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1. Convergence of Hyperbolic Neural Networks Under Riemannian Stochastic Gradient Descent

   https://doi.org/10.1007/s42967-023-00302-9  

   Whiting, Wes; Wang, Bao; Xin, Jack (October 2023, Communications on Applied Mathematics and Computation)

   Abstract We prove, under mild conditions, the convergence of a Riemannian gradient descent method for a hyperbolic neural network regression model, both in batch gradient descent and stochastic gradient descent. We also discuss a Riemannian version of the Adam algorithm. We show numerical simulations of these algorithms on various benchmarks. 

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2. Acceleration and Implicit Regularization in Gaussian Phase Retrieval

   Maunu, Tyler; Molina-Fructuoso, Martin (May 2024, Proceedings of Machine Learning Research)

   Dasgupta, Sanjoy; Mandt, Stephan; Li, Yingzhen (Ed.)

   We study accelerated optimization methods in the Gaussian phase retrieval problem. In this setting, we prove that gradient methods with Polyak or Nesterov momentum have similar implicit regularization to gradient descent. This implicit regularization ensures that the algorithms remain in a nice region, where the cost function is strongly convex and smooth despite being nonconvex in general. This ensures that these accelerated methods achieve faster rates of convergence than gradient descent. Experimental evidence demonstrates that the accelerated methods converge faster than gradient descent in practice. 

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3. Quantifying the Benefit of Using Differentiable Learning over Tangent Kernels

   Malach, Eran; Kamath, Pritish; Abbe, Emmanuel; Srebro, Nathan (July 2021, Proceedings of Machine Learning Research)

   null (Ed.)

   We study the relative power of learning with gradient descent on differentiable models, such as neural networks, versus using the corresponding tangent kernels. We show that under certain conditions, gradient descent achieves small error only if a related tangent kernel method achieves a non-trivial advantage over random guessing (a.k.a. weak learning), though this advantage might be very small even when gradient descent can achieve arbitrarily high accuracy. Complementing this, we show that without these conditions, gradient descent can in fact learn with small error even when no kernel method, in particular using the tangent kernel, can achieve a non-trivial advantage over random guessing. 

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4. Superpolynomial Lower Bounds for Learning One-Layer Neural Networks using Gradient Descent

   Goel, Surbhi; Gollakota, Aravind; Jin, Zhihan; Karmalkar, Sushrut; Klivans, Adam (July 2020, International Conference on Machine Learning)

   We prove the first superpolynomial lower bounds for learning one-layer neural networks with respect to the Gaussian distribution using gradient descent. We show that any classifier trained using gradient descent with respect to square-loss will fail to achieve small test error in polynomial time given access to samples labeled by a one-layer neural network. For classification, we give a stronger result, namely that any statistical query (SQ) algorithm (including gradient descent) will fail to achieve small test error in polynomial time. Prior work held only for gradient descent run with small batch sizes, required sharp activations, and applied to specific classes of queries. Our lower bounds hold for broad classes of activations including ReLU and sigmoid. The core of our result relies on a novel construction of a simple family of neural networks that are exactly orthogonal with respect to all spherically symmetric distributions. 

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5. Mirrorless Mirror Descent: A Natural Derivation of Mirror Descent

   Gunasekar, Suriya; Woodworth, Blake; Srebro, Nathan (January 2021, Proceedings of Machine Learning Research)

   null (Ed.)

   We present a direct (primal only) derivation of Mirror Descent as a “partial” discretization of gradient flow on a Riemannian manifold where the metric tensor is the Hessian of the Mirror Descent potential function. We contrast this discretization to Natural Gradient Descent, which is obtained by a “full” forward Euler discretization. This view helps shed light on the relationship between the methods and allows generalizing Mirror Descent to any Riemannian geometry in Rd, even when the metric tensor is not a Hessian, and thus there is no “dual.” 

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