FLamby: Datasets and Benchmarks for Cross-Silo Federated Learning in Realistic Healthcare Settings

Federated Learning (FL) is a novel approach enabling several clients holding sensitive data to collaboratively train machine learning models, without centralizing data. The cross-silo FL setting corresponds to the case of few ($2$--$50$) reliable clients, each holding medium to large datasets, and is typically found in applications such as healthcare, finance, or industry. While previous works have proposed representative datasets for cross-device FL, few realistic healthcare cross-silo FL datasets exist, thereby slowing algorithmic research in this critical application. In this work, we propose a novel cross-silo dataset suite focused on healthcare, FLamby (Federated Learning AMple Benchmark of Your cross-silo strategies), to bridge the gap between theory and practice of cross-silo FL. FLamby encompasses 7 healthcare datasets with natural splits, covering multiple tasks, modalities, and data volumes, each accompanied with baseline training code. As an illustration, we additionally benchmark standard FL algorithms on all datasets. Our flexible and modular suite allows researchers to easily download datasets, reproduce results and re-use the different components for their research. FLamby is available at~\url{www.github.com/owkin/flamby}.

SecureFedYJ: a safe feature Gaussianization protocol for Federated Learning

The Yeo-Johnson (YJ) transformation is a standard parametrized per-feature unidimensional transformation often used to Gaussianize features in machine learning. In this paper, we investigate the problem of applying the YJ transformation in a cross-silo Federated Learning setting under privacy constraints. For the first time, we prove that the YJ negative log-likelihood is in fact convex, which allows us to optimize it with exponential search. We numerically show that the resulting algorithm is more stable than the state-of-the-art approach based on the Brent minimization method. Building on this simple algorithm and Secure Multiparty Computation routines, we propose SecureFedYJ, a federated algorithm that performs a pooled-equivalent YJ transformation without leaking more information than the final fitted parameters do. Quantitative experiments on real data demonstrate that, in addition to being secure, our approach reliably normalizes features across silos as well as if data were pooled, making it a viable approach for safe federated feature Gaussianization.

Near-optimal estimation of smooth transport maps with kernel sums-of-squares

It was recently shown that under smoothness conditions, the squared Wasserstein distance between two distributions could be efficiently computed with appealing statistical error upper bounds. However, rather than the distance itself, the object of interest for applications such as generative modeling is the underlying optimal transport map. Hence, computational and statistical guarantees need to be obtained for the estimated maps themselves. In this paper, we propose the first tractable algorithm for which the statistical $L^2$ error on the maps nearly matches the existing minimax lower-bounds for smooth map estimation. Our method is based on solving the semi-dual formulation of optimal transport with an infinite-dimensional sum-of-squares reformulation, and leads to an algorithm which has dimension-free polynomial rates in the number of samples, with potentially exponentially dimension-dependent constants.

Learning PSD-valued functions using kernel sums-of-squares

Shape constraints such as positive semi-definiteness (PSD) for matrices or convexity for functions play a central role in many applications in machine learning and sciences, including metric learning, optimal transport, and economics. Yet, very few function models exist that enforce PSD-ness or convexity with good empirical performance and theoretical guarantees. In this paper, we introduce a kernel sum-of-squares model for functions that take values in the PSD cone, which extends kernel sums-of-squares models that were recently proposed to encode non-negative scalar functions. We provide a representer theorem for this class of PSD functions, show that it constitutes a universal approximator of PSD functions, and derive eigenvalue bounds in the case of subsampled equality constraints. We then apply our results to modeling convex functions, by enforcing a kernel sum-of-squares representation of their Hessian, and show that any smooth and strongly convex function may be thus represented. Finally, we illustrate our methods on a PSD matrix-valued regression task, and on scalar-valued convex regression.

A Note on Optimizing Distributions using Kernel Mean Embeddings

Kernel mean embeddings are a popular tool that consists in representing probability measures by their infinite-dimensional mean embeddings in a reproducing kernel Hilbert space. When the kernel is characteristic, mean embeddings can be used to define a distance between probability measures, known as the maximum mean discrepancy (MMD). A well-known advantage of mean embeddings and MMD is their low computational cost and low sample complexity. However, kernel mean embeddings have had limited applications to problems that consist in optimizing distributions, due to the difficulty of characterizing which Hilbert space vectors correspond to a probability distribution. In this note, we propose to leverage the kernel sums-of-squares parameterization of positive functions of Marteau-Ferey et al. [2020] to fit distributions in the MMD geometry. First, we show that when the kernel is characteristic, distributions with a kernel sum-of-squares density are dense. Then, we provide algorithms to optimize such distributions in the finite-sample setting, which we illustrate in a density fitting numerical experiment.

Dimension-free convergence rates for gradient Langevin dynamics in RKHS

Gradient Langevin dynamics (GLD) and stochastic GLD (SGLD) have attracted considerable attention lately, as a way to provide convergence guarantees in a non-convex setting. However, the known rates grow exponentially with the dimension of the space. In this work, we provide a convergence analysis of GLD and SGLD when the optimization space is an infinite dimensional Hilbert space. More precisely, we derive non-asymptotic, dimension-free convergence rates for GLD/SGLD when performing regularized non-convex optimization in a reproducing kernel Hilbert space. Amongst others, the convergence analysis relies on the properties of a stochastic differential equation, its discrete time Galerkin approximation and the geometric ergodicity of the associated Markov chains.

Missing Data Imputation using Optimal Transport

Missing data is a crucial issue when applying machine learning algorithms to real-world datasets. Starting from the simple assumption that two batches extracted randomly from the same dataset should share the same distribution, we leverage optimal transport distances to quantify that criterion and turn it into a loss function to impute missing data values. We propose practical methods to minimize these losses using end-to-end learning, that can exploit or not parametric assumptions on the underlying distributions of values. We evaluate our methods on datasets from the UCI repository, in MCAR, MAR and MNAR settings. These experiments show that OT-based methods match or out-perform state-of-the-art imputation methods, even for high percentages of missing values.