Zijun wu

Beijing Institute for Scientific and Engineering Computing (BISEC)

This article analyzes the stochastic runtime of the cross-entropy algorithm for the well-studied standard problems ONEMAX and LEADINGONES. We prove that the total number of solutions the algorithm needs to evaluate before reaching the optimal solution (i.e. its runtime) is bounded by a polynomial Q(n) in the problem size n with a probability growin...

We devote this paper to a theoretic analysis of deep neural networks from a game-theoretical perspective. We consider a general deep neural network D with linear activation functions f(x)=x+b. We show that the deep neural network can be transformed into a non-atomic congestion game, regardless whether it is fully connected or locally connected. Mor...

This paper concerns computing approximate pure Nash equilibria in weighted congestion games, which has been shown to be PLS-complete. With the help of $\hat{\Psi}$-game and approximate potential functions, we propose two algorithms based on best response dynamics, and prove that they efficiently compute $\frac{\rho}{1-\epsilon}$-approximate pure Na...

We analyze the convergence of the price of anarchy (PoA) of Nash equilibria in atomic congestion games with growing total demand T. When the cost functions are polynomials of the same degree, we obtain explicit rates for a rapid convergence of the PoAs of pure and mixed Nash equilibria to 1 in terms of 1/T and \(d_{max}/T\), where \(d_{max}\) is th...

As an important part of machine learning, deep learning has been intensively used in various fields relevant to data science. Despite of its popularity in practice, it is still of challenging to compute the optimal parameters of a deep neural network, which has been shown to be NP-hard. We devote the present paper to an analysis of deep neural netw...

The price of anarchy (PoA) is a standard measure for the inefficiency of selfish routing in the static Wardrop traffic model. Empirical studies and a recent analysis reveal a surprising property that the PoA tends to one when the total demand T gets large. These results are extended by a new framework for the limit analysis of the PoA in arbitrary...

This paper shows that the PoA in non-atomic congestion games is H{\"o}lder continuous w.r.t. combined disturbance on cost functions and demands. We then apply this result to the convergence analysis of the PoA.

This paper provides a comprehensive convergence analysis of the PoA of both pure and mixed Nash equilibria in atomic congestion games with unsplittable demands.

This article studies the user behavior in non-atomic congestion games. We consider non-atomic congestion games with continuous and non-decreasing functions and investigate the limit of the price of anarchy when the total user volume approaches infinity. We deepen the knowledge on {\em asymptotically well designed games} \cite{Wu2017Selfishness}, {\...

This article studies the user selfish behavior in non-atomic congestion games (NCG). We prove that the price of anarchy of general NCGs tends to 1 as number of users tends to infinity. This generalizes a recent result in the literature. Although our result is general, the proof appears simpler. For routing games with BPR travel time functions, we p...

This article analyzes the stochastic runtime of a Cross-Entropy algorithm mimicking an Max-Min Ant System with iteration-best reinforcement. It investigates the impact of magnitude of the sample size on the runtime to find optimal solutions for TSP instances. For simple TSP instances that have a {1,n}-valued distance function and a unique optimal s...

It is well-known that the quality of random number generators can often be improved by combining several generators, e.g. by summing or subtracting their results. In this paper we investigate the ratio of two random number generators as an alternative approach: the smaller of two input random numbers is divided by the larger, resulting in a rationa...

The discrete cross-entropy optimization algorithm iteratively samples solutions according to a probability density on the solution space. The density is adapted to the good solutions observed in the present sample before producing the next sample. The adaptation is controlled by a so-called smoothing parameter. We generalize this model by introduci...

Model-based search is an abstract framework that unifies the main features of a large class of heuristic procedures for combinatorial optimization, it includes ant algorithms, cross entropy and estimation of distribution algorithms. Properties shown for the model-based search therefore apply to all these algorithms. A crucial parameter for the long...