Author(s): Bradic, Jelena; Wager, Stefan; Zhu, Yinchu | Abstract: Many popular methods for building confidence intervals on causal effects under high-dimensional confounding require strong "ultra-sparsity" assumptions that may be difficult to validate in practice. To alleviate this difficulty, we here study a new method for average treatment effect estimation that yields asymptotically exact confidence intervals assuming that either the conditional response surface or the conditional probability of treatment allows for an ultra-sparse representation (but not necessarily both). This guarantee allows us to provide valid inference for average treatment effect in high dimensions under considerably more generality than available baselines. In addition, we showcase that our results are semi-parametrically efficient.
Our result is most closely related to a recent proposal by , who studied any linear functional whose Riesz representer admits an (approximate) linear representation. In another paper, considers theoretical results for estimators based on learning conditional mean function and the propensity score. In both papers, the key condition is that the product of @math -loss for learning the two nuisance parameters is @math , a condition referred to as rate double robustness; see Definition 2 in @cite_4 . Sufficient conditions for rate double robustness have been provided in these works in terms of sparsity levels. For example, Remark 5.2 of shows that rate double robustness is guaranteed when the product of two sparsity levels is @math , while Remark 7 of points out that under the assumption of bounded @math -norm of both parameters, rate double robustness holds whenever one of the sparsity levels is @math .
@cite_4
2127300249 2963653702 1964809923 2933844681
The problem of consistently estimating the sparsity pattern of a vector β* ∈ RP based on observations contaminated by noise arises in various contexts, including signal denoising, sparse approximation, compressed sensing, and model selection. We analyze the behavior of l1-constrained quadratic programming (QP), also referred to as the Lasso, for recovering the sparsity pattern. Our main result is to establish precise conditions on the problern dimension p, the number k of nonzero elements in β*, and the number of observations n that are necessary and sufficient for sparsity pattern recovery using the Lasso. We first analyze the case of observations made using deterministic design matrices and sub-Gaussian additive noise, and provide sufficient conditions for support recovery and l∞-error bounds, as well as results showing the necessity of incoherence and bounds on the minimum value. We then turn to the case of random designs, in which each row of the design is drawn from a N(0, Σ) ensemble. For a broad class of Gaussian ensembles satisfying mutual incoherence conditions, we compute explicit values of thresholds 0 0, if n > 2(θu + δ)k log(p - k), then the Lasso succeeds in recovering the sparsity pattern with probability converging to one for large problems, whereas for n < 2(θl - δ)k log(p - k), then the probability of successful recovery converges to zero. For the special case of the uniform Gaussian ensemble (Σ = I p×p), we show that θl = θu = 1, so that the precise threshold n = 2k log (p - k) is exactly determined. Convex sparsity-inducing regularizations are ubiquitous in high-dimensional machine learning, but solving the resulting optimization problems can be slow. To accelerate solvers, state-of-the-art approaches consist in reducing the size of the optimization problem at hand. In the context of regression, this can be achieved either by discarding irrelevant features (screening techniques) or by prioritizing features likely to be included in the support of the solution (working set techniques). Duality comes into play at several steps in these techniques. Here, we propose an extrapolation technique starting from a sequence of iterates in the dual that leads to the construction of improved dual points. This enables a tighter control of op-timality as used in stopping criterion, as well as better screening performance of Gap Safe rules. Finally, we propose a working set strategy based on an aggressive use of Gap Safe screening rules. Thanks to our new dual point construction, we show significant computational speedups on multiple real-world problems. This paper explores a class of empirical Bayes methods for level-dependent threshold selection in wavelet shrinkage. The prior considered for each wavelet coefficient is a mixture of an atom of probability at zero and a heavy-tailed density. The mixing weight, or sparsity parameter, for each level of the transform is chosen by marginal maximum likelihood. If estimation is carried out using the posterior median, this is a random thresholding procedure; the estimation can also be carried out using other thresholding rules with the same threshold. Details of the calculations needed for implementing the procedure are included. In practice, the estimates are quick to compute and there is software available. Simulations on the standard model functions show excellent performance, and applications to data drawn from various fields of application are used to explore the practical performance of the approach. By using a general result on the risk of the corresponding marginal maximum likelihood approach for a single sequence, overall bounds on the risk of the method are found subject to membership of the unknown function in one of a wide range of Besov classes, covering also the case of f of bounded variation. The rates obtained are optimal for any value of the parameter p in (0, ∞], simultaneously for a wide range of loss functions, each dominating the Lq norm of the σth derivative, with σ ≥ 0 and 0 < q ≤ 2. Attention is paid to the distinction between sampling the unknown function within white noise and sampling at discrete points, and between placing constraints on the function itself and on the discrete wavelet transform of its sequence of values at the observation points. Results for all relevant combinations of these scenarios are obtained. In some cases a key feature of the theory is a particular boundary-corrected wavelet basis, details of which are discussed. Overall, the approach described seems so far unique in combining the properties of fast computation, good theoretical properties and good performance in simulations and in practice. A key feature appears to be that the estimate of sparsity adapts to three different zones of estimation, first where the signal is not sparse enough for thresholding to be of benefit, second where an appropriately chosen threshold results in substantially improved estimation, and third where the signal is so sparse that the zero estimate gives the optimum accuracy rate. In this article we study a class of parameters with the so-called mixed bias property'. For parameters with this property, the bias of the semiparametric efficient one step estimator is equal to the mean of the product of the estimation errors of two nuisance functions. In non-parametric models, parameters with the mixed bias property admit so-called rate doubly robust estimators, i.e. estimators that are consistent and asymptotically normal when one succeeds in estimating both nuisance functions at sufficiently fast rates, with the possibility of trading off slower rates of convergence for the estimator of one of the nuisance functions with faster rates for the estimator of the other nuisance. We show that the class of parameters with the mixed bias property strictly includes two recently studied classes of parameters which, in turn, include many parameters of interest in causal inference. We characterize the form of parameters with the mixed bias property and of their influence functions. Furthermore, we derive two functional moment equations, each being solved at one of the two nuisance functions, as well as, two functional loss functions, each being minimized at one of the two nuisance functions. These loss functions can be used to derive loss based penalized estimators of the nuisance functions.