In the ever-evolving blockchain ecosystem, decentralized exchanges (DEXs) have seen significant growth, which, however, has also brought challenges of Maximal Extractable Value (MEV). DEXs offer a decentralized platform for cryptocurrency trading. Such trading mechanisms primarily include Constant Function Market Makers (CFMMs) and batch auctions.

Assortment optimization has received active explorations in the past few decades due to its practical importance. Despite the extensive literature dealing with optimization algorithms and latent score estimation, uncertainty quantification for the optimal assortment still needs to be explored and is of great practical significance. Instead of estimating and recovering the complete optimal offer set, decision-makers may only be interested in testing whether a given property holds true for the optimal assortment, such as whether they should include several products of interest in the optimal set, or how many categories of products the optimal set should include. This paper proposes a novel inferential framework for testing such properties. We consider the widely adopted multinomial logit (MNL) model, where we assume that each customer will purchase an item j within the offer set of products S with a probability proportional to the underlying preference score u*j associated with the product. For a full assortment of n products, our objective is to conduct a hypothesis test concerning a general optimal assortment property, given by:

A major challenge for ridesharing platforms is to guarantee profit and fairness simultaneously, especially in the presence of misaligned incentives of drivers and riders. We focus on the dispatching-pricing problem to maximize the total revenue while keeping both drivers and riders satisfied. We study the computational complexity of the problem, provide a novel two-phased pricing solution with revenue and fairness guarantees, extend it to stochastic settings and develop a dynamic (a.k.a., learning-while-doing) algorithm that actively collects data to learn the demand distribution during the scheduling process. We also conduct extensive experiments to demonstrate the effectiveness of our algorithms.

In this paper, we consider an unconstrained optimization model where the objective is a sum of a large number of possibly nonconvex functions, though overall the objective is assumed to be smooth and convex. Our bid to solving such model uses the framework of cubic regularization of Newton's method. As well known, the crux in cubic regularization is its utilization of the Hessian information, which may be computationally expensive for large-scale problems. To tackle this, we resort to approximating the Hessian matrix via sub-sampling. In particular, we propose to compute an approximated Hessian matrix by either uniformly or non-uniformly sub-sampling the components of the objective. Based upon such sampling strategy, we develop accelerated adaptive cubic regularization approaches and provide theoretical guarantees on global iteration complexity of $\O(\epsilon^{-1/3})$ with high probability, which matches that of the original accelerated cubic regularization methods Jiang et al. (2020) using the full Hessian information. Interestingly, we also show that in the worst case scenario our algorithm still achieves an $O(\epsilon^{-5/6}\log(\epsilon^{-1}))$ iteration complexity bound. The proof techniques are new to our knowledge and can be of independent interets. Experimental results on the regularized logistic regression problems demonstrate a clear effect of acceleration on several real data sets.

Linear contextual bandit is a class of sequential decision-making problems with important applications in recommendation systems, online advertising, healthcare, and other machine learning-related tasks. While there is much prior research, tight regret bounds of linear contextual bandit with infinite action sets remain open. In this paper, we consider the linear contextual bandit problem with (changing) infinite action sets. We prove a regret upper bound on the order of O(\sqrt{d^2T\log T}) \poly(\log\log T) where d is the domain dimension and T is the time horizon. Our upper bound matches the previous lower bound of \Omega(\sqrt{d^2 T\log T}) in [Li et al., 2019] up to iterated logarithmic terms.

Modern machine learning models (such as deep neural networks and boosting decision tree models) have become increasingly popular in financial market prediction, due to their superior capacity to extract complex non-linear patterns. However, since financial datasets have very low signal-to-noise ratio and are non-stationary, complex models are often very prone to overfitting and suffer from instability issues. Moreover, as various machine learning and data mining tools become more widely used in quantitative trading, many trading firms have been producing an increasing number of features (aka factors). Therefore, how to automatically select effective features becomes an imminent problem. To address these issues, we propose DoubleEnsemble, an ensemble framework leveraging learning trajectory based sample reweighting and shuffling based feature selection. Specifically, we identify the key samples based on the training dynamics on each sample and elicit key features based on the ablation impact of each feature via shuffling. Our model is applicable to a wide range of base models, capable of extracting complex patterns, while mitigating the overfitting and instability issues for financial market prediction. We conduct extensive experiments, including price prediction for cryptocurrencies and stock trading, using both DNN and gradient boosting decision tree as base models. Our experiment results demonstrate that DoubleEnsemble achieves a superior performance compared with several baseline methods.

In this paper, we study the dynamic assortment optimization problem over a finite selling season of length $T$. At each time period, the seller offers an arriving customer an assortment of substitutable products under a cardinality constraint, and the customer makes the purchase among offered products according to a discrete choice model. Most existing work associates each product with a real-valued fixed mean utility and assumes a multinomial logit choice (MNL) model. In many practical applications, feature/contextual information of products is readily available. In this paper, we incorporate the feature information by assuming a linear relationship between the mean utility and the feature. In addition, we allow the feature information of products to change over time so that the underlying choice model can also be non-stationary. To solve the dynamic assortment optimization under this changing contextual MNL model, we need to simultaneously learn the underlying unknown coefficient and make the decision on the assortment. To this end, we develop an upper confidence bound (UCB) based policy and establish the regret bound on the order of $\tilde{O}(d\sqrt{T})$, where $d$ is the dimension of the feature and $\tilde{O}$ suppresses logarithmic dependence. We further establish a lower bound $\Omega(d\sqrt{T}/{K})$, where $K$ is the cardinality constraint of an offered assortment, which is usually small. When $K$ is a constant, our policy is optimal up to logarithmic factors. In the exploitation phase of the UCB algorithm, we need to solve a combinatorial optimization problem for assortment optimization based on the learned information. We further develop an approximation algorithm and an efficient greedy heuristic. The effectiveness of the proposed policy is further demonstrated by our numerical studies.

In this paper we consider the dynamic assortment selection problem under an uncapacitated multinomial-logit (MNL) model. By carefully analyzing a revenue potential function, we show that a trisection based algorithm achieves an item-independent regret bound of O(sqrt(T log log T), which matches information theoretical lower bounds up to iterated logarithmic terms. Our proof technique draws tools from the unimodal/convex bandit literature as well as adaptive confidence parameters in minimax multi-armed bandit problems.

We study the active learning problem of top-k ranking from multi-wise comparisons under the popular multinomial logit model. Our goal is to identify the top-k items with high probability by adaptively querying sets for comparisons and observing the noisy output of the most preferred item from each comparison. To achieve this goal, we design a new active ranking algorithm without using any information about the underlying items’ preference scores. We also establish a matching lower bound on the sample complexity even when the set of preference scores is given to the algorithm. These two results together show that the proposed algorithm is nearly instance optimal (similar to instance optimal [12], but up to polylog factors). Our work extends the existing literature on rank aggregation in three directions. First, instead of studying a static problem with fixed data, we investigate the top-k ranking problem in an active learning setting. Second, we show our algorithm is nearly instance optimal, which is a much stronger theoretical guarantee. Finally, we extend the pairwise comparison to the multi-wise comparison, which has not been fully explored in ranking literature.

Motivated by applications in recommender systems, web search, social choice, and crowdsourcing, we consider the problem of identifying the set of top K items from noisy pairwise comparisons. In our setting, we are given r pairwise comparisons between each pair of n items, where each comparison has noise constrained by a very general noise model called the strong stochastic transitivity model. Our goal is to provide an optimal instance adaptive algorithm for the top-K ranking problem. In particular, we present a linear time algorithm that has a competitive ratio of Õ(√n)1; i.e., to solve any instance of top-K ranking, our algorithm needs at most Õ(√n) times as many samples needed as the best possible algorithm for that instance [in contrast, all previous known algorithms for the topK problem have competitive ratios of Ω̃(n) or worse]. We further show that this is tight (up to polylogarithmic factors): any algorithm for the top-K problem has competitive ratio of at least Ω̃(√n).

We study the problem of selecting K arms with the highest expected rewards in a stochastic n-armed bandit game. This problem has a wide range of applications, e.g., A/B testing, crowdsourcing, simulation optimization. Our goal is to develop a PAC algorithm, which, with probability at least $1-\delta$, identifies a set of K arms with the aggregate regret at most $\epsilon$. The notion of aggregate regret for multiple-arm identification was first introduced in Zhou et. al. (2014), which is defined as the difference of the averaged expected rewards between the selected set of arms and the best K arms. In contrast to Zhou et. al. (2014) that only provides instance-independent sample complexity, we introduce a new hardness parameter for characterizing the difficulty of any given instance. We further develop two algorithms and establish the corresponding sample complexity in terms of this hardness parameter. The derived sample complexity can be significantly smaller than state-of-the-art results for a large class of instances and matches the instance-independent lower bound up to a $\log(\epsilon^{-1})$ factor in the worst case. We also prove a lower bound result showing that the extra $\log(\epsilon^{-1})$ is necessary for instance-dependent algorithms using the introduced hardness parameter.

Motivated by applications in recommender systems, web search, social choice and crowdsourcing, we consider the problem of identifying the set of top K items from noisy pairwise comparisons. In our setting, we are non-actively given r pairwise comparisons between each pair of n items, where each comparison has noise constrained by a very general noise model called the strong stochastic transitivity (SST) model. We analyze the competitive ratio of algorithms for the top-K problem. In particular, we present a linear time algorithm for the top-K problem which has a competitive ratio of i.e. to solve any instance of top-K, our algorithm needs at most times as many samples needed as the best possible algorithm for that instance (in contrast, all previous known algorithms for the top-K problem have competitive ratios of Ω(n) or worse). We further show that this is tight: any algorithm for the top-K problem has competitive ratio at least

We study the problem of selecting K arms with the highest expected rewards in a stochastic N-armed bandit game. Instead of using existing evaluation metrics (e.g., misidentification probability or the metric in EXPLORE-K), we propose to use the aggregate regret, which is defined as the gap between the average reward of the optimal solution and that of our solution. Besides being a natural metric by itself, we argue that in many applications, such as our motivating example from crowdsourcing, the aggregate regret bound is more suitable. We propose a new PAC algorithm, which, with probability at least 1-δ, identifies a set of K arms with regret at most ε. We provide the sample complexity bound of our algorithm. To complement, we establish the lower bound and show that the sample complexity of our algorithm matches the lower bound. Finally, we report experimental results on both synthetic and real data sets, which demonstrates the superior performance of the proposed algorithm.

Stochastic gradient optimization is a class of widely used algorithms for training machine learning models. To optimize an objective, it uses the noisy gradient computed from the random data samples instead of the true gradient computed from the entire dataset. However, when the variance of the noisy gradient is large, the algorithm might spend much time bouncing around, leading to slower convergence and worse performance. In this paper, we develop a general approach of using control variate for variance reduction in stochastic gradient. Data statistics such as low-order moments (pre-computed or estimated online) is used to form the control variate. We demonstrate how to construct the control variate for two practical problems using stochastic gradient optimization. One is convex---the MAP estimation for logistic regression, and the other is non-convex---stochastic variational inference for latent Dirichlet allocation. On both problems, our approach shows faster convergence and better performance than the classical approach.

This paper considers a wide spectrum of regularized stochastic optimization problems where both the loss function and regularizer can be non-smooth. We develop a novel algorithm based on the regularized dual averaging (RDA) method, that can simultaneously achieve the optimal convergence rates for both convex and strongly convex loss. In particular, for strongly convex loss, it achieves the optimal rate of $O(\frac{1}{N}+\frac{1}{N^2})$ for $N$ iterations, which improves the best known rate $O(\frac{\log N }{N})$ of previous stochastic dual averaging algorithms. In addition, our method constructs the final solution directly from the proximal mapping instead of averaging of all previous iterates. For widely used sparsity-inducing regularizers (e.g., $\ell_1$-norm), it has the advantage of encouraging sparser solutions. We further develop a multi-stage extension using the proposed algorithm as a subroutine, which achieves the uniformly-optimal rate $O(\frac{1}{N}+\exp\{-N\})$ for strongly convex loss.

In this paper, we consider the multi-task sparse learning problem under the assumption that the dimensionality diverges with the sample size. The traditional l1/l2 multi-task lasso does not enjoy the oracle property unless a rather strong condition is enforced. Inspired by adaptive lasso, we propose a multi-stage procedure, adaptive multi-task lasso, to simultaneously conduct model estimation and variable selection across different tasks. Motivated by adaptive elastic-net, we further propose the adaptive multi-task elastic-net by adding another quadratic penalty to address the problem of collinearity. When the number of tasks is fixed, under weak assumptions, we establish the asymptotic oracle property for the proposed adaptive multi-task sparse learning methods including both adaptive multi-task lasso and elastic-net. In addition to the desirable asymptotic property, we show by simulations that adaptive sparse learning methods also achieve much improved finite sample performance. As a case study, we apply adaptive multi-task elastic-net to a cognitive science problem, where one wants to discover a compact semantic basis for predicting fMRI images. We show that adaptive multi-task sparse learning methods achieve superior performance and provide some insights into how the brain represents meanings of words.

Matrix factorization methods are extremely useful in many data mining tasks, yet their performances are often degraded by outliers. In this paper, we propose a novel robust matrix factorization algorithm that is insensitive to outliers. We directly formulate robust factorization as a matrix approximation problem with constraints on the rank of the matrix and the cardinality of the outlier set. Then, unlike existing methods that resort to convex relaxations, we solve this problem directly and efficiently. In addition, structural knowledge about the outliers can be incorporated to find outliers more effectively. We applied this method in anomaly detection tasks on various data sets. Empirical results show that this new algorithm is effective in robust modeling and anomaly detection, and our direct solution achieves superior performance over the state-of-the-art methods based on the L1-norm and the nuclear norm of matrices.

Latent semantic analysis (LSA), as one of the most popular unsupervised dimension reduction tools, has a wide range of applications in text mining and information retrieval. The key idea of LSA is to learn a projection matrix that maps the high dimensional vector space representations of documents to a lower dimensional latent space, i.e. so called latent topic space. In this paper, we propose a new model called Sparse LSA, which produces a sparse projection matrix via the ℓ1 regularization. Compared to the traditional LSA, Sparse LSA selects only a small number of relevant words for each topic and hence provides a compact representation of topic-word relationships. Moreover, Sparse LSA is computationally very efficient with much less memory usage for storing the projection matrix. Furthermore, we propose two important extensions of Sparse LSA: group structured Sparse LSA and non-negative Sparse LSA. We conduct experiments on several benchmark datasets and compare Sparse LSA and its extensions with several widely used methods, e.g. LSA, Sparse Coding and LDA. Empirical results suggest that Sparse LSA achieves similar performance gains to LSA, but is more efficient in projection computation, storage, and also well explain the topic-word relationships.

An important challenge in understanding climate change is to uncover the dependency relationships between various climate observations and forcing factors. Graphical lasso, a recently proposed L1 penalty based structure learning algorithm, has been proven successful for learning underlying dependency structures for the data drawn from a multivariate Gaussian distribution. However, climatological data often turn out to be non-Gaussian, e.g. cloud cover, precipitation, etc. In this paper, we examine nonparametric learning methods to address this challenge. In particular, we develop a methodology to learn dynamic graph structures from spatial-temporal data so that the graph structures at adjacent time or locations are similar. Experimental results demonstrate that our method not only recovers the underlying graph well but also captures the smooth variation properties on both synthetic data and climate data. An important challenge in understanding climate change is to uncover the dependency relationships between various climate observations and forcing factors. Graphical lasso, a recently proposed An important challenge in understanding climate change is to uncover the dependency relationships between various climate observations and forcing factors. Graphical lasso, a recently proposed L1 penalty based structure learning algorithm, has been proven successful for learning underlying dependency structures for the data drawn from a multivariate Gaussian distribution. However, climatological data often turn out to be non-Gaussian, e.g. cloud cover, precipitation, etc. In this paper, we examine nonparametric learning methods to address this challenge. In particular, we develop a methodology to learn dynamic graph structures from spatial-temporal data so that the graph structures at adjacent time or locations are similar. Experimental results demonstrate that our method not only recovers the underlying graph well but also captures the smooth variation properties on both synthetic data and climate data.

We study the retrieval task that ranks a set of objects for a given query in the pair wise preference learning framework. Recently researchers found out that raw features (e.g. words for text retrieval) and their pair wise features which describe relationships between two raw features (e.g. word synonymy or polysemy) could greatly improve the retrieval precision. However, most existing methods can not scale up to problems with many raw features (e.g. English vocabulary), due to the prohibitive computational cost on learning and the memory requirement to store a quadratic number of parameters. In this paper, we propose to learn a sparse representation of the pair wise features under the preference learning framework using the L1 regularization. Based on stochastic gradient descent, an online algorithm is devised to enforce the sparsity using a mini-batch shrinkage strategy. On multiple benchmark datasets, we show that our method achieves better performance with fast convergence, and takes much less memory on models with millions of parameters.

We propose a new nonparametric learning method based on multivariate dyadic regression trees (MDRTs). Unlike traditional dyadic decision trees (DDTs) or classification and regression trees (CARTs), MDRTs are constructed using penalized empirical risk minimization with a novel sparsity-inducing penalty. Theoretically, we show that MDRTs can simultaneously adapt to the unknown sparsity and smoothness of the true regression functions, and achieve the nearly optimal rates of convergence (in a minimax sense) for the class of $(\alpha, C)$-smooth functions. Empirically, MDRTs can simultaneously conduct function estimation and variable selection in high dimensions. To make MDRTs applicable for large-scale learning problems, we propose a greedy heuristics. The superior performance of MDRTs are demonstrated on both synthetic and real datasets.

Undirected graphical models encode in a graph $G$ the dependency structure of a random vector $Y$. In many applications, it is of interest to model $Y$ given another random vector $X$ as input. We refer to the problem of estimating the graph $G(x)$ of $Y$ conditioned on $X=x$ as ``graph-valued regression''. In this paper, we propose a semiparametric method for estimating $G(x)$ that builds a tree on the $X$ space just as in CART (classification and regression trees), but at each leaf of the tree estimates a graph. We call the method ``Graph-optimized CART'', or Go-CART. We study the theoretical properties of Go-CART using dyadic partitioning trees, establishing oracle inequalities on risk minimization and tree partition consistency. We also demonstrate the application of Go-CART to a meteorological dataset, showing how graph-valued regression can provide a useful tool for analyzing complex data.

Real-world relational data are seldom stationary, yet traditional collaborative filtering algorithms generally rely on this assumption. Motivated by our sales prediction problem, we propose a factor-based algorithm that is able to take time into account. By introducing additional factors for time, we formalize this problem as a tensor factorization with a special constraint on the time dimension. Further, we provide a fully Bayesian treatment to avoid tuning parameters and achieve automatic model complexity control. To learn the model we develop an efficient sampling procedure that is capable of analyzing large-scale data sets. This new algorithm, called Bayesian Probabilistic Tensor Factorization (BPTF), is evaluated on several real-world problems including sales prediction and movie recommendation. Empirical results demonstrate the superiority of our temporal model.

Many real world learning problems can be recast as multi-task learning problems which utilize correlations among different tasks to obtain better generalization performance than learning each task individually. The feature selection problem in multi-task setting has many applications in fields of computer vision, text classification and bio-informatics. Generally, it can be realized by solving a L-1-infinity regularized optimization problem. And the solution automatically yields the joint sparsity among different tasks. However, due to the nonsmooth nature of the L-1-infinity norm, there lacks an efficient training algorithm for solving such problem with general convex loss functions. In this paper, we propose an accelerated gradient method based on an ``optimal'' first order black-box method named after Nesterov and provide the convergence rate for smooth convex loss functions. For nonsmooth convex loss functions, such as hinge loss, our method still has fast convergence rate empirically. Moreover, by exploiting the structure of the L-1-infinity ball, we solve the black-box oracle in Nesterov's method by a simple sorting scheme. Our method is suitable for large-scale multi-task learning problem since it only utilizes the first order information and is very easy to implement. Experimental results show that our method significantly outperforms the most state-of-the-art methods in both convergence speed and learning accuracy.

This paper studies the forward greedy strategy in sparse nonparametric regression. For additive models, we propose an algorithm called additive forward regression; for general multivariate regression, we propose an algorithm called generalized forward regression. Both of them simultaneously conduct estimation and variable selection in nonparametric settings for the high dimensional sparse learning problem. Our main emphasis is empirical: on both simulated and real data, these two simple greedy methods can clearly outperform several state-of-the-art competitors, including the LASSO, a nonparametric version of the LASSO called the sparse additive model (SpAM) and a recently proposed adaptive parametric forward-backward algorithm called the Foba. Some theoretical justifications are also provided.