417 research outputs found
Anytime Guarantees for Reachability in Uncountable Markov Decision Processes
We consider the problem of approximating the reachability probabilities in Markov decision processes (MDP) with uncountable (continuous) state and action spaces. While there are algorithms that, for special classes of such MDP, provide a sequence of approximations converging to the true value in the limit, our aim is to obtain an algorithm with guarantees on the precision of the approximation.
As this problem is undecidable in general, assumptions on the MDP are necessary. Our main contribution is to identify sufficient assumptions that are as weak as possible, thus approaching the "boundary" of which systems can be correctly and reliably analyzed. To this end, we also argue why each of our assumptions is necessary for algorithms based on processing finitely many observations.
We present two solution variants. The first one provides converging lower bounds under weaker assumptions than typical ones from previous works concerned with guarantees. The second one then utilizes stronger assumptions to additionally provide converging upper bounds. Altogether, we obtain an anytime algorithm, i.e. yielding a sequence of approximants with known and iteratively improving precision, converging to the true value in the limit. Besides, due to the generality of our assumptions, our algorithms are very general templates, readily allowing for various heuristics from literature in contrast to, e.g., a specific discretization algorithm. Our theoretical contribution thus paves the way for future practical improvements without sacrificing correctness guarantees
Spurious, Emergent Laws in Number Worlds
We study some aspects of the emergence of logos from chaos on a basal model
of the universe using methods and techniques from algorithmic information and
Ramsey theories. Thereby an intrinsic and unusual mixture of meaningful and
spurious, emerging laws surfaces. The spurious, emergent laws abound, they can
be found almost everywhere. In accord with the ancient Greek theogony one could
say that logos, the Gods and the laws of the universe, originate from "the
void," or from chaos, a picture which supports the unresolvable/irreducible
lawless hypothesis. The analysis presented in this paper suggests that the
"laws" discovered in science correspond merely to syntactical correlations, are
local and not universal.Comment: 24 pages, invited contribution to "Contemporary Natural Philosophy
and Philosophies - Part 2" - Special Issue of the journal Philosophie
Metareasoning for Planning Under Uncertainty
The conventional model for online planning under uncertainty assumes that an
agent can stop and plan without incurring costs for the time spent planning.
However, planning time is not free in most real-world settings. For example, an
autonomous drone is subject to nature's forces, like gravity, even while it
thinks, and must either pay a price for counteracting these forces to stay in
place, or grapple with the state change caused by acquiescing to them. Policy
optimization in these settings requires metareasoning---a process that trades
off the cost of planning and the potential policy improvement that can be
achieved. We formalize and analyze the metareasoning problem for Markov
Decision Processes (MDPs). Our work subsumes previously studied special cases
of metareasoning and shows that in the general case, metareasoning is at most
polynomially harder than solving MDPs with any given algorithm that disregards
the cost of thinking. For reasons we discuss, optimal general metareasoning
turns out to be impractical, motivating approximations. We present approximate
metareasoning procedures which rely on special properties of the BRTDP planning
algorithm and explore the effectiveness of our methods on a variety of
problems.Comment: Extended version of IJCAI 2015 pape
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