Guarantees for Nonlinear Representation Learning: Non-identical Covariates, Dependent Data, Fewer Samples

• URL: http://arxiv.org/abs/2410.11227v1
• Date: Tue, 15 Oct 2024 03:20:19 GMT
• Title: Guarantees for Nonlinear Representation Learning: Non-identical Covariates, Dependent Data, Fewer Samples
• Authors: Thomas T. Zhang, Bruce D. Lee, Ingvar Ziemann, George J. Pappas, Nikolai Matni,
• Abstract summary: We study the sample-complexity of learning $T+1$ functions $f_star(t) circ g_star$ from a function class $mathcal F times mathcal G$. We show that as the number of tasks $T$ increases, both the sample requirement and risk bound converge to that of $r$-dimensional regression.
• Score: 24.45016514352055
• License:
• Abstract: 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 $\nu_{\mathrm{div}} \big(\frac{\mathrm{dim}(\mathcal F)}{N'} + \frac{\mathrm{C}(\mathcal G)}{N T} \big)$, where $C_{\mathrm{dep}}$ denotes the effect of data dependency, $\nu_{\mathrm{div}}$ 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.

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