arXiv subjects

Pan Xu

Publications and source records attributed to Pan Xu.

At least 73 records · Page 4Linked to original sources

arXiv2021

Group-level Fairness Maximization in Online Bipartite Matching

We consider the allocation of limited resources to heterogeneous customers who arrive in an online fashion. We would like to allocate the resources "fairly", so that no group of customers is marginalized in terms of their overall service rate. We study whether this is possible to do so in an online fashion, and if so, what a good online allocation policy is. We model this problem using online bipartite matching under stationary arrivals, a fundamental model in the literature typically studied under the objective of maximizing the total number of customers served. We instead study the objective of maximizing the minimum service rate across all groups, and propose two notions of fairness: long-run and short-run. For these fairness objectives, we analyze how competitive online algorithms can be, in comparison to offline algorithms which know the sequence of demands in advance. For long-run fairness, we propose two online heuristics (Sampling and Pooling) which establish asymptotic optimality in different regimes (no specialized supplies, no rare demand types, or imbalanced supply/demand). By contrast, outside all of these regimes, we show that the competitive ratio of online algorithms is between 0.632 and 0.732. For short-run fairness, we show for complete bipartite graphs that the competitive ratio of online algorithms is between 0.863 and 0.942; we also derive a probabilistic rejection algorithm which is asymptotically optimal in the total demand. Depending on the overall scarcity of resources, either our Sampling or Pooling heuristics could be desirable. The most difficult situation for online allocation occurs when the total supply is just enough to serve the total demand. We simulate our algorithms on a public ride-hailing dataset, which both demonstrates the efficacy of our heuristics and validates our managerial insights.

cs.DS↗

arXiv2021

Faster Convergence of Stochastic Gradient Langevin Dynamics for Non-Log-Concave Sampling

We provide a new convergence analysis of stochastic gradient Langevin dynamics (SGLD) for sampling from a class of distributions that can be non-log-concave. At the core of our approach is a novel conductance analysis of SGLD using an auxiliary time-reversible Markov Chain. Under certain conditions on the target distribution, we prove that $\tilde O(d^4ε^{-2})$ stochastic gradient evaluations suffice to guarantee $ε$-sampling error in terms of the total variation distance, where $d$ is the problem dimension. This improves existing results on the convergence rate of SGLD (Raginsky et al., 2017; Xu et al., 2018). We further show that provided an additional Hessian Lipschitz condition on the log-density function, SGLD is guaranteed to achieve $ε$-sampling error within $\tilde O(d^{15/4}ε^{-3/2})$ stochastic gradient evaluations. Our proof technique provides a new way to study the convergence of Langevin-based algorithms and sheds some light on the design of fast stochastic gradient-based sampling algorithms.

cs.LG↗

arXiv2020

Trading the System Efficiency for the Income Equality of Drivers in Rideshare

Several scientific studies have reported the existence of the income gap among rideshare drivers based on demographic factors such as gender, age, race, etc. In this paper, we study the income inequality among rideshare drivers due to discriminative cancellations from riders, and the tradeoff between the income inequality (called fairness objective) with the system efficiency (called profit objective). We proposed an online bipartite-matching model where riders are assumed to arrive sequentially following a distribution known in advance. The highlight of our model is the concept of acceptance rate between any pair of driver-rider types, where types are defined based on demographic factors. Specially, we assume each rider can accept or cancel the driver assigned to her, each occurs with a certain probability which reflects the acceptance degree from the rider type towards the driver type. We construct a bi-objective linear program as a valid benchmark and propose two LP-based parameterized online algorithms. Rigorous online competitive ratio analysis is offered to demonstrate the flexibility and efficiency of our online algorithms in balancing the two conflicting goals, promotions of fairness and profit. Experimental results on a real-world dataset are provided as well, which confirm our theoretical predictions.

cs.AI↗

arXiv2020

A Unified Model for the Two-stage Offline-then-Online Resource Allocation

With the popularity of the Internet, traditional offline resource allocation has evolved into a new form, called online resource allocation. It features the online arrivals of agents in the system and the real-time decision-making requirement upon the arrival of each online agent. Both offline and online resource allocation have wide applications in various real-world matching markets ranging from ridesharing to crowdsourcing. There are some emerging applications such as rebalancing in bike sharing and trip-vehicle dispatching in ridesharing, which involve a two-stage resource allocation process. The process consists of an offline phase and another sequential online phase, and both phases compete for the same set of resources. In this paper, we propose a unified model which incorporates both offline and online resource allocation into a single framework. Our model assumes non-uniform and known arrival distributions for online agents in the second online phase, which can be learned from historical data. We propose a parameterized linear programming (LP)-based algorithm, which is shown to be at most a constant factor of $1/4$ from the optimal. Experimental results on the real dataset show that our LP-based approaches outperform the LP-agnostic heuristics in terms of robustness and effectiveness.

cs.AI↗

arXiv2020

Neural Contextual Bandits with Deep Representation and Shallow Exploration

We study a general class of contextual bandits, where each context-action pair is associated with a raw feature vector, but the reward generating function is unknown. We propose a novel learning algorithm that transforms the raw feature vector using the last hidden layer of a deep ReLU neural network (deep representation learning), and uses an upper confidence bound (UCB) approach to explore in the last linear layer (shallow exploration). We prove that under standard assumptions, our proposed algorithm achieves $\tilde{O}(\sqrt{T})$ finite-time regret, where $T$ is the learning time horizon. Compared with existing neural contextual bandit algorithms, our approach is computationally much more efficient since it only needs to explore in the last layer of the deep neural network.

cs.LG↗

arXiv2020

Double Explore-then-Commit: Asymptotic Optimality and Beyond

We study the multi-armed bandit problem with subgaussian rewards. The explore-then-commit (ETC) strategy, which consists of an exploration phase followed by an exploitation phase, is one of the most widely used algorithms in a variety of online decision applications. Nevertheless, it has been shown in Garivier et al. (2016) that ETC is suboptimal in the asymptotic sense as the horizon grows, and thus, is worse than fully sequential strategies such as Upper Confidence Bound (UCB). In this paper, we show that a variant of ETC algorithm can actually achieve the asymptotic optimality for multi-armed bandit problems as UCB-type algorithms do and extend it to the batched bandit setting. Specifically, we propose a double explore-then-commit (DETC) algorithm that has two exploration and exploitation phases and prove that DETC achieves the asymptotically optimal regret bound. To our knowledge, DETC is the first non-fully-sequential algorithm that achieves such asymptotic optimality. In addition, we extend DETC to batched bandit problems, where (i) the exploration process is split into a small number of batches and (ii) the round complexity is of central interest. We prove that a batched version of DETC can achieve the asymptotic optimality with only a constant round complexity. This is the first batched bandit algorithm that can attain the optimal asymptotic regret bound and optimal round complexity simultaneously.

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arXiv2020

Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization

We present a unified framework to analyze the global convergence of Langevin dynamics based algorithms for nonconvex finite-sum optimization with $n$ component functions. At the core of our analysis is a direct analysis of the ergodicity of the numerical approximations to Langevin dynamics, which leads to faster convergence rates. Specifically, we show that gradient Langevin dynamics (GLD) and stochastic gradient Langevin dynamics (SGLD) converge to the almost minimizer within $\tilde O\big(nd/(λε) \big)$ and $\tilde O\big(d^7/(λ^5ε^5) \big)$ stochastic gradient evaluations respectively, where $d$ is the problem dimension, and $λ$ is the spectral gap of the Markov chain generated by GLD. Both results improve upon the best known gradient complexity results (Raginsky et al., 2017). Furthermore, for the first time we prove the global convergence guarantee for variance reduced stochastic gradient Langevin dynamics (SVRG-LD) to the almost minimizer within $\tilde O\big(\sqrt{n}d^5/(λ^4ε^{5/2})\big)$ stochastic gradient evaluations, which outperforms the gradient complexities of GLD and SGLD in a wide regime. Our theoretical analyses shed some light on using Langevin dynamics based algorithms for nonconvex optimization with provable guarantees.

stat.ML↗

arXiv2020

Stochastic Variance-Reduced Hamilton Monte Carlo Methods

We propose a fast stochastic Hamilton Monte Carlo (HMC) method, for sampling from a smooth and strongly log-concave distribution. At the core of our proposed method is a variance reduction technique inspired by the recent advance in stochastic optimization. We show that, to achieve $ε$ accuracy in 2-Wasserstein distance, our algorithm achieves $\tilde O(n+κ^{2}d^{1/2}/ε+κ^{4/3}d^{1/3}n^{2/3}/ε^{2/3})$ gradient complexity (i.e., number of component gradient evaluations), which outperforms the state-of-the-art HMC and stochastic gradient HMC methods in a wide regime. We also extend our algorithm for sampling from smooth and general log-concave distributions, and prove the corresponding gradient complexity as well. Experiments on both synthetic and real data demonstrate the superior performance of our algorithm.

stat.ML↗

arXiv2020

Stochastic Nested Variance Reduction for Nonconvex Optimization

We study finite-sum nonconvex optimization problems, where the objective function is an average of $n$ nonconvex functions. We propose a new stochastic gradient descent algorithm based on nested variance reduction. Compared with conventional stochastic variance reduced gradient (SVRG) algorithm that uses two reference points to construct a semi-stochastic gradient with diminishing variance in each iteration, our algorithm uses $K+1$ nested reference points to build a semi-stochastic gradient to further reduce its variance in each iteration. For smooth nonconvex functions, the proposed algorithm converges to an $ε$-approximate first-order stationary point (i.e., $\|\nabla F(\mathbf{x})\|_2\leq ε$) within $\tilde O(n\land ε^{-2}+ε^{-3}\land n^{1/2}ε^{-2})$ number of stochastic gradient evaluations. This improves the best known gradient complexity of SVRG $O(n+n^{2/3}ε^{-2})$ and that of SCSG $O(n\land ε^{-2}+ε^{-10/3}\land n^{2/3}ε^{-2})$. For gradient dominated functions, our algorithm also achieves better gradient complexity than the state-of-the-art algorithms. Thorough experimental results on different nonconvex optimization problems back up our theory.

cs.LG↗

arXiv2020

Improved Approximation Algorithms for Stochastic-Matching Problems

We consider the Stochastic Matching problem, which is motivated by applications in kidney exchange and online dating. In this problem, we are given an undirected graph. Each edge is assigned a known, independent probability of existence and a positive weight (or profit). We must probe an edge to discover whether or not it exists. Each node is assigned a positive integer called a timeout (or a patience). On this random graph we are executing a process, which probes the edges one-by-one and gradually constructs a matching. The process is constrained in two ways. First, if a probed edge exists, it must be added irrevocably to the matching (the query-commit model). Second, the timeout of a node $v$ upper-bounds the number of edges incident to $v$ that can be probed. The goal is to maximize the expected weight of the constructed matching. For this problem, Bansal et al. (Algorithmica 2012) provided a $0.33$-approximation algorithm for bipartite graphs and a $0.25$-approximation for general graphs. We improve the approximation factors to $0.39$ and $0.269$, respectively. The main technical ingredient in our result is a novel way of probing edges according to a not-uniformly-random permutation. Patching this method with an algorithm that works best for large-probability edges (plus additional ideas) leads to our improved approximation factors.

cs.DS↗

arXiv2020

MOTS: Minimax Optimal Thompson Sampling

Thompson sampling is one of the most widely used algorithms for many online decision problems, due to its simplicity in implementation and superior empirical performance over other state-of-the-art methods. Despite its popularity and empirical success, it has remained an open problem whether Thompson sampling can match the minimax lower bound $Ω(\sqrt{KT})$ for $K$-armed bandit problems, where $T$ is the total time horizon. In this paper, we solve this long open problem by proposing a variant of Thompson sampling called MOTS that adaptively clips the sampling instance of the chosen arm at each time step. We prove that this simple variant of Thompson sampling achieves the minimax optimal regret bound $O(\sqrt{KT})$ for finite time horizon $T$, as well as the asymptotic optimal regret bound for Gaussian rewards when $T$ approaches infinity. To our knowledge, MOTS is the first Thompson sampling type algorithm that achieves the minimax optimality for multi-armed bandit problems.

cs.LG↗

arXiv2020

Balancing the Tradeoff between Profit and Fairness in Rideshare Platforms During High-Demand Hours

Rideshare platforms, when assigning requests to drivers, tend to maximize profit for the system and/or minimize waiting time for riders. Such platforms can exacerbate biases that drivers may have over certain types of requests. We consider the case of peak hours when the demand for rides is more than the supply of drivers. Drivers are well aware of their advantage during the peak hours and can choose to be selective about which rides to accept. Moreover, if in such a scenario, the assignment of requests to drivers (by the platform) is made only to maximize profit and/or minimize wait time for riders, requests of a certain type (e.g. from a non-popular pickup location, or to a non-popular drop-off location) might never be assigned to a driver. Such a system can be highly unfair to riders. However, increasing fairness might come at a cost of the overall profit made by the rideshare platform. To balance these conflicting goals, we present a flexible, non-adaptive algorithm, \lpalg, that allows the platform designer to control the profit and fairness of the system via parameters $α$ and $β$ respectively. We model the matching problem as an online bipartite matching where the set of drivers is offline and requests arrive online. Upon the arrival of a request, we use \lpalg to assign it to a driver (the driver might then choose to accept or reject it) or reject the request. We formalize the measures of profit and fairness in our setting and show that by using \lpalg, the competitive ratios for profit and fairness measures would be no worse than $α/e$ and $β/e$ respectively. Extensive experimental results on both real-world and synthetic datasets confirm the validity of our theoretical lower bounds. Additionally, they show that $\lpalg$ under some choice of $(α, β)$ can beat two natural heuristics, Greedy and Uniform, on \emph{both} fairness and profit.

cs.AI↗

arXiv2020

A Finite-Time Analysis of Q-Learning with Neural Network Function Approximation

Q-learning with neural network function approximation (neural Q-learning for short) is among the most prevalent deep reinforcement learning algorithms. Despite its empirical success, the non-asymptotic convergence rate of neural Q-learning remains virtually unknown. In this paper, we present a finite-time analysis of a neural Q-learning algorithm, where the data are generated from a Markov decision process and the action-value function is approximated by a deep ReLU neural network. We prove that neural Q-learning finds the optimal policy with $O(1/\sqrt{T})$ convergence rate if the neural function approximator is sufficiently overparameterized, where $T$ is the number of iterations. To our best knowledge, our result is the first finite-time analysis of neural Q-learning under non-i.i.d. data assumption.

cs.LG↗

arXiv2019

Rank Aggregation via Heterogeneous Thurstone Preference Models

We propose the Heterogeneous Thurstone Model (HTM) for aggregating ranked data, which can take the accuracy levels of different users into account. By allowing different noise distributions, the proposed HTM model maintains the generality of Thurstone's original framework, and as such, also extends the Bradley-Terry-Luce (BTL) model for pairwise comparisons to heterogeneous populations of users. Under this framework, we also propose a rank aggregation algorithm based on alternating gradient descent to estimate the underlying item scores and accuracy levels of different users simultaneously from noisy pairwise comparisons. We theoretically prove that the proposed algorithm converges linearly up to a statistical error which matches that of the state-of-the-art method for the single-user BTL model. We evaluate the proposed HTM model and algorithm on both synthetic and real data, demonstrating that it outperforms existing methods.

cs.LG↗

arXiv2019

Mix and Match: Markov Chains & Mixing Times for Matching in Rideshare

Rideshare platforms such as Uber and Lyft dynamically dispatch drivers to match riders' requests. We model the dispatching process in rideshare as a Markov chain that takes into account the geographic mobility of both drivers and riders over time. Prior work explores dispatch policies in the limit of such Markov chains; we characterize when this limit assumption is valid, under a variety of natural dispatch policies. We give explicit bounds on convergence in general, and exact (including constants) convergence rates for special cases. Then, on simulated and real transit data, we show that our bounds characterize convergence rates -- even when the necessary theoretical assumptions are relaxed. Additionally these policies compare well against a standard reinforcement learning algorithm which optimizes for profit without any convergence properties.

cs.DS↗

arXiv2019

Algorithms to Approximate Column-Sparse Packing Problems

Column-sparse packing problems arise in several contexts in both deterministic and stochastic discrete optimization. We present two unifying ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain improved approximation algorithms for some well-known families of such problems. As three main examples, we attain the integrality gap, up to lower-order terms, for known LP relaxations for k-column sparse packing integer programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set packing (Bansal et al., Algorithmica, 2012), and go "half the remaining distance" to optimal for a major integrality-gap conjecture of Furedi, Kahn and Seymour on hypergraph matching (Combinatorica, 1993).

cs.DS↗

arXiv2019

Online Stochastic Matching: New Algorithms and Bounds

Online matching has received significant attention over the last 15 years due to its close connection to Internet advertising. As the seminal work of Karp, Vazirani, and Vazirani has an optimal (1 - 1/e) competitive ratio in the standard adversarial online model, much effort has gone into developing useful online models that incorporate some stochasticity in the arrival process. One such popular model is the "known I.I.D. model" where different customer-types arrive online from a known distribution. We develop algorithms with improved competitive ratios for some basic variants of this model with integral arrival rates, including (a) the case of general weighted edges, where we improve the best-known ratio of 0.667 due to Haeupler, Mirrokni and Zadimoghaddam to 0.705; and (b) the vertex-weighted case, where we improve the 0.7250 ratio of Jaillet and Lu to 0.7299. We also consider an extension of stochastic rewards, a variant where each edge has an independent probability of being present. For the setting of stochastic rewards with non-integral arrival rates, we present a simple optimal non-adaptive algorithm with a ratio of 1 - 1/e. For the special case where each edge is unweighted and has a uniform constant probability of being present, we improve upon 1 - 1/e by proposing a strengthened LP benchmark.

cs.DS↗

arXiv2019

Attenuate Locally, Win Globally: An Attenuation-based Framework for Online Stochastic Matching with Timeouts

Online matching problems have garnered significant attention in recent years due to numerous applications in e-commerce, online advertisements, ride-sharing, etc. Many of them capture the uncertainty in the real world by including stochasticity in both the arrival process and the matching process. The Online Stochastic Matching with Timeouts problem introduced by Bansal, et al., (Algorithmica, 2012) models matching markets (e.g., E-Bay, Amazon). Buyers arrive from an independent and identically distributed (i.i.d.) known distribution on buyer profiles and can be shown a list of items one at a time. Each buyer has some probability of purchasing each item and a limit (timeout) on the number of items they can be shown. Bansal et al., (Algorithmica, 2012) gave a 0.12-competitive algorithm which was improved by Adamczyk, et al., (ESA, 2015) to 0.24. We present an online attenuation framework that uses an algorithm for offline stochastic matching as a black box. On the upper bound side, we show that this framework, combined with a black-box adapted from Bansal et al., (Algorithmica, 2012), yields an online algorithm which nearly doubles the ratio to 0.46. On the lower bound side, we show that no algorithm can achieve a ratio better than 0.632 using the standard LP for this problem. This framework has a high potential for further improvements since new algorithms for offline stochastic matching can directly improve the ratio for the online problem. Our online framework also has the potential for a variety of extensions. For example, we introduce a natural generalization: Online Stochastic Matching with Two-sided Timeouts in which both online and offline vertices have timeouts. Our framework provides the first algorithm for this problem achieving a ratio of 0.30. We once again use the algorithm of Adamczyk et al., (ESA, 2015) as a black-box and plug-it into our framework.

cs.DS↗