On Correcting Inputs: Inverse Optimization for Online Structured Prediction

Algorithm designers typically assume that the input data is correct, and then proceed to find "optimal" or "sub-optimal" solutions using this input data. However this assumption of correct data does not always hold in practice, especially in the context of online learning systems where the objective is to learn appropriate feature weights given some training samples. Such scenarios necessitate the study of inverse optimization problems where one is given an input instance as well as a desired output and the task is to adjust the input data so that the given output is indeed optimal. Motivated by learning structured prediction models, in this paper we consider inverse optimization with a margin, i.e., we require the given output to be better than all other feasible outputs by a desired margin. We consider such inverse optimization problems for maximum weight matroid basis, matroid intersection, perfect matchings, minimum cost maximum flows, and shortest paths and derive the first known results for such problems with a non-zero margin. The effectiveness of these algorithmic approaches to online learning for structured prediction is also discussed.Comment: Conference version to appear in FSTTCS, 201

Integer Programming Formulations for the Shared Multicast Tree Problem

We study the shared multicast tree (SMT) problem in wireless networks. To support a multicast session between a set of network nodes, SMT aims to establish a wireless connection between them, such that the total energy consumption is minimized. All destinations of the multicast message must be connected, while non-destinations are optional nodes that can be used to relay messages. The objective function reflecting power consumption distinguishes SMT clearly from the traditional minimum Steiner tree problem. We develop two integer programming formulations for SMT. Both models are subsequently extended and strengthened. Theorems on relations between the LP bounds corresponding to the models are stated and proved. As the number of variables in the strongest formulations is a polynomial of degree four in the number of network nodes, the models are impractical for computing lower bounds in instances beyond a fairly small size, and therefore a constraint generation scheme is developed. Results from computational experiments with the models demonstrate good promise of the approaches taken.acceptedVersio

Contributions to the solution of the symmetric travelling salesman problem

Imperial Users onl

An Empirical Analysis of Approximation Algorithms for the Unweighted Tree Augmentation Problem

In this thesis, we perform an experimental study of approximation algorithms for the tree augmentation problem (TAP). TAP is a fundamental problem in network design. The goal of TAP is to add the minimum number of edges from a given edge set to a tree so that it becomes 2-edge connected. Formally, given a tree T = (V, E), where V denotes the set of vertices and E denotes the set of edges in the tree, and a set of edges (or links) L ⊆ V × V disjoint from E, the objective is to find a set of edges to add to the tree F ⊆ L such that the augmented tree (V, E ∪ F) is 2-edge connected. Our goal is to establish a baseline performance for each approximation algorithm on actual instances rather than worst-case instances. In particular, we are interested in whether the algorithms rank on practical instances is consistent with their worst-case guarantee rankings. We are also interested in whether preprocessing times, implementation difficulties, and running times justify the use of an algorithm in practice. We profiled and analyzed five approximation algorithms, viz., the Frederickson algorithm, the Nagamochi algorithm, the Even algorithm, the Adjiashivili algorithm, and the Grandoni algorithm. Additionally, we used an integer program and a simple randomized algorithm as benchmarks. The performance of each algorithm was measured using space, time, and quality comparison metrics. We found that the simple randomized is competitive with the approximation algorithms and that the algorithms rank according to their theoretical guarantees. The randomized algorithm is simpler to implement and understand. Furthermore, the randomized algorithm runs faster and uses less space than any of the more sophisticated approximation algorithms

A Lemke-like algorithm for the Multiclass Network Equilibrium Problem

14 pagesWe consider a nonatomic congestion game on a connected graph, with several classes of players. Each player wants to go from its origin vertex to its destination vertex at the minimum cost and all players of a given class share the same characteristics: cost functions on each arc, and origin-destination pair. Under some mild conditions, it is known that a Nash equilibrium exists, but the computation of an equilibrium in the multiclass case is an open problem for general functions. We consider the specific case where the cost functions are affine and propose an extension of Lemke's algorithm able to solve this problem. At the same time, it provides a constructive proof of the existence of an equilibrium in this case

Proximal Policy Gradient Arborescence for Quality Diversity Reinforcement Learning

Training generally capable agents that perform well in unseen dynamic environments is a long-term goal of robot learning. Quality Diversity Reinforcement Learning (QD-RL) is an emerging class of reinforcement learning (RL) algorithms that blend insights from Quality Diversity (QD) and RL to produce a collection of high performing and behaviorally diverse policies with respect to a behavioral embedding. Existing QD-RL approaches have thus far taken advantage of sample-efficient off-policy RL algorithms. However, recent advances in high-throughput, massively parallelized robotic simulators have opened the door for algorithms that can take advantage of such parallelism, and it is unclear how to scale existing off-policy QD-RL methods to these new data-rich regimes. In this work, we take the first steps to combine on-policy RL methods, specifically Proximal Policy Optimization (PPO), that can leverage massive parallelism, with QD, and propose a new QD-RL method with these high-throughput simulators and on-policy training in mind. Our proposed Proximal Policy Gradient Arborescence (PPGA) algorithm yields a 4x improvement over baselines on the challenging humanoid domain.Comment: Submitted to Neurips 202