Extra-Newton: A First Approach to Noise-Adaptive Accelerated Second-Order Methods

TL;DR

Extra-Newton introduces a universal, noise-adaptive second-order optimization algorithm that achieves optimal convergence rates for both stochastic and deterministic settings without prior knowledge of problem parameters.

Contribution

It is the first universal second-order method with adaptive step-size and global guarantees for both stochastic and deterministic convex optimization.

Findings

Achieves $O(\sigma / \sqrt{T})$ convergence with stochastic oracles.

Improves to $O(1 / T^3)$ convergence with deterministic oracles.

Operates without prior knowledge of smoothness, variance, or set diameter.

Abstract

This work proposes a universal and adaptive second-order method for minimizing second-order smooth, convex functions. Our algorithm achieves $O(\sigma / \sqrt{T})$ convergence when the oracle feedback is stochastic with variance $\sigma^2$, and improves its convergence to $O( 1 / T^3)$ with deterministic oracles, where $T$ is the number of iterations. Our method also interpolates these rates without knowing the nature of the oracle apriori, which is enabled by a parameter-free adaptive step-size that is oblivious to the knowledge of smoothness modulus, variance bounds and the diameter of the constrained set. To our knowledge, this is the first universal algorithm with such global guarantees within the second-order optimization literature.