This paper considers online convex optimization (OCO) problems - the paramount framework for online learning algorithm design. The loss function of learning task in OCO setting is based on streaming data so that OCO is a powerful tool to model large scale applications such as online recommender systems. Meanwhile, real-world data are usually of extreme high-dimensional due to modern feature engineering techniques so that the quadratic regression is impractical. Factorization Machine as well as its variants are efficient models for capturing feature interactions with low-rank matrix model but they can't fulfill the OCO setting due to their non-convexity. In this paper, We propose a projective quadratic regression (PQR) model. First, it can capture the import second-order feature information. Second, it is a convex model, so the requirements of OCO are fulfilled and the global optimal solution can be achieved. Moreover, existing modern online optimization methods such as Online Gradient Descent (OGD) or Follow-The-Regularized-Leader (FTRL) can be applied directly. In addition, by choosing a proper hyper-parameter, we show that it has the same order of space and time complexity as the linear model and thus can handle high-dimensional data. Experimental results demonstrate the performance of the proposed PQR model in terms of accuracy and efficiency by comparing with the state-of-the-art methods.

In the era of big data, it is highly desired to develop efficient machine learning algorithms to tackle massive data challenges such as storage bottleneck, algorithmic scalability, and interpretability. In this paper, we develop a novel efficient classification algorithm, called fast polynomial kernel classification (FPC), to conquer the scalability and storage challenges. Our main tools are a suitable selected feature mapping based on polynomial kernels and an alternating direction method of multipliers (ADMM) algorithm for a related non-smooth convex optimization problem. Fast learning rates as well as feasibility verifications including the convergence of ADMM and the selection of center points are established to justify theoretical behaviors of FPC. Our theoretical assertions are verified by a series of simulations and real data applications. The numerical results demonstrate that FPC significantly reduces the computational burden and storage memory of the existing learning schemes such as support vector machines and boosting, without sacrificing their generalization abilities much.

Optimal transport (OT) is a powerful tool for measuring the distance between two defined probability distributions. In this paper, we develop a new manifold named the coupling matrix manifold (CMM), where each point on CMM can be regarded as the transportation plan of the OT problem. We firstly explore the Riemannian geometry of CMM with the metric expressed by the Fisher information. These geometrical features of CMM have paved the way for developing numerical Riemannian optimization algorithms such as Riemannian gradient descent and Riemannian trust-region algorithms, forming a uniform optimization method for all types of OT problems. The proposed method is then applied to solve several OT problems studied by previous literature. The results of the numerical experiments illustrate that the optimization algorithms that are based on the method proposed in this paper are comparable to the classic ones, for example, the Sinkhorn algorithm, while outperforming other state-of-the-art algorithms without considering the geometry information, especially in the case of non-entropy optimal transport.

Reinforcement Learning (RL) has achieved state-of-the-art results in domains such as robotics and games. We build on this previous work by applying RL algorithms to a selection of canonical online stochastic optimization problems with a range of practical applications: Bin Packing, Newsvendor, and Vehicle Routing. While there is a nascent literature that applies RL to these problems, there are no commonly accepted benchmarks which can be used to compare proposed approaches rigorously in terms of performance, scale, or generalizability. This paper aims to fill that gap. For each problem we apply both standard approaches as well as newer RL algorithms and analyze results. In each case, the performance of the trained RL policy is competitive with or superior to the corresponding baselines, while not requiring much in the way of domain knowledge. This highlights the potential of RL in real-world dynamic resource allocation problems.

Last night on the train I read this nice paper by David Duvenaud and colleagues. So I thought it's time for a David Duvenaud birthday special (don't get too excited David, I won't make it an annual tradition...) I recently covered iMAML: the meta-learning algorithm that makes use of implicit gradients to sidestep backpropagating through the inner loop optimization in meta-learning/hyperparameter tuning. The method presented in (Lorraine et al, 2019) uses the same high-level idea, but introduces a different - on the surface less fiddly - approximation to the crucial inverse Hessian. I won't spend a lot of time introducing the whole meta-learning setup from scratch, you can use the previous post as a starting point. Many - though not all - meta-learning or hyperparameter optimization problems can be stated as nested optimization problems.

We introduce a general tensor model suitable for data analytic tasks for heterogeneous data sets, wherein there are joint low-rank structures within groups of observations, but also discriminative structures across different groups. To capture such complex structures, a double core tensor (DCOT) factorization model is introduced together with a family of smoothing loss functions. By leveraging the proposed smoothing function, the model accurately estimates the model factors, even in the presence of missing entries. A linearized ADMM method is employed to solve regularized versions of DCOT factorizations, that avoid large tensor operations and large memory storage requirements. Further, we establish theoretically its global convergence, together with consistency of the estimates of the model parameters. The effectiveness of the DCOT model is illustrated on several real-world examples including image completion, recommender systems, subspace clustering and detecting modules in heterogeneous Omics multi-modal data, since it provides more insightful decompositions than conventional tensor methods.

In this paper, we study Reinforcement Learning from Demonstrations (RLfD) that improves the exploration efficiency of Reinforcement Learning (RL) by providing expert demonstrations. Most of existing RLfD methods require demonstrations to be perfect and sufficient, which yet is unrealistic to meet in practice. To work on imperfect demonstrations, we first define an imperfect expert setting for RLfD in a formal way, and then point out that previous methods suffer from two issues in terms of optimality and convergence, respectively. Upon the theoretical findings we have derived, we tackle these two issues by regarding the expert guidance as a soft constraint on regulating the policy exploration of the agent, which eventually leads to a constrained optimization problem. We further demonstrate that such problem is able to be addressed efficiently by performing a local linear search on its dual form. Considerable empirical evaluations on a comprehensive collection of benchmarks indicate our method attains consistent improvement over other RLfD counterparts.

In an era of data driven digital transformation, a customer driven business strategy is essential for success. In the motion picture industry, movie exhibitors must compete to win share of consumers entertainment time (and wallet) against digital entertainment alternatives offered by mammoth sized, digital focused, competitors like Netflix, Amazon and Disney [1]. Customer loyalty, point-of-sale and digital payment p latforms produce rich insights that can leveraged to inform business operations and automate the decision-making pr ocess, effectively enabling movie exhibitors to compete using analytics and artificial intelligence. This study presen ts a new, customer driven, quantitative approach to movie scheduling that can be utilized by movie exhibitors to increase attendance and market share. The role of the exhibitor is to show films that are produced by movie st udios (see [2] for more details on the roles of the stakeholders in the movie industry). Exhibitors do not have d ecision making authority over the movies that are produced by the studios.

Learning models with discrete latent variables using stochastic gradient descent remains a challenge due to the high variance of gradient estimates. Modern variance reduction techniques mostly consider categorical distributions and have limited applicability when the number of possible outcomes becomes large. In this work, we consider models with latent permutations and propose control variates for the Plackett-Luce distribution. In particular, the control variates allow us to optimize black-box functions over permutations using stochastic gradient descent. To illustrate the approach, we consider a variety of causal structure learning tasks for continuous and discrete data. We show that our method outperforms competitive relaxation-based optimization methods and is also applicable to non-differentiable score functions.

Combinatorial optimization assumes that all parameters of the optimization problem, e.g. the weights in the objective function, are fixed. Often, these weights are mere estimates and increasingly machine learning techniques are used to for their estimation. Recently, Smart Predict and Optimize (SPO) has been proposed for problems with a linear objective function over the predictions, more specifically linear programming problems. It takes the regret of the predictions on the linear problem into account, by repeatedly solving it during learning. We investigate the use of SPO to solve more realistic discrete optimization problems. The main challenge is the repeated solving of the optimization problem. To this end, we investigate ways to relax the problem as well as warm-starting the learning and the solving. Our results show that even for discrete problems it often suffices to train by solving the relaxation in the SPO loss. Furthermore, this approach outperforms the state-of-the-art approach of Wilder, Dilkina, and Tambe. We experiment with weighted knapsack problems as well as complex scheduling problems, and show for the first time that a predict-and-optimize approach can successfully be used on large-scale combinatorial optimization problems. Introduction Combinatorial optimization aims to optimize an objective function over a set of feasible solutions defined on a discrete space. Numerous real-life decision-making problems can be formulated as combinatorial optimization problems (Korte et al. 2012; Trevisan 2011).