derive sensitivity lower bounds

Inferring sound lower bounds for least fixed points in quantitative verification of probabilistic programs remains challenging. Method: We propose the first lower-bound verification framework grounded in uniqueness conditions, generalizing ranking functions from non-probabilistic programs to the probabilistic setting. Our approach establishes a theoretical connection between generalized ranking supermartingales and uniqueness of fixed points, ensuring correctness of inferred lower bounds. The framework unifies verification of diverse quantitative properties—including weak pre-expectations, expected runtime, and higher-order moments—by integrating template-based constraint solving with weakest preexpectation semantics. Results: We implement an automated verification tool based on this framework. Experimental evaluation demonstrates significant improvements in both effectiveness and precision of lower-bound inference across a broad suite of probabilistic programs, including those with complex control flow and stochastic dynamics.

This work establishes the first polynomial sensitivity lower bounds for randomized approximation algorithms solving constraint satisfaction problems (CSPs), filling a key theoretical gap. To overcome the limitation of classical lower-bound techniques—which fail to preserve sensitivity—the authors innovatively adapt the PCP framework into a sensitivity-preserving variant, integrating Hamming distance metrics and analysis within the LOCAL model of distributed computing. The results yield tight polynomial sensitivity lower bounds for fundamental problems including Maximum Clique, Minimum Vertex Cover, and Maximum Cut. Concurrently, they imply tight round-complexity lower bounds for these problems in the LOCAL model. This is the first systematic demonstration of a deep connection between algorithmic sensitivity and distributed computational complexity, laying the foundation for a unified theory bridging the robustness of approximation algorithms and the scalability of distributed computation.

This study investigates the fundamental limits of output stability for approximation algorithms under infinitesimal input perturbations, formalized as sensitivity lower bounds. By transforming locally testable codes (LTCs) into constraint satisfaction problems (CSPs), the work introduces a general framework that, for the first time, leverages LTCs to establish linear sensitivity lower bounds for classic problems such as Max E3LIN2, and extends these results to settings like bipartite Max Cut. Combining complexity-theoretic reductions, analysis of Boolean function influences, and locality-preserving transformations from non-signaling models, the paper demonstrates that the sensitivity lower bounds for Max E3LIN2, Maximum Clique, and k-Cover tightly match their trivial upper bounds. This reveals an intrinsic barrier: even arbitrarily small input perturbations can induce substantial changes in near-optimal solutions.

This work addresses the long-standing bottleneck in prophet inequality research—manual construction of worst-case instances for tight competitive ratio proofs—by proposing the first unified framework that directly models tightness analysis as a computationally tractable optimization problem. Methodologically, it formalizes worst-case instance search as a convex optimization and linear programming problem subject to extremal probability distribution constraints, integrating tools from random-order theory and extremal probability analysis, thereby replacing the traditional decoupled paradigm of “algorithm analysis + counterexample construction.” Contributions include: (i) the first automated computation of tight competitive ratios; (ii) a unified derivation of tight bounds for multiple prophet inequality variants, yielding several new results; and (iii) rigorous verification of the optimality of several classical bounds. The framework significantly enhances the systematicity, scalability, and reliability of tightness proofs in online stochastic optimization.

This work establishes the first exponential lower bound on the resolution proof length for the Pigeonhole Principle (PHP) and Perfect Matching formulas over sparse, highly unbalanced expander graphs—bridging a long-standing theoretical gap between Ben-Sasson–Wigderson (for balanced constant-degree graphs) and Raz–Razborov (for highly unbalanced dense graphs). Methodologically, it extends Razborov’s pseudo-width technique to the previously intractable setting of highly unbalanced sparse graphs, integrating rigorous expansion analysis with novel combinatorial formula constructions to overcome the original technique’s reliance on graph balance and density. The result significantly broadens the applicability of pseudo-width methods and introduces a new paradigm for analyzing proof complexity over non-uniform graph structures.

Since the introduction of the Ideal Proof System (IPS) by Grochow and Pitassi (J. ACM 2018), a substantial body of work has established size lower bounds for IPS and its fragments. In particular, Forbes, Shpilka, Tzameret, and Wigderson (Theory Comput. 2021) developed the main lower-bound frameworks for restricted IPS fragments, namely functional lower bounds and the hard multiples method, while Alekseev, Grigoriev, Hirsch, and Tzameret (SIAM J. Comput. 2024) gave a general template for conditional lower bounds for full IPS. Yet all these lower bounds apply only to purely algebraic formulas over a field, that is, non-Boolean formulas not directly expressible in propositional logic. Proving lower bounds for CNF formulas has therefore remained a central open problem in this line of work. The current work resolves this question for IPS over read-once oblivious algebraic branching programs (roABPs) by proving lower bounds for refutations of CNF formulas in this system. Our approach is a rank-based feasible interpolation argument, following the method of Pudlák and Sgall (Proof Complexity and Feasible Arithmetic 1996) for monotone span programs, in which decomposing a given roABP refutation along a variable partition yields a low-dimensional space of polynomials from which we construct a span-program interpolant. We extend their result from Nullstellensatz refutations measured by degree to Nullstellensatz refutations measured by roABP size (i.e., roABP-IPS$_\text{LIN}$).

This work investigates the relationship between ordered structures—such as thresholds—and tree-like configurations, specifically 2-trees, within real-valued function classes exhibiting large sequential fat-shattering dimension. By integrating techniques from sequential fat-shattering dimension theory, stability analysis of functions, and order properties from combinatorial model theory, the paper introduces a more flexible framework for threshold extraction. This approach substantially improves upon existing bounds, resolving an open problem concerning the upper bound on the dual sequential fat-shattering dimension with at most a double-exponential dependence. In doing so, it corrects a previously flawed proof in the literature and strengthens related results by Anderson–Benedikt and Daskalakis–Golowich.

This work addresses the longstanding challenge of deriving strong lower bounds for the weak pigeonhole principle in proof complexity by introducing its algebraic generalization—the Weak Rank principle (WRank)—and constructing multiple encodings, including perfect matching and bamboo-tree CNF formulations. By innovatively designing a scalable lower-bound generator tailored for Polynomial Calculus Resolution over $F_2$ (PCR$_{F_2}$) and a novel pseudo-expectation method adapted to the Sherali–Adams system, the paper establishes the first exponential proof-length lower bounds for WRank in PCR$_{F_2}$, resolving a major open problem. It further demonstrates that circuit lower bound formulas admit no short proofs in this system. These results establish WRank as both necessary and sufficient for proving lower bounds against NC$^2$ and AC$^{0}[p]$, thereby cementing its central role in structured proof complexity analysis.

This work addresses the limitation imposed by strong global non-negativity, which hinders the automated synthesis of simple certificates for probabilistic programs, while naive relaxation of this condition compromises verification soundness. The paper introduces lazy Streett supermartingales and their lexicographic extensions, leveraging weak non-negativity within polynomial templates to enable sound verification of almost-sure satisfaction of ω-regular properties under a broad class of sampling distributions with bounded support. It is the first to generalize the weak non-negativity approach from termination analysis to general ω-regular specifications and provides a one-dimensional compositional characterization of lexicographic certificates. Experimental evaluation on 170 polynomial probabilistic program benchmarks demonstrates a 20.0–23.5 percentage point improvement in verification success rate over strong non-negativity baselines.

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.