Related papers: A quantum analogue of convex optimization
Many fundamental problems in machine learning can be formulated by the convex program \[ \min_{\theta\in R^d}\ \sum_{i=1}^{n}f_{i}(\theta), \] where each $f_i$ is a convex, Lipschitz function supported on a subset of $d_i$ coordinates of…
We consider the Schr\"odinger operator $$-\frac{d^2}{d x^2} + V \qquad \mbox{on an interval}~~[a,b]~\mbox{with Dirichlet boundary conditions},$$ where $V$ is bounded from below and prove a lower bound on the first eigenvalue $\lambda_1$ in…
The aim of this article is to prove a quantitative inequality for the first eigenvalue of a Schr\"odinger operator in the ball. More precisely, we optimize the first eigenvalue $\lambda(V)$ of the operator $\mathcal L_v:=-\Delta-V$ with…
Finite-sum optimization has wide applications in machine learning, covering important problems such as support vector machines, regression, etc. In this paper, we initiate the study of solving finite-sum optimization problems by quantum…
We explore whether quantum advantages can be found for the zeroth-order online convex optimization problem, which is also known as bandit convex optimization with multi-point feedback. In this setting, given access to zeroth-order oracles…
Assume that f is a strict convex function with a unique minimum in R^n. We divide the vector of n-variables to d groups of vector subvariables with d at least two. We assume that we can find the partial minimum of f with respect to each…
In machine learning, nonconvex optimization problems with multiple local optimums are often encountered. Graduated Optimization Algorithm (GOA) is a popular heuristic method to obtain global optimums of nonconvex problems through…
In this paper, we study private optimization problems for non-smooth convex functions $F(x)=\mathbb{E}_i f_i(x)$ on $\mathbb{R}^d$. We show that modifying the exponential mechanism by adding an $\ell_2^2$ regularizer to $F(x)$ and sampling…
In this paper, the proximal point algorithm for quasi-convex minimization problem in nonpositive curvature metric spaces is studied. We prove $\Delta$-convergence of the generated sequence to a critical point (which is defined in the text)…
In this paper, we propose an interior-point method for linearly constrained optimization problems (possibly nonconvex). The method - which we call the Hessian barrier algorithm (HBA) - combines a forward Euler discretization of Hessian…
While recent work suggests that quantum computers can speed up the solution of semidefinite programs, little is known about the quantum complexity of more general convex optimization. We present a quantum algorithm that can optimize a…
We consider the following eigenvalue optimization problem: Given a bounded domain $\Omega\subset\R^n$ and numbers $\alpha\geq 0$, $A\in [0,|\Omega|]$, find a subset $D\subset\Omega$ of area $A$ for which the first Dirichlet eigenvalue of…
This paper focuses on the problem of \emph{constrained} \emph{stochastic} optimization. A zeroth order Frank-Wolfe algorithm is proposed, which in addition to the projection-free nature of the vanilla Frank-Wolfe algorithm makes it gradient…
We propose a family of nonconvex optimization algorithms that are able to save gradient and negative curvature computations to a large extent, and are guaranteed to find an approximate local minimum with improved runtime complexity. At the…
Gradient descent ascent (GDA), the simplest single-loop algorithm for nonconvex minimax optimization, is widely used in practical applications such as generative adversarial networks (GANs) and adversarial training. Albeit its desirable…
Estimating the volume of a convex body is a central problem in convex geometry and can be viewed as a continuous version of counting. We present a quantum algorithm that estimates the volume of an $n$-dimensional convex body within…
The computation of time-optimal velocity profiles along prescribed paths, subject to generic acceleration constraints, is a crucial problem in robot trajectory planning, with particular relevance to autonomous racing. However, the existing…
Grover's algorithm is a fundamental quantum algorithm that offers a quadratic speedup for the unstructured search problem by alternately applying physically implementable oracle and diffusion operators. In this paper, we reformulate the…
We consider the problem of minimizing a continuous function given quantum access to a stochastic gradient oracle. We provide two new methods for the special case of minimizing a Lipschitz convex function. Each method obtains a dimension…
Minimizing a convex function over the spectrahedron, i.e., the set of all positive semidefinite matrices with unit trace, is an important optimization task with many applications in optimization, machine learning, and signal processing. It…