Sampling from the {G}ibbs Distribution in Congestion Games

Logit dynamics is a form of randomized game dynamics where players have a bias towards strategic deviations that give a higher improvement in cost. It is used extensively in practice. In congestion (or potential) games, the dynamics converges to the so-called Gibbs distribution over the set of all strategy profiles, when interpreted as a Markov chain. In general, logit dynamics might converge slowly to the Gibbs distribution, but beyond that, not much is known about their algorithmic aspects, nor that of the Gibbs distribution. In this work, we are interested in the following two questions for congestion games: i) Is there an efficient algorithm for sampling from the Gibbs distribution? ii) If yes, do there also exist natural randomized dynamics that converges quickly to the Gibbs distribution? We first study these questions in extension parallel congestion games, a well-studied special case of symmetric network congestion games. As our main result, we show that there is a simple variation on the logit dynamics (in which we in addition are allowed to randomly interchange the strategies of two players) that converges quickly to the Gibbs distribution in such games. This answers both questions above affirmatively. We also address the first question for the class of so-called capacitated $k$-uniform congestion games. To prove our results, we rely on the recent breakthrough work of Anari, Liu, Oveis-Gharan and Vinzant (2019) concerning the approximate sampling of the base of a matroid according to strongly log-concave probability distribution

Approximate Sampling and Counting of Graphs with Near-Regular Degree Intervals

The approximate uniform sampling of graphs with a given degree sequence is a well-known, extensively studied problem in theoretical computer science and has significant applications, e.g., in the analysis of social networks. In this work we study an extension of the problem, where degree intervals are specified rather than a single degree sequence. We are interested in sampling and counting graphs whose degree sequences satisfy the degree interval constraints. A natural scenario where this problem arises is in hypothesis testing on social networks that are only partially observed. In this work, we provide the first fully polynomial almost uniform sampler (FPAUS) as well as the first fully polynomial randomized approximation scheme (FPRAS) for sampling and counting, respectively, graphs with near-regular degree intervals, in which every node $i$ has a degree from an interval not too far away from a given $d \in \N$. In order to design our FPAUS, we rely on various state-of-the-art tools from Markov chain theory and combinatorics. In particular, we provide the first non-trivial algorithmic application of a breakthrough result of Liebenau and Wormald (2017) regarding an asymptotic formula for the number of graphs with a given near-regular degree sequence. Furthermore, we also make use of the recent breakthrough of Anari et al. (2019) on sampling a base of a matroid under a strongly log-concave probability distribution. As a more direct approach, we also study a natural Markov chain recently introduced by Rechner, Strowick and M\"uller-Hannemann (2018), based on three simple local operations: Switches, hinge flips, and additions/deletions of a single edge. We obtain the first theoretical results for this Markov chain by showing it is rapidly mixing for the case of near-regular degree intervals of size at most one

Primal and Dual Combinatorial Dimensions

We give tight bounds on the relation between the primal and dual of various combinatorial dimensions, such as the pseudo-dimension and fat-shattering dimension, for multi-valued function classes. These dimensional notions play an important role in the area of learning theory. We first review some (folklore) results that bound the dual dimension of a function class in terms of its primal, and after that give (almost) matching lower bounds. In particular, we give an appropriate generalization to multi-valued function classes of a well-known bound due to Assouad (1983), that relates the primal and dual VC-dimension of a binary function class

Secretary and Online Matching Problems with Machine Learned Advice

The classical analysis of online algorithms, due to its worst-case nature, can be quite pessimistic when the input instance at hand is far from worst-case. Often this is not an issue with machine learning approaches, which shine in exploiting patterns in past inputs in order to predict the future. However, such predictions, although usually accurate, can be arbitrarily poor. Inspired by a recent line of work, we augment three well-known online settings with machine learned predictions about the future, and develop algorithms that take them into account. In particular, we study the following online selection problems: (i) the classical secretary problem, (ii) online bipartite matching and (iii) the graphic matroid secretary problem. Our algorithms still come with a worst-case performance guarantee in the case that predictions are subpar while obtaining an improved competitive ratio (over the best-known classical online algorithm for each problem) when the predictions are sufficiently accurate. For each algorithm, we establish a trade-off between the competitive ratios obtained in the two respective cases

Tubular carcinoma and grade 1 (well-differentiated) invasive ductal carcinoma: Comparison of flat epithelial atypia and other intra-epithelial lesions

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/72926/1/j.1440-1827.2008.02280.x.pd

Sampling Hypergraphs with Given Degrees

There is a well-known connection between hypergraphs and bipartite graphs, obtained by treating the incidence matrix of the hypergraph as the biadjacency matrix of a bipartite graph. We use this connection to describe and analyse a rejection sampling algorithm for sampling simple uniform hypergraphs with a given degree sequence. Our algorithm uses, as a black box, an algorithm $\mathcal{A}$ for sampling bipartite graphs with given degrees, uniformly or nearly uniformly, in (expected) polynomial time. The expected runtime of the hypergraph sampling algorithm depends on the (expected) runtime of the bipartite graph sampling algorithm $\mathcal{A}$, and the probability that a uniformly random bipartite graph with given degrees corresponds to a simple hypergraph. We give some conditions on the hypergraph degree sequence which guarantee that this probability is bounded below by a constant

Observational constraints on the distribution and temperature dependence of H_2O_2 on the surface of Europa

We use Keck NIRSPEC to investigate the geographic distribution of hydrogen peroxide, a potentially biologically important oxidant, on the surface of Europa. Contrary to expectation, we see the highest abundances at low latitudes, potentially correlated with geologically young chaos terrain. We also use NASA IRTF SpeX spectra of Europa before and after eclipse to investigate the extent to which temperature controls equilibrium hydrogen peroxide concentrations on the surface. During eclipse, Europa's surface temperature falls 10-20 K. If temperature were a significant control on peroxide concentrations, then the hydrogen peroxide band strengths should be different pre- and post-eclipse. Ultimately, these investigations will help further our understanding of the surface, and perhaps subsurface, composition of Europa

Biocompatibility and degradation of the open-pored magnesium scaffolds LAE442 and La2

Porous magnesium implants are of particular interest for application as resorbable bone substitutes, due to their mechanical strength and a Young's modulus similar to bone. The objective of the present study was to compare the biocompatibility, bone and tissue ingrowth, and the degradation behaviour of scaffolds made from the magnesium alloys LAE442 (n= 40) and Mg-La2 (n= 40)in vivo. For this purpose, cylindrical magnesium scaffolds (diameter 4 mm, length 5 mm) with defined, interconnecting pores were produced by investment casting and coated with MgF2. The scaffolds were inserted into the cancellous part of the greater trochanter ossis femoris of rabbits. After implantation periods of 6, 12, 24 and 36 weeks, the bone-scaffold compounds were evaluated usingex vivo µCT80 images, histological examinations and energy dispersive x-ray spectroscopy analysis. The La2 scaffolds showed inhomogeneous and rapid degradation, with inferior osseointegration as compared to LAE442. For the early observation times, no bone and tissue could be observed in the pores of La2. Furthermore, the excessive amount of foreign body cells and fibrous capsule formation indicates insufficient biocompatibility of the La2 scaffolds. In contrast, the LAE442 scaffolds showed slow degradation and better osseointegration. Good vascularization, a moderate cellular response, bone and osteoid-like bone matrix at all implantation periods were observed in the pores of LAE442. In summary, porous LAE442 showed promise as a degradable scaffold for bone defect repair, based on its degradation behaviour and biocompatibility. However, further studies are needed to show it would have the necessary mechanical properties required over time for weight-bearing bone defects