Sublinear Algorithms for Wasserstein and Total Variation Distances: Applications to Fairness and Privacy Auditing

Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: I'm Tom, and with me are Jane, Lu, senior AI researcher at Tsinghua, Meng, lead engineer at a mysterious AI startup and Lalam, the in-house Large Language Model.

Jane: Today's paper: "Sublinear Algorithms for Wasserstein and Total Variation Distances".

Tom: The gist: The authors propose a generic framework to learn mergeable sublinear summaries of probability distributions from data streams, enabling sublinear time, space,

Jane: First, who's behind it and why it matters.

Title and authors: Tom: This paper, "Sublinear Algorithms for Wasserstein and Total Variation Distances: Applications to Fairness and Privacy Auditing," focuses on creating tools that can estimate these distribution distances using sublinear space relative to the number of observed samples.

Jane: And they're showing how this framework lets us compute mergeable summaries of distributions from multiple streams, which is a big deal for real-world data coming from different sources.

Lu: The authors are proposing algorithms called SPA and SCA that can approximate the PDF and CDF independently of the stream length.

Meng: This means they are aiming for sublinear space complexity relative to the number of samples, which directly addresses a major bottleneck in current methods.

Lalam: They use this reduction to compute mergeable summaries from the stream, allowing us to estimate Wasserstein and Total Variation distances between any two distributions even when the samples arrive from multiple sources.

The paper's summary: Tom: So, what does their summary of the work actually show us? They are showing that this framework allows for estimating Wasserstein and Total Variation distances between any two distributions while samples arrive in streams and from multiple sources.

Jane: It’s significant because existing methods often require linear space complexity and super-linear time complexity in terms of observed samples to get those Wasserstein distances when you don't know the exact parametric form of the data distribution.

Lu: They specifically use SCA to propose SWA, which PAC-estimates the Wasserstein distance by computing a sublinear summary of a distribution that is O n minus one/two close in Wasserstein distance using Oe, which is square root n, space for n samples <ref:2503.07775#pg3,a sublinear summary of a distribution that is O n>.

Meng: So they claim this SWA operates in the distributed or federated setting with a communication cost of Oe per round while still preserving that estimation guarantee.

Lalam: This sublinear complexity means we can estimate these distances much more efficiently than the previous methods that incurred a communication cost equal to the number of samples in the stream.

The paper's improvements: Tom: The paper points out several improvements they've made over existing literature, focusing on how they tackle those complexity issues we talked about earlier.

Jane: One major improvement is moving away from requiring linear space complexity for estimating the Wasserstein distance between distributions from samples. Instead, they achieve sublinear space complexity using algorithms like SWA which uses Oe, which is square root n, space for n samples while preserving estimation guarantees.

Lu: They also show that they can develop a universal estimator for any bounded-tail distribution; SWA can be used to compute Wp of mu n and mu hat n less than epsilon with probability at least one minus delta <ref:2503.07775#pg3>.

Meng: That means the required sample size is significantly lower than what's needed for exact estimation, which is good for practical applications where data collection is expensive.

Lalam: Another improvement involves using STVA to estimate the Total Variation distance, which they say is equivalent to estimating the HSD between output distributions, allowing for resource-efficient auditing by achieving a sublinear time, space, and communication complexity compared to the number of samples.

Conclusion: Tom: So wrapping things up on this paper about "Sublinear Algorithms for Wasserstein and Total Variation Distances: Applications to Fairness and Privacy Auditing," it seems the main implication is that we can estimate these critical distribution distances much more efficiently than before.

Jane: It means we can audit fairness and privacy in systems with data streams, even when we don't know the exact mathematical form of the underlying data distribution.

Lu: The authors established a generic framework for learning PDFs and CDFs that is independent of stream length, which opens up possibilities for analyzing complex continuous settings.

Meng: For me, the practical impact is that auditing fairness becomes much more feasible when dealing with real-time streams from many different sources where we can't afford linear space.

Lalam: And Lalam thinks this work is important because it provides a way to compute these distances in sublinear time and space, which should make the auditing process much less resource intensive overall.