We study minimal attention-only transformers under all-token corruption and show they admit a two-stage empirical Bayes interpretation. A single attention step computes a kernel-weighted posterior mean with respect to the empirical distribution defined by the context. Depth refines this distribution through particle dynamics (Stage 1), while a long-range skip-connection carries the noisy input as a query for posterior inference (Stage 2), revealing distinct statistical roles for depth and attention residuals. The framework isolates a minimal setting in which the context itself induces a depth-dependent energy landscape governing in-context inference. We show that effective denoising can emerge without an explicit noise schedule: a fixed kernel bandwidth and finite integration horizon suffice, yielding a principled depth-noise relationship. We further establish a posterior-mean recovery guarantee for a class of well-behaved priors, where the empirical estimator converges to the Bayes-optimal predictor under asymptotic conditions. Connecting these dynamics to reverse-diffusion limits, our results provide a statistical interpretation of attention as in-context inference via sample-based posterior estimation, without explicit density modeling.

We study recursive maximum likelihood estimation for stochastic interacting particle systems based on continuous observation of a single particle. In this regime, consistent estimation of the finite-particle log-likelihood is not possible, even in the limit as the number of particles $N\rightarrow\infty$ and the time horizon $t\rightarrow\infty$. We thus seek to optimise the stationary log-likelihood of the limiting mean-field system. We achieve this via a form of stochastic gradient estimate in continuous time, with stochastic gradient estimates computed using the continuous trajectory of the single observed particle, alongside a virtual interacting particle system and a virtual tangent interacting particle system, which are integrated with the online parameter estimate. For fixed numbers of real and virtual particles, we show that the resulting algorithms drive the gradient of a finite-particle surrogate objective to zero as $t\to\infty$. We then prove that, in the iterated limit $t\to\infty$ followed by $N,M\to\infty$, these surrogate gradients converge uniformly to the gradient of the stationary log-likelihood of the limiting mean-field system, yielding convergence to its stationary points. We illustrate the method on several numerical examples, including a model with quadratic confinement and interaction potentials, a model of interacting FitzHugh--Nagumo neurons, and a stochastic Kuramoto model.

Learning the potentials of interacting particle systems is a fundamental task across various scientific disciplines. A major challenge is that unlabeled data collected at discrete time points lack trajectory information due to limitations in data collection methods or privacy constraints. We address this challenge by introducing a trajectory-free self-test loss function that leverages the weak-form stochastic evolution equation of the empirical distribution. The loss function is quadratic in potentials, supporting parametric and nonparametric regression algorithms for robust estimation that scale to large, high-dimensional systems with big data. Systematic numerical tests show that our method outperforms baseline methods that regress on trajectories recovered via label matching, tolerating large observation time steps. We establish the convergence of parametric estimators as the sample size increases, providing a theoretical foundation for the proposed approach.

Transformers owe much of their empirical success in natural language processing to the self-attention blocks. Recent perspectives interpret attention blocks as interacting particle systems, whose mean-field limits correspond to gradient flows of interaction energy functionals on probability density spaces equipped with Wasserstein-$2$-type metrics. We extend this viewpoint by introducing accelerated attention blocks derived from inertial Nesterov-type dynamics on density spaces. In our proposed architecture, tokens carry both spatial (feature) and velocity variables. The time discretization and the approximation of accelerated density dynamics yield Hamiltonian momentum attention blocks, which constitute the proposed accelerated attention architectures. In particular, for linear self-attention, we show that the attention blocks approximate a Stein variational gradient flow, using a bilinear kernel, of a potential energy. In this setting, we prove that elliptically contoured probability distributions are preserved by the accelerated attention blocks. We present implementable particle-based algorithms and demonstrate that the proposed accelerated attention blocks converge faster than the classical attention blocks while preserving the number of oracle calls.

Neural Information Processing Systems http://nips.cc/

Providing a very low critical probabilitypc means that certification occurs when the simulation ends after alarge number of iterationsm. On the other hand, the projection of X onto any other direction orthogonal tog remains normal distributed. For each couple of parameters(N,T)we make1000 runs and count the number of false positive(i.e. the number of times the algorithm wrongfully asserted thatp < pc). Combining the latter proposal withT we obtain again a proposal reversible w.r.t. Step4: Conclusionbyinduction Let l0 be any critical level such thatπ0(h(X) > l0) > 0. We consider the following induction hypothesisatiterationk: Hk On the event, Lk l0, The probability that the two particle systems are equal tends exponentiallyfastto1whent + .

In particular, we study gradient descent, proximal gradient descent, coordinate gradient descent, proximal coordinate gradient, and Newton's methods as well as their Nesterov's accelerated variants in a unified framework motivated by a