SAM-on-the-Curve: Sharpness-Aware Mode Connectivity for Robust Weight-Space Interpolation

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-$\mathcal{K}$ 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 neural networks operations have a multiplicative nature rather than additive: halving or doubling a norm are analogous relatively but require unequal optimization distances when taking linear steps. Adaptive optimizers such as Adam normalize updates per coordinate, but update steps remain additive; weights with very different magnitudes receive similarly sized absolute changes, producing very different relative perturbations. We introduce \textbf{\method} (\textbf{\methodshort}), a weight reparameterization for neural networks that combines a sign-aware symmetric-exponential pathway with an identity-like linear pathway. The symmetric-exponential pathway is near-linear for small raw weights but increasingly curved at larger magnitudes. Additive updates in logarithmic space map to magnitude-proportional changes in effective weight space. The linear pathway provides a direct route through the transform that we hypothesize stabilizes optimization, while learnable scale, curvature, and offset parameters control balance between pathways and the curvature of the exponential pathway. These components create a curved parameter-space geometry that empirically improves speed of loss descent over standard linear parameterization. We also identify a useful \emph{mismatched initialization}: raw weights are chosen so a symmetric version of the transform matches Xavier statistics, but training uses an asymmetric forward transform that leaves positive weights at full strength while making negative weights smaller in magnitude; in small-model ablations, this improves early optimization and may act as a form of symmetry breaking. We train transformers on OpenWebText over nine width$\times$depth configurations, \methodshort reaches matched validation loss in 1.32--1.49$\times$ fewer training steps, with the largest widths seeing the biggest gains.

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.