Constrained second-order convex optimization algorithms are the method of choice when a high accuracy solution to a problem is needed, due to the quadratic convergence rates these methods enjoy when close to the optimum. These algorithms require the solution of a constrained quadratic subproblem at every iteration. In the case where the feasible region can only be accessed efficiently through a linear optimization oracle, and computing first-order information about the function, although possible, is costly, the coupling of constrained second-order and conditional gradient algorithms leads to competitive algorithms with solid theoretical guarantees and good numerical performance.

Generalized self-concordance is a key property present in the objective function of many important learning problems. We establish the convergence rate of a simple Frank-Wolfe variant that uses the open-loop step size strategy 𝛾 𝑡 = 2/(𝑡 + 2), obtaining a O (1/𝑡) convergence rate for this class of functions in terms of primal gap and Frank-Wolfe gap, where 𝑡 is the iteration count. This avoids the use of second-order information or the need to estimate local smoothness parameters of previous work. We also show improved convergence rates for various common cases, e.g., when the feasible region under consideration is uniformly convex or polyhedral.

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We present FrankWolfe.jl, an open-source implementation of several popular Frank–Wolfe and conditional gradients variants for first-order constrained optimization. The package is designed with flexibility and high performance in mind, allowing for easy extension and relying on few assumptions regarding the user-provided functions. It supports Julia’s unique multiple dispatch feature, and it interfaces smoothly with generic linear optimization formulations using MathOptInterface.jl.