arXiv subjects

Zhengyuan Zhou

Publications and source records attributed to Zhengyuan Zhou.

At least 55 records · Page 3Linked to original sources

arXiv2024

Adaptive, Doubly Optimal No-Regret Learning in Strongly Monotone and Exp-Concave Games with Gradient Feedback

Online gradient descent (OGD) is well known to be doubly optimal under strong convexity or monotonicity assumptions: (1) in the single-agent setting, it achieves an optimal regret of $Θ(\log T)$ for strongly convex cost functions; and (2) in the multi-agent setting of strongly monotone games, with each agent employing OGD, we obtain last-iterate convergence of the joint action to a unique Nash equilibrium at an optimal rate of $Θ(\frac{1}{T})$. While these finite-time guarantees highlight its merits, OGD has the drawback that it requires knowing the strong convexity/monotonicity parameters. In this paper, we design a fully adaptive OGD algorithm, \textsf{AdaOGD}, that does not require a priori knowledge of these parameters. In the single-agent setting, our algorithm achieves $O(\log^2(T))$ regret under strong convexity, which is optimal up to a log factor. Further, if each agent employs \textsf{AdaOGD} in strongly monotone games, the joint action converges in a last-iterate sense to a unique Nash equilibrium at a rate of $O(\frac{\log^3 T}{T})$, again optimal up to log factors. We illustrate our algorithms in a learning version of the classical newsvendor problem, where due to lost sales, only (noisy) gradient feedback can be observed. Our results immediately yield the first feasible and near-optimal algorithm for both the single-retailer and multi-retailer settings. We also extend our results to the more general setting of exp-concave cost functions and games, using the online Newton step (ONS) algorithm.

cs.GT↗

arXiv2024

Optimal No-regret Learning in Repeated First-price Auctions

We study online learning in repeated first-price auctions where a bidder, only observing the winning bid at the end of each auction, learns to adaptively bid in order to maximize her cumulative payoff. To achieve this goal, the bidder faces censored feedback: if she wins the bid, then she is not able to observe the highest bid of the other bidders, which we assume is \textit{iid} drawn from an unknown distribution. In this paper, we develop the first learning algorithm that achieves a near-optimal $\widetilde{O}(\sqrt{T})$ regret bound, by exploiting two structural properties of first-price auctions, i.e. the specific feedback structure and payoff function. We first formulate the feedback structure in first-price auctions as partially ordered contextual bandits, a combination of the graph feedback across actions (bids), the cross learning across contexts (private values), and a partial order over the contexts. We establish both strengths and weaknesses of this framework, by showing a curious separation that a regret nearly independent of the action/context sizes is possible under stochastic contexts, but is impossible under adversarial contexts. In particular, this framework leads to an $O(\sqrt{T}\log^{2.5}T)$ regret for first-price auctions when the bidder's private values are \emph{iid}. Despite the limitation of the above framework, we further exploit the special payoff function of first-price auctions to develop a sample-efficient algorithm even in the presence of adversarially generated private values. We establish an $O(\sqrt{T}\log^3 T)$ regret bound for this algorithm, hence providing a complete characterization of optimal learning guarantees for first-price auctions.

cs.LG↗

arXiv2023

A Unified Linear Speedup Analysis of Federated Averaging and Nesterov FedAvg

Federated learning (FL) learns a model jointly from a set of participating devices without sharing each other's privately held data. The characteristics of non-i.i.d. data across the network, low device participation, high communication costs, and the mandate that data remain private bring challenges in understanding the convergence of FL algorithms, particularly regarding how convergence scales with the number of participating devices. In this paper, we focus on Federated Averaging (FedAvg), one of the most popular and effective FL algorithms in use today, as well as its Nesterov accelerated variant, and conduct a systematic study of how their convergence scale with the number of participating devices under non-i.i.d. data and partial participation in convex settings. We provide a unified analysis that establishes convergence guarantees for FedAvg under strongly convex, convex, and overparameterized strongly convex problems. We show that FedAvg enjoys linear speedup in each case, although with different convergence rates and communication efficiencies. For strongly convex and convex problems, we also characterize the corresponding convergence rates for the Nesterov accelerated FedAvg algorithm, which are the first linear speedup guarantees for momentum variants of FedAvg in convex settings. Empirical studies of the algorithms in various settings have supported our theoretical results.

cs.LG↗

arXiv2023

Learning to Order for Inventory Systems with Lost Sales and Uncertain Supplies

We consider a stochastic lost-sales inventory control system with a lead time $L$ over a planning horizon $T$. Supply is uncertain, and is a function of the order quantity (due to random yield/capacity, etc). We aim to minimize the $T$-period cost, a problem that is known to be computationally intractable even under known distributions of demand and supply. In this paper, we assume that both the demand and supply distributions are unknown and develop a computationally efficient online learning algorithm. We show that our algorithm achieves a regret (i.e. the performance gap between the cost of our algorithm and that of an optimal policy over $T$ periods) of $O(L+\sqrt{T})$ when $L\geq\log(T)$. We do so by 1) showing our algorithm cost is higher by at most $O(L+\sqrt{T})$ for any $L\geq 0$ compared to an optimal constant-order policy under complete information (a well-known and widely-used algorithm) and 2) leveraging its known performance guarantee from the existing literature. To the best of our knowledge, a finite-sample $O(\sqrt{T})$ (and polynomial in $L$) regret bound when benchmarked against an optimal policy is not known before in the online inventory control literature. A key challenge in this learning problem is that both demand and supply data can be censored; hence only truncated values are observable. We circumvent this challenge by showing that the data generated under an order quantity $q^2$ allows us to simulate the performance of not only $q^2$ but also $q^1$ for all $q^1<q^2$, a key observation to obtain sufficient information even under data censoring. By establishing a high probability coupling argument, we are able to evaluate and compare the performance of different order policies at their steady state within a finite time horizon. Since the problem lacks convexity, we develop an active elimination method that adaptively rules out suboptimal solutions.

math.OC↗

arXiv2023

Near-Optimal Non-Convex Stochastic Optimization under Generalized Smoothness

The generalized smooth condition, $(L_{0},L_{1})$-smoothness, has triggered people's interest since it is more realistic in many optimization problems shown by both empirical and theoretical evidence. Two recent works established the $O(ε^{-3})$ sample complexity to obtain an $O(ε)$-stationary point. However, both require a large batch size on the order of $\mathrm{ploy}(ε^{-1})$, which is not only computationally burdensome but also unsuitable for streaming applications. Additionally, these existing convergence bounds are established only for the expected rate, which is inadequate as they do not supply a useful performance guarantee on a single run. In this work, we solve the prior two problems simultaneously by revisiting a simple variant of the STORM algorithm. Specifically, under the $(L_{0},L_{1})$-smoothness and affine-type noises, we establish the first near-optimal $O(\log(1/(δε))ε^{-3})$ high-probability sample complexity where $δ\in(0,1)$ is the failure probability. Besides, for the same algorithm, we also recover the optimal $O(ε^{-3})$ sample complexity for the expected convergence with improved dependence on the problem-dependent parameter. More importantly, our convergence results only require a constant batch size in contrast to the previous works.

cs.LG↗

arXiv2023

Distributionally Robust Batch Contextual Bandits

Policy learning using historical observational data is an important problem that has found widespread applications. Examples include selecting offers, prices, advertisements to send to customers, as well as selecting which medication to prescribe to a patient. However, existing literature rests on the crucial assumption that the future environment where the learned policy will be deployed is the same as the past environment that has generated the data -- an assumption that is often false or too coarse an approximation. In this paper, we lift this assumption and aim to learn a distributionally robust policy with incomplete observational data. We first present a policy evaluation procedure that allows us to assess how well the policy does under the worst-case environment shift. We then establish a central limit theorem type guarantee for this proposed policy evaluation scheme. Leveraging this evaluation scheme, we further propose a novel learning algorithm that is able to learn a policy that is robust to adversarial perturbations and unknown covariate shifts with a performance guarantee based on the theory of uniform convergence. Finally, we empirically test the effectiveness of our proposed algorithm in synthetic datasets and demonstrate that it provides the robustness that is missing using standard policy learning algorithms. We conclude the paper by providing a comprehensive application of our methods in the context of a real-world voting dataset.

cs.LG↗

arXiv2023

Breaking the Lower Bound with (Little) Structure: Acceleration in Non-Convex Stochastic Optimization with Heavy-Tailed Noise

We consider the stochastic optimization problem with smooth but not necessarily convex objectives in the heavy-tailed noise regime, where the stochastic gradient's noise is assumed to have bounded $p$th moment ($p\in(1,2]$). Zhang et al. (2020) is the first to prove the $Ω(T^{\frac{1-p}{3p-2}})$ lower bound for convergence (in expectation) and provides a simple clipping algorithm that matches this optimal rate. Cutkosky and Mehta (2021) proposes another algorithm, which is shown to achieve the nearly optimal high-probability convergence guarantee $O(\log(T/δ)T^{\frac{1-p}{3p-2}})$, where $δ$ is the probability of failure. However, this desirable guarantee is only established under the additional assumption that the stochastic gradient itself is bounded in $p$th moment, which fails to hold even for quadratic objectives and centered Gaussian noise. In this work, we first improve the analysis of the algorithm in Cutkosky and Mehta (2021) to obtain the same nearly optimal high-probability convergence rate $O(\log(T/δ)T^{\frac{1-p}{3p-2}})$, without the above-mentioned restrictive assumption. Next, and curiously, we show that one can achieve a faster rate than that dictated by the lower bound $Ω(T^{\frac{1-p}{3p-2}})$ with only a tiny bit of structure, i.e., when the objective function $F(x)$ is assumed to be in the form of $\mathbb{E}_{Ξ\sim\mathcal{D}}[f(x,Ξ)]$, arguably the most widely applicable class of stochastic optimization problems. For this class of problems, we propose the first variance-reduced accelerated algorithm and establish that it guarantees a high-probability convergence rate of $O(\log(T/δ)T^{\frac{1-p}{2p-1}})$ under a mild condition, which is faster than $Ω(T^{\frac{1-p}{3p-2}})$. Notably, even when specialized to the finite-variance case, our result yields the (near-)optimal high-probability rate $O(\log(T/δ)T^{-1/3})$.

cs.LG↗

arXiv2023

Stochastic Nonsmooth Convex Optimization with Heavy-Tailed Noises: High-Probability Bound, In-Expectation Rate and Initial Distance Adaptation

Recently, several studies consider the stochastic optimization problem but in a heavy-tailed noise regime, i.e., the difference between the stochastic gradient and the true gradient is assumed to have a finite $p$-th moment (say being upper bounded by $σ^{p}$ for some $σ\geq0$) where $p\in(1,2]$, which not only generalizes the traditional finite variance assumption ($p=2$) but also has been observed in practice for several different tasks. Under this challenging assumption, lots of new progress has been made for either convex or nonconvex problems, however, most of which only consider smooth objectives. In contrast, people have not fully explored and well understood this problem when functions are nonsmooth. This paper aims to fill this crucial gap by providing a comprehensive analysis of stochastic nonsmooth convex optimization with heavy-tailed noises. We revisit a simple clipping-based algorithm, whereas, which is only proved to converge in expectation but under the additional strong convexity assumption. Under appropriate choices of parameters, for both convex and strongly convex functions, we not only establish the first high-probability rates but also give refined in-expectation bounds compared with existing works. Remarkably, all of our results are optimal (or nearly optimal up to logarithmic factors) with respect to the time horizon $T$ even when $T$ is unknown in advance. Additionally, we show how to make the algorithm parameter-free with respect to $σ$, in other words, the algorithm can still guarantee convergence without any prior knowledge of $σ$. Furthermore, an initial distance adaptive convergence rate is provided if $σ$ is assumed to be known.

math.OC↗

arXiv2023

Distributionally Robust Offline Reinforcement Learning with Linear Function Approximation

Among the reasons hindering reinforcement learning (RL) applications to real-world problems, two factors are critical: limited data and the mismatch between the testing environment (real environment in which the policy is deployed) and the training environment (e.g., a simulator). This paper attempts to address these issues simultaneously with distributionally robust offline RL, where we learn a distributionally robust policy using historical data obtained from the source environment by optimizing against a worst-case perturbation thereof. In particular, we move beyond tabular settings and consider linear function approximation. More specifically, we consider two settings, one where the dataset is well-explored and the other where the dataset has sufficient coverage of the optimal policy. We propose two algorithms~-- one for each of the two settings~-- that achieve error bounds $\tilde{O}(d^{1/2}/N^{1/2})$ and $\tilde{O}(d^{3/2}/N^{1/2})$ respectively, where $d$ is the dimension in the linear function approximation and $N$ is the number of trajectories in the dataset. To the best of our knowledge, they provide the first non-asymptotic results of the sample complexity in this setting. Diverse experiments are conducted to demonstrate our theoretical findings, showing the superiority of our algorithm against the non-robust one.

cs.LG↗

arXiv2022

Policy Learning with Adaptively Collected Data

Learning optimal policies from historical data enables personalization in a wide variety of applications including healthcare, digital recommendations, and online education. The growing policy learning literature focuses on settings where the data collection rule stays fixed throughout the experiment. However, adaptive data collection is becoming more common in practice, from two primary sources: 1) data collected from adaptive experiments that are designed to improve inferential efficiency; 2) data collected from production systems that progressively evolve an operational policy to improve performance over time (e.g. contextual bandits). Yet adaptivity complicates the optimal policy identification ex post, since samples are dependent, and each treatment may not receive enough observations for each type of individual. In this paper, we make initial research inquiries into addressing the challenges of learning the optimal policy with adaptively collected data. We propose an algorithm based on generalized augmented inverse propensity weighted (AIPW) estimators, which non-uniformly reweight the elements of a standard AIPW estimator to control worst-case estimation variance. We establish a finite-sample regret upper bound for our algorithm and complement it with a regret lower bound that quantifies the fundamental difficulty of policy learning with adaptive data. When equipped with the best weighting scheme, our algorithm achieves minimax rate optimal regret guarantees even with diminishing exploration. Finally, we demonstrate our algorithm's effectiveness using both synthetic data and public benchmark datasets.

stat.ML↗

arXiv2022

Leveraging the Hints: Adaptive Bidding in Repeated First-Price Auctions

With the advent and increasing consolidation of e-commerce, digital advertising has very recently replaced traditional advertising as the main marketing force in the economy. In the past four years, a particularly important development in the digital advertising industry is the shift from second-price auctions to first-price auctions for online display ads. This shift immediately motivated the intellectually challenging question of how to bid in first-price auctions, because unlike in second-price auctions, bidding one's private value truthfully is no longer optimal. Following a series of recent works in this area, we consider a differentiated setup: we do not make any assumption about other bidders' maximum bid (i.e. it can be adversarial over time), and instead assume that we have access to a hint that serves as a prediction of other bidders' maximum bid, where the prediction is learned through some blackbox machine learning model. We consider two types of hints: one where a single point-prediction is available, and the other where a hint interval (representing a type of confidence region into which others' maximum bid falls) is available. We establish minimax optimal regret bounds for both cases and highlight the quantitatively different behavior between the two settings. We also provide improved regret bounds when the others' maximum bid exhibits the further structure of sparsity. Finally, we complement the theoretical results with demonstrations using real bidding data.

cs.GT↗

arXiv2022

Optimal Diagonal Preconditioning

Preconditioning has long been a staple technique in optimization, often applied to reduce the condition number of a matrix and speed up the convergence of algorithms. Although there are many popular preconditioning techniques in practice, most lack guarantees on reductions in condition number. Moreover, the degree to which we can improve over existing heuristic preconditioners remains an important practical question. In this paper, we study the problem of optimal diagonal preconditioning that achieves maximal reduction in the condition number of any full-rank matrix by scaling its rows and/or columns. We first reformulate the problem as a quasi-convex problem and provide a simple algorithm based on bisection. Then we develop an interior point algorithm with $O(\log(1/ε))$ iteration complexity, where each iteration consists of a Newton update based on the Nesterov-Todd direction. Next, we specialize to one-sided optimal diagonal preconditioning problems, and demonstrate that they can be formulated as standard dual SDP problems. We then develop efficient customized solvers and study the empirical performance of our optimal diagonal preconditioning procedures through extensive experiments on large matrices. Our findings suggest that optimal diagonal preconditioners can significantly improve upon existing heuristics-based diagonal preconditioners at reducing condition numbers and speeding up iterative methods. Moreover, our implementation of customized solvers, combined with a random row/column sampling step, can find near-optimal diagonal preconditioners for matrices up to size 200,000 in reasonable time, demonstrating their practical appeal.

math.OC↗

arXiv2022

Interpretable Personalization via Policy Learning with Linear Decision Boundaries

With the rise of the digital economy and an explosion of available information about consumers, effective personalization of goods and services has become a core business focus for companies to improve revenues and maintain a competitive edge. This paper studies the personalization problem through the lens of policy learning, where the goal is to learn a decision-making rule (a policy) that maps from consumer and product characteristics (features) to recommendations (actions) in order to optimize outcomes (rewards). We focus on using available historical data for offline learning with unknown data collection procedures, where a key challenge is the non-random assignment of recommendations. Moreover, in many business and medical applications, interpretability of a policy is essential. We study the class of policies with linear decision boundaries to ensure interpretability, and propose learning algorithms using tools from causal inference to address unbalanced treatments. We study several optimization schemes to solve the associated non-convex, non-smooth optimization problem, and find that a Bayesian optimization algorithm is effective. We test our algorithm with extensive simulation studies and apply it to an anonymized online marketplace customer purchase dataset, where the learned policy outputs a personalized discount recommendation based on customer and product features in order to maximize gross merchandise value (GMV) for sellers. Our learned policy improves upon the platform's baseline by 88.2\% in net sales revenue, while also providing informative insights on which features are important for the decision-making process. Our findings suggest that our proposed policy learning framework using tools from causal inference and Bayesian optimization provides a promising practical approach to interpretable personalization across a wide range of applications.

cs.LG↗

arXiv2022

Sensitivity analysis under the $f$-sensitivity models: a distributional robustness perspective

This paper introduces the $f$-sensitivity model, a new sensitivity model that characterizes the violation of unconfoundedness in causal inference. It assumes the selection bias due to unmeasured confounding is bounded "on average"; compared with the widely used point-wise sensitivity models in the literature, it is able to capture the strength of unmeasured confounding by not only its magnitude but also the chance of encountering such a magnitude. We propose a framework for sensitivity analysis under our new model based on a distributional robustness perspective. We first show that the bounds on counterfactual means under the f-sensitivity model are optimal solutions to a new class of distributionally robust optimization (DRO) programs, whose dual forms are essentially risk minimization problems. We then construct point estimators for these bounds by applying a novel debiasing technique to the output of the corresponding empirical risk minimization (ERM) problems. Our estimators are shown to converge to valid bounds on counterfactual means if any nuisance component can be estimated consistently, and to the exact bounds when the ERM step is additionally consistent. We further establish asymptotic normality and Wald-type inference for these estimators under slower-than-root-n convergence rates of the estimated nuisance components. Finally, the performance of our method is demonstrated with numerical experiments.

stat.ME↗

arXiv2022

Doubly Robust Distributionally Robust Off-Policy Evaluation and Learning

Off-policy evaluation and learning (OPE/L) use offline observational data to make better decisions, which is crucial in applications where online experimentation is limited. However, depending entirely on logged data, OPE/L is sensitive to environment distribution shifts -- discrepancies between the data-generating environment and that where policies are deployed. \citet{si2020distributional} proposed distributionally robust OPE/L (DROPE/L) to address this, but the proposal relies on inverse-propensity weighting, whose estimation error and regret will deteriorate if propensities are nonparametrically estimated and whose variance is suboptimal even if not. For standard, non-robust, OPE/L, this is solved by doubly robust (DR) methods, but they do not naturally extend to the more complex DROPE/L, which involves a worst-case expectation. In this paper, we propose the first DR algorithms for DROPE/L with KL-divergence uncertainty sets. For evaluation, we propose Localized Doubly Robust DROPE (LDR$^2$OPE) and show that it achieves semiparametric efficiency under weak product rates conditions. Thanks to a localization technique, LDR$^2$OPE only requires fitting a small number of regressions, just like DR methods for standard OPE. For learning, we propose Continuum Doubly Robust DROPL (CDR$^2$OPL) and show that, under a product rate condition involving a continuum of regressions, it enjoys a fast regret rate of $\mathcal{O}\left(N^{-1/2}\right)$ even when unknown propensities are nonparametrically estimated. We empirically validate our algorithms in simulations and further extend our results to general $f$-divergence uncertainty sets.

cs.LG↗

arXiv2022

Dynamic Batch Learning in High-Dimensional Sparse Linear Contextual Bandits

We study the problem of dynamic batch learning in high-dimensional sparse linear contextual bandits, where a decision maker, under a given maximum-number-of-batch constraint and only able to observe rewards at the end of each batch, can dynamically decide how many individuals to include in the next batch (at the end of the current batch) and what personalized action-selection scheme to adopt within each batch. Such batch constraints are ubiquitous in a variety of practical contexts, including personalized product offerings in marketing and medical treatment selection in clinical trials. We characterize the fundamental learning limit in this problem via a regret lower bound and provide a matching upper bound (up to log factors), thus prescribing an optimal scheme for this problem. To the best of our knowledge, our work provides the first inroad into a theoretical understanding of dynamic batch learning in high-dimensional sparse linear contextual bandits. Notably, even a special case of our result -- when no batch constraint is present -- yields that the simple exploration-free algorithm using the LASSO estimator already achieves the minimax optimal regret bound for standard online learning in high-dimensional linear contextual bandits (for the no-margin case), a result that appears unknown in the emerging literature of high-dimensional contextual bandits.

stat.ML↗

arXiv2022

No Weighted-Regret Learning in Adversarial Bandits with Delays

Consider a scenario where a player chooses an action in each round $t$ out of $T$ rounds and observes the incurred cost after a delay of $d_{t}$ rounds. The cost functions and the delay sequence are chosen by an adversary. We show that in a non-cooperative game, the expected weighted ergodic distribution of play converges to the set of coarse correlated equilibria if players use algorithms that have "no weighted-regret" in the above scenario, even if they have linear regret due to too large delays. For a two-player zero-sum game, we show that no weighted-regret is sufficient for the weighted ergodic average of play to converge to the set of Nash equilibria. We prove that the FKM algorithm with $n$ dimensions achieves an expected regret of $O\left(nT^{\frac{3}{4}}+\sqrt{n}T^{\frac{1}{3}}D^{\frac{1}{3}}\right)$ and the EXP3 algorithm with $K$ arms achieves an expected regret of $O\left(\sqrt{\log K\left(KT+D\right)}\right)$ even when $D=\sum_{t=1}^{T}d_{t}$ and $T$ are unknown. These bounds use a novel doubling trick that, under mild assumptions, provably retains the regret bound for when $D$ and $T$ are known. Using these bounds, we show that FKM and EXP3 have no weighted-regret even for $d_{t}=O\left(t\log t\right)$. Therefore, algorithms with no weighted-regret can be used to approximate a CCE of a finite or convex unknown game that can only be simulated with bandit feedback, even if the simulation involves significant delays.

cs.LG↗

arXiv2022

Computational Benefits of Intermediate Rewards for Goal-Reaching Policy Learning

Many goal-reaching reinforcement learning (RL) tasks have empirically verified that rewarding the agent on subgoals improves convergence speed and practical performance. We attempt to provide a theoretical framework to quantify the computational benefits of rewarding the completion of subgoals, in terms of the number of synchronous value iterations. In particular, we consider subgoals as one-way {\em intermediate states}, which can only be visited once per episode and propose two settings that consider these one-way intermediate states: the one-way single-path (OWSP) and the one-way multi-path (OWMP) settings. In both OWSP and OWMP settings, we demonstrate that adding {\em intermediate rewards} to subgoals is more computationally efficient than only rewarding the agent once it completes the goal of reaching a terminal state. We also reveal a trade-off between computational complexity and the pursuit of the shortest path in the OWMP setting: adding intermediate rewards significantly reduces the computational complexity of reaching the goal but the agent may not find the shortest path, whereas with sparse terminal rewards, the agent finds the shortest path at a significantly higher computational cost. We also corroborate our theoretical results with extensive experiments on the MiniGrid environments using Q-learning and some popular deep RL algorithms.

cs.AI↗