inequality manipulation

Classical mathematical inequalities lack a unified structural explanation. Method: This work pioneers the cross-domain transfer of the constraint modeling paradigm—originally developed for non-Shannon-type entropy inequalities in information theory—to algebra, probability theory, and functional analysis. Leveraging entropy-constrained modeling, group-theoretic and combinatorial structural analysis, generalization of probabilistic inequalities, and axiomatic derivation in inner-product spaces, we systematically reconstruct the AM–GM, Markov, and Cauchy–Schwarz inequalities. Contribution/Results: We establish a unified formal characterization of these three fundamental inequality classes and propose an extensible methodology for inequality discovery. This advances the foundational understanding of classical inequalities and inaugurates a novel, information-theoretic paradigm for systematically uncovering new mathematical constraint relations.

Existing generalization bounds fail to characterize approximate Bayesian inference algorithms—such as variational inference (VI)—that rely on stochastic gradient descent (SGD), leaving a critical theoretical gap. Method: This work systematically introduces algorithmic stability analysis into the approximate Bayesian inference framework. By quantifying how parameter perturbations during SGD affect the posterior approximation, it derives tight, verifiable, algorithm-specific generalization error bounds. Contribution/Results: The proposed bounds bridge the gap between traditional PAC-Bayes bounds and purely stability- or mutual-information-based bounds, while avoiding strong distributional or structural assumptions. Experiments on Bayesian deep learning models and real-world datasets demonstrate that the bounds are non-vacuous and effectively discriminate the generalization performance of different VI algorithms. This establishes a novel theoretical paradigm for analyzing approximate Bayesian inference methods.

This work addresses the poor scalability of traditional symbolic methods in automatically proving high-dimensional polynomial inequalities by introducing NSPI, a novel neurosymbolic framework that deeply integrates large language models (LLMs), symbolic computation, and formal verification in Lean. The approach leverages an LLM to generate sum-of-squares (SOS) decomposition conjectures, which are then refined into exact representations through symbolic optimization and formally verified within Lean, thereby establishing an end-to-end pipeline from heuristic discovery to rigorous proof. By synergistically combining the exploratory power of neural models with the mathematical rigor of symbolic and formal methods, NSPI achieves significantly improved efficiency and scalability, demonstrating strong performance on benchmarks involving up to ten variables.

Variational inequality (VI) problems commonly rely on stochastic methods assuming independent and identically distributed (i.i.d.) sampling, whereas practical implementations frequently adopt shuffling heuristics—randomly permuting data once and traversing it sequentially—despite the absence of rigorous theoretical foundations. Method: We propose a novel analytical framework tailored to non-i.i.d., sequential sampling, enabling the first rigorous convergence analysis of shuffling for VI problems. Contribution/Results: Our analysis establishes explicit iteration complexity upper bounds and proves that shuffling achieves convergence rates comparable to—or even better than—the optimal stochastic methods under both strongly monotone and monotone settings. The theoretical guarantees are validated across diverse benchmark VI problems, including saddle-point optimization and equilibrium computation. Empirical results demonstrate that shuffling significantly accelerates convergence over standard stochastic gradient methods, offering both computational efficiency and practical applicability.

There exists a significant gap between the theoretical convergence guarantees of deep learning optimization algorithms and their empirical performance, largely due to commonly adopted assumptions—such as Hessian boundedness—that lack empirical validation. Method: We introduce the first trajectory-aware measurement framework tightly aligned with key theoretical quantities, systematically evaluating the validity of mainstream assumptions across diverse architectures and datasets using large-scale training runs. Our framework quantifies dynamic properties—including gradient norms, Hessian spectral characteristics, and loss curvature—along optimization trajectories. Contribution/Results: We find that all examined theoretical assumptions fail to reliably predict actual convergence behavior and exhibit no robust correlation with optimization performance. This work uncovers a fundamental misalignment between theoretical modeling and practice, establishing the first reproducible benchmark for empirically calibrating and reconstructing optimization theory.

This work proposes a novel method for the automated discovery and verification of lower confidence bounds on the mean. By introducing a general relaxation framework parameterized by order statistics, the problem of finding optimal confidence bounds is formulated as a computationally tractable optimization problem, which unifies classical results such as Hoeffding’s inequality. The approach integrates mixed-integer linear programming with optimization relaxation theory to enable, for the first time, the automatic construction and formal verification of confidence bounds. In particular, when the order-statistic function is linear—as in the case of Hoeffding-type bounds—the method yields a mixed-integer linear program of linear size, allowing efficient approximation and rigorous validation of the target confidence bound.

This study investigates the stability of the Ingleton inequality under approximate conditional independence—specifically, when the conditional mutual information is small but nonzero—and quantifies the extent to which the inequality may be violated. To this end, we introduce a novel analytical framework that avoids tropicalization and dispenses with the intricate machinery of tropical probability spaces, thereby substantially simplifying the stability analysis. Within this framework, we derive explicit error bounds that improve upon existing estimates, resolve an open problem concerning the stability of the sum of two Ingleton expressions, and consequently construct a new family of entropy inequalities that confirm the stability of this sum.

This work proposes a dual-agent collaborative framework for automatically discovering convex relaxations to strengthen lower bounds in nonconvex optimization problems. An encoding agent generates tight constraints, while a theory agent validates their correctness through explicit dual feasible points and rigorous interval arithmetic. The approach pioneers the integration of large language model–driven autonomous research paradigms into convex relaxation construction, unifying automated lower-bound optimization with formal mathematical proof. The method achieves new state-of-the-art results on two classical optimization constants: improving $C_{6.2}$ from 1.28 to 1.2937 and $C_{6.5}$ from 0.379005 to 0.37912.

This work proposes a novel method to overcome the limitations of the classical absolute positivity criterion, which fails to handle nonlinear polynomial constraints involving universal quantifiers. Specifically, the approach addresses ∃∀ inequalities over the natural numbers by integrating monotonic algebra with well-founded order theory, thereby dispensing with the absolute positivity assumption. This advancement substantially broadens the class of constructible nonlinear polynomial interpretations. Experimental results demonstrate that the technique successfully solves constraint instances previously intractable to existing methods, thus extending the applicability of polynomial interpretations in termination and complexity analysis of term rewriting systems.

This work addresses the lack of machine-verifiable foundations in control theory for cyber-physical systems by developing an open-source formal library within the Lean interactive theorem prover. The library formalizes Lyapunov stability theory and the small-gain theorem, supporting continuous, discrete, and hybrid dynamical systems. A key contribution is a unified formulation of Lyapunov’s theorem applicable to both points and sets, alongside a relational definition of input–output systems that avoids well-posedness assumptions, enabling a fully formalized proof of the small-gain theorem. Leveraging mathematical tools such as neighborhood filters, the project establishes a scalable verification framework for control theory, laying the groundwork for trustworthy, machine-checked validation of cyber-physical systems.