Functional Risk Minimization (2412.21149v1)

Abstract: The field of Machine Learning has changed significantly since the 1970s. However, its most basic principle, Empirical Risk Minimization (ERM), remains unchanged. We propose Functional Risk Minimization~(FRM), a general framework where losses compare functions rather than outputs. This results in better performance in supervised, unsupervised, and RL experiments. In the FRM paradigm, for each data point $(x_i,y_i)$ there is function $f_{\theta_i}$ that fits it: $y_i = f_{\theta_i}(x_i)$. This allows FRM to subsume ERM for many common loss functions and to capture more realistic noise processes. We also show that FRM provides an avenue towards understanding generalization in the modern over-parameterized regime, as its objective can be framed as finding the simplest model that fits the training data.

• The paper presents FRM as an innovative approach that evaluates loss in function space to capture intrinsic data noise and variability better than traditional ERM.
• The paper leverages Functional Generative Models to model dataset variations, leading to improved generalization in tasks like linear least squares and reinforcement learning.
• The paper details experiments on structured data transformations, demonstrating FRM’s superior performance in real-world scenarios compared to conventional methods.

Functional Risk Minimization: A Novel Paradigm for Model Training

The foundational principle of Empirical Risk Minimization (ERM) has been a cornerstone in machine learning for several decades. However, the paper "Functional Risk Minimization" introduces Functional Risk Minimization (FRM), which posits an alternative approach to this well-established method. The authors, Ferran Alet et al., aim to address several limitations posed by the conventional ERM, especially in the context of modern, over-parameterized neural networks.

FRM Framework Overview

ERM operates under the premise of minimizing loss by comparing actual outputs to target outputs. However, this method faces challenges in the adaptive and over-parameterized landscapes of deep learning models where complex noise processes emerge. In contrast, FRM proposes a paradigm where losses are evaluated in the function space rather than the output space. Specifically, for each data point, a function $f_{\theta_i}$ is optimized to fit the noise and variability intrinsic to the data rather than solely the output error.

FRM's Approach and Theoretical Underpinnings

FRM is built upon Functional Generative Models (FGMs), wherein each datapoint $(x_i, y_i)$ is assumed to originate from its own function $f_{\theta_i}$ with form $y_i = f_{\theta_i}(x_i)$. This effectively models the variations within datasets through distributional changes across functions rather than changes in output space solely. The FRM framework leverages diverse function classes adaptable to the task's structural properties, thus offering a flexible model that can capture structured noise variations more aptly than traditional ERM-based approaches.

The authors provide a mathematical formulation demonstrating that common loss functions such as MSE and cross-entropy can be superseded by FRM under specific assumptions, illustrating the breadth of applicable scenarios. Furthermore, they highlight that this function-space perspective models real-world data complexity more naturally than classical noise assumptions often used in ERM.

Empirical Evidence and Experiments

Numerous experiments underscore the potential of FRM across various domains, including supervised, unsupervised, and reinforcement learning. Notably, when applied to linear least squares and Mountain Car reinforcement learning tasks, FRM resulted in superior model generalization, with reductions in test error even when training errors were lower for conventional ERM models.

Moreover, FRM's flexibility is showcased through a variational autoencoder experiment employing structural variability (e.g., translations and color changes in datasets such as MNIST). In scenarios where data presents structured variations embedded in function space, such as image data driven by transformations, FRM distinctly outperforms ERM by leveraging the intrinsic geometric properties modeled in the function space.

Implications and Future Perspectives

The promising results from FRM indicate potential shifts in how model training objectives might be structured, especially for complex, noise-driven datasets. This framework theoretically and empirically encourages considering the complexity and variability of datasets beyond simple output alterations. As such, FRM may lead to advancements in understanding and modeling over-parameterized models' generalization capabilities, providing a more nuanced understanding of training dynamics in modern machine learning ecosystems.

The paper suggests that FRM can be computationally intensive, especially when compared to ERM. However, the authors also note that advancements in optimization strategies, coupled with computational efficiency gains, will likely mitigate scalability concerns, broadening the applicability of FRM in real-world scenarios.

Ultimately, FRM stands as a compelling framework for rethinking loss landscape modeling, encouraging a more holistic view of functional adaptation within model training. As the field of machine learning continues to evolve, approaches like FRM could become instrumental in reconciling model expressivity with robust, generalized learning.