10 research outputs found
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
A PTAS for Euclidean TSP with Hyperplane Neighborhoods
In the Traveling Salesperson Problem with Neighborhoods (TSPN), we are given
a collection of geometric regions in some space. The goal is to output a tour
of minimum length that visits at least one point in each region. Even in the
Euclidean plane, TSPN is known to be APX-hard, which gives rise to studying
more tractable special cases of the problem. In this paper, we focus on the
fundamental special case of regions that are hyperplanes in the -dimensional
Euclidean space. This case contrasts the much-better understood case of
so-called fat regions.
While for an exact algorithm with running time is known,
settling the exact approximability of the problem for has been repeatedly
posed as an open question. To date, only an approximation algorithm with
guarantee exponential in is known, and NP-hardness remains open.
For arbitrary fixed , we develop a Polynomial Time Approximation Scheme
(PTAS) that works for both the tour and path version of the problem. Our
algorithm is based on approximating the convex hull of the optimal tour by a
convex polytope of bounded complexity. Such polytopes are represented as
solutions of a sophisticated LP formulation, which we combine with the
enumeration of crucial properties of the tour. As the approximation guarantee
approaches , our scheme adjusts the complexity of the considered polytopes
accordingly.
In the analysis of our approximation scheme, we show that our search space
includes a sufficiently good approximation of the optimum. To do so, we develop
a novel and general sparsification technique to transform an arbitrary convex
polytope into one with a constant number of vertices and, in turn, into one of
bounded complexity in the above sense. Hereby, we maintain important properties
of the polytope
Proximal Algorithms for Smoothed Online Convex Optimization with Predictions
We consider a smoothed online convex optimization (SOCO) problem with
predictions, where the learner has access to a finite lookahead window of
time-varying stage costs, but suffers a switching cost for changing its actions
at each stage. Based on the Alternating Proximal Gradient Descent (APGD)
framework, we develop Receding Horizon Alternating Proximal Descent (RHAPD) for
proximable, non-smooth and strongly convex stage costs, and RHAPD-Smooth
(RHAPD-S) for non-proximable, smooth and strongly convex stage costs. In
addition to outperforming gradient descent-based algorithms, while maintaining
a comparable runtime complexity, our proposed algorithms also allow us to solve
a wider range of problems. We provide theoretical upper bounds on the dynamic
regret achieved by the proposed algorithms, which decay exponentially with the
length of the lookahead window. The performance of the presented algorithms is
empirically demonstrated via numerical experiments on non-smooth regression,
dynamic trajectory tracking, and economic power dispatch problems.Comment: 28 pages, 13 figure
Online Optimization with Predictions and Non-convex Losses
We study online optimization in a setting where an online learner seeks to optimize a per-round hitting cost, which may be non-convex, while incurring a movement cost when changing actions between rounds. We ask: under what general conditions is it possible for an online learner to leverage predictions of future cost functions in order to achieve near-optimal costs? Prior work has provided near-optimal online algorithms for specific combinations of assumptions about hitting and switching costs, but no general results are known. In this work, we give two general sufficient conditions that specify a relationship between the hitting and movement costs which guarantees that a new algorithm, Synchronized Fixed Horizon Control (SFHC), achieves a 1+O(1/w) competitive ratio, where w is the number of predictions available to the learner. Our conditions do not require the cost functions to be convex, and we also derive competitive ratio results for non-convex hitting and movement costs. Our results provide the first constant, dimension-free competitive ratio for online non-convex optimization with movement costs. We also give an example of a natural problem, Convex Body Chasing (CBC), where the sufficient conditions are not satisfied and prove that no online algorithm can have a competitive ratio that converges to 1