152 research outputs found
The Role of Dimension in the Online Chasing Problem
Let be a metric space and -- a
collection of special objects. In the -chasing problem, an
online player receives a sequence of online requests and responds with a trajectory such that . This response incurs a movement cost ,
and the online player strives to minimize the competitive ratio -- the worst
case ratio over all input sequences between the online movement cost and the
optimal movement cost in hindsight. Under this setup, we call the
-chasing problem if there exists an
online algorithm with finite competitive ratio. In the case of Convex Body
Chasing (CBC) over real normed vector spaces, (Bubeck et al. 2019) proved the
chaseability of the problem. Furthermore, in the vector space setting, the
dimension of the ambient space appears to be the factor controlling the size of
the competitive ratio. Indeed, recently, (Sellke 2020) provided a
competitive online algorithm over arbitrary real normed vector spaces
, and we will shortly present a general strategy for
obtaining novel lower bounds of the form , for CBC
in the same setting. In this paper, we also prove that the
and dimensions of a metric space exert no control on the
hardness of ball chasing over the said metric space. More specifically, we show
that for any large enough , there exists a metric space
of doubling dimension and Assouad dimension such
that no online selector can achieve a finite competitive ratio in the general
ball chasing regime
Metrical Service Systems with Transformations
We consider a generalization of the fundamental online metrical service systems (MSS) problem where the feasible region can be transformed between requests. In this problem, which we call T-MSS, an algorithm maintains a point in a metric space and has to serve a sequence of requests. Each request is a map (transformation) : → between subsets and of the metric space. To serve it, the algorithm has to go to a point ∈ , paying the distance from its previous position. Then, the transformation is applied, modifying the algorithm’s state to ( ). Such transformations can model, e.g., changes to the environment that are outside of an algorithm’s control, and we therefore do not charge any additional cost to the algorithm when the transformation is applied. The transformations also allow to model requests occurring in the -taxi problem.
We show that for -Lipschitz transformations, the competitive ratio is Θ()−2 on -point metrics. Here, the upper bound is achieved by a deterministic algorithm and the lower bound holds even for randomized algorithms. For the -taxi problem, we prove a competitive ratio of Õ(( log )2). For chasing convex bodies, we show that even with contracting transformations no competitive algorithm exists.
The problem T-MSS has a striking connection to the following deep mathematical question: Given a finite metric space M, what is the required cardinality of an extension M̂ ⊇ M where each partial isometry on M extends to an automorphism? We give partial answers for special cases
Online learning for robust voltage control under uncertain grid topology
Voltage control generally requires accurate information about the grid's
topology in order to guarantee network stability. However, accurate topology
identification is challenging for existing methods, especially as the grid is
subject to increasingly frequent reconfiguration due to the adoption of
renewable energy. Further, running existing control mechanisms with incorrect
network information may lead to unstable control. In this work, we combine a
nested convex body chasing algorithm with a robust predictive controller to
achieve provably finite-time convergence to safe voltage limits in the online
setting where the network topology is initially unknown. Specifically, the
online controller does not know the true network topology and line parameters,
but instead learns them over time by narrowing down the set of network
topologies and line parameters that are consistent with its observations and
adjusting reactive power generation accordingly to keep voltages within desired
safety limits. We demonstrate the effectiveness of our approach in a case study
on a Southern California Edison 56-bus distribution system. Our experiments
show that in practical settings, the controller is indeed able to narrow the
set of consistent topologies quickly enough to make control decisions that
ensure stability in both linearized and realistic non-linear models of the
distribution grid.Comment: under submission. arXiv admin note: substantial text overlap with
arXiv:2206.1436
Better Bounds for Online Line Chasing
We study online competitive algorithms for the line chasing problem in Euclidean spaces R^d, where the input consists of an initial point P_0 and a sequence of lines X_1, X_2, ..., X_m, revealed one at a time. At each step t, when the line X_t is revealed, the algorithm must determine a point P_t in X_t. An online algorithm is called c-competitive if for any input sequence the path P_0, P_1..., P_m it computes has length at most c times the optimum path. The line chasing problem is a variant of a more general convex body chasing problem, where the sets X_t are arbitrary convex sets.
To date, the best competitive ratio for the line chasing problem was 28.1, even in the plane. We improve this bound by providing a simple 3-competitive algorithm for any dimension d. We complement this bound by a matching lower bound for algorithms that are memoryless in the sense of our algorithm, and a lower bound of 1.5358 for arbitrary algorithms. The latter bound also improves upon the previous lower bound of sqrt{2}~=1.412 for convex body chasing in 2 dimensions
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