Mode Combinability: Exploring Convex Combinations of Permutation Aligned Models (2308.11511v1)

Abstract: We explore element-wise convex combinations of two permutation-aligned neural network parameter vectors $\Theta_A$ and $\Theta_B$ of size $d$. We conduct extensive experiments by examining various distributions of such model combinations parametrized by elements of the hypercube $[0,1]^{d}$ and its vicinity. Our findings reveal that broad regions of the hypercube form surfaces of low loss values, indicating that the notion of linear mode connectivity extends to a more general phenomenon which we call mode combinability. We also make several novel observations regarding linear mode connectivity and model re-basin. We demonstrate a transitivity property: two models re-based to a common third model are also linear mode connected, and a robustness property: even with significant perturbations of the neuron matchings the resulting combinations continue to form a working model. Moreover, we analyze the functional and weight similarity of model combinations and show that such combinations are non-vacuous in the sense that there are significant functional differences between the resulting models.

• The paper generalizes linear mode connectivity to mode combinability by exploring convex combinations of permutation-aligned models, revealing expansive low-loss regions in neural parameter space.
• The paper demonstrates through empirical studies that aligned models maintain low-loss connectivity even with significant perturbations in neuron matching.
• The study highlights practical implications for transfer learning and network design by leveraging mode combinability to enable efficient model stitching and scalable architectures.

Exploring Mode Combinability in Permutation-Aligned Neural Models

This paper explores the domain of neural network optimization landscapes, particularly focusing on the concept of mode combinability. It extends the commonly known idea of linear mode connectivity (LMC) to a broader paradigm by exploring convex combinations of permutation-aligned model parameter vectors.

Overview and Contributions

The authors propose that the low-loss regions, typically observed along linear interpolations between trained models, are part of a more extensive phenomenon termed "mode combinability." This hypothesis is examined through comprehensive empirical studies. The key contributions of the paper are summarized as follows:

1. Generalizing LMC to Mode Combinability: The paper introduces the idea of element-wise convex combinations of neural network parameters, suggesting that vast areas of this parameter space demonstrate low-loss surfaces, thereby expanding the applicability of LMC.
2. Empirical Properties of Model Alignment: The paper examines the transitivity property, demonstrating that models aligned to a common reference model show low-loss connectivity. It also highlights the robustness of model combinations against perturbations in neuron matchings.
3. Functional and Weight Dissimilarities: It demonstrates the non-trivial functional differences between original models and their combinations, suggesting that mode combinability is not just a superficial phenomenon of averaging but an interplay with meaningful diversity in model functionality.

Experimental Insights

• Network Architecture and Setup: The experiments were conducted using ResNet-20 and a simplified non-residual convolutional network, Tiny-10, both trained on CIFAR-10. The models were re-based using permutation alignment techniques to explore vast parameter spaces effectively.
• Sampling Distributions: Various distributions were studied, including uniform sampling from sections of the unit hypercube, and sampling based on Bernoulli distribution, among others. These efforts substantiate the claim that the loss surface manifold is extensive, supporting effective parameter combinations beyond mere linear interpolations.
• Performance and Robustness: The aligned models exhibited high robustness. Empirical findings include minimal degradation in performance from sampled model combinations, even when disrupting a significant portion of neuron alignments.

Theoretical and Practical Implications

This extension of LMC to mode combinability has several implications:

• Theoretical Understanding of Neural Landscapes: Even though permutation symmetries have been acknowledged, this work highlights their practical utility in navigating and extending the flatness of the optimization basin.
• Model Stitching and Transfer Learning: The identity stitching between aligned models suggests potential for exploiting model combinability in transfer learning and aggregating knowledge across distinct but related tasks.
• Network Width and Scalability: The results emphasize network width's critical role in maintaining low-loss combinatory properties, informing future architecture design choices where model merging is requisite.

Future Developments

The paper opens avenues for novel research directions including:

• Further exploration of hyperparameter choices and their impact on the viability of model combinations.
• Application of mode combinability in larger-scale architectures and diverse datasets, particularly in multi-task or federated learning scenarios.
• Algorithmic improvements in handling alignment and interpolation across more complex model architectures.

By establishing a broader understanding of mode combinability, this research paves the way for innovative methodologies in model development, potentially enhancing the efficiency and effectiveness of neural network training and deployment strategies.