The Adaptive Lasso and Its Oracle Properties

By Hui Zou. 2006

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Subfields: Economics and Econometrics, Management Science and Operations Research, Statistics and Probability

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Citations in the corpus

Learning Credible Models (2017, methodology): “The most common forms of regularization, l 1 (LASSO) and l 2 θ represents the model parameters; β ∈ [ 0 , 1 ] is a hyperparameter that controls the tradeoff between the l 1 and l 2 norms; w is a set of non-negative weights for each feature; w ∗ | θ | [ i ] is the i th largest parameter sorted by mag”
Continual Learning with Node-Importance based Adaptive Group Sparse Regularization (2020, methodology): “Namely, for the unimportant nodes in G t−1 0 , we apply the group Lasso penalty (term (a)) as in [4], and for the important nodes in G\Gt−1 0 , we apply the group-sparsity based deviation penalty (term (b)) that adaptively penalizes (similarly as in [43, 35]) the deviation of θn` from θ̂ (t−1) n` de”
Regularization Paths for Generalized Linear Models via Coordinate Descent (2010, methodology): “Penalty rescaling would also allow, for example, our software to be used to implement the adaptive lasso (Zou 2006).”
CAM: Causal Additive Models, high-dimensional order search and penalized regression (2013, background): “The former is a well understood subject thanks to the broad literature in sparse regression and related techniques [17, 36, 41, 44, 47, 48], cf.”
Sparse additive models (2007, background): “Here, supp(β) = {j : βj 6= 0}. References include Wainwright (2006), Meinshausen and P. B¨uhlmann (2006), Zou (2005) and Zhao and Yu (2007).”
Stability selection (2008, methodology): “For the lasso, it amounts to the ‘neighbourhood stability’ condition (Meinshausen and Bühlmann, 2006), which is equivalent to the ‘irrepresentable condition’ (Zhao and Yu, 2006; Zou, 2006; Yuan and Lin, 2007).”
Nearly unbiased variable selection under minimax concave penalty (2010, background): “Zou (2006) proposed an adaptive LASSO with λ(2)ρ1(|βj |rj/λ) = λrj |βj |, where rj is a decreasing function of an initial estimate of βj .”
High-dimensional additive modeling (2008, methodology): “Similar to the adaptive lasso [34], we can also use different penalties for the different components, that is, use a penalty of the form J (fj ) = λ1 √ w1,j‖fj‖2n + λ2w2,j I (2)(fj ), where the weights w1,j and w2,j are ideally chosen in a data-adaptive way.”
A Simple Method for Detecting Interactions between a Treatment and a Large Number of Covariates (2012, methodology): “Therefore, one could also apply the adaptive Lasso procedure [Zou, 2006] with feature weights ŵj proportional to the reciprocal of the univariate “association strength” between the outcome Y and the jth component of W(Z).”
On the conditions used to prove oracle results for the Lasso (2009, background): “The name of the latter is inspired by the fact that this strengthened version is useful for the development of theory for the adaptive Lasso [24] which we do not show in this paper.”
PERFORMANCE GUARANTEES FOR INDIVIDUALIZED TREATMENT RULES (2011, methodology): “In fact, many other penalization methods such as SCAD [9] and l1 penalty with adaptive weights (adaptive Lasso; [37]) also have this property.”
Electronic medical records for discovery research in rheumatoid arthritis (2010, positive): “To avoid overfitting the model, we used the adaptive lasso procedure, which simultaneously identifies influential variables and provides stable estimates of the model parameters (25).”
Combinatorial Penalties: Structure preserved by convex relaxations (2017, background): “The following Theorem extends the results from [37] for the `1-norm.”
On the non‐negative garrotte estimator (2007, methodology): “In particular, Zou (2006) showed that a necessary condition for the lasso to be consistent in variable selection is max j ∈I {cov.”
Computational Statistics and Data Analysis a Note on Adaptive Group Lasso (neutral): “However, recent research suggests that the traditional lasso estimator may not be fully efficient (Fan and Li, 2001), and its model selection result could be inconsistent (Leng et al., 2006; Yuan and Lin, 2007; Zou, 2006).”
Detection of Differential Item Functioning Using the Lasso Approach (2015, background): “For instance, Zou (2006) introduced an adaptive lasso penalization in which the penalty parameter may vary depending on the parameters.”
$\ell_0$-penalized maximum likelihood for sparse directed acyclic graphs (2012, background): “In addition, the price to pay, in a standard regression problem with an l1-norm regularization, is a bias which should be further controlled with, for example, adaptive l1-norm regularization [Zou (2006)] or thresholding [van de Geer, Bühlmann and Zhou (2011)].”
A General Theory of Concave Regularization for High-Dimensional Sparse Estimation Problems (2011, methodology): “A more direct way to reduce the bias of the Lasso is via the adaptive Lasso procedure [50], which solves the following weighted 1 regularization problem for some α ∈ (0,1):”