data-dependent upper bounds

This work addresses the challenge of diagnosing why neural network training stalls—a phenomenon that may stem from convergence to global or local minima, insufficient model expressivity, or optimizer mismatch. The authors propose a “training-under-challenge” framework that enables reproducible and verifiable diagnosis and repair by constructing alternative solutions within the same certification class via pre-declared executable certificate programs and re-evaluating the objective. Central to this approach are the notions of challenge-closed optimality and challenge capacity modulus, integrated with a squared-loss block descent operator, a channel-gated ResNet-18 architecture, and output-direction coverage verification. In distillation tasks with known optima, eight internal challenges cover all 240 output directions, achieving residual bounds within 1.74–3.02× of the true optimality gap, effectively disentangling underutilization of the decoder from genuine representational insufficiency.

To address the trustworthiness challenge in verifying multi-objective reachability, invariance, and long-run average rewards simultaneously in Markov decision processes (MDPs), this paper introduces the first logical query framework supporting existential and universal quantification over multiple objectives. Methodologically, we design a certified linear programming–based algorithm that jointly generates independently verifiable mathematical certificates, diagnostic schedulers, and minimal counterexample systems. Our approach transcends conventional single-objective certification by unifying formal correctness guarantees with human-interpretable explanations. The implemented prototype tool demonstrates, across multiple benchmarks, certificate compactness (averaging <5% of the state space), witness inspectability (100% manually verifiable), and computational feasibility on models with up to 10⁶ states.

In data-driven algorithm tuning, utility functions often exhibit piecewise discontinuities and Pfaffian structure, undermining theoretical guarantees. Method: This paper proposes the Pfaffian GJ learning framework—a novel generalization of the classical GJ framework from rational functions to the broader class of Pfaffian functions. It integrates Pfaffian theory, geometric complexity analysis, and piecewise modeling techniques to establish rigorous theoretical guarantees for distributed and online learning. Contribution/Results: Unlike prior approaches requiring rationality assumptions on utility functions, our framework is the first to provide provably sound learning guarantees for non-rational—particularly Pfaffian—utility functions. It significantly extends the theoretical applicability boundary and delivers a formally verifiable foundation for automated configuration of diverse parameterized algorithms, including SAT solvers and scheduling policies. The framework ensures statistical consistency and convergence under mild regularity conditions on Pfaffian structures, thereby bridging a critical gap between practical algorithm tuning and foundational learning theory.

This work addresses the family of parametric optimization problems and proposes the first unified, data-driven framework for analyzing the generalization performance of both classical and learned optimizers. Methodologically: (1) it introduces PAC-Bayes theory to the analysis of learned optimizers, deriving verifiable, high-probability generalization upper bounds; (2) it establishes performance bounds for classical optimizers based on empirical convergence rates; and (3) it pioneers a learning paradigm that directly minimizes the PAC-Bayes bound during training. Evaluated on signal processing, control, and meta-learning tasks, the derived bounds are significantly tighter than conventional worst-case guarantees. Moreover, the theoretical generalization guarantees for learned optimizers consistently exceed the empirical performance of their non-learned baselines—thereby unifying theoretical rigor with practical efficacy.

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 work addresses the challenge of accelerating offline NP-hard optimization using machine learning while preserving worst-case correctness guarantees. The authors propose CASP, a framework that integrates learned predictions with polynomial-time verifiable certificates: a predicted solution is accepted only when accompanied by a valid certificate, ensuring correctness independent of prediction quality. CASP introduces a verifiable pruning mechanism that enables distribution-agnostic probably approximately correct (PAC) learnability and enhances solver efficiency by breaking symmetries in degenerate solution spaces. Experiments across five NP-hard problems demonstrate that unverified pruning can incur up to 26% optimality loss under distribution shift, whereas CASP achieves zero performance degradation while maintaining rigorous theoretical guarantees.

Combinatorial optimization problems are notoriously difficult to solve due to their exponentially growing solution spaces, despite the fact that feasibility can be verified in polynomial time. This work proposes Neural Certificate Pricing (NCP), a method that trains a neural network within an unsupervised learning framework to predict dual prices and employs a structured recovery layer to generate primal solutions. NCP is the first approach to embed certificate consistency directly into the neural architecture, enabling amortized separation. Theoretically, it establishes that first-order errors in price prediction induce only second-order degradation in objective value, thereby guaranteeing solution quality. Empirical results demonstrate that NCP matches or significantly outperforms existing neural baselines across three classes of combinatorial optimization problems, achieving higher computational efficiency and stronger out-of-distribution generalization.

This study addresses the problem of selecting optimal experiments to maximally reduce the uncertainty bounds of a target causal query when causal effects are only partially identifiable and experimental costs are constrained. The authors formulate this as a maximum utility optimization problem, evaluating each experiment by its worst-case reduction in bound width and leveraging the underlying causal graph structure to efficiently prune low-value candidates. They introduce a novel path-interception rule combined with an ID-algorithm-based identifiability check, enabling linear-time identification of zero-utility experiments and drastically reducing the otherwise super-exponential search space. Empirical results demonstrate that their method prunes 50–88% of candidate experiments on average over random graphs and bnlearn benchmark networks without solving polynomial programs, and successfully identifies the optimal intervention strategy for estimating the effect of physical activity on diabetes using NHANES data.

This study addresses the challenge of uncontrolled compensation costs arising from erroneous interventions in cross-population data-driven decision-making. To ensure affordability, the authors propose a geometry-inspired framework that incorporates a certification mechanism to tightly bound the probability of decision errors when intervention effects are substantial. For the first time, they integrate decision affordability with computational geometry, rigorously demonstrating that Delaunay interpolation yields optimal worst-case cost guarantees. By synergistically combining matching estimators with computational geometry theory, the approach not only refines target population selection but also informs efficient data collection strategies. Empirical validation on semi-synthetic datasets from development economics confirms the method’s significant advantages in controlling compensation costs while enhancing decision efficacy.

Current formal theorem provers excel at Olympiad-level mathematics but suffer performance degradation on undergraduate optimization problems—such as those involving convexity and optimality conditions—due to distributional shift. This work proposes a domain-specific transfer approach for optimization: leveraging a strong Olympiad-level prover, it constructs a large-scale optimization dataset via expert iteration and introduces a novel preference learning objective that combines perplexity weighting with penalties for non-progress steps to guide efficient proof search. Evaluated on a newly curated optimization benchmark, the method achieves state-of-the-art Pass@1 and Pass@32 performance among models of comparable scale, while maintaining competitive results on general theorem-proving tasks, thereby enabling effective cross-domain transfer without catastrophic forgetting.