arXiv subjects

Yinyu Ye

Publications and source records attributed to Yinyu Ye.

At least 109 records · Page 6Linked to original sources

arXiv2022

SOLNP+: A Derivative-Free Solver for Constrained Nonlinear Optimization

SOLNP+ is a derivative-free solver for constrained nonlinear optimization. It starts from SOLNP proposed in 1989 by Ye Ye with the main idea that uses finite difference to approximate the gradient. We incorporate the techniques of implicit filtering, new restart mechanism and modern quadratic programming solver into this new version with an ANSI C implementation. The algorithm exhibits a great advantage in running time and robustness under noise compared with the last version by MATLAB. SOLNP+ is free to download at https://github.com/COPT-Public/SOLNP_plus.

math.OC↗

arXiv2022

Fine-grained Correlation Loss for Regression

Regression learning is classic and fundamental for medical image analysis. It provides the continuous mapping for many critical applications, like the attribute estimation, object detection, segmentation and non-rigid registration. However, previous studies mainly took the case-wise criteria, like the mean square errors, as the optimization objectives. They ignored the very important population-wise correlation criterion, which is exactly the final evaluation metric in many tasks. In this work, we propose to revisit the classic regression tasks with novel investigations on directly optimizing the fine-grained correlation losses. We mainly explore two complementary correlation indexes as learnable losses: Pearson linear correlation (PLC) and Spearman rank correlation (SRC). The contributions of this paper are two folds. First, for the PLC on global level, we propose a strategy to make it robust against the outliers and regularize the key distribution factors. These efforts significantly stabilize the learning and magnify the efficacy of PLC. Second, for the SRC on local level, we propose a coarse-to-fine scheme to ease the learning of the exact ranking order among samples. Specifically, we convert the learning for the ranking of samples into the learning of similarity relationships among samples. We extensively validate our method on two typical ultrasound image regression tasks, including the image quality assessment and bio-metric measurement. Experiments prove that, with the fine-grained guidance in directly optimizing the correlation, the regression performances are significantly improved. Our proposed correlation losses are general and can be extended to more important applications.

cs.CV↗

arXiv2022

Fairer LP-based Online Allocation via Analytic Center

In this paper, we consider an online resource allocation problem where a decision maker accepts or rejects incoming customer requests irrevocably in order to maximize expected reward given limited resources. At each time, a new order/customer/bid is revealed with a request of some resource(s) and a reward. We consider a stochastic setting where all the orders are i.i.d. sampled from an unknown distribution. Such formulation arises from many classic applications such as the canonical (quantity-based) network revenue management problem and the Adwords problem. While the literature on the topic mainly focuses on regret minimization, our paper considers the \textit{fairness} aspect of the problem. On a high level, we define the fairness in a way that a fair online algorithm should treat similar agents/customers similarly, and the decision made for similar agents/customers should be consistent over time. To achieve this goal, we define the fair offline solution as the analytic center of the offline optimal solution set, and introduce \textit{cumulative unfairness} as the cumulative deviation from the online solutions to the fair offline solution over time. We propose a fair algorithm based on an interior-point LP solver and a mechanism that dynamically detects unfair resource spending. Our algorithm achieves cumulative unfairness on the scale of order $O(\log(T))$, while maintains the regret to be bounded without dependency on $T$. In addition, compared to the literature, our result is produced under less restrictive assumptions on the degeneracy of the underlying linear program.

cs.DS↗

arXiv2022

An Improved Analysis of LP-based Control for Revenue Management

In this paper, we study a class of revenue management problems where the decision maker aims to maximize the total revenue subject to budget constraints on multiple type of resources over a finite horizon. At each time, a new order/customer/bid is revealed with a request of some resource(s) and a reward, and the decision maker needs to either accept or reject the order. Upon the acceptance of the order, the resource request must be satisfied and the associated revenue (reward) can be collected. We consider a stochastic setting where all the orders are i.i.d. sampled, i.e., the reward-request pair at each time is drawn from an unknown distribution with finite support. The formulation contains many classic applications such as the quantity-based network revenue management problem and the Adwords problem. We focus on the classic LP-based adaptive algorithm and consider regret as the performance measure defined by the gap between the optimal objective value of the certainty-equivalent linear program (LP) and the expected revenue obtained by the online algorithm. Our contribution is two-fold: (i) when the underlying LP is nondegenerate, the algorithm achieves a problem-dependent regret upper bound that is independent of the horizon/number of time periods $T$; (ii) when the underlying LP is degenerate, the algorithm achieves a regret upper bound that scales on the order of $\sqrt{T}\log T$. To our knowledge, both results are new and improve the best existing bounds for the LP-based adaptive algorithm in the corresponding setting. We conclude with numerical experiments to further demonstrate our findings.

math.OC↗

arXiv2021

High-Dimensional Learning under ApproximateSparsity with Applications to Nonsmooth Estimation and Regularized Neural Networks

High-dimensional statistical learning (HDSL) has wide applications in data analysis, operations research, and decision-making. Despite the availability of multiple theoretical frameworks, most existing HDSL schemes stipulate the following two conditions: (a) the sparsity, and (b) the restricted strong convexity (RSC). This paper generalizes both conditions via the use of the folded concave penalty (FCP). More specifically, we consider an M-estimation problem where (i) the (conventional) sparsity is relaxed into the approximate sparsity and (ii) the RSC is completely absent. We show that the FCP-based regularization leads to poly-logarithmic sample complexity; the training data size is only required to be poly-logarithmic in the problem dimensionality. This finding can facilitate the analysis of two important classes of models that are currently less understood: the high-dimensional nonsmooth learning and the (deep) neural networks (NN). For both problems, we show that the poly-logarithmic sample complexity can be maintained. In particular, our results indicate that the generalizability of NNs under over-parameterization can be theoretically ensured with the aid of regularization.

math.ST↗

arXiv2021

An Adaptive State Aggregation Algorithm for Markov Decision Processes

Value iteration is a well-known method of solving Markov Decision Processes (MDPs) that is simple to implement and boasts strong theoretical convergence guarantees. However, the computational cost of value iteration quickly becomes infeasible as the size of the state space increases. Various methods have been proposed to overcome this issue for value iteration in large state and action space MDPs, often at the price, however, of generalizability and algorithmic simplicity. In this paper, we propose an intuitive algorithm for solving MDPs that reduces the cost of value iteration updates by dynamically grouping together states with similar cost-to-go values. We also prove that our algorithm converges almost surely to within $2\varepsilon / (1 - γ)$ of the true optimal value in the $\ell^\infty$ norm, where $γ$ is the discount factor and aggregated states differ by at most $\varepsilon$. Numerical experiments on a variety of simulated environments confirm the robustness of our algorithm and its ability to solve MDPs with much cheaper updates especially as the scale of the MDP problem increases.

cs.LG↗

arXiv2021

Distributed stochastic optimization with large delays

One of the most widely used methods for solving large-scale stochastic optimization problems is distributed asynchronous stochastic gradient descent (DASGD), a family of algorithms that result from parallelizing stochastic gradient descent on distributed computing architectures (possibly) asychronously. However, a key obstacle in the efficient implementation of DASGD is the issue of delays: when a computing node contributes a gradient update, the global model parameter may have already been updated by other nodes several times over, thereby rendering this gradient information stale. These delays can quickly add up if the computational throughput of a node is saturated, so the convergence of DASGD may be compromised in the presence of large delays. Our first contribution is that, by carefully tuning the algorithm's step-size, convergence to the critical set is still achieved in mean square, even if the delays grow unbounded at a polynomial rate. We also establish finer results in a broad class of structured optimization problems (called variationally coherent), where we show that DASGD converges to a global optimum with probability $1$ under the same delay assumptions. Together, these results contribute to the broad landscape of large-scale non-convex stochastic optimization by offering state-of-the-art theoretical guarantees and providing insights for algorithm design.

math.OC↗

arXiv2021

The Symmetry between Arms and Knapsacks: A Primal-Dual Approach for Bandits with Knapsacks

In this paper, we study the bandits with knapsacks (BwK) problem and develop a primal-dual based algorithm that achieves a problem-dependent logarithmic regret bound. The BwK problem extends the multi-arm bandit (MAB) problem to model the resource consumption associated with playing each arm, and the existing BwK literature has been mainly focused on deriving asymptotically optimal distribution-free regret bounds. We first study the primal and dual linear programs underlying the BwK problem. From this primal-dual perspective, we discover symmetry between arms and knapsacks, and then propose a new notion of sub-optimality measure for the BwK problem. The sub-optimality measure highlights the important role of knapsacks in determining algorithm regret and inspires the design of our two-phase algorithm. In the first phase, the algorithm identifies the optimal arms and the binding knapsacks, and in the second phase, it exhausts the binding knapsacks via playing the optimal arms through an adaptive procedure. Our regret upper bound involves the proposed sub-optimality measure and it has a logarithmic dependence on length of horizon $T$ and a polynomial dependence on $m$ (the numbers of arms) and $d$ (the number of knapsacks). To the best of our knowledge, this is the first problem-dependent logarithmic regret bound for solving the general BwK problem.

cs.LG↗

arXiv2021

Fisher Markets with Linear Constraints: Equilibrium Properties and Efficient Distributed Algorithms

The Fisher market is one of the most fundamental models for resource allocation problems in economic theory, wherein agents spend a budget of currency to buy goods that maximize their utilities, while producers sell capacity constrained goods in exchange for currency. However, the consideration of only two types of constraints, i.e., budgets of individual buyers and capacities of goods, makes Fisher markets less amenable for resource allocation settings when agents have additional linear constraints, e.g., knapsack and proportionality constraints. In this work, we introduce a modified Fisher market, where each agent may have additional linear constraints and show that this modification to classical Fisher markets fundamentally alters the properties of the market equilibrium as well as the optimal allocations. These properties of the modified Fisher market prompt us to introduce a budget perturbed social optimization problem (BP-SOP) and set prices based on the dual variables of BP-SOP's capacity constraints. To compute the budget perturbations, we develop a fixed point iterative scheme and validate its convergence through numerical experiments. Since this fixed point iterative scheme involves solving a centralized problem at each step, we propose a new class of distributed algorithms to compute equilibrium prices. In particular, we develop an Alternating Direction Method of Multipliers (ADMM) algorithm with strong convergence guarantees for Fisher markets with homogeneous linear constraints as well as for classical Fisher markets. In this algorithm, the prices are updated based on the tatonnement process, with a step size that is completely independent of the utilities of individual agents. Thus, our mechanism, both theoretically and computationally, overcomes a fundamental limitation of classical Fisher markets, which only consider capacity and budget constraints.

cs.GT↗

arXiv2021

Markets for Efficient Public Good Allocation with Social Distancing

Public goods are often either over-consumed in the absence of regulatory mechanisms, or remain completely unused, as in the Covid-19 pandemic, where social distance constraints are enforced to limit the number of people who can share public spaces. In this work, we plug this gap through market based mechanisms designed to efficiently allocate capacity constrained public goods. To design these mechanisms, we leverage the theory of Fisher markets, wherein each agent in the economy is endowed with an artificial currency budget that they can spend to avail public goods. While Fisher markets provide a strong methodological backbone to model resource allocation problems, their applicability is limited to settings involving two types of constraints - budgets of individual buyers and capacities of goods. Thus, we introduce a modified Fisher market, where each individual may have additional physical constraints, characterize its solution properties and establish the existence of a market equilibrium. Furthermore, to account for additional constraints we introduce a social convex optimization problem where we perturb the budgets of agents such that the KKT conditions of the perturbed social problem establishes equilibrium prices. Finally, to compute the budget perturbations we present a fixed point scheme and illustrate convergence guarantees through numerical experiments. Thus, our mechanism, both theoretically and computationally, overcomes a fundamental limitation of classical Fisher markets, which only consider capacity and budget constraints.

cs.GT↗

arXiv2021

Online Linear Programming: Dual Convergence, New Algorithms, and Regret Bounds

We study an online linear programming (OLP) problem under a random input model in which the columns of the constraint matrix along with the corresponding coefficients in the objective function are generated i.i.d. from an unknown distribution and revealed sequentially over time. Virtually all pre-existing online algorithms were based on learning the dual optimal solutions/prices of the linear programs (LP), and their analyses were focused on the aggregate objective value and solving the packing LP where all coefficients in the constraint matrix and objective are nonnegative. However, two major open questions were: (i) Does the set of LP optimal dual prices learned in the pre-existing algorithms converge to those of the "offline" LP, and (ii) Could the results be extended to general LP problems where the coefficients can be either positive or negative. We resolve these two questions by establishing convergence results for the dual prices under moderate regularity conditions for general LP problems. Specifically, we identify an equivalent form of the dual problem which relates the dual LP with a sample average approximation to a stochastic program. Furthermore, we propose a new type of OLP algorithm, Action-History-Dependent Learning Algorithm, which improves the previous algorithm performances by taking into account the past input data as well as decisions/actions already made. We derive an $O(\log n \log \log n)$ regret bound (under a locally strong convexity and smoothness condition) for the proposed algorithm, against the $O(\sqrt{n})$ bound for typical dual-price learning algorithms, where $n$ is the number of decision variables. Numerical experiments demonstrate the effectiveness of the proposed algorithm and the action-history-dependent design.

cs.DS↗

arXiv2020

Variance Reduced Value Iteration and Faster Algorithms for Solving Markov Decision Processes

In this paper we provide faster algorithms for approximately solving discounted Markov Decision Processes in multiple parameter regimes. Given a discounted Markov Decision Process (DMDP) with $|S|$ states, $|A|$ actions, discount factor $γ\in(0,1)$, and rewards in the range $[-M, M]$, we show how to compute an $ε$-optimal policy, with probability $1 - δ$ in time \[ \tilde{O}\left( \left(|S|^2 |A| + \frac{|S| |A|}{(1 - γ)^3} \right) \log\left( \frac{M}ε \right) \log\left( \frac{1}δ \right) \right) ~ . \] This contribution reflects the first nearly linear time, nearly linearly convergent algorithm for solving DMDPs for intermediate values of $γ$. We also show how to obtain improved sublinear time algorithms provided we can sample from the transition function in $O(1)$ time. Under this assumption we provide an algorithm which computes an $ε$-optimal policy with probability $1 - δ$ in time \[ \tilde{O} \left(\frac{|S| |A| M^2}{(1 - γ)^4 ε^2} \log \left(\frac{1}δ\right) \right) ~. \] Lastly, we extend both these algorithms to solve finite horizon MDPs. Our algorithms improve upon the previous best for approximately computing optimal policies for fixed-horizon MDPs in multiple parameter regimes. Interestingly, we obtain our results by a careful modification of approximate value iteration. We show how to combine classic approximate value iteration analysis with new techniques in variance reduction. Our fastest algorithms leverage further insights to ensure that our algorithms make monotonic progress towards the optimal value. This paper is one of few instances in using sampling to obtain a linearly convergent linear programming algorithm and we hope that the analysis may be useful more broadly.

cs.DS↗

arXiv2020

Distributionally Robust Local Non-parametric Conditional Estimation

Conditional estimation given specific covariate values (i.e., local conditional estimation or functional estimation) is ubiquitously useful with applications in engineering, social and natural sciences. Existing data-driven non-parametric estimators mostly focus on structured homogeneous data (e.g., weakly independent and stationary data), thus they are sensitive to adversarial noise and may perform poorly under a low sample size. To alleviate these issues, we propose a new distributionally robust estimator that generates non-parametric local estimates by minimizing the worst-case conditional expected loss over all adversarial distributions in a Wasserstein ambiguity set. We show that despite being generally intractable, the local estimator can be efficiently found via convex optimization under broadly applicable settings, and it is robust to the corruption and heterogeneity of the data. Experiments with synthetic and MNIST datasets show the competitive performance of this new class of estimators.

stat.ML↗

arXiv2020

Simple and Fast Algorithm for Binary Integer and Online Linear Programming

In this paper, we develop a simple and fast online algorithm for solving a class of binary integer linear programs (LPs) arisen in general resource allocation problem. The algorithm requires only one single pass through the input data and is free of doing any matrix inversion. It can be viewed as both an approximate algorithm for solving binary integer LPs and a fast algorithm for solving online LP problems. The algorithm is inspired by an equivalent form of the dual problem of the relaxed LP and it essentially performs (one-pass) projected stochastic subgradient descent in the dual space. We analyze the algorithm in two different models, stochastic input and random permutation, with minimal technical assumptions on the input data. The algorithm achieves $O\left(m \sqrt{n}\right)$ expected regret under the stochastic input model and $O\left((m+\log n)\sqrt{n}\right)$ expected regret under the random permutation model, and it achieves $O(m \sqrt{n})$ expected constraint violation under both models, where $n$ is the number of decision variables and $m$ is the number of constraints. The algorithm enjoys the same performance guarantee when generalized to a multi-dimensional LP setting which covers a wider range of applications. In addition, we employ the notion of permutational Rademacher complexity and derive regret bounds for two earlier online LP algorithms for comparison. Both algorithms improve the regret bound with a factor of $\sqrt{m}$ by paying more computational cost. Furthermore, we demonstrate how to convert the possibly infeasible solution to a feasible one through a randomized procedure. Numerical experiments illustrate the general applicability and effectiveness of the algorithms.

cs.DS↗

arXiv2020

A Mean-Field Theory for Learning the Schönberg Measure of Radial Basis Functions

We develop and analyze a projected particle Langevin optimization method to learn the distribution in the Schönberg integral representation of the radial basis functions from training samples. More specifically, we characterize a distributionally robust optimization method with respect to the Wasserstein distance to optimize the distribution in the Schönberg integral representation. To provide theoretical performance guarantees, we analyze the scaling limits of a projected particle online (stochastic) optimization method in the mean-field regime. In particular, we prove that in the scaling limits, the empirical measure of the Langevin particles converges to the law of a reflected Itô diffusion-drift process. Moreover, the drift is also a function of the law of the underlying process. Using Itô lemma for semi-martingales and Grisanov's change of measure for the Wiener processes, we then derive a Mckean-Vlasov type partial differential equation (PDE) with Robin boundary conditions that describes the evolution of the empirical measure of the projected Langevin particles in the mean-field regime. In addition, we establish the existence and uniqueness of the steady-state solutions of the derived PDE in the weak sense. We apply our learning approach to train radial kernels in the kernel locally sensitive hash (LSH) functions, where the training data-set is generated via a $k$-mean clustering method on a small subset of data-base. We subsequently apply our kernel LSH with a trained kernel for image retrieval task on MNIST data-set, and demonstrate the efficacy of our kernel learning approach. We also apply our kernel learning approach in conjunction with the kernel support vector machines (SVMs) for classification of benchmark data-sets.

math.ST↗

arXiv2020

An ADMM-Based Interior-Point Method for Large-Scale Linear Programming

We propose a new framework to implement interior point method (IPM) to solve very large linear programs (LP). Traditional IPMs typically use Newton's method to approximately solve a subproblem that aims to minimize a log-barrier penalty function at each iteration. Due its connection to Newton's method, IPM is often classified as second-order method -- a genre that is attached with stability and accuracy at the expense of scalability. Indeed, computing a Newton step amounts to solving a large linear system, which can be efficiently implemented if the input data are reasonably-sized and/or sparse and/or well-structured. However, in case the above premises fail, then the challenge still stands on the way for a traditional IPM. To deal with this challenge, one approach is to apply the iterative procedure, such as preconditioned conjugate gradient method, to solve the linear system. Since the linear system is different each iteration, it is difficult to find good pre-conditioner to achieve the overall solution efficiency. In this paper, an alternative approach is proposed. Instead of applying Newton's method, we resort to the alternating direction method of multipliers (ADMM) to approximately minimize the log-barrier penalty function at each iteration, under the framework of primal-dual path-following for a homogeneous self-dual embedded LP model. The resulting algorithm is an ADMM-Based Interior Point Method, abbreviated as ABIP in this paper. The new method inherits stability from IPM, and scalability from ADMM. Because of its self-dual embedding structure, ABIP is set to solve any LP without requiring prior knowledge about its feasibility. We conduct extensive numerical experiments testing ABIP with large-scale LPs from NETLIB and machine learning applications. The results demonstrate that ABIP compares favorably with existing LP solvers including SDPT3, MOSEK, DSDP-CG and SCS.

math.OC↗

arXiv2020

Computations and Complexities of Tarski's Fixed Points and Supermodular Games

We consider two models of computation for Tarski's order preserving function f related to fixed points in a complete lattice: the oracle function model and the polynomial function model. In both models, we find the first polynomial time algorithm for finding a Tarski's fixed point. In addition, we provide a matching oracle bound for determining the uniqueness in the oracle function model and prove it is Co-NP hard in the polynomial function model. The existence of the pure Nash equilibrium in supermodular games is proved by Tarski's fixed point theorem. Exploring the difference between supermodular games and Tarski's fixed point, we also develop the computational results for finding one pure Nash equilibrium and determining the uniqueness of the equilibrium in supermodular games.

cs.GT↗

arXiv2020

Managing Randomization in the Multi-Block Alternating Direction Method of Multipliers for Quadratic Optimization

The Alternating Direction Method of Multipliers (ADMM) has gained a lot of attention for solving large-scale and objective-separable constrained optimization. However, the two-block variable structure of the ADMM still limits the practical computational efficiency of the method, because one big matrix factorization is needed at least once even for linear and convex quadratic programming. This drawback may be overcome by enforcing a multi-block structure of the decision variables in the original optimization problem. Unfortunately, the multi-block ADMM, with more than two blocks, is not guaranteed to be convergent. On the other hand, two positive developments have been made: first, if in each cyclic loop one randomly permutes the updating order of the multiple blocks, then the method converges in expectation for solving any system of linear equations with any number of blocks. Secondly, such a randomly permuted ADMM also works for equality-constrained convex quadratic programming even when the objective function is not separable. The goal of this paper is twofold. First, we add more randomness into the ADMM by developing a randomly assembled cyclic ADMM (RAC-ADMM) where the decision variables in each block are randomly assembled. We discuss the theoretical properties of RAC-ADMM and show when random assembling helps and when it hurts, and develop a criterion to guarantee that it converges almost surely. Secondly, using the theoretical guidance on RAC-ADMM, we conduct multiple numerical tests on solving both randomly generated and large-scale benchmark quadratic optimization problems, which include continuous, and binary graph-partition and quadratic assignment, and selected machine learning problems. Our numerical tests show that the RAC-ADMM, with a variable-grouping strategy, could significantly improve the computation efficiency on solving most quadratic optimization problems.

math.OC↗