Sharp Minima Can Generalize For Deep Nets (1703.04933v2)

Abstract: Despite their overwhelming capacity to overfit, deep learning architectures tend to generalize relatively well to unseen data, allowing them to be deployed in practice. However, explaining why this is the case is still an open area of research. One standing hypothesis that is gaining popularity, e.g. Hochreiter & Schmidhuber (1997); Keskar et al. (2017), is that the flatness of minima of the loss function found by stochastic gradient based methods results in good generalization. This paper argues that most notions of flatness are problematic for deep models and can not be directly applied to explain generalization. Specifically, when focusing on deep networks with rectifier units, we can exploit the particular geometry of parameter space induced by the inherent symmetries that these architectures exhibit to build equivalent models corresponding to arbitrarily sharper minima. Furthermore, if we allow to reparametrize a function, the geometry of its parameters can change drastically without affecting its generalization properties.

• The paper challenges the prevailing assumption that flat minima ensure better generalization by demonstrating that sharp minima can also generalize well.
• It reveals that deep networks' non-Euclidean parameter space and reparameterization effects can artificially alter measures of flatness.
• The findings suggest that relying solely on flatness during optimization may be misguided, urging the need for more robust generalization metrics in deep learning.

Analysis of "Sharp Minima Can Generalize For Deep Nets"

The paper, "Sharp Minima Can Generalize For Deep Nets," authored by Laurent Dinh, Razvan Pascanu, Samy Bengio, and Yoshua Bengio, addresses a fundamental question in the field of deep learning: why do deep learning models, despite their high capacity for overfitting, generally exhibit good generalization to unseen data. The authors challenge the prevailing hypothesis that flat minima in the loss function landscape correlate with better generalization in deep networks.

Key Hypothesis and Argument

The hypothesis under scrutiny suggests that flat minima lead to better generalization because they provide robustness to perturbations in the parameter space. This idea has roots in the works of Hochreiter and Schmidhuber (1997) and has been recently revisited by Keskar et al. (2016). The prevailing thought is that flat minima, characterized by a wide region around a minimum where the loss remains relatively unchanged, imply a robust model that is less sensitive to the precise values of its parameters. This robustness is thought to contribute to better generalization.

Critical Issues with Flatness Arguments

The authors argue that the conventional notions of flatness are problematic and cannot be directly applied to deep models, particularly those with rectifier units. They highlight several issues:

1. Geometry of Parameter Space: Deep networks with rectifiers exhibit inherent symmetries due to the non-negative homogeneity of rectifier activation functions. This property implies that multiple parameter configurations can result in the same output, leading to a form of non-identifiability. The parameter spaces for such architectures are non-Euclidean, and the equivalent models could correspond to arbitrarily sharp minima.
2. Reparametrization: The paper demonstrates that the geometry of parameter space can be drastically changed through reparameterizations without affecting the generalization properties of the model. This observation undermines the idea that flatness in the original parameter space correlates with generalization.

Examination of Flatness Definitions

The paper scrutinizes common definitions of flatness and their implications in the context of deep networks:

• Volume $\epsilon$-flatness: Defined as the volume of the connected region around a minimum where the loss is roughly constant. The authors show that for deep rectified networks, every minimum has an infinitely large volume of approximately constant error, rendering this metric ineffective for distinguishing minima in terms of generalization.
• Hessian-Based Measures: Metrics like the spectral norm and trace of the Hessian matrix are often used to gauge flatness. The authors prove that through non-negative homogeneity transformations, one can adjust the Hessian’s eigenvalues without changing the model’s output, hence manipulating the perceived sharpness.
• $\epsilon$-sharpness: Defined as the maximum increase in loss within an $\epsilon$-neighborhood of a minimum. Similar to previous measures, this can also be manipulated through transformations in the parameter space, making it unreliable for judging generalization.

Practical and Theoretical Implications

The findings of this paper have significant implications:

• Training and Optimization: The concept that flat minima are inherently better is not justifiable in deep networks with rectifiers. Optimization methods that aim to find flatter minima might not necessarily lead to models that generalize better.
• Reevaluation of Generalization Metrics: Researchers should be cautious in using flatness as a proxy for generalization. Equivalently, different models’ generalization properties should not be judged solely based on the geometry of their minima in the parameter space.
• Model Reparametrization: Arbitrary reparametrizations can alter the perceived flatness of minima. This highlights the importance of considering the specific parametrization of models when evaluating their generalization properties.

Future Directions

This work suggests future research should focus on:

1. Robust Metrics for Generalization: Developing new metrics and methods that can predict generalization more reliably than the flatness of minima.
2. Understanding Non-Euclidean Geometry: Further examination of the geometric properties of parameter spaces for various deep learning architectures, and how these properties affect training dynamics and generalization.
3. Exploring Alternative Hypotheses: Identifying other factors beyond flatness that contribute to generalization, such as the role of implicit regularization introduced by optimization algorithms like stochastic gradient descent.

Conclusion

The paper "Sharp Minima Can Generalize For Deep Nets" challenges a key assumption in the deep learning community about the relationship between flat minima and generalization. Through rigorous analysis, the authors demonstrate the limitations of current flatness definitions and underscore the necessity of a deeper understanding of the geometry of learning models. This work paves the way for future developments in crafting more robust theories and metrics for understanding and improving generalization in deep learning.