18,854 research outputs found
A Unified Distributed Method for Constrained Networked Optimization via Saddle-Point Dynamics
This paper develops a unified distributed method for solving two classes of
constrained networked optimization problems, i.e., optimal consensus problem
and resource allocation problem with non-identical set constraints. We first
transform these two constrained networked optimization problems into a unified
saddle-point problem framework with set constraints. Subsequently, two
projection-based primal-dual algorithms via Optimistic Gradient Descent Ascent
(OGDA) method and Extra-gradient (EG) method are developed for solving
constrained saddle-point problems. It is shown that the developed algorithms
achieve exact convergence to a saddle point with an ergodic convergence rate
for general convex-concave functions. Based on the proposed
primal-dual algorithms via saddle-point dynamics, we develop unified
distributed algorithm design and convergence analysis for these two networked
optimization problems. Finally, two numerical examples are presented to
demonstrate the theoretical results
Quantum Algorithm for Finding the Negative Curvature Direction
Non-convex optimization is an essential problem in the field of machine learning. Optimization methods for non-convex problems can be roughly di- vided into first-order methods and second-order methods, depending on the or- der of the derivative to the objective function they used. Generally, to find the local minima, the second-order methods are applied to find the effective direction to escape the saddle point. Specifically, finding the Negative Curvature is considered as the subroutine to analyze the characteristic of the saddle point. However, the calculation of the Negative Curvature is expensive, which prevents the practical usage of second-order algorithms. In this thesis, we present an efficient quantum algorithm aiming to find the negative curvature direction for escaping the saddle point, which is a critical subroutine for many second-order non-convex optimization algorithms. We prove that our algorithm could produce the target state corresponding to the negative curvature direction with query complexity O ̃(polylog(d) ε^(-1)), where d is the dimension of the optimization function. The quantum negative curvature finding algorithm is exponentially faster than any known classical method, which takes time at least O(dε^(-1/2)). Moreover, we propose an efficient quantum algorithm to achieve the classical read-out of the target state. Our classical read-out algorithm runs exponentially faster on the degree of d than existing counterparts
Deep Learning without Poor Local Minima
In this paper, we prove a conjecture published in 1989 and also partially
address an open problem announced at the Conference on Learning Theory (COLT)
2015. With no unrealistic assumption, we first prove the following statements
for the squared loss function of deep linear neural networks with any depth and
any widths: 1) the function is non-convex and non-concave, 2) every local
minimum is a global minimum, 3) every critical point that is not a global
minimum is a saddle point, and 4) there exist "bad" saddle points (where the
Hessian has no negative eigenvalue) for the deeper networks (with more than
three layers), whereas there is no bad saddle point for the shallow networks
(with three layers). Moreover, for deep nonlinear neural networks, we prove the
same four statements via a reduction to a deep linear model under the
independence assumption adopted from recent work. As a result, we present an
instance, for which we can answer the following question: how difficult is it
to directly train a deep model in theory? It is more difficult than the
classical machine learning models (because of the non-convexity), but not too
difficult (because of the nonexistence of poor local minima). Furthermore, the
mathematically proven existence of bad saddle points for deeper models would
suggest a possible open problem. We note that even though we have advanced the
theoretical foundations of deep learning and non-convex optimization, there is
still a gap between theory and practice.Comment: In NIPS 2016. Selected for NIPS oral presentation (top 2%
submissions). ---- The final NIPS 2016 version: the results remain the sam
Inexact Model: A Framework for Optimization and Variational Inequalities
In this paper we propose a general algorithmic framework for first-order
methods in optimization in a broad sense, including minimization problems,
saddle-point problems and variational inequalities. This framework allows to
obtain many known methods as a special case, the list including accelerated
gradient method, composite optimization methods, level-set methods, proximal
methods. The idea of the framework is based on constructing an inexact model of
the main problem component, i.e. objective function in optimization or operator
in variational inequalities. Besides reproducing known results, our framework
allows to construct new methods, which we illustrate by constructing a
universal method for variational inequalities with composite structure. This
method works for smooth and non-smooth problems with optimal complexity without
a priori knowledge of the problem smoothness. We also generalize our framework
for strongly convex objectives and strongly monotone variational inequalities.Comment: 41 page
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