Tight Bounds for Blind Search on the Integers

We analyze a simple random process in which a token is moved in the interval $A=\{0,...,n\$: Fix a probability distribution$\mu$over$\{1,...,n\$. Initially, the token is placed in a random position in$A$. In round$t$, a random value$d$is chosen according to$\mu$. If the token is in position$a\geq d$, then it is moved to position$a-d$. Otherwise it stays put. Let$T$be the number of rounds until the token reaches position 0. We show tight bounds for the expectation of$T$for the optimal distribution$\mu$. More precisely, we show that$\min_\mu\{E_\mu(T)\=\Theta((\log n)^2)$. For the proof, a novel potential function argument is introduced. The research is motivated by the problem of approximating the minimum of a continuous function over$[0,1]$ with a ``blind'' optimization strategy

Asymptotic Bias of Stochastic Gradient Search

The asymptotic behavior of the stochastic gradient algorithm with a biased gradient estimator is analyzed. Relying on arguments based on the dynamic system theory (chain-recurrence) and the differential geometry (Yomdin theorem and Lojasiewicz inequality), tight bounds on the asymptotic bias of the iterates generated by such an algorithm are derived. The obtained results hold under mild conditions and cover a broad class of high-dimensional nonlinear algorithms. Using these results, the asymptotic properties of the policy-gradient (reinforcement) learning and adaptive population Monte Carlo sampling are studied. Relying on the same results, the asymptotic behavior of the recursive maximum split-likelihood estimation in hidden Markov models is analyzed, too.Comment: arXiv admin note: text overlap with arXiv:0907.102

The Folklore of Sorting Algorithms

The objective of this paper is to review the folklore knowledge seen in research work devoted on synthesis, optimization, and effectiveness of various sorting algorithms. We will examine sorting algorithms in the folklore lines and try to discover the tradeoffs between folklore and theorems. Finally, the folklore knowledge on complexity values of the sorting algorithms will be considered, verified and subsequently converged in to theorems