aiqbench analysis
This research demonstrates that benign loss landscapes can coexist with worst-case computational hardness in tree tensor networks, revealing that gradient descent failure arises from high-order degenerate saddle points rather than poor local minima. By embedding arbitrary read-once Boolean formulas, the study provides a rigorous framework linking landscape geometry to polynomial-time learnability limits. This insight clarifies why practical tasks succeed despite theoretical expressiveness bounds, offering new directions for analyzing optimization dynamics in expressive models.
Benign Loss Landscapes Can Coexist with Worst-Case Hardness
arXiv:2609.13057v1 Announce Type: new 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.
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.
Machine Learning (cs.LG); Machine Learning (stat.ML)
arXiv:2609.13057 [cs.LG]
(or arXiv:2609.13057v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.13057 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 298 words
