An Optimal High-Order Tensor Method for Convex Optimization

Bo Jiang, Haoyue Wang, Shuzhong Zhang
Proceedings of the Thirty-Second Conference on Learning Theory, PMLR 99:1799-1801, 2019.

Abstract

This paper is concerned with finding an optimal algorithm for minimizing a composite convex objective function. The basic setting is that the objective is the sum of two convex functions: the first function is smooth with up to the d-th order derivative information available, and the second function is possibly non-smooth, but its proximal tensor mappings can be computed approximately in an efficient manner. The problem is to find – in that setting – the best possible (optimal) iteration complexity for convex optimization. Along that line, for the smooth case (without the second non-smooth part in the objective), Nesterov (1983) proposed an optimal algorithm for the first-order methods (d=1) with iteration complexity O( 1 / k^2 ). A high-order tensor algorithm with iteration complexity of O( 1 / k^{d+1} ) was proposed by Baes (2009) and Nesterov (2018). In this paper, we propose a new high-order tensor algorithm for the general composite case, with the iteration complexity of O( 1 / k^{(3d+1)/2} ), which matches the lower bound for the d-th order methods as established in Nesterov (2018) and Shamir et al. (2018), and hence is optimal. Our approach is based on the Accelerated Hybrid Proximal Extragradient (A-HPE) framework proposed in Monteiro and Svaiter (2013), where a bisection procedure is installed for each A-HPE iteration. At each bisection step a proximal tensor subproblem is approximately solved, and the total number of bisection steps per A-HPE iteration is bounded by a logarithmic factor in the precision required.

Cite this Paper

BibTeX
@InProceedings{pmlr-v99-jiang19a, title = {An Optimal High-Order Tensor Method for Convex Optimization}, author = {Jiang, Bo and Wang, Haoyue and Zhang, Shuzhong}, booktitle = {Proceedings of the Thirty-Second Conference on Learning Theory}, pages = {1799--1801}, year = {2019}, editor = {Beygelzimer, Alina and Hsu, Daniel}, volume = {99}, series = {Proceedings of Machine Learning Research}, month = {25--28 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v99/jiang19a/jiang19a.pdf}, url = {https://proceedings.mlr.press/v99/jiang19a.html}, abstract = {This paper is concerned with finding an optimal algorithm for minimizing a composite convex objective function. The basic setting is that the objective is the sum of two convex functions: the first function is smooth with up to the d-th order derivative information available, and the second function is possibly non-smooth, but its proximal tensor mappings can be computed approximately in an efficient manner. The problem is to find – in that setting – the best possible (optimal) iteration complexity for convex optimization. Along that line, for the smooth case (without the second non-smooth part in the objective), Nesterov (1983) proposed an optimal algorithm for the first-order methods (d=1) with iteration complexity O( 1 / k^2 ). A high-order tensor algorithm with iteration complexity of O( 1 / k^{d+1} ) was proposed by Baes (2009) and Nesterov (2018). In this paper, we propose a new high-order tensor algorithm for the general composite case, with the iteration complexity of O( 1 / k^{(3d+1)/2} ), which matches the lower bound for the d-th order methods as established in Nesterov (2018) and Shamir et al. (2018), and hence is optimal. Our approach is based on the Accelerated Hybrid Proximal Extragradient (A-HPE) framework proposed in Monteiro and Svaiter (2013), where a bisection procedure is installed for each A-HPE iteration. At each bisection step a proximal tensor subproblem is approximately solved, and the total number of bisection steps per A-HPE iteration is bounded by a logarithmic factor in the precision required.} }
Endnote
%0 Conference Paper %T An Optimal High-Order Tensor Method for Convex Optimization %A Bo Jiang %A Haoyue Wang %A Shuzhong Zhang %B Proceedings of the Thirty-Second Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2019 %E Alina Beygelzimer %E Daniel Hsu %F pmlr-v99-jiang19a %I PMLR %P 1799--1801 %U https://proceedings.mlr.press/v99/jiang19a.html %V 99 %X This paper is concerned with finding an optimal algorithm for minimizing a composite convex objective function. The basic setting is that the objective is the sum of two convex functions: the first function is smooth with up to the d-th order derivative information available, and the second function is possibly non-smooth, but its proximal tensor mappings can be computed approximately in an efficient manner. The problem is to find – in that setting – the best possible (optimal) iteration complexity for convex optimization. Along that line, for the smooth case (without the second non-smooth part in the objective), Nesterov (1983) proposed an optimal algorithm for the first-order methods (d=1) with iteration complexity O( 1 / k^2 ). A high-order tensor algorithm with iteration complexity of O( 1 / k^{d+1} ) was proposed by Baes (2009) and Nesterov (2018). In this paper, we propose a new high-order tensor algorithm for the general composite case, with the iteration complexity of O( 1 / k^{(3d+1)/2} ), which matches the lower bound for the d-th order methods as established in Nesterov (2018) and Shamir et al. (2018), and hence is optimal. Our approach is based on the Accelerated Hybrid Proximal Extragradient (A-HPE) framework proposed in Monteiro and Svaiter (2013), where a bisection procedure is installed for each A-HPE iteration. At each bisection step a proximal tensor subproblem is approximately solved, and the total number of bisection steps per A-HPE iteration is bounded by a logarithmic factor in the precision required.
APA
Jiang, B., Wang, H. & Zhang, S.. (2019). An Optimal High-Order Tensor Method for Convex Optimization. Proceedings of the Thirty-Second Conference on Learning Theory, in Proceedings of Machine Learning Research 99:1799-1801 Available from https://proceedings.mlr.press/v99/jiang19a.html.

Related Material