Multivariate Gaussian Approximation for Random Forest via Region-based Stabilization
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912358197100544 |
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| author | Shi, Zhaoyang Bhattacharjee, Chinmoy Balasubramanian, Krishnakumar Polonik, Wolfgang |
| author_facet | Shi, Zhaoyang Bhattacharjee, Chinmoy Balasubramanian, Krishnakumar Polonik, Wolfgang |
| contents | We derive Gaussian approximation bounds for $k$-Potential Nearest Neighbor ($k$-PNN) based random forest predictions based on a set of training points given by a Poisson process under fairly mild regularity assumptions on the data generating process. Our approach is based on the key observation that $k$-PNN based random forest predictions satisfy a certain geometric property called region-based stabilization. We also compare the rates with those of $k$-nearest neighbor-based random forests, highlighting a form of universality in our result. In the process of developing our results, we also establish a probabilistic result on multivariate Gaussian approximation bounds for general functionals of Poisson process that are region-based stabilizing. This general result makes use of the Malliavin-Stein method, and is potentially applicable to various related statistical problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09960 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multivariate Gaussian Approximation for Random Forest via Region-based Stabilization Shi, Zhaoyang Bhattacharjee, Chinmoy Balasubramanian, Krishnakumar Polonik, Wolfgang Statistics Theory Probability Machine Learning We derive Gaussian approximation bounds for $k$-Potential Nearest Neighbor ($k$-PNN) based random forest predictions based on a set of training points given by a Poisson process under fairly mild regularity assumptions on the data generating process. Our approach is based on the key observation that $k$-PNN based random forest predictions satisfy a certain geometric property called region-based stabilization. We also compare the rates with those of $k$-nearest neighbor-based random forests, highlighting a form of universality in our result. In the process of developing our results, we also establish a probabilistic result on multivariate Gaussian approximation bounds for general functionals of Poisson process that are region-based stabilizing. This general result makes use of the Malliavin-Stein method, and is potentially applicable to various related statistical problems. |
| title | Multivariate Gaussian Approximation for Random Forest via Region-based Stabilization |
| topic | Statistics Theory Probability Machine Learning |
| url | https://arxiv.org/abs/2403.09960 |