🤖 AI Summary
This work addresses the theoretical analysis of algorithmic stability, aiming to characterize the sensitivity of combinatorial algorithms to input perturbations and to elucidate the fundamental trade-off between stability and solution quality—particularly generalization error. We propose the first unified stability framework encompassing randomized, iterative, and distributed algorithms. Within this framework, we establish tight equivalence conditions linking stability to uniform convergence and generalization bounds. Leveraging probabilistic inequalities, empirical process theory, and Lipschitz sensitivity decomposition, we derive sharp stability bounds for canonical algorithms including stochastic gradient descent (SGD) and empirical risk minimization (ERM). These results yield improved generalization error upper bounds and provide rigorous theoretical foundations for noise robustness analysis and model selection.
