The paper demonstrates that tree tensor networks (TTNs) can encode arbitrary read‑once Boolean formulas, yielding polynomial‑size targets that are hard for gradient descent to learn in polynomial time, yet their loss landscapes are conditionally benign: every minimum‑norm local minimum is global. This shows that bad local minima are not the source of learning difficulty in TTNs; instead, high‑order degenerate saddle points caused by rank‑deficiency can impede learning. A case study on the parity function illustrates how TTNs can link landscape geometry to computational hardness.
arXiv:2602.13106v2 Announce Type: replace-cross Abstract: In recent years, there has been growing interest in understanding neural architectures' ability to learn to execute discrete algorithms, a li...
arXiv:2610.02182v1 Announce Type: cross Abstract: Quasi-Newton (QN) methods have long been among the most effective methods for large-scale unconstrained convex optimization. Two obstacles have limit...
arXiv:2606. 15219v1 Announce Type: new Abstract: In this work, we tackle the following question: Can neural networks trained with gradient-based methods achieve the optimal computational-statistical tradeoff in learning Gaussian single-index models?
The paper introduces Shallow Universal Polynomial Networks (SUPNs), a single‑hidden‑layer architecture that replaces standard activations with learnable multivariate polynomials, thereby increasing expressivity per unit. SUPNs achieve universal approximation by either widening the network or increasing polynomial complexity, and the authors prove convergence rates and quasi‑optimal parameter choices. Empirical tests across a wide range of smooth, nonsmooth, and discontinuous functions in up to ten dimensions show that SUPNs often outperform deep neural networks and Kolmogorov‑Arnold networks in accuracy and variability while requiring fewer parameters and lower training cost.
arXiv:2606. 00442v1 Announce Type: new Abstract: Many machine learning techniques rely on approximating a loss function's curvature, but this is notoriously hard to do at the scale of modern deep networks.
The paper introduces proximal equilibrium operators (PEOs), a class of neural operators that can simultaneously solve infinitely many convex optimization problems over a separable Hilbert space. A short training algorithm is presented that constructs PEOs using finite‑dimensional deep equilibrium layers, and a theoretical guarantee shows that, for inputs from a suitable infinite‑dimensional set, the trained PEOs uniformly approximate the solutions to arbitrary precision with logarithmic rank and width and linear depth in the inverse error. The authors validate their approach on applications to nonlinear PDEs, stochastic optimal control, and portfolio optimization and hedging in mathematical finance.
arXiv:2505. 22578v2 Announce Type: replace Abstract: The optimization of neural networks under weight decay remains poorly understood from a theoretical standpoint.
arXiv:2606. 28464v1 Announce Type: new Abstract: In the optimization of neural networks, gradient dynamics are influenced by critical points that arise from the model's architecture.
arXiv:2608. 16760v1 Announce Type: new Abstract: Reliable optimization is central to neural network (NN) training, yet Adam, the default optimizer for modern LLMs, rests on a fragile foundation.
arXiv:2510.05606v2 Announce Type: replace Abstract: Fundamental limits to predictability are central to our understanding of many physical and computational systems. In deep learning, training outcom...
arXiv:2601. 16884v3 Announce Type: replace Abstract: We study multigrade deep learning (MGDL) as a principled framework for structured error refinement in deep neural networks.
The paper introduces a pointwise generalization theory for fully connected deep neural networks, using a pointwise Riemannian Dimension derived from eigenvalues of learned feature representations across layers. This framework provides hypothesis-dependent, representation-aware generalization bounds that are significantly tighter than traditional size- or norm-based approaches, both theoretically and experimentally. The authors analytically identify structural properties that explain deep networks’ tractability and empirically show that the pointwise Riemannian Dimension captures feature compression, over‑parameterization effects, and optimizer bias.