Efficient approaches for escaping higher order saddle points in non-convex optimization

Animashree Anandkumar, Rong Ge
29th Annual Conference on Learning Theory, PMLR 49:81-102, 2016.

Abstract

Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a \em local minimum, due to the existence of complicated saddle point structures in high dimensions. Many functions have \em degenerate saddle points such that the first and second order derivatives cannot distinguish them with local optima. In this paper we use higher order derivatives to escape these saddle points: we design the first efficient algorithm guaranteed to converge to a third order local optimum (while existing techniques are at most second order). We also show that it is NP-hard to extend this further to finding fourth order local optima.

Cite this Paper

BibTeX
@InProceedings{pmlr-v49-anandkumar16, title = {Efficient approaches for escaping higher order saddle points in non-convex optimization}, author = {Anandkumar, Animashree and Ge, Rong}, booktitle = {29th Annual Conference on Learning Theory}, pages = {81--102}, year = {2016}, editor = {Feldman, Vitaly and Rakhlin, Alexander and Shamir, Ohad}, volume = {49}, series = {Proceedings of Machine Learning Research}, address = {Columbia University, New York, New York, USA}, month = {23--26 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v49/anandkumar16.pdf}, url = {https://proceedings.mlr.press/v49/anandkumar16.html}, abstract = {Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a \em local minimum, due to the existence of complicated saddle point structures in high dimensions. Many functions have \em degenerate saddle points such that the first and second order derivatives cannot distinguish them with local optima. In this paper we use higher order derivatives to escape these saddle points: we design the first efficient algorithm guaranteed to converge to a third order local optimum (while existing techniques are at most second order). We also show that it is NP-hard to extend this further to finding fourth order local optima.} }
Endnote
%0 Conference Paper %T Efficient approaches for escaping higher order saddle points in non-convex optimization %A Animashree Anandkumar %A Rong Ge %B 29th Annual Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2016 %E Vitaly Feldman %E Alexander Rakhlin %E Ohad Shamir %F pmlr-v49-anandkumar16 %I PMLR %P 81--102 %U https://proceedings.mlr.press/v49/anandkumar16.html %V 49 %X Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a \em local minimum, due to the existence of complicated saddle point structures in high dimensions. Many functions have \em degenerate saddle points such that the first and second order derivatives cannot distinguish them with local optima. In this paper we use higher order derivatives to escape these saddle points: we design the first efficient algorithm guaranteed to converge to a third order local optimum (while existing techniques are at most second order). We also show that it is NP-hard to extend this further to finding fourth order local optima.
RIS
TY - CPAPER TI - Efficient approaches for escaping higher order saddle points in non-convex optimization AU - Animashree Anandkumar AU - Rong Ge BT - 29th Annual Conference on Learning Theory DA - 2016/06/06 ED - Vitaly Feldman ED - Alexander Rakhlin ED - Ohad Shamir ID - pmlr-v49-anandkumar16 PB - PMLR DP - Proceedings of Machine Learning Research VL - 49 SP - 81 EP - 102 L1 - http://proceedings.mlr.press/v49/anandkumar16.pdf UR - https://proceedings.mlr.press/v49/anandkumar16.html AB - Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a \em local minimum, due to the existence of complicated saddle point structures in high dimensions. Many functions have \em degenerate saddle points such that the first and second order derivatives cannot distinguish them with local optima. In this paper we use higher order derivatives to escape these saddle points: we design the first efficient algorithm guaranteed to converge to a third order local optimum (while existing techniques are at most second order). We also show that it is NP-hard to extend this further to finding fourth order local optima. ER -
APA
Anandkumar, A. & Ge, R.. (2016). Efficient approaches for escaping higher order saddle points in non-convex optimization. 29th Annual Conference on Learning Theory, in Proceedings of Machine Learning Research 49:81-102 Available from https://proceedings.mlr.press/v49/anandkumar16.html.

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