Bandit Online Optimization Over the Permutahedron

The permutahedron is the convex polytope with vertex set consisting of the vectors $(\pi(1),\dots, \pi(n))$ for all permutations (bijections) $\pi$ over $\{1,\dots, n\}$. We study a bandit game in which, at each step $t$, an adversary chooses a hidden weight weight vector $s_t$, a player chooses a vertex $\pi_t$ of the permutahedron and suffers an observed loss of $\sum_{i=1}^n \pi(i) s_t(i)$. A previous algorithm CombBand of Cesa-Bianchi et al (2009) guarantees a regret of $O(n\sqrt{T \log n})$ for a time horizon of $T$. Unfortunately, CombBand requires at each step an $n$-by-$n$ matrix permanent approximation to within improved accuracy as $T$ grows, resulting in a total running time that is super linear in $T$, making it impractical for large time horizons. We provide an algorithm of regret $O(n^{3/2}\sqrt{T})$ with total time complexity $O(n^3T)$. The ideas are a combination of CombBand and a recent algorithm by Ailon (2013) for online optimization over the permutahedron in the full information setting. The technical core is a bound on the variance of the Plackett-Luce noisy sorting process's "pseudo loss". The bound is obtained by establishing positive semi-definiteness of a family of 3-by-3 matrices generated from rational functions of exponentials of 3 parameters

Unusual Low-Temperature Phase in VO2_2 Nanoparticles

We present a systematic investigation of the crystal and electronic structure and the magnetic properties above and below the metal-insulator transition of ball-milled VO$_2$ nanoparticles and VO$_2$ microparticles. For this research, we performed a Rietveld analysis of synchrotron radiation x-ray diffraction data, O $K$ x-ray absorption spectroscopy, V $L_3$ resonant inelastic x-ray scattering, and magnetic susceptibility measurements. This study reveals an unusual low-temperature phase that involves the formation of an elongated and less-tilted V-V pair, a narrowed energy gap, and an induced paramagnetic contribution from the nanoparticles. We show that the change in the crystal structure is consistent with the change in the electronic states around the Fermi level, which leads us to suggest that the Peierls mechanism contributes to the energy splitting of the $a_{1g}$ state. Furthermore, we find that the high-temperature rutile structure of the nanoparticles is almost identical to that of the microparticles.Comment: 7 pages, 8 figures, 2 table