arXiv subjects

Don Li

Publications and source records attributed to Don Li.

3 recordsLinked to original sources

arXiv2026

Toward Well-Posed Problems in the Social Sciences: Hadamard's Criteria as Epistemic Guardrails

We develop an interdisciplinary framework for evaluating the epistemic robustness of social-scientific inquiry through Hadamard's criteria for well-posed problems: existence, uniqueness, and stability. Reinterpreting these criteria not as demands for deterministic certainty but as methodological guardrails, we show how they diagnose recurrent failures of identification and inference across quantitative and qualitative paradigms. In quantitative research (e.g., econometric modeling), instability manifests when substantive conclusions depend sensitively on model specifications or data filtering. In interpretivist qualitative research, non-uniqueness and non-falsifiability arise when theoretical frameworks are elastic enough to accommodate contradictory observations without pre-specified rejection criteria. We frame these failures as inverse problems where the mapping from empirical data to substantive claims fails to satisfy existence, uniqueness, or stability. Finally, we propose cross-paradigmatic safeguards: ex-ante falsification criteria, empirical boundary conditions, multiverse sensitivity auditing, and cross-observer validation, to ensure social-scientific claims remain appropriately constrained, falsifiable, and robust to perturbations in evidence and interpretation.

math.HO↗

arXiv2026

On the Condition Number Upper Bound of the L-BFGS Inverse Hessian Approximation Matrix with a Two-Sided Geometric Envelope Safeguarding Mechanism

The limited-memory BFGS (L-BFGS) algorithm is a cornerstone of large-scale optimization due to its linear memory and computational costs. However, in ill-conditioned or non-convex landscapes, the implicit inverse Hessian approximation can suffer from an exploding condition number, leading to numerical instability and degraded convergence. To address this, we propose Two-Sided L-BFGS, a safeguarded variant that dynamically constrains the condition number of the inverse Hessian operator via a two-sided geometric envelope. Moreover, we show that Two-Sided L-BFGS preserves accumulated curvature information and maintains standard $O(mn)$ memory and per-iteration time complexities. We prove that this geometric envelope yields a uniform bound on the condition number of every inverse Hessian approximation generated by the algorithm. By tracking the algebraic evolution of the extreme eigenvalues through $m$ consecutive quasi-Newton updates starting from a scaled identity matrix, the resulting bound is expressed explicitly as a function of the memory depth, problem dimension, and envelope hyperparameters. Moreover, we show that Two-Sided L-BFGS preserves asymptotic global convergence in non-convex regimes under standard smoothness and strong Wolfe line-search assumptions, matching the theoretical guarantees of L-BFGS variants utilizing the Li-Fukushima cautious update rule. Numerical experiments on high-dimensional optimization problems demonstrate that the proposed method maintains well-conditioned inverse Hessian approximations and improves robustness and convergence behavior on ill-conditioned benchmarks.

math.OC↗

arXiv2026

A Unified Lyapunov-IQC Framework for Uniform Stability of Smooth Quadratic First-Order Accelerated Optimizers

We develop a unified Lyapunov-integral quadratic constraint (IQC) framework for establishing uniform stability of first-order accelerated optimization algorithms in the $β$-smooth and $γ$-strongly convex regime. Classical analyses of uniform stability, such as the work of Hardt, Recht, and Singer for stochastic gradient descent (SGD), rely on direct coupling arguments and case-by-case control of iterate differences under random sampling. Extending such arguments to accelerated methods, such as Nesterov Accelerated Gradient (NAG), is complicated by the presence of higher-order state dynamics induced by momentum. We first extend this classical approach with the use of Lyapunov functions to provide a uniform stability bound for smooth quadratic NAG, and supplement this result with small-scale numerical experiments. We then extend this framework by modeling first-order accelerated optimizers as Lur'e-type feedback interconnections between a linear dynamical system and a (non-linear) gradient operator. $β$-Smoothness and $γ$-strong convexity are encoded a sector IQC inequality. Under this representation, uniform stability is certified via the existence of a quadratic Lyapunov function satisfying a finite-dimensional linear matrix inequality (LMI) in the form of a feasibility problem, which can be solved via semi-definite programming (SDP). We instantiate this framework for NAG and show how classical uniform stability bounds can be recovered via this framework. These results underscore a structural connection between optimization dynamics and robust control theory, providing a modular methodology for reliable and reproducible numerical certification of uniform stability and generalization behavior of first-order methods via convex optimization tools that is adaptable to increasingly complex optimization algorithms.

math.OC↗