AI 中文总结
该研究针对多面体上的凸β光滑函数优化，提出无参数分解不变条件梯度方法，在2级多面体和一般多面体下分别实现更优线性收敛速率，迭代次数显著少于现有方法。
AI 中文摘要
我们重新研究了由Garber和Meshi于2016年提出的分解不变条件梯度方法，用于在α二次增长条件下最小化实空间ℝⁿ中多面体上的凸且β光滑函数。针对2级多面体，我们设计了一种简单且无参数的二进步长规则，其线性收敛速率与最优面的维度成正比，而非像基于标准远离步的多面体条件梯度方法那样与环境维度成正比。对于一般多面体，在稍强的α_F面二次增长条件下，我们提出了一种方法，其达到ε近似解所需的迭代次数为n + βD²/(αr*²) + (d*+1)βD²/α_F · log(1/ε)，其中d*是最优面的维度，r*是最优集与不包含最优解的面之间的分离参数，D是多面体的直径。该方法同样无参数，仅依赖标准线搜索计算。这一结果在max{α/α_F(d*+1), 1/r*²} ≪ n的有意义区域内，优于之前达到ε近似所需迭代次数与βD²n/α · log(1/ε)成正比的条件梯度方法。
英文摘要
We revisit Decomposition-Invariant Conditional Gradient methods, originally introduced by Garber and Meshi in 2016, for minimizing a convex and $β$-smooth function over a polytope in $\reals^n$, under an $α$-quadratic growth condition. For 2-level polytopes we design a simple and parameter-free dyadic step-size rule that yields a linear convergence rate which scales with the dimension of the optimal face and not with the ambient dimension as in standard away-step-based conditional gradient methods for polytopes. For general polytopes, under a slightly stronger condition of $α_{\mathrm{F}}$-\textit{facial quadratic growth}, we introduce a method whose number of iterations to reach an $ε$-approximate solution is of the order $n+\frac{βD^2}{αr^{*2}} + \frac{(d^*+1)βD^2}{α_{\mathrm{F}}}\log(1/ε)$, where $d^*$ is the dimension of the optimal face, $r^*$ is a separation parameter between the optimal set and faces that do not contain an optimal solution, and $D$ is the diameter of the polytope. This method is also parameter-free and only relies on standard line-search computations. The second result improves upon previous conditional gradient methods, whose number of iterations to $ε$-approximation scales with $\frac{βD^2n}α\log(1/ε)$, in a meaningful regime $\max\{\fracα{α_{\rm F}}(d^*+1), \frac{1}{r^{*2}}\} \ll n$
