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Mengdi Wang

Publications and source records attributed to Mengdi Wang.

At least 253 records · Page 14Linked to original sources
arXiv2020

Generalized Leverage Score Sampling for Neural Networks

Leverage score sampling is a powerful technique that originates from theoretical computer science, which can be used to speed up a large number of fundamental questions, e.g. linear regression, linear programming, semi-definite programming, cutting plane method, graph sparsification, maximum matching and max-flow. Recently, it has been shown that leverage score sampling helps to accelerate kernel methods [Avron, Kapralov, Musco, Musco, Velingker and Zandieh 17]. In this work, we generalize the results in [Avron, Kapralov, Musco, Musco, Velingker and Zandieh 17] to a broader class of kernels. We further bring the leverage score sampling into the field of deep learning theory. $\bullet$ We show the connection between the initialization for neural network training and approximating the neural tangent kernel with random features. $\bullet$ We prove the equivalence between regularized neural network and neural tangent kernel ridge regression under the initialization of both classical random Gaussian and leverage score sampling.

cs.LG↗
arXiv2020

Improved Sample Complexity for Stochastic Compositional Variance Reduced Gradient

Convex composition optimization is an emerging topic that covers a wide range of applications arising from stochastic optimal control, reinforcement learning and multi-stage stochastic programming. Existing algorithms suffer from unsatisfactory sample complexity and practical issues since they ignore the convexity structure in the algorithmic design. In this paper, we develop a new stochastic compositional variance-reduced gradient algorithm with the sample complexity of $O((m+n)\log(1/ε)+1/ε^3)$ where $m+n$ is the total number of samples. Our algorithm is near-optimal as the dependence on $m+n$ is optimal up to a logarithmic factor. Experimental results on real-world datasets demonstrate the effectiveness and efficiency of the new algorithm.

math.OC↗
arXiv2020

Voting-Based Multi-Agent Reinforcement Learning for Intelligent IoT

The recent success of single-agent reinforcement learning (RL) in Internet of things (IoT) systems motivates the study of multi-agent reinforcement learning (MARL), which is more challenging but more useful in large-scale IoT. In this paper, we consider a voting-based MARL problem, in which the agents vote to make group decisions and the goal is to maximize the globally averaged returns. To this end, we formulate the MARL problem based on the linear programming form of the policy optimization problem and propose a distributed primal-dual algorithm to obtain the optimal solution. We also propose a voting mechanism through which the distributed learning achieves the same sublinear convergence rate as centralized learning. In other words, the distributed decision making does not slow down the process of achieving global consensus on optimality. Lastly, we verify the convergence of our proposed algorithm with numerical simulations and conduct case studies in practical multi-agent IoT systems.

cs.LG↗
arXiv2020

Variational Policy Gradient Method for Reinforcement Learning with General Utilities

In recent years, reinforcement learning (RL) systems with general goals beyond a cumulative sum of rewards have gained traction, such as in constrained problems, exploration, and acting upon prior experiences. In this paper, we consider policy optimization in Markov Decision Problems, where the objective is a general concave utility function of the state-action occupancy measure, which subsumes several of the aforementioned examples as special cases. Such generality invalidates the Bellman equation. As this means that dynamic programming no longer works, we focus on direct policy search. Analogously to the Policy Gradient Theorem \cite{sutton2000policy} available for RL with cumulative rewards, we derive a new Variational Policy Gradient Theorem for RL with general utilities, which establishes that the parametrized policy gradient may be obtained as the solution of a stochastic saddle point problem involving the Fenchel dual of the utility function. We develop a variational Monte Carlo gradient estimation algorithm to compute the policy gradient based on sample paths. We prove that the variational policy gradient scheme converges globally to the optimal policy for the general objective, though the optimization problem is nonconvex. We also establish its rate of convergence of the order $O(1/t)$ by exploiting the hidden convexity of the problem, and proves that it converges exponentially when the problem admits hidden strong convexity. Our analysis applies to the standard RL problem with cumulative rewards as a special case, in which case our result improves the available convergence rate.

cs.LG↗
arXiv2020

Picasso: A Sparse Learning Library for High Dimensional Data Analysis in R and Python

We describe a new library named picasso, which implements a unified framework of pathwise coordinate optimization for a variety of sparse learning problems (e.g., sparse linear regression, sparse logistic regression, sparse Poisson regression and scaled sparse linear regression) combined with efficient active set selection strategies. Besides, the library allows users to choose different sparsity-inducing regularizers, including the convex $\ell_1$, nonconvex MCP and SCAD regularizers. The library is coded in C++ and has user-friendly R and Python wrappers. Numerical experiments demonstrate that picasso can scale up to large problems efficiently.

stat.ML↗
arXiv2020

Concept Annotation for Intelligent Textbooks

With the increased popularity of electronic textbooks, there is a growing interests in developing a new generation of "intelligent textbooks", which have the ability to guide the readers according to their learning goals and current knowledge. The intelligent textbooks extend regular textbooks by integrating machine-manipulatable knowledge such as a knowledge map or a prerequisite-outcome relationship between sections, among which, the most popular integrated knowledge is a list of unique knowledge concepts associated with each section. With the help of this concept, multiple intelligent operations, such as content linking, content recommendation or student modeling, can be performed. However, annotating a reliable set of concepts to a textbook section is a challenge. Automatic unsupervised methods for extracting key-phrases as the concepts are known to have insufficient accuracy. Manual annotation by experts is considered as a preferred approach and can be used to produce both the target outcome and the labeled data for training supervised models. However, most researchers in education domain still consider the concept annotation process as an ad-hoc activity rather than an engineering task, resulting in low-quality annotated data. In this paper, we present a textbook knowledge engineering method to obtain reliable concept annotations. The outcomes of our work include a validated knowledge engineering procedure, a code-book for technical concept annotation, and a set of concept annotations for the target textbook, which could be used as gold standard in further research.

cs.IR↗
arXiv2020

Generalization Bounds for Stochastic Saddle Point Problems

This paper studies the generalization bounds for the empirical saddle point (ESP) solution to stochastic saddle point (SSP) problems. For SSP with Lipschitz continuous and strongly convex-strongly concave objective functions, we establish an $\mathcal{O}(1/n)$ generalization bound by using a uniform stability argument. We also provide generalization bounds under a variety of assumptions, including the cases without strong convexity and without bounded domains. We illustrate our results in two examples: batch policy learning in Markov decision process, and mixed strategy Nash equilibrium estimation for stochastic games. In each of these examples, we show that a regularized ESP solution enjoys a near-optimal sample complexity. To the best of our knowledge, this is the first set of results on the generalization theory of ESP.

math.OC↗
arXiv2020

Model-Based Reinforcement Learning with Value-Targeted Regression

This paper studies model-based reinforcement learning (RL) for regret minimization. We focus on finite-horizon episodic RL where the transition model $P$ belongs to a known family of models $\mathcal{P}$, a special case of which is when models in $\mathcal{P}$ take the form of linear mixtures: $P_θ = \sum_{i=1}^{d} θ_{i}P_{i}$. We propose a model based RL algorithm that is based on optimism principle: In each episode, the set of models that are `consistent' with the data collected is constructed. The criterion of consistency is based on the total squared error of that the model incurs on the task of predicting \emph{values} as determined by the last value estimate along the transitions. The next value function is then chosen by solving the optimistic planning problem with the constructed set of models. We derive a bound on the regret, which, in the special case of linear mixtures, the regret bound takes the form $\tilde{\mathcal{O}}(d\sqrt{H^{3}T})$, where $H$, $T$ and $d$ are the horizon, total number of steps and dimension of $θ$, respectively. In particular, this regret bound is independent of the total number of states or actions, and is close to a lower bound $Ω(\sqrt{HdT})$. For a general model family $\mathcal{P}$, the regret bound is derived using the notion of the so-called Eluder dimension proposed by Russo & Van Roy (2014).

cs.LG↗
arXiv2020

Visual Data Analysis and Simulation Prediction for COVID-19

The COVID-19 (formerly, 2019-nCoV) epidemic has become a global health emergency, as such, WHO declared PHEIC. China has taken the most hit since the outbreak of the virus, which could be dated as far back as late November by some experts. It was not until January 23rd that the Wuhan government finally recognized the severity of the epidemic and took a drastic measure to curtain the virus spread by closing down all transportation connecting the outside world. In this study, we seek to answer a few questions: How did the virus get spread from the epicenter Wuhan city to the rest of the country? To what extent did the measures, such as, city closure and community quarantine, help controlling the situation? More importantly, can we forecast any significant future development of the event had some of the conditions changed? By collecting and visualizing publicly available data, we first show patterns and characteristics of the epidemic development; we then employ a mathematical model of disease transmission dynamics to evaluate the effectiveness of some epidemic control measures, and more importantly, to offer a few tips on preventive measures.

physics.med-ph↗
arXiv2020

Cautious Reinforcement Learning via Distributional Risk in the Dual Domain

We study the estimation of risk-sensitive policies in reinforcement learning problems defined by a Markov Decision Process (MDPs) whose state and action spaces are countably finite. Prior efforts are predominately afflicted by computational challenges associated with the fact that risk-sensitive MDPs are time-inconsistent. To ameliorate this issue, we propose a new definition of risk, which we call caution, as a penalty function added to the dual objective of the linear programming (LP) formulation of reinforcement learning. The caution measures the distributional risk of a policy, which is a function of the policy's long-term state occupancy distribution. To solve this problem in an online model-free manner, we propose a stochastic variant of primal-dual method that uses Kullback-Lieber (KL) divergence as its proximal term. We establish that the number of iterations/samples required to attain approximately optimal solutions of this scheme matches tight dependencies on the cardinality of the state and action spaces, but differs in its dependence on the infinity norm of the gradient of the risk measure. Experiments demonstrate the merits of this approach for improving the reliability of reward accumulation without additional computational burdens.

stat.ML↗
arXiv2020

Sketching Transformed Matrices with Applications to Natural Language Processing

Suppose we are given a large matrix $A=(a_{i,j})$ that cannot be stored in memory but is in a disk or is presented in a data stream. However, we need to compute a matrix decomposition of the entry-wisely transformed matrix, $f(A):=(f(a_{i,j}))$ for some function $f$. Is it possible to do it in a space efficient way? Many machine learning applications indeed need to deal with such large transformed matrices, for example word embedding method in NLP needs to work with the pointwise mutual information (PMI) matrix, while the entrywise transformation makes it difficult to apply known linear algebraic tools. Existing approaches for this problem either need to store the whole matrix and perform the entry-wise transformation afterwards, which is space consuming or infeasible, or need to redesign the learning method, which is application specific and requires substantial remodeling. In this paper, we first propose a space-efficient sketching algorithm for computing the product of a given small matrix with the transformed matrix. It works for a general family of transformations with provable small error bounds and thus can be used as a primitive in downstream learning tasks. We then apply this primitive to a concrete application: low-rank approximation. We show that our approach obtains small error and is efficient in both space and time. We complement our theoretical results with experiments on synthetic and real data.

cs.DS↗
arXiv2020

Minimax-Optimal Off-Policy Evaluation with Linear Function Approximation

This paper studies the statistical theory of batch data reinforcement learning with function approximation. Consider the off-policy evaluation problem, which is to estimate the cumulative value of a new target policy from logged history generated by unknown behavioral policies. We study a regression-based fitted Q iteration method, and show that it is equivalent to a model-based method that estimates a conditional mean embedding of the transition operator. We prove that this method is information-theoretically optimal and has nearly minimal estimation error. In particular, by leveraging contraction property of Markov processes and martingale concentration, we establish a finite-sample instance-dependent error upper bound and a nearly-matching minimax lower bound. The policy evaluation error depends sharply on a restricted $χ^2$-divergence over the function class between the long-term distribution of the target policy and the distribution of past data. This restricted $χ^2$-divergence is both instance-dependent and function-class-dependent. It characterizes the statistical limit of off-policy evaluation. Further, we provide an easily computable confidence bound for the policy evaluator, which may be useful for optimistic planning and safe policy improvement.

cs.LG↗
arXiv2019

Spectral State Compression of Markov Processes

Model reduction of Markov processes is a basic problem in modeling state-transition systems. Motivated by the state aggregation approach rooted in control theory, we study the statistical state compression of a discrete-state Markov chain from empirical trajectories. Through the lens of spectral decomposition, we study the rank and features of Markov processes, as well as properties like representability, aggregability, and lumpability. We develop spectral methods for estimating the transition matrix of a low-rank Markov model, estimating the leading subspace spanned by Markov features, and recovering latent structures like state aggregation and lumpable partition of the state space. We prove statistical upper bounds for the estimation errors and nearly matching minimax lower bounds. Numerical studies are performed on synthetic data and a dataset of New York City taxi trips.

stat.ML↗
arXiv2019

Continuous Control with Contexts, Provably

A fundamental challenge in artificial intelligence is to build an agent that generalizes and adapts to unseen environments. A common strategy is to build a decoder that takes the context of the unseen new environment as input and generates a policy accordingly. The current paper studies how to build a decoder for the fundamental continuous control task, linear quadratic regulator (LQR), which can model a wide range of real-world physical environments. We present a simple algorithm for this problem, which uses upper confidence bound (UCB) to refine the estimate of the decoder and balance the exploration-exploitation trade-off. Theoretically, our algorithm enjoys a $\widetilde{O}\left(\sqrt{T}\right)$ regret bound in the online setting where $T$ is the number of environments the agent played. This also implies after playing $\widetilde{O}\left(1/ε^2\right)$ environments, the agent is able to transfer the learned knowledge to obtain an $ε$-suboptimal policy for an unseen environment. To our knowledge, this is first provably efficient algorithm to build a decoder in the continuous control setting. While our main focus is theoretical, we also present experiments that demonstrate the effectiveness of our algorithm.

cs.LG↗
arXiv2019

Learning low-dimensional state embeddings and metastable clusters from time series data

This paper studies how to find compact state embeddings from high-dimensional Markov state trajectories, where the transition kernel has a small intrinsic rank. In the spirit of diffusion map, we propose an efficient method for learning a low-dimensional state embedding and capturing the process's dynamics. This idea also leads to a kernel reshaping method for more accurate nonparametric estimation of the transition function. State embedding can be used to cluster states into metastable sets, thereby identifying the slow dynamics. Sharp statistical error bounds and misclassification rate are proved. Experiment on a simulated dynamical system shows that the state clustering method indeed reveals metastable structures. We also experiment with time series generated by layers of a Deep-Q-Network when playing an Atari game. The embedding method identifies game states to be similar if they share similar future events, even though their raw data are far different.

cs.LG↗
arXiv2019

State Aggregation Learning from Markov Transition Data

State aggregation is a popular model reduction method rooted in optimal control. It reduces the complexity of engineering systems by mapping the system's states into a small number of meta-states. The choice of aggregation map often depends on the data analysts' knowledge and is largely ad hoc. In this paper, we propose a tractable algorithm that estimates the probabilistic aggregation map from the system's trajectory. We adopt a soft-aggregation model, where each meta-state has a signature raw state, called an anchor state. This model includes several common state aggregation models as special cases. Our proposed method is a simple two-step algorithm: The first step is spectral decomposition of empirical transition matrix, and the second step conducts a linear transformation of singular vectors to find their approximate convex hull. It outputs the aggregation distributions and disaggregation distributions for each meta-state in explicit forms, which are not obtainable by classical spectral methods. On the theoretical side, we prove sharp error bounds for estimating the aggregation and disaggregation distributions and for identifying anchor states. The analysis relies on a new entry-wise deviation bound for singular vectors of the empirical transition matrix of a Markov process, which is of independent interest and cannot be deduced from existing literature. The application of our method to Manhattan traffic data successfully generates a data-driven state aggregation map with nice interpretations.

cs.LG↗
arXiv2019

Characterizing Deep Learning Training Workloads on Alibaba-PAI

Modern deep learning models have been exploited in various domains, including computer vision (CV), natural language processing (NLP), search and recommendation. In practical AI clusters, workloads training these models are run using software frameworks such as TensorFlow, Caffe, PyTorch and CNTK. One critical issue for efficiently operating practical AI clouds, is to characterize the computing and data transfer demands of these workloads, and more importantly, the training performance given the underlying software framework and hardware configurations. In this paper, we characterize deep learning training workloads from Platform of Artificial Intelligence (PAI) in Alibaba. We establish an analytical framework to investigate detailed execution time breakdown of various workloads using different training architectures, to identify performance bottleneck. Results show that weight/gradient communication during training takes almost 62% of the total execution time among all our workloads on average. The computation part, involving both GPU computing and memory access, are not the biggest bottleneck based on collective behavior of the workloads. We further evaluate attainable performance of the workloads on various potential software/hardware mappings, and explore implications on software architecture selection and hardware configurations. We identify that 60% of PS/Worker workloads can be potentially sped up when ported to the AllReduce architecture exploiting the high-speed NVLink for GPU interconnect, and on average 1.7X speedup can be achieved when Ethernet bandwidth is upgraded from 25 Gbps to 100 Gbps.

cs.PF↗