Using Taylor-Approximated Gradients to Improve the Frank-Wolfe Method for Empirical Risk Minimization
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arXiv
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| Format: | Preprint |
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2022
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| author | Xiong, Zikai Freund, Robert M. |
| author_facet | Xiong, Zikai Freund, Robert M. |
| contents | The Frank-Wolfe method has become increasingly useful in statistical and machine learning applications, due to the structure-inducing properties of the iterates, and especially in settings where linear minimization over the feasible set is more computationally efficient than projection. In the setting of Empirical Risk Minimization -- one of the fundamental optimization problems in statistical and machine learning -- the computational effectiveness of Frank-Wolfe methods typically grows linearly in the number of data observations $n$. This is in stark contrast to the case for typical stochastic projection methods. In order to reduce this dependence on $n$, we look to second-order smoothness of typical smooth loss functions (least squares loss and logistic loss, for example) and we propose amending the Frank-Wolfe method with Taylor series-approximated gradients, including variants for both deterministic and stochastic settings. Compared with current state-of-the-art methods in the regime where the optimality tolerance $\varepsilon$ is sufficiently small, our methods are able to simultaneously reduce the dependence on large $n$ while obtaining optimal convergence rates of Frank-Wolfe methods, in both the convex and non-convex settings. We also propose a novel adaptive step-size approach for which we have computational guarantees. Last of all, we present computational experiments which show that our methods exhibit very significant speed-ups over existing methods on real-world datasets for both convex and non-convex binary classification problems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_13933 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Using Taylor-Approximated Gradients to Improve the Frank-Wolfe Method for Empirical Risk Minimization Xiong, Zikai Freund, Robert M. Machine Learning Optimization and Control The Frank-Wolfe method has become increasingly useful in statistical and machine learning applications, due to the structure-inducing properties of the iterates, and especially in settings where linear minimization over the feasible set is more computationally efficient than projection. In the setting of Empirical Risk Minimization -- one of the fundamental optimization problems in statistical and machine learning -- the computational effectiveness of Frank-Wolfe methods typically grows linearly in the number of data observations $n$. This is in stark contrast to the case for typical stochastic projection methods. In order to reduce this dependence on $n$, we look to second-order smoothness of typical smooth loss functions (least squares loss and logistic loss, for example) and we propose amending the Frank-Wolfe method with Taylor series-approximated gradients, including variants for both deterministic and stochastic settings. Compared with current state-of-the-art methods in the regime where the optimality tolerance $\varepsilon$ is sufficiently small, our methods are able to simultaneously reduce the dependence on large $n$ while obtaining optimal convergence rates of Frank-Wolfe methods, in both the convex and non-convex settings. We also propose a novel adaptive step-size approach for which we have computational guarantees. Last of all, we present computational experiments which show that our methods exhibit very significant speed-ups over existing methods on real-world datasets for both convex and non-convex binary classification problems. |
| title | Using Taylor-Approximated Gradients to Improve the Frank-Wolfe Method for Empirical Risk Minimization |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2208.13933 |