A framework for bilevel optimization that enables stochastic and global variance reduction algorithms

• URL: http://arxiv.org/abs/2201.13409v1
• Date: Mon, 31 Jan 2022 18:17:25 GMT
• Title: A framework for bilevel optimization that enables stochastic and global variance reduction algorithms
• Authors: Mathieu Dagr\'eou, Pierre Ablin, Samuel Vaiter, Thomas Moreau
• Abstract summary: Bilevel optimization is a problem of minimizing a value function which involves the arg-minimum of another function. We introduce a novel framework, in which the solution of the inner problem, the solution of the linear system, and the main variable evolve at the same time. We demonstrate that SABA, an adaptation of the celebrated SAGA algorithm in our framework, has $O(frac1T)$ convergence rate, and that it achieves linear convergence under Polyak-Lojasciewicz assumption.
• Score: 17.12280360174073
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Bilevel optimization, the problem of minimizing a value function which involves the arg-minimum of another function, appears in many areas of machine learning. In a large scale setting where the number of samples is huge, it is crucial to develop stochastic methods, which only use a few samples at a time to progress. However, computing the gradient of the value function involves solving a linear system, which makes it difficult to derive unbiased stochastic estimates. To overcome this problem we introduce a novel framework, in which the solution of the inner problem, the solution of the linear system, and the main variable evolve at the same time. These directions are written as a sum, making it straightforward to derive unbiased estimates. The simplicity of our approach allows us to develop global variance reduction algorithms, where the dynamics of all variables is subject to variance reduction. We demonstrate that SABA, an adaptation of the celebrated SAGA algorithm in our framework, has $O(\frac1T)$ convergence rate, and that it achieves linear convergence under Polyak-Lojasciewicz assumption. This is the first stochastic algorithm for bilevel optimization that verifies either of these properties. Numerical experiments validate the usefulness of our method.

Related papers

• A Stochastic Approach to Bi-Level Optimization for Hyperparameter Optimization and Meta Learning [74.80956524812714]
  We tackle the general differentiable meta learning problem that is ubiquitous in modern deep learning. These problems are often formalized as Bi-Level optimizations (BLO) We introduce a novel perspective by turning a given BLO problem into a ii optimization, where the inner loss function becomes a smooth distribution, and the outer loss becomes an expected loss over the inner distribution.
  arXiv  Detail & Related papers  (2024-10-14T12:10:06Z)
• 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)
• Continuation Newton methods with deflation techniques for global optimization problems [3.705839280172101]
  A global minimum point of an optimization problem is of interest in engineering. In this article, we consider a new memetic algorithm for this nonlinear largescale problem. According to our numerical experiments, new algorithm works well for unconstrained unconstrained problems.
  arXiv  Detail & Related papers  (2021-07-29T09:53:49Z)
• Adaptive Importance Sampling for Finite-Sum Optimization and Sampling with Decreasing Step-Sizes [4.355567556995855]
  We propose Avare, a simple and efficient algorithm for adaptive importance sampling for finite-sum optimization and sampling with decreasing step-sizes. Under standard technical conditions, we show that Avare achieves $mathcalO(T2/3)$ and $mathcalO(T5/6)$ dynamic regret for SGD and SGLD respectively when run with $mathcalO(T5/6)$ step sizes.
  arXiv  Detail & Related papers  (2021-03-23T00:28:15Z)
• A Momentum-Assisted Single-Timescale Stochastic Approximation Algorithm for Bilevel Optimization [112.59170319105971]
  We propose a new algorithm -- the Momentum- Single-timescale Approximation (MSTSA) -- for tackling problems. MSTSA allows us to control the error in iterations due to inaccurate solution to the lower level subproblem.
  arXiv  Detail & Related papers  (2021-02-15T07:10:33Z)
• A Two-Timescale Framework for Bilevel Optimization: Complexity Analysis and Application to Actor-Critic [142.1492359556374]
  Bilevel optimization is a class of problems which exhibit a two-level structure. We propose a two-timescale approximation (TTSA) algorithm for tackling such a bilevel problem. We show that a two-timescale natural actor-critic policy optimization algorithm can be viewed as a special case of our TTSA framework.
  arXiv  Detail & Related papers  (2020-07-10T05:20:02Z)
• Aligning Partially Overlapping Point Sets: an Inner Approximation Algorithm [80.15123031136564]
  We propose a robust method to align point sets where there is no prior information about the value of the transformation. Our algorithm does not need regularization on transformation, and thus can handle the situation where there is no prior information about the values of the transformations. Experimental results demonstrate the better robustness of the proposed method over state-of-the-art algorithms.
  arXiv  Detail & Related papers  (2020-07-05T15:23:33Z)
• Hybrid Variance-Reduced SGD Algorithms For Nonconvex-Concave Minimax Problems [26.24895953952318]
  We develop an algorithm to solve a class of non-gence minimax problems. They can also work with both single or two mini-batch derivatives.
  arXiv  Detail & Related papers  (2020-06-27T03:05:18Z)
• Gradient Free Minimax Optimization: Variance Reduction and Faster Convergence [120.9336529957224]
  In this paper, we denote the non-strongly setting on the magnitude of a gradient-free minimax optimization problem. We show that a novel zeroth-order variance reduced descent algorithm achieves the best known query complexity.
  arXiv  Detail & Related papers  (2020-06-16T17:55:46Z)
• Optimizing generalization on the train set: a novel gradient-based framework to train parameters and hyperparameters simultaneously [0.0]
  Generalization is a central problem in Machine Learning. We present a novel approach based on a new measure of risk that allows us to develop novel fully automatic procedures for generalization.
  arXiv  Detail & Related papers  (2020-06-11T18:04:36Z)
• Towards Better Understanding of Adaptive Gradient Algorithms in Generative Adversarial Nets [71.05306664267832]
  Adaptive algorithms perform gradient updates using the history of gradients and are ubiquitous in training deep neural networks. In this paper we analyze a variant of OptimisticOA algorithm for nonconcave minmax problems. Our experiments show that adaptive GAN non-adaptive gradient algorithms can be observed empirically.
  arXiv  Detail & Related papers  (2019-12-26T22:10:10Z)

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.