Neural Information Processing Systems http://nips.cc/

When data is distributed across multiple devices, communication cost often becomes a bottleneck of modern machine learning tasks [38]. This is even more so in federated learning type settings, where communication occurs over bandwidth-limited wireless links [31].

We consider the problem of estimating a $d$-dimensional discrete distribution from its samples observed under a $b$-bit communication constraint. In contrast to most previous results that largely focus on the global minimax error, we study the local behavior of the estimation error and provide \emph{pointwise} bounds that depend on the target distribution $p$. In particular, we show that the $\ell_2$ error decays with $O\left(\frac{\lVert p\rVert_{1/2}}{n2^b}\vee \frac{1}{n}\right)$ when $n$ is sufficiently large, hence it is governed by the \emph{half-norm} of $p$ instead of the ambient dimension $d$. For the achievability result, we propose a two-round sequentially interactive estimation scheme that achieves this error rate uniformly over all $p$. Our scheme is based on a novel local refinement idea, where we first use a standard global minimax scheme to localize $p$ and then use the remaining samples to locally refine our estimate.We also develop a new local minimax lower bound with (almost) matching $\ell_2$ error, showing that any interactive scheme must admit a $\Omega\left( \frac{\lVert p \rVert_{{(1+\delta)}/{2}}}{n2^b}\right)$ $\ell_2$ error for any $\delta > 0$ when $n$ is sufficiently large. The lower bound is derived by first finding the best parametric sub-model containing $p$, and then upper bounding the quantized Fisher information under this model. Our upper and lower bounds together indicate that the $\mathsf{H}_{1/2}(p) = \log(\lVert p \rVert_{{1}/{2}})$ bits of communication is both sufficient and necessary to achieve the optimal (centralized) performance, where $\mathsf{H}_{{1}/{2}}(p)$ is the R\'enyi entropy of order $2$. Therefore, under the $\ell_2$ loss, the correct measure of the local communication complexity at $p$ is its R\'enyi entropy.

The optimal transport (OT) map is a geometry-driven transformation between high-dimensional probability distributions which underpins a wide range of tasks in statistics, applied probability, and machine learning. However, existing statistical theory for OT map estimation is quite restricted, hinging on Brenier's theorem (quadratic cost, absolutely continuous source) to guarantee existence and uniqueness of a deterministic OT map, on which various additional regularity assumptions are imposed to obtain quantitative error bounds. In many real-world problems these conditions fail or cannot be certified, in which case optimal transportation is possible only via stochastic maps that can split mass. To broaden the scope of map estimation theory to such settings, this work introduces a novel metric for evaluating the transportation quality of stochastic maps. Under this metric, we develop computationally efficient map estimators with near-optimal finite-sample risk bounds, subject to easy-to-verify minimal assumptions. Our analysis further accommodates common forms of adversarial sample contamination, yielding estimators with robust estimation guarantees. Empirical experiments are provided which validate our theory and demonstrate the utility of the proposed framework in settings where existing theory fails. These contributions constitute the first general-purpose theory for map estimation, compatible with a wide spectrum of real-world applications where optimal transport may be intrinsically stochastic.

Neural Information Processing Systems http://nips.cc/

Open-vocabulary learning requires modeling the data distribution in open environments, which consists of both seen-class and unseen-class data. Existing methods estimate the distribution in open environments using seen-class data, where the absence of unseen classes makes the estimation error inherently unidentifiable. Intuitively, learning beyond the seen classes is crucial for distribution estimation to bound the estimation error. We theoretically demonstrate that the distribution can be effectively estimated by generating unseen-class data, through which the estimation error is upper-bounded. Building on this theoretical insight, we propose a novel open-vocabulary learning method, which generates unseen-class data for estimating the distribution in open environments. The method consists of a class-domain-wise data generation pipeline and a distribution alignment algorithm. The data generation pipeline generates unseen-class data under the guidance of a hierarchical semantic tree and domain information inferred from the seen-class data, facilitating accurate distribution estimation. With the generated data, the distribution alignment algorithm estimates and maximizes the posterior probability to enhance generalization in open-vocabulary learning. Extensive experiments on $11$ datasets demonstrate that our method outperforms baseline approaches by up to $14\%$, highlighting its effectiveness and superiority.

When data is distributed across multiple devices, communication cost often becomes a bottleneck of modern machine learning tasks [38]. This is even more so in federated learning type settings, where communication occurs over bandwidth-limited wireless links [31].

Neural Information Processing Systems http://nips.cc/