**Comparison-Based Algorithm (CBA)**

Stochastic optimization finds a wide range of applications in operations research and management science. However, existing stochastic optimization techniques often require the information of random samples (e.g., demands in newsvendor problem) or the objective values at the sampled points (e.g., the lost sales cost), which might not be available in practice. In this paper, we consider a new setup for stochastic optimization, in which the decision maker can only access to comparison information between a random sample and two chosen points in each iteration. We propose a comparison-based algorithm (CBA) to solve such problems in single dimension with convex objective functions. Particularly, the CBA properly chooses the two points in each iteration and constructs an unbiased gradient estimate for the original problem. We show that the CBA achieves the same convergence rate as the optimal stochastic gradient methods (with the samples observed). We also consider extensions of our approach to high dimensional quadratic problems as well as problems with non-convex objective functions. Numerical experiments show that the CBA performs well in test problems. … **Domain Partitioning Network (DoPaNet)**

Standard adversarial training involves two agents, namely a generator and a discriminator, playing a mini-max game. However, even if the players converge to an equilibrium, the generator may only recover a part of the target data distribution, in a situation commonly referred to as mode collapse. In this work, we present the Domain Partitioning Network (DoPaNet), a new approach to deal with mode collapse in generative adversarial learning. We employ multiple discriminators, each encouraging the generator to cover a different part of the target distribution. To ensure these parts do not overlap and collapse into the same mode, we add a classifier as a third agent in the game. The classifier decides which discriminator the generator is trained against for each sample. Through experiments on toy examples and real images, we show the merits of DoPaNet in covering the real distribution and its superiority with respect to the competing methods. Besides, we also show that we can control the modes from which samples are generated using DoPaNet. … **True Asymptotic Natural Gradient Optimization (TANGO)**

We introduce a simple algorithm, True Asymptotic Natural Gradient Optimization (TANGO), that converges to a true natural gradient descent in the limit of small learning rates, without explicit Fisher matrix estimation. For quadratic models the algorithm is also an instance of averaged stochastic gradient, where the parameter is a moving average of a ‘fast’, constant-rate gradient descent. TANGO appears as a particular de-linearization of averaged SGD, and is sometimes quite different on non-quadratic models. This further connects averaged SGD and natural gradient, both of which are arguably optimal asymptotically. In large dimension, small learning rates will be required to approximate the natural gradient well. Still, this shows it is possible to get arbitrarily close to exact natural gradient descent with a lightweight algorithm. … **Deep Expander Network (X-Net)**

Deep Neural Networks, while being unreasonably effective for several vision tasks, have their usage limited by the computational and memory requirements, both during training and inference stages. Analyzing and improving the connectivity patterns between layers of a network has resulted in several compact architectures like GoogleNet, ResNet and DenseNet-BC. In this work, we utilize results from graph theory to develop an efficient connection pattern between consecutive layers. Specifically, we use {\it expander graphs} that have excellent connectivity properties to develop a sparse network architecture, the deep expander network (X-Net). The X-Nets are shown to have high connectivity for a given level of sparsity. We also develop highly efficient training and inference algorithms for such networks. Experimental results show that we can achieve the similar or better accuracy as DenseNet-BC with two-thirds the number of parameters and FLOPs on several image classification benchmarks. We hope that this work motivates other approaches to utilize results from graph theory to develop efficient network architectures. …

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**07**
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Feb 2020

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