    # Statistically Optimal First Order Algorithms: A Proof via Orthogonalization

We consider a class of statistical estimation problems in which we are given a random data matrix X∈ℝ^n× d (and possibly some labels y∈ℝ^n) and would like to estimate a coefficient vector θ∈ℝ^d (or possibly a constant number of such vectors). Special cases include low-rank matrix estimation and regularized estimation in generalized linear models (e.g., sparse regression). First order methods proceed by iteratively multiplying current estimates by X or its transpose. Examples include gradient descent or its accelerated variants. Celentano, Montanari, Wu proved that for any constant number of iterations (matrix vector multiplications), the optimal first order algorithm is a specific approximate message passing algorithm (known as `Bayes AMP'). The error of this estimator can be characterized in the high-dimensional asymptotics n,d→∞, n/d→δ, and provides a lower bound to the estimation error of any first order algorithm. Here we present a simpler proof of the same result, and generalize it to broader classes of data distributions and of first order algorithms, including algorithms with non-separable nonlinearities. Most importantly, the new proof technique does not require to construct an equivalent tree-structured estimation problem, and is therefore susceptible of a broader range of applications.

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