Optimal Approximation -- Smoothness Tradeoffs for Soft-Max Functions
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866915699814825984 |
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| author | Epasto, Alessandro Mahdian, Mohammad Mirrokni, Vahab Zampetakis, Manolis |
| author_facet | Epasto, Alessandro Mahdian, Mohammad Mirrokni, Vahab Zampetakis, Manolis |
| contents | A soft-max function has two main efficiency measures: (1) approximation - which corresponds to how well it approximates the maximum function, (2) smoothness - which shows how sensitive it is to changes of its input. Our goal is to identify the optimal approximation-smoothness tradeoffs for different measures of approximation and smoothness. This leads to novel soft-max functions, each of which is optimal for a different application. The most commonly used soft-max function, called exponential mechanism, has optimal tradeoff between approximation measured in terms of expected additive approximation and smoothness measured with respect to Rényi Divergence. We introduce a soft-max function, called "piecewise linear soft-max", with optimal tradeoff between approximation, measured in terms of worst-case additive approximation and smoothness, measured with respect to $\ell_q$-norm. The worst-case approximation guarantee of the piecewise linear mechanism enforces sparsity in the output of our soft-max function, a property that is known to be important in Machine Learning applications [Martins et al. '16, Laha et al. '18] and is not satisfied by the exponential mechanism. Moreover, the $\ell_q$-smoothness is suitable for applications in Mechanism Design and Game Theory where the piecewise linear mechanism outperforms the exponential mechanism. Finally, we investigate another soft-max function, called power mechanism, with optimal tradeoff between expected \textit{multiplicative} approximation and smoothness with respect to the Rényi Divergence, which provides improved theoretical and practical results in differentially private submodular optimization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_11450 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Optimal Approximation -- Smoothness Tradeoffs for Soft-Max Functions Epasto, Alessandro Mahdian, Mohammad Mirrokni, Vahab Zampetakis, Manolis Machine Learning Data Structures and Algorithms A soft-max function has two main efficiency measures: (1) approximation - which corresponds to how well it approximates the maximum function, (2) smoothness - which shows how sensitive it is to changes of its input. Our goal is to identify the optimal approximation-smoothness tradeoffs for different measures of approximation and smoothness. This leads to novel soft-max functions, each of which is optimal for a different application. The most commonly used soft-max function, called exponential mechanism, has optimal tradeoff between approximation measured in terms of expected additive approximation and smoothness measured with respect to Rényi Divergence. We introduce a soft-max function, called "piecewise linear soft-max", with optimal tradeoff between approximation, measured in terms of worst-case additive approximation and smoothness, measured with respect to $\ell_q$-norm. The worst-case approximation guarantee of the piecewise linear mechanism enforces sparsity in the output of our soft-max function, a property that is known to be important in Machine Learning applications [Martins et al. '16, Laha et al. '18] and is not satisfied by the exponential mechanism. Moreover, the $\ell_q$-smoothness is suitable for applications in Mechanism Design and Game Theory where the piecewise linear mechanism outperforms the exponential mechanism. Finally, we investigate another soft-max function, called power mechanism, with optimal tradeoff between expected \textit{multiplicative} approximation and smoothness with respect to the Rényi Divergence, which provides improved theoretical and practical results in differentially private submodular optimization. |
| title | Optimal Approximation -- Smoothness Tradeoffs for Soft-Max Functions |
| topic | Machine Learning Data Structures and Algorithms |
| url | https://arxiv.org/abs/2010.11450 |