Freight transportation marketplace rates are typically challenging to forecast accurately. In this work, we have developed a novel statistical technique based on signature transforms and have built a predictive and adaptive model to forecast these marketplace rates. Our technique is based on two key elements of the signature transform: one being its universal nonlinearity property, which linearizes the feature space and hence translates the forecasting problem into linear regression, and the other being the signature kernel, which allows for comparing computationally efficiently similarities between time series data. Combined, it allows for efficient feature generation and precise identification of seasonality and regime switching in the forecasting process.

When designing algorithms for finite-time-horizon episodic reinforcement learning problems, a common approach is to introduce a fictitious discount factor and use stationary policies for approximations. Empirically, it has been shown that the fictitious discount factor helps reduce variance, and stationary policies serve to save the per-iteration computational cost. Theoretically, however, there is no existing work on convergence analysis for algorithms with this fictitious discount recipe. This paper takes the first step towards analyzing these algorithms. It focuses on two vanilla policy gradient (VPG) variants: the first being a widely used variant with discounted advantage estimations (DAE), the second with an additional fictitious discount factor in the score functions of the policy gradient estimators. Non-asymptotic convergence guarantees are established for both algorithms, and the additional discount factor is shown to reduce the bias introduced in DAE and thus improve the algorithm convergence asymptotically. A key ingredient of our analysis is to connect three settings of Markov decision processes (MDPs): the finite-time-horizon, the average reward and the discounted settings. To our best knowledge, this is the first theoretical guarantee on fictitious discount algorithms for the episodic reinforcement learning of finite-time-horizon MDPs, which also leads to the (first) global convergence of policy gradient methods for finite-time-horizon episodic reinforcement learning.

We study finite-time horizon continuous-time linear-quadratic reinforcement learning problems in an episodic setting, where both the state and control coefficients are unknown to the controller. We first propose a least-squares algorithm based on continuous-time observations and controls, and establish a logarithmic regret bound of magnitude $\mathcal{O}((\ln M)(\ln\ln M) )$, with $M$ being the number of learning episodes. The analysis consists of two components: perturbation analysis, which exploits the regularity and robustness of the associated Riccati differential equation; and parameter estimation error, which relies on sub-exponential properties of continuous-time least-squares estimators. We further propose a practically implementable least-squares algorithm based on discrete-time observations and piecewise constant controls, which achieves similar logarithmic regret with an additional term depending explicitly on the time stepsizes used in the algorithm.

Wasserstein Generative Adversarial Networks (WGANs) provide a versatile class of models, which have attracted great attention in various applications. However, this framework has two main drawbacks: (i) Wasserstein-1 (or Earth-Mover) distance is restrictive such that WGANs cannot always fit data geometry well; (ii) It is difficult to achieve fast training of WGANs. In this paper, we propose a new class of Relaxed Wasserstein (RW) distances by generalizing Wasserstein-1 distance with Bregman cost functions. We show that RW distances achieve nice statistical properties while not sacrificing the computational tractability. Combined with the GANs framework, we develop Relaxed WGANs (RWGANs) which are not only statistically flexible but can be approximated efficiently using heuristic approaches. Experiments on real images demonstrate that the RWGAN with Kullback-Leibler (KL) cost function outperforms other competing approaches, e.g., WGANs, even with gradient penalty.

This paper presents a general mean-field game (GMFG) framework for simultaneous learning and decision-making in stochastic games with a large population. It first establishes the existence of a unique Nash Equilibrium to this GMFG, and explains that naively combining Q-learning with the fixed-point approach in classical MFGs yields unstable algorithms. It then proposes a Q-learning algorithm with Boltzmann policy (GMF-Q), with analysis of convergence property and computational complexity. The experiments on repeated Ad auction problems demonstrate that this GMF-Q algorithm is efficient and robust in terms of convergence and learning accuracy. Moreover, its performance is superior in convergence, stability, and learning ability, when compared with existing algorithms for multi-agent reinforcement learning.

Topic sparsity refers to the observation that individual documents usually focus on several salient topics instead of covering a wide variety of topics, and a real topic adopts a narrow range of terms instead of a wide coverage of the vocabulary. Understanding this topic sparsity is especially important for analyzing user-generated web content and social media, which are featured in the form of extremely short posts and discussions. As topic sparsity of individual documents in online social media increases, so does the difficulty of analyzing the online text sources using traditional methods. In this paper, we propose two novel neural models by providing sparse posterior distributions over topics based on the Gaussian sparsemax construction, enabling efficient training by stochastic backpropagation. We construct an inference network conditioned on the input data and infer the variational distribution with the relaxed Wasserstein (RW) divergence. Unlike existing works based on Gaussian softmax construction and Kullback-Leibler (KL) divergence, our approaches can identify latent topic sparsity with training stability, predictive performance, and topic coherence. Experiments on different genres of large text corpora have demonstrated the effectiveness of our models as they outperform both probabilistic and neural methods.

We consider the problem of predicting a random variable X from observations, denoted by a random variable Z. It is well known that the conditional expectation E[X|Z] is the optimal L/sup 2/ predictor (also known as "the least-mean-square error" predictor) of X, among all (Borel measurable) functions of Z. In this orrespondence, we provide necessary and sufficient conditions for the general loss functions under which the conditional expectation is the unique optimal predictor. We show that E[X|Z] is the optimal predictor for all Bregman loss functions (BLFs), of which the L/sup 2/ loss function is a special case. Moreover, under mild conditions, we show that the BLFs are exhaustive, i.e., if for every random variable X, the infimum of E[F(X,y)] over all constants y is attained by the expectation E[X], then F is a BLF.

This paper provides necessary and sufficient conditions for general loss functions under which the conditional expectation is the unique optimal predictor of a random variable. Further, using such loss functions, we give an exact characterization of the difference between the two sides of Jensen's inequality.