The Implicit Bias of Gradient Descent on Separable Data

We examine gradient descent on unregularized logistic regression problems, with homogeneous linear predictors on linearly separable datasets. We show the predictor converges to the direction of the max-margin (hard margin SVM) solution. The result also generalizes to other monotone decreasing loss functions with an infimum at infinity, to multi-class problems, and to training a weight layer in a deep network in a certain restricted setting. Furthermore, we show this convergence is very slow, and only logarithmic in the convergence of the loss itself. This can help explain the benefit of continuing to optimize the logistic or cross-entropy loss even after the training error is zero and the training loss is extremely small, and, as we show, even if the validation loss increases. Our methodology can also aid in understanding implicit regularization n more complex models and with other optimization methods.Comment: Final JMLR version, with improved discussions over v3. Main improvements in journal version over conference version (v2 appeared in ICLR): We proved the measure zero case for main theorem (with implications for the rates), and the multi-class cas

Summarizing First-Person Videos from Third Persons' Points of Views

Video highlight or summarization is among interesting topics in computer vision, which benefits a variety of applications like viewing, searching, or storage. However, most existing studies rely on training data of third-person videos, which cannot easily generalize to highlight the first-person ones. With the goal of deriving an effective model to summarize first-person videos, we propose a novel deep neural network architecture for describing and discriminating vital spatiotemporal information across videos with different points of view. Our proposed model is realized in a semi-supervised setting, in which fully annotated third-person videos, unlabeled first-person videos, and a small number of annotated first-person ones are presented during training. In our experiments, qualitative and quantitative evaluations on both benchmarks and our collected first-person video datasets are presented.Comment: 16+10 pages, ECCV 201