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In this work, we aim to characterize the statistical complexity of realizable regression both in the PAC learning setting and the online learning setting. Previous work had established the sufficiency of finiteness of the fat shattering dimension for PAC learnability and the necessity of finiteness of the scaled Natarajan dimension, but little progress had been made towards a more complete characterization since the work of Simon (SICOMP '97). To this end, we first introduce a minimax instance optimal learner for realizable regression and propose a novel dimension that both qualitatively and quantitatively characterizes which classes of real-valued predictors are learnable. We then identify a combinatorial dimension related to the Graph dimension that characterizes ERM learnability in the realizable setting. Finally, we establish a necessary condition for learnability based on a combinatorial dimension related to the DS dimension, and conjecture that it may also be sufficient in this context. Additionally, in the context of online learning we provide a dimension that characterizes the minimax instance optimal cumulative loss up to a constant factor and design an optimal online learner for realizable regression, thus resolving an open question raised by Daskalakis and Golowich in STOC '22.

This report presents an automatic evaluation of the general machine translation task of the Seventh Conference on Machine Translation (WMT22). It evaluates a total of 185 systems for 21 translation directions including high-resource to low-resource language pairs and from closely related to distant languages. This large-scale automatic evaluation highlights some of the current limits of state-of-the-art machine translation systems. It also shows how automatic metrics, namely chrF, BLEU, and COMET, can complement themselves to mitigate their own limits in terms of interpretability and accuracy.