39,173 research outputs found
Asymptotically optimal sequential anomaly identification with ordering sampling rules
The problem of sequential anomaly detection and identification is considered
in the presence of a sampling constraint. Specifically, multiple data streams
are generated by distinct sources and the goal is to quickly identify those
that exhibit ``anomalous'' behavior, when it is not possible to sample every
source at each time instant. Thus, in addition to a stopping rule, which
determines when to stop sampling, and a decision rule, which indicates which
sources to identify as anomalous upon stopping, one needs to specify a sampling
rule that determines which sources to sample at each time instant. The focus of
this work is on ordering sampling rules, which sample the data sources, among
those currently estimated as anomalous (resp. non-anomalous), for which the
corresponding local test statistics have the smallest (resp. largest) values.
It is shown that with an appropriate design, which is specified explicitly, an
ordering sampling rule leads to the optimal expected time for stopping, among
all policies that satisfy the same sampling and error constraints, to a
first-order asymptotic approximation as the false positive and false negative
error rates under control both go to zero. This is the first asymptotic
optimality result for ordering sampling rules when multiple sources can be
sampled per time instant. Moreover, this is established under a general setup
where the number of anomalies is not required to be a priori known. A novel
proof technique is introduced, which unifies different versions of the problem
regarding the homogeneity of the sources and prior information on the number of
anomalies
What Do Our Choices Say About Our Preferences?
Taking online decisions is a part of everyday life. Think of buying a house,
parking a car or taking part in an auction. We often take those decisions
publicly, which may breach our privacy - a party observing our choices may
learn a lot about our preferences. In this paper we investigate the online
stopping algorithms from the privacy preserving perspective, using a
mathematically rigorous differential privacy notion.
In differentially private algorithms there is usually an issue of balancing
the privacy and utility. In this regime, in most cases, having both optimality
and high level of privacy at the same time is impossible. We propose a natural
mechanism to achieve a controllable trade-off, quantified by a parameter,
between the accuracy of the online algorithm and its privacy. Depending on the
parameter, our mechanism can be optimal with weaker differential privacy or
suboptimal, yet more privacy-preserving. We conduct a detailed accuracy and
privacy analysis of our mechanism applied to the optimal algorithm for the
classical secretary problem. Thereby the classical notions from two distinct
areas - optimal stopping and differential privacy - meet for the first time.Comment: 22 pages, 6 figure
Optimal Composition Ordering Problems for Piecewise Linear Functions
In this paper, we introduce maximum composition ordering problems. The input
is real functions and a constant
. We consider two settings: total and partial compositions. The
maximum total composition ordering problem is to compute a permutation
which maximizes , where .
The maximum partial composition ordering problem is to compute a permutation
and a nonnegative integer which maximize
.
We propose time algorithms for the maximum total and partial
composition ordering problems for monotone linear functions , which
generalize linear deterioration and shortening models for the time-dependent
scheduling problem. We also show that the maximum partial composition ordering
problem can be solved in polynomial time if is of form
for some constants , and . We
finally prove that there exists no constant-factor approximation algorithm for
the problems, even if 's are monotone, piecewise linear functions with at
most two pieces, unless P=NP.Comment: 19 pages, 4 figure
- β¦