Data-driven algorithm design frames hyperparameter tuning as a statistical learning problem, but establishing generalization guarantees remains challenging due to the implicit, non-smooth dependence of model performance on hyperparameters. Existing multi-dimensional bounds under piecewise-polynomial assumptions remain theoretically loose and lack comprehensive lower bounds. We resolve this by establishing tight pseudo-dimension bounds for multi-dimensional data-driven tuning. First, we refine the learning-theoretic upper bound using real algebraic geometry; by analyzing invariant connected sign cells during block elimination rather than isolated sign vectors, we avoid topological over-counting to derive strictly sharper sample complexities. Second, we present a multi-regime lower-bound framework that disentangles combinatorial and algebraic capacities. By constructing shattered problem instances across distinct regimes, we prove our upper bounds are tightly saturated. Finally, we extend our topological framework to accommodate general bi-level validation-loss tuning and broader semi-algebraic applications.
Recent research has developed practical, parallelizable first-order methods for large scale linear programming, but performance is highly dependent on hyperparameter selection. We derive generalization guarantees for hyperparameter tuning within (cu)PDLP, a state-of-the-art first-order LP solver designed for modern hardware. First, we pin down the behavior of PDHG, the primal-dual hybrid gradient algorithm that underlies PDLP, as a function of its step size and primal weight, leading to linear sample complexity guarantees for learning those parameters. We then conduct a structural analysis of PDLP, which augments PDHG with several specialized techniques like preconditioning, adaptive step sizes, averaging, adaptive restarts, and smoothed primal weight updates. Our analysis captures the behavior of the solution trajectory as a function of the hyperparameters and leverages recent advances in data-driven algorithm design to obtain polynomial sample complexity guarantees for learning those hyperparameters. Finally, we conduct proof-of-concept experiments that demonstrate the need for data-driven PDLP parameter tuning. Our results showcase the versatility of the data-driven algorithm design toolkit for principled hyperparameter tuning within solver-grade implementations of complex modern optimization algorithms.