Benignity of loss landscape with weight decay requires both large overparametrization and initialization (2505.22578v1)

Abstract: The optimization of neural networks under weight decay remains poorly understood from a theoretical standpoint. While weight decay is standard practice in modern training procedures, most theoretical analyses focus on unregularized settings. In this work, we investigate the loss landscape of the $\ell_2$-regularized training loss for two-layer ReLU networks. We show that the landscape becomes benign -- i.e., free of spurious local minima -- under large overparametrization, specifically when the network width $m$ satisfies $m \gtrsim \min(n^d, 2^n)$, where $n$ is the number of data points and $d$ the input dimension. More precisely in this regime, almost all constant activation regions contain a global minimum and no spurious local minima. We further show that this level of overparametrization is not only sufficient but also necessary via the example of orthogonal data. Finally, we demonstrate that such loss landscape results primarily hold relevance in the large initialization regime. In contrast, for small initializations -- corresponding to the feature learning regime -- optimization can still converge to spurious local minima, despite the global benignity of the landscape.

• The paper demonstrates that ℓ2-regularized two-layer ReLU networks achieve benign loss landscapes when high overparametrization (m ≳ min(n^d, 2^n)) and large initialization are used.
• It reveals that, under optimal regimes, nearly all constant activation regions contain a global minimum, effectively eliminating spurious local minima and enabling efficient optimization.
• Findings emphasize that improper (small) initialization can lead to suboptimal convergence even with a benign landscape, highlighting the need for carefully balanced parameter settings.

Benignity of Loss Landscape with Weight Decay: Requirements for Overparametrization and Initialization

The paper "Benignity of loss landscape with weight decay requires both large overparametrization and initialization" presents a detailed analysis of the conditions under which the loss landscape of $\ell_2$-regularized two-layer ReLU neural networks becomes benign. This essentially means the landscape is devoid of spurious local minima, facilitating optimization to reach global minima efficiently.

Overparametrization and Initialization

A core focus of this work is delineating the parameters for achieving a benign loss landscape. The researchers establish that for a neural network's loss landscape to become benign—specifically for $\ell_2$-regularized training of two-layer ReLU networks—a large degree of overparametrization is essential. They show that the network width $m$ needs to satisfy $m \gtrsim \min(n^d, 2^n)$, where $n$ is the number of data points and $d$ is the input dimension. Moreover, large initialization is critical in this framework. It is particularly needed when considering the landscape’s applicability in regimes that are relevant to typical initialization scales employed in practice.

Loss Landscapes in Large Overparametrization

The paper demonstrates that in a regime of large overparametrization, almost all constant activation regions contain a global minimum, and no spurious local minima are present. This landscape property ensures the benignity that is crucial for efficient optimization. Importantly, the analysis indicates that such large overparametrization isn't just beneficial but necessary. For instance, in the presence of orthogonal data, the same level of overparametrization is required to prevent convergence to non-optimal solutions.

Initialization Regimes and Their Impact

The benignity of the landscape alone does not guarantee optimal outcomes across all settings. The paper highlights that while large initialization scales (e.g., in the NTK regime) align well with the benign landscape leading to effective convergence behaviors, small initialization can result in suboptimal solutions despite a globally benign landscape. In such cases, the initialization regime plays a pivotal role in determining the trajectory of the optimization process and can lead to convergence towards suboptimal solutions, even when the landscape does not have spurious minima.

Practical Implications and Future Directions

This insight into the requisite overparametrization and initialization characteristics offers a deeper understanding for both academic and practical applications. Practitioners must consider these settings to harness the benignity of loss landscapes effectively, particularly for tasks involving large neural network architectures and datasets.

However, the stipulation that overparametrization must be exceedingly large ($m \gtrsim \min(n^d, 2^n)$) may not align with practical constraints or objectives, such as minimizing computational overhead or model size. Thus, future work can explore adaptive or hybrid methods that leverage the principles outlined while maintaining efficiency. Furthermore, expanding this analysis to other network architectures and types of regularization could yield additional insights applicable in broader contexts of machine learning and deep learning.

Conclusion

The paper convincingly argues that achieving a benign loss landscape in $\ell_2$-regularized two-layer ReLU networks prescribing both significant overparametrization and careful consideration of initialization. This ensures effective convergence to global minima, a critical factor for neural network optimization in high-dimensional and complex data regimes. The findings direct researchers and practitioners towards better model design and initialization strategies, underscoring the interconnectedness of theoretical landscape properties and practical training dynamics in neural networks.