Kolmogorov-Arnold Networks: Approximation and Learning Guarantees for Functions and their Derivatives

Inspired by the Kolmogorov-Arnold superposition theorem, Kolmogorov-Arnold Networks (KANs) have recently emerged as an improved backbone for most deep learning frameworks, promising more adaptivity than their multilayer perception (MLP) predecessor by allowing for trainable spline-based activation functions. In this paper, we probe the theoretical foundations of the KAN architecture by showing that it can optimally approximate any Besov function in $B^{s}_{p,q}(\mathcal{X})$ on a bounded open, or even fractal, domain $\mathcal{X}$ in $\mathbb{R}^d$ at the optimal approximation rate with respect to any weaker Besov norm $B^{\alpha}_{p,q}(\mathcal{X})$; where $\alpha < s$. We complement our approximation guarantee with a dimension-free estimate on the sample complexity of a residual KAN model when learning a function of Besov regularity from $N$ i.i.d. noiseless samples. Our KAN architecture incorporates contemporary deep learning wisdom by leveraging residual/skip connections between layers.

Generative Market Equilibrium Models with Stable Adversarial Learning via Reinforcement

We present a general computational framework for solving continuous-time financial market equilibria under minimal modeling assumptions while incorporating realistic financial frictions, such as trading costs, and supporting multiple interacting agents. Inspired by generative adversarial networks (GANs), our approach employs a novel generative deep reinforcement learning framework with a decoupling feedback system embedded in the adversarial training loop, which we term as the \emph{reinforcement link}. This architecture stabilizes the training dynamics by incorporating feedback from the discriminator. Our theoretically guided feedback mechanism enables the decoupling of the equilibrium system, overcoming challenges that hinder conventional numerical algorithms. Experimentally, our algorithm not only learns but also provides testable predictions on how asset returns and volatilities emerge from the endogenous trading behavior of market participants, where traditional analytical methods fall short. The design of our model is further supported by an approximation guarantee.

Guiding Two-Layer Neural Network Lipschitzness via Gradient Descent Learning Rate Constraints

We demonstrate that applying an eventual decay to the learning rate (LR) in empirical risk minimization (ERM), where the mean-squared-error loss is minimized using standard gradient descent (GD) for training a two-layer neural network with Lipschitz activation functions, ensures that the resulting network exhibits a high degree of Lipschitz regularity, that is, a small Lipschitz constant. Moreover, we show that this decay does not hinder the convergence rate of the empirical risk, now measured with the Huber loss, toward a critical point of the non-convex empirical risk. From these findings, we derive generalization bounds for two-layer neural networks trained with GD and a decaying LR with a sub-linear dependence on its number of trainable parameters, suggesting that the statistical behaviour of these networks is independent of overparameterization. We validate our theoretical results with a series of toy numerical experiments, where surprisingly, we observe that networks trained with constant step size GD exhibit similar learning and regularity properties to those trained with a decaying LR. This suggests that neural networks trained with standard GD may already be highly regular learners.

Keep It Light! Simplifying Image Clustering Via Text-Free Adapters

Many competitive clustering pipelines have a multi-modal design, leveraging large language models (LLMs) or other text encoders, and text-image pairs, which are often unavailable in real-world downstream applications. Additionally, such frameworks are generally complicated to train and require substantial computational resources, making widespread adoption challenging. In this work, we show that in deep clustering, competitive performance with more complex state-of-the-art methods can be achieved using a text-free and highly simplified training pipeline. In particular, our approach, Simple Clustering via Pre-trained models (SCP), trains only a small cluster head while leveraging pre-trained vision model feature representations and positive data pairs. Experiments on benchmark datasets including CIFAR-10, CIFAR-20, CIFAR-100, STL-10, ImageNet-10, and ImageNet-Dogs, demonstrate that SCP achieves highly competitive performance. Furthermore, we provide a theoretical result explaining why, at least under ideal conditions, additional text-based embeddings may not be necessary to achieve strong clustering performance in vision.

Is In-Context Universality Enough? MLPs are Also Universal In-Context

The success of transformers is often linked to their ability to perform in-context learning. Recent work shows that transformers are universal in context, capable of approximating any real-valued continuous function of a context (a probability measure over $\mathcal{X}\subseteq \mathbb{R}^d$) and a query $x\in \mathcal{X}$. This raises the question: Does in-context universality explain their advantage over classical models? We answer this in the negative by proving that MLPs with trainable activation functions are also universal in-context. This suggests the transformer's success is likely due to other factors like inductive bias or training stability.

Neural Operators Can Play Dynamic Stackelberg Games

Dynamic Stackelberg games are a broad class of two-player games in which the leader acts first, and the follower chooses a response strategy to the leader's strategy. Unfortunately, only stylized Stackelberg games are explicitly solvable since the follower's best-response operator (as a function of the control of the leader) is typically analytically intractable. This paper addresses this issue by showing that the \textit{follower's best-response operator} can be approximately implemented by an \textit{attention-based neural operator}, uniformly on compact subsets of adapted open-loop controls for the leader. We further show that the value of the Stackelberg game where the follower uses the approximate best-response operator approximates the value of the original Stackelberg game. Our main result is obtained using our universal approximation theorem for attention-based neural operators between spaces of square-integrable adapted stochastic processes, as well as stability results for a general class of Stackelberg games.

Scalable Message Passing Neural Networks: No Need for Attention in Large Graph Representation Learning

We propose Scalable Message Passing Neural Networks (SMPNNs) and demonstrate that, by integrating standard convolutional message passing into a Pre-Layer Normalization Transformer-style block instead of attention, we can produce high-performing deep message-passing-based Graph Neural Networks (GNNs). This modification yields results competitive with the state-of-the-art in large graph transductive learning, particularly outperforming the best Graph Transformers in the literature, without requiring the otherwise computationally and memory-expensive attention mechanism. Our architecture not only scales to large graphs but also makes it possible to construct deep message-passing networks, unlike simple GNNs, which have traditionally been constrained to shallow architectures due to oversmoothing. Moreover, we provide a new theoretical analysis of oversmoothing based on universal approximation which we use to motivate SMPNNs. We show that in the context of graph convolutions, residual connections are necessary for maintaining the universal approximation properties of downstream learners and that removing them can lead to a loss of universality.

A Comprehensive Analysis on the Learning Curve in Kernel Ridge Regression

This paper conducts a comprehensive study of the learning curves of kernel ridge regression (KRR) under minimal assumptions. Our contributions are three-fold: 1) we analyze the role of key properties of the kernel, such as its spectral eigen-decay, the characteristics of the eigenfunctions, and the smoothness of the kernel; 2) we demonstrate the validity of the Gaussian Equivalent Property (GEP), which states that the generalization performance of KRR remains the same when the whitened features are replaced by standard Gaussian vectors, thereby shedding light on the success of previous analyzes under the Gaussian Design Assumption; 3) we derive novel bounds that improve over existing bounds across a broad range of setting such as (in)dependent feature vectors and various combinations of eigen-decay rates in the over/underparameterized regimes.

Simultaneously Solving FBSDEs with Neural Operators of Logarithmic Depth, Constant Width, and Sub-Linear Rank

Forward-backwards stochastic differential equations (FBSDEs) are central in optimal control, game theory, economics, and mathematical finance. Unfortunately, the available FBSDE solvers operate on \textit{individual} FBSDEs, meaning that they cannot provide a computationally feasible strategy for solving large families of FBSDEs as these solvers must be re-run several times. \textit{Neural operators} (NOs) offer an alternative approach for \textit{simultaneously solving} large families of FBSDEs by directly approximating the solution operator mapping \textit{inputs:} terminal conditions and dynamics of the backwards process to \textit{outputs:} solutions to the associated FBSDE. Though universal approximation theorems (UATs) guarantee the existence of such NOs, these NOs are unrealistically large. We confirm that ``small'' NOs can uniformly approximate the solution operator to structured families of FBSDEs with random terminal time, uniformly on suitable compact sets determined by Sobolev norms, to any prescribed error $\varepsilon>0$ using a depth of $\mathcal{O}(\log(1/\varepsilon))$, a width of $\mathcal{O}(1)$, and a sub-linear rank; i.e. $\mathcal{O}(1/\varepsilon^r)$ for some $r<1$. This result is rooted in our second main contribution, which shows that convolutional NOs of similar depth, width, and rank can approximate the solution operator to a broad class of Elliptic PDEs. A key insight here is that the convolutional layers of our NO can efficiently encode the Green's function associated to the Elliptic PDEs linked to our FBSDEs. A byproduct of our analysis is the first theoretical justification for the benefit of lifting channels in NOs: they exponentially decelerate the growth rate of the NO's rank.

Inverse Entropic Optimal Transport Solves Semi-supervised Learning via Data Likelihood Maximization

Learning conditional distributions $\pi^*(\cdot|x)$ is a central problem in machine learning, which is typically approached via supervised methods with paired data $(x,y) \sim \pi^*$. However, acquiring paired data samples is often challenging, especially in problems such as domain translation. This necessitates the development of $\textit{semi-supervised}$ models that utilize both limited paired data and additional unpaired i.i.d. samples $x \sim \pi^*_x$ and $y \sim \pi^*_y$ from the marginal distributions. The usage of such combined data is complex and often relies on heuristic approaches. To tackle this issue, we propose a new learning paradigm that integrates both paired and unpaired data $\textbf{seamlessly}$ through the data likelihood maximization techniques. We demonstrate that our approach also connects intriguingly with inverse entropic optimal transport (OT). This finding allows us to apply recent advances in computational OT to establish a $\textbf{light}$ learning algorithm to get $\pi^*(\cdot|x)$. Furthermore, we demonstrate through empirical tests that our method effectively learns conditional distributions using paired and unpaired data simultaneously.