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Information-based complexity and information-based optimization
This is an article that will appear in the Encyclopedia of Optimization (Kluwer, 2000). It concerns optimization in two senses. The first is that information-based complexity (IBC) is the study of the minimal computational resources to solve continuous mathematical problems. The second is that the computational complexity of optimization problems is one of the areas studied in IBC. We discuss IBC and information-based optimization in turn
A measure of statistical complexity based on predictive information
We introduce an information theoretic measure of statistical structure,
called 'binding information', for sets of random variables, and compare it with
several previously proposed measures including excess entropy, Bialek et al.'s
predictive information, and the multi-information. We derive some of the
properties of the binding information, particularly in relation to the
multi-information, and show that, for finite sets of binary random variables,
the processes which maximises binding information are the 'parity' processes.
Finally we discuss some of the implications this has for the use of the binding
information as a measure of complexity.Comment: 4 pages, 3 figure
Information-based complexity, feedback and dynamics in convex programming
We study the intrinsic limitations of sequential convex optimization through
the lens of feedback information theory. In the oracle model of optimization,
an algorithm queries an {\em oracle} for noisy information about the unknown
objective function, and the goal is to (approximately) minimize every function
in a given class using as few queries as possible. We show that, in order for a
function to be optimized, the algorithm must be able to accumulate enough
information about the objective. This, in turn, puts limits on the speed of
optimization under specific assumptions on the oracle and the type of feedback.
Our techniques are akin to the ones used in statistical literature to obtain
minimax lower bounds on the risks of estimation procedures; the notable
difference is that, unlike in the case of i.i.d. data, a sequential
optimization algorithm can gather observations in a {\em controlled} manner, so
that the amount of information at each step is allowed to change in time. In
particular, we show that optimization algorithms often obey the law of
diminishing returns: the signal-to-noise ratio drops as the optimization
algorithm approaches the optimum. To underscore the generality of the tools, we
use our approach to derive fundamental lower bounds for a certain active
learning problem. Overall, the present work connects the intuitive notions of
information in optimization, experimental design, estimation, and active
learning to the quantitative notion of Shannon information.Comment: final version; to appear in IEEE Transactions on Information Theor
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