Statistical Gains from Looped Estimation under Parameter Budgets

Growing memory demands in artificial intelligence motivate learning with fewer trainable parameters. We ask whether a looped estimator, which repeatedly applies one fitted operator with parameters shared across iterations, can improve statistical accuracy under a common parameter budget. Its conventional untied counterpart uses separate parameters at each iteration. For general likelihood models, we establish an upper bound on squared Hellinger risk for looped sieve maximum likelihood and a minimax lower bound over the tuned untied family. These bounds reveal a parameter--iteration--accuracy tradeoff: repeated computation can improve approximation without adding parameters, while increasing computational cost and fitted-class complexity. For targets of known Hölder smoothness, looped residual feedforward networks and a specified post-layer-normalized Transformer attain the minimax polynomial rate up to logarithmic factors with a fixed number of bounded real parameters. At sufficiently large fixed budgets, looped worst-case risk vanishes as sample size grows, whereas optimal worst-case untied risk remains bounded away from zero. Under specified growing-budget conditions, the loop-to-untied risk ratio also tends to zero. Gaussian and Laplace regression, binary response, and energy-based density estimation illustrate the theory.