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Gradient-Normalized Smoothness for Optimization with Approximate Hessians

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.5Download Report PDF
optimizationsecond-order methodsHessian approximations
Abstract:

In this work, we develop new optimization algorithms that use approximate second-order information combined with the gradient regularization technique to achieve fast global convergence rates for both convex and non-convex objectives. The key innovation of our analysis is a novel notion called Gradient-Normalized Smoothness, which characterizes the maximum radius of a ball around the current point that yields a good relative approximation of the gradient field. Our theory establishes a natural intrinsic connection between Hessian approximation and the linearization of the gradient. Importantly, Gradient-Normalized Smoothness does not depend on the specific problem class of the objective functions, while effectively translating local information about the gradient field and Hessian approximation into the global behavior of the method. This new concept equips approximate second-order algorithms with universal global convergence guarantees, recovering state-of-the-art rates for functions with Hölder-continuous Hessians and third derivatives, quasi-self-concordant functions, as well as smooth classes in first-order optimization. These rates are achieved automatically and extend to broader classes, such as generalized self-concordant functions. We demonstrate direct applications of our results for global linear rates in logistic regression and softmax problems with approximate Hessians, as well as in non-convex optimization using Fisher and Gauss-Newton approximations.

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Overview

Overall Novelty Assessment

The paper introduces Gradient-Normalized Smoothness as a unifying framework for approximate second-order optimization, aiming to establish universal convergence guarantees across convex and non-convex settings. It resides in the 'Convergence Theory for Approximate Hessian Methods' leaf, which contains four papers total—a moderately populated node within the broader 'Theoretical Foundations and Convergence Analysis' branch. This placement indicates the work targets foundational convergence analysis rather than algorithm-specific implementation or domain applications, situating it in a core but not overcrowded research direction.

The taxonomy reveals neighboring leaves focused on 'Second-Order Approximation Theory' (three papers on mathematical frameworks for curvature approximation) and 'Optimality Conditions and Theoretical Characterizations' (three papers on KKT conditions and sequential optimality). The paper's emphasis on connecting Hessian approximation quality to gradient field linearization bridges these areas, potentially drawing on approximation theory while contributing convergence guarantees. Nearby branches address practical Hessian construction techniques and algorithmic variants, suggesting the work complements rather than duplicates existing implementation-focused studies.

Among twenty-three candidates examined, the Gradient-Normalized Smoothness concept (ten candidates, zero refutations) and universal convergence theory (three candidates, zero refutations) appear relatively novel within the limited search scope. The gradient-regularized Newton method with approximate Hessians (ten candidates, one refutation) shows overlap with prior work, indicating this algorithmic component may have precedent. The statistics suggest the conceptual framework is less anticipated than the specific algorithmic instantiation, though the search examined only top-ranked semantic matches rather than exhaustive coverage.

Based on the limited literature search, the work's novelty appears concentrated in its theoretical lens—Gradient-Normalized Smoothness as a problem-agnostic convergence criterion—rather than in the algorithmic mechanics. The analysis does not capture the full breadth of optimization theory, and deeper investigation of gradient regularization techniques or alternative smoothness characterizations could reveal additional overlaps. The taxonomy context suggests the contribution fills a gap in unifying convergence analysis across problem classes, though the extent of this gap remains partially characterized.

Taxonomy

Core-task Taxonomy Papers
50
3
Claimed Contributions
23
Contribution Candidate Papers Compared
1
Refutable Paper

Research Landscape Overview

Core task: optimization with approximate second-order information. This field addresses the challenge of exploiting curvature information to accelerate convergence without incurring the prohibitive cost of exact Hessian computation and inversion. The taxonomy organizes the landscape into six main branches. Theoretical Foundations and Convergence Analysis establishes rigorous guarantees for methods that rely on inexact or sampled curvature, examining conditions under which approximate second-order steps preserve superlinear or fast linear rates. Hessian Approximation Techniques explores diverse strategies—from low-rank factorizations and Kronecker structures to stochastic subsampling and quasi-Newton updates—that trade off fidelity against computational expense. Algorithm Design and Variants encompasses the spectrum of practical schemes, including trust-region methods, line-search Newton variants, and hybrid approaches that blend first- and second-order information. Implementation Strategies and Computational Efficiency focuses on making these methods scalable through parallelization, memory-efficient data structures, and adaptive preconditioning. Specialized Problem Classes tailors approximate Hessian techniques to settings such as constrained optimization, saddle-point problems, and manifold optimization, while Application Domains demonstrates their impact in neural network training, inverse problems, and large-scale scientific computing. Recent work has intensified around making second-order methods practical for deep learning and high-dimensional settings. A dense cluster of studies investigates how to construct cheap yet informative curvature approximations—ranging from diagonal or block-diagonal Hessian estimates to Kronecker-factored structures—that can be computed and inverted in near-linear time. Another active line examines the interplay between stochastic sampling and convergence guarantees, asking how subsampled Hessians or gradient covariances affect iteration complexity and generalization. The original paper ```json[0] sits squarely within the convergence theory branch, contributing rigorous analysis of how approximation errors propagate through Newton-type iterations. It complements foundational work on inexact Newton methods Inexact Hessian Newton[7] and recent studies on gradient-normalized smoothness Gradient-Normalized Smoothness[3], which together clarify when and why approximate second-order information suffices for fast convergence. By contrast, neighboring efforts such as Symplectic Stiefel Optimization[4] and Saddlepoint Topology Optimization[39] emphasize specialized geometric or constrained settings, highlighting the breadth of contexts in which approximate curvature plays a pivotal role.

Claimed Contributions

Gradient-Normalized Smoothness concept

The authors introduce Gradient-Normalized Smoothness, a new local characterization that measures how well the gradient field can be linearly approximated in a neighborhood of the current point. This concept unifies the treatment of Hessian approximation errors and Taylor approximation errors, providing a problem-class-free framework for analyzing second-order methods.

10 retrieved papers
Universal convergence theory for approximate second-order methods

The authors develop a unified convergence theory that automatically adapts to different problem classes and Hessian approximation quality. Their framework recovers existing state-of-the-art rates for various smoothness classes and extends to broader function classes such as generalized self-concordant functions, all without requiring prior knowledge of problem-specific parameters.

3 retrieved papers
Gradient-regularized Newton method with approximate Hessians

The authors propose Algorithm 1, which combines gradient regularization with approximate Hessian information. The method uses a universal step-size rule based on Gradient-Normalized Smoothness that automatically adjusts to both the objective's smoothness class and the Hessian approximation quality, achieving fast convergence without problem-specific tuning.

10 retrieved papers
Can Refute

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Gradient-Normalized Smoothness concept

The authors introduce Gradient-Normalized Smoothness, a new local characterization that measures how well the gradient field can be linearly approximated in a neighborhood of the current point. This concept unifies the treatment of Hessian approximation errors and Taylor approximation errors, providing a problem-class-free framework for analyzing second-order methods.

Contribution

Universal convergence theory for approximate second-order methods

The authors develop a unified convergence theory that automatically adapts to different problem classes and Hessian approximation quality. Their framework recovers existing state-of-the-art rates for various smoothness classes and extends to broader function classes such as generalized self-concordant functions, all without requiring prior knowledge of problem-specific parameters.

Contribution

Gradient-regularized Newton method with approximate Hessians

The authors propose Algorithm 1, which combines gradient regularization with approximate Hessian information. The method uses a universal step-size rule based on Gradient-Normalized Smoothness that automatically adjusts to both the objective's smoothness class and the Hessian approximation quality, achieving fast convergence without problem-specific tuning.

Table of Contents

Gradient-Normalized Smoothness for Optimization with Approximate Hessians | Novelty Validation