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Piunovskiy, Aleksey B - One of the best experts on this subject based on the ideXlab platform.
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Sufficiency of Deterministic Policies for Atomless Discounted and Uniformly Absorbing MDPs with Multiple Criteria
'Society for Industrial & Applied Mathematics (SIAM)', 2019Co-Authors: Feinberg, Eugene A, Piunovskiy, Aleksey BAbstract:This paper studies Markov decision processes (MDPs) with atomless initial state distributions and atomless transition probabilities. Such MDPs are called atomless. The initial state distribution is considered to be fixed. We show that for discounted MDPs with bounded one-step reward vector-functions, for each policy there exists a deterministic (that is, nonrandomized and stationary) policy with the same performance vector. This fact is proved in the paper for a more general class of uniformly absorbing MDPs with expected total rewards, and then it is extended under certain assumptions to MDPs with unbounded rewards. For problems with multiple criteria and constraints, the results of this paper imply that for atomless MDPs studied in this paper it is sufficient to consider only deterministic policies, while without the atomless assumption it is well-known that randomized policies can outperform deterministic ones. We also provide an example of an MDP demonstrating that if a vector measure is defined on a Standard Borel Space, then Lyapunov's convexity theorem is a special case of the described results
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Sufficiency of Deterministic Policies for Atomless Discounted and Uniformly Absorbing MDPs with Multiple Criteria
2018Co-Authors: Feinberg, Eugene A, Piunovskiy, Aleksey BAbstract:This paper studies Markov Decision Processes (MDPs) with atomless initial state distributions and atomless transition probabilities. Such MDPs are called atomless. The initial state distribution is considered to be fixed. We show that for discounted MDPs with bounded one-step reward vector-functions, for each policy there exists a deterministic (that is, nonrandomized and stationary) policy with the same performance vector. This fact is proved in the paper for a more general class of uniformly absorbing MDPs with expected total costs, and then it is extended under certain assumptions to MDPs with unbounded rewards. For problems with multiple criteria and constraints, the results of this paper imply that for atomless MDPs studied in this paper it is sufficient to consider only deterministic policies, while without the atomless assumption it is well-known that randomized policies can outperform deterministic ones. We also provide an example of an MDP demonstrating that, if a vector measure is defined on a Standard Borel Space, then Lyapunov's convexity theorem is a special case of the described results
Michael Hochman - One of the best experts on this subject based on the ideXlab platform.
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every Borel automorphism without finite invariant measures admits a two set generator
Journal of the European Mathematical Society, 2018Co-Authors: Michael HochmanAbstract:We show that if an automorphism of a Standard Borel Space does not admit finite invariant measures, then it has a two-set generator modulo the σ-ideal generated by wandering sets. This implies that if the entropies of invariant probability measures of a Borel system are all less than log k, then the system admits a k-set generator, and that a wide class of hyperbolic-like systems are classified completely at the Borel level by entropy and periodic points counts.
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isomorphism and embedding of Borel systems on full sets
Acta Applicandae Mathematicae, 2013Co-Authors: Michael HochmanAbstract:A Borel system consists of a measurable automorphism of a Standard Borel Space. We consider Borel embeddings and isomorphisms between such systems modulo null sets, i.e. sets which have measure zero for every invariant probability measure. For every t>0 we show that in this category, up to isomorphism, there exists a unique free Borel system (Y,S) which is strictly t-universal in the sense that all invariant measures on Y have entropy
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isomorphism and embedding of Borel systems on full sets
arXiv: Dynamical Systems, 2010Co-Authors: Michael HochmanAbstract:A Borel system consists of a measurable automorphism of a Standard Borel Space. We consider Borel embeddings and isomorphisms between such systems modulo null sets, i.e. sets which have measure zero for every invariant probability measure. For every t>0 we show that in this category there exists a unique free Borel system (Y,S) which is strictly t-universal in the sense that all invariant measures on Y have entropy
Feinberg, Eugene A - One of the best experts on this subject based on the ideXlab platform.
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Sufficiency of Deterministic Policies for Atomless Discounted and Uniformly Absorbing MDPs with Multiple Criteria
'Society for Industrial & Applied Mathematics (SIAM)', 2019Co-Authors: Feinberg, Eugene A, Piunovskiy, Aleksey BAbstract:This paper studies Markov decision processes (MDPs) with atomless initial state distributions and atomless transition probabilities. Such MDPs are called atomless. The initial state distribution is considered to be fixed. We show that for discounted MDPs with bounded one-step reward vector-functions, for each policy there exists a deterministic (that is, nonrandomized and stationary) policy with the same performance vector. This fact is proved in the paper for a more general class of uniformly absorbing MDPs with expected total rewards, and then it is extended under certain assumptions to MDPs with unbounded rewards. For problems with multiple criteria and constraints, the results of this paper imply that for atomless MDPs studied in this paper it is sufficient to consider only deterministic policies, while without the atomless assumption it is well-known that randomized policies can outperform deterministic ones. We also provide an example of an MDP demonstrating that if a vector measure is defined on a Standard Borel Space, then Lyapunov's convexity theorem is a special case of the described results
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Sufficiency of Deterministic Policies for Atomless Discounted and Uniformly Absorbing MDPs with Multiple Criteria
2018Co-Authors: Feinberg, Eugene A, Piunovskiy, Aleksey BAbstract:This paper studies Markov Decision Processes (MDPs) with atomless initial state distributions and atomless transition probabilities. Such MDPs are called atomless. The initial state distribution is considered to be fixed. We show that for discounted MDPs with bounded one-step reward vector-functions, for each policy there exists a deterministic (that is, nonrandomized and stationary) policy with the same performance vector. This fact is proved in the paper for a more general class of uniformly absorbing MDPs with expected total costs, and then it is extended under certain assumptions to MDPs with unbounded rewards. For problems with multiple criteria and constraints, the results of this paper imply that for atomless MDPs studied in this paper it is sufficient to consider only deterministic policies, while without the atomless assumption it is well-known that randomized policies can outperform deterministic ones. We also provide an example of an MDP demonstrating that, if a vector measure is defined on a Standard Borel Space, then Lyapunov's convexity theorem is a special case of the described results
Vladimir Pestov - One of the best experts on this subject based on the ideXlab platform.
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some extremely amenable groups related to operator algebras and ergodic theory
Journal of The Institute of Mathematics of Jussieu, 2007Co-Authors: Thierry Giordano, Vladimir PestovAbstract:A topological group G is called extremely amenable if every continu- ous action of G on a compact Space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neu- mann algebra is approximately finite-dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group, which strengthens a result by de la Harpe. As a consequence, a C � -algebra A is nuclear if and only if the unitary group U(A) with the relative weak topology is strongly amenable in the sense of Glasner. We prove that the group of automorphisms of a Lebesgue Space with a non-atomic measure is extremely amenable with the weak topology and establish a similar result for groups of non-singular transformations. As a consequence, we prove extreme amenability of the groups of isometries of L p (0,1), 1 ≤ p < ∞, extending a classical result of Gromov and Milman (p = 2). We show that a measure class preserving equivalence relation R on a Standard Borel Space is amenable if and only if the full group (R), equipped with the uniform topology, is extremely amenable. Finally, we give natural examples of concentration to a nontriv- ial Space in the sense of Gromov occuring in the automorphism groups of injective factors of type III.
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some extremely amenable groups related to operator algebras and ergodic theory
arXiv: Operator Algebras, 2004Co-Authors: Thierry Giordano, Vladimir PestovAbstract:A topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact Space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann algebra is approximately finite-dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group, which strengthens a result by de la Harpe. As a consequence, a $C^\ast$-algebra $A$ is nuclear if and only if the unitary group $U(A)$ with the relative weak topology is strongly amenable in the sense of Glasner. We prove that the group of automorphisms of a Lebesgue Space with a non-atomic measure is extremely amenable with the weak topology and establish a similar result for groups of non-singular transformations. As a consequence, we prove extreme amenability of the groups of isometries of $L^p(0,1)$, $1\leq p<\infty$, extending a classical result of Gromov and Milman ($p=2$). We show that a measure class preserving equivalence relation $\mathcal R$ on a Standard Borel Space is amenable if and only if the full group $[{\mathcal R}]$, equipped with the uniform topology, is extremely amenable. Finally, we give natural examples of concentration to a nontrivial Space in the sense of Gromov occuring in the automorphism groups of injective factors of type $III$.
Sasyk Román - One of the best experts on this subject based on the ideXlab platform.
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Non-classification of free Araki-Woods factors and $\tau$-invariants
'European Mathematical Society Publishing House', 2019Co-Authors: Sasyk Román, Törnquist Asger, Vaes StefaanAbstract:We define the Standard Borel Space of free Araki-Woods factors and prove that their isomorphism relation is not classifiable by countable structures. We also prove that equality of $\tau$-topologies, arising as invariants of type III factors, as well as coycle and outer conjugacy of actions of abelian groups on free product factors are not classifiable by countable structures.Comment: v3: minor changes, final version, to appear in Groups, Geometry, and Dynamic
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Non-classification of free Araki-Woods factors and τ-invariants
'European Mathematical Society Publishing House', 2019Co-Authors: Sasyk Román, Törnquist Asger, Vaes StefanAbstract:We define the Standard Borel Space of free Araki-Woods factors and prove that their isomorphism relation is not classifiable by countable structures. We also prove that equality of τ-topologies, arising as invariants of type III factors, as well as coycle and outer conjugacy of actions of abelian groups on free product factors are not classifiable by countable structures.Fil: Sasyk, Roman. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Katholikie Universiteit Leuven; Bélgica. Universidad de Copenhagen; DinamarcaFil: Törnquist, Asger. University of Copenhagen. Department of Mathematics; DinamarcaFil: Vaes, Stefan. Katholikie Universiteit Leuven; Bélgic
