1,335 research outputs found
Simple Analysis of Sparse, Sign-Consistent JL
Allen-Zhu, Gelashvili, Micali, and Shavit construct a sparse, sign-consistent Johnson-Lindenstrauss distribution, and prove that this distribution yields an essentially optimal dimension for the correct choice of sparsity. However, their analysis of the upper bound on the dimension and sparsity requires a complicated combinatorial graph-based argument similar to Kane and Nelson\u27s analysis of sparse JL. We present a simple, combinatorics-free analysis of sparse, sign-consistent JL that yields the same dimension and sparsity upper bounds as the original analysis. Our analysis also yields dimension/sparsity tradeoffs, which were not previously known.
As with previous proofs in this area, our analysis is based on applying Markov\u27s inequality to the pth moment of an error term that can be expressed as a quadratic form of Rademacher variables. Interestingly, we show that, unlike in previous work in the area, the traditionally used Hanson-Wright bound is not strong enough to yield our desired result. Indeed, although the Hanson-Wright bound is known to be optimal for gaussian degree-2 chaos, it was already shown to be suboptimal for Rademachers. Surprisingly, we are able to show a simple moment bound for quadratic forms of Rademachers that is sufficiently tight to achieve our desired result, which given the ubiquity of moment and tail bounds in theoretical computer science, is likely to be of broader interest
Individual Fairness in Pipelines
It is well understood that a system built from individually fair components
may not itself be individually fair. In this work, we investigate individual
fairness under pipeline composition. Pipelines differ from ordinary sequential
or repeated composition in that individuals may drop out at any stage, and
classification in subsequent stages may depend on the remaining "cohort" of
individuals. As an example, a company might hire a team for a new project and
at a later point promote the highest performer on the team. Unlike other
repeated classification settings, where the degree of unfairness degrades
gracefully over multiple fair steps, the degree of unfairness in pipelines can
be arbitrary, even in a pipeline with just two stages.
Guided by a panoply of real-world examples, we provide a rigorous framework
for evaluating different types of fairness guarantees for pipelines. We show
that na\"{i}ve auditing is unable to uncover systematic unfairness and that, in
order to ensure fairness, some form of dependence must exist between the design
of algorithms at different stages in the pipeline. Finally, we provide
constructions that permit flexibility at later stages, meaning that there is no
need to lock in the entire pipeline at the time that the early stage is
constructed
Independence and concurrent separation logic
A compositional Petri net-based semantics is given to a simple language
allowing pointer manipulation and parallelism. The model is then applied to
give a notion of validity to the judgements made by concurrent separation logic
that emphasizes the process-environment duality inherent in such rely-guarantee
reasoning. Soundness of the rules of concurrent separation logic with respect
to this definition of validity is shown. The independence information retained
by the Petri net model is then exploited to characterize the independence of
parallel processes enforced by the logic. This is shown to permit a refinement
operation capable of changing the granularity of atomic actions
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