Boosting with early stopping: Convergence and consistency

Boosting is one of the most significant advances in machine learning for classification and regression. In its original and computationally flexible version, boosting seeks to minimize empirically a loss function in a greedy fashion. The resulting estimator takes an additive function form and is built iteratively by applying a base estimator (or learner) to updated samples depending on the previous iterations. An unusual regularization technique, early stopping, is employed based on CV or a test set. This paper studies numerical convergence, consistency and statistical rates of convergence of boosting with early stopping, when it is carried out over the linear span of a family of basis functions. For general loss functions, we prove the convergence of boosting's greedy optimization to the infinimum of the loss function over the linear span. Using the numerical convergence result, we find early-stopping strategies under which boosting is shown to be consistent based on i.i.d. samples, and we obtain bounds on the rates of convergence for boosting estimators. Simulation studies are also presented to illustrate the relevance of our theoretical results for providing insights to practical aspects of boosting. As a side product, these results also reveal the importance of restricting the greedy search step-sizes, as known in practice through the work of Friedman and others. Moreover, our results lead to a rigorous proof that for a linearly separable problem, AdaBoost with \epsilon\to0 step-size becomes an L^1-margin maximizer when left to run to convergence.Comment: Published at http://dx.doi.org/10.1214/009053605000000255 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Parity violation in pre-inflationary bounce

The power suppression at largest scale in the CMB TT powerspectrum might imply the occurrence of a pre-inflationary bounce. We calculate the circularly polarized gravitational wave, leaded by the gravitational Chern-Simons term which universally appears in particle physics and string theory, in an inflationary background with the pre-inflationary bounce. The circularly polarized gravitational wave will induce TB and EB correlations at CMB last scatting surface. We find that although the BB powerspectrum at largest scale is also suppressed, the TB and EB powerspectra at corresponding scales may be enhanced.Comment: 15 pages, 4 figures,the final versio

A magnetically driven origin for the low luminosity GRB 170817A associated with GW170817

The gamma-ray burst GRB 170817A associated with GW170817 is subluminous and subenergetic compared with other typical short GRBs. It may be due to a relativistic jet viewed off-axis, or a structured jet, or cocoon emission. Giant flares from magnetars may possibly be ruled out. However, the luminosity and energetics of GRB 170817A is coincident with that of magnetar giant flares. After the coalescence of the binary neutron star, a hypermassive neutron star may be formed. The hypermassive neutron star may have magnetar-strength magnetic field. During the collapse of the hypermassive neutron star, the magnetic field energy will also be released. This giant-flare-like event may explain the the luminosity and energetics of GRB 170817A. Bursts with similar luminosity and energetics are expected in future neutron star-neutron star or neutron star-black hole mergers.Comment: 6 pages, 1 figure, accepted in Research in Astronomy and Astrophysic

Current Chinese bryological literature (4)

According to our collections of literature, about 400 scientific papers dealing with Chinese bryophytes have been published in China and abroad during 1990’s. Among these, more than 50 % were published in different scientific journals in China and often written in Chinese with English abstract, which are not well known and assessable for foreign bryologists. Therefore, in addition to previous Chinese literature I-III (Cao et al. 1990, Li et Zhang 1993, 1994), we present the fourth part of Chinese literature herewith. It is hoped that this up-dated list will provide useful information for all people who are interested in bryological research