New Computational Guarantees for Solving Convex Optimization Problems with First Order Methods, via a Function Growth Condition Measure

Motivated by recent work of Renegar, we present new computational methods and associated computational guarantees for solving convex optimization problems using first-order methods. Our problem of interest is the general convex optimization problem $f^* = \min_{x \in Q} f(x)$, where we presume knowledge of a strict lower bound $f_{\mathrm{slb}} < f^*$. [Indeed, $f_{\mathrm{slb}}$ is naturally known when optimizing many loss functions in statistics and machine learning (least-squares, logistic loss, exponential loss, total variation loss, etc.) as well as in Renegar's transformed version of the standard conic optimization problem; in all these cases one has $f_{\mathrm{slb}} = 0 < f^*$.] We introduce a new functional measure called the growth constant $G$ for $f(\cdot)$, that measures how quickly the level sets of $f(\cdot)$ grow relative to the function value, and that plays a fundamental role in the complexity analysis. When $f(\cdot)$ is non-smooth, we present new computational guarantees for the Subgradient Descent Method and for smoothing methods, that can improve existing computational guarantees in several ways, most notably when the initial iterate $x^0$ is far from the optimal solution set. When $f(\cdot)$ is smooth, we present a scheme for periodically restarting the Accelerated Gradient Method that can also improve existing computational guarantees when $x^0$ is far from the optimal solution set, and in the presence of added structure we present a scheme using parametrically increased smoothing that further improves the associated computational guarantees.