Yanjun Han

Yanjun Han, Philippe Rigollet, George Stepaniants

Feature alignment methods are used in many scientific disciplines for data pooling, annotation, and comparison. As an instance of a permutation learning problem, feature alignment presents significant statistical and computational challenges. In this work, we propose the covariance alignment model to study and compare various alignment methods and establish a minimax lower bound for covariance alignment that has a non-standard dimension scaling because of the presence of a nuisance parameter. This lower bound is in fact minimax optimal and is achieved by a natural quasi MLE. However, this estimator involves a search over all permutations which is computationally infeasible even when the problem has moderate size. To overcome this limitation, we show that the celebrated Gromov-Wasserstein algorithm from optimal transport which is more amenable to fast implementation even on large-scale problems is also minimax optimal. These results give the first statistical justification for the deployment of the Gromov-Wasserstein algorithm in practice.

Yuxiao Wen, Yanjun Han, Zhengyuan Zhou

We study regret minimization in repeated first-price auctions (FPAs), where a bidder observes only the realized outcome after each auction -- win or loss. This setup reflects practical scenarios in online display advertising where the actual value of an impression depends on the difference between two potential outcomes, such as clicks or conversion rates, when the auction is won versus lost. We analyze three outcome models: (1) adversarial outcomes without features, (2) linear potential outcomes with features, and (3) linear treatment effects in features. For each setting, we propose algorithms that jointly estimate private values and optimize bidding strategies, achieving near-optimal regret bounds. Notably, our framework enjoys a unique feature that the treatments are also actively chosen, and hence eliminates the need for the overlap condition commonly required in causal inference.

Yuxiao Wen, Yanjun Han, Zhengyuan Zhou

We consider contextual bandits with graph feedback, a class of interactive learning problems with richer structures than vanilla contextual bandits, where taking an action reveals the rewards for all neighboring actions in the feedback graph under all contexts. Unlike the multi-armed bandits setting where a growing literature has painted a near-complete understanding of graph feedback, much remains unexplored in the contextual bandits counterpart. In this paper, we make inroads into this inquiry by establishing a regret lower bound $Ω(\sqrt{β_M(G) T})$, where $M$ is the number of contexts, $G$ is the feedback graph, and $β_M(G)$ is our proposed graph-theoretic quantity that characterizes the fundamental learning limit for this class of problems. Interestingly, $β_M(G)$ interpolates between $α(G)$ (the independence number of the graph) and $\mathsf{m}(G)$ (the maximum acyclic subgraph (MAS) number of the graph) as the number of contexts $M$ varies. We also provide algorithms that achieve near-optimal regret for important classes of context sequences and/or feedback graphs, such as transitively closed graphs that find applications in auctions and inventory control. In particular, with many contexts, our results show that the MAS number essentially characterizes the statistical complexity for contextual bandits, as opposed to the independence number in multi-armed bandits.

Yuzhou Gu, Yanjun Han, Jian Qian

We study the evolution of information in interactive decision making through the lens of a stochastic multi-armed bandit problem. Focusing on a fundamental example where a unique optimal arm outperforms the rest by a fixed margin, we characterize the optimal success probability and mutual information over time. Our findings reveal distinct growth phases in mutual information -- initially linear, transitioning to quadratic, and finally returning to linear -- highlighting curious behavioral differences between interactive and non-interactive environments. In particular, we show that optimal success probability and mutual information can be decoupled, where achieving optimal learning does not necessarily require maximizing information gain. These findings shed new light on the intricate interplay between information and learning in interactive decision making.

Nived Rajaraman, Yanjun Han, Lin F. Yang et al.

We study the statistical limits of Imitation Learning (IL) in episodic Markov Decision Processes (MDPs) with a state space $\mathcal{S}$. We focus on the known-transition setting where the learner is provided a dataset of $N$ length-$H$ trajectories from a deterministic expert policy and knows the MDP transition. We establish an upper bound $O(|\mathcal{S}|H^{3/2}/N)$ for the suboptimality using the Mimic-MD algorithm in Rajaraman et al (2020) which we prove to be computationally efficient. In contrast, we show the minimax suboptimality grows as $Ω( H^{3/2}/N)$ when $|\mathcal{S}|\geq 3$ while the unknown-transition setting suffers from a larger sharp rate $Θ(|\mathcal{S}|H^2/N)$ (Rajaraman et al (2020)). The lower bound is established by proving a two-way reduction between IL and the value estimation problem of the unknown expert policy under any given reward function, as well as building connections with linear functional estimation with subsampled observations. We further show that under the additional assumption that the expert is optimal for the true reward function, there exists an efficient algorithm, which we term as Mimic-Mixture, that provably achieves suboptimality $O(1/N)$ for arbitrary 3-state MDPs with rewards only at the terminal layer. In contrast, no algorithm can achieve suboptimality $O(\sqrt{H}/N)$ with high probability if the expert is not constrained to be optimal. Our work formally establishes the benefit of the expert optimal assumption in the known transition setting, while Rajaraman et al (2020) showed it does not help when transitions are unknown.

Xi Chen, Yanjun Han, Yining Wang

In this paper we study the adversarial combinatorial bandit with a known non-linear reward function, extending existing work on adversarial linear combinatorial bandit. {The adversarial combinatorial bandit with general non-linear reward is an important open problem in bandit literature, and it is still unclear whether there is a significant gap from the case of linear reward, stochastic bandit, or semi-bandit feedback.} We show that, with $N$ arms and subsets of $K$ arms being chosen at each of $T$ time periods, the minimax optimal regret is $\widetildeΘ_{d}(\sqrt{N^d T})$ if the reward function is a $d$-degree polynomial with $d< K$, and $Θ_K(\sqrt{N^K T})$ if the reward function is not a low-degree polynomial. {Both bounds are significantly different from the bound $O(\sqrt{\mathrm{poly}(N,K)T})$ for the linear case, which suggests that there is a fundamental gap between the linear and non-linear reward structures.} Our result also finds applications to adversarial assortment optimization problem in online recommendation. We show that in the worst-case of adversarial assortment problem, the optimal algorithm must treat each individual $\binom{N}{K}$ assortment as independent.

Yanjun Han, Zhengyuan Zhou, Aaron Flores et al.

First-price auctions have very recently swept the online advertising industry, replacing second-price auctions as the predominant auction mechanism on many platforms. This shift has brought forth important challenges for a bidder: how should one bid in a first-price auction, where unlike in second-price auctions, it is no longer optimal to bid one's private value truthfully and hard to know the others' bidding behaviors? In this paper, we take an online learning angle and address the fundamental problem of learning to bid in repeated first-price auctions, where both the bidder's private valuations and other bidders' bids can be arbitrary. We develop the first minimax optimal online bidding algorithm that achieves an $\widetilde{O}(\sqrt{T})$ regret when competing with the set of all Lipschitz bidding policies, a strong oracle that contains a rich set of bidding strategies. This novel algorithm is built on the insight that the presence of a good expert can be leveraged to improve performance, as well as an original hierarchical expert-chaining structure, both of which could be of independent interest in online learning. Further, by exploiting the product structure that exists in the problem, we modify this algorithm--in its vanilla form statistically optimal but computationally infeasible--to a computationally efficient and space efficient algorithm that also retains the same $\widetilde{O}(\sqrt{T})$ minimax optimal regret guarantee. Additionally, through an impossibility result, we highlight that one is unlikely to compete this favorably with a stronger oracle (than the considered Lipschitz bidding policies). Finally, we test our algorithm on three real-world first-price auction datasets obtained from Verizon Media and demonstrate our algorithm's superior performance compared to several existing bidding algorithms.

Yanjun Han, Zhengqing Zhou, Zhengyuan Zhou et al.

We study the sequential batch learning problem in linear contextual bandits with finite action sets, where the decision maker is constrained to split incoming individuals into (at most) a fixed number of batches and can only observe outcomes for the individuals within a batch at the batch's end. Compared to both standard online contextual bandits learning or offline policy learning in contexutal bandits, this sequential batch learning problem provides a finer-grained formulation of many personalized sequential decision making problems in practical applications, including medical treatment in clinical trials, product recommendation in e-commerce and adaptive experiment design in crowdsourcing. We study two settings of the problem: one where the contexts are arbitrarily generated and the other where the contexts are \textit{iid} drawn from some distribution. In each setting, we establish a regret lower bound and provide an algorithm, whose regret upper bound nearly matches the lower bound. As an important insight revealed therefrom, in the former setting, we show that the number of batches required to achieve the fully online performance is polynomial in the time horizon, while for the latter setting, a pure-exploitation algorithm with a judicious batch partition scheme achieves the fully online performance even when the number of batches is less than logarithmic in the time horizon. Together, our results provide a near-complete characterization of sequential decision making in linear contextual bandits when batch constraints are present.

Yanjun Han, Zhengyuan Zhou, Tsachy Weissman

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.

Jayadev Acharya, Clément L. Canonne, Yanjun Han et al.

We study goodness-of-fit of discrete distributions in the distributed setting, where samples are divided between multiple users who can only release a limited amount of information about their samples due to various information constraints. Recently, a subset of the authors showed that having access to a common random seed (i.e., shared randomness) leads to a significant reduction in the sample complexity of this problem. In this work, we provide a complete understanding of the interplay between the amount of shared randomness available, the stringency of information constraints, and the sample complexity of the testing problem by characterizing a tight trade-off between these three parameters. We provide a general distributed goodness-of-fit protocol that as a function of the amount of shared randomness interpolates smoothly between the private- and public-coin sample complexities. We complement our upper bound with a general framework to prove lower bounds on the sample complexity of this testing problems under limited shared randomness. Finally, we instantiate our bounds for the two archetypal information constraints of communication and local privacy, and show that our sample complexity bounds are optimal as a function of all the parameters of the problem, including the amount of shared randomness. A key component of our upper bounds is a new primitive of domain compression, a tool that allows us to map distributions to a much smaller domain size while preserving their pairwise distances, using a limited amount of randomness.

Zijun Gao, Yanjun Han, Zhimei Ren et al.

In this paper, we study the multi-armed bandit problem in the batched setting where the employed policy must split data into a small number of batches. While the minimax regret for the two-armed stochastic bandits has been completely characterized in \cite{perchet2016batched}, the effect of the number of arms on the regret for the multi-armed case is still open. Moreover, the question whether adaptively chosen batch sizes will help to reduce the regret also remains underexplored. In this paper, we propose the BaSE (batched successive elimination) policy to achieve the rate-optimal regrets (within logarithmic factors) for batched multi-armed bandits, with matching lower bounds even if the batch sizes are determined in an adaptive manner.

Leighton Pate Barnes, Yanjun Han, Ayfer Ozgur

We consider the problem of learning high-dimensional, nonparametric and structured (e.g. Gaussian) distributions in distributed networks, where each node in the network observes an independent sample from the underlying distribution and can use $k$ bits to communicate its sample to a central processor. We consider three different models for communication. Under the independent model, each node communicates its sample to a central processor by independently encoding it into $k$ bits. Under the more general sequential or blackboard communication models, nodes can share information interactively but each node is restricted to write at most $k$ bits on the final transcript. We characterize the impact of the communication constraint $k$ on the minimax risk of estimating the underlying distribution under $\ell^2$ loss. We develop minimax lower bounds that apply in a unified way to many common statistical models and reveal that the impact of the communication constraint can be qualitatively different depending on the tail behavior of the score function associated with each model. A key ingredient in our proofs is a geometric characterization of Fisher information from quantized samples.

Yanjun Han, Jiantao Jiao, Tsachy Weissman

We present \emph{Local Moment Matching (LMM)}, a unified methodology for symmetric functional estimation and distribution estimation under Wasserstein distance. We construct an efficiently computable estimator that achieves the minimax rates in estimating the distribution up to permutation, and show that the plug-in approach of our unlabeled distribution estimator is "universal" in estimating symmetric functionals of discrete distributions. Instead of doing best polynomial approximation explicitly as in existing literature of functional estimation, the plug-in approach conducts polynomial approximation implicitly and attains the optimal sample complexity for the entropy, power sum and support size functionals.

Yanjun Han, Jiantao Jiao, Chuan-Zheng Lee et al.

Estimating the entropy based on data is one of the prototypical problems in distribution property testing and estimation. For estimating the Shannon entropy of a distribution on $S$ elements with independent samples, [Paninski2004] showed that the sample complexity is sublinear in $S$, and [Valiant--Valiant2011] showed that consistent estimation of Shannon entropy is possible if and only if the sample size $n$ far exceeds $\frac{S}{\log S}$. In this paper we consider the problem of estimating the entropy rate of a stationary reversible Markov chain with $S$ states from a sample path of $n$ observations. We show that: (1) As long as the Markov chain mixes not too slowly, i.e., the relaxation time is at most $O(\frac{S}{\ln^3 S})$, consistent estimation is achievable when $n \gg \frac{S^2}{\log S}$. (2) As long as the Markov chain has some slight dependency, i.e., the relaxation time is at least $1+Ω(\frac{\ln^2 S}{\sqrt{S}})$, consistent estimation is impossible when $n \lesssim \frac{S^2}{\log S}$. Under both assumptions, the optimal estimation accuracy is shown to be $Θ(\frac{S^2}{n \log S})$. In comparison, the empirical entropy rate requires at least $Ω(S^2)$ samples to be consistent, even when the Markov chain is memoryless. In addition to synthetic experiments, we also apply the estimators that achieve the optimal sample complexity to estimate the entropy rate of the English language in the Penn Treebank and the Google One Billion Words corpora, which provides a natural benchmark for language modeling and relates it directly to the widely used perplexity measure.

Jiantao Jiao, Weihao Gao, Yanjun Han

We analyze the Kozachenko--Leonenko (KL) nearest neighbor estimator for the differential entropy. We obtain the first uniform upper bound on its performance over Hölder balls on a torus without assuming any conditions on how close the density could be from zero. Accompanying a new minimax lower bound over the Hölder ball, we show that the KL estimator is achieving the minimax rates up to logarithmic factors without cognizance of the smoothness parameter $s$ of the Hölder ball for $s\in (0,2]$ and arbitrary dimension $d$, rendering it the first estimator that provably satisfies this property.

Yanjun Han, Jiantao Jiao, Rajarshi Mukherjee

We provide a complete picture of asymptotically minimax estimation of $L_r$-norms (for any $r\ge 1$) of the mean in Gaussian white noise model over Nikolskii-Besov spaces. In this regard, we complement the work of Lepski, Nemirovski and Spokoiny (1999), who considered the cases of $r=1$ (with poly-logarithmic gap between upper and lower bounds) and $r$ even (with asymptotically sharp upper and lower bounds) over Hölder spaces. We additionally consider the case of asymptotically adaptive minimax estimation and demonstrate a difference between even and non-even $r$ in terms of an investigator's ability to produce asymptotically adaptive minimax estimators without paying a penalty.

Jiantao Jiao, Yanjun Han

We analyze bias correction methods using jackknife, bootstrap, and Taylor series. We focus on the binomial model, and consider the problem of bias correction for estimating $f(p)$, where $f \in C[0,1]$ is arbitrary. We characterize the supremum norm of the bias of general jackknife and bootstrap estimators for any continuous functions, and demonstrate the in delete-$d$ jackknife, different values of $d$ may lead to drastically different behaviors in jackknife. We show that in the binomial model, iterating the bootstrap bias correction infinitely many times may lead to divergence of bias and variance, and demonstrate that the bias properties of the bootstrap bias corrected estimator after $r-1$ rounds are of the same order as that of the $r$-jackknife estimator if a bounded coefficients condition is satisfied.

Jiantao Jiao, Yanjun Han, Irena Fischer-Hwang et al.

We show through case studies that it is easier to estimate the fundamental limits of data processing than to construct explicit algorithms to achieve those limits. Focusing on binary classification, data compression, and prediction under logarithmic loss, we show that in the finite space setting, when it is possible to construct an estimator of the limits with vanishing error with $n$ samples, it may require at least $n\ln n$ samples to construct an explicit algorithm to achieve the limits.

Sihan Li, Jiantao Jiao, Yanjun Han et al.

The Residual Network (ResNet), proposed in He et al. (2015), utilized shortcut connections to significantly reduce the difficulty of training, which resulted in great performance boosts in terms of both training and generalization error. It was empirically observed in He et al. (2015) that stacking more layers of residual blocks with shortcut 2 results in smaller training error, while it is not true for shortcut of length 1 or 3. We provide a theoretical explanation for the uniqueness of shortcut 2. We show that with or without nonlinearities, by adding shortcuts that have depth two, the condition number of the Hessian of the loss function at the zero initial point is depth-invariant, which makes training very deep models no more difficult than shallow ones. Shortcuts of higher depth result in an extremely flat (high-order) stationary point initially, from which the optimization algorithm is hard to escape. The shortcut 1, however, is essentially equivalent to no shortcuts, which has a condition number exploding to infinity as the number of layers grows. We further argue that as the number of layers tends to infinity, it suffices to only look at the loss function at the zero initial point. Extensive experiments are provided accompanying our theoretical results. We show that initializing the network to small weights with shortcut 2 achieves significantly better results than random Gaussian (Xavier) initialization, orthogonal initialization, and shortcuts of deeper depth, from various perspectives ranging from final loss, learning dynamics and stability, to the behavior of the Hessian along the learning process.

Jiantao Jiao, Kartik Venkat, Yanjun Han et al.

Maximum likelihood is the most widely used statistical estimation technique. Recent work by the authors introduced a general methodology for the construction of estimators for functionals in parametric models, and demonstrated improvements - both in theory and in practice - over the maximum likelihood estimator (MLE), particularly in high dimensional scenarios involving parameter dimension comparable to or larger than the number of samples. This approach to estimation, building on results from approximation theory, is shown to yield minimax rate-optimal estimators for a wide class of functionals, implementable with modest computational requirements. In a nutshell, a message of this recent work is that, for a wide class of functionals, the performance of these essentially optimal estimators with $n$ samples is comparable to that of the MLE with $n \ln n$ samples. In the present paper, we highlight the applicability of the aforementioned methodology to statistical problems beyond functional estimation, and show that it can yield substantial gains. For example, we demonstrate that for learning tree-structured graphical models, our approach achieves a significant reduction of the required data size compared with the classical Chow--Liu algorithm, which is an implementation of the MLE, to achieve the same accuracy. The key step in improving the Chow--Liu algorithm is to replace the empirical mutual information with the estimator for mutual information proposed by the authors. Further, applying the same replacement approach to classical Bayesian network classification, the resulting classifiers uniformly outperform the previous classifiers on 26 widely used datasets.