Deep Neural Networks: Multi-Classification and Universal Approximation

We demonstrate that a ReLU deep neural network with a width of $2$ and a depth of $2N+4M-1$ layers can achieve finite sample memorization for any dataset comprising $N$ elements in $\mathbb{R}^d$, where $d\ge1,$ and $M$ classes, thereby ensuring accurate classification. By modeling the neural network as a time-discrete nonlinear dynamical system, we interpret the memorization property as a problem of simultaneous or ensemble controllability. This problem is addressed by constructing the network parameters inductively and explicitly, bypassing the need for training or solving any optimization problem. Additionally, we establish that such a network can achieve universal approximation in $L^p(\Omega;\mathbb{R}_+)$, where $\Omega$ is a bounded subset of $\mathbb{R}^d$ and $p\in[1,\infty)$, using a ReLU deep neural network with a width of $d+1$. We also provide depth estimates for approximating $W^{1,p}$ functions and width estimates for approximating $L^p(\Omega;\mathbb{R}^m)$ for $m\geq1$. Our proofs are constructive, offering explicit values for the biases and weights involved.

Clustering in pure-attention hardmax transformers and its role in sentiment analysis

Transformers are extremely successful machine learning models whose mathematical properties remain poorly understood. Here, we rigorously characterize the behavior of transformers with hardmax self-attention and normalization sublayers as the number of layers tends to infinity. By viewing such transformers as discrete-time dynamical systems describing the evolution of points in a Euclidean space, and thanks to a geometric interpretation of the self-attention mechanism based on hyperplane separation, we show that the transformer inputs asymptotically converge to a clustered equilibrium determined by special points called leaders. We then leverage this theoretical understanding to solve sentiment analysis problems from language processing using a fully interpretable transformer model, which effectively captures `context' by clustering meaningless words around leader words carrying the most meaning. Finally, we outline remaining challenges to bridge the gap between the mathematical analysis of transformers and their real-life implementation.

FedADMM-InSa: An Inexact and Self-Adaptive ADMM for Federated Learning

Federated learning (FL) is a promising framework for learning from distributed data while maintaining privacy. The development of efficient FL algorithms encounters various challenges, including heterogeneous data and systems, limited communication capacities, and constrained local computational resources. Recently developed FedADMM methods show great resilience to both data and system heterogeneity. However, they still suffer from performance deterioration if the hyperparameters are not carefully tuned. To address this issue, we propose an inexact and self-adaptive FedADMM algorithm, termed FedADMM-InSa. First, we design an inexactness criterion for the clients' local updates to eliminate the need for empirically setting the local training accuracy. This inexactness criterion can be assessed by each client independently based on its unique condition, thereby reducing the local computational cost and mitigating the undesirable straggle effect. The convergence of the resulting inexact ADMM is proved under the assumption of strongly convex loss functions. Additionally, we present a self-adaptive scheme that dynamically adjusts each client's penalty parameter, enhancing algorithm robustness by mitigating the need for empirical penalty parameter choices for each client. Extensive numerical experiments on both synthetic and real-world datasets are conducted. As validated by some numerical tests, our proposed algorithm can reduce the clients' local computational load significantly and also accelerate the learning process compared to the vanilla FedADMM.

Interplay between depth and width for interpolation in neural ODEs

Neural ordinary differential equations (neural ODEs) have emerged as a natural tool for supervised learning from a control perspective, yet a complete understanding of their optimal architecture remains elusive. In this work, we examine the interplay between their width $p$ and number of layer transitions $L$ (effectively the depth $L+1$). Specifically, we assess the model expressivity in terms of its capacity to interpolate either a finite dataset $D$ comprising $N$ pairs of points or two probability measures in $\mathbb{R}^d$ within a Wasserstein error margin $\varepsilon>0$. Our findings reveal a balancing trade-off between $p$ and $L$, with $L$ scaling as $O(1+N/p)$ for dataset interpolation, and $L=O\left(1+(p\varepsilon^d)^{-1}\right)$ for measure interpolation. In the autonomous case, where $L=0$, a separate study is required, which we undertake focusing on dataset interpolation. We address the relaxed problem of $\varepsilon$-approximate controllability and establish an error decay of $\varepsilon\sim O(\log(p)p^{-1/d})$. This decay rate is a consequence of applying a universal approximation theorem to a custom-built Lipschitz vector field that interpolates $D$. In the high-dimensional setting, we further demonstrate that $p=O(N)$ neurons are likely sufficient to achieve exact control.

Optimized classification with neural ODEs via separability

Classification of $N$ points becomes a simultaneous control problem when viewed through the lens of neural ordinary differential equations (neural ODEs), which represent the time-continuous limit of residual networks. For the narrow model, with one neuron per hidden layer, it has been shown that the task can be achieved using $O(N)$ neurons. In this study, we focus on estimating the number of neurons required for efficient cluster-based classification, particularly in the worst-case scenario where points are independently and uniformly distributed in $[0,1]^d$. Our analysis provides a novel method for quantifying the probability of requiring fewer than $O(N)$ neurons, emphasizing the asymptotic behavior as both $d$ and $N$ increase. Additionally, under the sole assumption that the data are in general position, we propose a new constructive algorithm that simultaneously classifies clusters of $d$ points from any initial configuration, effectively reducing the maximal complexity to $O(N/d)$ neurons.

Approximate and Weighted Data Reconstruction Attack in Federated Learning

Federated Learning (FL) is a distributed learning paradigm that enables multiple clients to collaborate on building a machine learning model without sharing their private data. Although FL is considered privacy-preserved by design, recent data reconstruction attacks demonstrate that an attacker can recover clients' training data based on the parameters shared in FL. However, most existing methods fail to attack the most widely used horizontal Federated Averaging (FedAvg) scenario, where clients share model parameters after multiple local training steps. To tackle this issue, we propose an interpolation-based approximation method, which makes attacking FedAvg scenarios feasible by generating the intermediate model updates of the clients' local training processes. Then, we design a layer-wise weighted loss function to improve the data quality of reconstruction. We assign different weights to model updates in different layers concerning the neural network structure, with the weights tuned by Bayesian optimization. Finally, experimental results validate the superiority of our proposed approximate and weighted attack (AWA) method over the other state-of-the-art methods, as demonstrated by the substantial improvement in different evaluation metrics for image data reconstructions.

Turnpike in optimal control of PDEs, ResNets, and beyond

The \emph{turnpike property} in contemporary macroeconomics asserts that if an economic planner seeks to move an economy from one level of capital to another, then the most efficient path, as long as the planner has enough time, is to rapidly move stock to a level close to the optimal stationary or constant path, then allow for capital to develop along that path until the desired term is nearly reached, at which point the stock ought to be moved to the final target. Motivated in part by its nature as a resource allocation strategy, over the past decade, the turnpike property has also been shown to hold for several classes of partial differential equations arising in mechanics. When formalized mathematically, the turnpike theory corroborates the insights from economics: for an optimal control problem set in a finite-time horizon, optimal controls and corresponding states, are close (often exponentially), during most of the time, except near the initial and final time, to the optimal control and corresponding state for the associated stationary optimal control problem. In particular, the former are mostly constant over time. This fact provides a rigorous meaning to the asymptotic simplification that some optimal control problems appear to enjoy over long time intervals, allowing the consideration of the corresponding stationary problem for computing and applications. We review a slice of the theory developed over the past decade --the controllability of the underlying system is an important ingredient, and can even be used to devise simple turnpike-like strategies which are nearly optimal--, and present several novel applications, including, among many others, the characterization of Hamilton-Jacobi-Bellman asymptotics, and stability estimates in deep learning via residual neural networks.

Large-time asymptotics in deep learning

It is by now well-known that practical deep supervised learning may roughly be cast as an optimal control problem for a specific discrete-time, nonlinear dynamical system called an artificial neural network. In this work, we consider the continuous-time formulation of the deep supervised learning problem, and study the latter's behavior when the final time horizon increases, a fact that can be interpreted as increasing the number of layers in the neural network setting.When considering the classical regularized empirical risk minimization problem, we show that, in long time, the optimal states converge to zero training error, namely approach the zero training error regime, whilst the optimal control parameters approach, on an appropriate scale, minimal norm parameters with corresponding states precisely in the zero training error regime. This result provides an alternative theoretical underpinning to the notion that neural networks learn best in the overparametrized regime, when seen from the large layer perspective. We also propose a learning problem consisting of minimizing a cost with a state tracking term, and establish the well-known turnpike property, which indicates that the solutions of the learning problem in long time intervals consist of three pieces, the first and the last of which being transient short-time arcs, and the middle piece being a long-time arc staying exponentially close to the optimal solution of an associated static learning problem. This property in fact stipulates a quantitative estimate for the number of layers required to reach the zero training error regime. Both of the aforementioned asymptotic regimes are addressed in the context of continuous-time and continuous space-time neural networks, the latter taking the form of nonlinear, integro-differential equations, hence covering residual neural networks with both fixed and possibly variable depths.