FeDXL: Provable Federated Learning for Deep X-Risk Optimization

In this paper, we tackle a novel federated learning (FL) problem for optimizing a family of X-risks, to which no existing FL algorithms are applicable. In particular, the objective has the form of $\mathbb E_{z\sim S_1} f(\mathbb E_{z'\sim S_2} \ell(w; z, z'))$, where two sets of data $S_1, S_2$ are distributed over multiple machines, $\ell(\cdot)$ is a pairwise loss that only depends on the prediction outputs of the input data pairs $(z, z')$, and $f(\cdot)$ is possibly a non-linear non-convex function. This problem has important applications in machine learning, e.g., AUROC maximization with a pairwise loss, and partial AUROC maximization with a compositional loss. The challenges for designing an FL algorithm lie in the non-decomposability of the objective over multiple machines and the interdependency between different machines. To address the challenges, we propose an active-passive decomposition framework that decouples the gradient's components with two types, namely active parts and passive parts, where the active parts depend on local data that are computed with the local model and the passive parts depend on other machines that are communicated/computed based on historical models and samples. Under this framework, we develop two provable FL algorithms (FeDXL) for handling linear and nonlinear $f$, respectively, based on federated averaging and merging. We develop a novel theoretical analysis to combat the latency of the passive parts and the interdependency between the local model parameters and the involved data for computing local gradient estimators. We establish both iteration and communication complexities and show that using the historical samples and models for computing the passive parts do not degrade the complexities. We conduct empirical studies of FeDXL for deep AUROC and partial AUROC maximization, and demonstrate their performance compared with several baselines

Distribution specificities of long-period comets' perihelia. Hypothesis of the large planetary body on the periphery of the Solar System

The present paper reviews selected aspects of the Guliyev's hypothesis about the massive celestial body at a distance of 250-400 AU from the Sun as well as the factor of comets transfer. It is shown, that the conjecture of the point around which cometary perihelia might be concentrated, is not consistent. On the issue of perihelia distribution, priority should be given to the assumption that there is a plane or planes around which the concentration takes place. A total of 24 comet groups were investigated. In almost all cases there are detected two types of planes or zones: the first one is very close to the ecliptic, another one is about perpendicular to it and has the parameters: ip = 86{\deg}, {\Omega}p = 271.7{\deg}. The existence of the first area appears to be related to the influence of giant planets. The Guliyev's hypothesis says that there is a massive perturber in the second zone, at a distance of 250-400 AU. It shows that number of aphelia and distant nodes of cometary orbits in this interval significantly exceeds the expected background. Analysis of the angular parameters of the comets, calculated relative to the second plane (reference point is the ascending node of a large circle) displays clear patterns: shortage of comets near i' = 180{\deg}, excess of them near B'= 0{\deg} (ecliptic latitude of perihelion) and shortage near B'=-90{\deg}. The analysis also shows irregularity of distant nodes, overpopulation of perihelion longitudes in the range 350{\deg}-20{\deg}. Plotted distributions of aphelia N(Q) and distant cometary nodes clearly indicate a perturbation of the natural course near 300 AU. On the basis of collected cometary data, we have estimated orbital elements of the hypothetical planetary body: a = 337 AU; e = 0.14; {\omega} = 57{\deg}; {\Omega} = 272.7{\deg}; i = 86{\deg}