Current stateof-the-art first-order algorithms find an approximate Nash equilibrium using Õ(κ x κ y) [Tseng, 1995] or Õ(min{κ x κy, κ x κ y }) [Alkousa et al., 2019] gradient evaluations, where κ x and κ y are the condition numbers for the strong-convexity and strong-concavity assumptions. A gap remains between these results and the best existing lower bound Ω( κ x κ y) due to Zhang et al. [2019]. This paper presents the first algorithm with Õ( κ x κ y) gradient complexity, matching the lower bound up to logarithmic factors. Our new algorithm is designed based on an accelerated proximal point method and an accelerated solver for minimax proximal steps. It can be easily extended to the settings of strongly-convex-concave, convex-concave, nonconvex-strongly-concave, and nonconvexconcave functions. This paper also presents algorithms that match or outperform all existing methods in these settings in terms of gradient complexity, up to logarithmic factors.

In machine learning, boosting is one of the most popular methods that designed to combine multiple base learners to a superior one. The well-known Boosted Decision Tree classifier, has been widely adopted in many areas. In the big data era, the data held by individual and entities, like personal images, browsing history and census information, are more likely to contain sensitive information. The privacy concern raises when such data leaves the hand of the owners and be further explored or mined. Such privacy issue demands that the machine learning algorithm should be privacy aware. Recently, Local Differential Privacy is proposed as an effective privacy protection approach, which offers a strong guarantee to the data owners, as the data is perturbed before any further usage, and the true values never leave the hands of the owners. Thus the machine learning algorithm with the private data instances is of great value and importance. In this paper, we are interested in developing the privacy-preserving boosting algorithm that a data user is allowed to build a classifier without knowing or deriving the exact value of each data samples. Our experiments demonstrate the effectiveness of the proposed boosting algorithm and the high utility of the learned classifiers.

Mobile crowdsensing has gained significant attention in recent years and has become a critical paradigm for emerging Internet of Things applications. The sensing devices continuously generate a significant quantity of data, which provide tremendous opportunities to develop innovative intelligent applications. To utilize these data to train machine learning models while not compromising user privacy, federated learning has become a promising solution. However, there is little understanding of whether federated learning algorithms are guaranteed to converge. We reconsider model averaging in federated learning and formulate it as a gradient-based method with biased gradients. This novel perspective assists analysis of its convergence rate and provides a new direction for more acceleration. We prove for the first time that the federated averaging algorithm is guaranteed to converge for non-convex problems, without imposing additional assumptions. We further propose a novel accelerated federated learning algorithm and provide a convergence guarantee. Simulated federated learning experiments are conducted to train deep neural networks on benchmark datasets, and experimental results show that our proposed method converges faster than previous approaches.

Regret minimization is treated as the golden rule in the traditional study of online learning. However, regret minimization algorithms tend to converge to the static optimum, thus being suboptimal for changing environments. To address this limitation, new performance measures, including dynamic regret and adaptive regret have been proposed to guide the design of online algorithms. The former one aims to minimize the global regret with respect to a sequence of changing comparators, and the latter one attempts to minimize every local regret with respect to a fixed comparator. Existing algorithms for dynamic regret and adaptive regret are developed independently, and only target one performance measure. In this paper, we bridge this gap by proposing novel online algorithms that are able to minimize the dynamic regret and adaptive regret simultaneously. In fact, our theoretical guarantee is even stronger in the sense that one algorithm is able to minimize the dynamic regret over any interval.

Causal inference is perhaps one of the most fundamental concepts in science, beginning originally from the works of some of the ancient philosophers, through today, but also weaved strongly in current work from statisticians, machine learning experts, and scientists from many other fields. This paper takes the perspective of information flow, which includes the Nobel prize winning work on Granger-causality, and the recently highly popular transfer entropy, these being probabilistic in nature. Our main contribution will be to develop analysis tools that will allow a geometric interpretation of information flow as a causal inference indicated by positive transfer entropy. We will describe the effective dimensionality of an underlying manifold as projected into the outcome space that summarizes information flow. Therefore contrasting the probabilistic and geometric perspectives, we will introduce a new measure of causal inference based on the fractal correlation dimension conditionally applied to competing explanations of future forecasts, which we will write $GeoC_{y\rightarrow x}$. This avoids some of the boundedness issues that we show exist for the transfer entropy, $T_{y\rightarrow x}$. We will highlight our discussions with data developed from synthetic models of successively more complex nature: then include the H\'{e}non map example, and finally a real physiological example relating breathing and heart rate function. Keywords: Causal Inference; Transfer Entropy; Differential Entropy; Correlation Dimension; Pinsker's Inequality; Frobenius-Perron operator.

In display advertising, predicting the conversion rate, that is, the probability that a user takes a predefined action on an advertiser's website, such as purchasing goods is fundamental in estimating the value of displaying the advertisement. However, there is a relatively long time delay between a click and its resultant conversion. Because of the delayed feedback, some positive instances at the training period are labeled as negative because some conversions have not yet occurred when training data are gathered. As a result, the conditional label distributions differ between the training data and the production environment. This situation is referred to as a feedback shift. We address this problem by using an importance weight approach typically used for covariate shift correction. We prove its consistency for the feedback shift. Results in both offline and online experiments show that our proposed method outperforms the existing method.

The gradient boosting algorithm constructs a regression estimator using a linear combination of simple "base learners". In order to obtain a robust non-parametric regression estimator that is scalable to high dimensional problems we propose a robust boosting algorithm based on a two-stage approach, similar to what is done for robust linear regression: we first minimize a robust residual scale estimator, and then improve its efficiency by optimizing a bounded loss function. Unlike previous proposals, our algorithm does not need to compute an ad-hoc residual scale estimator in each step. Since our loss functions are typically non-convex, we propose initializing our algorithm with an $L_1$ regression tree, which is fast to compute. We also introduce a robust variable importance metric for variable selection that is calculated via a permutation procedure. Through simulated and real data experiments, we compare our method against gradient boosting with squared loss and other robust boosting methods in the literature. With clean data, our method works equally well as gradient boosting with the squared loss. With symmetric and asymmetrically contaminated data, we show that our proposed method outperforms in terms of prediction error and variable selection accuracy.

We provide a deterministic space-efficient algorithm for estimating ridge regression. For $n$ data points with $d$ features and a large enough regularization parameter, we provide a solution within $\varepsilon$ L$_2$ error using only $O(d/\varepsilon)$ space. This is the first $o(d^2)$ space algorithm for this classic problem. The algorithm sketches the covariance matrix by variants of Frequent Directions, which implies it can operate in insertion-only streams and a variety of distributed data settings. In comparisons to randomized sketching algorithms on synthetic and real-world datasets, our algorithm has less empirical error using less space and similar time.

We propose a new embedding method which is particularly well-suited for settings where the sample size greatly exceeds the ambient dimension. Our technique consists of partitioning the space into simplices and then embedding the data points into features corresponding to the simplices' barycentric coordinates. We then train a linear classifier in the rich feature space obtained from the simplices. The decision boundary may be highly non-linear, though it is linear within each simplex (and hence piecewise-linear overall). Further, our method can approximate any convex body. We give generalization bounds based on empirical margin and a novel hybrid sample compression technique. An extensive empirical evaluation shows that our method consistently outperforms a range of popular kernel embedding methods.

In reinforcement learning, an agent learns to reach a set of goals by means of an external reward signal. In the natural world, intelligent organisms learn from internal drives, bypassing the need for external signals, which is beneficial for a wide range of tasks. Motivated by this observation, we propose to formulate an intrinsic objective as the mutual information between the goal states and the controllable states. This objective encourages the agent to take control of its environment. Subsequently, we derive a surrogate objective of the proposed reward function, which can be optimized efficiently. Lastly, we evaluate the developed framework in different robotic manipulation and navigation tasks and demonstrate the efficacy of our approach. A video showing experimental results is available at \url{https://youtu.be/CT4CKMWBYz0}.