Geometric structure of Deep Learning networks and construction of global ${\mathcal L}^2$ minimizers (2309.10639v4)

Abstract: In this paper, we explicitly determine local and global minimizers of the $\mathcal{L}^2$ cost function in underparametrized Deep Learning (DL) networks; our main goal is to shed light on their geometric structure and properties. We accomplish this by a direct construction, without invoking the gradient descent flow at any point of this work. We specifically consider $L$ hidden layers, a ReLU ramp activation function, an $\mathcal{L}^2$ Schatten class (or Hilbert-Schmidt) cost function, input and output spaces $\mathbb{R}^Q$ with equal dimension $Q\geq1$, and hidden layers also defined on $\mathbb{R}^{Q}$; the training inputs are assumed to be sufficiently clustered. The training input size $N$ can be arbitrarily large - thus, we are considering the underparametrized regime. More general settings are left to future work. We construct an explicit family of minimizers for the global minimum of the cost function in the case $L\geq Q$, which we show to be degenerate. Moreover, we determine a set of $2^Q-1$ distinct degenerate local minima of the cost function. In the context presented here, the concatenation of hidden layers of the DL network is reinterpreted as a recursive application of a {\em truncation map} which "curates" the training inputs by minimizing their noise to signal ratio.

• The paper explicitly constructs a family of global minimizers for underparameterized networks through the use of truncation maps.
• It identifies 2^Q - 1 distinct degenerate local minima, deepening insights on network optimization landscapes.
• The work reinterprets hidden layers as rank-preserving transformations, offering new perspectives for neural network design and training.

Geometric Structure of Deep Learning Networks and Construction of Global C2 Minimizers

The paper "Geometric Structure of Deep Learning Networks and Construction of Global C2 Minimizers," authored by Thomas Chen and Patricia Muñoz Ewald, presents a mathematical investigation into the local and global minimization of the C2 cost function in underparameterized deep learning networks. The work primarily focuses on enhancing the understanding of the geometric structures and properties of such networks' minimizers and does so through explicit construction, eschewing the typical approach of leveraging gradient descent flows.

Key Contributions and Insights

The authors explore deep learning networks characterized by L hidden layers, ReLU activations, and a C2 Schatten class cost function. They focus on networks where input and output spaces, as well as hidden layers, share the same dimension (Q), thus operating in an underparameterized regime with potentially large input size (N). The paper assumes well-clustered training inputs.

1. Explicit Family of Minimizers: For configurations where $L \geq Q$, the authors explicitly construct a family of global minimizers, showing that these minimizers are degenerate. This construction reveals a robust geometric insight, in which hidden layers can be interpreted as recursive applications of a "truncation map," effectively reducing each input cluster to a distinct point.
2. Distinct Local Minima: The paper identifies and explicitly defines $2^Q - 1$ distinct degenerate local minima for the considered cost function, expanding the understanding of local minima beyond conventional expectations.
3. Impact of Truncation Maps: The reinterpretation of hidden layers as truncation maps offers a new perspective on the structural similarities between deep learning networks and concepts from renormalization group theory in quantum field analysis. This mapping approach effectively curates the input data by minimizing noise, refining the inputs' signal-to-noise ratio.
4. Rank-Preserving Transformations: The construction utilizes specific conditions to ensure the transformations used (via truncation maps) are rank-preserving, crucially maintaining the dimensions' integrity while minimizing the loss function.

Practical and Theoretical Implications

The findings have significant implications for the theoretical landscape of neural networks by providing new perspectives on the geometric structure of neural network minimizers. Practically, understanding the intrinsic geometry of these systems can enhance network design and training methods, potentially improving performance by informing better initialization strategies or architectures.

Speculation on Future Developments

Given the novel insights presented, future exploration may extend into more varied neural network architectures, potentially those incorporating non-ReLU activations or differing dimensional settings. Additionally, the insights regarding degenerate minima and geometric structures may inform the development of new optimization algorithms that step beyond gradient descent-based techniques.

In conclusion, this paper advances the understanding of deep learning networks' geometric underpinnings, offering explicit constructions of minimizers and reinterpreting hidden layer operations. The work presents a mathematically rigorous foundation that not only enhances theoretical perspectives but also suggests practical advancements in training strategies and network design.