arXiv subjects

Qi Meng

Publications and source records attributed to Qi Meng.

65 records · Page 4Linked to original sources

arXiv2019

Compact $sssc\bar{c}$ pentaquark states predicted by a quark model

Several compact $sssc\bar c$ pentaquark resonances are predicted in a potential quark model. The Hamiltonian is the best available one, which reproduces the masses of the low-lying charmed and strange hadrons well. Full five-body calculations are carried out by the use of the Gaussian expansion method, and the relevant baryon-meson thresholds are taken into account explicitly. Employing the real scaling method, we predict four sharp resonances, $J^P=1/2^-$ ($E=5180$ MeV, $Γ=20$ MeV), $5/2^-$ (5645 MeV, 30 MeV), $5/2^-$ (5670 MeV, 50 MeV), and $1/2^+$ (5360 MeV, 80 MeV). These are the candidates of compact pentaquark resonance states from the current best quark model, which should be confirmed either by experiments or lattice QCD calculations.

nucl-th↗

arXiv2019

Reinforcement Learning with Dynamic Boltzmann Softmax Updates

Value function estimation is an important task in reinforcement learning, i.e., prediction. The Boltzmann softmax operator is a natural value estimator and can provide several benefits. However, it does not satisfy the non-expansion property, and its direct use may fail to converge even in value iteration. In this paper, we propose to update the value function with dynamic Boltzmann softmax (DBS) operator, which has good convergence property in the setting of planning and learning. Experimental results on GridWorld show that the DBS operator enables better estimation of the value function, which rectifies the convergence issue of the softmax operator. Finally, we propose the DBS-DQN algorithm by applying dynamic Boltzmann softmax updates in deep Q-network, which outperforms DQN substantially in 40 out of 49 Atari games.

cs.LG↗

arXiv2019

Positively Scale-Invariant Flatness of ReLU Neural Networks

It was empirically confirmed by Keskar et al.\cite{SharpMinima} that flatter minima generalize better. However, for the popular ReLU network, sharp minimum can also generalize well \cite{SharpMinimacan}. The conclusion demonstrates that the existing definitions of flatness fail to account for the complex geometry of ReLU neural networks because they can't cover the Positively Scale-Invariant (PSI) property of ReLU network. In this paper, we formalize the PSI causes problem of existing definitions of flatness and propose a new description of flatness - \emph{PSI-flatness}. PSI-flatness is defined on the values of basis paths \cite{GSGD} instead of weights. Values of basis paths have been shown to be the PSI-variables and can sufficiently represent the ReLU neural networks which ensure the PSI property of PSI-flatness. Then we study the relation between PSI-flatness and generalization theoretically and empirically. First, we formulate a generalization bound based on PSI-flatness which shows generalization error decreasing with the ratio between the largest basis path value and the smallest basis path value. That is to say, the minimum with balanced values of basis paths will more likely to be flatter and generalize better. Finally. we visualize the PSI-flatness of loss surface around two learned models which indicates the minimum with smaller PSI-flatness can indeed generalize better.

cs.LG↗

arXiv2018

Target Transfer Q-Learning and Its Convergence Analysis

Q-learning is one of the most popular methods in Reinforcement Learning (RL). Transfer Learning aims to utilize the learned knowledge from source tasks to help new tasks to improve the sample complexity of the new tasks. Considering that data collection in RL is both more time and cost consuming and Q-learning converges slowly comparing to supervised learning, different kinds of transfer RL algorithms are designed. However, most of them are heuristic with no theoretical guarantee of the convergence rate. Therefore, it is important for us to clearly understand when and how will transfer learning help RL method and provide the theoretical guarantee for the improvement of the sample complexity. In this paper, we propose to transfer the Q-function learned in the source task to the target of the Q-learning in the new task when certain safe conditions are satisfied. We call this new transfer Q-learning method target transfer Q-Learning. The safe conditions are necessary to avoid the harm to the new tasks and thus ensure the convergence of the algorithm. We study the convergence rate of the target transfer Q-learning. We prove that if the two tasks are similar with respect to the MDPs, the optimal Q-functions in the source and new RL tasks are similar which means the error of the transferred target Q-function in new MDP is small. Also, the convergence rate analysis shows that the target transfer Q-Learning will converge faster than Q-learning if the error of the transferred target Q-function is smaller than the current Q-function in the new task. Based on our theoretical results, we design the safe condition as the Bellman error of the transferred target Q-function is less than the current Q-function. Our experiments are consistent with our theoretical founding and verified the effectiveness of our proposed target transfer Q-learning method.

cs.LG↗

arXiv2018

Capacity Control of ReLU Neural Networks by Basis-path Norm

Recently, path norm was proposed as a new capacity measure for neural networks with Rectified Linear Unit (ReLU) activation function, which takes the rescaling-invariant property of ReLU into account. It has been shown that the generalization error bound in terms of the path norm explains the empirical generalization behaviors of the ReLU neural networks better than that of other capacity measures. Moreover, optimization algorithms which take path norm as the regularization term to the loss function, like Path-SGD, have been shown to achieve better generalization performance. However, the path norm counts the values of all paths, and hence the capacity measure based on path norm could be improperly influenced by the dependency among different paths. It is also known that each path of a ReLU network can be represented by a small group of linearly independent basis paths with multiplication and division operation, which indicates that the generalization behavior of the network only depends on only a few basis paths. Motivated by this, we propose a new norm \emph{Basis-path Norm} based on a group of linearly independent paths to measure the capacity of neural networks more accurately. We establish a generalization error bound based on this basis path norm, and show it explains the generalization behaviors of ReLU networks more accurately than previous capacity measures via extensive experiments. In addition, we develop optimization algorithms which minimize the empirical risk regularized by the basis-path norm. Our experiments on benchmark datasets demonstrate that the proposed regularization method achieves clearly better performance on the test set than the previous regularization approaches.

cs.LG↗

arXiv2018

Differential Equations for Modeling Asynchronous Algorithms

Asynchronous stochastic gradient descent (ASGD) is a popular parallel optimization algorithm in machine learning. Most theoretical analysis on ASGD take a discrete view and prove upper bounds for their convergence rates. However, the discrete view has its intrinsic limitations: there is no characterization of the optimization path and the proof techniques are induction-based and thus usually complicated. Inspired by the recent successful adoptions of stochastic differential equations (SDE) to the theoretical analysis of SGD, in this paper, we study the continuous approximation of ASGD by using stochastic differential delay equations (SDDE). We introduce the approximation method and study the approximation error. Then we conduct theoretical analysis on the convergence rates of ASGD algorithm based on the continuous approximation. There are two methods: moment estimation and energy function minimization can be used to analyze the convergence rates. Moment estimation depends on the specific form of the loss function, while energy function minimization only leverages the convex property of the loss function, and does not depend on its specific form. In addition to the convergence analysis, the continuous view also helps us derive better convergence rates. All of this clearly shows the advantage of taking the continuous view in gradient descent algorithms.

stat.ML↗

arXiv2018

Calculation of Dissociation Temperature of Nucleon Using Gaussian Expansion Method

The first study of the dissociation temperature of nucleon in hot QCD medium in the framework of constituent quark model is presented. The temperature-dependent potential energy of the three quark system, taking as the internal energy of the system are obtained from the free energy of the system, and the temperature-dependent free energy is derived based on Debye-Hückel theory. The lattice QCD results of the free energy for heavy three-quark system are employed and extended to the light three-quark system. The Schrödinger equation for nucleon is solved with the help of Gaussian expansion method and the dissociation temperature of the nucleon is determined according to the temperature dependence of binding energy and radii. Comparing with the dissociation temperature of $J/ψ$, the dissociation temperature of nucleon is higher. So, nucleon is more difficult to melt than Charmonium.

hep-ph↗

arXiv2017

Convergence Analysis of Distributed Stochastic Gradient Descent with Shuffling

When using stochastic gradient descent to solve large-scale machine learning problems, a common practice of data processing is to shuffle the training data, partition the data across multiple machines if needed, and then perform several epochs of training on the re-shuffled (either locally or globally) data. The above procedure makes the instances used to compute the gradients no longer independently sampled from the training data set. Then does the distributed SGD method have desirable convergence properties in this practical situation? In this paper, we give answers to this question. First, we give a mathematical formulation for the practical data processing procedure in distributed machine learning, which we call data partition with global/local shuffling. We observe that global shuffling is equivalent to without-replacement sampling if the shuffling operations are independent. We prove that SGD with global shuffling has convergence guarantee in both convex and non-convex cases. An interesting finding is that, the non-convex tasks like deep learning are more suitable to apply shuffling comparing to the convex tasks. Second, we conduct the convergence analysis for SGD with local shuffling. The convergence rate for local shuffling is slower than that for global shuffling, since it will lose some information if there's no communication between partitioned data. Finally, we consider the situation when the permutation after shuffling is not uniformly distributed (insufficient shuffling), and discuss the condition under which this insufficiency will not influence the convergence rate. Our theoretical results provide important insights to large-scale machine learning, especially in the selection of data processing methods in order to achieve faster convergence and good speedup. Our theoretical findings are verified by extensive experiments on logistic regression and deep neural networks.

stat.ML↗

arXiv2016

A Communication-Efficient Parallel Algorithm for Decision Tree

Decision tree (and its extensions such as Gradient Boosting Decision Trees and Random Forest) is a widely used machine learning algorithm, due to its practical effectiveness and model interpretability. With the emergence of big data, there is an increasing need to parallelize the training process of decision tree. However, most existing attempts along this line suffer from high communication costs. In this paper, we propose a new algorithm, called \emph{Parallel Voting Decision Tree (PV-Tree)}, to tackle this challenge. After partitioning the training data onto a number of (e.g., $M$) machines, this algorithm performs both local voting and global voting in each iteration. For local voting, the top-$k$ attributes are selected from each machine according to its local data. Then, globally top-$2k$ attributes are determined by a majority voting among these local candidates. Finally, the full-grained histograms of the globally top-$2k$ attributes are collected from local machines in order to identify the best (most informative) attribute and its split point. PV-Tree can achieve a very low communication cost (independent of the total number of attributes) and thus can scale out very well. Furthermore, theoretical analysis shows that this algorithm can learn a near optimal decision tree, since it can find the best attribute with a large probability. Our experiments on real-world datasets show that PV-Tree significantly outperforms the existing parallel decision tree algorithms in the trade-off between accuracy and efficiency.

cs.LG↗

arXiv2016

Generalization Error Bounds for Optimization Algorithms via Stability

Many machine learning tasks can be formulated as Regularized Empirical Risk Minimization (R-ERM), and solved by optimization algorithms such as gradient descent (GD), stochastic gradient descent (SGD), and stochastic variance reduction (SVRG). Conventional analysis on these optimization algorithms focuses on their convergence rates during the training process, however, people in the machine learning community may care more about the generalization performance of the learned model on unseen test data. In this paper, we investigate on this issue, by using stability as a tool. In particular, we decompose the generalization error for R-ERM, and derive its upper bound for both convex and non-convex cases. In convex cases, we prove that the generalization error can be bounded by the convergence rate of the optimization algorithm and the stability of the R-ERM process, both in expectation (in the order of $\mathcal{O}((1/n)+\mathbb{E}ρ(T))$, where $ρ(T)$ is the convergence error and $T$ is the number of iterations) and in high probability (in the order of $\mathcal{O}\left(\frac{\log{1/δ}}{\sqrt{n}}+ρ(T)\right)$ with probability $1-δ$). For non-convex cases, we can also obtain a similar expected generalization error bound. Our theorems indicate that 1) along with the training process, the generalization error will decrease for all the optimization algorithms under our investigation; 2) Comparatively speaking, SVRG has better generalization ability than GD and SGD. We have conducted experiments on both convex and non-convex problems, and the experimental results verify our theoretical findings.

stat.ML↗

arXiv2016

Asynchronous Stochastic Proximal Optimization Algorithms with Variance Reduction

Regularized empirical risk minimization (R-ERM) is an important branch of machine learning, since it constrains the capacity of the hypothesis space and guarantees the generalization ability of the learning algorithm. Two classic proximal optimization algorithms, i.e., proximal stochastic gradient descent (ProxSGD) and proximal stochastic coordinate descent (ProxSCD) have been widely used to solve the R-ERM problem. Recently, variance reduction technique was proposed to improve ProxSGD and ProxSCD, and the corresponding ProxSVRG and ProxSVRCD have better convergence rate. These proximal algorithms with variance reduction technique have also achieved great success in applications at small and moderate scales. However, in order to solve large-scale R-ERM problems and make more practical impacts, the parallel version of these algorithms are sorely needed. In this paper, we propose asynchronous ProxSVRG (Async-ProxSVRG) and asynchronous ProxSVRCD (Async-ProxSVRCD) algorithms, and prove that Async-ProxSVRG can achieve near linear speedup when the training data is sparse, while Async-ProxSVRCD can achieve near linear speedup regardless of the sparse condition, as long as the number of block partitions are appropriately set. We have conducted experiments on a regularized logistic regression task. The results verified our theoretical findings and demonstrated the practical efficiency of the asynchronous stochastic proximal algorithms with variance reduction.

cs.LG↗