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One of the key challenges in decentralized and federated learning is to design algorithms that efficiently deal with highly heterogeneous data distributions across agents. In this paper, we revisit the analysis of Decentralized Stochastic Gradient Descent algorithm (D-SGD) under data heterogeneity. We exhibit the key role played by a new quantity, called \emph{neighborhood heterogeneity}, on the convergence rate of D-SGD. By coupling the communication topology and the heterogeneity, our analysis sheds light on the poorly understood interplay between these two concepts in decentralized learning. We then argue that neighborhood heterogeneity provides a natural criterion to learn data-dependent topologies that reduce (and can even eliminate) the otherwise detrimental effect of data heterogeneity on the convergence time of D-SGD. For the important case of classification with label skew, we formulate the problem of learning such a good topology as a tractable optimization problem that we solve with a Frank-Wolfe algorithm. As illustrated over a set of simulated and real-world experiments, our approach provides a principled way to design a sparse topology that balances the convergence speed and the per-iteration communication costs of D-SGD under data heterogeneity.

Fluid-structure systems occur in a range of scientific and engineering applications. The immersed boundary(IB) method is a widely recognized and effective modeling paradigm for simulating fluid-structure interaction(FSI) in such systems, but a difficulty of the IB formulation is that the pressure and viscous stress are generally discontinuous at the interface. The conventional IB method regularizes these discontinuities, which typically yields low-order accuracy at these interfaces. The immersed interface method(IIM) is an IB-like approach to FSI that sharply imposes stress jump conditions, enabling higher-order accuracy, but prior applications of the IIM have been largely restricted to methods that rely on smooth representations of the interface geometry. This paper introduces an IIM that uses only a C0 representation of the interface,such as those provided by standard nodal Lagrangian FE methods. Verification examples for models with prescribed motion demonstrate that the method sharply resolves stress discontinuities along the IB while avoiding the need for analytic information of the interface geometry. We demonstrate that only the lowest-order jump conditions for the pressure and velocity gradient are required to realize global 2nd-order accuracy. Specifically,we show 2nd-order global convergence rate along with nearly 2nd-order local convergence in the Eulerian velocity, and between 1st-and 2nd-order global convergence rates along with 1st-order local convergence for the Eulerian pressure. We also show 2nd-order local convergence in the interfacial displacement and velocity along with 1st-order local convergence in the fluid traction. As a demonstration of the method's ability to tackle complex geometries,this approach is also used to simulate flow in an anatomical model of the inferior vena cava.

We present a data-driven approach to characterizing nonidentifiability of a model's parameters and illustrate it through dynamic as well as steady kinetic models. By employing Diffusion Maps and their extensions, we discover the minimal combinations of parameters required to characterize the output behavior of a chemical system: a set of effective parameters for the model. Furthermore, we introduce and use a Conformal Autoencoder Neural Network technique, as well as a kernel-based Jointly Smooth Function technique, to disentangle the redundant parameter combinations that do not affect the output behavior from the ones that do. We discuss the interpretability of our data-driven effective parameters, and demonstrate the utility of the approach both for behavior prediction and parameter estimation. In the latter task, it becomes important to describe level sets in parameter space that are consistent with a particular output behavior. We validate our approach on a model of multisite phosphorylation, where a reduced set of effective parameters (nonlinear combinations of the physical ones) has previously been established analytically.

Gradient Descent (GD) is a powerful workhorse of modern machine learning thanks to its scalability and efficiency in high-dimensional spaces. Its ability to find local minimisers is only guaranteed for losses with Lipschitz gradients, where it can be seen as a 'bona-fide' discretisation of an underlying gradient flow. Yet, many ML setups involving overparametrised models do not fall into this problem class, which has motivated research beyond the so-called "Edge of Stability", where the step-size crosses the admissibility threshold inversely proportional to the Lipschitz constant above. Perhaps surprisingly, GD has been empirically observed to still converge regardless of local instability. In this work, we study a local condition for such an unstable convergence around a local minima in a low dimensional setting. We then leverage these insights to establish global convergence of a two-layer single-neuron ReLU student network aligning with the teacher neuron in a large learning rate beyond the Edge of Stability under population loss. Meanwhile, while the difference of norms of the two layers is preserved by gradient flow, we show that GD above the edge of stability induces a balancing effect, leading to the same norms across the layers.

In this paper we provide a rigorous convergence analysis for the renowned particle swarm optimization method by using tools from stochastic calculus and the analysis of partial differential equations. Based on a time-continuous formulation of the particle dynamics as a system of stochastic differential equations, we establish convergence to a global minimizer of a possibly nonconvex and nonsmooth objective function in two steps. First, we prove consensus formation of an associated mean-field dynamics by analyzing the time-evolution of the variance of the particle distribution. We then show that this consensus is close to a global minimizer by employing the asymptotic Laplace principle and a tractability condition on the energy landscape of the objective function. These results allow for the usage of memory mechanisms, and hold for a rich class of objectives provided certain conditions of well-preparation of the hyperparameters and the initial datum. In a second step, at least for the case without memory effects, we provide a quantitative result about the mean-field approximation of particle swarm optimization, which specifies the convergence of the interacting particle system to the associated mean-field limit. Combining these two results allows for global convergence guarantees of the numerical particle swarm optimization method with provable polynomial complexity. To demonstrate the applicability of the method we propose an efficient and parallelizable implementation, which is tested in particular on a competitive and well-understood high-dimensional benchmark problem in machine learning.

Functional linear and single-index models are core regression methods in functional data analysis and are widely used methods for performing regression when the covariates are observed random functions coupled with scalar responses in a wide range of applications. In the existing literature, however, the construction of associated estimators and the study of their theoretical properties is invariably carried out on a case-by-case basis for specific models under consideration. In this work, we provide a unified methodological and theoretical framework for estimating the index in functional linear and single-index models; in the later case the proposed approach does not require the specification of the link function. In terms of methodology, we show that the reproducing kernel Hilbert space (RKHS) based functional linear least-squares estimator, when viewed through the lens of an infinite-dimensional Gaussian Stein's identity, also provides an estimator of the index of the single-index model. On the theoretical side, we characterize the convergence rates of the proposed estimators for both linear and single-index models. Our analysis has several key advantages: (i) we do not require restrictive commutativity assumptions for the covariance operator of the random covariates on one hand and the integral operator associated with the reproducing kernel on the other hand; and (ii) we also allow for the true index parameter to lie outside of the chosen RKHS, thereby allowing for index mis-specification as well as for quantifying the degree of such index mis-specification. Several existing results emerge as special cases of our analysis.

In this work, we provide a fundamental unified convergence theorem used for deriving expected and almost sure convergence results for a series of stochastic optimization methods. Our unified theorem only requires to verify several representative conditions and is not tailored to any specific algorithm. As a direct application, we recover expected and almost sure convergence results of the stochastic gradient method (SGD) and random reshuffling (RR) under more general settings. Moreover, we establish new expected and almost sure convergence results for the stochastic proximal gradient method (prox-SGD) and stochastic model-based methods (SMM) for nonsmooth nonconvex optimization problems. These applications reveal that our unified theorem provides a plugin-type convergence analysis and strong convergence guarantees for a wide class of stochastic optimization methods.

We propose a co-variance corrected random batch method for interacting particle systems. By establishing a certain entropic central limit theorem, we provide entropic convergence guarantees for the law of the entire trajectories of all particles of the proposed method to the law of the trajectories of the discrete time interacting particle system whenever the batch size $B \gg (\alpha n)^{\frac{1}{3}}$ (where $n$ is the number of particles and $\alpha$ is the time discretization parameter). This in turn implies that the outputs of these methods are nearly \emph{statistically indistinguishable} when $B$ is even moderately large. Previous works mainly considered convergence in Wasserstein distance with required stringent assumptions on the potentials or the bounds had an exponential dependence on the time horizon. This work makes minimal assumptions on the interaction potentials and in particular establishes that even when the particle trajectories diverge to infinity, they do so in the same way for both the methods. Such guarantees are very useful in light of the recent advances in interacting particle based algorithms for sampling.

A recent goal in the theory of deep learning is to identify how neural networks can escape the "lazy training," or Neural Tangent Kernel (NTK) regime, where the network is coupled with its first order Taylor expansion at initialization. While the NTK is minimax optimal for learning dense polynomials (Ghorbani et al, 2021), it cannot learn features, and hence has poor sample complexity for learning many classes of functions including sparse polynomials. Recent works have thus aimed to identify settings where gradient based algorithms provably generalize better than the NTK. One such example is the "QuadNTK" approach of Bai and Lee (2020), which analyzes the second-order term in the Taylor expansion. Bai and Lee (2020) show that the second-order term can learn sparse polynomials efficiently; however, it sacrifices the ability to learn general dense polynomials. In this paper, we analyze how gradient descent on a two-layer neural network can escape the NTK regime by utilizing a spectral characterization of the NTK (Montanari and Zhong, 2020) and building on the QuadNTK approach. We first expand upon the spectral analysis to identify "good" directions in parameter space in which we can move without harming generalization. Next, we show that a wide two-layer neural network can jointly use the NTK and QuadNTK to fit target functions consisting of a dense low-degree term and a sparse high-degree term -- something neither the NTK nor the QuadNTK can do on their own. Finally, we construct a regularizer which encourages our parameter vector to move in the "good" directions, and show that gradient descent on the regularized loss will converge to a global minimizer, which also has low test error. This yields an end to end convergence and generalization guarantee with provable sample complexity improvement over both the NTK and QuadNTK on their own.

We study the $L^1$-approximation of the log-Heston SDE at equidistant time points by Euler-type methods. We establish the convergence order $ 1/2-\epsilon$ for $\epsilon >0$ arbitrarily small, if the Feller index $\nu$ of the underlying CIR process satisfies $\nu > 1$. Thus, we recover the standard convergence order of the Euler scheme for SDEs with globally Lipschitz coefficients. Moreover, we discuss the case $\nu \leq 1$ and illustrate our findings by several numerical examples.