Benign loss landscapes coexist with learning difficulty in tree tensor networks
Benign Loss Landscapes Can Coexist with Worst-Case Hardness
Machine Learning
Summary
Deep learning models sometimes face tasks that are easy to evaluate but very hard to learn using gradient-based methods. The authors study a math-based model called tree tensor networks, which can represent complex logical formulas that are hard to learn efficiently. They found that, surprisingly, bad local solutions are not the cause of this learning difficulty. Instead, the problem comes from subtle flat regions in the learning landscape, revealing new insights into when and how learning can be hard.
What this means in practice
- •For machine learning engineers: Identify potential learning bottlenecks in complex model architectures by analyzing loss landscape geometry beyond local minima.
- •For cryptographers: Use insights about hard-to-learn functions embedded in tree tensor networks to design systems resistant to gradient-based attacks.
A theory result. No direct application yet.
Authors
Zach Furman, Stephan Wäldchen, Yangda Bei, Liam Hodgkinson
Abstract
Deep neural networks are expressive enough to contain worst-case targets that can be evaluated in polynomial time but cannot be learned in polynomial time by gradient descent. For practical tasks they nonetheless learn well, raising the question of what non-generic structure of real-world targets enables this. Existing surrogate models cannot pose this question because they either lack hard-to-learn targets entirely (deep linear networks) or cannot evaluate such targets efficiently (kernel methods, infinite-width limits). We study tree tensor networks (TTNs), a model class that generalizes deep linear networks and Tucker decompositions. We show they embed arbitrary read-once Boolean formulas, and thus contain polynomial-size targets that cannot be learned by gradient descent in polynomial time under the same mechanism as neural networks. Despite this, we prove that their loss landscapes are conditionally benign for every realizable target: every local minimum that is minimum-norm is global. Thus, surprisingly, bad local minima are not what distinguishes between typical and worst-case problems in TTNs. Instead, learning difficulty in TTNs can arise from high-order degenerate saddle points, which we show are caused by rank-deficiency. This is explored through a case study of the parity function, illustrating the potential for TTNs to relate landscape geometry to computational hardness.