arXiv subjects

Martin J. Wainwright

Publications and source records attributed to Martin J. Wainwright.

At least 37 records · Page 2Linked to original sources

arXiv2024

Optimal and instance-dependent guarantees for Markovian linear stochastic approximation

We study stochastic approximation procedures for approximately solving a $d$-dimensional linear fixed point equation based on observing a trajectory of length $n$ from an ergodic Markov chain. We first exhibit a non-asymptotic bound of the order $t_{\mathrm{mix}} \tfrac{d}{n}$ on the squared error of the last iterate of a standard scheme, where $t_{\mathrm{mix}}$ is a mixing time. We then prove a non-asymptotic instance-dependent bound on a suitably averaged sequence of iterates, with a leading term that matches the local asymptotic minimax limit, including sharp dependence on the parameters $(d, t_{\mathrm{mix}})$ in the higher order terms. We complement these upper bounds with a non-asymptotic minimax lower bound that establishes the instance-optimality of the averaged SA estimator. We derive corollaries of these results for policy evaluation with Markov noise -- covering the TD($λ$) family of algorithms for all $λ\in [0, 1)$ -- and linear autoregressive models. Our instance-dependent characterizations open the door to the design of fine-grained model selection procedures for hyperparameter tuning (e.g., choosing the value of $λ$ when running the TD($λ$) algorithm).

math.OC↗

arXiv2024

Taming "data-hungry" reinforcement learning? Stability in continuous state-action spaces

We introduce a novel framework for analyzing reinforcement learning (RL) in continuous state-action spaces, and use it to prove fast rates of convergence in both off-line and on-line settings. Our analysis highlights two key stability properties, relating to how changes in value functions and/or policies affect the Bellman operator and occupation measures. We argue that these properties are satisfied in many continuous state-action Markov decision processes, and demonstrate how they arise naturally when using linear function approximation methods. Our analysis offers fresh perspectives on the roles of pessimism and optimism in off-line and on-line RL, and highlights the connection between off-line RL and transfer learning.

cs.LG↗

arXiv2023

A decorrelation method for general regression adjustment in randomized experiments

We study regression adjustment with general function class approximations for estimating the average treatment effect in the design-based setting. Standard regression adjustment involves bias due to sample re-use, and this bias leads to behavior that is sub-optimal in the sample size, and/or imposes restrictive assumptions. Our main contribution is to introduce a novel decorrelation-based approach that circumvents these issues. We prove guarantees, both asymptotic and non-asymptotic, relative to the oracle functions that are targeted by a given regression adjustment procedure. We illustrate our method by applying it to various high-dimensional and non-parametric problems, exhibiting improved sample complexity and weakened assumptions relative to known approaches.

stat.ME↗

arXiv2023

Revisiting minimum description length complexity in overparameterized models

Complexity is a fundamental concept underlying statistical learning theory that aims to inform generalization performance. Parameter count, while successful in low-dimensional settings, is not well-justified for overparameterized settings when the number of parameters is more than the number of training samples. We revisit complexity measures based on Rissanen's principle of minimum description length (MDL) and define a novel MDL-based complexity (MDL-COMP) that remains valid for overparameterized models. MDL-COMP is defined via an optimality criterion over the encodings induced by a good Ridge estimator class. We provide an extensive theoretical characterization of MDL-COMP for linear models and kernel methods and show that it is not just a function of parameter count, but rather a function of the singular values of the design or the kernel matrix and the signal-to-noise ratio. For a linear model with $n$ observations, $d$ parameters, and i.i.d. Gaussian predictors, MDL-COMP scales linearly with $d$ when $d n$. For kernel methods, we show that MDL-COMP informs minimax in-sample error, and can decrease as the dimensionality of the input increases. We also prove that MDL-COMP upper bounds the in-sample mean squared error (MSE). Via an array of simulations and real-data experiments, we show that a data-driven Prac-MDL-COMP informs hyper-parameter tuning for optimizing test MSE with ridge regression in limited data settings, sometimes improving upon cross-validation and (always) saving computational costs. Finally, our findings also suggest that the recently observed double decent phenomenons in overparameterized models might be a consequence of the choice of non-ideal estimators.

cs.LG↗

arXiv2023

Doubly High-Dimensional Contextual Bandits: An Interpretable Model for Joint Assortment-Pricing

Key challenges in running a retail business include how to select products to present to consumers (the assortment problem), and how to price products (the pricing problem) to maximize revenue or profit. Instead of considering these problems in isolation, we propose a joint approach to assortment-pricing based on contextual bandits. Our model is doubly high-dimensional, in that both context vectors and actions are allowed to take values in high-dimensional spaces. In order to circumvent the curse of dimensionality, we propose a simple yet flexible model that captures the interactions between covariates and actions via a (near) low-rank representation matrix. The resulting class of models is reasonably expressive while remaining interpretable through latent factors, and includes various structured linear bandit and pricing models as particular cases. We propose a computationally tractable procedure that combines an exploration/exploitation protocol with an efficient low-rank matrix estimator, and we prove bounds on its regret. Simulation results show that this method has lower regret than state-of-the-art methods applied to various standard bandit and pricing models. Real-world case studies on the assortment-pricing problem, from an industry-leading instant noodles company to an emerging beauty start-up, underscore the gains achievable using our method. In each case, we show at least three-fold gains in revenue or profit by our bandit method, as well as the interpretability of the latent factor models that are learned.

stat.ML↗

arXiv2023

Challenges of the inconsistency regime: Novel debiasing methods for missing data models

We study semi-parametric estimation of the population mean when data is observed missing at random (MAR) in the $n < p$ "inconsistency regime", in which neither the outcome model nor the propensity/missingness model can be estimated consistently. Consider a high-dimensional linear-GLM specification in which the number of confounders is proportional to the sample size. In the case $n > p$, past work has developed theory for the classical AIPW estimator in this model and established its variance inflation and asymptotic normality when the outcome model is fit by ordinary least squares. Ordinary least squares is no longer feasible in the case $n < p$ studied here, and we also demonstrate that a number of classical debiasing procedures become inconsistent. This challenge motivates our development and analysis of a novel procedure: we establish that it is consistent for the population mean under proportional asymptotics allowing for $n < p$, and also provide confidence intervals for the linear model coefficients. Providing such guarantees in the inconsistency regime requires a new debiasing approach that combines penalized M-estimates of both the outcome and propensity/missingness models in a non-standard way.

math.ST↗

arXiv2023

Optimally tackling covariate shift in RKHS-based nonparametric regression

We study the covariate shift problem in the context of nonparametric regression over a reproducing kernel Hilbert space (RKHS). We focus on two natural families of covariate shift problems defined using the likelihood ratios between the source and target distributions. When the likelihood ratios are uniformly bounded, we prove that the kernel ridge regression (KRR) estimator with a carefully chosen regularization parameter is minimax rate-optimal (up to a log factor) for a large family of RKHSs with regular kernel eigenvalues. Interestingly, KRR does not require full knowledge of likelihood ratios apart from an upper bound on them. In striking contrast to the standard statistical setting without covariate shift, we also demonstrate that a naive estimator, which minimizes the empirical risk over the function class, is strictly sub-optimal under covariate shift as compared to KRR. We then address the larger class of covariate shift problems where the likelihood ratio is possibly unbounded yet has a finite second moment. Here, we propose a reweighted KRR estimator that weights samples based on a careful truncation of the likelihood ratios. Again, we are able to show that this estimator is minimax rate-optimal, up to logarithmic factors.

math.ST↗

arXiv2023

When is the estimated propensity score better? High-dimensional analysis and bias correction

Anecdotally, using an estimated propensity score is superior to the true propensity score in estimating the average treatment effect based on observational data. However, this claim comes with several qualifications: it holds only if propensity score model is correctly specified and the number of covariates $d$ is small relative to the sample size $n$. We revisit this phenomenon by studying the inverse propensity score weighting (IPW) estimator based on a logistic model with a diverging number of covariates. We first show that the IPW estimator based on the estimated propensity score is consistent and asymptotically normal with smaller variance than the oracle IPW estimator (using the true propensity score) if and only if $n \gtrsim d^2$. We then propose a debiased IPW estimator that achieves the same guarantees in the regime $n \gtrsim d^{3/2}$. Our proofs rely on a novel non-asymptotic decomposition of the IPW error along with careful control of the higher order terms.

stat.ME↗

arXiv2023

Noisy recovery from random linear observations: Sharp minimax rates under elliptical constraints

Estimation problems with constrained parameter spaces arise in various settings. In many of these problems, the observations available to the statistician can be modelled as arising from the noisy realization of the image of a random linear operator; an important special case is random design regression. We derive sharp rates of estimation for arbitrary compact elliptical parameter sets and demonstrate how they depend on the distribution of the random linear operator. Our main result is a functional that characterizes the minimax rate of estimation in terms of the noise level, the law of the random operator, and elliptical norms that define the error metric and the parameter space. This nonasymptotic result is sharp up to an explicit universal constant, and it becomes asymptotically exact as the radius of the parameter space is allowed to grow. We demonstrate the generality of the result by applying it to both parametric and nonparametric regression problems, including those involving distribution shift or dependent covariates.

math.ST↗

arXiv2023

Near-optimal inference in adaptive linear regression

When data is collected in an adaptive manner, even simple methods like ordinary least squares can exhibit non-normal asymptotic behavior. As an undesirable consequence, hypothesis tests and confidence intervals based on asymptotic normality can lead to erroneous results. We propose a family of online debiasing estimators to correct these distributional anomalies in least squares estimation. Our proposed methods take advantage of the covariance structure present in the dataset and provide sharper estimates in directions for which more information has accrued. We establish an asymptotic normality property for our proposed online debiasing estimators under mild conditions on the data collection process and provide asymptotically exact confidence intervals. We additionally prove a minimax lower bound for the adaptive linear regression problem, thereby providing a baseline by which to compare estimators. There are various conditions under which our proposed estimators achieve the minimax lower bound. We demonstrate the usefulness of our theory via applications to multi-armed bandit, autoregressive time series estimation, and active learning with exploration.

math.ST↗

arXiv2023

Kernel-based off-policy estimation without overlap: Instance optimality beyond semiparametric efficiency

We study optimal procedures for estimating a linear functional based on observational data. In many problems of this kind, a widely used assumption is strict overlap, i.e., uniform boundedness of the importance ratio, which measures how well the observational data covers the directions of interest. When it is violated, the classical semi-parametric efficiency bound can easily become infinite, so that the instance-optimal risk depends on the function class used to model the regression function. For any convex and symmetric function class $\mathcal{F}$, we derive a non-asymptotic local minimax bound on the mean-squared error in estimating a broad class of linear functionals. This lower bound refines the classical semi-parametric one, and makes connections to moduli of continuity in functional estimation. When $\mathcal{F}$ is a reproducing kernel Hilbert space, we prove that this lower bound can be achieved up to a constant factor by analyzing a computationally simple regression estimator. We apply our general results to various families of examples, thereby uncovering a spectrum of rates that interpolate between the classical theories of semi-parametric efficiency (with $\sqrt{n}$-consistency) and the slower minimax rates associated with non-parametric function estimation.

math.ST↗

arXiv2022

Optimal variance-reduced stochastic approximation in Banach spaces

We study the problem of estimating the fixed point of a contractive operator defined on a separable Banach space. Focusing on a stochastic query model that provides noisy evaluations of the operator, we analyze a variance-reduced stochastic approximation scheme, and establish non-asymptotic bounds for both the operator defect and the estimation error, measured in an arbitrary semi-norm. In contrast to worst-case guarantees, our bounds are instance-dependent, and achieve the local asymptotic minimax risk non-asymptotically. For linear operators, contractivity can be relaxed to multi-step contractivity, so that the theory can be applied to problems like average reward policy evaluation problem in reinforcement learning. We illustrate the theory via applications to stochastic shortest path problems, two-player zero-sum Markov games, as well as policy evaluation and $Q$-learning for tabular Markov decision processes.

math.ST↗

arXiv2022

Policy evaluation from a single path: Multi-step methods, mixing and mis-specification

We study non-parametric estimation of the value function of an infinite-horizon $γ$-discounted Markov reward process (MRP) using observations from a single trajectory. We provide non-asymptotic guarantees for a general family of kernel-based multi-step temporal difference (TD) estimates, including canonical $K$-step look-ahead TD for $K = 1, 2, \ldots$ and the TD$(λ)$ family for $λ\in [0,1)$ as special cases. Our bounds capture its dependence on Bellman fluctuations, mixing time of the Markov chain, any mis-specification in the model, as well as the choice of weight function defining the estimator itself, and reveal some delicate interactions between mixing time and model mis-specification. For a given TD method applied to a well-specified model, its statistical error under trajectory data is similar to that of i.i.d. sample transition pairs, whereas under mis-specification, temporal dependence in data inflates the statistical error. However, any such deterioration can be mitigated by increased look-ahead. We complement our upper bounds by proving minimax lower bounds that establish optimality of TD-based methods with appropriately chosen look-ahead and weighting, and reveal some fundamental differences between value function estimation and ordinary non-parametric regression.

stat.ML↗

arXiv2022

Krylov-Bellman boosting: Super-linear policy evaluation in general state spaces

We present and analyze the Krylov-Bellman Boosting (KBB) algorithm for policy evaluation in general state spaces. It alternates between fitting the Bellman residual using non-parametric regression (as in boosting), and estimating the value function via the least-squares temporal difference (LSTD) procedure applied with a feature set that grows adaptively over time. By exploiting the connection to Krylov methods, we equip this method with two attractive guarantees. First, we provide a general convergence bound that allows for separate estimation errors in residual fitting and LSTD computation. Consistent with our numerical experiments, this bound shows that convergence rates depend on the restricted spectral structure, and are typically super-linear. Second, by combining this meta-result with sample-size dependent guarantees for residual fitting and LSTD computation, we obtain concrete statistical guarantees that depend on the sample size along with the complexity of the function class used to fit the residuals. We illustrate the behavior of the KBB algorithm for various types of policy evaluation problems, and typically find large reductions in sample complexity relative to the standard approach of fitted value iterationn.

stat.ML↗

arXiv2022

Bellman Residual Orthogonalization for Offline Reinforcement Learning

We propose and analyze a reinforcement learning principle that approximates the Bellman equations by enforcing their validity only along an user-defined space of test functions. Focusing on applications to model-free offline RL with function approximation, we exploit this principle to derive confidence intervals for off-policy evaluation, as well as to optimize over policies within a prescribed policy class. We prove an oracle inequality on our policy optimization procedure in terms of a trade-off between the value and uncertainty of an arbitrary comparator policy. Different choices of test function spaces allow us to tackle different problems within a common framework. We characterize the loss of efficiency in moving from on-policy to off-policy data using our procedures, and establish connections to concentrability coefficients studied in past work. We examine in depth the implementation of our methods with linear function approximation, and provide theoretical guarantees with polynomial-time implementations even when Bellman closure does not hold.

cs.LG↗

arXiv2022

Off-policy estimation of linear functionals: Non-asymptotic theory for semi-parametric efficiency

The problem of estimating a linear functional based on observational data is canonical in both the causal inference and bandit literatures. We analyze a broad class of two-stage procedures that first estimate the treatment effect function, and then use this quantity to estimate the linear functional. We prove non-asymptotic upper bounds on the mean-squared error of such procedures: these bounds reveal that in order to obtain non-asymptotically optimal procedures, the error in estimating the treatment effect should be minimized in a certain weighted $L^2$-norm. We analyze a two-stage procedure based on constrained regression in this weighted norm, and establish its instance-dependent optimality in finite samples via matching non-asymptotic local minimax lower bounds. These results show that the optimal non-asymptotic risk, in addition to depending on the asymptotically efficient variance, depends on the weighted norm distance between the true outcome function and its approximation by the richest function class supported by the sample size.

math.ST↗

arXiv2022

A Diffusion Process Perspective on Posterior Contraction Rates for Parameters

We analyze the posterior contraction rates of parameters in Bayesian models via the Langevin diffusion process, in particular by controlling moments of the stochastic process and taking limits. Analogous to the non-asymptotic analysis of statistical M-estimators and stochastic optimization algorithms, our contraction rates depend on the structure of the population log-likelihood function, and stochastic perturbation bounds between the population and sample log-likelihood functions. Convergence rates are determined by a non-linear equation that relates the population-level structure to stochastic perturbation terms, along with a term characterizing the diffusive behavior. Based on this technique, we also prove non-asymptotic versions of a Bernstein-von-Mises guarantee for the posterior. We illustrate this general theory by deriving posterior convergence rates for various concrete examples, as well as approximate posterior distributions computed using Langevin sampling procedures.

math.ST↗

arXiv2022

Stabilizing Q-learning with Linear Architectures for Provably Efficient Learning

The $Q$-learning algorithm is a simple and widely-used stochastic approximation scheme for reinforcement learning, but the basic protocol can exhibit instability in conjunction with function approximation. Such instability can be observed even with linear function approximation. In practice, tools such as target networks and experience replay appear to be essential, but the individual contribution of each of these mechanisms is not well understood theoretically. This work proposes an exploration variant of the basic $Q$-learning protocol with linear function approximation. Our modular analysis illustrates the role played by each algorithmic tool that we adopt: a second order update rule, a set of target networks, and a mechanism akin to experience replay. Together, they enable state of the art regret bounds on linear MDPs while preserving the most prominent feature of the algorithm, namely a space complexity independent of the number of step elapsed. We show that the performance of the algorithm degrades very gracefully under a novel and more permissive notion of approximation error. The algorithm also exhibits a form of instance-dependence, in that its performance depends on the "effective" feature dimension.

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