Related papers: Quantum algorithm for finding the negative curvatu…
We initiate the study of quantum algorithms for escaping from saddle points with provable guarantee. Given a function $f\colon\mathbb{R}^{n}\to\mathbb{R}$, our quantum algorithm outputs an $\epsilon$-approximate second-order stationary…
We consider escaping saddle points of nonconvex problems where only the function evaluations can be accessed. Although a variety of works have been proposed, the majority of them require either second or first-order information, and only a…
Recent results suggest that quantum computers possess the potential to speed up nonconvex optimization problems. However, a crucial factor for the implementation of quantum optimization algorithms is their robustness against experimental…
In this paper, we propose algorithms that exploit negative curvature for solving noisy nonlinear nonconvex unconstrained optimization problems. We consider both deterministic and stochastic inexact settings, and develop two-step algorithms…
This paper considers the projection-free sparse convex optimization problem for the vector domain and the matrix domain, which covers a large number of important applications in machine learning and data science. For the vector domain…
Optimizing non-convex functions is of primary importance in the vast majority of machine learning algorithms. Even though many gradient descent based algorithms have been studied, successive convex approximation based algorithms have been…
Gradient descent and its variants are widely used in machine learning. However, oracle access of gradient may not be available in many applications, limiting the direct use of gradient descent. This paper proposes a method of estimating…
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…
We investigate quantum algorithms for classification, a fundamental problem in machine learning, with provable guarantees. Given $n$ $d$-dimensional data points, the state-of-the-art (and optimal) classical algorithm for training…
We develop a quantum-classical hybrid algorithm for function optimization that is particularly useful in the training of neural networks since it makes use of particular aspects of high-dimensional energy landscapes. Due to a recent…
Escaping saddle points is a central research topic in nonconvex optimization. In this paper, we propose a simple gradient-based algorithm such that for a smooth function $f\colon\mathbb{R}^n\to\mathbb{R}$, it outputs an…
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…
Machine learning problems such as neural network training, tensor decomposition, and matrix factorization, require local minimization of a nonconvex function. This local minimization is challenged by the presence of saddle points, of which…
We propose stochastic optimization algorithms that can find local minima faster than existing algorithms for nonconvex optimization problems, by exploiting the third-order smoothness to escape non-degenerate saddle points more efficiently.…
In this paper, we give a sharp analysis for Stochastic Gradient Descent (SGD) and prove that SGD is able to efficiently escape from saddle points and find an $(\epsilon, O(\epsilon^{0.5}))$-approximate second-order stationary point in…
This paper considers the problem for finding the $(\delta,\epsilon)$-Goldstein stationary point of Lipschitz continuous objective, which is a rich function class to cover a great number of important applications. We construct a zeroth-order…
Gradient descent is a fundamental algorithm in both theory and practice for continuous optimization. Identifying its quantum counterpart would be appealing to both theoretical and practical quantum applications. A conventional approach to…
Recently, there has been a surge of interest for quantum computation for its ability to exponentially speed up algorithms, including machine learning algorithms. However, Tang suggested that the exponential speed up can also be done on a…
We present a quantum algorithm to solve dynamic programming problems with convex value functions. For linear discrete-time systems with a $d$-dimensional state space of size $N$, the proposed algorithm outputs a quantum-mechanical…
In this paper, we identify a family of nonconvex continuous optimization instances, each $d$-dimensional instance with $2^d$ local minima, to demonstrate a quantum-classical performance separation. Specifically, we prove that the recently…