text: One fundamental goal in any learning algorithm is to mitigate its risk for overfitting. Mathematically, this requires that the learning algorithm enjoys a small generalization risk, which is defined either in expectation or in probability. Both types of generalization are commonly used in the literature. For instance, generalization in expectation has been used to analyze algorithms, such as ridge regression and SGD, whereas generalization in probability is used in the VC theory, among others. Recently, a third notion of generalization has been studied, called uniform generalization, which requires that the generalization risk vanishes uniformly in expectation across all bounded parametric losses. It has been shown that uniform generalization is, in fact, equivalent to an information-theoretic stability constraint, and that it recovers classical results in learning theory. It is achievable under various settings, such as sample compression schemes, finite hypothesis spaces, finite domains, and differential privacy. However, the relationship between uniform generalization and concentration remained unknown. In this paper, we answer this question by proving that, while a generalization in expectation does not imply a generalization in probability, a uniform generalization in expectation does imply concentration. We establish a chain rule for the uniform generalization risk of the composition of hypotheses and use it to derive a large deviation bound. Finally, we prove that the bound is tight.
  @cite_5: Adaptivity is an important feature of data analysis---the choice of questions to ask about a dataset often depends on previous interactions with the same dataset. However, statistical validity is typically studied in a nonadaptive model, where all questions are specified before the dataset is drawn. Recent work by (STOC, 2015) and Hardt and Ullman (FOCS, 2014) initiated the formal study of this problem, and gave the first upper and lower bounds on the achievable generalization error for adaptive data analysis. Specifically, suppose there is an unknown distribution @math and a set of @math independent samples @math is drawn from @math . We seek an algorithm that, given @math as input, accurately answers a sequence of adaptively chosen queries about the unknown distribution @math . How many samples @math must we draw from the distribution, as a function of the type of queries, the number of queries, and the desired level of accuracy? In this work we make two new contributions: (i) We give upper bounds on the number of samples @math that are needed to answer statistical queries. The bounds improve and simplify the work of (STOC, 2015), and have been applied in subsequent work by those authors (Science, 2015, NIPS, 2015). (ii) We prove the first upper bounds on the number of samples required to answer more general families of queries. These include arbitrary low-sensitivity queries and an important class of optimization queries. As in , our algorithms are based on a connection with algorithmic stability in the form of differential privacy. We extend their work by giving a quantitatively optimal, more general, and simpler proof of their main theorem that stability implies low generalization error. We also study weaker stability guarantees such as bounded KL divergence and total variation distance.
  @cite_1: The problem of characterizing learnability is the most basic question of statistical learning theory. A fundamental and long-standing answer, at least for the case of supervised classification and regression, is that learnability is equivalent to uniform convergence of the empirical risk to the population risk, and that if a problem is learnable, it is learnable via empirical risk minimization. In this paper, we consider the General Learning Setting (introduced by Vapnik), which includes most statistical learning problems as special cases. We show that in this setting, there are non-trivial learning problems where uniform convergence does not hold, empirical risk minimization fails, and yet they are learnable using alternative mechanisms. Instead of uniform convergence, we identify stability as the key necessary and sufficient condition for learnability. Moreover, we show that the conditions for learnability in the general setting are significantly more complex than in supervised classification and regression.
summary: Besides deriving a concentration bound, we also establish that our bound is tight. This tightness result is inspired by the work of @cite_5 (Lemma 7.4) and @cite_1 (Example 3), where similar results are established for differential privacy and learnability respectively. In Section , we combine techniques from both works to show that our concentration bound is indeed tight.