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We study problem-dependent rates, i.e., generalization errors that scalenear-optimally with the variance, the effective loss, or the gradient normsevaluated at the "best hypothesis." We introduce a principled framework dubbed"uniform localized convergence," and characterize sharp problem-dependent ratesfor central statistical learning problems. From a methodological viewpoint, ourframework resolves several fundamental limitations of existing uniformconvergence and localization analysis approaches. It also provides improvementsand some level of unification in the study of localized complexities, one-sideduniform inequalities, and sample-based iterative algorithms. In the so-called"slow rate" regime, we provides the first (moment-penalized) estimator thatachieves the optimal variance-dependent rate for general "rich" classes; wealso establish improved loss-dependent rate for standard empirical riskminimization. In the "fast rate" regime, we establish finite-sampleproblem-dependent bounds that are comparable to precise asymptotics. Inaddition, we show that iterative algorithms like gradient descent andfirst-order Expectation-Maximization can achieve optimal generalization errorin several representative problems across the areas of non-convex learning,stochastic optimization, and learning with missing data.

Towards Optimal Problem Dependent Generalization Error Bounds in Statistical Learning Theory