In this section we show how the refined upper bound on the regret of the EXP algorithm proved using the potential function approach (KL divergence) also gives us a better bound for the expert game setup with bandit feedback. Last lecture we showed how in the case of expert prediction with bandit feedback using the Exp3 algorithm, the regret is upper bounded by T 2/3n1/3 using a rough upper boun...

We are concerned with the perturbation of a multiple eigenvalue μ of the Hermitian matrix A = diag(μI,A22) when it undergoes an off-diagonal perturbation E whose columns have widely varying magnitudes. When some of E’s columns are much smaller than the others, some copies of μ are much less sensitive than any existing bound suggests. We explain this phenomenon by establishing individual perturb...

This paper examines the Stark effect, as a first order perturbation of manifestly covariant hydrogen-like bound states. These bound states are solutions to a relativistic Schrödinger equation with invariant evolution parameter, and represent mass eigenstates whose eigenvalues correspond to the well-known energy spectrum of the nonrelativistic theory. In analogy to the nonrelativistic case, the ...

We present a randomized algorithm for estimating the pth moment Fp of the frequency vector of a data stream in the general update (turnstile) model to within a multiplicative factor of 1±ǫ, for p > 2, with high constant confidence. For 0 < ǫ ≤ 1, the algorithm uses space O(nǫ+ nǫ log(n)) words. This improves over the current bound of O(nǫ log(n)) words by Andoni et. al. in [2]. Our space upper ...

We establish new conditions for obtaining uniform bounds on the moments of discrete-time stochastic processes. Our results require a weak negative drift criterion along with state-dependent restriction centered conditional process. They, in particular, generalize main result [22] which requires constant bound averaged one-step jumps The feature our make them suitable large class multiplicative-...

Let (X, d) be any finite metric space with n elements. We show that there are two pairs of distinct elements in X that determine two nearly equal distances in the sense that their ratio differs from 1 by at most 9 logn n2 . This bound (apart for the multiplicative constant) is best possible and we construct a metric space that attains this bound. We discus related questions and consider in part...

Recently the author introduced a semidefinite upper bound on the square of the stability number of a graph, the inverse theta number, which is proved here to be multiplicative with respect to the strong graph product, hence to be an upper bound for the square of the Shannon capacity of the graph. We also describe a heuristic algorithm for the stable set problem based on semidefinite programming...

Nonnegative matrix factorization (NMF) can be formulated as a minimization problem with bound constraints. Although bound-constrained optimization has been studied extensively in both theory and practice, so far no study has formally applied its techniques to NMF. In this letter, we propose two projected gradient methods for NMF, both of which exhibit strong optimization properties. We discuss ...

A many-valued modal logic is introduced that combines the usual Kripke frame semantics of the modal logic K with connectives interpreted locally at worlds by lattice and group operations over the real numbers. A labelled tableau system is provided and a coNEXPTIME upper bound obtained for checking validity in the logic. Focussing on the modal-multiplicative fragment, the labelled tableau system...