Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA Novel Learnable Gradient Descent Type Algorithm for Non-convex Non-smooth Inverse Problems
123
0
0.0
(
0
)
Added by
Qingchao Zhang
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Optimization algorithms for solving nonconvex inverse problem have attracted significant interests recently. However, existing methods require the nonconvex regularization to be smooth or simple to ensure convergence. In this paper, we propose a novel gradient descent type algorithm, by leveraging the idea of residual learning and Nesterovs smoothing technique, to solve inverse problems consisting of general nonconvex and nonsmooth regularization with provable convergence. Moreover, we develop a neural network architecture intimating this algorithm to learn the nonlinear sparsity transformation adaptively from training data, which also inherits the convergence to accommodate the general nonconvex structure of this learned transformation. Numerical results demonstrate that the proposed network outperforms the state-of-the-art methods on a variety of different image reconstruction problems in terms of efficiency and accuracy.
rate research
Read More
We study the first-order convex optimization problem, where we have black-box access to a (not necessarily smooth) function $f:mathbb{R}^n to mathbb{R}$ and its (sub)gradient. Our goal is to find an $epsilon$-approximate minimum of $f$ starting from a point that is distance at most $R$ from the true minimum. If $f$ is $G$-Lipschitz, then the classic gradient descent algorithm solves this problem with $O((GR/epsilon)^{2})$ queries. Importantly, the number of queries is independent of the dimension $n$ and gradient descent is optimal in this regard: No deterministic or randomized algorithm can achieve better complexity that is still independent of the dimension $n$. In this paper we reprove the randomized lower bound of $Omega((GR/epsilon)^{2})$ using a simpler argument than previous lower bounds. We then show that although the function family used in the lower bound is hard for randomized algorithms, it can be solved using $O(GR/epsilon)$ quantum queries. We then show an improved lower bound against quantum algorithms using a different set of instances and establish our main result that in general even quantum algorithms need $Omega((GR/epsilon)^2)$ queries to solve the problem. Hence there is no quantum speedup over gradient descent for black-box first-order convex optimization without further assumptions on the function family.
We consider the problem of finding local minimizers in non-convex and non-smooth optimization. Under the assumption of strict saddle points, positive results have been derived for first-order methods. We present the first known results for the non-smooth case, which requires different analysis and a different algorithm.
In this paper we propose several adaptive gradient methods for stochastic optimization. Unlike AdaGrad-type of methods, our algorithms are based on Armijo-type line search and they simultaneously adapt to the unknown Lipschitz constant of the gradient and variance of the stochastic approximation for the gradient. We consider an accelerated and non-accelerated gradient descent for convex problems and gradient descent for non-convex problems. In the experiments we demonstrate superiority of our methods to existing adaptive methods, e.g. AdaGrad and Adam.
We study adversary-resilient stochastic distributed optimization, in which $m$ machines can independently compute stochastic gradients, and cooperate to jointly optimize over their local objective functions. However, an $alpha$-fraction of the machines are $textit{Byzantine}$, in that they may behave in arbitrary, adversarial ways. We consider a variant of this procedure in the challenging $textit{non-convex}$ case. Our main result is a new algorithm SafeguardSGD which can provably escape saddle points and find approximate local minima of the non-convex objective. The algorithm is based on a new concentration filtering technique, and its sample and time complexity bounds match the best known theoretical bounds in the stochastic, distributed setting when no Byzantine machines are present. Our algorithm is very practical: it improves upon the performance of all prior methods when training deep neural networks, it is relatively lightweight, and it is the first method to withstand two recently-proposed Byzantine attacks.
Inspired by the decomposition in the hybrid quantum-classical optimization algorithm we introduced in arXiv:1902.04215, we propose here a new (fully classical) approach to solving certain non-convex integer programs using Graver bases. This method is well suited when (a) the constraint matrix $A$ has a special structure so that its Graver basis can be computed systematically, (b) several feasible solutions can also be constructed easily and (c) the objective function can be viewed as many convex functions quilted together. Classes of problems that satisfy these conditions include Cardinality Boolean Quadratic Problems (CBQP), Quadratic Semi-Assignment Problems (QSAP) and Quadratic Assignment Problems (QAP). Our Graver Augmented Multi-seed Algorithm (GAMA) utilizes augmentation along Graver basis elements (the improvement direction is obtained by comparing objective function values) from these multiple initial feasible solutions. We compare our approach with a best-in-class commercially available solver (Gurobi). Sensitivity analysis indicates that the rate at which GAMA slows down as the problem size increases is much lower than that of Gurobi. We find that for several instances of practical relevance, GAMA not only vastly outperforms in terms of time to find the optimal solution (by two or three orders of magnitude), but also finds optimal solutions within minutes when the commercial solver is not able to do so in 4 or 10 hours (depending on the problem class) in several cases.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
