709 research outputs found
Online Optimization with Memory and Competitive Control
This paper presents competitive algorithms for a novel class of online optimization problems with memory. We consider a setting where the learner seeks to minimize the sum of a hitting cost and a switching cost that depends on the previous p decisions. This setting generalizes Smoothed Online Convex Optimization. The proposed approach, Optimistic Regularized Online Balanced Descent, achieves a constant, dimension-free competitive ratio. Further, we show a connection between online optimization with memory and online control with adversarial disturbances. This connection, in turn, leads to a new constant-competitive policy for a rich class of online control problems
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
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
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