@article{pearlmutter1994fast,
  title={Fast exact multiplication by the Hessian},
  author={Pearlmutter, Barak A},
  journal={Neural computation},
  volume={6},
  number={1},
  pages={147--160},
  year={1994},
  publisher={MIT Press}
}
@book{saadNumericalMethodsLarge2011,
	address = {Philadelphia},
	edition = {Rev. ed},
	series = {Classics in applied mathematics},
	title = {Numerical methods for large eigenvalue problems},
	isbn = {978-1-61197-072-2},
	number = {66},
	publisher = {Society for Industrial and Applied Mathematics},
	author = {Saad, Y.},
	year = {2011},
	keywords = {Eigenvalues, Nonsymmetric matrices},
	file = {Saad - 2011 - Numerical methods for large eigenvalue problems.pdf:/Users/brent/Zotero/storage/U5E3WWNH/Saad - 2011 - Numerical methods for large eigenvalue problems.pdf:application/pdf},
}

@book{golub_matrix_2013,
	address = {Baltimore},
	edition = {Fourth},
	series = {Johns {Hopkins} studies in the mathematical sciences},
	title = {Matrix computations},
	language = {en},
	publisher = {The Johns Hopkins University Press},
	author = {Golub, Gene H. and Van Loan, Charles F.},
	year = {2013},
	keywords = {Matrices, Data processing},
}


@book{nesterovLecturesConvexOptimization2018,
	title = {Lectures on convex optimization},
	volume = {137},
	publisher = {Springer},
	author = {Nesterov, Yurii},
	year = {2018},
	file = {Nesterov - 2018 - Lectures on convex optimization.pdf:/Users/brent/Zotero/storage/8HX869QC/Nesterov - 2018 - Lectures on convex optimization.pdf:application/pdf;Snapshot:/Users/brent/Zotero/storage/M6WPS6GV/10.html:text/html},
}

@inproceedings{schulman2015trust,
  title={Trust region policy optimization},
  author={Schulman, John and Levine, Sergey and Abbeel, Pieter and Jordan, Michael and Moritz, Philipp},
  booktitle={International conference on machine learning},
  pages={1889--1897},
  year={2015},
  organization={PMLR}
}

@misc{he2023homogeneous,
      title={Homogeneous Second-Order Descent Framework: A Fast Alternative to Newton-Type Methods}, 
      author={Chang He and Yuntian Jiang and Chuwen Zhang and Dongdong Ge and Bo Jiang and Yinyu Ye},
      year={2023},
      eprint={2306.17516},
      archivePrefix={arXiv},
      primaryClass={math.OC}
}

@misc{garage,
 author = {The garage contributors},
 title = {Garage: A toolkit for reproducible reinforcement learning research},
 year = {2019},
 publisher = {GitHub},
 journal = {GitHub repository},
 howpublished = {\url{https://github.com/rlworkgroup/garage}},
 commit = {be070842071f736eb24f28e4b902a9f144f5c97b}
}

@article{paszke2019pytorch,
  title={Pytorch: An imperative style, high-performance deep learning library},
  author={Paszke, Adam and Gross, Sam and Massa, Francisco and Lerer, Adam and Bradbury, James and Chanan, Gregory and Killeen, Trevor and Lin, Zeming and Gimelshein, Natalia and Antiga, Luca and others},
  journal={Advances in neural information processing systems},
  volume={32},
  year={2019}
}

@inproceedings{huang2020momentum,
  title={Momentum-based policy gradient methods},
  author={Huang, Feihu and Gao, Shangqian and Pei, Jian and Huang, Heng},
  booktitle={International conference on machine learning},
  pages={4422--4433},
  year={2020},
  organization={PMLR}
}

@article{williams1992simple,
  title={Simple statistical gradient-following algorithms for connectionist reinforcement learning},
  author={Williams, Ronald J},
  journal={Machine learning},
  volume={8},
  pages={229--256},
  year={1992},
  publisher={Springer}
}

@article{schulman2017proximal,
  title={Proximal policy optimization algorithms},
  author={Schulman, John and Wolski, Filip and Dhariwal, Prafulla and Radford, Alec and Klimov, Oleg},
  journal={arXiv preprint arXiv:1707.06347},
  year={2017}
}

@article{kuczynski_estimating_1992,
  title    = {Estimating the {Largest} {Eigenvalue} by the {Power} and {Lanczos} {Algorithms} with a {Random} {Start}},
  volume   = {13},
  language = {en},
  number   = {4},
  urldate  = {2022-10-13},
  journal  = {SIAM Journal on Matrix Analysis and Applications},
  author   = {Kuczyński, J. and Woźniakowski, H.},
  month    = oct,
  year     = {1992},
  pages    = {1094--1122},
}

@article{carmon_accelerated_2018,
  title    = {Accelerated {Methods} for {NonConvex} {Optimization}},
  volume   = {28},
  issn     = {1052-6234, 1095-7189},
  url      = {https://epubs.siam.org/doi/10.1137/17M1114296},
  doi      = {10.1137/17M1114296},
  abstract = {We present an accelerated gradient method for nonconvex optimization problems with Lipschitz continuous ﬁrst and second derivatives. In a time O( −7/4 log(1/ )), the method ﬁnds an -stationary point, meaning a point x such that ∇f (x) ≤ . The method improves upon the O( −2) complexity of gradient descent and provides the additional second-order guarantee that λmin(∇2f (x)) − 1/2 for the computed x. Furthermore, our method is Hessian free, i.e., it only requires gradient computations, and is therefore suitable for large-scale applications.},
  language = {en},
  number   = {2},
  urldate  = {2022-07-05},
  journal  = {SIAM Journal on Optimization},
  author   = {Carmon, Yair and Duchi, John C. and Hinder, Oliver and Sidford, Aaron},
  month    = jan,
  year     = {2018},
  pages    = {1751--1772},
  file     = {Carmon et al. - 2018 - Accelerated Methods for NonConvex Optimization.pdf:/Users/brent/Zotero/storage/7PHAG97H/Carmon et al. - 2018 - Accelerated Methods for NonConvex Optimization.pdf:application/pdf;Full Text:/Users/brent/Zotero/storage/2X6MKWKV/Carmon et al. - 2018 - Accelerated methods for nonconvex optimization.pdf:application/pdf;Snapshot:/Users/brent/Zotero/storage/PPCR5859/17M1114296.html:text/html}
}
@article{pardalos_quadratic_1991,
	title = {Quadratic programming with one negative eigenvalue is {NP}-hard},
	volume = {1},
	language = {en},
	number = {1},
	urldate = {2023-07-31},
	journal = {Journal of Global Optimization},
	author = {Pardalos, Panos M. and Vavasis, Stephen A.},
	month = mar,
	year = {1991},
	keywords = {Global optimization, NP-hard, quadratic programming},
	pages = {15--22},
}

@article{agarwal2021theory,
  title={On the Theory of Policy Gradient Methods: Optimality, Approximation, and Distribution Shift.},
  author={Agarwal, Alekh and Kakade, Sham M and Lee, Jason D and Mahajan, Gaurav},
  journal={J. Mach. Learn. Res.},
  volume={22},
  number={98},
  pages={1--76},
  year={2021}
}
@article{liu2020improved,
  title={An improved analysis of (variance-reduced) policy gradient and natural policy gradient methods},
  author={Liu, Yanli and Zhang, Kaiqing and Basar, Tamer and Yin, Wotao},
  journal={Advances in Neural Information Processing Systems},
  volume={33},
  pages={7624--7636},
  year={2020}
}
@article{fang2018spider,
  title={Spider: Near-optimal non-convex optimization via stochastic path-integrated differential estimator},
  author={Fang, Cong and Li, Chris Junchi and Lin, Zhouchen and Zhang, Tong},
  journal={Advances in neural information processing systems},
  volume={31},
  year={2018}
}

@inproceedings{kohler2017sub,
  title={Sub-sampled cubic regularization for non-convex optimization},
  author={Kohler, Jonas Moritz and Lucchi, Aurelien},
  booktitle={International Conference on Machine Learning},
  pages={1895--1904},
  year={2017},
  organization={PMLR}
}
@article{tripuraneni2018stochastic,
  title={Stochastic cubic regularization for fast nonconvex optimization},
  author={Tripuraneni, Nilesh and Stern, Mitchell and Jin, Chi and Regier, Jeffrey and Jordan, Michael I},
  journal={Advances in neural information processing systems},
  volume={31},
  year={2018}
}

@article{cartis_adaptive_2011,
	title = {Adaptive cubic regularisation methods for unconstrained optimization. {Part} {I}: motivation, convergence and numerical results},
	volume = {127},
	issn = {0025-5610, 1436-4646},
	shorttitle = {Adaptive cubic regularisation methods for unconstrained optimization. {Part} {I}},
	url = {http://link.springer.com/10.1007/s10107-009-0286-5},
	doi = {10.1007/s10107-009-0286-5},
	abstract = {An Adaptive Regularisation framework using Cubics (ARC) was proposed for unconstrained optimization and analysed in Cartis, Gould \& Toint (Part I, 2007). In this companion paper, we further the analysis by providing worst-case global iteration complexity bounds for ARC and a second-order variant to achieve approximate first-order, and for the latter even second-order, criticality of the iterates. In particular, the second-order ARC algorithm requires at most O(ǫ−3/2) iterations to drive the objective’s gradient below the desired accuracy ǫ, and O(ǫ−3), to reach approximate nonnegative curvature in a subspace. The orders of these bounds match those proved by Nesterov \& Polyak (Math. Programming 108(1), 2006, pp 177-205) for their Algorithm 3.3 which minimizes the cubic model globally on each iteration. Our approach is more general, and relevant to practical (large-scale) calculations, as ARC allows the cubic model to be solved only approximately and may employ approximate Hessians.},
	language = {en},
	number = {2},
	urldate = {2023-05-16},
	journal = {Mathematical Programming},
	author = {Cartis, Coralia and Gould, Nicholas I. M. and Toint, Philippe L.},
	month = apr,
	year = {2011},
	note = {150 citations (Crossref) [2023-05-16]},
	pages = {245--295},
	file = {Cartis et al_2011_Adaptive cubic regularisation methods for unconstrained optimization.pdf:C\:\\Users\\Tank\\Zotero\\storage\\ZYY7XYLX\\Cartis et al_2011_Adaptive cubic regularisation methods for unconstrained optimization.pdf:application/pdf},
}

@misc{song_inexact_2019,
	title = {Inexact {Proximal} {Cubic} {Regularized} {Newton} {Methods} for {Convex} {Optimization}},
	url = {http://arxiv.org/abs/1902.02388},
	abstract = {In this paper, we use Proximal Cubic regularized Newton Methods (PCNM) to optimize the sum of a smooth convex function and a non-smooth convex function, where we use inexact gradient and Hessian, and an inexact subsolver for the cubic regularized second-order subproblem. We propose inexact variants of PCNM and accelerated PCNM respectively, and show that both variants can achieve the same convergence rate as in the exact case, provided that the errors in the inexact gradient, Hessian and subsolver decrease at appropriate rates. Meanwhile, in the online stochastic setting where data comes endlessly, we give the overall complexity of the proposed algorithms and show that they are as competitive as the stochastic gradient descent. Moreover, we give the overall complexity of the proposed algorithms in the finite-sum setting and show that it is as competitive as the state of the art variance reduced algorithms. Finally, we propose an efficient algorithm for the cubic regularized second-order subproblem, which can converge to an enough small neighborhood of the optimal solution in a superlinear rate.},
	urldate = {2023-05-16},
	publisher = {arXiv},
	author = {Song, Chaobing and Liu, Ji and Jiang, Yong},
	month = feb,
	year = {2019},
	note = {arXiv:1902.02388 [math]},
	keywords = {Mathematics - Optimization and Control, /unread},
	file = {Song et al_2019_Inexact Proximal Cubic Regularized Newton Methods for Convex Optimization.pdf:C\:\\Users\\Tank\\Zotero\\storage\\LNNAEF33\\Song et al_2019_Inexact Proximal Cubic Regularized Newton Methods for Convex Optimization.pdf:application/pdf},
}

@article{zhang_adaptive_2022,
	title = {Adaptive {Stochastic} {Variance} {Reduction} for {Subsampled} {Newton} {Method} with {Cubic} {Regularization}},
	volume = {4},
	issn = {2575-1484, 2575-1492},
	url = {https://pubsonline.informs.org/doi/10.1287/ijoo.2021.0058},
	doi = {10.1287/ijoo.2021.0058},
	abstract = {The cubic regularized Newton method of Nesterov and Polyak has become increasingly popular for nonconvex optimization because of its capability of finding an approximate local solution with a second order guarantee and its low iteration complexity. Several recent works extend this method to the setting of minimizing the average of N smooth functions by replacing the exact gradients and Hessians with subsampled approximations. It is shown that the total Hessian sample complexity can be reduced to be sublinear in N per iteration by leveraging stochastic variance reduction techniques. We present an adaptive variance reduction scheme for a subsampled Newton method with cubic regularization and show that the expected Hessian sample complexity is [Formula: see text] for finding an [Formula: see text]-approximate local solution (in terms of first and second order guarantees, respectively). Moreover, we show that the same Hessian sample complexity is retained with fixed sample sizes if exact gradients are used. The techniques of our analysis are different from previous works in that we do not rely on high probability bounds based on matrix concentration inequalities. Instead, we derive and utilize new bounds on the third and fourth order moments of the average of random matrices, which are of independent interest on their own.},
	language = {en},
	number = {1},
	urldate = {2023-05-16},
	journal = {INFORMS Journal on Optimization},
	author = {Zhang, Junyu and Xiao, Lin and Zhang, Shuzhong},
	month = jan,
	year = {2022},
	note = {0 citations (Crossref) [2023-05-16]},
	keywords = {/unread},
	pages = {45--64},
	file = {Zhang et al_2022_Adaptive Stochastic Variance Reduction for Subsampled Newton Method with Cubic.pdf:C\:\\Users\\Tank\\Zotero\\storage\\DZHKLZCV\\Zhang et al_2022_Adaptive Stochastic Variance Reduction for Subsampled Newton Method with Cubic.pdf:application/pdf},
}

@misc{zhou_stochastic_2018,
	title = {Stochastic {Variance}-{Reduced} {Cubic} {Regularized} {Newton} {Method}},
	url = {http://arxiv.org/abs/1802.04796},
	abstract = {We propose a stochastic variance-reduced cubic regularized Newton method for non-convex optimization. At the core of our algorithm is a novel semi-stochastic gradient along with a semi-stochastic Hessian, which are speciﬁcally designed for√cubic regularization method. We show that our algorithm is guaranteed to converge to an ( , )-approximately local minimum within O(n4/5/ 3/2) second-order oracle calls, which outperforms the state-of-the-art cubic regularization algorithms including subsampled cubic regularization. Our work also sheds light on the application of variance reduction technique to high-order non-convex optimization methods. Thorough experiments on various non-convex optimization problems support our theory.},
	language = {en},
	urldate = {2023-05-16},
	publisher = {arXiv},
	author = {Zhou, Dongruo and Xu, Pan and Gu, Quanquan},
	month = feb,
	year = {2018},
	note = {22 citations (Semantic Scholar/arXiv) [2023-05-16]
arXiv:1802.04796 [cs, math]},
	keywords = {Computer Science - Machine Learning, Mathematics - Optimization and Control},
	file = {Zhou et al. - 2018 - Stochastic Variance-Reduced Cubic Regularized Newt.pdf:C\:\\Users\\Tank\\Zotero\\storage\\5LP4QJ27\\Zhou et al. - 2018 - Stochastic Variance-Reduced Cubic Regularized Newt.pdf:application/pdf},
}

@misc{kohler_sub-sampled_2017,
	title = {Sub-sampled {Cubic} {Regularization} for {Non}-convex {Optimization}},
	url = {http://arxiv.org/abs/1705.05933},
	abstract = {We consider the minimization of non-convex functions that typically arise in machine learning. Speciﬁcally, we focus our attention on a variant of trust region methods known as cubic regularization. This approach is particularly attractive because it escapes strict saddle points and it provides stronger convergence guarantees than ﬁrst- and second-order as well as classical trust region methods. However, it suffers from a high computational complexity that makes it impractical for large-scale learning. Here, we propose a novel method that uses sub-sampling to lower this computational cost. By the use of concentration inequalities we provide a sampling scheme that gives sufﬁciently accurate gradient and Hessian approximations to retain the strong global and local convergence guarantees of cubically regularized methods. To the best of our knowledge this is the ﬁrst work that gives global convergence guarantees for a sub-sampled variant of cubic regularization on non-convex functions. Furthermore, we provide experimental results supporting our theory.},
	language = {en},
	urldate = {2023-05-16},
	publisher = {arXiv},
	author = {Kohler, Jonas Moritz and Lucchi, Aurelien},
	month = jul,
	year = {2017},
	note = {114 citations (Semantic Scholar/arXiv) [2023-05-16]
arXiv:1705.05933 [cs, math, stat]},
	keywords = {Computer Science - Machine Learning, Mathematics - Optimization and Control, Statistics - Machine Learning},
	file = {Kohler and Lucchi - 2017 - Sub-sampled Cubic Regularization for Non-convex Op.pdf:C\:\\Users\\Tank\\Zotero\\storage\\M2EX2QGS\\Kohler and Lucchi - 2017 - Sub-sampled Cubic Regularization for Non-convex Op.pdf:application/pdf},
}

@misc{wang_stochastic_2018,
	title = {Stochastic {Variance}-{Reduced} {Cubic} {Regularization} for {Nonconvex} {Optimization}},
	url = {http://arxiv.org/abs/1802.07372},
	abstract = {Cubic regularization (CR) is an optimization method with emerging popularity due to its capability to escape saddle points and converge to second-order stationary solutions for nonconvex optimization. However, CR encounters a high sample complexity issue for ﬁnite-sum problems with a large data size. In this paper, we propose a stochastic variance-reduced cubicregularization (SVRC) method under random sampling, and study its convergence guarantee as well as sample complexity. We show that the iteration complexity of SVRC for achieving a second-order stationary solution within ǫ accuracy is O(ǫ−3/2), which matches the state-of-art result on CR types of methods. Moreover, our proposed variance reduction scheme signiﬁcantly reduces the per-iteration sample complexity. The resulting total Hessian sample complexity of our SVRC is O(N 2/3ǫ−3/2), which outperforms the state-of-art result by a factor of O(N 2/15). We also study our SVRC under random sampling without replacement scheme, which yields a lower per-iteration sample complexity, and hence justiﬁes its practical applicability.},
	language = {en},
	urldate = {2023-05-16},
	publisher = {arXiv},
	author = {Wang, Zhe and Zhou, Yi and Liang, Yingbin and Lan, Guanghui},
	month = oct,
	year = {2018},
	note = {40 citations (Semantic Scholar/arXiv) [2023-05-16]
arXiv:1802.07372 [cs, math, stat]},
	keywords = {Computer Science - Machine Learning, Mathematics - Optimization and Control, /unread, Statistics - Machine Learning},
	file = {Wang et al. - 2018 - Stochastic Variance-Reduced Cubic Regularization f.pdf:C\:\\Users\\Tank\\Zotero\\storage\\KIX5BUH2\\Wang et al. - 2018 - Stochastic Variance-Reduced Cubic Regularization f.pdf:application/pdf},
}

@article{tripuraneni_stochastic_nodate,
	title = {Stochastic {Cubic} {Regularization} for {Fast} {Nonconvex} {Optimization}},
	abstract = {This paper proposes a stochastic variant of a classic algorithm—the cubicregularized Newton method [Nesterov and Polyak, 2006]. The proposed algorithm efﬁciently escapes saddle points and ﬁnds approximate local minima for general smooth, nonconvex functions in only O˜(✏ 3.5) stochastic gradient and stochastic Hessian-vector product evaluations. The latter can be computed as efﬁciently as stochastic gradients. This improves upon the O˜(✏ 4) rate of stochastic gradient descent. Our rate matches the best-known result for ﬁnding local minima without requiring any delicate acceleration or variance-reduction techniques.},
	language = {en},
	author = {Tripuraneni, Nilesh and Stern, Mitchell and Jin, Chi and Regier, Jeffrey and Jordan, Michael I},
	keywords = {/unread},
	file = {Tripuraneni et al. - Stochastic Cubic Regularization for Fast Nonconvex.pdf:C\:\\Users\\Tank\\Zotero\\storage\\VCSS2HW4\\Tripuraneni et al. - Stochastic Cubic Regularization for Fast Nonconvex.pdf:application/pdf},
}


@inproceedings{arjevani_second-order_2020,
  title={Second-order information in non-convex stochastic optimization: Power and limitations},
  author={Arjevani, Yossi and Carmon, Yair and Duchi, John C and Foster, Dylan J and Sekhari, Ayush and Sridharan, Karthik},
  booktitle={Conference on Learning Theory},
  pages={242--299},
  year={2020},
  organization={PMLR}
}

@misc{zhou_stochastic_2019,
	title = {Stochastic {Recursive} {Variance}-{Reduced} {Cubic} {Regularization} {Methods}},
	url = {http://arxiv.org/abs/1901.11518},
	abstract = {Stochastic Variance-Reduced Cubic regularization (SVRC) algorithms have received increasing attention due to its improved gradient/Hessian complexities (i.e., number of queries to stochastic gradient/Hessian oracles) to find local minima for nonconvex finite-sum optimization. However, it is unclear whether existing SVRC algorithms can be further improved. Moreover, the semi-stochastic Hessian estimator adopted in existing SVRC algorithms prevents the use of Hessian-vector product-based fast cubic subproblem solvers, which makes SVRC algorithms computationally intractable for high-dimensional problems. In this paper, we first present a Stochastic Recursive Variance-Reduced Cubic regularization method (SRVRC) using a recursively updated semi-stochastic gradient and Hessian estimators. It enjoys improved gradient and Hessian complexities to find an \$({\textbackslash}epsilon, {\textbackslash}sqrt\{{\textbackslash}epsilon\})\$-approximate local minimum, and outperforms the state-of-the-art SVRC algorithms. Built upon SRVRC, we further propose a Hessian-free SRVRC algorithm, namely SRVRC\$\_\{{\textbackslash}text\{free\}\}\$, which only requires stochastic gradient and Hessian-vector product computations, and achieves \${\textbackslash}tilde O(dn{\textbackslash}epsilon{\textasciicircum}\{-2\} {\textbackslash}land d{\textbackslash}epsilon{\textasciicircum}\{-3\})\$ runtime complexity, where \$n\$ is the number of component functions in the finite-sum structure, \$d\$ is the problem dimension, and \${\textbackslash}epsilon\$ is the optimization precision. This outperforms the best-known runtime complexity \${\textbackslash}tilde O(d{\textbackslash}epsilon{\textasciicircum}\{-3.5\})\$ achieved by stochastic cubic regularization algorithm proposed in Tripuraneni et al. 2018.},
	urldate = {2023-05-16},
	publisher = {arXiv},
	author = {Zhou, Dongruo and Gu, Quanquan},
	month = oct,
	year = {2019},
	note = {arXiv:1901.11518 [cs, math, stat]},
	keywords = {Computer Science - Machine Learning, Mathematics - Optimization and Control, Statistics - Machine Learning},
	file = {arXiv.org Snapshot:C\:\\Users\\Tank\\Zotero\\storage\\EFF7L6W3\\1901.html:text/html;Zhou_Gu_2019_Stochastic Recursive Variance-Reduced Cubic Regularization Methods.pdf:C\:\\Users\\Tank\\Zotero\\storage\\DGW5WJ2V\\Zhou_Gu_2019_Stochastic Recursive Variance-Reduced Cubic Regularization Methods.pdf:application/pdf},
}

@misc{xu_newton-type_2019,
	title = {Newton-{Type} {Methods} for {Non}-{Convex} {Optimization} {Under} {Inexact} {Hessian} {Information}},
	url = {http://arxiv.org/abs/1708.07164},
	abstract = {We consider variants of trust-region and cubic regularization methods for non-convex optimization, in which the Hessian matrix is approximated. Under mild conditions on the inexact Hessian, and using approximate solution of the corresponding sub-problems, we provide iteration complexity to achieve \$ {\textbackslash}epsilon \$-approximate second-order optimality which have shown to be tight. Our Hessian approximation conditions constitute a major relaxation over the existing ones in the literature. Consequently, we are able to show that such mild conditions allow for the construction of the approximate Hessian through various random sampling methods. In this light, we consider the canonical problem of finite-sum minimization, provide appropriate uniform and non-uniform sub-sampling strategies to construct such Hessian approximations, and obtain optimal iteration complexity for the corresponding sub-sampled trust-region and cubic regularization methods.},
	urldate = {2023-05-16},
	publisher = {arXiv},
	author = {Xu, Peng and Roosta, Fred and Mahoney, Michael W.},
	month = may,
	year = {2019},
	note = {160 citations (Semantic Scholar/arXiv) [2023-05-16]
arXiv:1708.07164 [cs, math, stat]},
	keywords = {Computer Science - Machine Learning, Mathematics - Optimization and Control, /unread, Statistics - Machine Learning, Computer Science - Computational Complexity},
}

@misc{shen_stochastic_2019,
	title = {A {Stochastic} {Trust} {Region} {Method} for {Non}-convex {Minimization}},
	url = {http://arxiv.org/abs/1903.01540},
	abstract = {We target the problem of finding a local minimum in non-convex finite-sum minimization. Towards this goal, we first prove that the trust region method with inexact gradient and Hessian estimation can achieve a convergence rate of order \${\textbackslash}mathcal\{O\}(1/\{k{\textasciicircum}\{2/3\}\})\$ as long as those differential estimations are sufficiently accurate. Combining such result with a novel Hessian estimator, we propose the sample-efficient stochastic trust region (STR) algorithm which finds an \$({\textbackslash}epsilon, {\textbackslash}sqrt\{{\textbackslash}epsilon\})\$-approximate local minimum within \${\textbackslash}mathcal\{O\}(\{{\textbackslash}sqrt\{n\}\}/\{{\textbackslash}epsilon{\textasciicircum}\{1.5\}\})\$ stochastic Hessian oracle queries. This improves state-of-the-art result by \${\textbackslash}mathcal\{O\}(n{\textasciicircum}\{1/6\})\$. Experiments verify theoretical conclusions and the efficiency of STR.},
	urldate = {2023-05-16},
	publisher = {arXiv},
	author = {Shen, Zebang and Zhou, Pan and Fang, Cong and Ribeiro, Alejandro},
	month = mar,
	year = {2019},
	note = {8 citations (Semantic Scholar/arXiv) [2023-05-16]
arXiv:1903.01540 [math, stat]},
	keywords = {Mathematics - Optimization and Control, /unread, Statistics - Machine Learning},
}

@article{nesterov_accelerating_2007,
	title = {Accelerating the cubic regularization of {Newton}’s method on convex problems},
	volume = {112},
	issn = {0025-5610, 1436-4646},
	url = {http://link.springer.com/10.1007/s10107-006-0089-x},
	doi = {10.1007/s10107-006-0089-x},
	language = {en},
	number = {1},
	urldate = {2023-05-16},
	journal = {Mathematical Programming},
	author = {Nesterov, Yu.},
	month = jul,
	year = {2007},
	note = {227 citations (Semantic Scholar/DOI) [2023-05-16]},
	keywords = {/unread},
	pages = {159--181},
	file = {Accelerating the cubic regularization of Newton’s method on convex problems.pdf:C\:\\Users\\Tank\\Zotero\\storage\\DXKT7ZGS\\Accelerating the cubic regularization of Newton’s method on convex problems.pdf:application/pdf},
}

@misc{kamzolov_accelerated_2023,
	title = {Accelerated {Adaptive} {Cubic} {Regularized} {Quasi}-{Newton} {Methods}},
	url = {http://arxiv.org/abs/2302.04987},
	doi = {10.48550/arXiv.2302.04987},
	abstract = {In this paper, we propose Cubic Regularized Quasi-Newton Methods for (strongly) star-convex and Accelerated Cubic Regularized Quasi-Newton for convex optimization. The proposed class of algorithms combines the global convergence of the Cubic Newton with the affordability of constructing the Hessian approximation via a Quasi-Newton update. To construct these methods we introduce two novel concepts of Hessian inexactness and propose the Adaptive Inexact Cubic Regularized Newton algorithm and its accelerated version, which adjust to the approximation error. We show that different Quasi-Newton updates satisfy these approximation conditions and provide a complexity-efficient way to solve the cubic subproblem. We provide global convergence rates with explicit dependence on Hessian approximation error. By combining cubic regularization, acceleration technique, and adaptivity, our algorithms show good practical performance, especially from the points where classical Newton diverges.},
	urldate = {2023-05-16},
	publisher = {arXiv},
	author = {Kamzolov, Dmitry and Ziu, Klea and Agafonov, Artem and Takáč, Martin},
	month = feb,
	year = {2023},
	note = {0 citations (Semantic Scholar/arXiv) [2023-05-16]
arXiv:2302.04987 [math]},
	keywords = {Mathematics - Optimization and Control},
	file = {arXiv.org Snapshot:C\:\\Users\\Tank\\Zotero\\storage\\24QHAXUA\\2302.html:text/html;Kamzolov et al_2023_Accelerated Adaptive Cubic Regularized Quasi-Newton Methods.pdf:C\:\\Users\\Tank\\Zotero\\storage\\R3BPDQCS\\Kamzolov et al_2023_Accelerated Adaptive Cubic Regularized Quasi-Newton Methods.pdf:application/pdf},
}

@article{agarwal_second-order_2017,
	title = {Second-{Order} {Stochastic} {Optimization} for {Machine} {Learning} in {Linear} {Time}},
	abstract = {First-order stochastic methods are the state-of-the-art in large-scale machine learning optimization owing to eﬃcient per-iteration complexity. Second-order methods, while able to provide faster convergence, have been much less explored due to the high cost of computing the second-order information. In this paper we develop second-order stochastic methods for optimization problems in machine learning that match the per-iteration cost of gradient based methods, and in certain settings improve upon the overall running time over popular ﬁrst-order methods. Furthermore, our algorithm has the desirable property of being implementable in time linear in the sparsity of the input data.},
	language = {en},
	author = {Agarwal, Naman and Bullins, Brian and Hazan, Elad},
	month = nov,
	year = {2017},
	file = {Agarwal et al. - Second-Order Stochastic Optimization for Machine L.pdf:C\:\\Users\\Tank\\Zotero\\storage\\KE38AF8T\\Agarwal et al. - Second-Order Stochastic Optimization for Machine L.pdf:application/pdf},
}

@article{byrd_stochastic_2016,
	title = {A {Stochastic} {Quasi}-{Newton} {Method} for {Large}-{Scale} {Optimization}},
	volume = {26},
	issn = {1052-6234, 1095-7189},
	url = {http://epubs.siam.org/doi/10.1137/140954362},
	doi = {10.1137/140954362},
	language = {en},
	number = {2},
	urldate = {2023-05-20},
	journal = {SIAM Journal on Optimization},
	author = {Byrd, R. H. and Hansen, S. L. and Nocedal, Jorge and Singer, Y.},
	month = jan,
	year = {2016},
	keywords = {/unread},
	pages = {1008--1031},
	file = {Byrd et al_2016_A Stochastic Quasi-Newton Method for Large-Scale Optimization.pdf:C\:\\Users\\Tank\\Zotero\\storage\\TT96KS8D\\Byrd et al_2016_A Stochastic Quasi-Newton Method for Large-Scale Optimization.pdf:application/pdf},
}
@misc{zhangHomogeneousSecondOrderDescent2022,
  title = {A Homogeneous Second-Order Descent Method for Nonconvex Optimization},
  author = {Zhang, Chuwen and Ge, Dongdong and He, Chang and Jiang, Bo and Jiang, Yuntian and Xue, Chenyu and Ye, Yinyu},
  year = {2022},
  eprint = {2211.08212},
  archivePrefix = {arxiv},
}

@misc{moritz_linearly-convergent_2016,
	title = {A {Linearly}-{Convergent} {Stochastic} {L}-{BFGS} {Algorithm}},
	url = {http://arxiv.org/abs/1508.02087},
	abstract = {We propose a new stochastic L-BFGS algorithm and prove a linear convergence rate for strongly convex and smooth functions. Our algorithm draws heavily from a recent stochastic variant of L-BFGS proposed in Byrd et al. (2014) as well as a recent approach to variance reduction for stochastic gradient descent from Johnson and Zhang (2013). We demonstrate experimentally that our algorithm performs well on large-scale convex and non-convex optimization problems, exhibiting linear convergence and rapidly solving the optimization problems to high levels of precision. Furthermore, we show that our algorithm performs well for a wide-range of step sizes, often differing by several orders of magnitude.},
	urldate = {2023-05-20},
	publisher = {arXiv},
	author = {Moritz, Philipp and Nishihara, Robert and Jordan, Michael I.},
	month = apr,
	year = {2016},
	note = {arXiv:1508.02087 [cs, math, stat]},
	keywords = {Computer Science - Machine Learning, Mathematics - Optimization and Control, Statistics - Machine Learning, Statistics - Computation, Mathematics - Numerical Analysis},
	file = {arXiv.org Snapshot:C\:\\Users\\Tank\\Zotero\\storage\\5L89G3DK\\1508.html:text/html;Moritz et al_2016_A Linearly-Convergent Stochastic L-BFGS Algorithm.pdf:C\:\\Users\\Tank\\Zotero\\storage\\E3D3L9GJ\\Moritz et al_2016_A Linearly-Convergent Stochastic L-BFGS Algorithm.pdf:application/pdf},
}

@article{mokhtari_res_2014,
	title = {{RES}: {Regularized} {Stochastic} {BFGS} {Algorithm}},
	volume = {62},
	issn = {1053-587X, 1941-0476},
	shorttitle = {{RES}},
	url = {http://arxiv.org/abs/1401.7625},
	doi = {10.1109/TSP.2014.2357775},
	abstract = {RES, a regularized stochastic version of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method is proposed to solve convex optimization problems with stochastic objectives. The use of stochastic gradient descent algorithms is widespread, but the number of iterations required to approximate optimal arguments can be prohibitive in high dimensional problems. Application of second order methods, on the other hand, is impracticable because computation of objective function Hessian inverses incurs excessive computational cost. BFGS modifies gradient descent by introducing a Hessian approximation matrix computed from finite gradient differences. RES utilizes stochastic gradients in lieu of deterministic gradients for both, the determination of descent directions and the approximation of the objective function's curvature. Since stochastic gradients can be computed at manageable computational cost RES is realizable and retains the convergence rate advantages of its deterministic counterparts. Convergence results show that lower and upper bounds on the Hessian egeinvalues of the sample functions are sufficient to guarantee convergence to optimal arguments. Numerical experiments showcase reductions in convergence time relative to stochastic gradient descent algorithms and non-regularized stochastic versions of BFGS. An application of RES to the implementation of support vector machines is developed.},
	number = {23},
	urldate = {2023-05-20},
	journal = {IEEE Transactions on Signal Processing},
	author = {Mokhtari, Aryan and Ribeiro, Alejandro},
	month = dec,
	year = {2014},
	note = {arXiv:1401.7625 [cs, math, stat]},
	keywords = {Computer Science - Machine Learning, Mathematics - Optimization and Control, /unread, Statistics - Machine Learning},
	pages = {6089--6104},
	file = {arXiv.org Snapshot:C\:\\Users\\Tank\\Zotero\\storage\\PJ2IBRZK\\1401.html:text/html;Mokhtari_Ribeiro_2014_RES.pdf:C\:\\Users\\Tank\\Zotero\\storage\\LPD8CEN4\\Mokhtari_Ribeiro_2014_RES.pdf:application/pdf},
}

@inproceedings{erdogdu_convergence_2015,
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	volume = {28},
	url = {https://proceedings.neurips.cc/paper_files/paper/2015/file/404dcc91b2aeaa7caa47487d1483e48a-Paper.pdf},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {Curran Associates, Inc.},
	author = {Erdogdu, Murat A and Montanari, Andrea},
	editor = {Cortes, C. and Lawrence, N. and Lee, D. and Sugiyama, M. and Garnett, R.},
	year = {2015},
	file = {Erdogdu_Montanari_2015_Convergence rates of sub-sampled Newton methods.pdf:C\:\\Users\\Tank\\Zotero\\storage\\X9I898TE\\Erdogdu_Montanari_2015_Convergence rates of sub-sampled Newton methods.pdf:application/pdf},
}

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  title={A Homogenous Second-Order Descent Method for Nonconvex Optimization},
  author={Zhang, Chuwen and Ge, Dongdong and He, Chang and Jiang, Bo and Jiang, Yuntian and Xue, Chenyu and Ye, Yinyu},
  journal={arXiv preprint arXiv:2211.08212},
  year={2022}
}

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