Gaussian mixture models (GMMs) are fundamental to machine learning due to their flexibility as approximating densities. However, uncertainty quantification of GMMs remains a challenge as differential entropy lacks a closed form.

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

Gaussian mixture models (GMMs) are fundamental to machine learning due to their flexibility as approximating densities. However, uncertainty quantification of GMMs remains a challenge as differential entropy lacks a closed form. This paper explores polynomial approximations, specifically Taylor and Legendre, to the GMM entropy from a theoretical and practical perspective. We provide new analysis of a widely used approach due to Huber et al.(2008) and show that the series diverges under simple conditions. Motivated by this divergence we provide a novel Taylor series that is provably convergent to the true entropy of any GMM.

Transformer models have gained significant attention due to their power in machine learning tasks. Their extensive deployment has raised concerns about the potential leakage of sensitive information during inference. However, when being applied to Transformers, existing approaches based on secure two-party computation (2PC) bring about efficiency limitations in two folds: (1) resource-intensive matrix multiplications in linear layers, and (2) complex non-linear activation functions like $\mathsf{GELU}$ and $\mathsf{Softmax}$. This work presents a new two-party inference framework $\mathsf{Nimbus}$ for Transformer models. Specifically, we propose a new 2PC paradigm to securely compute matrix multiplications based on an outer-product insight, which achieves $2.9\times \sim 12.5\times$ performance improvements compared to the state-of-the-art (SOTA) protocol. Furthermore, through a new observation of utilizing the input distribution, we propose an approach of low-degree polynomial approximation for $\mathsf{GELU}$ and $\mathsf{Softmax}$, which improves the performance of the SOTA polynomial approximation by $2.9\times \sim 4.0\times$, where the average accuracy loss of our approach is 0.08\% compared to the non-2PC inference without privacy. Compared with the SOTA two-party inference, $\mathsf{Nimbus}$ improves the end-to-end performance of $BERT_{base}$ inference by $2.7\times \sim 4.7\times$ across different network settings.

This work estimates safe invariant subsets of the Region of Attraction (ROA) for a seven-state vehicle-with-driver system, capturing both asymptotic stability and the influence of state-safety bounds along the system trajectory. Safe sets are computed by optimizing Lyapunov functions through an original iterative Sum-of-Squares (SOS) procedure. The method is first demonstrated on a two-state benchmark, where it accurately recovers a prescribed safe region as the 1-level set of a polynomial Lyapunov function. We then describe the distinguishing characteristics of the studied vehicle-with-driver system: the control dynamics mimic human driver behavior through a delayed preview-tracking model that, with suitable parameter choices, can also emulate digital controllers. To enable SOS optimization, a polynomial approximation of the nonlinear vehicle model is derived, together with its operating-envelope constraints. The framework is then applied to understeering and oversteering scenarios, and the estimated safe sets are compared with reference boundaries obtained from exhaustive simulations. The results show that SOS techniques can efficiently deliver Lyapunov-defined safe regions, supporting their potential use for real-time safety assessment, for example as a supervisory layer for active vehicle control.

A classical result of Carleman, based on the theory of quasianalytic functions, shows that polynomials are dense in $L^2(μ)$ for any $μ$ such that the moments $\int x^k dμ$ do not grow too rapidly as $k \to \infty$. In this work, we develop a fairly tight quantitative analogue of the underlying Denjoy-Carleman theorem via complex analysis, and show that this allows for nonasymptotic control of the rate of approximation by polynomials for any smooth function with polynomial growth at infinity. In many cases, this allows us to establish $L^2$ approximation-theoretic results for functions over general classes of distributions (e.g., multivariate sub-Gaussian or sub-exponential distributions) which were previously known only in special cases. As one application, we show that the Paley--Wiener class of functions bandlimited to $[-Ω,Ω]$ admits superexponential rates of approximation over all strictly sub-exponential distributions, which leads to a new characterization of the class. As another application, we solve an open problem recently posed by Chandrasekaran, Klivans, Kontonis, Meka and Stavropoulos on the smoothed analysis of learning, and also obtain quantitative improvements to their main results and applications.

Agnostic learning of Boolean halfspaces is a fundamental problem in computational learning theory, but it is known to be computationally hard even for weak learning. Recent work [CKKMK24] proposed smoothed analysis as a way to bypass such hardness, but existing frameworks rely on additive Gaussian perturbations, making them unsuitable for discrete domains. We introduce a new smoothed agnostic learning framework for Boolean inputs, where perturbations are modeled via random bit flips. This defines a natural discrete analogue of smoothed optimality generalizing the Gaussian case. Under strictly subexponential assumptions on the input distribution, we give an efficient algorithm for learning halfspaces in this model, with runtime and sample complexity approximately n raised to a poly(1/(sigma * epsilon)) factor. Previously, such algorithms were known only with strong structural assumptions for the discrete hypercube, for example, independent coordinates or symmetric distributions. Our result provides the first computationally efficient guarantee for smoothed agnostic learning of halfspaces over the Boolean hypercube, bridging the gap between worst-case intractability and practical learnability in discrete settings.

Recent advances in recording technologies have allowed neuroscientists to record simultaneous spiking activity from hundreds to thousands of neurons in multiple brain regions. Such large-scale recordings pose a major challenge to existing statistical methods for neural data analysis. Here we develop highly scalable approximate inference methods for Poisson generalized linear models (GLMs) that require only a single pass over the data. Our approach relies on a recently proposed method for obtaining approximate sufficient statistics for GLMs using polynomial approximations [Huggins et al., 2017], which we adapt to the Poisson GLM setting. We focus on inference using quadratic approximations to nonlinear terms in the Poisson GLM log-likelihood with Gaussian priors, for which we derive closed-form solutions to the approximate maximum likelihood and MAP estimates, posterior distribution, and marginal likelihood. We introduce an adaptive procedure to select the polynomial approximation interval and show that the resulting method allows for efficient and accurate inference and regularization of high-dimensional parameters. We use the quadratic estimator to fit a fully-coupled Poisson GLM to spike train data recorded from 831 neurons across five regions of the mouse brain for a duration of 41 minutes, binned at 1 ms resolution. Across all neurons, this model is fit to over 2 billion spike count bins and identifies fine-timescale statistical dependencies between neurons within and across cortical and subcortical areas.