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We consider a general class of regression models with normally distributedcovariates, and the associated nonconvex problem of fitting these models fromdata. We develop a general recipe for analyzing the convergence of iterativealgorithms for this task from a random initialization. In particular, providedeach iteration can be written as the solution to a convex optimization problemsatisfying some natural conditions, we leverage Gaussian comparison theorems toderive a deterministic sequence that provides sharp upper and lower bounds onthe error of the algorithm with sample-splitting. Crucially, this deterministicsequence accurately captures both the convergence rate of the algorithm and theeventual error floor in the finite-sample regime, and is distinct from thecommonly used "population" sequence that results from taking theinfinite-sample limit. We apply our general framework to derive severalconcrete consequences for parameter estimation in popular statistical modelsincluding phase retrieval and mixtures of regressions. Provided the sample sizescales near-linearly in the dimension, we show sharp global convergence ratesfor both higher-order algorithms based on alternating updates and first-orderalgorithms based on subgradient descent. These corollaries, in turn, yieldmultiple consequences, including: (a) Proof that higher-order algorithms canconverge significantly faster than their first-order counterparts (andsometimes super-linearly), even if the two share the same population update and(b) Intricacies in super-linear convergence behavior for higher-orderalgorithms, which can be nonstandard (e.g., with exponent 3/2) and sensitive tothe noise level in the problem. We complement these results with extensivenumerical experiments, which show excellent agreement with our theoreticalpredictions.

Sharp global convergence guarantees for iterative nonconvex optimization: A Gaussian process perspective