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1. Newton-LESS: Sparsification without Trade-offs for the Sketched Newton Up-date

   Derezinski, Michal; Lacotte, Jonathan; Pilanci, Mert; Mahoney, Michael W. (January 2021, Advances in neural information processing systems)

   In second-order optimization, a potential bottleneck can be computing the Hessian matrix of the optimized function at every iteration. Randomized sketching has emerged as a powerful technique for constructing estimates of the Hessian which can be used to perform approximate Newton steps. This involves multiplication by a random sketching matrix, which introduces a trade-off between the computational cost of sketching and the convergence rate of the optimization algorithm. A theoretically desirable but practically much too expensive choice is to use a dense Gaussian sketching matrix, which produces unbiased estimates of the exact Newton step and which offers strong problem-independent convergence guarantees. We show that the Gaussian sketching matrix can be drastically sparsified, significantly reducing the computational cost of sketching, without substantially affecting its convergence properties. This approach, called Newton LESS, is based on a recently introduced sketching technique: LEverage Score Sparsified (LESS) embeddings. We prove that Newton-LESS enjoys nearly the same problem-independent local convergence rate as Gaussian embeddings, not just up to constant factors but even down to lower order terms, for a large class of optimization tasks. In particular, this leads to a new state-of-the-art convergence result for an iterative least squares solver. Finally, we extend LESS embeddings to include uniformly sparsified random sign matrices which can be implemented efficiently and which perform well in numerical experiments. 

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2. Stochastic Quasi-Newton Methods

   https://doi.org/10.1109/JPROC.2020.3023660  

   Mokhtari, Aryan; Ribeiro, Alejandro (November 2020, Proceedings of the IEEE)

   null (Ed.)
3. Newton-Anderson at Singular Points

   https://doi.org/10.4208/ijnam2023-1029  

   Dallas, Matt; Pollock, Sara (October 2023, International Journal of Numerical Analysis and Modeling)

   In this paper we develop convergence and acceleration theory for Anderson acceleration applied to Newton’s method for nonlinear systems in which the Jacobian is singular at a solution. For these problems, the standard Newton algorithm converges linearly in a region about the solution; and, it has been previously observed that Anderson acceleration can substantially improve convergence without additional a priori knowledge, and with little additional computation cost. We present an analysis of the Newton-Anderson algorithm in this context, and introduce a novel and theoretically supported safeguarding strategy. The convergence results are demonstrated with the Chandrasekhar H-equation and a variety of benchmark examples. 

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4. Stochastic Newton Proximal Extragradient Method

   Jiang, Ruichen; Dereziński, Michał; Mokhtari, Aryan (December 2024, Advances in Neural Information Processing Systems (NeurIPS 2024))

   Stochastic second-order methods are known to achieve fast local convergence in strongly convex optimization by relying on noisy Hessian estimates to precondition the gradient. Yet, most of these methods achieve superlinear convergence only when the stochastic Hessian noise diminishes, requiring an increase in the per-iteration cost as time progresses. Recent work in \cite{na2022hessian} addressed this issue via a Hessian averaging scheme that achieves a superlinear convergence rate without increasing the per-iteration cost. However, the considered method exhibits a slow global convergence rate, requiring up to ~O(κ^2) iterations to reach the superlinear rate of ~O((1/t)^{t/2}), where κ is the problem's condition number. In this paper, we propose a novel stochastic Newton proximal extragradient method that significantly improves these bounds, achieving a faster global linear rate and reaching the same fast superlinear rate in ~O(κ) iterations. We achieve this by developing a novel extension of the Hybrid Proximal Extragradient (HPE) framework, which simultaneously achieves fast global and local convergence rates for strongly convex functions with access to a noisy Hessian oracle. 

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5. Stochastic Newton Proximal Extragradient Method

   Jiang, Ruichen; Dereziński, Michał; Mokhtari, Aryan (November 2024, https://doi.org/10.48550/arXiv.2406.01478)

   Stochastic second-order methods accelerate local convergence in strongly convex optimization by using noisy Hessian estimates to precondition gradients. However, they typically achieve superlinear convergence only when Hessian noise diminishes, which increases per-iteration costs. Prior work [arXiv:2204.09266] introduced a Hessian averaging scheme that maintains low per-iteration cost while achieving superlinear convergence, but with slow global convergence, requiring 𝑂 ~ ( 𝜅 2 ) O ~ (κ 2 ) iterations to reach the superlinear rate of 𝑂 ~ ( ( 1 / 𝑡 ) 𝑡 / 2 ) O ~ ((1/t) t/2 ), where 𝜅 κ is the condition number. This paper proposes a stochastic Newton proximal extragradient method that improves these bounds, delivering faster global linear convergence and achieving the same fast superlinear rate in only 𝑂 ~ ( 𝜅 ) O ~ (κ) iterations. The method extends the Hybrid Proximal Extragradient (HPE) framework, yielding improved global and local convergence guarantees for strongly convex functions with access to a noisy Hessian oracle. 

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