Interior Point Methods with a Gradient Oracle

Abstract: We provide an interior point method based on quasi-Newton iterations, which only requires first-order access to a strongly self-concordant barrier function. To achieve this, we extend the techniques of Dunagan-Harvey [STOC '07] to maintain a preconditioner, while using only first-order information. We measure the quality of this preconditioner in terms of its relative excentricity to the unknown Hessian matrix, and we generalize these techniques to convex functions with a slowly-changing Hessian. We combine this with an interior point method to show that, given first-order access to an appropriate barrier function for a convex set $K$, we can solve well-conditioned linear optimization problems over $K$ to $\varepsilon$ precision in time $$\widetilde{O}\left(\left(\mathcal{T}+n<sup>{2}\right)\sqrt{n\nu}\log\left(1/\varepsilon\right)\right)$$, where $\nu$ is the self-concordance parameter of the barrier function, and $\mathcal{T}$ is the time required to make a gradient query. As a consequence we show that: $\bullet$ Linear optimization over $n$-dimensional convex sets can be solved in time $$\widetilde{O}\left(\left(\mathcal{T}n+n<sup>{3}\right)\log\left(1/\varepsilon\right)\right)$$. This parallels the running time achieved by state of the art algorithms for cutting plane methods, when replacing separation oracles with first-order oracles for an appropriate barrier function. $\bullet$ We can solve semidefinite programs involving $m\geq n$ matrices in $\mathbb{R}^{n\times n}$ in time $$\widetilde{O}\left(mn<sup>{4}+m<sup>{1.25}n<sup>{3.5}\log\left(1/\varepsilon\right)\right)$$, improving over the state of the art algorithms, in the case where $$m=\Omega\left(n<sup>{\frac{3.5}{\omega-1.25}}\right)$$. Along the way we develop a host of tools allowing us to control the evolution of our potential functions, using techniques from matrix analysis and Schur convexity.