2019-06-25-jiang19a.md

abstract	section	title	layout	series	id	month	tex_title	firstpage	lastpage	page	order	cycles	bibtex_author	author	date	address	publisher	container-title	volume	genre	issued	pdf	extras
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.	contributed	An Optimal High-Order Tensor Method for Convex Optimization	inproceedings	Proceedings of Machine Learning Research	jiang19a	0	An Optimal High-Order Tensor Method for Convex Optimization	1799	1801	1799-1801	1799	false	Jiang, Bo and Wang, Haoyue and Zhang, Shuzhong	given family Bo Jiang given family Haoyue Wang given family Shuzhong Zhang	2019-06-25		PMLR	Proceedings of the Thirty-Second Conference on Learning Theory	99	inproceedings	date-parts 2019 6 25	http://proceedings.mlr.press/v99/jiang19a/jiang19a.pdf