Absence of spurious solutions far from ground truth: A low-rank analysis with high-order losses

Ziye Ma, Ying Chen, Javad Lavaei, Somayeh Sojoudi
Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR 238:1603-1611, 2024.

Abstract

Matrix sensing problems exhibit pervasive non-convexity, plaguing optimization with a proliferation of suboptimal spurious solutions. Avoiding convergence to these critical points poses a major challenge. This work provides new theoretical insights that help demystify the intricacies of the non-convex landscape. In this work, we prove that under certain conditions, critical points sufficiently distant from the ground truth matrix exhibit favorable geometry by being strict saddle points rather than troublesome local minima. Moreover, we introduce the notion of higher-order losses for the matrix sensing problem and show that the incorporation of such losses into the objective function amplifies the negative curvature around those distant critical points. This implies that increasing the complexity of the objective function via high-order losses accelerates the escape from such critical points and acts as a desirable alternative to increasing the complexity of the optimization problem via over-parametrization. By elucidating key characteristics of the non-convex optimization landscape, this work makes progress towards a comprehensive framework for tackling broader machine learning objectives plagued by non-convexity.

Cite this Paper

BibTeX
@InProceedings{pmlr-v238-ma24a, title = { Absence of spurious solutions far from ground truth: A low-rank analysis with high-order losses }, author = {Ma, Ziye and Chen, Ying and Lavaei, Javad and Sojoudi, Somayeh}, booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics}, pages = {1603--1611}, year = {2024}, editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen}, volume = {238}, series = {Proceedings of Machine Learning Research}, month = {02--04 May}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v238/ma24a/ma24a.pdf}, url = {https://proceedings.mlr.press/v238/ma24a.html}, abstract = { Matrix sensing problems exhibit pervasive non-convexity, plaguing optimization with a proliferation of suboptimal spurious solutions. Avoiding convergence to these critical points poses a major challenge. This work provides new theoretical insights that help demystify the intricacies of the non-convex landscape. In this work, we prove that under certain conditions, critical points sufficiently distant from the ground truth matrix exhibit favorable geometry by being strict saddle points rather than troublesome local minima. Moreover, we introduce the notion of higher-order losses for the matrix sensing problem and show that the incorporation of such losses into the objective function amplifies the negative curvature around those distant critical points. This implies that increasing the complexity of the objective function via high-order losses accelerates the escape from such critical points and acts as a desirable alternative to increasing the complexity of the optimization problem via over-parametrization. By elucidating key characteristics of the non-convex optimization landscape, this work makes progress towards a comprehensive framework for tackling broader machine learning objectives plagued by non-convexity. } }
Endnote
%0 Conference Paper %T Absence of spurious solutions far from ground truth: A low-rank analysis with high-order losses %A Ziye Ma %A Ying Chen %A Javad Lavaei %A Somayeh Sojoudi %B Proceedings of The 27th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2024 %E Sanjoy Dasgupta %E Stephan Mandt %E Yingzhen Li %F pmlr-v238-ma24a %I PMLR %P 1603--1611 %U https://proceedings.mlr.press/v238/ma24a.html %V 238 %X Matrix sensing problems exhibit pervasive non-convexity, plaguing optimization with a proliferation of suboptimal spurious solutions. Avoiding convergence to these critical points poses a major challenge. This work provides new theoretical insights that help demystify the intricacies of the non-convex landscape. In this work, we prove that under certain conditions, critical points sufficiently distant from the ground truth matrix exhibit favorable geometry by being strict saddle points rather than troublesome local minima. Moreover, we introduce the notion of higher-order losses for the matrix sensing problem and show that the incorporation of such losses into the objective function amplifies the negative curvature around those distant critical points. This implies that increasing the complexity of the objective function via high-order losses accelerates the escape from such critical points and acts as a desirable alternative to increasing the complexity of the optimization problem via over-parametrization. By elucidating key characteristics of the non-convex optimization landscape, this work makes progress towards a comprehensive framework for tackling broader machine learning objectives plagued by non-convexity.
APA
Ma, Z., Chen, Y., Lavaei, J. & Sojoudi, S.. (2024). Absence of spurious solutions far from ground truth: A low-rank analysis with high-order losses . Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 238:1603-1611 Available from https://proceedings.mlr.press/v238/ma24a.html.

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