Affine-invariant Cubic Newton with Weak Learners

Gradient Boosting Decision Trees (GBDTs) dominate tabular machine learning, with modern implementations like XGBoost, LightGBM, and CatBoost being based on Newton boosting: a second-order descent step in the space of decision trees. Despite its empirical success, the global convergence of Newton boosting is poorly understood compared to first-order boosting. In this paper, we introduce Restricted Newton Descent, which studies convex optimization with Newton's method on Hilbert spaces with inexact iterates, based on the concepts of cosine angle and weak gradient edge. Within this framework, we recover Newton boosting with GBDTs and classical finite-dimensional theory as special cases. We first prove that vanilla Newton boosting achieves a linear rate of convergence for smooth, strongly convex losses that satisfy a Hessian-dominance condition. To handle general convex losses with Lipschitz Hessians, we extend a recent gradient regularized Newton scheme to the restricted weak learner setting. This scheme minimally modifies the classical algorithm by introducing an adaptive $\ell_2$-regularization term proportional to the square root of the gradient norm at each iteration. We establish a $\mathcal{O}(\frac{1}{k^2})$ rate for this scheme, thereby obtaining a globally convergent second-order GBDT algorithm with a rate matching that of first-order boosting with Nesterov momentum. In numerical experiments, we show that our scheme converges while vanilla Newton boosting may diverge.

We study stochastic Cubic Newton methods for solving general possibly non-convex minimization problems. We propose a new framework, which we call the helper framework, that provides a unified view of the stochastic and variance-reduced second-order algorithms equipped with global complexity guarantees. It can also be applied to learning with auxiliary information. Our helper framework offers the algorithm designer high flexibility for constructing and analyzing the stochastic Cubic Newton methods, allowing arbitrary size batches, and the use of noisy and possibly biased estimates of the gradients and Hessians, incorporating both the variance reduction and the lazy Hessian updates. We recover the best-known complexities for the stochastic and variance-reduced Cubic Newton, under weak assumptions on the noise. A direct consequence of our theory is the new lazy stochastic second-order method, which significantly improves the arithmetic complexity for large dimension problems. We also establish complexity bounds for the classes of gradient-dominated objectives, that include convex and strongly convex problems. For Auxiliary Learning, we show that using a helper (auxiliary function) can outperform training alone if a given similarity measure is small.

Overparameterized networks often have ill-conditioned, saddle rich loss landscapes. They are usually trained with first-order methods that struggle on those landscapes, while many second-order methods stay infeasible at scale. Some scalable cubic methods replace the Hessian by a computationally cheap approximation and drop the exact block curvature. This paper introduces ARC-block, a blockwise adaptive regularization with cubics (ARC) optimizer. For each parameter tensor it minimizes a cubic model over the true block Hessian, with a per-block adaptive cubic constant and a monotone guard on the full loss. Large tensors are handled matrix-free in a Lanczos-built Krylov subspace. ARC-block's CubicKrylov subsolver's descent is monotone with a per-block cubic decrease bound. For exact steps, and for Krylov steps that meet the Cartis--Gould--Toint residual test, we provide an $\mathcal{O}(\sqrt{B}\varepsilon^{-3/2})$ sweep bound above a cross-block coupling floor. Another subsolver, ARC-$\varphi_1$, uses a trust-controlled exponential relaxation step. Among the cubic Newton methods evaluated here, these are the only ones whose steps stay exact on every parameter block (within the Krylov subspace), including on a $91.4$M-parameter implicit neural representation (INR). Run to convergence on FINER 2D image-fitting INR network, ARC-$\varphi_1$ reaches $133.5$ dB peak signal-to-noise ratio (PSNR), with no post peak collapse. Tuned Adam peaks at $78.2$ dB at $2{,}629$ s (step ${\sim}625$k, about $44$ minutes), further steps degrade its PSNR. On the same benchmark, SOAP, L-BFGS, and the scalable diagonal cubic method AdaCubic stay at or below Adam's $78.2$ dB peak. Over the benchmark's 20 images, run to a plateau rule, ARC-$\varphi_1$'s per-image PSNR has median $132.4$ and mean $131.5\pm3.3$ dB against tuned Adam's $72.8\pm4.5$ dB.