Jian Qian

Dylan J. Foster, Noah Golowich, Jian Qian et al. · mit

We consider the problem of interactive decision making, encompassing structured bandits and reinforcement learning with general function approximation. Recently, Foster et al. (2021) introduced the Decision-Estimation Coefficient, a measure of statistical complexity that lower bounds the optimal regret for interactive decision making, as well as a meta-algorithm, Estimation-to-Decisions, which achieves upper bounds in terms of the same quantity. Estimation-to-Decisions is a reduction, which lifts algorithms for (supervised) online estimation into algorithms for decision making. In this paper, we show that by combining Estimation-to-Decisions with a specialized form of optimistic estimation introduced by Zhang (2022), it is possible to obtain guarantees that improve upon those of Foster et al. (2021) by accommodating more lenient notions of estimation error. We use this approach to derive regret bounds for model-free reinforcement learning with value function approximation, and give structural results showing when it can and cannot help more generally.

Haochuan Li, Jian Qian, Yi Tian et al.

Classical analysis of convex and non-convex optimization methods often requires the Lipshitzness of the gradient, which limits the analysis to functions bounded by quadratics. Recent work relaxed this requirement to a non-uniform smoothness condition with the Hessian norm bounded by an affine function of the gradient norm, and proved convergence in the non-convex setting via gradient clipping, assuming bounded noise. In this paper, we further generalize this non-uniform smoothness condition and develop a simple, yet powerful analysis technique that bounds the gradients along the trajectory, thereby leading to stronger results for both convex and non-convex optimization problems. In particular, we obtain the classical convergence rates for (stochastic) gradient descent and Nesterov's accelerated gradient method in the convex and/or non-convex setting under this general smoothness condition. The new analysis approach does not require gradient clipping and allows heavy-tailed noise with bounded variance in the stochastic setting.

Ali Jadbabaie, Haochuan Li, Jian Qian et al.

In this paper, we study a linear bandit optimization problem in a federated setting where a large collection of distributed agents collaboratively learn a common linear bandit model. Standard federated learning algorithms applied to this setting are vulnerable to Byzantine attacks on even a small fraction of agents. We propose a novel algorithm with a robust aggregation oracle that utilizes the geometric median. We prove that our proposed algorithm is robust to Byzantine attacks on fewer than half of agents and achieves a sublinear $\tilde{\mathcal{O}}({T^{3/4}})$ regret with $\mathcal{O}(\sqrt{T})$ steps of communication in $T$ steps. Moreover, we make our algorithm differentially private via a tree-based mechanism. Finally, if the level of corruption is known to be small, we show that using the geometric median of mean oracle for robust aggregation further improves the regret bound.

Jian Qian, Miao Sun, Ashley Lee et al.

Depth completion aims to predict dense depth maps with sparse depth measurements from a depth sensor. Currently, Convolutional Neural Network (CNN) based models are the most popular methods applied to depth completion tasks. However, despite the excellent high-end performance, they suffer from a limited representation area. To overcome the drawbacks of CNNs, a more effective and powerful method has been presented: the Transformer, which is an adaptive self-attention setting sequence-to-sequence model. While the standard Transformer quadratically increases the computational cost from the key-query dot-product of input resolution which improperly employs depth completion tasks. In this work, we propose a different window-based Transformer architecture for depth completion tasks named Sparse-to-Dense Transformer (SDformer). The network consists of an input module for the depth map and RGB image features extraction and concatenation, a U-shaped encoder-decoder Transformer for extracting deep features, and a refinement module. Specifically, we first concatenate the depth map features with the RGB image features through the input model. Then, instead of calculating self-attention with the whole feature maps, we apply different window sizes to extract the long-range depth dependencies. Finally, we refine the predicted features from the input module and the U-shaped encoder-decoder Transformer module to get the enriching depth features and employ a convolution layer to obtain the dense depth map. In practice, the SDformer obtains state-of-the-art results against the CNN-based depth completion models with lower computing loads and parameters on the NYU Depth V2 and KITTI DC datasets.

Spencer Compton, Gábor Lugosi, Jaouad Mourtada et al.

We study density estimation in Kullback-Leibler divergence: given an i.i.d. sample from an unknown density $p$, the goal is to construct an estimator $\widehat p$ such that $\mathrm{KL}(p,\widehat p)$ is small with high probability. We consider two settings involving a finite dictionary of $M$ densities: (i) model aggregation, where $p$ belongs to the dictionary, and (ii) convex aggregation (mixture density estimation), where $p$ is a mixture of densities from the dictionary. Crucially, we make no assumption on the base densities: their ratios may be unbounded and their supports may differ. For both problems, we identify the best possible high-probability guarantees in terms of the dictionary size, sample size, and confidence level. These optimal rates are higher than those achievable when density ratios are bounded by absolute constants; for mixture density estimation, they match existing lower bounds in the special case of discrete distributions. Our analysis of the mixture case hinges on two new covering results. First, we provide a sharp, distribution-free upper bound on the local Hellinger entropy of the class of mixtures of $M$ distributions. Second, we prove an optimal ratio covering theorem for convex sets: for every convex compact set $K\subset \mathbb{R}_+^d$, there exists a subset $A\subset K$ with at most $2^{8d}$ elements such that each element of $K$ is coordinate-wise dominated by an element of $A$ up to a universal constant factor. This geometric result is of independent interest; notably, it yields new cardinality estimates for $\varepsilon$-approximate Pareto sets in multi-objective optimization when the attainable set of objective vectors is convex.

Jian Qian, Bingyu Xie, Biao Wan et al.

Time series generation is a crucial research topic in the area of decision-making systems, which can be particularly important in domains like autonomous driving, healthcare, and, notably, robotics. Recent approaches focus on learning in the data space to model time series information. However, the data space often contains limited observations and noisy features. In this paper, we propose TimeLDM, a novel latent diffusion model for high-quality time series generation. TimeLDM is composed of a variational autoencoder that encodes time series into an informative and smoothed latent content and a latent diffusion model operating in the latent space to generate latent information. We evaluate the ability of our method to generate synthetic time series with simulated and real-world datasets and benchmark the performance against existing state-of-the-art methods. Qualitatively and quantitatively, we find that the proposed TimeLDM persistently delivers high-quality generated time series. For example, TimeLDM achieves new state-of-the-art results on the simulated benchmarks and an average improvement of 55% in Discriminative score with all benchmarks. Further studies demonstrate that our method yields more robust outcomes across various lengths of time series data generation. Especially, for the Context-FID score and Discriminative score, TimeLDM realizes significant improvements of 80% and 50%, respectively. The code will be released after publication.

Jian Qian, Miao Sun, Sifan Zhou et al.

In-context learning (ICL) leverages in-context examples as prompts for the predictions of Large Language Models (LLMs). These prompts play a crucial role in achieving strong performance. However, the selection of suitable prompts from a large pool of labeled examples often entails significant annotation costs. To address this challenge, we propose Sub-SA (Submodular Selective Annotation), a submodule-based selective annotation method. The aim of Sub-SA is to reduce annotation costs while improving the quality of in-context examples and minimizing the time consumption of the selection process. In Sub-SA, we design a submodular function that facilitates effective subset selection for annotation and demonstrates the characteristics of monotonically and submodularity from the theoretical perspective. Specifically, we propose RPR (Reward and Penalty Regularization) to better balance the diversity and representativeness of the unlabeled dataset attributed to a reward term and a penalty term, respectively. Consequently, the selection for annotations can be effectively addressed with a simple yet effective greedy search algorithm based on the submodular function. Finally, we apply the similarity prompt retrieval to get the examples for ICL.

Muheng Li, Jian Qian, Wenlong Mou

Test-time scaling has emerged as a critical avenue for enhancing the reasoning capabilities of Large Language Models (LLMs). Though the straight-forward ''best-of-$N$'' (BoN) strategy has already demonstrated significant improvements in performance, it lacks principled guidance on the choice of $N$, budget allocation, and multi-stage decision-making, thereby leaving substantial room for optimization. While many works have explored such optimization, rigorous theoretical guarantees remain limited. In this work, we propose new methodologies to predict and improve scaling properties via tail-guided search. By estimating the tail distribution of rewards, our method predicts the scaling law of LLMs without the need for exhaustive evaluations. Leveraging this prediction tool, we introduce Scaling-Law Guided (SLG) Search, a new test-time algorithm that dynamically allocates compute to identify and exploit intermediate states with the highest predicted potential. We theoretically prove that SLG achieves vanishing regret compared to perfect-information oracles, and achieves expected rewards that would otherwise require a polynomially larger compute budget required when using BoN. Empirically, we validate our framework across different LLMs and reward models, confirming that tail-guided allocation consistently achieves higher reward yields than Best-of-$N$ under identical compute budgets. Our code is available at https://github.com/PotatoJnny/Scaling-Law-Guided-search.

Muheng Li, Jian Qian, Wenlong Mou

Large language models are increasingly deployed with test-time strategies: sample $N$ responses, score them with a reward model or verifier, and return the best. This deployment rule exposes a mismatch in post-training: standard objectives optimize the mean reward of a single response, whereas best-of-$N$ performance is governed by the upper tail of the reward distribution. Recent test-time-aware objectives partly address this mismatch, but typically assume that training can use the same per-prompt rollout budget as deployment, which is impractical when post-training must cover many prompts while deployment can allocate much larger per-prompt test-time compute. We study this budget-mismatch regime, where only $m\ll N$ per-prompt rollouts are available during training but the target objective is best-of-$N$ deployment. Under structural assumptions on the reward tails, we show that the policy gradient of the best-of-$N$ objective can be approximated from a much smaller rollout group by extrapolating upper-tail statistics. This yields a family of Tail-Extrapolated estimators for best-of-$N$-oriented post-training: a simple direct estimator, Tail-Extrapolated Advantage (TEA), and a fixed-order debiased Prefix-TEA estimator based on moment cancellation. Experiments on instruction-following tasks show that TEA and Prefix-TEA improve best-of-$N$ performance across different language models, reward models and datasets under various training and test-time budget settings.

Fan Chen, Jian Qian, Alexander Rakhlin et al.

Self-normalized martingale inequalities lie at the heart of confidence ellipsoids for online least squares and, more broadly, many bandit and reinforcement-learning results. Yet existing vector and scalar results typically rely on bounded covariates and an explicit regularization matrix, producing bounds that are \emph{not scale-invariant}: although the self-normalized quantity is scale-invariant by definition, its standard upper bounds are not. We characterize when scale-invariant upper bounds on self-normalized martingales are possible. Without further assumptions, we prove that nontrivial scale-invariant bounds exist only in dimension $d=1$; moreover, in $d=1$ we obtain $O(\log T)$ scale-invariant self-normalized bounds without any assumptions on the covariates. In contrast, for $d>1$ we show that no nontrivial scale-invariant bound can hold in full generality. We then connect this dichotomy to \emph{doubly-uniform} regret in online linear regression (i.e., regret bounds that are simultaneously independent of the covariate scale and the comparator norm) and use it to resolve the open question of Gaillard, Gerchinovitz, Huard, and Stoltz, \emph{``Uniform regret bounds over $\mathbb{R}^d$ for the sequential linear regression problem with the square loss''} (ALT 2019): in $d=1$ we give an explicit algorithm with $O(\log T)$ doubly-uniform regret, whereas for $d>1$ sublinear doubly-uniform regret is impossible. Finally, under a natural \emph{smoothness} condition (bounded Radon--Nikodym derivatives of the conditional covariate laws with respect to a fixed base measure), we recover sublinear regret for $d>1$ without bounded covariates and derive a self-normalized concentration inequality free of the usual regularization penalties, yielding arguably a first natural scale-invariant bound for adaptive, non-i.i.d. vector martingales.

Yujie Shen, Zihan Wang, Jian Qian et al.

Training data reconstruction from KKT conditions has shown striking empirical success, yet it remains unclear when the resulting KKT equations have unique solutions and, even in identifiable regimes, how to reliably recover solutions by optimization. This work hereby focuses on these two complementary questions: identifiability and optimization. On the identifiability side, we discuss the sufficient conditions for KKT system of two-layer networks with polynomial activations to uniquely determine the training data, providing a theoretical explanation of when and why reconstruction is possible. On the optimization side, we introduce sample splitting, a curvature-aware refinement step applicable to general reconstruction objectives (not limited to KKT-based formulations): it creates additional descent directions to escape poor stationary points and refine solutions. Experiments demonstrate that augmenting several existing reconstruction methods with sample splitting consistently improves reconstruction performance.

Haichen Hu, Jian Qian, David Simchi-Levi

Reinforcement learning (RL) in large environments often suffers from severe computational bottlenecks, as conventional regret minimization algorithms require repeated, costly calls to planning and statistical estimation oracles. While recent advances have explored offline oracle-efficient algorithms, their computational complexity typically scales with the cardinality of the state and action spaces, rendering them intractable for large-scale or continuous environments. In this paper, we address this fundamental limitation by studying offline oracle-efficient episodic RL through the lens of log-barrier and log-determinant regularization. Specifically, for tabular Markov Decision Processes (MDPs), we propose a novel algorithm that achieves the optimal $\tilde{O}(\sqrt{T})$ regret bound while requiring only $O(H\log\log T)$ calls to both the offline statistical estimation and planning oracles when $T$ is known and $O(H\log T)$ calls when $T$ is unknown. Crucially, this oracle complexity is entirely independent of the size of the state and action spaces. This strict independence drastically reduces the planning oracle complexity, representing a substantial improvement over existing offline oracle-efficient algorithms (Qian et al., 2024). Furthermore, we demonstrate the versatility of our framework by generalizing the algorithm to linear MDPs featuring infinite state spaces and arbitrary action spaces. We prove that this generalized approach successfully attains meaningful sub-linear regret. Consequently, our work yields the first doubly oracle-efficient (i.e., efficient with respect to both statistical estimation and policy optimization) regret minimization algorithm capable of solving MDPs with infinite state and action spaces, significantly expanding the boundaries of computationally tractable RL.

Zeyu Jia, Jian Qian, Alexander Rakhlin et al.

We consider realizable contextual bandits with general function approximation, investigating how small reward variance can lead to better-than-minimax regret bounds. Unlike in minimax bounds, we show that the eluder dimension $d_\text{elu}$$-$a complexity measure of the function class$-$plays a crucial role in variance-dependent bounds. We consider two types of adversary: (1) Weak adversary: The adversary sets the reward variance before observing the learner's action. In this setting, we prove that a regret of $Ω(\sqrt{\min\{A,d_\text{elu}\}Λ}+d_\text{elu})$ is unavoidable when $d_{\text{elu}}\leq\sqrt{AT}$, where $A$ is the number of actions, $T$ is the total number of rounds, and $Λ$ is the total variance over $T$ rounds. For the $A\leq d_\text{elu}$ regime, we derive a nearly matching upper bound $\tilde{O}(\sqrt{AΛ}+d_\text{elu})$ for the special case where the variance is revealed at the beginning of each round. (2) Strong adversary: The adversary sets the reward variance after observing the learner's action. We show that a regret of $Ω(\sqrt{d_\text{elu}Λ}+d_\text{elu})$ is unavoidable when $\sqrt{d_\text{elu}Λ}+d_\text{elu}\leq\sqrt{AT}$. In this setting, we provide an upper bound of order $\tilde{O}(d_\text{elu}\sqrtΛ+d_\text{elu})$. Furthermore, we examine the setting where the function class additionally provides distributional information of the reward, as studied by Wang et al. (2024). We demonstrate that the regret bound $\tilde{O}(\sqrt{d_\text{elu}Λ}+d_\text{elu})$ established in their work is unimprovable when $\sqrt{d_{\text{elu}}Λ}+d_\text{elu}\leq\sqrt{AT}$. However, with a slightly different definition of the total variance and with the assumption that the reward follows a Gaussian distribution, one can achieve a regret of $\tilde{O}(\sqrt{AΛ}+d_\text{elu})$.

Dylan J. Foster, Yanjun Han, Jian Qian et al. · mit

$ $The classical theory of statistical estimation aims to estimate a parameter of interest under data generated from a fixed design ("offline estimation"), while the contemporary theory of online learning provides algorithms for estimation under adaptively chosen covariates ("online estimation"). Motivated by connections between estimation and interactive decision making, we ask: is it possible to convert offline estimation algorithms into online estimation algorithms in a black-box fashion? We investigate this question from an information-theoretic perspective by introducing a new framework, Oracle-Efficient Online Estimation (OEOE), where the learner can only interact with the data stream indirectly through a sequence of offline estimators produced by a black-box algorithm operating on the stream. Our main results settle the statistical and computational complexity of online estimation in this framework. $\bullet$ Statistical complexity. We show that information-theoretically, there exist algorithms that achieve near-optimal online estimation error via black-box offline estimation oracles, and give a nearly-tight characterization for minimax rates in the OEOE framework. $\bullet$ Computational complexity. We show that the guarantees above cannot be achieved in a computationally efficient fashion in general, but give a refined characterization for the special case of conditional density estimation: computationally efficient online estimation via black-box offline estimation is possible whenever it is possible via unrestricted algorithms. Finally, we apply our results to give offline oracle-efficient algorithms for interactive decision making.

Jian Qian, Alexander Rakhlin, Nikita Zhivotovskiy

We revisit the sequential variants of linear regression with the squared loss, classification problems with hinge loss, and logistic regression, all characterized by unbounded losses in the setup where no assumptions are made on the magnitude of design vectors and the norm of the optimal vector of parameters. The key distinction from existing results lies in our assumption that the set of design vectors is known in advance (though their order is not), a setup sometimes referred to as transductive online learning. While this assumption seems similar to fixed design regression or denoising, we demonstrate that the sequential nature of our algorithms allows us to convert our bounds into statistical ones with random design without making any additional assumptions about the distribution of the design vectors--an impossibility for standard denoising results. Our key tools are based on the exponential weights algorithm with carefully chosen transductive (design-dependent) priors, which exploit the full horizon of the design vectors. Our classification regret bounds have a feature that is only attributed to bounded losses in the literature: they depend solely on the dimension of the parameter space and on the number of rounds, independent of the design vectors or the norm of the optimal solution. For linear regression with squared loss, we further extend our analysis to the sparse case, providing sparsity regret bounds that additionally depend on the magnitude of the response variables. We argue that these improved bounds are specific to the transductive setting and unattainable in the worst-case sequential setup. Our algorithms, in several cases, have polynomial time approximations and reduce to sampling with respect to log-concave measures instead of aggregating over hard-to-construct $\varepsilon$-covers of classes.

Jian Qian, Teck Lun Goh, Bingyu Xie et al.

Biological signals, such as electroencephalograms (EEGs) and electrocardiograms (ECGs), play a pivotal role in numerous clinical practices, such as diagnosing brain and cardiac arrhythmic diseases. Existing methods for biosignal classification rely on Attention-based frameworks with dense Feed Forward layers, which lead to inefficient learning, high computational overhead, and suboptimal performance. In this work, we introduce BioMamba, a Spectro-Temporal Embedding strategy applied to the Bidirectional Mamba framework with Sparse Feed Forward layers to enable effective learning of biosignal sequences. By integrating these three key components, BioMamba effectively addresses the limitations of existing methods. Extensive experiments demonstrate that BioMamba significantly outperforms state-of-the-art methods with marked improvement in classification performance. The advantages of the proposed BioMamba include (1) Reliability: BioMamba consistently delivers robust results, confirmed across six evaluation metrics. (2) Efficiency: We assess both model and training efficiency, the BioMamba demonstrates computational effectiveness by reducing model size and resource consumption compared to existing approaches. (3) Generality: With the capacity to effectively classify a diverse set of tasks, BioMamba demonstrates adaptability and effectiveness across various domains and applications.

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.

Jian Qian, Jiachen Xu

Leave-one-out (LOO) prediction provides a principled, data-dependent measure of generalization, yet guarantees in fully transductive settings remain poorly understood beyond specialized models. We introduce Median of Level-Set Aggregation (MLSA), a general aggregation procedure based on empirical-risk level sets around the ERM. For arbitrary fixed datasets and losses satisfying a mild monotonicity condition, we establish a multiplicative oracle inequality for the LOO error of the form \[ LOO_S(\hat{h}) \;\le\; C \cdot \frac{1}{n} \min_{h\in H} L_S(h) \;+\; \frac{Comp(S,H,\ell)}{n}, \qquad C>1. \] The analysis is based on a local level-set growth condition controlling how the set of near-optimal empirical-risk minimizers expands as the tolerance increases. We verify this condition in several canonical settings. For classification with VC classes under the 0-1 loss, the resulting complexity scales as $O(d \log n)$, where $d$ is the VC dimension. For finite hypothesis and density classes under bounded or log loss, it scales as $O(\log |H|)$ and $O(\log |P|)$, respectively. For logistic regression with bounded covariates and parameters, a volumetric argument based on the empirical covariance matrix yields complexity scaling as $O(d \log n)$ up to problem-dependent factors.

Jian Qian, Shu Ge

Despite the widespread use of boosting in structured prediction, a general theoretical understanding of aggregation beyond scalar losses remains incomplete. We study vector-valued and conditional density prediction under general divergences and identify stability conditions under which aggregation amplifies weak guarantees into strong ones. We formalize this stability property as \emph{$(α,β)$-boostability}. We show that geometric median aggregation achieves $(α,β)$-boostability for a broad class of divergences, with tradeoffs that depend on the underlying geometry. For vector-valued prediction and conditional density estimation, we characterize boostability under common divergences ($\ell_1$, $\ell_2$, $\TV$, and $\Hel$) with geometric median, revealing a sharp distinction between dimension-dependent and dimension-free regimes. We further show that while KL divergence is not directly boostable via geometric median aggregation, it can be handled indirectly through boostability under Hellinger distance. Building on these structural results, we propose a generic boosting framework \textsc{GeoMedBoost} based on exponential reweighting and geometric-median aggregation. Under a weak learner condition and $(α,β)$-boostability, we obtain exponential decay of the empirical divergence exceedance error. Our framework recovers classical algorithms such as \textsc{MedBoost}, \textsc{AdaBoost}, and \textsc{SAMME} as special cases, and provides a unified geometric view of boosting for structured prediction.

Wenlong Mou, Jian Qian

Bootstrapping and rollout are two fundamental principles for value function estimation in reinforcement learning (RL). We introduce a novel class of Bellman operators, called subgraph Bellman operators, that interpolate between bootstrapping and rollout methods. Our estimator, derived by solving the fixed point of the empirical subgraph Bellman operator, combines the strengths of the bootstrapping-based temporal difference (TD) estimator and the rollout-based Monte Carlo (MC) methods. Specifically, the error upper bound of our estimator approaches the optimal variance achieved by TD, with an additional term depending on the exit probability of a selected subset of the state space. At the same time, the estimator exhibits the finite-sample adaptivity of MC, with sample complexity depending only on the occupancy measure of this subset. We complement the upper bound with an information-theoretic lower bound, showing that the additional term is unavoidable given a reasonable sample size. Together, these results establish subgraph Bellman estimators as an optimal and adaptive framework for reconciling TD and MC methods in policy evaluation.

Dylan J. Foster, Sham M. Kakade, Jian Qian et al.

A fundamental challenge in interactive learning and decision making, ranging from bandit problems to reinforcement learning, is to provide sample-efficient, adaptive learning algorithms that achieve near-optimal regret. This question is analogous to the classical problem of optimal (supervised) statistical learning, where there are well-known complexity measures (e.g., VC dimension and Rademacher complexity) that govern the statistical complexity of learning. However, characterizing the statistical complexity of interactive learning is substantially more challenging due to the adaptive nature of the problem. The main result of this work provides a complexity measure, the Decision-Estimation Coefficient, that is proven to be both necessary and sufficient for sample-efficient interactive learning. In particular, we provide: 1. a lower bound on the optimal regret for any interactive decision making problem, establishing the Decision-Estimation Coefficient as a fundamental limit. 2. a unified algorithm design principle, Estimation-to-Decisions (E2D), which transforms any algorithm for supervised estimation into an online algorithm for decision making. E2D attains a regret bound that matches our lower bound up to dependence on a notion of estimation performance, thereby achieving optimal sample-efficient learning as characterized by the Decision-Estimation Coefficient. Taken together, these results constitute a theory of learnability for interactive decision making. When applied to reinforcement learning settings, the Decision-Estimation Coefficient recovers essentially all existing hardness results and lower bounds. More broadly, the approach can be viewed as a decision-theoretic analogue of the classical Le Cam theory of statistical estimation; it also unifies a number of existing approaches -- both Bayesian and frequentist.

Avrim Blum, Steve Hanneke, Jian Qian et al.

We study the problem of robust learning under clean-label data-poisoning attacks, where the attacker injects (an arbitrary set of) correctly-labeled examples to the training set to fool the algorithm into making mistakes on specific test instances at test time. The learning goal is to minimize the attackable rate (the probability mass of attackable test instances), which is more difficult than optimal PAC learning. As we show, any robust algorithm with diminishing attackable rate can achieve the optimal dependence on $ε$ in its PAC sample complexity, i.e., $O(1/ε)$. On the other hand, the attackable rate might be large even for some optimal PAC learners, e.g., SVM for linear classifiers. Furthermore, we show that the class of linear hypotheses is not robustly learnable when the data distribution has zero margin and is robustly learnable in the case of positive margin but requires sample complexity exponential in the dimension. For a general hypothesis class with bounded VC dimension, if the attacker is limited to add at most $t>0$ poison examples, the optimal robust learning sample complexity grows almost linearly with $t$.

Xuedong Shang, Han Shao, Jian Qian

Multi-armed bandits are widely applied in scenarios like recommender systems, for which the goal is to maximize the click rate. However, more factors should be considered, e.g., user stickiness, user growth rate, user experience assessment, etc. In this paper, we model this situation as a problem of $K$-armed bandit with multiple losses. We define relative loss vector of an arm where the $i$-th entry compares the arm and the optimal arm with respect to the $i$-th loss. We study two goals: (a) finding the arm with the minimum $\ell^\infty$-norm of relative losses with a given confidence level (which refers to fixed-confidence best-arm identification); (b) minimizing the $\ell^\infty$-norm of cumulative relative losses (which refers to regret minimization). For goal (a), we derive a problem-dependent sample complexity lower bound and discuss how to achieve matching algorithms. For goal (b), we provide a regret lower bound of $Ω(T^{2/3})$ and provide a matching algorithm.

Yi Tian, Jian Qian, Suvrit Sra

We study minimax optimal reinforcement learning in episodic factored Markov decision processes (FMDPs), which are MDPs with conditionally independent transition components. Assuming the factorization is known, we propose two model-based algorithms. The first one achieves minimax optimal regret guarantees for a rich class of factored structures, while the second one enjoys better computational complexity with a slightly worse regret. A key new ingredient of our algorithms is the design of a bonus term to guide exploration. We complement our algorithms by presenting several structure-dependent lower bounds on regret for FMDPs that reveal the difficulty hiding in the intricacy of the structures.

Jian Qian, Ronan Fruit, Matteo Pirotta et al.

We investigate concentration inequalities for Dirichlet and Multinomial random variables.

Matthew Schlegel, Wesley Chung, Daniel Graves et al.

Importance sampling (IS) is a common reweighting strategy for off-policy prediction in reinforcement learning. While it is consistent and unbiased, it can result in high variance updates to the weights for the value function. In this work, we explore a resampling strategy as an alternative to reweighting. We propose Importance Resampling (IR) for off-policy prediction, which resamples experience from a replay buffer and applies standard on-policy updates. The approach avoids using importance sampling ratios in the update, instead correcting the distribution before the update. We characterize the bias and consistency of IR, particularly compared to Weighted IS (WIS). We demonstrate in several microworlds that IR has improved sample efficiency and lower variance updates, as compared to IS and several variance-reduced IS strategies, including variants of WIS and V-trace which clips IS ratios. We also provide a demonstration showing IR improves over IS for learning a value function from images in a racing car simulator.

Jian Qian, Ronan Fruit, Matteo Pirotta et al.

We introduce and analyse two algorithms for exploration-exploitation in discrete and continuous Markov Decision Processes (MDPs) based on exploration bonuses. SCAL$^+$ is a variant of SCAL (Fruit et al., 2018) that performs efficient exploration-exploitation in any unknown weakly-communicating MDP for which an upper bound C on the span of the optimal bias function is known. For an MDP with $S$ states, $A$ actions and $Γ\leq S$ possible next states, we prove that SCAL$^+$ achieves the same theoretical guarantees as SCAL (i.e., a high probability regret bound of $\widetilde{O}(C\sqrt{ΓSAT})$), with a much smaller computational complexity. Similarly, C-SCAL$^+$ exploits an exploration bonus to achieve sublinear regret in any undiscounted MDP with continuous state space. We show that C-SCAL$^+$ achieves the same regret bound as UCCRL (Ortner and Ryabko, 2012) while being the first implementable algorithm with regret guarantees in this setting. While optimistic algorithms such as UCRL, SCAL or UCCRL maintain a high-confidence set of plausible MDPs around the true unknown MDP, SCAL$^+$ and C-SCAL$^+$ leverage on an exploration bonus to directly plan on the empirically estimated MDP, thus being more computationally efficient.