Recent literature has demonstrated promising results for training Generative Adversarial Networks by employing a set of discriminators, in contrast to the traditional game involving one generator against a single adversary. Such methods perform single-objective optimization on some simple consolidation of the losses, e.g. an arithmetic average. In this work, we revisit the multiple-discriminator setting by framing the simultaneous minimization of losses provided by different models as a multi-objective optimization problem. Specifically, we evaluate the performance of multiple gradient descent and the hypervolume maximization algorithm on a number of different datasets. Moreover, we argue that the previously proposed methods and hypervolume maximization can all be seen as variations of multiple gradient descent in which the update direction can be computed efficiently. Our results indicate that hypervolume maximization presents a better compromise between sample quality and computational cost than previous methods.

We study the problem of minimizing the average ofa very large number of smooth functions, which is of key importance in training supervised learning models. One of the most celebrated methods in this context is the SAGA algorithm of Defazio et al. (2014). Despite years of research onthe topic, a general-purpose version of SAGA--one that would include arbitrary importance samplingand minibatching schemes--does not exist. We remedy this situation and propose a general and flexible variant of SAGA following the arbitrary sampling paradigm. We perform an iteration complexity analysis of the method, largely possible due to the construction of new stochastic Lyapunov functions. We establish linear convergencerates in the smooth and strongly convex regime, and under a quadratic functional growth condition (i.e., in a regime not assuming strong convexity). Our rates match those of the primal-dual method Quartz (Qu et al., 2015) for which an arbitrary sampling analysis is available, which makes a significant step towards closing the gap in our understanding of complexity of primal anddual methods for finite sum problems.

Bayesian optimization is a sample-efficient method for finding a global optimum of an expensive-to-evaluate black-box function. A global solution is found by accumulating a pair of query point and corresponding function value, repeating these two procedures: (i) learning a surrogate model for the objective function using the data observed so far; (ii) the maximization of an acquisition function to determine where next to query the objective function. Convergence guarantees are only valid when the global optimizer of the acquisition function is found and selected as the next query point. In practice, however, local optimizers of acquisition functions are also used, since searching the exact optimizer of the acquisition function is often a non-trivial or time-consuming task. In this paper we present an analysis on the behavior of local optimizers of acquisition functions, in terms of instantaneous regrets over global optimizers. We also present the performance analysis when multi-started local optimizers are used to find the maximum of the acquisition function. Numerical experiments confirm the validity of our theoretical analysis.

We present a novel algorithm for overcomplete independent components analysis (ICA), where the number of latent sources k exceeds the dimension p of observed variables. Previous algorithms either suffer from high computational complexity or make strong assumptions about the form of the mixing matrix. Our algorithm does not make any sparsity assumption yet enjoys favorable computational and theoretical properties. Our algorithm consists of two main steps: (a) estimation of the Hessians of the cumulant generating function (as opposed to the fourth and higher order cumulants used by most algorithms) and (b) a novel semi-definite programming (SDP) relaxation for recovering a mixing component. We show that this relaxation can be efficiently solved with a projected accelerated gradient descent method, which makes the whole algorithm computationally practical. Moreover, we conjecture that the proposed program recovers a mixing component at the rate k < p^2/4 and prove that a mixing component can be recovered with high probability when k < (2 - epsilon) p log p when the original components are sampled uniformly at random on the hyper sphere. Experiments are provided on synthetic data and the CIFAR-10 dataset of real images.

Fairness by decision-makers is believed to be auditable by third parties. In this study, we show that this is not always true. We consider the following scenario. Imagine a decision-maker who discloses a subset of his dataset with decisions to make his decisions auditable. If he is corrupt, and he deliberately selects a subset that looks fair even though the overall decision is unfair, can we identify this decision-maker's fraud? We answer this question negatively. We first propose a sampling method that produces a subset whose distribution is biased from the original (to pretend to be fair); however, its differentiation from uniform sampling is difficult. We call such a sampling method as stealthily biased sampling, which is formulated as a Wasserstein distance minimization problem, and is solved through a minimum-cost flow computation. We proved that the stealthily biased sampling minimizes an upper-bound of the indistinguishability. We conducted experiments to see that the stealthily biased sampling is, in fact, difficult to detect.

We report on an empirical study of the main strategies for conditional quantile estimation in the context of stochastic computer experiments. To ensure adequate diversity, six metamodels are presented, divided into three categories based on order statistics, functional approaches, and those of Bayesian inspiration. The metamodels are tested on several problems characterized by the size of the training set, the input dimension, the quantile order and the value of the probability density function in the neighborhood of the quantile. The metamodels studied reveal good contrasts in our set of 480 experiments, enabling several patterns to be extracted. Based on our results, guidelines are proposed to allow users to select the best method for a given problem.

To overcome the oscillation problem in the classical momentum-based optimizer, recent work associates it with the proportional-integral (PI) controller, and artificially adds D term producing a PID controller. It suppresses oscillation with the sacrifice of introducing extra hyper-parameter. In this paper, we start by analyzing: why momentum-based method oscillates about the optimal point? and answering that: the fluctuation problem relates to the lag effect of integral (I) term. Inspired by the conditional integration idea in classical control society, we propose SPI-Optimizer, an integral-Separated PI controller based optimizer WITHOUT introducing extra hyperparameter. It separates momentum term adaptively when the inconsistency of current and historical gradient direction occurs. Extensive experiments demonstrate that SPIOptimizer generalizes well on popular network architectures to eliminate the oscillation, and owns competitive performance with faster convergence speed (up to 40% epochs reduction ratio ) and more accurate classification result on MNIST, CIFAR10, and CIFAR100 (up to 27.5% error reduction ratio) than the state-of-the-art methods.

Next-generation wireless networks (NGWN) have a substantial potential in terms of supporting a broad range of complex compelling applications both in military and civilian fields, where the users are able to enjoy high-rate, low-latency, low-cost and reliable information services. Achieving this ambitious goal requires new radio techniques for adaptive learning and intelligent decision making because of the complex heterogeneous nature of the network structures and wireless services. Machine learning algorithms have great success in supporting big data analytics, efficient parameter estimation and interactive decision making. Hence, in this article, we review the thirty-year history of machine learning by elaborating on supervised learning, unsupervised learning, reinforcement learning and deep learning, respectively. Furthermore, we investigate their employment in the compelling applications of NGWNs, including heterogeneous networks (HetNets), cognitive radios (CR), Internet of things (IoT), machine to machine networks (M2M), and so on. This article aims for assisting the readers in clarifying the motivation and methodology of the various machine learning algorithms, so as to invoke them for hitherto unexplored services as well as scenarios of future wireless networks.

Recently, researchers proposed various low-precision gradient compression, for efficient communication in large-scale distributed optimization. Based on these work, we try to reduce the communication complexity from a new direction. We pursue an ideal bijective mapping between two spaces of gradient distribution, so that the mapped gradient carries greater information entropy after the compression. In our setting, all servers should share a reference gradient in advance, and they communicate via the normalized gradients, which are the subtraction or quotient, between current gradients and the reference. To obtain a reference vector that yields a stronger signal-to-noise ratio, dynamically in each iteration, we extract and fuse information from the past trajectory in hindsight, and search for an optimal reference for compression. We name this to be the trajectory-based normalized gradients (TNG). It bridges the research from different societies, like coding, optimization, systems, and learning. It is easy to implement and can universally combine with existing algorithms. Our experiments on benchmarking hard non-convex functions, convex problems like logistic regression demonstrate that TNG is more compression-efficient for communication of distributed optimization of general functions.

Quantum annealing (QA) is a generic method for solving optimization problems using fictitious quantum fluctuation. The current device performing QA involves controlling the transverse field; it is classically simulatable by using the standard technique for mapping the quantum spin systems to the classical ones. In this sense, the current system for QA is not powerful despite utilizing quantum fluctuation. Hence, we developed a system with a time-dependent Hamiltonian consisting of a combination of the formulated Ising model and the "driver" Hamiltonian with only quantum fluctuation. In the previous study, for a fully connected spin model, quantum fluctuation can be addressed in a relatively simple way. We proved that the fully connected antiferromagnetic interaction can be transformed into a fluctuating transverse field and is thus classically simulatable at sufficiently low temperatures. Using the fluctuating transverse field, we established several ways to simulate part of the nonstoquastic Hamiltonian on classical computers. We formulated a message-passing algorithm in the present study. This algorithm is capable of assessing the performance of QA with part of the nonstoquastic Hamiltonian having a large number of spins. In other words, we developed a different approach for simulating the nonstoquastic Hamiltonian without using the quantum Monte Carlo technique. Our results were validated by comparison to the results obtained by the replica method.