Xuwei Yang

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.

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.

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.

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.

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).

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

Most industry-standard generative AIs and feature encoders are proprietary, offering only black-box access: their outputs are observable, but their internal parameters and architectures remain hidden from the end-user. This black-box access is especially limiting when constructing mixture-of-expert type ensemble models since the user cannot optimize each proprietary AI's internal parameters. Our problem naturally lends itself to a non-competitive game-theoretic lens where each proprietary AI (agent) is inherently competing against the other AI agents, with this competition arising naturally due to their obliviousness of the AI's to their internal structure. In contrast, the user acts as a central planner trying to synchronize the ensemble of competing AIs. We show the existence of the unique Nash equilibrium in the online setting, which we even compute in closed-form by eliciting a feedback mechanism between any given time series and the sequence generated by each (proprietary) AI agent. Our solution is implemented as a decentralized, federated-learning algorithm in which each agent optimizes their structure locally on their machine without ever releasing any internal structure to the others. We obtain refined expressions for pre-trained models such as transformers, random feature models, and echo-state networks. Our ``proprietary federated learning'' algorithm is implemented on a range of real-world and synthetic time-series benchmarks. It achieves orders-of-magnitude improvements in predictive accuracy over natural benchmarks, of which there are surprisingly few due to this natural problem still being largely unexplored.