18,270 research outputs found
On the decomposition of Generalized Additive Independence models
The GAI (Generalized Additive Independence) model proposed by Fishburn is a
generalization of the additive utility model, which need not satisfy mutual
preferential independence. Its great generality makes however its application
and study difficult. We consider a significant subclass of GAI models, namely
the discrete 2-additive GAI models, and provide for this class a decomposition
into nonnegative monotone terms. This decomposition allows a reduction from
exponential to quadratic complexity in any optimization problem involving
discrete 2-additive models, making them usable in practice
Folner tilings for actions of amenable groups
We show that every probability-measure-preserving action of a countable
amenable group G can be tiled, modulo a null set, using finitely many finite
subsets of G ("shapes") with prescribed approximate invariance so that the
collection of tiling centers for each shape is Borel. This is a dynamical
version of the Downarowicz--Huczek--Zhang tiling theorem for countable amenable
groups and strengthens the Ornstein--Weiss Rokhlin lemma. As an application we
prove that, for every countably infinite amenable group G, the crossed product
of a generic free minimal action of G on the Cantor set is Z-stable.Comment: Minor revisions. Final versio
Online Convex Optimization for Sequential Decision Processes and Extensive-Form Games
Regret minimization is a powerful tool for solving large-scale extensive-form
games. State-of-the-art methods rely on minimizing regret locally at each
decision point. In this work we derive a new framework for regret minimization
on sequential decision problems and extensive-form games with general compact
convex sets at each decision point and general convex losses, as opposed to
prior work which has been for simplex decision points and linear losses. We
call our framework laminar regret decomposition. It generalizes the CFR
algorithm to this more general setting. Furthermore, our framework enables a
new proof of CFR even in the known setting, which is derived from a perspective
of decomposing polytope regret, thereby leading to an arguably simpler
interpretation of the algorithm. Our generalization to convex compact sets and
convex losses allows us to develop new algorithms for several problems:
regularized sequential decision making, regularized Nash equilibria in
extensive-form games, and computing approximate extensive-form perfect
equilibria. Our generalization also leads to the first regret-minimization
algorithm for computing reduced-normal-form quantal response equilibria based
on minimizing local regrets. Experiments show that our framework leads to
algorithms that scale at a rate comparable to the fastest variants of
counterfactual regret minimization for computing Nash equilibrium, and
therefore our approach leads to the first algorithm for computing quantal
response equilibria in extremely large games. Finally we show that our
framework enables a new kind of scalable opponent exploitation approach
Decomposition Strategies for Constructive Preference Elicitation
We tackle the problem of constructive preference elicitation, that is the
problem of learning user preferences over very large decision problems,
involving a combinatorial space of possible outcomes. In this setting, the
suggested configuration is synthesized on-the-fly by solving a constrained
optimization problem, while the preferences are learned itera tively by
interacting with the user. Previous work has shown that Coactive Learning is a
suitable method for learning user preferences in constructive scenarios. In
Coactive Learning the user provides feedback to the algorithm in the form of an
improvement to a suggested configuration. When the problem involves many
decision variables and constraints, this type of interaction poses a
significant cognitive burden on the user. We propose a decomposition technique
for large preference-based decision problems relying exclusively on inference
and feedback over partial configurations. This has the clear advantage of
drastically reducing the user cognitive load. Additionally, part-wise inference
can be (up to exponentially) less computationally demanding than inference over
full configurations. We discuss the theoretical implications of working with
parts and present promising empirical results on one synthetic and two
realistic constructive problems.Comment: Accepted at the Thirty-Second AAAI Conference on Artificial
Intelligence (AAAI-18
Laboratory Games and Quantum Behaviour: The Normal Form with a Separable State Space
The subjective expected utility (SEU) criterion is formulated for a particular four-person âlaboratory gameâ that a Bayesian rational decision maker plays with Nature, Chance, and an Experimenter who influences what quantum behaviour is observable by choosing an orthonormal basis in a separable complex Hilbert space of latent variables. Nature chooses a state in this basis, along with an observed data series governing Chance's random choice of consequence. When Gleason's theorem holds, imposing quantum equivalence implies that the expected likelihood of any data series w.r.t. prior beliefs equals the trace of the product of appropriate subjective density and likelihood operators.
The continuous behavior of the numeraire portfolio under small changes in information structure, probabilistic views and investment constraints
The numeraire portfolio in a financial market is the unique positive wealth
process that makes all other nonnegative wealth processes, when deflated by it,
supermartingales. The numeraire portfolio depends on market characteristics,
which include: (a) the information flow available to acting agents, given by a
filtration; (b) the statistical evolution of the asset prices and, more
generally, the states of nature, given by a probability measure; and (c)
possible restrictions that acting agents might be facing on available
investment strategies, modeled by a constraints set. In a financial market with
continuous-path asset prices, we establish the stable behavior of the numeraire
portfolio when each of the aforementioned market parameters is changed in an
infinitesimal way.Comment: 16 pages; revised versio
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