523 research outputs found
Online Learning: Beyond Regret
We study online learnability of a wide class of problems, extending the
results of (Rakhlin, Sridharan, Tewari, 2010) to general notions of performance
measure well beyond external regret. Our framework simultaneously captures such
well-known notions as internal and general Phi-regret, learning with
non-additive global cost functions, Blackwell's approachability, calibration of
forecasters, adaptive regret, and more. We show that learnability in all these
situations is due to control of the same three quantities: a martingale
convergence term, a term describing the ability to perform well if future is
known, and a generalization of sequential Rademacher complexity, studied in
(Rakhlin, Sridharan, Tewari, 2010). Since we directly study complexity of the
problem instead of focusing on efficient algorithms, we are able to improve and
extend many known results which have been previously derived via an algorithmic
construction
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Set Theory
This workshop included selected talks on pure set theory and its applications, simultaneously showing diversity and coherence of the subject
Advances in Zero-Sum Dynamic Games
International audienceThe survey presents recent results in the theory of two-person zero-sum repeated games and their connections with differential and continuous-time games. The emphasis is made on the following(1) A general model allows to deal simultaneously with stochastic and informational aspects.(2) All evaluations of the stage payoffs can be covered in the same framework (and not only the usual Cesà ro and Abel means).(3) The model in discrete time can be seen and analyzed as a discretization of a continuous time game. Moreover, tools and ideas from repeated games are very fruitful for continuous time games and vice versa.(4) Numerous important conjectures have been answered (some in the negative).(5) New tools and original models have been proposed. As a consequence, the field (discrete versus continuous time, stochastic versus incomplete information models) has a much more unified structure, and research is extremely active
Examples of covering properties of boundary points of space-times
The problem of classifying boundary points of space-time, for example
singularities, regular points and points at infinity, is an unexpectedly subtle
one. Due to the fact that whether or not two boundary points are identified or
even "nearby" is dependant on the way the space-time is embedded, difficulties
occur when singularities are thought of as an inherently local aspect of a
space-time, as an analogy with electromagnetism would imply. The completion of
a manifold with respect to a pseudo-Riemannian metric can be defined
intrinsically, [SS94]. This is done via an equivalence relation, formalising
which boundary sets cover other sets. This paper works through the
possibilities, providing examples to show that all covering relations not
immediately ruled out by the definitions are possible
Approachability in unknown games: Online learning meets multi-objective optimization
In the standard setting of approachability there are two players and a target
set. The players play repeatedly a known vector-valued game where the first
player wants to have the average vector-valued payoff converge to the target
set which the other player tries to exclude it from this set. We revisit this
setting in the spirit of online learning and do not assume that the first
player knows the game structure: she receives an arbitrary vector-valued reward
vector at every round. She wishes to approach the smallest ("best") possible
set given the observed average payoffs in hindsight. This extension of the
standard setting has implications even when the original target set is not
approachable and when it is not obvious which expansion of it should be
approached instead. We show that it is impossible, in general, to approach the
best target set in hindsight and propose achievable though ambitious
alternative goals. We further propose a concrete strategy to approach these
goals. Our method does not require projection onto a target set and amounts to
switching between scalar regret minimization algorithms that are performed in
episodes. Applications to global cost minimization and to approachability under
sample path constraints are considered
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