226 research outputs found
Predicting {0, 1}-Functions on Randomly Drawn Points
AbstractWe consider the problem of predicting {0, 1}-valued functions on Rn and smaller domains, based on their values on randomly drawn points. Our model is related to Valiant′s PAC learning model, but does not require the hypotheses used for prediction to be represented in any specified form. In our main result we show how to construct prediction strategies that are optimal to within a constant factor for any reasonable class F of target functions. This result is based on new combinatorial results about classes of functions of finite VC dimension. We also discuss more computationally efficient algorithms for predicting indicator functions of axis-parallel rectangles, more general intersection closed concept classes, and halfspaces in Rn. These are also optimal to within a constant factor. Finally, we compare the general performance of prediction strategies derived by our method to that of those derived from methods in PAC learning theory
Braess's Paradox in Wireless Networks: The Danger of Improved Technology
When comparing new wireless technologies, it is common to consider the effect
that they have on the capacity of the network (defined as the maximum number of
simultaneously satisfiable links). For example, it has been shown that giving
receivers the ability to do interference cancellation, or allowing transmitters
to use power control, never decreases the capacity and can in certain cases
increase it by , where is the
ratio of the longest link length to the smallest transmitter-receiver distance
and is the maximum transmission power. But there is no reason to
expect the optimal capacity to be realized in practice, particularly since
maximizing the capacity is known to be NP-hard. In reality, we would expect
links to behave as self-interested agents, and thus when introducing a new
technology it makes more sense to compare the values reached at game-theoretic
equilibria than the optimum values.
In this paper we initiate this line of work by comparing various notions of
equilibria (particularly Nash equilibria and no-regret behavior) when using a
supposedly "better" technology. We show a version of Braess's Paradox for all
of them: in certain networks, upgrading technology can actually make the
equilibria \emph{worse}, despite an increase in the capacity. We construct
instances where this decrease is a constant factor for power control,
interference cancellation, and improvements in the SINR threshold (),
and is when power control is combined with interference
cancellation. However, we show that these examples are basically tight: the
decrease is at most O(1) for power control, interference cancellation, and
improved , and is at most when power control is
combined with interference cancellation
Competing with stationary prediction strategies
In this paper we introduce the class of stationary prediction strategies and
construct a prediction algorithm that asymptotically performs as well as the
best continuous stationary strategy. We make mild compactness assumptions but
no stochastic assumptions about the environment. In particular, no assumption
of stationarity is made about the environment, and the stationarity of the
considered strategies only means that they do not depend explicitly on time; we
argue that it is natural to consider only stationary strategies even for highly
non-stationary environments.Comment: 20 page
Do Prices Coordinate Markets?
Walrasian equilibrium prices can be said to coordinate markets: They support
a welfare optimal allocation in which each buyer is buying bundle of goods that
is individually most preferred. However, this clean story has two caveats.
First, the prices alone are not sufficient to coordinate the market, and buyers
may need to select among their most preferred bundles in a coordinated way to
find a feasible allocation. Second, we don't in practice expect to encounter
exact equilibrium prices tailored to the market, but instead only approximate
prices, somehow encoding "distributional" information about the market. How
well do prices work to coordinate markets when tie-breaking is not coordinated,
and they encode only distributional information?
We answer this question. First, we provide a genericity condition such that
for buyers with Matroid Based Valuations, overdemand with respect to
equilibrium prices is at most 1, independent of the supply of goods, even when
tie-breaking is done in an uncoordinated fashion. Second, we provide
learning-theoretic results that show that such prices are robust to changing
the buyers in the market, so long as all buyers are sampled from the same
(unknown) distribution
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