Stochastic linear optimization never overfits with quadratically-bounded losses on general data

This work provides test error bounds for iterative fixed point methods on linear predictors — specifically, stochastic and batch mirror descent (MD), and stochastic temporal difference learning (TD) — with two core contributions: (a) a single proof technique which gives high probability guarantees despite the absence of projections, regularization, or any equivalents, even when optima have large or infinite norm, for quadratically-bounded losses (e.g., providing unified treatment of squared and logistic losses); (b) locally-adapted rates which depend not on global problem structure (such as conditions numbers and maximum margins), but rather on properties of low norm predictors which may suffer some small excess test error. The proof technique is an elementary and versatile coupling argument, and is demonstrated here in the following settings: stochastic MD under realizability; stochastic MD for general Markov data; batch MD for general IID data; stochastic MD on heavy-tailed data (still without projections); stochastic TD on approximately mixing Markov chains (all prior stochastic TD bounds are in expectation).