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This paper proposes a novel multi-agent reinforcement learning (MARL) method to learn multiple coordinated agents under directed acyclic graph (DAG) constraints. Unlike existing MARL approaches, our method explicitly exploits the DAG structure between agents to achieve more effective learning performance. Theoretically, we propose a novel surrogate value function based on a MARL model with synthetic rewards (MARLM-SR) and prove that it serves as a lower bound of the optimal value function. Computationally, we propose a practical training algorithm that exploits new notion of leader agent and reward generator and distributor agent to guide the decomposed follower agents to better explore the parameter space in environments with DAG constraints. Empirically, we exploit four DAG environments including a real-world scheduling for one of Intel's high volume packaging and test factory to benchmark our methods and show it outperforms the other non-DAG approaches.

Sequences are often not received in their entirety at once, but instead, received incrementally over time, element by element. Early predictions yielding a higher benefit, one aims to classify a sequence as accurately as possible, as soon as possible, without having to wait for the last element. For this early sequence classification, we introduce our novel classifier-induced stopping. While previous methods depend on exploration during training to learn when to stop and classify, ours is a more direct, supervised approach. Our classifier-induced stopping achieves an average Pareto frontier AUC increase of 11.8% over multiple experiments., Comment: 12 pages, 6 figures

We introduce a mini-batch stochastic variance-reduced algorithm to solve finite-sum scale invariant problems which cover several examples in machine learning and statistics such as principal component analysis (PCA) and estimation of mixture proportions. The algorithm is a stochastic generalization of scale invariant power iteration, specializing to power iteration when full-batch is used for the PCA problem. In convergence analysis, we show the expectation of the optimality gap decreases at a linear rate under some conditions on the step size, epoch length, batch size and initial iterate. Numerical experiments on the non-negative factorization problem with the Kullback-Leibler divergence using real and synthetic datasets demonstrate that the proposed stochastic approach not only converges faster than state-of-the-art deterministic algorithms but also produces excellent quality robust solutions.

We study a decentralized multi-agent multi-armed bandit problem in which multiple clients are connected by time dependent random graphs provided by an environment. The reward distributions of each arm vary across clients and rewards are generated independently over time by an environment based on distributions that include both sub-exponential and sub-gaussian distributions. Each client pulls an arm and communicates with neighbors based on the graph provided by the environment. The goal is to minimize the overall regret of the entire system through collaborations. To this end, we introduce a novel algorithmic framework, which first provides robust simulation methods for generating random graphs using rapidly mixing Markov chains or the random graph model, and then combines an averaging-based consensus approach with a newly proposed weighting technique and the upper confidence bound to deliver a UCB-type solution. Our algorithms account for the randomness in the graphs, removing the conventional doubly stochasticity assumption, and only require the knowledge of the number of clients at initialization. We derive optimal instance-dependent regret upper bounds of order $\log{T}$ in both sub-gaussian and sub-exponential environments, and a nearly optimal mean-gap independent regret upper bound of order $\sqrt{T}\log T$ up to a $\log T$ factor. Importantly, our regret bounds hold with high probability and capture graph randomness, whereas prior works consider expected regret under assumptions and require more stringent reward distributions., Comment: 14 pages, under review

Federated learning is a distributed machine learning paradigm where multiple data owners (clients) collaboratively train one machine learning model while keeping data on their own devices. The heterogeneity of client datasets is one of the most important challenges of federated learning algorithms. Studies have found performance reduction with standard federated algorithms, such as FedAvg, on non-IID data. Many existing works on handling non-IID data adopt the same aggregation framework as FedAvg and focus on improving model updates either on the server side or on clients. In this work, we tackle this challenge in a different view by introducing redistribution rounds that delay the aggregation. We perform experiments on multiple tasks and show that the proposed framework significantly improves the performance on non-IID data.

In a recurrent setting, conventional approaches to neural architecture search find and fix a general model for all data samples and time steps. We propose a novel algorithm that can dynamically search for the structure of cells in a recurrent neural network model. Based on a combination of recurrent and recursive neural networks, our algorithm is able to construct customized cell structures for each data sample and time step, allowing for a more efficient architecture search than existing models. Experiments on three common datasets show that the algorithm discovers high-performance cell architectures and achieves better prediction accuracy compared to the GRU structure for language modelling and sentiment analysis.

Feature acquisition algorithms address the problem of acquiring informative features while balancing the costs of acquisition to improve the learning performances of ML models. Previous approaches have focused on calculating the expected utility values of features to determine the acquisition sequences. Other approaches formulated the problem as a Markov Decision Process (MDP) and applied reinforcement learning based algorithms. In comparison to previous approaches, we focus on 1) formulating the feature acquisition problem as a MDP and applying Monte Carlo Tree Search, 2) calculating the intermediary rewards for each acquisition step based on model improvements and acquisition costs and 3) simultaneously optimizing model improvement and acquisition costs with multi-objective Monte Carlo Tree Search. With Proximal Policy Optimization and Deep Q-Network algorithms as benchmark, we show the effectiveness of our proposed approach with experimental study., Comment: 13 pages, 7 figures

Stochastic optimization algorithms using exponential moving averages of the past gradients, such as ADAM, RMSProp and AdaGrad, have been having great successes in many applications, especially in training deep neural networks. ADAM in particular stands out as efficient and robust. Despite of its outstanding performance, ADAM has been proved to be divergent for some specific problems. We revisit the divergent question and provide divergent examples under stronger conditions such as in expectation or high probability. Under a variance reduction assumption, we show that an ADAM-type algorithm converges, which means that it is the variance of gradients that causes the divergence of original ADAM. To this end, we propose a variance reduced version of ADAM and provide a convergent analysis of the algorithm. Numerical experiments show that the proposed algorithm has as good performance as ADAM. Our work suggests a new direction for fixing the convergence issues.

Certain type of documents such as tweets are collected by specifying a set of keywords. As topics of interest change with time it is beneficial to adjust keywords dynamically. The challenge is that these need to be specified ahead of knowing the forthcoming documents and the underlying topics. The future topics should mimic past topics of interest yet there should be some novelty in them. We develop a keyword-based topic model that dynamically selects a subset of keywords to be used to collect future documents. The generative process first selects keywords and then the underlying documents based on the specified keywords. The model is trained by using a variational lower bound and stochastic gradient optimization. The inference consists of finding a subset of keywords where given a subset the model predicts the underlying topic-word matrix for the unknown forthcoming documents. We compare the keyword topic model against a benchmark model using viral predictions of tweets combined with a topic model. The keyword-based topic model outperforms this sophisticated baseline model by 67%.

In model serving, having one fixed model during the entire often life-long inference process is usually detrimental to model performance, as data distribution evolves over time, resulting in lack of reliability of the model trained on historical data. It is important to detect changes and retrain the model in time. The existing methods generally have three weaknesses: 1) using only classification error rate as signal, 2) assuming ground truth labels are immediately available after features from samples are received and 3) unable to decide what data to use to retrain the model when change occurs. We address the first problem by utilizing six different signals to capture a wide range of characteristics of data, and we address the second problem by allowing lag of labels, where labels of corresponding features are received after a lag in time. For the third problem, our proposed method automatically decides what data to use to retrain based on the signals. Extensive experiments on structured and unstructured data for different type of data changes establish that our method consistently outperforms the state-of-the-art methods by a large margin.

Many models such as Long Short Term Memory (LSTMs), Gated Recurrent Units (GRUs) and transformers have been developed to classify time series data with the assumption that events in a sequence are ordered. On the other hand, fewer models have been developed for set based inputs, where order does not matter. There are several use cases where data is given as partially-ordered sequences because of the granularity or uncertainty of time stamps. We introduce a novel transformer based model for such prediction tasks, and benchmark against extensions of existing order invariant models. We also discuss how transition probabilities between events in a sequence can be used to improve model performance. We show that the transformer-based equal-time model outperforms extensions of existing set models on three data sets.

Existing methods of pruning deep neural networks focus on removing unnecessary parameters of the trained network and fine tuning the model afterwards to find a good solution that recovers the initial performance of the trained model. Unlike other works, our method pays special attention to the quality of the solution in the compressed model and inference computation time by pruning neurons. The proposed algorithm directs the parameters of the compressed model toward a flatter solution by exploring the spectral radius of Hessian which results in better generalization on unseen data. Moreover, the method does not work with a pre-trained network and performs training and pruning simultaneously. Our result shows that it improves the state-of-the-art results on neuron compression. The method is able to achieve very small networks with small accuracy degradation across different neural network models., Comment: 15 pages, 5 figures

We develop a regularization method which finds flat minima during the training of deep neural networks and other machine learning models. These minima generalize better than sharp minima, allowing models to better generalize to real word test data, which may be distributed differently from the training data. Specifically, we propose a method of regularized optimization to reduce the spectral radius of the Hessian of the loss function. Additionally, we derive algorithms to efficiently perform this optimization on neural networks and prove convergence results for these algorithms. Furthermore, we demonstrate that our algorithm works effectively on multiple real world applications in multiple domains including healthcare. In order to show our models generalize well, we introduce different methods of testing generalizability., Comment: 12 pages

In inference, open-set classification is to either classify a sample into a known class from training or reject it as an unknown class. Existing deep open-set classifiers train explicit closed-set classifiers, in some cases disjointly utilizing reconstruction, which we find dilutes the latent representation's ability to distinguish unknown classes. In contrast, we train our model to cooperatively learn reconstruction and perform class-based clustering in the latent space. With this, our Gaussian mixture variational autoencoder (GMVAE) achieves more accurate and robust open-set classification results, with an average F1 improvement of 29.5%, through extensive experiments aided by analytical results., Comment: 12 pages including 8 figures and 4 tables, plus 6 pages of supplementary material

We present a stochastic variance-reduced heavy ball power iteration algorithm for solving PCA and provide a convergence analysis for it. The algorithm is an extension of heavy ball power iteration, incorporating a step size so that progress can be controlled depending on the magnitude of the variance of stochastic gradients. The algorithm works with any size of the mini-batch, and if the step size is appropriately chosen, it attains global linear convergence to the first eigenvector of the covariance matrix in expectation. The global linear convergence result in expectation is analogous to those of stochastic variance-reduced gradient methods for convex optimization but due to non-convexity of PCA, it has never been shown for previous stochastic variants of power iteration since it requires very different techniques. We provide the first such analysis and stress that our framework can be used to establish convergence of the previous stochastic algorithms for any initial vector and in expectation. Experimental results show that the algorithm attains acceleration in a large batch regime, outperforming benchmark algorithms especially when the eigen-gap is small.

Power iteration has been generalized to solve many interesting problems in machine learning and statistics. Despite its striking success, theoretical understanding of when and how such an algorithm enjoys good convergence property is limited. In this work, we introduce a new class of optimization problems called scale invariant problems and prove that they can be efficiently solved by scale invariant power iteration (SCI-PI) with a generalized convergence guarantee of power iteration. By deriving that a stationary point is an eigenvector of the Hessian evaluated at the point, we show that scale invariant problems indeed resemble the leading eigenvector problem near a local optimum. Also, based on a novel reformulation, we geometrically derive SCI-PI which has a general form of power iteration. The convergence analysis shows that SCI-PI attains local linear convergence with a rate being proportional to the top two eigenvalues of the Hessian at the optimum. Moreover, we discuss some extended settings of scale invariant problems and provide similar convergence results for them. In numerical experiments, we introduce applications to independent component analysis, Gaussian mixtures, and non-negative matrix factorization. Experimental results demonstrate that SCI-PI is competitive to state-of-the-art benchmark algorithms and often yield better solutions.

Nowadays, online learning is an appealing learning paradigm, which is of great interest in practice due to the recent emergence of large scale applications such as online advertising placement and online web ranking. Standard online learning assumes a finite number of samples while in practice data is streamed infinitely. In such a setting gradient descent with a diminishing learning rate does not work. We first introduce regret with rolling window, a new performance metric for online streaming learning, which measures the performance of an algorithm on every fixed number of contiguous samples. At the same time, we propose a family of algorithms based on gradient descent with a constant or adaptive learning rate and provide very technical analyses establishing regret bound properties of the algorithms. We cover the convex setting showing the regret of the order of the square root of the size of the window in the constant and dynamic learning rate scenarios. Our proof is applicable also to the standard online setting where we provide the first analysis of the same regret order (the previous proofs have flaws). We also study a two layer neural network setting with ReLU activation. In this case we establish that if initial weights are close to a stationary point, the same square root regret bound is attainable. We conduct computational experiments demonstrating a superior performance of the proposed algorithms.

Deep recurrent neural networks perform well on sequence data and are the model of choice. However, it is a daunting task to decide the structure of the networks, i.e. the number of layers, especially considering different computational needs of a sequence. We propose a layer flexible recurrent neural network with adaptive computation time, and expand it to a sequence to sequence model. Different from the adaptive computation time model, our model has a dynamic number of transmission states which vary by step and sequence. We evaluate the model on a financial data set and Wikipedia language modeling. Experimental results show the performance improvement of 7\% to 12\% and indicate the model's ability to dynamically change the number of layers along with the computational steps., Comment: 11 pages, 5 figures

There are time series that are amenable to recurrent neural network (RNN) solutions when treated as sequences, but some series, e.g. asynchronous time series, provide a richer variation of feature types than current RNN cells take into account. In order to address such situations, we introduce a unified RNN that handles five different feature types, each in a different manner. Our RNN framework separates sequential features into two groups dependent on their frequency, which we call sparse and dense features, and which affect cell updates differently. Further, we also incorporate time features at the sequential level that relate to the time between specified events in the sequence and are used to modify the cell's memory state. We also include two types of static (whole sequence level) features, one related to time and one not, which are combined with the encoder output. The experiments show that the modeling framework proposed does increase performance compared to standard cells.

Recurrent neural networks and sequence to sequence models require a predetermined length for prediction output length. Our model addresses this by allowing the network to predict a variable length output in inference. A new loss function with a tailored gradient computation is developed that trades off prediction accuracy and output length. The model utilizes a function to determine whether a particular output at a time should be evaluated or not given a predetermined threshold. We evaluate the model on the problem of predicting the prices of securities. We find that the model makes longer predictions for more stable securities and it naturally balances prediction accuracy and length.

As the size of modern data sets exceeds the disk and memory capacities of a single computer, machine learning practitioners have resorted to parallel and distributed computing. Given that optimization is one of the pillars of machine learning and predictive modeling, distributed optimization methods have recently garnered ample attention, in particular when either observations or features are distributed, but not both. We propose a general stochastic algorithm where observations, features, and gradient components can be sampled in a double distributed setting, i.e., with both features and observations distributed. Very technical analyses establish convergence properties of the algorithm under different conditions on the learning rate (diminishing to zero or constant). Computational experiments in Spark demonstrate a superior performance of our algorithm versus a benchmark in early iterations of the algorithm, which is due to the stochastic components of the algorithm., Comment: 11 figures, 41 pages

The in-game economies of massively multi-player online games (MMOGs) are complex systems that have to be carefully designed and managed. This paper presents the results of an analysis of auction house data from the MMOG Glitch, across a 14 month time period, the entire lifetime of the game. The data comprise almost 3 million data points, over 20,000 unique players and more than 650 products. Furthermore, an interactive visualization, based on Sankey flow diagrams, is presented which shows the proportion of the different clusters across each time bin, as well as the flow of players between clusters. The diagram allows evaluation of migration of players between clusters as a function of time, as well as churn analysis. The presented work provides a template analysis and visualization model for progression-based or temporal-based analysis of player behavior broadly applicable to games.

In this paper, we predict the likelihood of a player making a shot in basketball from multiagent trajectories. Previous approaches to similar problems center on hand-crafting features to capture domain specific knowledge. Although intuitive, recent work in deep learning has shown this approach is prone to missing important predictive features. To circumvent this issue, we present a convolutional neural network (CNN) approach where we initially represent the multiagent behavior as an image. To encode the adversarial nature of basketball, we use a multi-channel image which we then feed into a CNN. Additionally, to capture the temporal aspect of the trajectories we "fade" the player trajectories. We find that this approach is superior to a traditional FFN model. By using gradient ascent to create images using an already trained CNN, we discover what features the CNN filters learn. Last, we find that a combined CNN+FFN is the best performing network with an error rate of 39%.

As the size of modern data sets exceeds the disk and memory capacities of a single computer, machine learning practitioners have resorted to parallel and distributed computing. Given that optimization is one of the pillars of machine learning and predictive modeling, distributed optimization methods have recently garnered ample attention in the literature. Although previous research has mostly focused on settings where either the observations, or features of the problem at hand are stored in distributed fashion, the situation where both are partitioned across the nodes of a computer cluster (doubly distributed) has barely been studied. In this work we propose two doubly distributed optimization algorithms. The first one falls under the umbrella of distributed dual coordinate ascent methods, while the second one belongs to the class of stochastic gradient/coordinate descent hybrid methods. We conduct numerical experiments in Spark using real-world and simulated data sets and study the scaling properties of our methods. Our empirical evaluation of the proposed algorithms demonstrates the out-performance of a block distributed ADMM method, which, to the best of our knowledge is the only other existing doubly distributed optimization algorithm.