Momentum Residual Neural Networks (2102.07870v3)

Abstract: The training of deep residual neural networks (ResNets) with backpropagation has a memory cost that increases linearly with respect to the depth of the network. A way to circumvent this issue is to use reversible architectures. In this paper, we propose to change the forward rule of a ResNet by adding a momentum term. The resulting networks, momentum residual neural networks (Momentum ResNets), are invertible. Unlike previous invertible architectures, they can be used as a drop-in replacement for any existing ResNet block. We show that Momentum ResNets can be interpreted in the infinitesimal step size regime as second-order ordinary differential equations (ODEs) and exactly characterize how adding momentum progressively increases the representation capabilities of Momentum ResNets. Our analysis reveals that Momentum ResNets can learn any linear mapping up to a multiplicative factor, while ResNets cannot. In a learning to optimize setting, where convergence to a fixed point is required, we show theoretically and empirically that our method succeeds while existing invertible architectures fail. We show on CIFAR and ImageNet that Momentum ResNets have the same accuracy as ResNets, while having a much smaller memory footprint, and show that pre-trained Momentum ResNets are promising for fine-tuning models.

• The paper introduces a momentum term in ResNets that transforms forward dynamics to enable invertible computation and reduced memory usage.
• The approach leverages a second-order ODE framework to enhance representational capabilities and achieve universal approximation.
• Empirical results on CIFAR-10, CIFAR-100, and ImageNet show maintained accuracy with significantly lower memory requirements.

An Examination of Momentum Residual Neural Networks

The paper "Momentum Residual Neural Networks" presents notable advancements in the domain of deep learning by introducing a novel architecture: Momentum Residual Neural Networks (Momentum ResNets). The core innovation lies in incorporating a momentum term into the forward pass of traditional Residual Networks (ResNets), resulting in an invertible network with reduced memory footprint.

Technical Advancements

Momentum ResNets aim to address the memory-intensive nature of deep learning architectures, especially during backpropagation. Conventional ResNets require storing activations at each layer, which can become prohibitive as network depth increases. By constructing networks that are invertible, Momentum ResNets enable on-the-fly recomputation of layer activations, eschewing the need for storage and effectively reducing memory requirements.

Forward Rule Modification: In a typical ResNet, the forward pass is given by $x_{n+1} = x_n + f(x_n, \theta_n)$. The Momentum ResNet modifies this to: $\begin{array}{r@{\hspace{1mm}l} v_{n+1} & = \gamma v_n + (1-\gamma) f(x_n,\theta_n) \ x_{n+1} & = x_n + v_{n+1} \end{array}$ Here, $\gamma$ is a momentum term that can be adjusted to fine-tune the network's representation capacity and memory savings. The incorporation of this momentum term transforms the dynamics of the network, allowing for reversible computations.

Theoretical Insights

Momentum ResNets can be interpreted through the lens of continuous mathematics as second-order ordinary differential equations (ODEs), in contrast to the first-order ODE framework within which typical ResNets are understood. This second-order behavior, facilitated by the momentum term, enhances the model's ability to represent complex functions.

Universality and Representation Capabilities: The paper argues that Momentum ResNets offer a richer representation framework compared to ResNets or first-order neural ODEs, capable of universal approximation in the linear case. The authors provide theoretical evidence that by increasing the momentum term, the set of representable mappings grows larger, even encompassing function sets that first-order models cannot achieve.

Empirical Validation

Empirical evidence from experiments conducted on CIFAR-10, CIFAR-100, and ImageNet datasets corroborates the theoretical claims. The paper demonstrates that Momentum ResNets achieve comparable classification accuracy to ResNets while maintaining a significantly lower memory footprint. Additionally, the flexibility in the momentum term allows pre-trained ResNet models to be seamlessly converted to Momentum ResNet counterparts, facilitating model fine-tuning without extensive re-training.

Learning to Optimize: In settings where convergence to a fixed point is desirable, such as in optimization tasks, Momentum ResNets exhibit superior performance over invertible architectures like RevNets. This capability is attributed to the stable fixed point introduced by the momentum term, an improvement highlighted by experiments within the Learned-ISTA framework.

Implications and Future Directions

The introduction of Momentum ResNets offers profound implications for the deployment of deep learning models in memory-constrained environments. Practically, it enables the scaling of deep learning applications where hardware limitations previously restricted network depth. Theoretically, it aligns deep learning more closely with continuous mathematics frameworks, promoting further exploration into dynamical systems and their potential in creating more efficient neural architectures.

Future research could explore further integration of numerical approaches from differential equations, the exploration of non-linear dynamics in Momentum ResNets, and the adaptation of this framework to other neural network paradigms beyond image classification, such as natural language processing and reinforcement learning. Overall, Momentum ResNets represent a significant stride towards optimizing memory efficiency in deep learning architectures without compromising accuracy or computational feasibility.