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.

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.

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. The resulting neural-network PDE model, being a function of the neural network parameters, can be calibrated to available 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-network 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-network 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 (1) 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 (2) the nonlinearity of the limit PDE system, which leads to a non-convex optimization problem, even in the infinite-width hidden layer limit (unlike in typical neual 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.

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.

Gradient is All You Need?

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, offering a novel explanation for the success of stochastic relaxations of gradient descent. On the other side, contrary to the conventional wisdom for which zero-order methods ought to be inefficient or not to possess generalization abilities, our results unveil an intrinsic gradient descent nature of such heuristics. This viewpoint furthermore complements previous insights into the working principles of CBO, which describe the dynamics in the mean-field limit through a nonlinear nonlocal partial differential equation that allows to alleviate complexities of the nonconvex function landscape. Our proofs leverage a completely nonsmooth analysis, which combines a novel quantitative version of the Laplace principle (log-sum-exp trick) and the minimizing movement scheme (proximal iteration). In doing so, we furnish useful and precise insights that explain how stochastic perturbations of gradient descent overcome energy barriers and reach deep levels of nonconvex functions. Instructive numerical illustrations support the provided theoretical insights.

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, flexibel 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.