[1903.01777] A New Approach to Adaptive Data Analysis and Learning via Maximal Leakage
(5) From this we can derive that if each of the Ai has a bounded leakage (and thus, generalizes), even the adaptive composition of the whole sequence will have bounded leakage (although, with a worse bound) and potentially maintain the generalization guarantees and avoid overfitting

Abstract: There is an increasing concern that most current published research findings
are false. The main cause seems to lie in the fundamental disconnection between
theory and practice in data analysis. While the former typically relies on
statistical independence, the latter is an inherently adaptive process: new
hypotheses are formulated based on the outcomes of previous analyses. A recent
line of work tries to mitigate these issues by enforcing constraints, such as
differential privacy, that compose adaptively while degrading gracefully and
thus provide statistical guarantees even in adaptive contexts. Our contribution
consists in the introduction of a new approach, based on the concept of Maximal
Leakage, an information-theoretic measure of leakage of information. The main
result allows us to compare the probability of an event happening when
adaptivity is considered with respect to the non-adaptive scenario. The bound
we derive represents a generalization of the bounds used in non-adaptive
scenarios (e.g., McDiarmid's inequality for $c$-sensitive functions, false
discovery error control via significance level, etc.), and allows us to
replicate or even improve, in certain regimes, the results obtained using
Max-Information or Differential Privacy. In contrast with the line of work
started by Dwork et al., our results do not rely on Differential Privacy but
are, in principle, applicable to every algorithm that has a bounded leakage,
including the differentially private algorithms and the ones with a short
description length.