Learning Sparse Additive Models with Interactions in High Dimensions

Hemant Tyagi, Anastasios Kyrillidis, Bernd Gärtner, Andreas Krause
Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, PMLR 51:111-120, 2016.

Abstract

A function f: \mathbbR^d →\mathbbR is referred to as a Sparse Additive Model (SPAM), if it is of the form f(x) = \sum_l ∈S \phi_l(x_l), where S ⊂[d], |S| ≪d. Assuming \phi_l’s and S to be unknown, the problem of estimating f from its samples has been studied extensively. In this work, we consider a generalized SPAM, allowing for second order interaction terms. For some S_1 ⊂[d], S_2 ⊂[d] \choose 2, the function f is assumed to be of the form: f(x) = \sum_p ∈S_1 \phi_p (x_p) + \sum_(l,l’) ∈S_2 \phi_l,l’ (x_l, x_l’). Assuming \phi_p, \phi_(l,l’), S_1 and S_2 to be unknown, we provide a randomized algorithm that queries f and exactly recovers S_1,S_2. Consequently, this also enables us to estimate the underlying \phi_p, \phi_l,l’. We derive sample complexity bounds for our scheme and also extend our analysis to include the situation where the queries are corrupted with noise – either stochastic, or arbitrary but bounded. Lastly, we provide simulation results on synthetic data, that validate our theoretical findings.

Cite this Paper

BibTeX
@InProceedings{pmlr-v51-tyagi16, title = {Learning Sparse Additive Models with Interactions in High Dimensions}, author = {Tyagi, Hemant and Kyrillidis, Anastasios and Gärtner, Bernd and Krause, Andreas}, booktitle = {Proceedings of the 19th International Conference on Artificial Intelligence and Statistics}, pages = {111--120}, year = {2016}, editor = {Gretton, Arthur and Robert, Christian C.}, volume = {51}, series = {Proceedings of Machine Learning Research}, address = {Cadiz, Spain}, month = {09--11 May}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v51/tyagi16.pdf}, url = {https://proceedings.mlr.press/v51/tyagi16.html}, abstract = {A function f: \mathbbR^d →\mathbbR is referred to as a Sparse Additive Model (SPAM), if it is of the form f(x) = \sum_l ∈S \phi_l(x_l), where S ⊂[d], |S| ≪d. Assuming \phi_l’s and S to be unknown, the problem of estimating f from its samples has been studied extensively. In this work, we consider a generalized SPAM, allowing for second order interaction terms. For some S_1 ⊂[d], S_2 ⊂[d] \choose 2, the function f is assumed to be of the form: f(x) = \sum_p ∈S_1 \phi_p (x_p) + \sum_(l,l’) ∈S_2 \phi_l,l’ (x_l, x_l’). Assuming \phi_p, \phi_(l,l’), S_1 and S_2 to be unknown, we provide a randomized algorithm that queries f and exactly recovers S_1,S_2. Consequently, this also enables us to estimate the underlying \phi_p, \phi_l,l’. We derive sample complexity bounds for our scheme and also extend our analysis to include the situation where the queries are corrupted with noise – either stochastic, or arbitrary but bounded. Lastly, we provide simulation results on synthetic data, that validate our theoretical findings.} }
Endnote
%0 Conference Paper %T Learning Sparse Additive Models with Interactions in High Dimensions %A Hemant Tyagi %A Anastasios Kyrillidis %A Bernd Gärtner %A Andreas Krause %B Proceedings of the 19th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2016 %E Arthur Gretton %E Christian C. Robert %F pmlr-v51-tyagi16 %I PMLR %P 111--120 %U https://proceedings.mlr.press/v51/tyagi16.html %V 51 %X A function f: \mathbbR^d →\mathbbR is referred to as a Sparse Additive Model (SPAM), if it is of the form f(x) = \sum_l ∈S \phi_l(x_l), where S ⊂[d], |S| ≪d. Assuming \phi_l’s and S to be unknown, the problem of estimating f from its samples has been studied extensively. In this work, we consider a generalized SPAM, allowing for second order interaction terms. For some S_1 ⊂[d], S_2 ⊂[d] \choose 2, the function f is assumed to be of the form: f(x) = \sum_p ∈S_1 \phi_p (x_p) + \sum_(l,l’) ∈S_2 \phi_l,l’ (x_l, x_l’). Assuming \phi_p, \phi_(l,l’), S_1 and S_2 to be unknown, we provide a randomized algorithm that queries f and exactly recovers S_1,S_2. Consequently, this also enables us to estimate the underlying \phi_p, \phi_l,l’. We derive sample complexity bounds for our scheme and also extend our analysis to include the situation where the queries are corrupted with noise – either stochastic, or arbitrary but bounded. Lastly, we provide simulation results on synthetic data, that validate our theoretical findings.
RIS
TY - CPAPER TI - Learning Sparse Additive Models with Interactions in High Dimensions AU - Hemant Tyagi AU - Anastasios Kyrillidis AU - Bernd Gärtner AU - Andreas Krause BT - Proceedings of the 19th International Conference on Artificial Intelligence and Statistics DA - 2016/05/02 ED - Arthur Gretton ED - Christian C. Robert ID - pmlr-v51-tyagi16 PB - PMLR DP - Proceedings of Machine Learning Research VL - 51 SP - 111 EP - 120 L1 - http://proceedings.mlr.press/v51/tyagi16.pdf UR - https://proceedings.mlr.press/v51/tyagi16.html AB - A function f: \mathbbR^d →\mathbbR is referred to as a Sparse Additive Model (SPAM), if it is of the form f(x) = \sum_l ∈S \phi_l(x_l), where S ⊂[d], |S| ≪d. Assuming \phi_l’s and S to be unknown, the problem of estimating f from its samples has been studied extensively. In this work, we consider a generalized SPAM, allowing for second order interaction terms. For some S_1 ⊂[d], S_2 ⊂[d] \choose 2, the function f is assumed to be of the form: f(x) = \sum_p ∈S_1 \phi_p (x_p) + \sum_(l,l’) ∈S_2 \phi_l,l’ (x_l, x_l’). Assuming \phi_p, \phi_(l,l’), S_1 and S_2 to be unknown, we provide a randomized algorithm that queries f and exactly recovers S_1,S_2. Consequently, this also enables us to estimate the underlying \phi_p, \phi_l,l’. We derive sample complexity bounds for our scheme and also extend our analysis to include the situation where the queries are corrupted with noise – either stochastic, or arbitrary but bounded. Lastly, we provide simulation results on synthetic data, that validate our theoretical findings. ER -
APA
Tyagi, H., Kyrillidis, A., Gärtner, B. & Krause, A.. (2016). Learning Sparse Additive Models with Interactions in High Dimensions. Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 51:111-120 Available from https://proceedings.mlr.press/v51/tyagi16.html.

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