Improving Equivariant Model Training via Constraint Relaxation (2408.13242v2)

Abstract: Equivariant neural networks have been widely used in a variety of applications due to their ability to generalize well in tasks where the underlying data symmetries are known. Despite their successes, such networks can be difficult to optimize and require careful hyperparameter tuning to train successfully. In this work, we propose a novel framework for improving the optimization of such models by relaxing the hard equivariance constraint during training: We relax the equivariance constraint of the network's intermediate layers by introducing an additional non-equivariant term that we progressively constrain until we arrive at an equivariant solution. By controlling the magnitude of the activation of the additional relaxation term, we allow the model to optimize over a larger hypothesis space containing approximate equivariant networks and converge back to an equivariant solution at the end of training. We provide experimental results on different state-of-the-art network architectures, demonstrating how this training framework can result in equivariant models with improved generalization performance. Our code is available at https://github.com/StefanosPert/Equivariant_Optimization_CR

• The paper introduces a novel framework that relaxes hard equivariance constraints to explore a broader hypothesis space during GCNN training.
• It employs a scheduled relaxation parameter and Lie derivative regularization to balance approximation with strict model equivariance.
• Empirical tests on point cloud classification, Nbody simulation, and molecular dynamics demonstrate improved generalization and efficiency.

Improving Equivariant Model Training via Constraint Relaxation

Recent advancements in neural network (NN) design have increasingly leveraged symmetries inherent in data through group equivariant convolutional neural networks (GCNNs). These networks, distinguished by their capacity to model transformations within a defined symmetry group, are prominent in diverse applications ranging from molecular biology to computer vision. While this explicit symmetry incorporation leads to enhanced generalization and data efficiency, it intrinsically complicates the model optimization process due to stringent equivariance constraints.

The paper by Pertigkiozoglou et al. addresses these optimization challenges head-on by introducing a novel framework for GCNN training called constraint relaxation. This method relaxes the hard constraint of equivariance during training, allowing the neural network to explore a broader hypothesis space that includes approximate equivariant mappings. Post-training, the relaxed models are projected back onto the space of strictly equivariant models. This framework provides GCNNs with a more flexible optimization landscape, promising to deliver models that blend optimal performance with rigorous symmetry constraints.

Framework Overview

The core idea centers on modifying the training of GCNNs to temporarily relax their strict equivariance constraints. Specifically, during training, an additional non-equivariant term is introduced to the network's intermediate layers. The magnitude of this term is controlled and progressively constrained, gradually guiding the model towards a strict equivariant solution by the time training is complete.

In formal terms, for a function $f$ that must be equivariant to a group $G$, the paper proposes the following structure for the network's intermediate layers:

$f(x) = f_e(x) + \theta W x$

where $f_e \in H$ (an equivariant function space), $$W \in \mathbb{R}^{|V_{\text{out}}| \times |V_{\text{in}}|}$$, and $\theta$ is a scheduled relaxation parameter. Initially, $\theta$ is set to a higher value, promoting a wider search space for optimization. As training progresses, $\theta$ is decreased, converging the network back into the strictly equivariant function space $H$.

Regularization via Lie Derivatives

A significant contribution of this work is the use of Lie derivative regularization to measure how closely a network approximates equivariance. For GCNN layers, the Lie derivative with respect to a group generator $A$ is defined as:

$$\mathcal{L}_A(W) = -d\rho_{\text{out}}(A)W + W d\rho_{\text{in}}(A)$$

where $\rho_{\text{in}}$ and $\rho_{\text{out}}$ are representations of the group $G$ acting on input and output spaces, respectively. During training, the norm of this Lie derivative is minimized to ensure that the network remains in close approximation to an equivariant model even when the constraint is relaxed.

Empirical Validation

The efficacy of this framework was demonstrated across several tasks:

1. Point Cloud Classification: The authors utilized the Vector Neurons PointNet and DGCNN architectures on the ModelNet40 dataset. The relaxed training framework significantly improved test accuracy, especially for smaller, potentially under-parameterized networks.
2. Nbody Simulation: Training SEGNNs to predict particle positions in a simulated physical system, the framework again showed improvements in generalization. Importantly, the performance gains were more pronounced for smaller networks and with larger training datasets.
3. Molecular Dynamics: Applying the framework within the Equiformer architecture for molecular energy and force prediction tasks yielded enhanced predictive accuracy on the MD17 dataset, demonstrating the utility of the proposed method in data-scarce regimes.
4. Approximate Equivariance: The method was also tested on approximately equivariant models, such as steerable CNNs for smoke flow prediction. By introducing the relaxed training framework, the authors demonstrated improved performance in settings with rotational and scale symmetries.

Implications and Future Directions

This work holds several practical and theoretical implications for the field of machine learning:

• Enhanced Optimization for Equivariant Networks: By providing a more tractable optimization landscape, it leads to the construction of higher-performing GCNNs without sacrificing the benefits of hardcoded symmetries.
• Scalability Across Domains: The framework's ability to improve training across varied domains (from physical simulations to molecular dynamics) highlights its general applicability.
• Potential for Broader Approximate Equivariance: The ability to incorporate relaxation into approximately equivariant networks suggests that this framework could potentially unify and optimize a broader class of symmetry-respecting models.

Future research might focus on extending this relaxation and projection methodology to more complicated symmetry groups beyond matrix Lie and discrete finite groups. Additionally, theoretical analysis to quantify the optimization error and further understand the dynamics of this relaxation framework could provide deeper insights.

In summary, by relaxing and progressively enforcing equivariance constraints during training, this paper proposes a method to surmount existing optimization challenges of GCNNs, broadening the horizon for their application and efficacy.