Second-order Conditional Gradient Sliding
This work addresses the need for high-accuracy solutions in optimization problems where the feasible region is accessed via a linear optimization oracle and first-order information is costly, offering an incremental improvement over existing methods.
The paper tackles the problem of constrained second-order convex optimization by introducing the Second-Order Conditional Gradient Sliding (SOCGS) algorithm, which achieves quadratic convergence in primal gap after a finite number of iterations and requires O(log(log 1/ε)) first-order and Hessian oracle calls and O(log(1/ε) log(log 1/ε)) linear minimization oracle calls for an ε-optimal solution.
Constrained second-order convex optimization algorithms are the method of choice when a high accuracy solution to a problem is needed, due to their local quadratic convergence. These algorithms require the solution of a constrained quadratic subproblem at every iteration. We present the \emph{Second-Order Conditional Gradient Sliding} (SOCGS) algorithm, which uses a projection-free algorithm to solve the constrained quadratic subproblems inexactly. When the feasible region is a polytope the algorithm converges quadratically in primal gap after a finite number of linearly convergent iterations. Once in the quadratic regime the SOCGS algorithm requires $\mathcal{O}(\log(\log 1/\varepsilon))$ first-order and Hessian oracle calls and $\mathcal{O}(\log (1/\varepsilon) \log(\log1/\varepsilon))$ linear minimization oracle calls to achieve an $\varepsilon$-optimal solution. This algorithm is useful when the feasible region can only be accessed efficiently through a linear optimization oracle, and computing first-order information of the function, although possible, is costly.