Weight-Sharing Regularization

Abstract: Weight-sharing is ubiquitous in deep learning. Motivated by this, we propose a "weight-sharing regularization" penalty on the weights $w \in \mathbb{R}^d$ of a neural network, defined as $\mathcal{R}(w) = \frac{1}{d - 1}\sum_{i > j}^d |w_i - w_j|$. We study the proximal mapping of $\mathcal{R}$ and provide an intuitive interpretation of it in terms of a physical system of interacting particles. We also parallelize existing algorithms for $\operatorname{prox}_\mathcal{R}$ (to run on GPU) and find that one of them is fast in practice but slow ($O(d)$) for worst-case inputs. Using the physical interpretation, we design a novel parallel algorithm which runs in $O(\log^3 d)$ when sufficient processors are available, thus guaranteeing fast training. Our experiments reveal that weight-sharing regularization enables fully connected networks to learn convolution-like filters even when pixels have been shuffled while convolutional neural networks fail in this setting. Our code is available on github.

• The paper introduces weight-sharing regularization, a technique that enforces similar weight values across network components to mitigate overfitting.
• It rigorously derives the proximal mapping of the regularization function using a physical analogy and adapts parallel algorithms for GPU efficiency.
• Empirical results show that the method enables fully connected networks to learn convolution-like filters, even under pixel permutation challenges.

An Expert Overview of "Weight-Sharing Regularization"

The paper "Weight-Sharing Regularization" discusses a novel regularization technique for neural networks that encourages the sharing of weights across different components of the network. This method, termed weight-sharing regularization, introduces a penalty on the network's weights, defined specifically as $$\mathcal{R}(w) = \frac{1}{d-1}\sum_{i > j}^d |w_i - w_j|$$. This regularization is inspired by the widespread adoption of weight-sharing mechanisms in deep learning architectures such as CNNs and transformers.

Key Contributions

The paper makes several technical contributions of note:

1. Proximal Mapping of $\mathcal{R}$: The authors rigorously derive the proximal mapping for the weight-sharing regularization function $\mathcal{R}$. They employ a physical analogy using a system of interacting particles to provide an intuitive understanding of the proximal operation. This perspective not only illuminates the conceptual framework behind the regularization but also aids in algorithm development.
2. Parallel Algorithms: The authors provide parallel adaptations of existing algorithms for computing $\prox_{\mathcal{R}}$, adapted for execution on modern GPU architectures. One resulting algorithm achieves a depth of $O(\log^3 d)$ with sufficient computational resources, highlighting potential efficiency improvements for large-scale models.
3. Learning Convolution-like Filters: Experimentally, weight-sharing regularization demonstrated the capability of enabling fully connected networks to learn convolution-like filters. This ability was particularly evident even under pixel permutations, where traditional convolutional neural networks (CNNs) were unable to discern such patterns.

The contributions also extend to the empirical validation of the introduced concept across well-known benchmarks, such as MNIST and CIFAR10, demonstrating practical improvements in learning and generalization.

Implications and Future Directions

The introduction of weight-sharing regularization presents significant implications for both theoretical research and practical implementations in machine learning:

• Generalization Performance: By enforcing similarity among weights, the regularization technique holds promise for combating overfitting in fully connected neural networks—a common challenge that typically requires heuristic or architecture-specific solutions like dropout.
• Architecture-Independent Learning: The ability to learn convolution-like structures without explicit architectural constraints opens new avenues for generic architectures to self-discover efficient connectivity patterns, fostering potential advancements in neural architecture search (NAS).

In future research, it could be beneficial to explore different forms and generalizations of the weight-sharing penalty, particularly in multi-dimensional cases or where other weight correlation structures are desired. Additionally, the balance between computational efficiency and model accuracy will likely be a focal point, especially for handling large-scale datasets and models. Implementations of these algorithms in various deep learning frameworks can lead to broader community adoption, stimulating further exploration and optimization.

In conclusion, the exploration and successful application of weight-sharing regularization represent a notable advancement in the field of regularization techniques, promising to enhance the flexibility and performance of neural network models in a variety of learning tasks.