Anastasis Kratsios

Mikhail Persiianov, Arip Asadulaev, Nikita Andreev et al. · eth-zurich

Learning conditional distributions $π^*(\cdot|x)$ is a central problem in machine learning, which is typically approached via supervised methods with paired data $(x,y) \sim π^*$. 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 $x \sim π^*_x$ and $y \sim π^*_y$ 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 called $\textbf{EBiEOT}$ that integrates both paired and unpaired data seamlessly using 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 an $\textit{end-to-end}$ learning algorithm to get $π^*(\cdot|x)$. In addition, we derive the universal approximation property, demonstrating that our approach can theoretically recover true conditional distributions with arbitrarily small error. Finally, we demonstrate through empirical tests that our method effectively learns conditional distributions using paired and unpaired data simultaneously. The code of $\texttt{EBiEOT}$ is available at https://github.com/MuXauJl11110/EBiEOT.

Anastasis Kratsios, Giulia Livieri, Philipp Schmocker · eth-zurich

We address fundamental challenges in representing and computing $\mathbb{R}^{d}$-valued predictable square-integrable processes over $[0,T]$, collected in the space $\mathcal{H}^2_T(\mathbb{R}^{d})$. These processes are central to continuous-time stochastic control, reinforcement learning, and mathematical finance. Although Wiener-chaos expansions offer strong theoretical tools, traditional computational methods are hindered by the need for large chaos dictionaries and high-order iterated integrals. To overcome these obstacles, we introduce NeuralChaos -- a neural operator architecture that produces elements of $\mathcal{H}^2_T(\mathbb{R}^{d})$ using only finitely many evaluations of the driving Brownian motion, while preserving predictability and square-integrability. We prove that NeuralChaos is dense in $\mathcal{H}^2_T(\mathbb{R}^{d})$ and achieves the best $N$-term chaoslet approximation rates for compressible and Malliavin--Sobolev regular processes. Moreover, compressibility is shown to be typical for processes from $\mathcal{H}^2_T(\mathbb{R}^{d})$ under non-degenerate sub-Gaussian sampling. In contrast, we show that finite-dimensional Markovian neural SDE models constitute a meagre and Gaussian-null subset in $\mathcal{H}^2_T(\mathbb{R}^{d})$, regardless of discretization, whereas compressible processes are generic. Numerical experiments on a stochastic optimal control problem and dynamic hedging highlight the practical effectiveness of our approach. Our results enable more efficient and expressive modelling in stochastic analysis and mathematical finance.

J. Antonio Lara Benitez, Takashi Furuya, Florian Faucher et al. · eth-zurich

Despite their remarkable success in approximating a wide range of operators defined by PDEs, existing neural operators (NOs) do not necessarily perform well for all physics problems. We focus here on high-frequency waves to highlight possible shortcomings. To resolve these, we propose a subfamily of NOs enabling an enhanced empirical approximation of the nonlinear operator mapping wave speed to solution, or boundary values for the Helmholtz equation on a bounded domain. The latter operator is commonly referred to as the ''forward'' operator in the study of inverse problems. Our methodology draws inspiration from transformers and techniques such as stochastic depth. Our experiments reveal certain surprises in the generalization and the relevance of introducing stochastic depth. Our NOs show superior performance as compared with standard NOs, not only for testing within the training distribution but also for out-of-distribution scenarios. To delve into this observation, we offer an in-depth analysis of the Rademacher complexity associated with our modified models and prove an upper bound tied to their stochastic depth that existing NOs do not satisfy. Furthermore, we obtain a novel out-of-distribution risk bound tailored to Gaussian measures on Banach spaces, again relating stochastic depth with the bound. We conclude by proposing a hypernetwork version of the subfamily of NOs as a surrogate model for the mentioned forward operator.

Songyan Hou, Parnian Kassraie, Anastasis Kratsios et al. · eth-zurich

Existing generalization bounds fail to explain crucial factors that drive the generalization of modern neural networks. Since such bounds often hold uniformly over all parameters, they suffer from over-parametrization and fail to account for the strong inductive bias of initialization and stochastic gradient descent. As an alternative, we propose a novel optimal transport interpretation of the generalization problem. This allows us to derive instance-dependent generalization bounds that depend on the local Lipschitz regularity of the learned prediction function in the data space. Therefore, our bounds are agnostic to the parametrization of the model and work well when the number of training samples is much smaller than the number of parameters. With small modifications, our approach yields accelerated rates for data on low-dimensional manifolds and guarantees under distribution shifts. We empirically analyze our generalization bounds for neural networks, showing that the bound values are meaningful and capture the effect of popular regularization methods during training.

Yicen Li, Jose Antonio Lara Benitez, Ruiyang Hong et al. · eth-zurich

Operator learning is reshaping scientific computing by amortizing inference across infinite families of problems. While neural operators (NOs) are increasingly well understood for regression, far less is known for classification and its unsupervised analogue: clustering. We prove that sample-based neural operators can learn any finite collection of classes in an infinite-dimensional reproducing kernel Hilbert space, even when the classes are neither convex nor connected, under mild kernel sampling assumptions. Our universal clustering theorem shows that any $K$ closed classes can be approximated to arbitrary precision by NO-parameterized classes in the upper Kuratowski topology on closed sets, a notion that can be interpreted as disallowing false-positive misclassifications. Building on this, we develop an NO-powered clustering pipeline for functional data and apply it to unlabeled families of ordinary differential equation (ODE) trajectories. Discretized trajectories are lifted by a fixed pre-trained encoder into a continuous feature map and mapped to soft assignments by a lightweight trainable head. Experiments on diverse synthetic ODE benchmarks show that the resulting practical SNO recovers latent dynamical structure in regimes where classical methods fail, providing evidence consistent with our universal clustering theory.

Tin Sum Cheng, Aurelien Lucchi, Ivan Dokmanić et al.

Existing statistical learning guarantees for general kernel regressors often yield loose bounds when used with finite-rank kernels. Yet, finite-rank kernels naturally appear in several machine learning problems, e.g.\ when fine-tuning a pre-trained deep neural network's last layer to adapt it to a novel task when performing transfer learning. We address this gap for finite-rank kernel ridge regression (KRR) by deriving sharp non-asymptotic upper and lower bounds for the KRR test error of any finite-rank KRR. Our bounds are tighter than previously derived bounds on finite-rank KRR, and unlike comparable results, they also remain valid for any regularization parameters.

Anastasis Kratsios, Chong Liu, Matti Lassas et al.

Motivated by the developing mathematics of deep learning, we build universal functions approximators of continuous maps between arbitrary Polish metric spaces $\mathcal{X}$ and $\mathcal{Y}$ using elementary functions between Euclidean spaces as building blocks. Earlier results assume that the target space $\mathcal{Y}$ is a topological vector space. We overcome this limitation by ``randomization'': our approximators output discrete probability measures over $\mathcal{Y}$. When $\mathcal{X}$ and $\mathcal{Y}$ are Polish without additional structure, we prove very general qualitative guarantees; when they have suitable combinatorial structure, we prove quantitative guarantees for Hölder-like maps, including maps between finite graphs, solution operators to rough differential equations between certain Carnot groups, and continuous non-linear operators between Banach spaces arising in inverse problems. In particular, we show that the required number of Dirac measures is determined by the combinatorial structure of $\mathcal{X}$ and $\mathcal{Y}$. For barycentric $\mathcal{Y}$, including Banach spaces, $\mathbb{R}$-trees, Hadamard manifolds, or Wasserstein spaces on Polish metric spaces, our approximators reduce to $\mathcal{Y}$-valued functions. When the Euclidean approximators are neural networks, our constructions generalize transformer networks, providing a new probabilistic viewpoint of geometric deep learning.

Xuwei Yang, Anastasis Kratsios, Florian Krach et al.

We propose an optimal iterative scheme for federated transfer learning, where a central planner has access to datasets ${\cal D}_1,\dots,{\cal D}_N$ for the same learning model $f_θ$. Our objective is to minimize the cumulative deviation of the generated parameters $\{θ_i(t)\}_{t=0}^T$ across all $T$ iterations from the specialized parameters $θ^\star_{1},\ldots,θ^\star_N$ obtained for each dataset, while respecting the loss function for the model $f_{θ(T)}$ produced by the algorithm upon halting. We only allow for continual communication between each of the specialized models (nodes/agents) and the central planner (server), at each iteration (round). For the case where the model $f_θ$ is a finite-rank kernel regression, we derive explicit updates for the regret-optimal algorithm. By leveraging symmetries within the regret-optimal algorithm, we further develop a nearly regret-optimal heuristic that runs with $\mathcal{O}(Np^2)$ fewer elementary operations, where $p$ is the dimension of the parameter space. Additionally, we investigate the adversarial robustness of the regret-optimal algorithm showing that an adversary which perturbs $q$ training pairs by at-most $\varepsilon>0$, across all training sets, cannot reduce the regret-optimal algorithm's regret by more than $\mathcal{O}(\varepsilon q \bar{N}^{1/2})$, where $\bar{N}$ is the aggregate number of training pairs. To validate our theoretical findings, we conduct numerical experiments in the context of American option pricing, utilizing a randomly generated finite-rank kernel.

Ruiyang Hong, Anastasis Kratsios · eth-zurich

The foundations of deep learning are supported by the seemingly opposing perspectives of approximation or learning theory. The former advocates for large/expressive models that need not generalize, while the latter considers classes that generalize but may be too small/constrained to be universal approximators. Motivated by real-world deep learning implementations that are both expressive and statistically reliable, we ask: "Is there a class of neural networks that is both large enough to be universal but structured enough to generalize?" This paper constructively provides a positive answer to this question by identifying a highly structured class of ReLU multilayer perceptions (MLPs), which are optimal function approximators and are statistically well-behaved. We show that any $(L,α)$-Hölder function from $[0,1]^d$ to $[-n,n]$ can be approximated to a uniform $\mathcal{O}(1/n)$ error on $[0,1]^d$ with a sparsely connected ReLU MLP with the same Hölder exponent $α$ and coefficient $L$, of width $\mathcal{O}(dn^{d/α})$, depth $\mathcal{O}(\log(d))$, with $\mathcal{O}(dn^{d/α})$ nonzero parameters, and whose weights and biases take values in $\{0,\pm 1/2\}$ except in the first and last layers which instead have magnitude at-most $n$. Further, our class of MLPs achieves a near-optimal sample complexity of $\mathcal{O}(\log(N)/\sqrt{N})$ when given $N$ i.i.d. normalized sub-Gaussian training samples. We achieve this through a new construction that perfectly fits together linear pieces using Kuhn triangulations, along with a new proof technique which shows that our construction preserves the regularity of not only the Hölder functions, but also any uniformly continuous function. Our results imply that neural networks can solve the McShane extension problem on suitable finite sets.

Haitz Sáez de Ocáriz Borde, Anastasis Kratsios · eth-zurich

The inductive bias of a graph neural network (GNN) is largely encoded in its specified graph. Latent graph inference relies on latent geometric representations to dynamically rewire or infer a GNN's graph to maximize the GNN's predictive downstream performance, but it lacks solid theoretical foundations in terms of embedding-based representation guarantees. This paper addresses this issue by introducing a trainable deep learning architecture, coined neural snowflake, that can adaptively implement fractal-like metrics on $\mathbb{R}^d$. We prove that any given finite weights graph can be isometrically embedded by a standard MLP encoder. Furthermore, when the latent graph can be represented in the feature space of a sufficiently regular kernel, we show that the combined neural snowflake and MLP encoder do not succumb to the curse of dimensionality by using only a low-degree polynomial number of parameters in the number of nodes. This implementation enables a low-dimensional isometric embedding of the latent graph. We conduct synthetic experiments to demonstrate the superior metric learning capabilities of neural snowflakes when compared to more familiar spaces like Euclidean space. Additionally, we carry out latent graph inference experiments on graph benchmarks. Consistently, the neural snowflake model achieves predictive performance that either matches or surpasses that of the state-of-the-art latent graph inference models. Importantly, this performance improvement is achieved without requiring random search for optimal latent geometry. Instead, the neural snowflake model achieves this enhancement in a differentiable manner.

Anastasis Kratsios, Ruiyang Hong, Haitz Sáez de Ocáriz Borde · eth-zurich

We study the representation capacity of deep hyperbolic neural networks (HNNs) with a ReLU activation function. We establish the first proof that HNNs can $\varepsilon$-isometrically embed any finite weighted tree into a hyperbolic space of dimension $d$ at least equal to $2$ with prescribed sectional curvature $κ<0$, for any $\varepsilon> 1$ (where $\varepsilon=1$ being optimal). We establish rigorous upper bounds for the network complexity on an HNN implementing the embedding. We find that the network complexity of HNN implementing the graph representation is independent of the representation fidelity/distortion. We contrast this result against our lower bounds on distortion which any ReLU multi-layer perceptron (MLP) must exert when embedding a tree with $L>2^d$ leaves into a $d$-dimensional Euclidean space, which we show at least $Ω(L^{1/d})$; independently of the depth, width, and (possibly discontinuous) activation function defining the MLP.

Anastasis Kratsios, Behnoosh Zamanlooy · eth-zurich

We study the problem of approximating compactly-supported integrable functions while implementing their support set using feedforward neural networks. Our first main result transcribes this "structured" approximation problem into a universality problem. We do this by constructing a refinement of the usual topology on the space $L^1_{\operatorname{loc}}(\mathbb{R}^d,\mathbb{R}^D)$ of locally-integrable functions in which compactly-supported functions can only be approximated in $L^1$-norm by functions with matching discretized support. We establish the universality of ReLU feedforward networks with bilinear pooling layers in this refined topology. Consequentially, we find that ReLU feedforward networks with bilinear pooling can approximate compactly supported functions while implementing their discretized support. We derive a quantitative uniform version of our universal approximation theorem on the dense subclass of compactly-supported Lipschitz functions. This quantitative result expresses the depth, width, and the number of bilinear pooling layers required to construct this ReLU network via the target function's regularity, the metric capacity and diameter of its essential support, and the dimensions of the inputs and output spaces. Conversely, we show that polynomial regressors and analytic feedforward networks are not universal in this space.

Yicen Li, Ruiyang Hong, Anastasis Kratsios et al.

Classical clustering methods usually return either a finite partition of the observed data or a finite dendrogram over it. This finite-sample view is inadequate when the hierarchy of interest is a recursive geometric object with fine-scale refinements that continue beyond the levels directly observed. We introduce classification fields: infinite-depth hierarchical cluster structures on $\mathbb{R}^d$ generated by a local parent-to-child refinement rule. A classification field generator maps each parent centre to an ordered, bounded, and separated tuple of child residuals. Together with a root and a scale factor, this rule recursively generates cluster centres, Voronoi cells, and a metric DAG encoding the hierarchy. Given only a finite prefix of such a hierarchy, we learn a classification field predictor that approximates the generator and can be rolled out to unseen depths. We prove exponential truncation convergence in the completed cell metric and ReLU realizability with width $O(\varepsilon^{-γ})$ and depth $\widetilde O(\varepsilon^{-3γ/2})$, where $γ=\log K/(-\log s)$, up to finite-window aspect-ratio factors. The approximation holds at the level of the induced compact metric structures, measured in the completed cell-metric Hausdorff distance. Experimental validation on matched CFG-generated hierarchies, IFS fractals, and image-induced recursive clustering hierarchies shows that learned predictors preserve ordered child slots, unordered geometry, and hierarchy-level path metrics under recursive rollout. These results support the claim that finite hierarchical observations can reveal local refinement rules capable of generating substantially deeper classification fields.

Haitz Sáez de Ocáriz Borde, Anastasis Kratsios, Marc T. Law et al. · eth-zurich

We propose a class of trainable deep learning-based geometries called Neural Spacetimes (NSTs), which can universally represent nodes in weighted directed acyclic graphs (DAGs) as events in a spacetime manifold. While most works in the literature focus on undirected graph representation learning or causality embedding separately, our differentiable geometry can encode both graph edge weights in its spatial dimensions and causality in the form of edge directionality in its temporal dimensions. We use a product manifold that combines a quasi-metric (for space) and a partial order (for time). NSTs are implemented as three neural networks trained in an end-to-end manner: an embedding network, which learns to optimize the location of nodes as events in the spacetime manifold, and two other networks that optimize the space and time geometries in parallel, which we call a neural (quasi-)metric and a neural partial order, respectively. The latter two networks leverage recent ideas at the intersection of fractal geometry and deep learning to shape the geometry of the representation space in a data-driven fashion, unlike other works in the literature that use fixed spacetime manifolds such as Minkowski space or De Sitter space to embed DAGs. Our main theoretical guarantee is a universal embedding theorem, showing that any $k$-point DAG can be embedded into an NST with $1+\mathcal{O}(\log(k))$ distortion while exactly preserving its causal structure. The total number of parameters defining the NST is sub-cubic in $k$ and linear in the width of the DAG. If the DAG has a planar Hasse diagram, this is improved to $\mathcal{O}(\log(k)) + 2)$ spatial and 2 temporal dimensions. We validate our framework computationally with synthetic weighted DAGs and real-world network embeddings; in both cases, the NSTs achieve lower embedding distortions than their counterparts using fixed spacetime geometries.

Anastasis Kratsios, Valentin Debarnot, Ivan Dokmanić

We study representations of data from an arbitrary metric space $\mathcal{X}$ in the space of univariate Gaussian mixtures with a transport metric (Delon and Desolneux 2020). We derive embedding guarantees for feature maps implemented by small neural networks called \emph{probabilistic transformers}. Our guarantees are of memorization type: we prove that a probabilistic transformer of depth about $n\log(n)$ and width about $n^2$ can bi-Hölder embed any $n$-point dataset from $\mathcal{X}$ with low metric distortion, thus avoiding the curse of dimensionality. We further derive probabilistic bi-Lipschitz guarantees, which trade off the amount of distortion and the probability that a randomly chosen pair of points embeds with that distortion. If $\mathcal{X}$'s geometry is sufficiently regular, we obtain stronger, bi-Lipschitz guarantees for all points in the dataset. As applications, we derive neural embedding guarantees for datasets from Riemannian manifolds, metric trees, and certain types of combinatorial graphs. When instead embedding into multivariate Gaussian mixtures, we show that probabilistic transformers can compute bi-Hölder embeddings with arbitrarily small distortion.

Anastasis Kratsios, Gregory Cousins, Haitz Sáez de Ocáriz Borde et al.

We show that, in a precise sense, a broad class of feedforward neural networks learn (have finite sample complexity) in the PAC model: every fixed finite feedforward architecture whose layers are definable in an o-minimal structure has finite sample complexity in the agnostic PAC setting, even with unbounded parameters. This covers standard fixed-size MLPs, CNNs, GNNs, and transformers with fixed sequence length, together with the operations and layers typically used in such architectures, including linear projections, residual connections, attention mechanisms, pooling layers, normalization layers, and admissible positional encodings. Hence, distribution-free learnability for modern non-recurrent architectures is not an exceptional property of particular activations or architecture-specific VC arguments, but a consequence of tame feedforward computation. Our results reposition finite-sample PAC learnability as a baseline rather than a differentiator: they shift the focus of architectural comparison toward inductive biases, symmetries and geometric priors, scalability, and optimization behaviour.

Blanka Horvath, Anastasis Kratsios, Yannick Limmer et al.

The use of attention-based deep learning models in stochastic filtering, e.g. transformers and deep Kalman filters, has recently come into focus; however, the potential for these models to solve stochastic filtering problems remains largely unknown. The paper provides an affirmative answer to this open problem in the theoretical foundations of machine learning by showing that a class of continuous-time transformer models, called \textit{filterformers}, can approximately implement the conditional law of a broad class of non-Markovian and conditionally Gaussian signal processes given noisy continuous-time (possibly non-Gaussian) measurements. Our approximation guarantees hold uniformly over sufficiently regular compact subsets of continuous-time paths, where the worst-case 2-Wasserstein distance between the true optimal filter and our deep learning model quantifies the approximation error. Our construction relies on two new customizations of the standard attention mechanism: The first can losslessly adapt to the characteristics of a broad range of paths since we show that the attention mechanism implements bi-Lipschitz embeddings of sufficiently regular sets of paths into low-dimensional Euclidean spaces; thus, it incurs no ``dimension reduction error''. The latter attention mechanism is tailored to the geometry of Gaussian measures in the $2$-Wasserstein space. Our analysis relies on new stability estimates of robust optimal filters in the conditionally Gaussian setting.

Anastasis Kratsios

Convex functionals are ubiquitous in applied analysis, appearing as value functions, risk measures, super-hedging prices, and loss functionals in machine learning. In many applications, however, the functional is only observed through finitely many exact pointwise evaluations. We ask whether a convex functional on a separable Hilbert space $H$ can be reconstructed, up to arbitrary uniform accuracy, by an explicit formula which preserves convexity and Lipschitz regularity and is finitely computable. We answer this affirmatively. For every compact convex $C\subseteq H$, every $L$-Lipschitz convex functional $ρ:C\to\mathbb{R}$, and every $\varepsilon>0$, we construct an explicit finite-sample reconstruction which is convex, $L$-Lipschitz, and uniformly $\varepsilon$-accurate on $C$. The construction uses only finitely many linear measurements $\langle b,\cdot\rangle_H$, with $b$ lying in a finite-dimensional subspace of $H$, and is exactly implementable by a $\operatorname{ReLU}$-MLP. Building on this, we introduce convex neural functionals (CNFs), a structured trainable architecture class containing our reconstruction, whose every admissible parameter configuration is automatically convex and Lipschitz, providing a principled foundation for learning convex functionals from finite data.

Xuwei Yang, David B. Emerson, Fatemeh Tavakoli et al.

In the modern age of large-scale AI, federated learning has become an increasingly important tool for training large populations of AI agents; however, its computational and communication costs can rapidly fail to scale with the number of agents. This is precisely where decentralized agentic strategies shine: each agent acts autonomously, using only its own state together with a minimal summary of the ensemble, namely the mean-field. We derive the unique optimal decentralized policy in closed form. Optimality is characterized through a worst-client/minimax criterion: minimizing the under-performer regret, namely the maximal online cost incurred by the weakest agent in the ensemble. We further prove that the resulting decentralized policy asymptotically converges, in the large-population limit, to the Nash-optimal centralized policy, whose direct computation is not scalable. We use an online weighting mechanism to optimize the server-computed mixture of client predictions, thereby improving the mean prediction in addition to the previously optimized weakest-client prediction. Numerical experiments verify our theoretical guarantees and demonstrate that our decentralized policy typically outperforms natural greedy decentralized baselines.

Anastasis Kratsios, A. Martina Neuman, Philipp Petersen

We compare in-context learning with fixed queries and agentic learning with adaptive queries for uniform approximation of task families. We consider two settings: an unrestricted regime, where querying and approximation are arbitrary functions, and a realizable regime, where we require these operations to be implemented by ReLU neural networks. In both settings, adaptivity never hinders approximation performance. However, this advantage can change when one passes from the unrestricted regime to the realizable regime. We identify four distinct approximation scenarios, each witnessed by an explicit task family: (a) no advantage of adaptivity; (b) an advantage in the unrestricted regime that persists under ReLU realizability; (c) an advantage that arises only under realizability; and (d) an advantage that disappears under realizability. This demonstrates that representational constraints interact profoundly with the effect of adaptivity.

Anastasis Kratsios, Cody Hyndman

We consider the problem of simultaneously approximating the conditional distribution of market prices and their log returns with a single machine learning model. We show that an instance of the GDN model of Kratsios and Papon (2022) solves this problem without having prior assumptions on the market's "clipped" log returns, other than that they follow a generalized Ornstein-Uhlenbeck process with a priori unknown dynamics. We provide universal approximation guarantees for these conditional distributions and contingent claims with a Lipschitz payoff function.

Hossein Rouhvarzi, Anastasis Kratsios

Incremental flow-based denoising models have reshaped generative modelling, but their empirical advantage still lacks a rigorous approximation-theoretic foundation. We show that incremental generation is necessary and sufficient for universal flow-based generation on the largest natural class of self-maps of $[0,1]^d$ compatible with denoising pipelines, namely the orientation-preserving homeomorphisms of $[0,1]^d$. All our guarantees are uniform on the underlying maps and hence imply approximation both samplewise and in distribution. Using a new topological-dynamical argument, we first prove an impossibility theorem: the class of all single-step autonomous flows, independently of the architecture, width, depth, or Lipschitz activation of the underlying neural network, is meagre and therefore not universal in the space of orientation-preserving homeomorphisms of $[0,1]^d$. By exploiting algebraic properties of autonomous flows, we conversely show that every orientation-preserving Lipschitz homeomorphism on $[0,1]^d$ can be approximated at rate $\mathcal{O}(n^{-1/d})$ by a composition of at most $K_d$ such flows, where $K_d$ depends only on the dimension. Under additional smoothness assumptions, the approximation rate can be made dimension-free, and $K_d$ can be chosen uniformly over the class being approximated. Finally, by linearly lifting the domain into one higher dimension, we obtain structured universal approximation results for continuous functions and for probability measures on $[0,1]^d$, the latter realized as pushforwards of empirical measures with vanishing $1$-Wasserstein error.

Takashi Furuya, Anastasis Kratsios, Dylan Possamaï et al.

We introduce a mild generative variant of the classical neural operator model, which leverages Kolmogorov--Arnold networks to solve infinite families of second-order backward stochastic differential equations ($2$BSDEs) on regular bounded Euclidean domains with random terminal time. Our first main result shows that the solution operator associated with a broad range of $2$BSDE families is approximable by appropriate neural operator models. We then identify a structured subclass of (infinite) families of $2$BSDEs whose neural operator approximation requires only a polynomial number of parameters in the reciprocal approximation rate, as opposed to the exponential requirement in general worst-case neural operator guarantees.

Anastasis Kratsios, Takashi Furuya, Jose Antonio Lara Benitez et al.

In this paper, we construct a mixture of neural operators (MoNOs) between function spaces whose complexity is distributed over a network of expert neural operators (NOs), with each NO satisfying parameter scaling restrictions. Our main result is a \textit{distributed} universal approximation theorem guaranteeing that any Lipschitz non-linear operator between $L^2([0,1]^d)$ spaces can be approximated uniformly over the Sobolev unit ball therein, to any given $\varepsilon>0$ accuracy, by an MoNO while satisfying the constraint that: each expert NO has a depth, width, and rank of $\mathcal{O}(\varepsilon^{-1})$. Naturally, our result implies that the required number of experts must be large, however, each NO is guaranteed to be small enough to be loadable into the active memory of most computers for reasonable accuracies $\varepsilon$. During our analysis, we also obtain new quantitative expression rates for classical NOs approximating uniformly continuous non-linear operators uniformly on compact subsets of $L^2([0,1]^d)$.

Tin Sum Cheng, Aurelien Lucchi, Anastasis Kratsios et al.

We derive new bounds for the condition number of kernel matrices, which we then use to enhance existing non-asymptotic test error bounds for kernel ridgeless regression (KRR) in the over-parameterized regime for a fixed input dimension. For kernels with polynomial spectral decay, we recover the bound from previous work; for exponential decay, our bound is non-trivial and novel. Our contribution is two-fold: (i) we rigorously prove the phenomena of tempered overfitting and catastrophic overfitting under the sub-Gaussian design assumption, closing an existing gap in the literature; (ii) we identify that the independence of the features plays an important role in guaranteeing tempered overfitting, raising concerns about approximating KRR generalization using the Gaussian design assumption in previous literature.

Guillermo Alvarez, Ibrahim Ekren, Anastasis Kratsios et al.

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.

Haitz Sáez de Ocáriz Borde, Artem Lukoianov, Anastasis Kratsios et al.

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.

Tin Sum Cheng, Aurelien Lucchi, Anastasis Kratsios et al.

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.

Anastasis Kratsios, Takashi Furuya · eth-zurich

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.

Nils Detering, Luca Galimberti, Anastasis Kratsios et al. · eth-zurich

Since their introduction by Kipf and Welling in $2017$, a primary use of graph convolutional networks is transductive node classification, where missing labels are inferred within a single observed graph and its feature matrix. Despite the widespread use of the network model, the statistical foundations of transductive learning remain limited, as standard inference frameworks typically rely on multiple independent samples rather than a single graph. In this work, we address these gaps by developing new concentration-of-measure tools that leverage the geometric regularities of large graphs via low-dimensional metric embeddings. The emergent regularities are captured using a random graph model; however, the methods remain applicable to deterministic graphs once observed. We establish two principal learning results. The first concerns arbitrary deterministic $k$-vertex graphs, and the second addresses random graphs that share key geometric properties with an Erdős-Rényi graph $\mathbf{G}=\mathbf{G}(k,p)$ in the regime $p \in \mathcal{O}((\log (k)/k)^{1/2})$. The first result serves as the basis for and illuminates the second. We then extend these results to the graph convolutional network setting, where additional challenges arise. Lastly, our learning guarantees remain informative even with a few labelled nodes $N$ and achieve the optimal nonparametric rate $\mathcal{O}(N^{-1/2})$ as $N$ grows.

Dena Firoozi, Anastasis Kratsios, Xuwei Yang

Traditional mean-field game (MFG) solvers operate on an instance-by-instance basis, which becomes infeasible when many related problems must be solved (e.g., for seeking a robust description of the solution under perturbations of the dynamics or utilities, or in settings involving continuum-parameterized agents.). We overcome this by training neural operators (NOs) to learn the rules-to-equilibrium map from the problem data (``rules'': dynamics and cost functionals) of LQ MFGs defined on separable Hilbert spaces to the corresponding equilibrium strategy. Our main result is a statistical guarantee: an NO trained on a small number of randomly sampled rules reliably solves unseen LQ MFG variants, even in infinite-dimensional settings. The number of NO parameters needed remains controlled under appropriate rule sampling during training. Our guarantee follows from three results: (i) local-Lipschitz estimates for the highly nonlinear rules-to-equilibrium map; (ii) a universal approximation theorem using NOs with a prespecified Lipschitz regularity (unlike traditional NO results where the NO's Lipschitz constant can diverge as the approximation error vanishes); and (iii) new sample-complexity bounds for $L$-Lipschitz learners in infinite dimensions, directly applicable as the Lipschitz constants of our approximating NOs are controlled in (ii).

Anastasis Kratsios, Ariel Neufeld, Philipp Schmocker · eth-zurich

Neural operators (NOs) are a class of deep learning models designed to simultaneously solve infinitely many related problems by casting them into an infinite-dimensional space, whereon these NOs operate. A significant gap remains between theory and practice: worst-case parameter bounds from universal approximation theorems suggest that NOs may require an unrealistically large number of parameters to solve most operator learning problems, which stands in direct opposition to a slew of experimental evidence. This paper closes that gap for a specific class of {NOs}, generative {equilibrium operators} (GEOs), using (realistic) finite-dimensional deep equilibrium layers, when solving families of convex optimization problems over a separable Hilbert space $X$. Here, the inputs are smooth, convex loss functions on $X$, and outputs are the associated (approximate) solutions to the optimization problem defined by each input loss. We show that when the input losses lie in suitable infinite-dimensional compact sets, our GEO can uniformly approximate the corresponding solutions to arbitrary precision, with rank, depth, and width growing only logarithmically in the reciprocal of the approximation error. We then validate both our theoretical results and the trainability of GEOs on three applications: (1) nonlinear PDEs, (2) stochastic optimal control problems, and (3) hedging problems in mathematical finance under liquidity constraints.

Anastasis Kratsios, Tin Sum Cheng, Aurelien Lucchi et al.

Low-Rank Adaptation (LoRA) has emerged as a widely adopted parameter-efficient fine-tuning (PEFT) technique for foundation models. Recent work has highlighted an inherent asymmetry in the initialization of LoRA's low-rank factors, which has been present since its inception and was presumably derived experimentally. This paper focuses on providing a comprehensive theoretical characterization of asymmetric LoRA with frozen random factors. First, while existing research provides upper-bound generalization guarantees based on averages over multiple experiments, the behaviour of a single fine-tuning run with specific random factors remains an open question. We address this by investigating the concentration of the typical LoRA generalization gap around its mean. Our main upper bound reveals a sample complexity of $\tilde{\mathcal{O}}\left(\frac{\sqrt{r}}{\sqrt{N}}\right)$ with high probability for rank $r$ LoRAs trained on $N$ samples. Additionally, we also determine the fundamental limits in terms of sample efficiency, establishing a matching lower bound of $\mathcal{O}\left(\frac{1}{\sqrt{N}}\right)$. By more closely reflecting the practical scenario of a single fine-tuning run, our findings offer crucial insights into the reliability and practicality of asymmetric LoRA.

Anastasis Kratsios, Haitz Sáez de Ocáriz Borde, Takashi Furuya et al. · eth-zurich

Mixture-of-Experts (MoEs) can scale up beyond traditional deep learning models by employing a routing strategy in which each input is processed by a single "expert" deep learning model. This strategy allows us to scale up the number of parameters defining the MoE while maintaining sparse activation, i.e., MoEs only load a small number of their total parameters into GPU VRAM for the forward pass depending on the input. In this paper, we provide an approximation and learning-theoretic analysis of mixtures of expert MLPs with (P)ReLU activation functions. We first prove that for every error level $\varepsilon>0$ and every Lipschitz function $f:[0,1]^n\to \mathbb{R}$, one can construct a MoMLP model (a Mixture-of-Experts comprising of (P)ReLU MLPs) which uniformly approximates $f$ to $\varepsilon$ accuracy over $[0,1]^n$, while only requiring networks of $\mathcal{O}(\varepsilon^{-1})$ parameters to be loaded in memory. Additionally, we show that MoMLPs can generalize since the entire MoMLP model has a (finite) VC dimension of $\tilde{O}(L\max\{nL,JW\})$, if there are $L$ experts and each expert has a depth and width of $J$ and $W$, respectively.

Wenhao Li, Anastasis Kratsios, Hrad Ghoukasian et al.

The proliferation of agentic systems has thrust the reasoning capabilities of AI into the forefront of contemporary machine learning. While it is known that there \emph{exist} neural networks which can reason through any Boolean task $f:\{0,1\}^B\to\{0,1\}$, in the sense that they emulate Boolean circuits with fan-in $2$ and fan-out $1$ gates, trained models have been repeatedly demonstrated to fall short of these theoretical ideals. This raises the question: \textit{Can one exhibit a deep learning model which \textbf{certifiably} always reasons and can \textbf{universally} reason through any Boolean task?} Moreover, such a model should ideally require few parameters to solve simple Boolean tasks. We answer this question affirmatively by exhibiting a deep learning architecture which parameterizes distributions over Boolean circuits with the guarantee that, for every parameter configuration, a sample is almost surely a valid Boolean circuit (and hence admits an intrinsic circuit-level certificate). We then prove a universality theorem: for any Boolean $f:\{0,1\}^B\to\{0,1\}$, there exists a parameter configuration under which the sampled circuit computes $f$ with arbitrarily high probability. When $f$ is an $\mathcal{O}(\log B)$-junta, the required number of parameters scales linearly with the input dimension $B$. Empirically, on a controlled truth-table completion benchmark aligned with our setting, the proposed architecture trains reliably and achieves high exact-match accuracy while preserving the predicted structure: every internal unit is Boolean-valued on $\{0,1\}^B$. Matched MLP baselines reach comparable accuracy, but only about $10\%$ of hidden units admit a Boolean representation; i.e.\ are two-valued over the Boolean cube.

Anastasis Kratsios, Giulia Livieri, A. Martina Neuman

We study the statistical behaviour of reasoning probes in a stylized model of looped reasoning, given by Boolean circuits whose computational graph is a perfect $ν$-ary tree ($ν\ge 2$) and whose output is appended to the input and fed back iteratively for subsequent computation rounds. A reasoning probe has access to a sampled subset of internal computation nodes, possibly without covering the entire graph, and seeks to infer which $ν$-ary Boolean gate is executed at each queried node, representing uncertainty via a probability distribution over a fixed collection of $\mathtt{m}$ admissible $ν$-ary gates. This partial observability induces a generalization problem, which we analyze in a realizable, transductive setting. We show that, when the reasoning probe is parameterized by a graph convolutional network (GCN)-based hypothesis class and queries $N$ nodes, the worst-case generalization error attains the optimal rate $\mathcal{O}(\sqrt{\log(2/δ)}/\sqrt{N})$ with probability at least $1-δ$, for $δ\in (0,1)$. Our analysis combines snowflake metric embedding techniques with tools from statistical optimal transport. A key insight is that this optimal rate is achievable independently of graph size, owing to the existence of a low-distortion one-dimensional snowflake embedding of the induced graph metric. As a consequence, our results provide a sharp characterization of how structural properties of the computational graph govern the statistical efficiency of reasoning under partial access.

Anastasis Kratsios, Tin Sum Cheng, Daniel Roy

At its core, machine learning seeks to train models that reliably generalize beyond noisy observations; however, the theoretical vacuum in which state-of-the-art universal approximation theorems (UATs) operate isolates them from this goal, as they assume noiseless data and allow network parameters to be chosen freely, independent of algorithmic realism. This paper bridges that gap by introducing an architecture-specific randomized training algorithm that constructs a uniform approximator from $N$ noisy training samples on the $d$-dimensional cube $[0,1]^d$. Our trained neural networks attain the minimax-optimal quantity of \textit{trainable} (non-random) parameters, subject to logarithmic factors which vanish under the idealized noiseless sampling assumed in classical UATs. Additionally, our trained models replicate key behaviours of real-world neural networks, absent in standard UAT constructions, by: (1) exhibiting sub-linear parametric complexity when fine-tuning on structurally related and favourable out-of-distribution tasks, (2) exactly interpolating the training data, and (3) maintaining reasonable Lipschitz regularity (after the initial clustering attention layer). These properties bring state-of-the-art UATs closer to practical machine learning, shifting the central open question from algorithmic implementability with noisy samples to whether stochastic gradient descent can achieve comparable guarantees.

Reza Arabpour, Haitz Sáez de Ocáriz Borde, Anastasis Kratsios · eth-zurich

Low-Rank Adapters (LoRAs) have transformed the fine-tuning of Large Language Models (LLMs) by enabling parameter-efficient updates. However, their widespread adoption remains limited by the reliance on GPU-based training. In this work, we propose a theoretically grounded approach to LoRA fine-tuning designed specifically for users with limited computational resources, particularly those restricted to standard laptop CPUs. Our method learns a meta-operator that maps any input dataset, represented as a probability distribution, to a set of LoRA weights by leveraging a large bank of pre-trained adapters for the Mistral-7B-Instruct-v0.2 model. Instead of performing new gradient-based updates, our pipeline constructs adapters via lightweight combinations of existing LoRAs directly on CPU. While the resulting adapters do not match the performance of GPU-trained counterparts, they consistently outperform the base Mistral model on downstream tasks, offering a practical and accessible alternative to traditional GPU-based fine-tuning.

Anastasis Kratsios, Xiaofei Shi, Qiang Sun et al. · eth-zurich

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.

Yicen Li, Haitz Sáez de Ocáriz Borde, Anastasis Kratsios et al. · eth-zurich

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.

Takashi Furuya, Anastasis Kratsios · eth-zurich

Forward-backwards stochastic differential equations (FBSDEs) play an important role in optimal control, game theory, economics, mathematical finance, and in reinforcement learning. 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 decoupled 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. Upon making only a few simple theoretically-guided tweaks to the standard convolutional NO build, 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 using a logarithmic depth, a constant width, and a polynomial rank in the reciprocal approximation error. This result is rooted in our second result, and main contribution to the NOs for PDE literature, showing that our convolutional NOs of similar depth and width but grow only \textit{quadratically} (at a dimension-free rate) when uniformly approximating the solution operator of the associated class of semilinear Elliptic PDEs to these families of FBSDEs. A key insight into how NOs work we uncover is that the convolutional layers of our NO can approximately implement the fixed point iteration used to prove the existence of a unique solution to these semilinear Elliptic PDEs.

Raeid Saqur, Anastasis Kratsios, Florian Krach et al.

We propose MoE-F - a formalized mechanism for combining $N$ pre-trained Large Language Models (LLMs) for online time-series prediction by adaptively forecasting the best weighting of LLM predictions at every time step. Our mechanism leverages the conditional information in each expert's running performance to forecast the best combination of LLMs for predicting the time series in its next step. Diverging from static (learned) Mixture of Experts (MoE) methods, our approach employs time-adaptive stochastic filtering techniques to combine experts. By framing the expert selection problem as a finite state-space, continuous-time Hidden Markov model (HMM), we can leverage the Wohman-Shiryaev filter. Our approach first constructs N parallel filters corresponding to each of the $N$ individual LLMs. Each filter proposes its best combination of LLMs, given the information that they have access to. Subsequently, the N filter outputs are optimally aggregated to maximize their robust predictive power, and this update is computed efficiently via a closed-form expression, generating our ensemble predictor. Our contributions are: **(I)** the MoE-F plug-and-play filtering harness algorithm, **(II)** theoretical optimality guarantees of the proposed filtering-based gating algorithm (via optimality guarantees for its parallel Bayesian filtering and its robust aggregation steps), and **(III)** empirical evaluation and ablative results using state-of-the-art foundational and MoE LLMs on a real-world __Financial Market Movement__ task where MoE-F attains a remarkable 17\% absolute and 48.5\% relative F1 measure improvement over the next best performing individual LLM expert predicting short-horizon market movement based on streaming news. Further, we provide empirical evidence of substantial performance gains in applying MoE-F over specialized models in the long-horizon time-series forecasting domain.

Anastasis Kratsios, A. Martina Neuman, Gudmund Pammer · eth-zurich

Machine learning models with inputs in a Euclidean space $\mathbb{R}^d$, when implemented on digital computers, generalize, and their generalization gap converges to $0$ at a rate of $c/N^{1/2}$ concerning the sample size $N$. However, the constant $c>0$ obtained through classical methods can be large in terms of the ambient dimension $d$ and machine precision, posing a challenge when $N$ is small to realistically large. In this paper, we derive a family of generalization bounds $\{c_m/N^{1/(2\vee m)}\}_{m=1}^{\infty}$ tailored for learning models on digital computers, which adapt to both the sample size $N$ and the so-called geometric representation dimension $m$ of the discrete learning problem. Adjusting the parameter $m$ according to $N$ results in significantly tighter generalization bounds for practical sample sizes $N$, while setting $m$ small maintains the optimal dimension-free worst-case rate of $\mathcal{O}(1/N^{1/2})$. Notably, $c_{m}\in \mathcal{O}(m^{1/2})$ for learning models on discretized Euclidean domains. Furthermore, our adaptive generalization bounds are formulated based on our new non-asymptotic result for concentration of measure in finite metric spaces, established via leveraging metric embedding arguments.

Beatrice Acciaio, Anastasis Kratsios, Gudmund Pammer

Several problems in stochastic analysis are defined through their geometry, and preserving that geometric structure is essential to generating meaningful predictions. Nevertheless, how to design principled deep learning (DL) models capable of encoding these geometric structures remains largely unknown. We address this open problem by introducing a universal causal geometric DL framework in which the user specifies a suitable pair of metric spaces $\mathscr{X}$ and $\mathscr{Y}$ and our framework returns a DL model capable of causally approximating any ``regular'' map sending time series in $\mathscr{X}^{\mathbb{Z}}$ to time series in $\mathscr{Y}^{\mathbb{Z}}$ while respecting their forward flow of information throughout time. Suitable geometries on $\mathscr{Y}$ include various (adapted) Wasserstein spaces arising in optimal stopping problems, a variety of statistical manifolds describing the conditional distribution of continuous-time finite state Markov chains, and all Fréchet spaces admitting a Schauder basis, e.g. as in classical finance. Suitable spaces $\mathscr{X}$ are compact subsets of any Euclidean space. Our results all quantitatively express the number of parameters needed for our DL model to achieve a given approximation error as a function of the target map's regularity and the geometric structure both of $\mathscr{X}$ and of $\mathscr{Y}$. Even when omitting any temporal structure, our universal approximation theorems are the first guarantees that Hölder functions, defined between such $\mathscr{X}$ and $\mathscr{Y}$ can be approximated by DL models.

Anastasis Kratsios, Behnoosh Zamanlooy, Tianlin Liu et al.

Many practical problems need the output of a machine learning model to satisfy a set of constraints, $K$. Nevertheless, there is no known guarantee that classical neural network architectures can exactly encode constraints while simultaneously achieving universality. We provide a quantitative constrained universal approximation theorem which guarantees that for any non-convex compact set $K$ and any continuous function $f:\mathbb{R}^n\rightarrow K$, there is a probabilistic transformer $\hat{F}$ whose randomized outputs all lie in $K$ and whose expected output uniformly approximates $f$. Our second main result is a "deep neural version" of Berge's Maximum Theorem (1963). The result guarantees that given an objective function $L$, a constraint set $K$, and a family of soft constraint sets, there is a probabilistic transformer $\hat{F}$ that approximately minimizes $L$ and whose outputs belong to $K$; moreover, $\hat{F}$ approximately satisfies the soft constraints. Our results imply the first universal approximation theorem for classical transformers with exact convex constraint satisfaction. They also yield that a chart-free universal approximation theorem for Riemannian manifold-valued functions subject to suitable geodesically convex constraints.

Anastasis Kratsios

We introduce a deep learning model that can universally approximate regular conditional distributions (RCDs). The proposed model operates in three phases: first, it linearizes inputs from a given metric space $\mathcal{X}$ to $\mathbb{R}^d$ via a feature map, then a deep feedforward neural network processes these linearized features, and then the network's outputs are then transformed to the $1$-Wasserstein space $\mathcal{P}_1(\mathbb{R}^D)$ via a probabilistic extension of the attention mechanism of Bahdanau et al.\ (2014). Our model, called the \textit{probabilistic transformer (PT)}, can approximate any continuous function from $\mathbb{R}^d $ to $\mathcal{P}_1(\mathbb{R}^D)$ uniformly on compact sets, quantitatively. We identify two ways in which the PT avoids the curse of dimensionality when approximating $\mathcal{P}_1(\mathbb{R}^D)$-valued functions. The first strategy builds functions in $C(\mathbb{R}^d,\mathcal{P}_1(\mathbb{R}^D))$ which can be efficiently approximated by a PT, uniformly on any given compact subset of $\mathbb{R}^d$. In the second approach, given any function $f$ in $C(\mathbb{R}^d,\mathcal{P}_1(\mathbb{R}^D))$, we build compact subsets of $\mathbb{R}^d$ whereon $f$ can be efficiently approximated by a PT.

Anastasis Kratsios, Leonie Papon

This paper addresses the growing need to process non-Euclidean data, by introducing a geometric deep learning (GDL) framework for building universal feedforward-type models compatible with differentiable manifold geometries. We show that our GDL models can approximate any continuous target function uniformly on compact sets of a controlled maximum diameter. We obtain curvature-dependent lower-bounds on this maximum diameter and upper-bounds on the depth of our approximating GDL models. Conversely, we find that there is always a continuous function between any two non-degenerate compact manifolds that any "locally-defined" GDL model cannot uniformly approximate. Our last main result identifies data-dependent conditions guaranteeing that the GDL model implementing our approximation breaks "the curse of dimensionality." We find that any "real-world" (i.e. finite) dataset always satisfies our condition and, conversely, any dataset satisfies our requirement if the target function is smooth. As applications, we confirm the universal approximation capabilities of the following GDL models: Ganea et al. (2018)'s hyperbolic feedforward networks, the architecture implementing Krishnan et al. (2015)'s deep Kalman-Filter, and deep softmax classifiers. We build universal extensions/variants of: the SPD-matrix regressor of Meyer et al. (2011), and Fletcher (2003)'s Procrustean regressor. In the Euclidean setting, our results imply a quantitative version of Kidger and Lyons (2020)'s approximation theorem and a data-dependent version of Yarotsky and Zhevnerchuk (2019)'s uncursed approximation rates.

Philippe Casgrain, Anastasis Kratsios

The need for fast and robust optimization algorithms are of critical importance in all areas of machine learning. This paper treats the task of designing optimization algorithms as an optimal control problem. Using regret as a metric for an algorithm's performance, we study the existence, uniqueness and consistency of regret-optimal algorithms. By providing first-order optimality conditions for the control problem, we show that regret-optimal algorithms must satisfy a specific structure in their dynamics which we show is equivalent to performing dual-preconditioned gradient descent on the value function generated by its regret. Using these optimal dynamics, we provide bounds on their rates of convergence to solutions of convex optimization problems. Though closed-form optimal dynamics cannot be obtained in general, we present fast numerical methods for approximating them, generating optimization algorithms which directly optimize their long-term regret. Lastly, these are benchmarked against commonly used optimization algorithms to demonstrate their effectiveness.

Anastasis Kratsios, Behnoosh Zamanlooy

Most stochastic gradient descent algorithms can optimize neural networks that are sub-differentiable in their parameters; however, this implies that the neural network's activation function must exhibit a degree of continuity which limits the neural network model's uniform approximation capacity to continuous functions. This paper focuses on the case where the discontinuities arise from distinct sub-patterns, each defined on different parts of the input space. We propose a new discontinuous deep neural network model trainable via a decoupled two-step procedure that avoids passing gradient updates through the network's only and strategically placed, discontinuous unit. We provide approximation guarantees for our architecture in the space of bounded continuous functions and universal approximation guarantees in the space of piecewise continuous functions which we introduced herein. We present a novel semi-supervised two-step training procedure for our discontinuous deep learning model, tailored to its structure, and we provide theoretical support for its effectiveness. The performance of our model and trained with the propose procedure is evaluated experimentally on both real-world financial datasets and synthetic datasets.

Anastasis Kratsios, Behnoosh Zamanlooy

Most $L^p$-type universal approximation theorems guarantee that a given machine learning model class $\mathscr{F}\subseteq C(\mathbb{R}^d,\mathbb{R}^D)$ is dense in $L^p_μ(\mathbb{R}^d,\mathbb{R}^D)$ for any suitable finite Borel measure $μ$ on $\mathbb{R}^d$. Unfortunately, this means that the model's approximation quality can rapidly degenerate outside some compact subset of $\mathbb{R}^d$, as any such measure is largely concentrated on some bounded subset of $\mathbb{R}^d$. This paper proposes a generic solution to this approximation theoretic problem by introducing a canonical transformation which "upgrades $\mathscr{F}$'s approximation property" in the following sense. The transformed model class, denoted by $\mathscr{F}\text{-tope}$, is shown to be dense in $L^p_{μ,\text{strict}}(\mathbb{R}^d,\mathbb{R}^D)$ which is a topological space whose elements are locally $p$-integrable functions and whose topology is much finer than usual norm topology on $L^p_μ(\mathbb{R}^d,\mathbb{R}^D)$; here $μ$ is any suitable $σ$-finite Borel measure $μ$ on $\mathbb{R}^d$. Next, we show that if $\mathscr{F}$ is any family of analytic functions then there is always a strict "gap" between $\mathscr{F}\text{-tope}$'s expressibility and that of $\mathscr{F}$, since we find that $\mathscr{F}$ can never dense in $L^p_{μ,\text{strict}}(\mathbb{R}^d,\mathbb{R}^D)$. In the general case, where $\mathscr{F}$ may contain non-analytic functions, we provide an abstract form of these results guaranteeing that there always exists some function space in which $\mathscr{F}\text{-tope}$ is dense but $\mathscr{F}$ is not, while, the converse is never possible. Applications to feedforward networks, convolutional neural networks, and polynomial bases are explored.