The Role of Dimension in the Online Chasing Problem

Let $(X, d)$ be a metric space and $\mathcal{C} \subseteq 2^X$ -- a collection of special objects. In the $(X,d,\mathcal{C})$-chasing problem, an online player receives a sequence of online requests $\{B_t\}_{t=1}^T \subseteq \mathcal{C}$ and responds with a trajectory $\{x_t\}_{t=1}^T$ such that $x_t \in B_t$. This response incurs a movement cost $\sum_{t=1}^T d(x_t, x_{t-1})$, 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 $(X,d,\mathcal{C})$-chasing problem $\textit{chaseable}$ 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 $d-$competitive online algorithm over arbitrary real normed vector spaces $(\mathbb{R}^d, ||\cdot||)$, and we will shortly present a general strategy for obtaining novel lower bounds of the form $\Omega(d^c), \enspace c > 0$, for CBC in the same setting. In this paper, we also prove that the $\textit{doubling}$ and $\textit{Assouad}$ 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 $\rho \in \mathbb{R}$, there exists a metric space $(X,d)$ of doubling dimension $\Theta(\rho)$ and Assouad dimension $\rho$ such that no online selector can achieve a finite competitive ratio in the general ball chasing regime

Nested convex bodies are chaseable

In the Convex Body Chasing problem, we are given an initial point v0 2 Rd and an online sequence of n convex bodies F1; : : : ; Fn. When we receive Fi, we are required to move inside Fi. Our goal is to minimize the total distance traveled. This fundamental online problem was first studied by Friedman and Linial (DCG 1993). They proved an ( p d) lower bound on the competitive ratio, and conjectured that a competitive ratio depending only on d is possible. However, despite much interest in the problem, the conjecture remains wide open. We consider the setting in which the convex bodies are nested: F1 : : : Fn. The nested setting is closely related to extending the online LP framework of Buchbinder and Naor (ESA 2005) to arbitrary linear constraints. Moreover, this setting retains much of the difficulty of the general setting and captures an essential obstacle in resolving Friedman and Linial's conjecture. In this work, we give a f(d)competitive algorithm for chasing nested convex bodies in Rd

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