We study correlation bounds under pairwise independent distributions for functions with no large Fourier coefficients. Functions in which all Fourier coefficients are bounded by$δ$are called$δ$-$uniform$. The search for such bounds is motivated by their potential applicability to hardness of approximation, derandomization, and additive combinatorics.

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Consensus optimization has received considerable attention in recent years. A number of decentralized algorithms have been proposed for convex consensus optimization. However, to the behaviors or consensus nonconvex optimization, our understanding is more limited. When we lose convexity, we cannot hope that our algorithms always return global solutions though they sometimes still do. Somewhat surprisingly, the decentralized consensus algorithms, DGD and Prox-DGD, retain most other properties that are known in the convex setting. In particular, when diminishing (or constant) step sizes are used, we can prove convergence to a (or a neighborhood of) consensus stationary solution under some regular assumptions. It is worth noting that Prox-DGD can handle nonconvex nonsmooth functions if their proximal operators can be computed. Such functions include SCAD, MCP, and lq quasinorms, q ∈ [0, 1). Similarly, Prox-DGD can take the constraint to a nonconvex set with an easy projection. To establish these properties, we have to introduce a completely different line of analysis, as well as modify existing proofs that were used in the convex setting.

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In this paper we derive the stochastic differentials of the conditional central moments of the nonlinear filtering problems, especially those of the polynomial filtering problem, and develop a novel suboptimal method by solving this evolution equation. The basic idea is to augment the state of the original nonlinear system by including the original states' conditional central moments such that the augmented states form a so-called bilinear system after truncating. During our derivation, it is clear to see that the stochastic differentials of the conditional central moments of the linear filtering problem (i.e., $f$, $g$ and $h$ are all at most degree one polynomials) form a closed system automatically without truncation. This gives one reason for the existence of optimal filtering for linear problems. On the contrary, the conditional central moments form an infinite dimensional system, in general. To reduce it to a closed-form, we let all the high enough central moments to be zero, as one did in the Carleman approach \cite{GMP}. Consequently, a novel suboptimal method is developed by dealing with the bilinear system. Numerical simulation is performed for the cubic sensor problem to illustrate the accuracy and numerical stability.