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.

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.

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.