Second-order Conditional Gradient Sliding

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 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 O(log(log 1/varepsilon)) first-order and Hessian oracle calls and 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.