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 $\widetilde{O}\left(\left(\mathcal{T}+n^{2}\right)\sqrt{n\nu}\log\left(1/\varepsilon\right)\right)$, 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 $\widetilde{O}\left(\left(\mathcal{T}n+n^{3}\right)\log\left(1/\varepsilon\right)\right)$. 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 $\widetilde{O}\left(mn^{4}+m^{1.25}n^{3.5}\log\left(1/\varepsilon\right)\right)$, improving over the state of the art algorithms, in the case where $m=\Omega\left(n^{\frac{3.5}{\omega-1.25}}\right)$. 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.

翻译：我们提出了一种基于拟牛顿迭代的内点方法，该方法仅需对强自和谐障碍函数进行一次梯度访问。为实现这一目标，我们扩展了Dunagan-Harvey [STOC '07] 的技术，在仅使用一阶信息的同时维护预条件子。我们通过该预条件子与未知海森矩阵的相对离心率来衡量其质量，并将这些技术推广到海森矩阵缓慢变化的凸函数上。结合内点方法，我们证明：给定对凸集$K$的适当障碍函数的一次梯度访问，可在$\widetilde{O}\left(\left(\mathcal{T}+n^{2}\right)\sqrt{n\nu}\log\left(1/\varepsilon\right)\right)$时间内求解$K$上良态线性优化问题至$\varepsilon$精度，其中$\nu$为障碍函数的自和谐参数，$\mathcal{T}$为单次梯度查询所需时间。由此得到：$\bullet$ 对$n$维凸集上的线性优化可于$\widetilde{O}\left(\left(\mathcal{T}n+n^{3}\right)\log\left(1/\varepsilon\right)\right)$时间内求解，该运行时间与当前最优割平面法在用一阶预言替代投影预言时的结果相当。$\bullet$ 对涉及$\mathbb{R}^{n\times n}$中$m\geq n$个矩阵的半定规划，我们可在$\widetilde{O}\left(mn^{4}+m^{1.25}n^{3.5}\log\left(1/\varepsilon\right)\right)$时间内求解，在$m=\Omega\left(n^{\frac{3.5}{\omega-1.25}}\right)$情形下优于现有最优算法。在此过程中，我们利用矩阵分析与舒尔凸性技术开发了一系列控制势函数演化的工具。