arXiv subjects

Conor McMeel

Publications and source records attributed to Conor McMeel.

4 recordsLinked to original sources

arXiv2022

Uncertainty quantification for subgradient descent, with applications to relaxations of discrete problems

We consider the problem of minimizing a convex function that depends on an uncertain parameter $θ$. The uncertainty in the objective function means that the optimum, $x^*(θ)$, is also a function of $θ$. We propose an efficient method to compute $x^*(θ)$ and its statistics. We use a chaos expansion of $x^*(θ)$ along a truncated basis and study a restarted subgradient method that compute the optimal coefficients. We establish the convergence rate of the method as the number of basis functions increases, and hence the dimensionality of the optimization problem is increased. We give a non-asymptotic convergence rate for subgradient descent, building on earlier work that looked at gradient and accelerated gradient descent. Additionally, this work explicitly deals with the issue of projections, and suggests a method to deal with non-trivial projections. We show how this algorithm can be used to quantify uncertainty in discrete problems by utilising the (convex) Lovasz Extension for the min s,t-cut graph problem.

math.OC↗

arXiv2021

Uncertainty Quantification for Gradient and Accelerated Gradient Descent Methods on Strongly Convex Functions

We consider the problem of minimizing a strongly convex function that depends on an uncertain parameter $θ$. The uncertainty in the objective function means that the optimum, $x^*(θ)$, is also a function of $θ$. We propose an efficient method to compute $x^*(θ)$ and its statistics. We use a chaos expansion of $x^*(θ)$ along a truncated basis and study first-order methods that compute the optimal coefficients. We establish the convergence rate of the method as the number of basis functions, and hence the dimensionality of the optimization problem is increased. We give the first non-asymptotic rates for the gradient descent and the accelerated gradient descent methods. Our analysis exploits convexity and does not rely on a diminishing step-size strategy. As a result, it is much faster than the state-of-the-art both in theory and in our preliminary numerical experiments. A surprising side-effect of our analysis is that the proposed method also acts as a variance reduction technique to the problem of estimating $x^*(θ)$.

math.OC↗

arXiv2020

Sensitivity Analysis of Submodular Function Maximization

We study the recently introduced idea of worst-case sensitivity for monotone submodular maximization with cardinality constraint $k$, which captures the degree to which the output argument changes on deletion of an element in the input. We find that for large classes of algorithms that non-trivial sensitivity of $o(k)$ is not possible, even with bounded curvature, and that these results also hold in the distributed framework. However, we also show that in the regime $k = Ω(n)$ that we can obtain $O(1)$ sensitivity for sufficiently low curvature.

cs.DS↗

arXiv2019

Majorisation-minimisation algorithms for minimising the difference between lattice submodular functions

We consider the problem of minimising functions represented as a difference of lattice submodular functions. We propose analogues to the SupSub, SubSup and ModMod routines for lattice submodular functions. We show that our majorisation-minimisation algorithms produce iterates that monotonically decrease, and that we converge to a local minimum. We also extend additive hardness results, and show that a broad range of functions can be expressed as the difference of submodular functions.

cs.DS↗