arXiv subjects

Konstantin Riedl

Publications and source records attributed to Konstantin Riedl.

17 recordsLinked to original sources

arXiv2026

On the Diverse Dynamical Behaviors Arising in Deep Linear Transformers

We study the inference-time behavior of deep linear encoder-only transformers through the lens of interacting particle systems. In this perspective, tokens are modeled as particles that interact dynamically through successive linear self-attention layers. We show that in embedding dimension two, for any key, query, and value matrices, the dynamics can be reformulated as a generalized Kuramoto-type model with pure second-harmonic coupling. This formulation is amenable to Watanabe--Strogatz theory which reveals the dynamics are intrinsically low-dimensional regardless of the parameter matrices. For a class of token initializations associated with the Ott--Antonsen (OA) manifold, we show that the parameter matrices induce a diverse variety of long-time behaviors in linear transformers, including clustering, oscillations, and bifurcations. The oscillations and bifurcations are characterized by uncovering a hidden Hamiltonian structure in the dynamics. By establishing a structural stability result, we further show that dynamics initialized near the OA manifold exhibit the same long-time behavior as those initialized exactly on the manifold. Motivated by our theory in dimension two, we conduct numerical experiments for analogous parameter regimes in higher-dimensional transformers. Our numerical experiments suggest that the long-time behaviors characterized in our theoretical results persist in higher dimensions.

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arXiv2026

Exploiting Structure with Anisotropic Consensus-Based Optimization

Anisotropic consensus-based optimization (CBO), a multi-agent metaheuristic derivative-free optimization method, which reliably finds global minima of nonsmooth and nonconvex objective functions while being amenable to rigorous theoretical analysis, automatically detects and exploits additively separable structures of high-dimensional objective functions. This enables the algorithm to mitigate the curse of dimensionality where the objective function decomposes additively into lower-dimensional components. In this paper, we show this property proving that the computational complexity of anisotropic CBO depends exponentially only on the intrinsic dimension $\mathbf{d}$ of the objective function, rather than the ambient dimension $D\gg\mathbf{d}$. Additionally, we demonstrate that the computational complexity depends only on tractability conditions of the lower-dimensional components rather than on the full energy landscape, allowing for a more refined description of the objective function and algorithmic complexity, as the objective function landscape is captured directly at the level of the individual components. Our results highlight the effectiveness of anisotropic CBO for additively separable objective functions provided sufficient alignment between the structure of the anisotropic noise and the separability structure of the objective. This motivates the design of an enhanced algorithm that learns during optimization how to effectively explore the loss landscape by aligning the noise with the structure of the objective function, which we leave for future research. Numerical experiments validate our theoretical results, accentuating the influence of the intrinsic dimensionality $\mathbf{d}$, the level of separability, and the complexity and non-convexity of the objective within the separable components on performance and computational complexity of the anisotropic CBO algorithm.

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arXiv2026

Convergence Analysis of Newton's Method for Neural Networks in the Overparameterized Limit

A convergence analysis is developed for the regularized Newton method for training neural networks (NNs) in the overparameterized limit. As the number of hidden units tends to infinity, the NN training dynamics converge in probability to the solution of a deterministic limit equation involving a ``Newton neural tangent kernel'' (NNTK). Explicit rates characterizing this convergence are provided and, in the infinite-width limit, we prove that the NN converges exponentially fast to the target data (i.e., a global minimizer with zero loss). We show that this convergence is uniform across the frequency spectrum, addressing the spectral bias inherent in gradient descent. The eigenvalues of the NTK for gradient descent accumulate at zero, leading to slow convergence for target data with high-frequency components. In contrast, the NNTK has uniformly lower bounded eigenvalues if the regularization parameter is selected appropriately, allowing Newton's method to converge more quickly for data with high-frequency components. Mathematical challenges that need to be addressed in our analysis include the implicit parameter update of the Newton method with a potentially indefinite Hessian matrix and the fact that the dimension of this linear system of equations tends to infinity as the NN width grows. This complicates deriving the training dynamics in the overparameterized limit as well as proving the convergence of the finite-width dynamics thereto. The analysis identifies a scaling formula for selecting the regularization parameter, which we show can vanish at a suitable rate as the number of hidden units becomes larger. We prove that, for sufficiently large numbers of hidden units, the regularized Hessian remains positive definite during training and the Newton updates for individual NN parameters converge to zero, showing that the model behaves as a linearization around the initialization.

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arXiv2026

Convergence of Consensus-Based Particle Methods for Nonconvex Bi-Level Optimization

In this paper, we study a consensus-based optimization method for nonconvex bi-level optimization, where the objective is to minimize an upper-level function over the set of global minimizers of a lower-level problem. The proposed approach is derivative-free, and constructs its consensus point via smooth quantile selection combined with a Gibbs-type Laplace approximation. We establish convergence guarantees for both the associated \textit{mean-field} dynamics and its \textit{finite-particle} approximation. In particular, under suitable assumptions on smooth quantile localization, error bounds, and stability, we show that the mean-field law reaches any arbitrary prescribed Wasserstein neighborhood of the target bi-level solution with an explicit exponential rate up to the hitting time. Numerical experiments on a two-dimensional constrained problem and neural network training further support the theoretical results.

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arXiv2026

Quantifying Concentration Phenomena of Mean-Field Transformers in the Low-Temperature Regime

Transformers with self-attention modules as their core components have become an integral architecture in modern large language and foundation models. In this paper, we study the evolution of tokens in deep encoder-only transformers at inference time which is described in the large-token limit by a mean-field continuity equation. Leveraging ideas from the convergence analysis of interacting multi-particle systems, with particles corresponding to tokens, we prove that the token distribution rapidly concentrates onto the push-forward of the initial distribution under a projection map induced by the key, query, and value matrices, and remains metastable for moderate times. Specifically, we show that the Wasserstein distance of the two distributions scales like $\sqrt{{\log(β+1)}/β}\exp(Ct)+\exp(-ct)$ in terms of the temperature parameter $β^{-1}\to 0$ and inference time $t\geq 0$. For the proof, we establish Lyapunov-type estimates for the zero-temperature equation, identify its limit as $t\to\infty$, and employ a stability estimate in Wasserstein space together with a quantitative Laplace principle to couple the two equations. Our result implies that for time scales of order $\logβ$ the token distribution concentrates at the identified limiting distribution. Numerical experiments confirm this and, beyond that, complement our theory by showing that for finite $β$ and large $t$ the dynamics enter a different terminal phase, dominated by the spectrum of the value matrix.

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arXiv2026

Gradient is All You Need? How Consensus-Based Optimization can be Interpreted as a Stochastic Relaxation of Gradient Descent

In this paper, we provide a novel analytical perspective on the theoretical understanding of gradient-based learning algorithms by interpreting consensus-based optimization (CBO), a recently proposed multi-particle derivative-free optimization method, as a stochastic relaxation of gradient descent. Remarkably, we observe that through communication of the particles, CBO exhibits a stochastic gradient descent (SGD)-like behavior despite solely relying on evaluations of the objective function. The fundamental value of such link between CBO and SGD lies in the fact that CBO is provably globally convergent to global minimizers for ample classes of nonsmooth and nonconvex objective functions. Hence, on the one side, we offer a novel explanation for the success of stochastic relaxations of gradient descent by furnishing useful and precise insights that explain how problem-tailored stochastic perturbations of gradient descent (like the ones induced by CBO) overcome energy barriers and reach deep levels of nonconvex functions. On the other side, and contrary to the conventional wisdom for which derivative-free methods ought to be inefficient or not to possess generalization abilities, our results unveil an intrinsic gradient descent nature of heuristics. Instructive numerical illustrations support the provided theoretical insights.

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arXiv2025

Strong Global Convergence of the Consensus-Based Optimization Algorithm

Consensus-based optimization (CBO) is a multi-agent metaheuristic derivative-free optimization algorithm that has proven to be capable of globally minimizing nonconvex nonsmooth functions across a diverse range of applications while being amenable to theoretical analysis. The method leverages an interplay between exploration of the energy landscape of the objective function through a system of interacting particles subject to stochasticity and exploitation of the particles' positions through the computation of a global consensus about the location of the minimizer based on the Laplace principle. In this paper, we prove strong mean square convergence of the practical numerical time-discrete CBO algorithm to the global minimizer for a rich class of objective functions. For CBO with both isotropic and anisotropic diffusion, our convergence result features conditions on the choice of the hyperparameters as well as explicit rates of convergence in the time discretization step size $Δt$ and the number of particles $N$. By interpreting the time-discrete algorithm at the continuous-time level through a system of stochastic differential equations (SDEs), our proof strategy combines traditional finite-time convergence theory for numerical methods applied to SDEs with careful considerations due to the fact that the CBO coefficients do not satisfy a global Lipschitz condition. To accomodate the latter, we adopt a recently proposed generalization of Sznitman's classical argument, which allows to discard an event of small probability, controllable through fine moment estimates for the particle systems.

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arXiv2025

Global Convergence of Adjoint-Optimized Neural PDEs

Many engineering and scientific fields have recently become interested in modeling terms in partial differential equations (PDEs) with neural networks, which requires solving the inverse problem of learning neural network terms from observed data in order to approximate missing or unresolved physics in the PDE model. The resulting neural-network PDE model, being a function of the neural network parameters, can be calibrated to the available ground truth data by optimizing over the PDE using gradient descent, where the gradient is evaluated in a computationally efficient manner by solving an adjoint PDE. These neural PDE models have emerged as an important research area in scientific machine learning. In this paper, we study the convergence of the adjoint gradient descent optimization method for training neural PDE models in the limit where both the number of hidden units and the training time tend to infinity. Specifically, for a general class of nonlinear parabolic PDEs with a neural network embedded in the source term, we prove convergence of the trained neural-network PDE solution to the target data (i.e., a global minimizer). The global convergence proof poses a unique mathematical challenge that is not encountered in finite-dimensional neural network convergence analyses due to (i) the neural network training dynamics involving a non-local neural network kernel operator in the infinite-width hidden layer limit where the kernel lacks a spectral gap for its eigenvalues and (ii) the nonlinearity of the limit PDE system, which leads to a non-convex optimization problem in the neural network function even in the infinite-width hidden layer limit (unlike in typical neural network training cases where the optimization problem becomes convex in the large neuron limit). The theoretical results are illustrated and empirically validated by numerical studies.

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arXiv2025

Defending Against Diverse Attacks in Federated Learning Through Consensus-Based Bi-Level Optimization

Adversarial attacks pose significant challenges in many machine learning applications, particularly in the setting of distributed training and federated learning, where malicious agents seek to corrupt the training process with the goal of jeopardizing and compromising the performance and reliability of the final models. In this paper, we address the problem of robust federated learning in the presence of such attacks by formulating the training task as a bi-level optimization problem. We conduct a theoretical analysis of the resilience of consensus-based bi-level optimization (CB$^2$O), an interacting multi-particle metaheuristic optimization method, in adversarial settings. Specifically, we provide a global convergence analysis of CB$^2$O in mean-field law in the presence of malicious agents, demonstrating the robustness of CB$^2$O against a diverse range of attacks. Thereby, we offer insights into how specific hyperparameter choices enable to mitigate adversarial effects. On the practical side, we extend CB$^2$O to the clustered federated learning setting by proposing FedCB$^2$O, a novel interacting multi-particle system, and design a practical algorithm that addresses the demands of real-world applications. Extensive experiments demonstrate the robustness of the FedCB$^2$O algorithm against label-flipping attacks in decentralized clustered federated learning scenarios, showcasing its effectiveness in practical contexts.

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arXiv2025

CB$^2$O: Consensus-Based Bi-Level Optimization

Bi-level optimization problems, where one wishes to find the global minimizer of an upper-level objective function over the globally optimal solution set of a lower-level objective, arise in a variety of scenarios throughout science and engineering, machine learning, and artificial intelligence. In this paper, we propose and investigate, analytically and experimentally, consensus-based bi-level optimization (CB$^2$O), a multi-particle metaheuristic derivative-free optimization method designed to solve bi-level optimization problems when both objectives may be nonconvex. Our method leverages within the computation of the consensus point a carefully designed particle selection principle implemented through a suitable choice of a quantile on the level of the lower-level objective, together with a Laplace principle-type approximation w.r.t. the upper-level objective function, to ensure that the bi-level optimization problem is solved in an intrinsic manner. We give an existence proof of solutions to a corresponding mean-field dynamics, for which we first establish the stability of our consensus point w.r.t. a combination of Wasserstein and $L^2$ perturbations, and consecutively resort to PDE considerations extending the classical Picard iteration to construct a solution. For such solution, we provide a global convergence analysis in mean-field law showing that the solution of the associated nonlinear nonlocal Fokker-Planck equation converges exponentially fast to the unique solution of the bi-level optimization problem provided suitable choices of the hyperparameters. The practicability and efficiency of our CB$^2$O algorithm is demonstrated through extensive numerical experiments in the settings of constrained global optimization, sparse representation learning, and robust (clustered) federated learning.

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arXiv2024

CBX: Python and Julia packages for consensus-based interacting particle methods

We introduce CBXPy and ConsensusBasedX.jl, Python and Julia implementations of consensus-based interacting particle systems (CBX), which generalise consensus-based optimization methods (CBO) for global, derivative-free optimisation. The raison d'être of our libraries is twofold: on the one hand, to offer high-performance implementations of CBX methods that the community can use directly, while on the other, providing a general interface that can accommodate and be extended to further variations of the CBX family. Python and Julia were selected as the leading high-level languages in terms of usage and performance, as well as their popularity among the scientific computing community. Both libraries have been developed with a common ethos, ensuring a similar API and core functionality, while leveraging the strengths of each language and writing idiomatic code.

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arXiv2024

Consensus-Based Optimization Methods Converge Globally

In this paper, we study consensus-based optimization (CBO), which is a multi-agent metaheuristic derivative-free optimization method that can globally minimize nonconvex nonsmooth functions and is amenable to theoretical analysis. Based on an experimentally supported intuition that, on average, CBO performs a gradient descent of the squared Euclidean distance to the global minimizer, we devise a novel technique for proving the convergence to the global minimizer in mean-field law for a rich class of objective functions. The result unveils internal mechanisms of CBO that are responsible for the success of the method. In particular, we prove that CBO performs a convexification of a large class of optimization problems as the number of optimizing agents goes to infinity. Furthermore, we improve prior analyses by requiring mild assumptions about the initialization of the method and by covering objectives that are merely locally Lipschitz continuous. As a core component of this analysis, we establish a quantitative nonasymptotic Laplace principle, which may be of independent interest. From the result of CBO convergence in mean-field law, it becomes apparent that the hardness of any global optimization problem is necessarily encoded in the rate of the mean-field approximation, for which we provide a novel probabilistic quantitative estimate. The combination of these results allows to obtain probabilistic global convergence guarantees of the numerical CBO method.

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arXiv2024

Convergence of Anisotropic Consensus-Based Optimization in Mean-Field Law

In this paper we study anisotropic consensus-based optimization (CBO), a multi-agent metaheuristic derivative-free optimization method capable of globally minimizing nonconvex and nonsmooth functions in high dimensions. CBO is based on stochastic swarm intelligence, and inspired by consensus dynamics and opinion formation. Compared to other metaheuristic algorithms like particle swarm optimization, CBO is of a simpler nature and therefore more amenable to theoretical analysis. By adapting a recently established proof technique, we show that anisotropic CBO converges globally with a dimension-independent rate for a rich class of objective functions under minimal assumptions on the initialization of the method. Moreover, the proof technique reveals that CBO performs a convexification of the optimization problem as the number of agents goes to infinity, thus providing an insight into the internal CBO mechanisms responsible for the success of the method. To motivate anisotropic CBO from a practical perspective, we further test the method on a complicated high-dimensional benchmark problem, which is well understood in the machine learning literature.

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arXiv2024

Consensus-Based Optimization with Truncated Noise

Consensus-based optimization (CBO) is a versatile multi-particle metaheuristic optimization method suitable for performing nonconvex and nonsmooth global optimizations in high dimensions. It has proven effective in various applications while at the same time being amenable to a theoretical convergence analysis. In this paper, we explore a variant of CBO, which incorporates truncated noise in order to enhance the well-behavedness of the statistics of the law of the dynamics. By introducing this additional truncation in the noise term of the CBO dynamics, we achieve that, in contrast to the original version, higher moments of the law of the particle system can be effectively bounded. As a result, our proposed variant exhibits enhanced convergence performance, allowing in particular for wider flexibility in choosing the noise parameter of the method as we confirm experimentally. By analyzing the time-evolution of the Wasserstein-$2$ distance between the empirical measure of the interacting particle system and the global minimizer of the objective function, we rigorously prove convergence in expectation of the proposed CBO variant requiring only minimal assumptions on the objective function and on the initialization. Numerical evidences demonstrate the benefit of truncating the noise in CBO.

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arXiv2023

Leveraging Memory Effects and Gradient Information in Consensus-Based Optimization: On Global Convergence in Mean-Field Law

In this paper we study consensus-based optimization (CBO), a versatile, flexible and customizable optimization method suitable for performing nonconvex and nonsmooth global optimizations in high dimensions. CBO is a multi-particle metaheuristic, which is effective in various applications and at the same time amenable to theoretical analysis thanks to its minimalistic design. The underlying dynamics, however, is flexible enough to incorporate different mechanisms widely used in evolutionary computation and machine learning, as we show by analyzing a variant of CBO which makes use of memory effects and gradient information. We rigorously prove that this dynamics converges to a global minimizer of the objective function in mean-field law for a vast class of functions under minimal assumptions on the initialization of the method. The proof in particular reveals how to leverage further, in some applications advantageous, forces in the dynamics without loosing provable global convergence. To demonstrate the benefit of the herein investigated memory effects and gradient information in certain applications, we present numerical evidence for the superiority of this CBO variant in applications such as machine learning and compressed sensing, which en passant widen the scope of applications of CBO.

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arXiv2023

Consensus-Based Optimization for Saddle Point Problems

In this paper, we propose consensus-based optimization for saddle point problems (CBO-SP), a novel multi-particle metaheuristic derivative-free optimization method capable of provably finding global Nash equilibria. Following the idea of swarm intelligence, the method employs a group of interacting particles, which perform a minimization over one variable and a maximization over the other. This paradigm permits a passage to the mean-field limit, which makes the method amenable to theoretical analysis and allows to obtain rigorous convergence guarantees under reasonable assumptions about the initialization and the objective function, which most notably include nonconvex-nonconcave objectives.

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arXiv2022

On the Global Convergence of Particle Swarm Optimization Methods

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.

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