Characterizing the implicit bias via a primal-dual analysis

Ziwei Ji, Matus Telgarsky
Proceedings of the 32nd International Conference on Algorithmic Learning Theory, PMLR 132:772-804, 2021.

Abstract

This paper shows that the implicit bias of gradient descent on linearly separable data is exactly characterized by the optimal solution of a dual optimization problem given by a smoothed margin, even for general losses. This is in contrast to prior results, which are often tailored to exponentially-tailed losses. For the exponential loss specifically, with $n$ training examples and $t$ gradient descent steps, our dual analysis further allows us to prove an $O\del{\ln(n)/\ln(t)}$ convergence rate to the $\ell_2$ maximum margin direction, when a constant step size is used. This rate is tight in both $n$ and $t$, which has not been presented by prior work. On the other hand, with a properly chosen but aggressive step size schedule, we prove $O(1/t)$ rates for both $\ell_2$ margin maximization and implicit bias, whereas prior work (including all first-order methods for the general hard-margin linear SVM problem) proved $\widetilde{O}(1/\sqrt{t})$ margin rates, or $O(1/t)$ margin rates to a suboptimal margin, with an implied (slower) bias rate. Our key observations include that gradient descent on the primal variable naturally induces a mirror descent update on the dual variable, and that the dual objective in this setting is smooth enough to give a faster rate.

Cite this Paper

BibTeX
@InProceedings{pmlr-v132-ji21a, title = {Characterizing the implicit bias via a primal-dual analysis}, author = {Ji, Ziwei and Telgarsky, Matus}, booktitle = {Proceedings of the 32nd International Conference on Algorithmic Learning Theory}, pages = {772--804}, year = {2021}, editor = {Feldman, Vitaly and Ligett, Katrina and Sabato, Sivan}, volume = {132}, series = {Proceedings of Machine Learning Research}, month = {16--19 Mar}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v132/ji21a/ji21a.pdf}, url = {https://proceedings.mlr.press/v132/ji21a.html}, abstract = { This paper shows that the implicit bias of gradient descent on linearly separable data is exactly characterized by the optimal solution of a dual optimization problem given by a smoothed margin, even for general losses. This is in contrast to prior results, which are often tailored to exponentially-tailed losses. For the exponential loss specifically, with $n$ training examples and $t$ gradient descent steps, our dual analysis further allows us to prove an $O\del{\ln(n)/\ln(t)}$ convergence rate to the $\ell_2$ maximum margin direction, when a constant step size is used. This rate is tight in both $n$ and $t$, which has not been presented by prior work. On the other hand, with a properly chosen but aggressive step size schedule, we prove $O(1/t)$ rates for both $\ell_2$ margin maximization and implicit bias, whereas prior work (including all first-order methods for the general hard-margin linear SVM problem) proved $\widetilde{O}(1/\sqrt{t})$ margin rates, or $O(1/t)$ margin rates to a suboptimal margin, with an implied (slower) bias rate. Our key observations include that gradient descent on the primal variable naturally induces a mirror descent update on the dual variable, and that the dual objective in this setting is smooth enough to give a faster rate. } }
Endnote
%0 Conference Paper %T Characterizing the implicit bias via a primal-dual analysis %A Ziwei Ji %A Matus Telgarsky %B Proceedings of the 32nd International Conference on Algorithmic Learning Theory %C Proceedings of Machine Learning Research %D 2021 %E Vitaly Feldman %E Katrina Ligett %E Sivan Sabato %F pmlr-v132-ji21a %I PMLR %P 772--804 %U https://proceedings.mlr.press/v132/ji21a.html %V 132 %X This paper shows that the implicit bias of gradient descent on linearly separable data is exactly characterized by the optimal solution of a dual optimization problem given by a smoothed margin, even for general losses. This is in contrast to prior results, which are often tailored to exponentially-tailed losses. For the exponential loss specifically, with $n$ training examples and $t$ gradient descent steps, our dual analysis further allows us to prove an $O\del{\ln(n)/\ln(t)}$ convergence rate to the $\ell_2$ maximum margin direction, when a constant step size is used. This rate is tight in both $n$ and $t$, which has not been presented by prior work. On the other hand, with a properly chosen but aggressive step size schedule, we prove $O(1/t)$ rates for both $\ell_2$ margin maximization and implicit bias, whereas prior work (including all first-order methods for the general hard-margin linear SVM problem) proved $\widetilde{O}(1/\sqrt{t})$ margin rates, or $O(1/t)$ margin rates to a suboptimal margin, with an implied (slower) bias rate. Our key observations include that gradient descent on the primal variable naturally induces a mirror descent update on the dual variable, and that the dual objective in this setting is smooth enough to give a faster rate.
APA
Ji, Z. & Telgarsky, M.. (2021). Characterizing the implicit bias via a primal-dual analysis. Proceedings of the 32nd International Conference on Algorithmic Learning Theory, in Proceedings of Machine Learning Research 132:772-804 Available from https://proceedings.mlr.press/v132/ji21a.html.

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