Guarantees for Nonlinear Representation Learning: Non-identical Covariates, Dependent Data, Fewer Samples
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912073267544064 |
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| author | Zhang, Thomas T. Lee, Bruce D. Ziemann, Ingvar Pappas, George J. Matni, Nikolai |
| author_facet | Zhang, Thomas T. Lee, Bruce D. Ziemann, Ingvar Pappas, George J. Matni, Nikolai |
| contents | A driving force behind the diverse applicability of modern machine learning is the ability to extract meaningful features across many sources. However, many practical domains involve data that are non-identically distributed across sources, and statistically dependent within its source, violating vital assumptions in existing theoretical studies. Toward addressing these issues, we establish statistical guarantees for learning general $\textit{nonlinear}$ representations from multiple data sources that admit different input distributions and possibly dependent data. Specifically, we study the sample-complexity of learning $T+1$ functions $f_\star^{(t)} \circ g_\star$ from a function class $\mathcal F \times \mathcal G$, where $f_\star^{(t)}$ are task specific linear functions and $g_\star$ is a shared nonlinear representation. A representation $\hat g$ is estimated using $N$ samples from each of $T$ source tasks, and a fine-tuning function $\hat f^{(0)}$ is fit using $N'$ samples from a target task passed through $\hat g$. We show that when $N \gtrsim C_{\mathrm{dep}} (\mathrm{dim}(\mathcal F) + \mathrm{C}(\mathcal G)/T)$, the excess risk of $\hat f^{(0)} \circ \hat g$ on the target task decays as $ν_{\mathrm{div}} \big(\frac{\mathrm{dim}(\mathcal F)}{N'} + \frac{\mathrm{C}(\mathcal G)}{N T} \big)$, where $C_{\mathrm{dep}}$ denotes the effect of data dependency, $ν_{\mathrm{div}}$ denotes an (estimatable) measure of $\textit{task-diversity}$ between the source and target tasks, and $\mathrm C(\mathcal G)$ denotes the complexity of the representation class $\mathcal G$. In particular, our analysis reveals: as the number of tasks $T$ increases, both the sample requirement and risk bound converge to that of $r$-dimensional regression as if $g_\star$ had been given, and the effect of dependency only enters the sample requirement, leaving the risk bound matching the iid setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11227 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Guarantees for Nonlinear Representation Learning: Non-identical Covariates, Dependent Data, Fewer Samples Zhang, Thomas T. Lee, Bruce D. Ziemann, Ingvar Pappas, George J. Matni, Nikolai Machine Learning Systems and Control A driving force behind the diverse applicability of modern machine learning is the ability to extract meaningful features across many sources. However, many practical domains involve data that are non-identically distributed across sources, and statistically dependent within its source, violating vital assumptions in existing theoretical studies. Toward addressing these issues, we establish statistical guarantees for learning general $\textit{nonlinear}$ representations from multiple data sources that admit different input distributions and possibly dependent data. Specifically, we study the sample-complexity of learning $T+1$ functions $f_\star^{(t)} \circ g_\star$ from a function class $\mathcal F \times \mathcal G$, where $f_\star^{(t)}$ are task specific linear functions and $g_\star$ is a shared nonlinear representation. A representation $\hat g$ is estimated using $N$ samples from each of $T$ source tasks, and a fine-tuning function $\hat f^{(0)}$ is fit using $N'$ samples from a target task passed through $\hat g$. We show that when $N \gtrsim C_{\mathrm{dep}} (\mathrm{dim}(\mathcal F) + \mathrm{C}(\mathcal G)/T)$, the excess risk of $\hat f^{(0)} \circ \hat g$ on the target task decays as $ν_{\mathrm{div}} \big(\frac{\mathrm{dim}(\mathcal F)}{N'} + \frac{\mathrm{C}(\mathcal G)}{N T} \big)$, where $C_{\mathrm{dep}}$ denotes the effect of data dependency, $ν_{\mathrm{div}}$ denotes an (estimatable) measure of $\textit{task-diversity}$ between the source and target tasks, and $\mathrm C(\mathcal G)$ denotes the complexity of the representation class $\mathcal G$. In particular, our analysis reveals: as the number of tasks $T$ increases, both the sample requirement and risk bound converge to that of $r$-dimensional regression as if $g_\star$ had been given, and the effect of dependency only enters the sample requirement, leaving the risk bound matching the iid setting. |
| title | Guarantees for Nonlinear Representation Learning: Non-identical Covariates, Dependent Data, Fewer Samples |
| topic | Machine Learning Systems and Control |
| url | https://arxiv.org/abs/2410.11227 |