Organizations: University of Electronic Science and Technology of China · Delft University of Technology
Abstract
The loss landscape of Deep Neural Networks (DNNs) exhibits highly complex and non-convex properties. Recent studies have revealed the phenomenon of mode connectivity, demonstrating that independently trained network modes can be connected via a continuous low-loss path. However, existing mode connectivity research is predominantly confined to classifier-based models, leaving it an open question whether similar geometric properties exist in modern complex models. In this paper, we extend the boundaries of mode connectivity to generative and contrastive domains (specifically DDPM and NanoCLIP). Addressing the unique architecture of DDPM and CLIP, we propose an architecture-aware connection building algorithm. Extensive empirical results demonstrate for the first time that we successfully discover mode connectivity between independently trained DDPM and NanoCLIP modes. Our work provides a novel perspective for understanding the geometric properties of the loss landscapes in modern generative and contrastive models.
Mode connectivity has been widely studied, yet the role of the optimizer remains underexplored. We revisit it through optimizer-induced implicit regularization, asking how connectivity behaves when restricted to solutions constrained by a given optimizer. For two-layer ReLU networks, we show that solutions from a single optimizer -- AdamW, Muon, or others in the Lion-K family -- form a connected set at sufficiently large width, a result not implied by prior work. We then characterize how optimizer-induced regions interact: at large width two different regions can be disjoint or overlap depending on regularization, while in our small-width example AdamW and Muon converge to disconnected zero-loss components separated by a provable loss barrier. Empirically, in GPT-2 pretraining, we observe same-optimizer paths preserve each model's spectrum while cross-optimizer paths traverse a smooth transition. Our results reveal optimizer-dependent structure beyond classical mode connectivity literature.
Many striking phenomena in deep learning, such as linear mode connectivity and the structured behavior of training dynamics, are closely tied to parameter symmetries: transformations that leave the realized function unchanged. Despite growing attention to parameter symmetries, the exact interplay between parameters, data, and representations remains underexplored. To investigate this, we develop a theoretical framework of effective function classes, i.e., the set of functions a neuron can realize on its input support, and the norm cost of realizing them. We then formalize effective symmetry breaking via neuron identifiability across independent training runs. Our analysis shows that neural networks can admit large families of approximately equivalent solutions even in structurally asymmetric models. We further show that neuron identifiability enables representation merging without prior alignment, and characterize when such merging admits a linear low-loss path. These findings highlight the role of effective function classes in affecting the loss landscape.
Vincent Bürgin, Daniel Herbst, Ya-Wei Eileen Lin +1
Machine Unlearning aims to remove undesired information from trained models without full retraining from scratch. Despite recent progress, the loss landscape and optimization geometry of unlearning are poorly understood. In this paper, we study machine unlearning through the lens of mode connectivity--the phenomenon that independently trained models can often be connected by smooth low-loss paths in parameter space. We introduce {\em mode connectivity in unlearning} (MCU) and evaluate it across a range of settings, including curriculum learning, second-order optimization, and connectivity across different unlearning methods. We find that many unlearned models lie in connected basins with smooth retain/forget behavior, while changes in training dynamics can move solutions into different basins. MCU also reveals that models within the same basin can differ substantially on privacy metrics, and that unlearning progresses nonlinearly from the original model to the unlearned model. In addition, linear connectivity suggests that most approximate unlearning methods are mechanistically distinct from retraining. Finally, MCU-based ensembling can improve generalization and robustness to relearning attacks, and MCU smoothness correlates with unlearning difficulty. To our knowledge, this is the first study of machine unlearning through the lens of mode connectivity.