Towards moderate overparameterization: global convergence guarantees for training shallow neural networks (1902.04674v1)

Abstract: Many modern neural network architectures are trained in an overparameterized regime where the parameters of the model exceed the size of the training dataset. Sufficiently overparameterized neural network architectures in principle have the capacity to fit any set of labels including random noise. However, given the highly nonconvex nature of the training landscape it is not clear what level and kind of overparameterization is required for first order methods to converge to a global optima that perfectly interpolate any labels. A number of recent theoretical works have shown that for very wide neural networks where the number of hidden units is polynomially large in the size of the training data gradient descent starting from a random initialization does indeed converge to a global optima. However, in practice much more moderate levels of overparameterization seems to be sufficient and in many cases overparameterized models seem to perfectly interpolate the training data as soon as the number of parameters exceed the size of the training data by a constant factor. Thus there is a huge gap between the existing theoretical literature and practical experiments. In this paper we take a step towards closing this gap. Focusing on shallow neural nets and smooth activations, we show that (stochastic) gradient descent when initialized at random converges at a geometric rate to a nearby global optima as soon as the square-root of the number of network parameters exceeds the size of the training data. Our results also benefit from a fast convergence rate and continue to hold for non-differentiable activations such as Rectified Linear Units (ReLUs).

An Analytical Study on Overparameterized Neural Networks and Global Convergence

The paper "Towards moderate overparameterization: global convergence guarantees for training shallow neural networks" provides an analytical exploration of the conditions necessary for achieving global convergence in overparameterized neural network architectures. The authors, Samet Oymak and Mahdi Soltanolkotabi, specifically focus on the capacities of shallow neural networks with smooth activations, and extend their analysis to non-differentiable activations, such as Rectified Linear Units (ReLUs).

Modern neural networks are often characterized by a parameter space that exceeds the size of the training dataset, leading to various convergence dynamics, particularly when gradient descent and its stochastic variants are used. The impressive empirical success in fitting random labels suggests an underlying mechanism in overparameterized regimes that the current framework aims to elucidate.

Core Contributions

1. Moderate Overparameterization and Convergence: The authors demonstrate that for shallow neural networks, a geometric convergence rate to a global optima is attainable when the square root of the parameters exceeds the size of the training data. Here, even non-differentiable networks exhibit this fast convergence, extending similar findings for differentiable functions. The convergence holds robust against initialized weights drawn randomly from certain distributions.
2. Quantification and Gap Bridging: The work quantitatively bridges the gap between existing theoretical criteria and empirical observations. It establishes that overparameterized models, initially requiring polynomially large parameter spaces relative to the data size for theoretical convergence, can achieve the same convergence under substantially lower parameter counts in practice.
3. Impact of Activation Functions: The paper rigorously distinguishes the behaviors of networks based on the smoothness of activation functions. For smooth functions, convergence can be guaranteed under less stringent conditions compared to setups involving ReLU activations, which demand relatively larger overparameterization levels.
4. Theoretical Frameworks and Extensions: Through comprehensive mathematical tools, including random matrix theory, Hermite polynomial expansions, and spectral analyses, the authors offer scalable and broadly applicable techniques for interactive overparameterization learning problems. The proposed methods have potential implications for enhancing generalization assurances in high-dimensional parameter spaces.
5. SGD and Practical Implications: The authors derive conditions for stochastic gradient descent, providing evidence that this ubiquitous optimization technique performs comparably to deterministic gradient descent concerning convergence rates, even when parameter dimensionality just modestly exceeds data size. This finding ties into practical implementations, reinforcing relevance in real-world, large-scale computations.

Implications and Future Research Directions

Practically, this paper sheds light on designing neural network architectures with effectively utilized overparameterization, offering a precise understanding that could ultimately inform decisions around network width configurations in algorithm design. Theoretically, the research sets the stage for further explorations into the interplay between overparameterization and generalization. Expanding upon these results might involve investigating deeper networks and potentially uncovering relationships with other loss functions and different optimization landscapes.

The current exploration critically advances our comprehension of neural network learning dynamics in overparameterized settings, particularly emphasizing structured approaches to achieve zero training error effectively. Future studies could aim to refine the results further, possibly extending frameworks to deep learning architectures and untangling the complexities involved in generalized architectures. Continued focus on harmonizing theoretical underpinnings with empirical realities will likely persist as a fertile research avenue in machine learning, potentially catalyzing the development of more robust, efficient, and adaptable neural network systems.