Abstract: We explore a new learning setting in which each randomly generated sample gives rise to an additional deterministic sample, called side information, that is also classified by the oracle. Hence a learning algorithm utilizing side information chooses from a smaller and more accurate set of concepts, and is expected to operate more efficiently. In general, side information learning utilizes dependent data and the training space differs from the evaluation space, as the the output of the algorithm will only need to classify a typical observation. We analyze a simple problem of learning open intervals and compare exact learning rates with and without side information. Many times side information yields exponentially better learning rates, but in some cases this improvement may vanish.
Abstract:
This paper studies bounded-velocity control of a Brownian
motion when discretionary stopping, or `leaving', is allowed.
The goal is to choose a control law and a stopping time in order to
minimize the expected sum of a running and a termination
cost, when both costs increase as a function of distance
from the origin. There are two versions of
this problem: the
When no discretionary stopping is allowed, the partially-observed case has been solved by Benes, Karatzas and Rishel (1991) and Karatzas and Ocone (1993). The full solution when stopping is allowed remains open. In this paper, we obtain lower bounds on the optimal stopping region using stopping regions of related, fully-observed problems.
Abstract: Suppose that the nonlinear filtering equations are solved using an incorrect initial condition. It is known that the relative entropy of the actual conditional distribution with respect to this incorrectly initialized filter is a positive supermartingale. In this paper, we study the filtering of diffusion signals. Using the Kushner-Stratonovich equations, we decompose the relative entropy supermartingale into decreasing and local martingale terms, and we derive an entropy bound on information and error measures of the difference between conditional distribution and incorrectly initialized filter.
Abstract: This paper proves exponential asymptotic stability of discrete-time filters for the estimation of solutions to stochastic difference equations, when the observation noise is bounded. No assumption is made on the ergodicity of the signal. The proof uses the Hilbert projective metric, introduced into filter stability analysis by Atar and Zeitouni. It is shown that when the signal noise is sufficiently regular, boundedness of the observation noise implies that the filter update operation is, on average, a strict contraction with respect to the Hilbert metric. Asymptotic stability then follows.
Abstract: We show that a cadlag, local martingale has conditionally independent increments and symmetric jumps if and only if its law is invariant under integral transformations which preserve quadratic variation.