Lower Complexity Bounds of Finite-Sum Optimization Problems: The Results and Construction

• URL: http://arxiv.org/abs/2103.08280v1
• Date: Mon, 15 Mar 2021 11:20:31 GMT
• Title: Lower Complexity Bounds of Finite-Sum Optimization Problems: The Results and Construction
• Authors: Yuze Han, Guangzeng Xie, Zhihua Zhang
• Abstract summary: We consider Proximal Incremental First-order (PIFO) algorithms which have access to gradient and proximal oracle for each individual component. We develop a novel approach for constructing adversarial problems, which partitions the tridiagonal matrix of classical examples into $n$ groups.
• Score: 18.65143269806133
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The contribution of this paper includes two aspects. First, we study the lower bound complexity for the minimax optimization problem whose objective function is the average of $n$ individual smooth component functions. We consider Proximal Incremental First-order (PIFO) algorithms which have access to gradient and proximal oracle for each individual component. We develop a novel approach for constructing adversarial problems, which partitions the tridiagonal matrix of classical examples into $n$ groups. This construction is friendly to the analysis of incremental gradient and proximal oracle. With this approach, we demonstrate the lower bounds of first-order algorithms for finding an $\varepsilon$-suboptimal point and an $\varepsilon$-stationary point in different settings. Second, we also derive the lower bounds of minimization optimization with PIFO algorithms from our approach, which can cover the results in \citep{woodworth2016tight} and improve the results in \citep{zhou2019lower}.

Related papers

• Self-concordant Smoothing for Large-Scale Convex Composite Optimization [0.0]
  We introduce a notion of self-concordant smoothing for minimizing the sum of two convex functions, one of which is smooth and the other may be nonsmooth. We prove the convergence of two resulting algorithms: Prox-N-SCORE, a proximal Newton algorithm and Prox-GGN-SCORE, a proximal generalized Gauss-Newton algorithm.
  arXiv  Detail & Related papers  (2023-09-04T19:47:04Z)
• Stochastic Nested Compositional Bi-level Optimization for Robust Feature Learning [11.236838268731804]
  We develop and analyze algorithms for solving nested bi-level optimization problems. Our proposed algorithm does not rely on matrix complexity or mini-batches.
  arXiv  Detail & Related papers  (2023-07-11T15:52:04Z)
• Gradient-free optimization of highly smooth functions: improved analysis and a new algorithm [87.22224691317766]
  This work studies problems with zero-order noisy oracle information under the assumption that the objective function is highly smooth. We consider two kinds of zero-order projected gradient descent algorithms.
  arXiv  Detail & Related papers  (2023-06-03T17:05:13Z)
• A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimization [77.8485863487028]
  In this paper we consider finding an approximate second-order stationary point (SOSP) that minimizes a twice different subject general non conic optimization. In particular, we propose a Newton-CG based-augmentedconjugate method for finding an approximate SOSP.
  arXiv  Detail & Related papers  (2023-01-10T20:43:29Z)
• A Projection-free Algorithm for Constrained Stochastic Multi-level Composition Optimization [12.096252285460814]
  We propose a projection-free conditional gradient-type algorithm for composition optimization. We show that the number of oracles and the linear-minimization oracle required by the proposed algorithm, are of order $mathcalO_T(epsilon-2)$ and $mathcalO_T(epsilon-3)$ respectively.
  arXiv  Detail & Related papers  (2022-02-09T06:05:38Z)
• Efficiently Escaping Saddle Points in Bilevel Optimization [48.925688192913]
  Bilevel optimization is one of the problems in machine learning. Recent developments in bilevel optimization converge on the first fundamental nonaptature multi-step analysis.
  arXiv  Detail & Related papers  (2022-02-08T07:10:06Z)
• DoCoM: Compressed Decentralized Optimization with Near-Optimal Sample Complexity [25.775517797956237]
  This paper proposes the Doubly Compressed Momentum-assisted tracking algorithm $ttDoCoM$ for communication. We show that our algorithm outperforms several state-of-the-art algorithms in practice.
  arXiv  Detail & Related papers  (2022-02-01T07:27:34Z)
• Stochastic Multi-level Composition Optimization Algorithms with Level-Independent Convergence Rates [12.783783498844022]
  We study smooth multi-level composition optimization problems, where the objective function is a nested composition of $T$ functions. We show that the first algorithm, which is a generalization of citeGhaRuswan20 to the $T$ level case, can achieve a sample complexity of $mathcalO (1/epsilon$6) This is the first time that such an online algorithm designed for the (un) multi-level setting, obtains the same sample complexity under standard assumptions.
  arXiv  Detail & Related papers  (2020-08-24T15:57:50Z)
• 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)
• 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)
• 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)

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.