Efficient approximate solves for shifted-power subproblems

Develop efficiently implementable approximate solves for the exact multi-center shifted-power subproblems required by accumulative power regularization while preserving the overall near-optimal oracle complexity.

Background

The accumulative power-regularization method achieves the paper’s near-optimal gradient-norm complexity by solving a sequence of increasingly regularized objectives. Each inner iteration requires an exact multi-center shifted-power call involving several shifted power regularizers centered at approximate solutions from earlier stages.

The paper identifies the computational implementation of these exact subproblem solves as the principal remaining algorithmic issue. The unresolved challenge is to replace them with approximate solves without degrading the overall oracle complexity. This is distinct from the paper’s theoretical complexity results, which establish the desired rates assuming access to the exact shifted-power calls.