A mean-square bound for the lattice discrepancy of bodies of rotation with flat points on the boundary

Let B denote a three-dimensional body of rotation, with respect to one coordinate axis, whose boundary is sufficiently smooth and of bounded nonzero Gaussian curvature throughout, except for the two boundary points on the axis of rotation, where the curvature may vanish. For a large real variable t, we are interested in the number A(t) of integer points in the linearly dilated body tB, in particular in the lattice discrepancy P(t) = A(t) - volume(tB). We are able to evaluate the contribution of the boundary points of curvature zero to P(t), with a remainder that is fairly small in mean-square.Comment: 16 page

Memory distribution in complex fitness landscapes

In a co-evolutionary context, the survive probability of individual elements of a system depends on their relation with their neighbors. The natural selection process depends on the whole population, which is determined by local events between individuals. Particular characteristics assigned to each individual, as larger memory, usually improve the individual fitness, but an agent possess also endogenous characteristics that induce to re-evaluate her fitness landscape and choose the best-suited kind of interaction, inducing a non absolute value of the outcomes of the interaction. In this work, a novel model with agents combining memory and rational choice is introduced, where individual choices in a complex fitness landscape induce changes in the distribution of the number of agents as a function of the time. In particular, the tail of this distribution is fat compared with distributions for agents interacting only with memory.Comment: 6 pages, 3 figures, submited to Physica

Hilbert C*-modules and amenable actions

We study actions of discrete groups on Hilbert $C^*$-modules induced from topological actions on compact Hausdorff spaces. We show non-amenability of actions of non-amenable and non-a-T-menable groups, provided there exists a quasi-invariant probability measure which is sufficiently close to being invariant.Comment: Final version, to appear in Studia Mathematic

Query Complexity of Derivative-Free Optimization

This paper provides lower bounds on the convergence rate of Derivative Free Optimization (DFO) with noisy function evaluations, exposing a fundamental and unavoidable gap between the performance of algorithms with access to gradients and those with access to only function evaluations. However, there are situations in which DFO is unavoidable, and for such situations we propose a new DFO algorithm that is proved to be near optimal for the class of strongly convex objective functions. A distinctive feature of the algorithm is that it uses only Boolean-valued function comparisons, rather than function evaluations. This makes the algorithm useful in an even wider range of applications, such as optimization based on paired comparisons from human subjects, for example. We also show that regardless of whether DFO is based on noisy function evaluations or Boolean-valued function comparisons, the convergence rate is the same

Active Learning for Undirected Graphical Model Selection

This paper studies graphical model selection, i.e., the problem of estimating a graph of statistical relationships among a collection of random variables. Conventional graphical model selection algorithms are passive, i.e., they require all the measurements to have been collected before processing begins. We propose an active learning algorithm that uses junction tree representations to adapt future measurements based on the information gathered from prior measurements. We prove that, under certain conditions, our active learning algorithm requires fewer scalar measurements than any passive algorithm to reliably estimate a graph. A range of numerical results validate our theory and demonstrates the benefits of active learning.Comment: AISTATS 201

Neighboring suboptimal control for vehicle guidance

The neighboring optimal feedback control law is developed for systems with a piecewise linear control for the case where the optimal control is obtained by nonlinear programming techniques. To develop the control perturbation for a given deviation from the nominal path, the second variation is minimized subject to the constraint that the final conditions be satisfied (neighboring suboptimal control). This process leads to a feedback relationship between the control perturbation and the measured deviation from the nominal state. Neighboring suboptimal control is applied to the lunar launch problem. Two approaches, single optimization and multiple optimization for calculating the gains are used, and the gains are tested in a guidance simulation with a mismatch in the acceleration of gravity. Both approaches give acceptable results, but multiple optimization keeps the perturbed path closer to the nominal path