A Robust Adaptive Stochastic Gradient Method for Deep Learning

Stochastic gradient algorithms are the main focus of large-scale optimization problems and led to important successes in the recent advancement of the deep learning algorithms. The convergence of SGD depends on the careful choice of learning rate and the amount of the noise in stochastic estimates of the gradients. In this paper, we propose an adaptive learning rate algorithm, which utilizes stochastic curvature information of the loss function for automatically tuning the learning rates. The information about the element-wise curvature of the loss function is estimated from the local statistics of the stochastic first order gradients. We further propose a new variance reduction technique to speed up the convergence. In our experiments with deep neural networks, we obtained better performance compared to the popular stochastic gradient algorithms.Comment: IJCNN 2017 Accepted Paper, An extension of our paper, "ADASECANT: Robust Adaptive Secant Method for Stochastic Gradient

Optimistic Robust Optimization With Applications To Machine Learning

Robust Optimization has traditionally taken a pessimistic, or worst-case viewpoint of uncertainty which is motivated by a desire to find sets of optimal policies that maintain feasibility under a variety of operating conditions. In this paper, we explore an optimistic, or best-case view of uncertainty and show that it can be a fruitful approach. We show that these techniques can be used to address a wide variety of problems. First, we apply our methods in the context of robust linear programming, providing a method for reducing conservatism in intuitive ways that encode economically realistic modeling assumptions. Second, we look at problems in machine learning and find that this approach is strongly connected to the existing literature. Specifically, we provide a new interpretation for popular sparsity inducing non-convex regularization schemes. Additionally, we show that successful approaches for dealing with outliers and noise can be interpreted as optimistic robust optimization problems. Although many of the problems resulting from our approach are non-convex, we find that DCA or DCA-like optimization approaches can be intuitive and efficient

Generative Modeling through the Semi-dual Formulation of Unbalanced Optimal Transport

Optimal Transport (OT) problem investigates a transport map that bridges two distributions while minimizing a given cost function. In this regard, OT between tractable prior distribution and data has been utilized for generative modeling tasks. However, OT-based methods are susceptible to outliers and face optimization challenges during training. In this paper, we propose a novel generative model based on the semi-dual formulation of Unbalanced Optimal Transport (UOT). Unlike OT, UOT relaxes the hard constraint on distribution matching. This approach provides better robustness against outliers, stability during training, and faster convergence. We validate these properties empirically through experiments. Moreover, we study the theoretical upper-bound of divergence between distributions in UOT. Our model outperforms existing OT-based generative models, achieving FID scores of 2.97 on CIFAR-10 and 5.80 on CelebA-HQ-256.Comment: 23 pages, 15 figure

Optimal Pose and Shape Estimation for Category-level 3D Object Perception

We consider a category-level perception problem, where one is given 3D sensor data picturing an object of a given category (e.g. a car), and has to reconstruct the pose and shape of the object despite intra-class variability (i.e. different car models have different shapes). We consider an active shape model, where -- for an object category -- we are given a library of potential CAD models describing objects in that category, and we adopt a standard formulation where pose and shape estimation are formulated as a non-convex optimization. Our first contribution is to provide the first certifiably optimal solver for pose and shape estimation. In particular, we show that rotation estimation can be decoupled from the estimation of the object translation and shape, and we demonstrate that (i) the optimal object rotation can be computed via a tight (small-size) semidefinite relaxation, and (ii) the translation and shape parameters can be computed in closed-form given the rotation. Our second contribution is to add an outlier rejection layer to our solver, hence making it robust to a large number of misdetections. Towards this goal, we wrap our optimal solver in a robust estimation scheme based on graduated non-convexity. To further enhance robustness to outliers, we also develop the first graph-theoretic formulation to prune outliers in category-level perception, which removes outliers via convex hull and maximum clique computations; the resulting approach is robust to 70%-90% outliers. Our third contribution is an extensive experimental evaluation. Besides providing an ablation study on a simulated dataset and on the PASCAL3D+ dataset, we combine our solver with a deep-learned keypoint detector, and show that the resulting approach improves over the state of the art in vehicle pose estimation in the ApolloScape datasets