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
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.