Research

We study a continuous-time opinion model in which agents interact in pairs and each agent has a fixed anchor. At each interaction epoch, both opinions are updated according to a possibly nonlinear rule with random inputs. Under suitable stability and moment assumptions, we establish uniform-in-time propagation of chaos: as the population grows, the joint law of any fixed number of agents converges, uniformly over time, to the corresponding product of the law solving a nonlinear Boltzmann equation. We first obtain a graphical estimate in total variation on finite time intervals, assuming only that the interaction rule is measurable. We extend this approximation uniformly over time by showing that both the finite system and its nonlinear limit converge exponentially fast to their respective stationary laws. We also prove uniform-in-time convergence in Wasserstein distance of order two under a contraction condition in mean square. For this result, we assume either a bounded state space or a drift condition that keeps moments of some order greater than two uniformly bounded.

We examine two applications. For an anchored Friedkin-Johnsen model with random coefficients, we derive an affine stochastic recurrence for the stationary deviation of an opinion from its anchor. We use this recurrence to give conditions for sub-Gaussian or power-law tails and to characterize the exponent in the power-law case. We also study a model of biased assimilation, in which agents favor information that agrees with their current opinions. With the anchor and incoming evidence held at neutral values, we determine when an interaction moves an opinion farther from the neutral value, even if it starts close to it. Simulations illustrate the tail predictions and show how a single peak in the opinion distribution can split into two.

For stochastic gradient descent (SGD) with a constant stepsize \(\alpha\), the invariant law of the iterates, centered at a minimizer, describes the behavior of the algorithm over long time horizons. In the strongly convex case, this invariant law has the familiar \(\sqrt{\alpha}\) scaling and a Gaussian limit as \(\alpha \downarrow 0\). We show that this behavior changes fundamentally for convex objectives \(H\) with flat minima and (sub)quadratic tails.

More specifically, we study SGD with Markovian noise generated by a contractive driving chain. For every sufficiently small constant stepsize \(\alpha\), we prove existence, uniqueness, and geometric convergence to an augmented invariant law in a Wasserstein distance induced by an \(\alpha\)-dependent metric.

When the minimizer \(x_{\star}\) has local flatness exponent \(m \geq 2\), meaning that \(\nabla^2 H(x) \asymp \lVert x-x_{\star}\rVert^{m-2} I_d\) as \(x \to x_{\star}\), we obtain a contraction bound with factor \(1-c\alpha^{m-1}\), where \(c>0\) is a constant. This recovers the factor \(1-c\alpha\) in the quadratic case \(m=2\). We then analyze the small-stepsize scaling limit.

We show that the invariant law concentrates on the scale \(\alpha^{1/m}\) and that the rescaled iterates converge weakly to the stationary distribution of the stochastic differential equation \[\mathrm{d}Y_t=-h_0(Y_t)\,\mathrm{d}t+\Sigma^{1/2}\,\mathrm{d}B_t,\] where \(h_0\) is the limiting drift at the minimizer and \(\Sigma\) denotes the asymptotic covariance. This recovers the Gaussian limit when \(m=2\) and gives generally non-Gaussian stationary limits in the flat case \(m>2\). Finally, we give corresponding results for coordinate-separable objectives with unequal flatness exponents.

Derived uniform Gaussian local and integro-local asymptotics for the location of a point selected with probability proportional to its weight in a dynamic version of the Neyman contagious point process. The results cover one-dimensional and multivariate lattice, non-lattice, and mixed settings, extending classical local limit theorems from i.i.d. sums to independent, non-identically distributed summands generated by the model.