We develop a randomized Newton method capable of solving learning problems with huge dimensional feature spaces, which is a common setting in applications such as medical imaging, genomics and seismology. Our method leverages randomized sketching in a new way, by finding the Newton direction constrained to the space spanned by a random sketch. We develop a simple global linear convergence theory that holds for practically all sketching techniques, which gives the practitioners the freedom to design custom sketching approaches suitable for particular applications. We perform numerical experiments which demonstrate the efficiency of our method as compared to accelerated gradient descent and the full Newton method. Our method can be seen as a refinement and a randomized extension of the results of Karimireddy, Stich, and Jaggi (2019).

We propose a framework which makes it feasible to directly train deep neural networks with respect to popular families of task-specific non-decomposable performance measures such as AUC, multi-class AUC, $F$-measure and others. A common feature of the optimization model that emerges from these tasks is that it involves solving a Linear Programs (LP) during training where representations learned by upstream layers characterize the constraints or the feasible set. The constraint matrix is not only large but the constraints are also modified at each iteration. We show how adopting a set of ingenious ideas proposed by Mangasarian for 1-norm SVMs -- which advocates for solving LPs with a generalized Newton method -- provides a simple and effective solution that can be run on the GPU. In particular, this strategy needs little unrolling, which makes it more efficient during backward pass. Further, even when the constraint matrix is too large to fit on the GPU memory (say large minibatch settings), we show that running the Newton method in a lower dimensional space yields accurate gradients for training, by utilizing a statistical concept called {\em sufficient} dimension reduction. While a number of specialized algorithms have been proposed for the models that we describe here, our module turns out to be applicable without any specific adjustments or relaxations. We describe each use case, study its properties and demonstrate the efficacy of the approach over alternatives which use surrogate lower bounds and often, specialized optimization schemes. Frequently, we achieve superior computational behavior and performance improvements on common datasets used in the literature.

In this paper, we present the first stepsize schedule for Newton method resulting in fast global and local convergence guarantees. In particular, we a) prove an $\mathcal O \left( 1/{k^2} \right)$ global rate, which matches the state-of-the-art global rate of cubically regularized Newton method of Polyak and Nesterov (2006) and of regularized Newton method of Mishchenko (2021), and the later variant of Doikov and Nesterov (2021), b) prove a local quadratic rate, which matches the best-known local rate of second-order methods, and c) our stepsize formula is simple, explicit, and does not require solving any subproblem. Our convergence proofs hold under affine-invariant assumptions closely related to the notion of self-concordance. Finally, our method has competitive performance when compared to existing baselines which share the same fast global convergence guarantees.

Classical results indicate that under certain assumptions, Newton's method

Multitask reinforcement learning (MTRL) suffers from scalability issues when the number of tasks or trajectories grows large.

These methods can be broadly categorized into two types: function learning and operator learning approaches. In function learning, the goal is to directly learn the solution.

However, these methods typically reach superlinear convergence only when the stochastic Hessian noise diminishes, increasing per-iteration costs over time.

Empirically, we verify the computational efficiency of our methods.

Recent work in deep learning has opened new possibilities for solving classical algorithmic tasks using end-to-end learned models. In this work, we investigate the fundamental task of solving linear systems, particularly those that are ill-conditioned. Existing numerical methods for ill-conditioned systems often require careful parameter tuning, preconditioning, or domain-specific expertise to ensure accuracy and stability. In this work, we propose Algebraformer, a Transformer-based architecture that learns to solve linear systems end-to-end, even in the presence of severe ill-conditioning. Our model leverages a novel encoding scheme that enables efficient representation of matrix and vector inputs, with a memory complexity of $O(n^2)$, supporting scalable inference. We demonstrate its effectiveness on application-driven linear problems, including interpolation tasks from spectral methods for boundary value problems and acceleration of the Newton method. Algebraformer achieves competitive accuracy with significantly lower computational overhead at test time, demonstrating that general-purpose neural architectures can effectively reduce complexity in traditional scientific computing pipelines.