High-Dimensional Procrustes Matching via Tree Counts

Suppose we observe two sets of $n$ Gaussian vectors in $\mathbb{R}^d$, with the promise that, after applying a permutation of $[n]$ and a rotation of $\mathbb{R}^d$, the two sets are $ρ$-correlated. The Procrustes matching problem asks us to recover the unknown permutation of $[n]$ that aligns the two sets. The problem is well-studied in the low-dimensional regime $d=O(\log n)$, but the high-dimensional regime $d\gg \log n$ has remained largely uncharted: prior matching guarantees require nearly perfect correlation $ρ=1-o(1)$, even for information-theoretic recovery. Our main result is a polynomial-time algorithm for exact recovery at constant correlation. The algorithm works by computing and comparing weighted counts of a specially chosen family of ``wide'' trees. So long as $d\ge \mathrm{polylog}(n)$, the algorithm succeeds with high probability for any $ρ^2>\sqrtα$, where $α\approx 0.338$ is Otter's tree-counting constant. We complement this algorithmic result with an improved information-theoretic guarantee, showing that exact recovery is possible when $ρ^2 \gtrsim \max\{\log n/d,\sqrt{\log n/n}\}$. We also carry out a low-degree advantage calculation, which suggests that the condition $ρ^2 > \sqrtα$ is necessary for any tree-counting algorithm.

Collaborative Learning with Shared Linear Representations: Statistical Rates and Optimal Algorithms

Collaborative learning enables multiple clients to learn shared feature representations across local data distributions, with the goal of improving model performance and reducing overall sample complexity. While empirical evidence shows the success of collaborative learning, a theoretical understanding of the optimal statistical rate remains lacking, even in linear settings. In this paper, we identify the optimal statistical rate when clients share a common low-dimensional linear representation. Specifically, we design a spectral estimator with local averaging that approximates the optimal solution to the least squares problem. We establish a minimax lower bound to demonstrate that our estimator achieves the optimal error rate. Notably, the optimal rate reveals two distinct phases. In typical cases, our rate matches the standard rate based on the parameter counting of the linear representation. However, a statistical penalty arises in collaborative learning when there are too many clients or when local datasets are relatively small. Furthermore, our results, unlike existing ones, show that, at a system level, collaboration always reduces overall sample complexity compared to independent client learning. In addition, at an individual level, we provide a more precise characterization of when collaboration benefits a client in transfer learning and private fine-tuning.

Understanding Generalization of Federated Learning via Stability: Heterogeneity Matters

Generalization performance is a key metric in evaluating machine learning models when applied to real-world applications. Good generalization indicates the model can predict unseen data correctly when trained under a limited number of data. Federated learning (FL), which has emerged as a popular distributed learning framework, allows multiple devices or clients to train a shared model without violating privacy requirements. While the existing literature has studied extensively the generalization performances of centralized machine learning algorithms, similar analysis in the federated settings is either absent or with very restrictive assumptions on the loss functions. In this paper, we aim to analyze the generalization performances of federated learning by means of algorithmic stability, which measures the change of the output model of an algorithm when perturbing one data point. Three widely-used algorithms are studied, including FedAvg, SCAFFOLD, and FedProx, under convex and non-convex loss functions. Our analysis shows that the generalization performances of models trained by these three algorithms are closely related to the heterogeneity of clients' datasets as well as the convergence behaviors of the algorithms. Particularly, in the i.i.d. setting, our results recover the classical results of stochastic gradient descent (SGD).