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Paths and Ambient Spaces in Neural Loss Landscapes

Daniel Dold, Julius Kobialka, Nicolai Palm, Emanuel Sommer, David Rügamer, Oliver Dürr

TL;DR

Paths and ambient spaces in neural loss landscapes introduces a Bezier-parameterized loss path $b_{m{eta}}$ and a $K$-dimensional loss tunnel to study low-loss regions in high-dimensional neural networks. It advances subspace inference by embedding the path into an ambient tunnel and designing a principled tunnel prior to improve MCMC sampling, with a Rotation Minimizing Frame for stable tunnel construction. The work presents scaling laws, entropy–energy dynamics, and extensive experiments on synthetic data, UCI benchmarks, and MNIST, demonstrating improved uncertainty quantification and sampling conditioning in subspaces. Overall, the approach offers a practical framework for exploring neural loss landscapes and enhancing Bayesian inference through structured, geometry-aware priors and ambient-space lifting.

Abstract

Understanding the structure of neural network loss surfaces, particularly the emergence of low-loss tunnels, is critical for advancing neural network theory and practice. In this paper, we propose a novel approach to directly embed loss tunnels into the loss landscape of neural networks. Exploring the properties of these loss tunnels offers new insights into their length and structure and sheds light on some common misconceptions. We then apply our approach to Bayesian neural networks, where we improve subspace inference by identifying pitfalls and proposing a more natural prior that better guides the sampling procedure.

Paths and Ambient Spaces in Neural Loss Landscapes

TL;DR

Paths and ambient spaces in neural loss landscapes introduces a Bezier-parameterized loss path and a -dimensional loss tunnel to study low-loss regions in high-dimensional neural networks. It advances subspace inference by embedding the path into an ambient tunnel and designing a principled tunnel prior to improve MCMC sampling, with a Rotation Minimizing Frame for stable tunnel construction. The work presents scaling laws, entropy–energy dynamics, and extensive experiments on synthetic data, UCI benchmarks, and MNIST, demonstrating improved uncertainty quantification and sampling conditioning in subspaces. Overall, the approach offers a practical framework for exploring neural loss landscapes and enhancing Bayesian inference through structured, geometry-aware priors and ambient-space lifting.

Abstract

Understanding the structure of neural network loss surfaces, particularly the emergence of low-loss tunnels, is critical for advancing neural network theory and practice. In this paper, we propose a novel approach to directly embed loss tunnels into the loss landscape of neural networks. Exploring the properties of these loss tunnels offers new insights into their length and structure and sheds light on some common misconceptions. We then apply our approach to Bayesian neural networks, where we improve subspace inference by identifying pitfalls and proposing a more natural prior that better guides the sampling procedure.

Paper Structure

This paper contains 46 sections, 2 theorems, 32 equations, 21 figures, 3 tables, 2 algorithms.

Table of Contents

  1. INTRODUCTION AND RELATED WORK
  2. Our Contributions
  3. LOSS PATHS AND TUNNELS
  4. Notation and Objective
  5. Path Optimization
  6. Practical Implementation
  7. Path Characteristics and Dynamics
  8. Tunnel Embedding and Description
  9. Tunnel Symmetries
  10. ADVANCING SUBSPACE INFERENCE
  11. NUMERICAL EXPERIMENTS
  12. Scaling Behavior in Loss Landscapes
  13. Volume vs. Tunnel Sampling
  14. Tunnel Smoothness
  15. Regression Benchmarks
  16. ...and 31 more sections

Key Result

Theorem 1

If each $\bm{\theta}\in\bm{\Lambda}$ is permutation symmetry-free, an $\epsilon$-tube, $\epsilon>0$, exists around the path $b_{\bm{\Lambda}}$ that also contains no permutation symmetries.

Figures (21)

  • Figure 1: Negative log-likelihood (NLL; y-axis) along the path, comparing the method proposed in izmailov2020subspace to our approach across five different data splits and initializations (shaded regions represent one std.dev.).
  • Figure 2: Scaling behavior of path characteristics. Panel (A) shows the center of mass $\| \bar{\bm{\theta}} \|$, and Panel (B) shows the end-to-end distance $R_e$, both following square-root scaling with the effective time constant. Symbols denote different values of $\sigma$: $\circ \, \sigma = 0.1, \, \triangle \, \sigma = 1, \, \square \, \sigma = 5, \, \lozenge \, \sigma = 10$, though some values are only partially visible due to overplotting.
  • Figure 3: Frenet–Serret Frame (left) and our RMF implementation (right) on an exemplary 2D Bézier curve.
  • Figure 4: Comparison of a prior in the "volume space" (left), in the tunnel space with a uniform prior on $t$ (center) and in the tunnel space with adjusted prior through $s\sim U(0,\mathcal{S})$ (right).
  • Figure 5: Corresponding to \ref{['fig:scaling_law_plot1']}, Panels A and B show $\|\bar{\bm{\theta}}\|$ and the end-to-end distance ($R_e$), respectively. Panel C presents the training loss function, while Panel D depicts the Bézier curve length ($\mathcal{S}$). The lower panels include a weight decay of 0.1, which restricts free diffusion.
  • ...and 16 more figures

Theorems & Definitions (5)

  • Definition 1: Loss Path
  • Definition 2: Loss Tunnel
  • Theorem 1: informal
  • Theorem 2: Permutation Symmetries
  • proof : Proof of \ref{['thm:perm-inv']}