prove lipschitz continuity

This work addresses the challenge of verifying Lipschitz constants in conventional neural networks, which typically relies on computationally expensive methods or overly loose trivial bounds that fail to guarantee adversarial robustness and generalization. The authors propose a novel “verification-by-training” paradigm that integrates structural design to directly optimize and tighten trivial Lipschitz bounds during training, thereby circumventing complex post-hoc verification. Key innovations include norm-saturating polynomial activations (polyactivations), unbiased sinusoidal layers, and extensions to non-Euclidean norms, collectively eliminating three major sources of bound looseness. On MNIST, the resulting networks achieve Lipschitz bounds several orders of magnitude lower than existing approaches, with less than 10% error relative to the true Lipschitz constant, significantly enhancing both robustness and generalization performance.

Verifying stability of black-box nonlinear control systems is challenging when no prior dynamical model is available. Method: This paper proposes a model-free, data-driven stability verification method that directly learns the Lie derivative of a Lyapunov function—bypassing explicit system dynamics approximation. It integrates region-wise sampling-based validation with a counterexample-guided inductive synthesis (CEGIS) framework, underpinned by Lipschitz-based error bounds to ensure provably terminating synthesis. Contribution/Results: The approach certifies regional stability for 2D and 3D systems using only thousands of samples—requiring fewer than 0.01% of the samples needed by state-of-the-art black-box methods. It guarantees soundness and completeness within bounded regions, enables certified termination, and supports visualization of hard-to-verify stable regions.

Existing methods for estimating the Lipschitz constant of deep neural networks suffer from a trade-off between accuracy and scalability: semidefinite programming (SDP)-based approaches incur prohibitive computational cost and poor scalability, whereas closed-form methods yield overly conservative bounds. This work introduces a novel class of closed-form, scalable Lipschitz upper bounds. By generalizing the parameterization of the LipSDP feasible region and integrating matrix norm inequalities with layer-wise propagation bound optimization, our method enables cooperative utilization of multiple parameterized feasible points—without invoking an SDP solver—for the first time. The approach unifies and generalizes ECLipsE-Fast, achieving significantly tighter bounds and higher computational efficiency on large-scale networks. It supports real-time robustness verification for models with up to hundreds of millions of parameters.

This work addresses the lack of explicit characterizations of Lipschitz constants for feature maps induced by integral kernels—a gap that hinders robustness and stability guarantees in kernel methods. Building on functional analysis, probability integral transforms, and kernel theory, the study investigates the Lipschitz regularity of such feature maps under differentiability conditions, establishing sufficient conditions for continuity and deriving explicit formulas for the associated constants. For the first time, closed-form expressions of Lipschitz constants are provided for Gaussian kernels, ReLU random neural network kernels, and translation-invariant kernels with cosine activation, revealing an equivalence between this Lipschitz property and the existence of the second moment of the weight distribution. Numerical experiments confirm the convergence behavior of these constants in finite-width networks, and the paper concludes by posing open questions regarding their asymptotic properties.

This work addresses the neglect of input-space regularity mechanisms in theoretical analyses of deep neural network generalization error bounds, focusing specifically on the dynamic evolution of the empirical Lipschitz constant during the double-descent phenomenon. Methodologically, we conduct a systematic analysis of SGD training trajectories, gradient magnitude estimation, loss landscape curvature approximation, and phase segmentation of double descent. Our key contribution is the first empirical demonstration that the Lipschitz constant exhibits pronounced non-monotonic surge-and-decay behavior near the critical transition regime—precisely synchronized with peaks and troughs in test error. Furthermore, we establish that, near the critical point, the norm of parameter-space gradients tightly couples with the input-space Lipschitz constant; moreover, both model complexity and optimization dynamics are jointly characterized by loss curvature and the Euclidean distance of parameters from initialization. This work provides a novel geometric perspective and quantifiable mechanistic framework for understanding double descent.

This study investigates the computational complexity of computing the Lipschitz constant of the solution mapping for multiparametric quadratic programs—a quantity essential for optimization-based control analysis. By leveraging computational complexity theory, APX-hardness analysis, and parameterized complexity techniques, the work establishes for the first time that this problem is not only NP-hard but also APX-hard, even in the scalar parameter case. Nevertheless, the problem becomes polynomial-time solvable when either the number of constraints or the number of decision variables is fixed. These theoretical findings demonstrate that the intrinsic difficulty stems from the number of constraints and decision variables rather than the dimensionality of the parameters. Numerical experiments corroborate the validity of these conclusions.

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 study addresses the conditions under which the strong law of large numbers holds for locally Lipschitz stochastic functions under a Lipschitz pseudometric, overcoming limitations present in existing literature. By introducing the o-minimal structure assumption from model theory into this probabilistic framework—an approach not previously explored—the work integrates local Lipschitz analysis with pseudometric techniques to substantially broaden the class of admissible functions. The main contribution lies in establishing the validity of the strong law of large numbers for a wide class of functions, including those definable in o-minimal structures, while simultaneously ensuring uniform convergence of Clarke subdifferentials and finite-sample identifiability of solutions.

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.

Existing quadratic constraint approaches for characterizing neural network activation functions are overly conservative, limiting the precision of reachability and safety analyses. This work proposes a domain-dependent framework for verifiable quadratic inequalities: it generates candidate constraints via local sampling and employs sum-of-squares (SOS) certificates to ensure global validity, yielding tight and sound quadratic representations for scalar nonlinearities. The method transcends the limitations of conventional sector or slope bounds by incorporating neuron-wise dependencies and local bound refinement—particularly for ReLU networks—to reduce conservatism. It is compatible with convex quadratic programming, semialgebraic set descriptions, and integral quadratic constraint (IQC) techniques. Experiments demonstrate that the framework significantly improves analysis accuracy for smooth activations such as tanh and extends effectively to systems involving saturation-type nonlinearities.