Non-Asymptotic Convergence Analysis of Inexact Gradient Methods for Machine Learning Without Strong Convexity

Many recent applications in machine learning and data fitting call for the algorithmic solution of structured smooth convex optimization problems. Although the gradient descent method is a natural choice for this task, it requires exact gradient computations and hence can be inefficient when the problem size is large or the gradient is difficult to evaluate. Therefore, there has been much interest in inexact gradient methods (IGMs), in which an efficiently computable approximate gradient is used to perform the update in each iteration. Currently, non-asymptotic linear convergence results for IGMs are typically established under the assumption that the objective function is strongly convex, which is not satisfied in many applications of interest; while linear convergence results that do not require the strong convexity assumption are usually asymptotic in nature. In this paper, we combine the best of these two types of results and establish---under the standard assumption that the gradient approximation errors decrease linearly to zero---the non-asymptotic linear convergence of IGMs when applied to a class of structured convex optimization problems. Such a class covers settings where the objective function is not necessarily strongly convex and includes the least squares and logistic regression problems. We believe that our techniques will find further applications in the non-asymptotic convergence analysis of other first-order methods

Two-Phase Learning for Weakly Supervised Object Localization

Weakly supervised semantic segmentation and localiza- tion have a problem of focusing only on the most important parts of an image since they use only image-level annota- tions. In this paper, we solve this problem fundamentally via two-phase learning. Our networks are trained in two steps. In the first step, a conventional fully convolutional network (FCN) is trained to find the most discriminative parts of an image. In the second step, the activations on the most salient parts are suppressed by inference conditional feedback, and then the second learning is performed to find the area of the next most important parts. By combining the activations of both phases, the entire portion of the tar- get object can be captured. Our proposed training scheme is novel and can be utilized in well-designed techniques for weakly supervised semantic segmentation, salient region detection, and object location prediction. Detailed experi- ments demonstrate the effectiveness of our two-phase learn- ing in each task.Comment: Accepted at ICCV 201