We show that social learning is not useful in a model of team binary decision making by voting, where each vote carries equal weight. Specifically, we consider Bayesian binary hypothesis testing where agents have any conditionally-independent observation distribution and their local decisions are fused by any L-out-of-N fusion rule. The agents make local decisions sequentially, with each allowed to use its own private signal and all precedent local decisions. Though social learning generally occurs in that precedent local decisions affect an agent's belief, optimal team performance is obtained when all precedent local decisions are ignored. Thus, social learning is futile, and secret ballots are optimal. This contrasts with typical studies of social learning because we include a fusion center rather than concentrating on the performance of the latest-acting agents.

This paper studies the zero-noise limit of high-dimensional small-noise diffusion processes governed by the stochastic differential equation (SDE): \[ dX_{t}^{\varepsilon }=b(X_{t}^{\varepsilon })\,dt+\varepsilon \,dW_{t}, \quad X_{0}^{\varepsilon }=0, \quad \varepsilon >0, \] where drift $b$ is measurable and bounded. The associated ordinary differential equation (ODE) $\dot{x}_{t}=b(x_{t})$ may have multiple Filippov solutions due to lack of Lipschitz continuity, while non-degenerate additive noise ensures unique strong solutions for each $\varepsilon >0$. Integrating the Stroock-Varadhan support theorem, comparison theorem for diffusion processes, law of the iterated logarithm (LIL) for Brownian motion, and Hausdorff dimension from geometric measure theory, we analyze the weak limit distribution $μ^{0}=\lim_{\varepsilon \rightarrow 0}\mathcal{L}(X_{t}^{\varepsilon })$. We find instantaneous escape Filippov solutions dominate the zero-noise limit, with the support of $μ^{0}$ being the closure of points reached by these solutions at fixed $t$ (delayed solutions are geometrically negligible). The comparison theorem verifies uniform weak convergence under small drift perturbations; LIL quantifies $X_{t}^{\varepsilon }$ fluctuations as $\varepsilon \rightarrow 0$; Hausdorff dimension analysis shows the support has dimension strictly less than ambient space dimension $d$, making $μ^{0}$ singular with respect to the Lebesgue measure. The compact support set's structure depends only on drift dynamics and instantaneous escape solutions, not Brownian motion or $d$. Our work unifies probabilistic limit theory, geometric measure theory, ODE non-uniqueness and differential inclusion theory, providing a comprehensive framework for high-dimensional non-unique systems' zero-noise limit and new insights into singular limit distributions in stochastic analysis.

Gaussian and quadratic approximations of message passing algorithms on graphs have attracted considerable recent attention due to their computational simplicity, analytic tractability, and wide applicability in optimization and statistical inference problems. This paper presents a systematic framework for incorporating such approximate message passing (AMP) methods in general graphical models. The key concept is a partition of dependencies of a general graphical model into strong and weak edges, with the weak edges representing interactions through aggregates of small, linearizable couplings of variables. AMP approximations based on the Central Limit Theorem can be readily applied to aggregates of many weak edges and integrated with standard message passing updates on the strong edges. The resulting algorithm, which we call hybrid generalized approximate message passing (HyGAMP), can yield significantly simpler implementations of sum-product and max-sum loopy belief propagation. By varying the partition of strong and weak edges, a performance--complexity trade-off can be achieved. Group sparsity and multinomial logistic regression problems are studied as examples of the proposed methodology.