arXiv subjects

Yinyu Ye

Publications and source records attributed to Yinyu Ye.

At least 127 records · Page 7Linked to original sources

arXiv2020

On a Randomized Multi-Block ADMM for Solving Selected Machine Learning Problems

The Alternating Direction Method of Multipliers (ADMM) has now days gained tremendous attentions for solving large-scale machine learning and signal processing problems due to the relative simplicity. However, the two-block structure of the classical ADMM still limits the size of the real problems being solved. When one forces a more-than-two-block structure by variable-splitting, the convergence speed slows down greatly as observed in practice. Recently, a randomly assembled cyclic multi-block ADMM (RAC-MBADMM) was developed by the authors for solving general convex and nonconvex quadratic optimization problems where the number of blocks can go greater than two so that each sub-problem has a smaller size and can be solved much more efficiently. In this paper, we apply this method to solving few selected machine learning problems related to convex quadratic optimization, such as Linear Regression, LASSO, Elastic-Net, and SVM. We prove that the algorithm would converge in expectation linearly under the standard statistical data assumptions. We use our general-purpose solver to conduct multiple numerical tests, solving both synthetic and large-scale bench-mark problems. Our results show that RAC-MBADMM could significantly outperform, in both solution time and quality, other optimization algorithms/codes for solving these machine learning problems, and match up the performance of the best tailored methods such as Glmnet or LIBSVM. In certain problem regions RAC-MBADMM even achieves a superior performance than that of the tailored methods.

math.OC↗

arXiv2020

Interior-Point Methods Strike Back: Solving the Wasserstein Barycenter Problem

Computing the Wasserstein barycenter of a set of probability measures under the optimal transport metric can quickly become prohibitive for traditional second-order algorithms, such as interior-point methods, as the support size of the measures increases. In this paper, we overcome the difficulty by developing a new adapted interior-point method that fully exploits the problem's special matrix structure to reduce the iteration complexity and speed up the Newton procedure. Different from regularization approaches, our method achieves a well-balanced tradeoff between accuracy and speed. A numerical comparison on various distributions with existing algorithms exhibits the computational advantages of our approach. Moreover, we demonstrate the practicality of our algorithm on image benchmark problems including MNIST and Fashion-MNIST.

math.OC↗

arXiv2019

Solving Discounted Stochastic Two-Player Games with Near-Optimal Time and Sample Complexity

In this paper, we settle the sampling complexity of solving discounted two-player turn-based zero-sum stochastic games up to polylogarithmic factors. Given a stochastic game with discount factor $γ\in(0,1)$ we provide an algorithm that computes an $ε$-optimal strategy with high-probability given $\tilde{O}((1 - γ)^{-3} ε^{-2})$ samples from the transition function for each state-action-pair. Our algorithm runs in time nearly linear in the number of samples and uses space nearly linear in the number of state-action pairs. As stochastic games generalize Markov decision processes (MDPs) our runtime and sample complexities are optimal due to Azar et al (2013). We achieve our results by showing how to generalize a near-optimal Q-learning based algorithms for MDP, in particular Sidford et al (2018), to two-player strategy computation algorithms. This overcomes limitations of standard Q-learning and strategy iteration or alternating minimization based approaches and we hope will pave the way for future reinforcement learning results by facilitating the extension of MDP results to multi-agent settings with little loss.

cs.LG↗

arXiv2019

On the behavior of Lagrange multipliers in convex and non-convex infeasible interior point methods

We analyze sequences generated by interior point methods (IPMs) in convex and nonconvex settings. We prove that moving the primal feasibility at the same rate as the barrier parameter $μ$ ensures the Lagrange multiplier sequence remains bounded, provided the limit point of the primal sequence has a Lagrange multiplier. This result does not require constraint qualifications. We also guarantee the IPM finds a solution satisfying strict complementarity if one exists. On the other hand, if the primal feasibility is reduced too slowly, then the algorithm converges to a point of minimal complementarity; if the primal feasibility is reduced too quickly and the set of Lagrange multipliers is unbounded, then the norm of the Lagrange multiplier tends to infinity. Our theory has important implications for the design of IPMs. Specifically, we show that IPOPT, an algorithm that does not carefully control primal feasibility has practical issues with the dual multipliers values growing to unnecessarily large values. Conversely, the one-phase IPM of \citet*{hinder2018one}, an algorithm that controls primal feasibility as our theory suggests, has no such issue.

math.OC↗

arXiv2019

Toward Solving 2-TBSG Efficiently

2-TBSG is a two-player game model which aims to find Nash equilibriums and is widely utilized in reinforced learning and AI. Inspired by the fact that the simplex method for solving the deterministic discounted Markov decision processes (MDPs) is strongly polynomial independent of the discounted factor, we are trying to answer an open problem whether there is a similar algorithm for 2-TBSG. We develop a simplex strategy iteration where one player updates its strategy with a simplex step while the other player finds an optimal counterstrategy in turn, and a modified simplex strategy iteration. Both of them belong to a class of geometrically converging algorithms. We establish the strongly polynomial property of these algorithms by considering a strategy combined from the current strategy and the equilibrium strategy. Moreover, we present a method to transform general 2-TBSGs into special 2-TBSGs where each state has exactly two actions.

cs.GT↗

arXiv2019

Near-Optimal Time and Sample Complexities for Solving Discounted Markov Decision Process with a Generative Model

In this paper we consider the problem of computing an $ε$-optimal policy of a discounted Markov Decision Process (DMDP) provided we can only access its transition function through a generative sampling model that given any state-action pair samples from the transition function in $O(1)$ time. Given such a DMDP with states $S$, actions $A$, discount factor $γ\in(0,1)$, and rewards in range $[0, 1]$ we provide an algorithm which computes an $ε$-optimal policy with probability $1 - δ$ where \emph{both} the time spent and number of sample taken are upper bounded by \[ O\left[\frac{|S||A|}{(1-γ)^3 ε^2} \log \left(\frac{|S||A|}{(1-γ)δε} \right) \log\left(\frac{1}{(1-γ)ε}\right)\right] ~. \] For fixed values of $ε$, this improves upon the previous best known bounds by a factor of $(1 - γ)^{-1}$ and matches the sample complexity lower bounds proved in Azar et al. (2013) up to logarithmic factors. We also extend our method to computing $ε$-optimal policies for finite-horizon MDP with a generative model and provide a nearly matching sample complexity lower bound.

math.OC↗

arXiv2019

On the Efficiency of Random Permutation for ADMM and Coordinate Descent

Random permutation is observed to be powerful for optimization algorithms: for multi-block ADMM (alternating direction method of multipliers), while the classical cyclic version divergence, the randomly permuted version converges in practice; for BCD (block coordinate descent), the randomly permuted version is typically faster than other versions. In this paper, we provide strong theoretical evidence that random permutation has positive effects on ADMM and BCD, by analyzing randomly permuted ADMM (RP-ADMM) for solving linear systems of equations, and randomly permuted BCD (RP-BCD) for solving unconstrained quadratic problems. First, we prove that RP-ADMM converges in expectation for solving systems of linear equations. The key technical result is that the spectrum of the expected update matrix of RP-BCD lies in $(-1/3, 1)$, instead of the typical range $(-1, 1)$. Second, we establish expected convergence rates of RP-ADMM for solving linear sytems and RP-BCD for solving unconstrained quadratic problems. This expected rate of RP-BCD is $O(n)$ times better than the worst-case rate of cyclic BCD, thus establishing a gap of at least $O(n)$ between RP-BCD and cyclic BCD. To analyze RP-BCD, we propose a conjecture of a new matrix AM-GM (algebraic mean-geometric mean) inequality, and prove a weaker version of it.

math.OC↗

arXiv2018

Exact Semidefinite Formulations for a Class of (Random and Non-Random) Nonconvex Quadratic Programs

We study a class of quadratically constrained quadratic programs (QCQPs), called {\em diagonal QCQPs\/}, which contain no off-diagonal terms $x_j x_k$ for $j \ne k$, and we provide a sufficient condition on the problem data guaranteeing that the basic Shor semidefinite relaxation is exact. Our condition complements and refines those already present in the literature and can be checked in polynomial time. We then extend our analysis from diagonal QCQPs to general QCQPs, i.e., ones with no particular structure. By reformulating a general QCQP into diagonal form, we establish new, polynomial-time-checkable sufficient conditions for the semidefinite relaxations of general QCQPs to be exact. Finally, these ideas are extended to show that a class of random general QCQPs has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables. To the best of our knowledge, this is the first result establishing the exactness of the semidefinite relaxation for random general QCQPs.

math.OC↗

arXiv2018

Worst-case Complexity of Cyclic Coordinate Descent: $O(n^2)$ Gap with Randomized Version

This paper concerns the worst-case complexity of cyclic coordinate descent (C-CD) for minimizing a convex quadratic function, which is equivalent to Gauss-Seidel method and can be transformed to Kaczmarz method and projection onto convex sets (POCS). We observe that the known provable complexity of C-CD can be $O(n^2)$ times slower than randomized coordinate descent (R-CD), but no example was rigorously proven to exhibit such a large gap. In this paper we show that the gap indeed exists. We prove that there exists an example for which C-CD takes at least $O(n^4 κ_{\text{CD}} \log\frac{1}ε)$ operations, where $κ_{\text{CD}}$ is related to Demmel's condition number and it determines the convergence rate of R-CD. It implies that in the worst case C-CD can indeed be $O(n^2)$ times slower than R-CD, which has complexity $O( n^2 κ_{\text{CD}} \log\frac{1}ε)$. Note that for this example, the gap exists for any fixed update order, not just a particular order. Based on the example, we establish several almost tight complexity bounds of C-CD for quadratic problems. One difficulty with the analysis is that the spectral radius of a non-symmetric iteration matrix does not necessarily constitute a \textit{lower bound} for the convergence rate. An immediate consequence is that for Gauss-Seidel method, Kaczmarz method and POCS, there is also an $O(n^2) $ gap between the cyclic versions and randomized versions (for solving linear systems). We also show that the classical convergence rate of POCS by Smith, Solmon and Wager [1] is always worse and sometimes can be infinitely times worse than our bound.

math.OC↗

arXiv2018

A one-phase interior point method for nonconvex optimization

The work of Wachter and Biegler suggests that infeasible-start interior point methods (IPMs) developed for linear programming cannot be adapted to nonlinear optimization without significant modification, i.e., using a two-phase or penalty method. We propose an IPM that, by careful initialization and updates of the slack variables, is guaranteed to find a first-order certificate of local infeasibility, local optimality or unboundedness of the (shifted) feasible region. Our proposed algorithm differs from other IPM methods for nonconvex programming because we reduce primal feasibility at the same rate as the barrier parameter. This gives an algorithm with more robust convergence properties and closely resembles successful algorithms from linear programming. We implement the algorithm and compare with IPOPT on a subset of CUTEst problems. Our algorithm requires a similar median number of iterations, but fails on only 9% of the problems compared with 16% for IPOPT. Experiments on infeasible variants of the CUTEst problems indicate superior performance for detecting infeasibility. The code for our implementation can be found at https://github.com/ohinder/OnePhase .

math.OC↗

arXiv2017

Strong NP-Hardness for Sparse Optimization with Concave Penalty Functions

Consider the regularized sparse minimization problem, which involves empirical sums of loss functions for $n$ data points (each of dimension $d$) and a nonconvex sparsity penalty. We prove that finding an $\mathcal{O}(n^{c_1}d^{c_2})$-optimal solution to the regularized sparse optimization problem is strongly NP-hard for any $c_1, c_2\in [0,1)$ such that $c_1+c_2<1$. The result applies to a broad class of loss functions and sparse penalty functions. It suggests that one cannot even approximately solve the sparse optimization problem in polynomial time, unless P $=$ NP.

math.OC↗

arXiv2017

A Derandomized Algorithm for RP-ADMM with Symmetric Gauss-Seidel Method

For multi-block alternating direction method of multipliers(ADMM), where the objective function can be decomposed into multiple block components, we show that with block symmetric Gauss-Seidel iteration, the algorithm will converge quickly. The method will apply a block symmetric Gauss-Seidel iteration in the primal update and a linear correction that can be derived in view of Richard iteration. We also establish the linear convergence rate for linear systems.

math.OC↗

arXiv2017

Extended ADMM and BCD for Nonseparable Convex Minimization Models with Quadratic Coupling Terms: Convergence Analysis and Insights

In this paper, we establish the convergence of the proximal alternating direction method of multipliers (ADMM) and block coordinate descent (BCD) for nonseparable minimization models with quadratic coupling terms. The novel convergence results presented in this paper answer several open questions that have been the subject of considerable discussion. We firstly extend the 2-block proximal ADMM to linearly constrained convex optimization with a coupled quadratic objective function, an area where theoretical understanding is currently lacking, and prove that the sequence generated by the proximal ADMM converges in point-wise manner to a primal-dual solution pair. Moreover, we apply randomly permuted ADMM (RPADMM) to nonseparable multi-block convex optimization, and prove its expected convergence for a class of nonseparable quadratic programming problems. When the linear constraint vanishes, the 2-block proximal ADMM and RPADMM reduce to the 2-block cyclic proximal BCD method and randomly permuted BCD (RPBCD). Our study provides the first iterate convergence result for 2-block cyclic proximal BCD without assuming the boundedness of the iterates. We also theoretically establish the expected iterate convergence result concerning multi-block RPBCD for convex quadratic optimization. In addition, we demonstrate that RPBCD may have a worse convergence rate than cyclic proximal BCD for 2-block convex quadratic minimization problems. Although the results on RPADMM and RPBCD are restricted to quadratic minimization models, they provide some interesting insights: 1) random permutation makes ADMM and BCD more robust for multi-block convex minimization problems; 2) cyclic BCD may outperform RPBCD for "nice" problems, and therefore RPBCD should be applied with caution when solving general convex optimization problems.

math.OC↗

arXiv2017

Optimality condition and complexity analysis for linearly-constrained optimization without differentiability on the boundary

In this paper we consider the minimization of a continuous function that is potentially not differentiable or not twice differentiable on the boundary of the feasible region. By exploiting an interior point technique, we present first- and second-order optimality conditions for this problem that reduces to classical ones when the derivative on the boundary is available. For this type of problems, existing necessary conditions often rely on the notion of subdifferential or become non-trivially weaker than the KKT condition in the (twice-)differentiable counterpart problems. In contrast, this paper presents a new set of first- and second-order necessary conditions that are derived without the use of subdifferential and reduces to exactly the KKT condition when (twice-)differentiability holds. As a result, these conditions are stronger than some existing ones considered for the discussed minimization problem when only non-negativity constraints are present. To solve for these optimality conditions in the special but important case of linearly constrained problems, we present two novel interior trust-region point algorithms and show that their worst-case computational efficiency in achieving the potentially stronger optimality conditions match the best known complexity bounds. Since this work considers a more general problem than the literature, our results also indicate that best known complexity bounds hold for a wider class of nonlinear programming problems.

cs.CC↗

arXiv2015

Market Making with Model Uncertainty

Pari-mutuel markets are trading platforms through which the common market maker simultaneously clears multiple contingent claims markets. This market has several distinctive properties that began attracting the attention of the financial industry in the 2000s. For example, the platform aggregates liquidity from the individual contingent claims market into the common pool while shielding the market maker from potential financial loss. The contribution of this paper is two-fold. First, we provide a new economic interpretation of the market-clearing strategy of a pari-mutuel market that is well known in the literature. The pari-mutuel auctioneer is shown to be equivalent to the market maker with extreme ambiguity aversion for the future contingent event. Second, based on this theoretical understanding, we present a new market-clearing algorithm called the Knightian Pari-mutuel Mechanism (KPM). The KPM retains many interesting properties of pari-mutuel markets while explicitly controlling for the market maker's ambiguity aversion. In addition, the KPM is computationally efficient in that it is solvable in polynomial time.

q-fin.TR↗

arXiv2014

Likelihood Robust Optimization for Data-driven Problems

We consider optimal decision-making problems in an uncertain environment. In particular, we consider the case in which the distribution of the input is unknown, yet there is abundant historical data drawn from the distribution. In this paper, we propose a new type of distributionally robust optimization model called the likelihood robust optimization (LRO) model for this class of problems. In contrast to previous work on distributionally robust optimization that focuses on certain parameters (e.g., mean, variance, etc.) of the input distribution, we exploit the historical data and define the accessible distribution set to contain only those distributions that make the observed data achieve a certain level of likelihood. Then we formulate the targeting problem as one of optimizing the expected value of the objective function under the worst-case distribution in that set. Our model avoids the over-conservativeness of some prior robust approaches by ruling out unrealistic distributions while maintaining robustness of the solution for any statistically likely outcomes. We present statistical analyses of our model using Bayesian statistics and empirical likelihood theory. Specifically, we prove the asymptotic behavior of our distribution set and establish the relationship between our model and other distributionally robust models. To test the performance of our model, we apply it to the newsvendor problem and the portfolio selection problem. The test results show that the solutions of our model indeed have desirable performance.

math.OC↗

arXiv2014

Online Allocation Rules in Display Advertising

Efficient allocation of impressions to advertisers in display advertising has a significant impact on advertisers' utility and the browsing experience of users. The problem becomes particularly challenging in the presence of advertisers with limited budgets as this creates a complex interaction among advertisers in the optimal impression assignment. In this paper, we study online impression allocation in display advertising with budgeted advertisers. That is, upon arrival of each impression, cost and revenue vectors are revealed and the impression should be assigned to an advertiser almost immediately. Without any assumption on the distribution/arrival of impressions, we propose a framework to capture the risk to the ad network for each possible allocation; impressions are allocated to advertisers such that the risk of ad network is minimized. In practice, this translates to starting with an initial estimate of dual prices and updating them according to the belief of the ad network toward the future demand and remaining budgets. We apply our algorithms to a real data set, and we empirically show that Kullback-Leibler divergence risk measure has the best performance in terms of revenue and balanced budget delivery.

math.OC↗

arXiv2014

A Dynamic Near-Optimal Algorithm for Online Linear Programming

A natural optimization model that formulates many online resource allocation and revenue management problems is the online linear program (LP) in which the constraint matrix is revealed column by column along with the corresponding objective coefficient. In such a model, a decision variable has to be set each time a column is revealed without observing the future inputs and the goal is to maximize the overall objective function. In this paper, we provide a near-optimal algorithm for this general class of online problems under the assumption of random order of arrival and some mild conditions on the size of the LP right-hand-side input. Specifically, our learning-based algorithm works by dynamically updating a threshold price vector at geometric time intervals, where the dual prices learned from the revealed columns in the previous period are used to determine the sequential decisions in the current period. Due to the feature of dynamic learning, the competitiveness of our algorithm improves over the past study of the same problem. We also present a worst-case example showing that the performance of our algorithm is near-optimal.

cs.DS↗