Optimal prediction for sparse linear models? Lower bounds for coordinate-separable M-estimators

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For the problem of high-dimensional sparse linear regression, it is known that an ℓ₀-based estimator can achieve a 1/n "fast" rate on the prediction error without any conditions on the design matrix, whereas in absence of restrictive conditions on the design matrix, popular polynomial-time methods only guarantee the 1/√n "slow" rate. In this paper, we show that the slow rate is intrinsic to a broad class of M-estimators. In particular, for estimators based on minimizing a least-squares cost function together with a (possibly non-convex) coordinate-wise separable regularizer, there is always a "bad" local optimum such that the associated prediction error is lower bounded by a constant multiple of 1/√n. For convex regularizers, this lower bound applies to all global optima. The theory is applicable to many popular estimators, including convex ℓ₁-based methods as well as M-estimators based on nonconvex regularizers, including the SCAD penalty or the MCP regularizer. In addition, for a broad class of nonconvex regularizers, we show that the bad local optima are very common, in that a broad class of local minimization algorithms with random initialization will typically converge to a bad solution.