have,
which gives a sublinear rate of convergence with respect to the optimality condition.
To show the efficiency of the Algorithm 1, we conduct a small numerical experiment to compare with (Khamaru and Wainwright, 2018, Algorithm 2). The authors proposed multiple approaches for DC programs; the prox-type algorithm (Algorithm 2) did particularly well for subset selection, (Khamaru and Wainwright, 2018, Figure 2).
We generate Lasso simulation data with variables of dimension , and samples. The number of nonzero elements in true generating variable is 10. We take , and apply both Algorithm 1 and (Khamaru and Wainwright, 2018, Algorithm 2). Result are shown in Figure 1. The per-iteration progress of the methods is comparable, but Algorithm (1) continues at a linear rate to a lower value of the objective, while (Khamaru and Wainwright, 2018, Algorithm 2) cannot get past a certain local minimum. This comparison is very brief; we leave a detailed study comparing Algorithm 1 with DC-based algorithms left to future work focusing on algorithms, along with further analysis of Algorithm 1 and its variants under the Kurdyka-Lojasiewicz assumption Attouch et al. (2013).
5 Experimental Results
Simulations for Lasso.
We run two experiments with the least squares linear regression. For all simulations we consider the regularization parameter range for all comparison methods, and we fix MCP and SCAD parameters to be 2.5 and 3.0, respectively (Since the results are not sensitive to them). We use two classes of matrices and introduced in Loh and Wainwright (2017). For each simulation, we compare the probability of recovering the correct support and the -error as a function of iterations , and we check the consistency of stationary points.
In our first simulation, we generate i.i.d. observations from where , with = 0.7. Note that this specific choice of satisfies the incoherence condition. We give non-zero values at only random positions with distribution
, and the response variables are generated by, where . Figure 2 summarizes our first simulation results. In the first row, we set respectively and increases the number of samples, . We observe that the probability of correct support recovery for trimmed Lasso is higher than standard Lasso with any number of samples in all cases. In the second row, the first graph shows that our estimator with Trimmed -penalty converges to the same stationary point with the correct support regardless of initializations, which agrees with our Corollary 3. Moreover, we can check that our Trimmed -penalty is superior to MCP and SCAD penalties in terms of -error plots.
In our second simulations, we replace covariance matrix with , which does not satisfy incoherence condition. ( is a matrix with ’s on the diagonal, ’s in the first positions of the row and column, and ’s everywhere else.) We use . Figure 3 illustrates the second simulation results. Since does not satisfy the incoherence conditions vanilla Lasso fails all the time, we focus on comparisons between Trimmed -penalty and other non-convex penalties, MCP and SCAD. In the first row, we can observe that Trimmed Lasso slightly outperforms MCP and SCAD penalties in terms of probability of successful support recovery for all cases. As in first simulations, we can see in second row that our estimator with Trimmed -penalty has smallest -errors among nonconvex penalties and recovers correct support consistently.
Due to space constraints, experiments on sparse Gaussian Graphical Models and on real data are provided as supplementary materials.
Simulations for Gaussian Graphical Models.
We consider the “diamond” graph example described in Ravikumar et al. (2011) (section 3.1.1) to assess the performance of Graphical Trimmed Lasso when the incoherence condition holds and when it is violated. Specifically, we consider a graph , with vertex set and with all edges except . We consider a family of true covariance matrices with diagonal entries for all ; off-diagonal elements for all edges ; ; and finally the entry corresponding to the non-edge is set as We analyze the performance of Graphical Trimmed Lasso under two settings: As discussed in Ravikumar et al. (2011), if the incoherence condition is satisfied ; if it is violated. Under both settings, we report the probability of successful support recovery based on 100 replicate experiments for and and compare it with Graphical Lasso, Graphical SCAD and Graphical MCP (The MCP and SCAD parameters were set to 2.5 and 3.0 as varying these did not affect the results signifcantly). For each method and replicate experiment, success is declared if the true support is recovered for at least one value of along the solution path. We can see that for a wide range of values for the trimming parameter, Graphical Trimmed Lasso outperforms SCAD and MCP alternatives regardless of whether the incoherence condition holds or not. In addition its probability of success is always superior to that of vanilla Graphical Lasso, which fails to recover the true support when the incoherence condition is violated.
6 Concluding Remarks
In this paper we have studied high-dimensional M-estimators with trimmed penalty. By leaving the largest parameter entries penalty-free, these estimators alleviate the bias incurred by the vanilla penalty. Our theoretical results in terms of support recovery and error bounds hold for any local optimum and are competitive with other non-convex approaches. In addition they indicate a perhaps surprising robustness of the procedure with respect to the trimming parameter These findings were corroborated by extensive simulation experiments. As future work we plan to generalize our study to the trimming of other decomposable regularizers such as mixed norms and to unsupervised approaches such as convex clustering with fusion penalties Radchenko and Mukherjee (2017).
We have also developed a provably convergent customized algorithm for the trimmed problem. The algorithm and analysis technique are based on problem structure rather than a simple DC structure, and appears to give promising numerical results. We expect that the approach will be useful for more general regularizes, and thorough comparison to DC based approaches is left to future work.
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Appendix A Sparse graphical models
We also derive the corollary for the trimmed Graphical lasso (5). Following the proof strategy derived in Loh and Wainwright (2017), we assume throughout that the sample size scales with the row sparsity of true parameter, inverse covariance , which is milder condition than other works ( scaling with , the number of non zero entries of ): Consider the program (5) where the ’s are drawn from a sub-Gaussian, and a sample size is greater than . Suppose further that we choose and satisfying
for any selection of ,
Further suppose that is lower bounded by for some constant . Then with high probability at least , any local minimum of (4) has the following property:
for every pair , we have ,
if , all are successfully estimated as zero and we have
if , at least the smallest entries in have exactly zero and we have
Note that the condition (1) is the incoherence condition studied in Ravikumar et al. (2011), and results are consistent with the sparse linear model case above and comparable to counterparts for or regularized Glasso Loh and Wainwright (2017).
Appendix B Proofs
b.1 Proof of Theorem 3
We extend the standard PDW technique Wainwright (2009c); Yang et al. (2015); Loh and Wainwright (2017) for the trimmed regularizers. For any fixed , we construct a primal and dual witness pair with the strict dual feasibility. Specifically, given the fixed , consider the following program:
Note that the program (19) is convex (under 1) where the regularizer is only effective over entries in (fixed) . We construct the primal and dual pair following (6) and (7). The following lemma can guarantee under the strict dual feasibility that any solution of (19) has the same sparsity structure on with . Moreover, since the restricted program (7) is strictly convex as shown in the lemma below, we can conclude that is the unique minimum point of the restricted program (19) given .
Suppose that there exists a primal optimal solution for (19) with associated sub-gradient (or dual) such that . Then any optimal solution of (19) will satisfy for all . The lemma can be directly achieved by the basic property of convex optimization problem, as developed in existing works using PDW Wainwright (2009c); Yang et al. (2015). Note that even though the original problem with the trimmed regularizer is not convex, (19) given is convex. Therefore, by complementary slackness, we have . Therefore, any optimal solution of (19) will satisfy for all since the associated (absolute) sub-gradient is strictly smaller than 1 by the assumption in the statement.
Now from the definition of , we have
Since both and are zero vectors, we obtain
Therefore, under the assumption on in the statement, the selection of in which there exists some s.t. , , and , yields contradictory solution with (2). Under the strict dual feasibility condition for this specific choice of (along with Lemma B.1) can guarantee that there is no local minimum for that choice of . Hence, (B.1) can guarantee that for every pair such that and , we have (since ). Note that for any valid selection of , this statement holds. This immediately implies that any local minimum of (2) satisfies this property as well, as in the statement.
Finally turning to the bound when , we have since all entries in are not penalized as shown above. In this case, becomes zero vector (since is empty in the construction of ), and the bound in (B.1) will be tighter as
b.2 Proof of Corollary 3
The proof our corollary is similar to that of Corollary 1 of Loh and Wainwright (2017), who derive the result for -amenable regularizers. Here we only describe the parts that need to be modified from Loh and Wainwright (2017).
In order to utilize theorems in the main paper, we need to establish the RSC condition 2 and the strict dual feasibility (8). First, the RSC is known to hold w.h.p as shown in several previous works such as Lemma B.2.
In order to show the remaining strict dual feasibility condition of our PDW construction, we consider (20) (by the zero-subgradient and the definition of ) in the block form:
By simple manipulation, we can obtain
where is an orthogonal project matrix on : .
For any , we define such that . Then we have
Hence by the sub-Gaussian tail bounds followed by a union bound, we can conclude that
with probability at least for all selections of . We can establish have strict dual feasibility for any selection of w.h.p, provided , and now turn to bounds. From (10), we have
Then for , we define such that . Since for any selection of , is bounded as follows:
Similarly by the sub-Gaussian tail bound and a union bound over , we can obtain
with probability at least .
b.3 Proof of Corollary A
As in the proof of Corollary 3, the proof procedure is quite similar to that of Corollary 4 of Loh and Wainwright (2017). Deriving upper bounds on in Loh and Wainwright (2017) can be seamlessly extendable to upper bounds on for any selection of . mainly because the required upper bounds are related to entry-wise maximum on the true support but entry-wise maximum in this case is uniformly upper bounded for all entries.
Specifically, it computes the upper bound of from the fact that . This actually holds for any selection of . Similarly, it computes the upper bound of by Hölder’s inequality and the definition of matrix induced norms: , which clearly holds for any index beyond . Finally, is shown to be upper bounded by the fact that .
The remaining proof of this result directly follows similar lines to the proof of Corollary 4 in Loh and Wainwright (2017).
b.4 Proof of Theorem 4
From Algorithm 1, we obtain the relation
from the proximal gradient steps.
At -th iteration, we have