Accelerated Algorithms for Convex and Non-Convex Optimization on Manifolds

• URL: http://arxiv.org/abs/2010.08908v1
• Date: Sun, 18 Oct 2020 02:48:22 GMT
• Title: Accelerated Algorithms for Convex and Non-Convex Optimization on Manifolds
• Authors: Lizhen Lin, Bayan Saparbayeva, Michael Minyi Zhang, David B. Dunson
• Abstract summary: We propose a scheme for solving convex and non- optimization problems on distance. Our proposed algorithm adapts to the level of complexity in the objective function.
• Score: 9.632674803757475
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose a general scheme for solving convex and non-convex optimization problems on manifolds. The central idea is that, by adding a multiple of the squared retraction distance to the objective function in question, we "convexify" the objective function and solve a series of convex sub-problems in the optimization procedure. One of the key challenges for optimization on manifolds is the difficulty of verifying the complexity of the objective function, e.g., whether the objective function is convex or non-convex, and the degree of non-convexity. Our proposed algorithm adapts to the level of complexity in the objective function. We show that when the objective function is convex, the algorithm provably converges to the optimum and leads to accelerated convergence. When the objective function is non-convex, the algorithm will converge to a stationary point. Our proposed method unifies insights from Nesterov's original idea for accelerating gradient descent algorithms with recent developments in optimization algorithms in Euclidean space. We demonstrate the utility of our algorithms on several manifold optimization tasks such as estimating intrinsic and extrinsic Fr\'echet means on spheres and low-rank matrix factorization with Grassmann manifolds applied to the Netflix rating data set.

Related papers

• Accelerating Cutting-Plane Algorithms via Reinforcement Learning Surrogates [49.84541884653309]
  A current standard approach to solving convex discrete optimization problems is the use of cutting-plane algorithms. Despite the existence of a number of general-purpose cut-generating algorithms, large-scale discrete optimization problems continue to suffer from intractability. We propose a method for accelerating cutting-plane algorithms via reinforcement learning.
  arXiv  Detail & Related papers  (2023-07-17T20:11:56Z)
• Linearization Algorithms for Fully Composite Optimization [61.20539085730636]
  This paper studies first-order algorithms for solving fully composite optimization problems convex compact sets. We leverage the structure of the objective by handling differentiable and non-differentiable separately, linearizing only the smooth parts.
  arXiv  Detail & Related papers  (2023-02-24T18:41:48Z)
• Faster Riemannian Newton-type Optimization by Subsampling and Cubic Regularization [3.867143522757309]
  This work is on constrained large-scale non-constrained optimization where the constraint set implies a manifold structure. We propose a new second-order saddleian optimization algorithm, aiming at improving convergence and reducing computational cost.
  arXiv  Detail & Related papers  (2023-02-22T00:37:44Z)
• Efficient Gradient Approximation Method for Constrained Bilevel Optimization [2.0305676256390934]
  Bilevel optimization has been developed with large-scale high-dimensional data. This paper considers a constrained bilevel problem with convex and non-differentiable approximations.
  arXiv  Detail & Related papers  (2023-02-03T19:34:56Z)
• Adaptive Zeroth-Order Optimisation of Nonconvex Composite Objectives [1.7640556247739623]
  We analyze algorithms for zeroth-order entropy composite objectives, focusing on dependence on dimensionality. This is achieved by exploiting low dimensional structure of the decision set using the mirror descent method with an estimation alike function. To improve the gradient, we replace the classic sampling method based on Rademacher and show that the mini-batch method copes with non-Eucli geometry.
  arXiv  Detail & Related papers  (2022-08-09T07:36:25Z)
• An Algebraically Converging Stochastic Gradient Descent Algorithm for Global Optimization [14.336473214524663]
  A key component in the algorithm is the randomness based on the value of the objective function. We prove the convergence of the algorithm with an algebra and tuning in the parameter space. We present several numerical examples to demonstrate the efficiency and robustness of the algorithm.
  arXiv  Detail & Related papers  (2022-04-12T16:27:49Z)
• Zeroth-Order Hybrid Gradient Descent: Towards A Principled Black-Box Optimization Framework [100.36569795440889]
  This work is on the iteration of zero-th-order (ZO) optimization which does not require first-order information. We show that with a graceful design in coordinate importance sampling, the proposed ZO optimization method is efficient both in terms of complexity as well as as function query cost.
  arXiv  Detail & Related papers  (2020-12-21T17:29:58Z)
• Sequential Subspace Search for Functional Bayesian Optimization Incorporating Experimenter Intuition [63.011641517977644]
  Our algorithm generates a sequence of finite-dimensional random subspaces of functional space spanned by a set of draws from the experimenter's Gaussian Process. Standard Bayesian optimisation is applied on each subspace, and the best solution found used as a starting point (origin) for the next subspace. We test our algorithm in simulated and real-world experiments, namely blind function matching, finding the optimal precipitation-strengthening function for an aluminium alloy, and learning rate schedule optimisation for deep networks.
  arXiv  Detail & Related papers  (2020-09-08T06:54:11Z)
• Exploiting Higher Order Smoothness in Derivative-free Optimization and Continuous Bandits [99.70167985955352]
  We study the problem of zero-order optimization of a strongly convex function. We consider a randomized approximation of the projected gradient descent algorithm. Our results imply that the zero-order algorithm is nearly optimal in terms of sample complexity and the problem parameters.
  arXiv  Detail & Related papers  (2020-06-14T10:42:23Z)
• Convergence of adaptive algorithms for weakly convex constrained optimization [59.36386973876765]
  We prove the $mathcaltilde O(t-1/4)$ rate of convergence for the norm of the gradient of Moreau envelope. Our analysis works with mini-batch size of $1$, constant first and second order moment parameters, and possibly smooth optimization domains.
  arXiv  Detail & Related papers  (2020-06-11T17:43:19Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.