Previous studies on two-timescale stochastic approximation (SA) mainly focused on bounding mean-squared errors under diminishing stepsize schemes. In this work, we investigate {\it constant} stpesize schemes through the lens of Markov processes, proving that the iterates of both timescales converge to a unique joint stationary distribution in Wasserstein metric. We derive explicit geometric and non-asymptotic convergence rates, as well as the variance and bias introduced by constant stepsizes in the presence of Markovian noise. Specifically, with two constant stepsizes $\alpha < \beta$, we show that the biases scale linearly with both stepsizes as $\Theta(\alpha)+\Theta(\beta)$ up to higher-order terms, while the variance of the slower iterate (resp., faster iterate) scales only with its own stepsize as $O(\alpha)$ (resp., $O(\beta)$). Unlike previous work, our results require no additional assumptions such as $\beta^2 \ll \alpha$ nor extra dependence on dimensions. These fine-grained characterizations allow tail-averaging and extrapolation techniques to reduce variance and bias, improving mean-squared error bound to $O(\beta^4 + \frac{1}{t})$ for both iterates.

We consider off-policy selection and learning in contextual bandits, where the learner aims to select or train a reward-maximizing policy using data collected by a fixed behavior policy. Our contribution is two-fold. First, we propose a novel off-policy selection method that leverages a new betting-based confidence bound applied to an inverse propensity weight sequence. Our theoretical analysis reveals that this method achieves a significantly improved, variance-adaptive guarantee over prior work. Second, we propose a novel and generic condition on the optimization objective for off-policy learning that strikes a different balance between bias and variance. One special case, which we call freezing, tends to induce low variance, which is preferred in small-data regimes. Our analysis shows that it matches the best existing guarantees. In our empirical study, our selection method outperforms existing methods, and freezing exhibits improved performance in small-sample regimes.

Contextual multi-armed bandits are a popular choice to model sequential decision-making. E.g., in a healthcare application we may perform various tests to asses a patient condition (exploration) and then decide on the best treatment to give (exploitation). When human design strategies, they aim for the exploration to be fast, since the patient’s health is at stake, and easy to interpret for a physician overseeing the process. However, common bandit algorithms are nothing like that: The regret caused by exploration scales with $\sqrt{H}$ over $H$ rounds and decision strategies are based on opaque statistical considerations. In this paper, we use an original classification view to meta learn interpretable and fast exploration plans for a fixed collection of bandits $\mathbb{M}$. The plan is prescribed by an interpretable decision tree probing decisions’ payoff to classify the test bandit. The test regret of the plan in the stochastic and contextual setting scales with $O (\lambda^{-2} C_{\lambda} (\mathbb{M}) \log^2 (MH))$, being $M$ the size of $\mathbb{M}$, $\lambda$ a separation parameter over the bandits, and $C_\lambda (\mathbb{M})$ a novel classification-coefficient that fundamentally links meta learning bandits with classification. Through a nearly matching lower bound, we show that $C_\lambda (\mathbb{M})$ inherently captures the complexity of the setting.

We study offline policy optimization for infinite-horizon average-reward Markov decision processes (MDPs) with large or infinite state spaces. Specifically, we propose a pessimistic actor-critic algorithm that uses a computationally efficient linear function class for value function estimation. At the core of our method is a critic that computes a pessimistic estimate of the average reward under the current policy, as well as the corresponding policy gradient, by solving a fixed-point Bellman equation, rather than solving a successive sequence of regression problems as in finite horizon settings. This procedure reduces to solving a second-order cone program, which is computationally tractable. Our theoretical analysis is based on a weak partial data coverage assumption, which requires only that the offline data aligns well with the expected feature vector of a comparator policy. Under this condition, we show that our algorithm achieves the optimal sample complexity of O(\varepsilon^{-2}) for learning a near-optimal policy, up to model misspecification errors.

In this work, we study first-order algorithms for solving Bilevel Optimization (BO) where the objective functions are smooth but possibly nonconvex in both levels and the variables are restricted to closed convex sets. As a first step, we study the landscape of BO through the lens of penalty methods, in which the upper- and lower-level objectives are combined in a weighted sum with penalty parameter $\sigma > 0$. In particular, we establish a strong connection between the penalty function and the hyper-objective by explicitly characterizing the conditions under which the values and derivatives of the two must be $O(\sigma)$-close. A by-product of our analysis is the explicit formula for the gradient of hyper-objective when the lower-level problem has multiple solutions under minimal conditions, which could be of independent interest. Next, viewing the penalty formulation as $O(\sigma)$-approximation of the original BO, we propose first-order algorithms that find an $\epsilon$-stationary solution by optimizing the penalty formulation with $\sigma = O(\epsilon)$. When the perturbed lower-level problem uniformly satisfies the {\it small-error} proximal error-bound (EB) condition, we propose a first-order algorithm that converges to an $\epsilon$-stationary point of the penalty function using in total $O(\epsilon^{-7})$ accesses to first-order stochastic gradient oracles. Under an additional assumption on stochastic oracles, we show that the algorithm can be implemented in a fully {\it single-loop} manner, {\it i.e.,} with $O(1)$ samples per iteration, and achieves the improved oracle-complexity of $O(\epsilon^{-5})$.

In many interactive decision-making problems, there is contextual side information that remains fixed within the course of an interaction. This problem has been studied quite extensively under the assumption the context is fully observed, as well as in the opposing limit when the context is unobserved, a special type of POMDP also referred to as a Latent MDP (LMDP). In this work, we consider a class of decision problems that interpolates between the settings, namely, between the case the context is fully observed, and the case the context is unobserved. We refer to this class of decision problems as *LMDPs with prospective side information*. In such an environment an agent receives additional, weakly revealing, information on the latent context at the beginning of each episode. We show that, surprisingly, this problem is not captured by contemporary POMDP settings and is not solved by RL algorithms designed for partially observed environments. We then establish that any sample efficient algorithm must suffer at least $\Omega(K^{2/3})$-regret, as opposed to standard $\Omega(\sqrt{K})$ lower bounds. We design an algorithm with a matching upper bound that depends only polynomially on the problem parameters. This establishes exponential improvement in the sample complexity relative to the existing LMDP lower bound, when prospective information is not given in prior work.

We consider the problem of finding stationary points in Bilevel optimization when the lower-level problem is unconstrained and strongly convex. The problem has been extensively studied in recent years; the main technical challenge is to keep track of lower-level solutions $y^*(x)$ in response to the changes in the upper-level variables $x$. Subsequently, all existing approaches tie their analyses to a genie algorithm that knows lower-level solutions and, therefore, need not query any points far from them. We consider a dual question to such approaches: suppose we have an oracle, which we call $y^*$-aware, that returns an $O(\epsilon)$-estimate of the lower-level solution, in addition to first-order gradient estimators locally unbiased within the $\Theta(\epsilon)$-ball around $y^*(x)$. We study the complexity of finding stationary points with such an $y^*$-aware oracle: we propose a simple first-order method that converges to an $\epsilon$ stationary point using $O(\epsilon^{-6}), O(\epsilon^{-4})$ access to first-order $y^*$-aware oracles. Our upper bounds also apply to standard unbiased first-order oracles, improving the best-known complexity of first-order methods by $O(\epsilon)$ with minimal assumptions. We then provide the matching $\Omega(\epsilon^{-6})$, $\Omega(\epsilon^{-4})$ lower bounds without and with an additional smoothness assumption, respectively. Our results imply that any approach that simulates an algorithm with an $y^*$-aware oracle must suffer the same lower bounds.

In many real-world decision problems there is partially observed, hidden or latent information that remains fixed throughout an interaction. Such decision problems can be modeled as Latent Markov Decision Processes (LMDPs), where a latent variable is selected at the beginning of an interaction and is not disclosed to the agent initially. In last decade, there has been significant progress in designing learning algorithms for solving LMDPs under different structural assumptions. However, for general LMDPs, there is no known learning algorithm that provably matches the existing lower bound. We effectively resolve this open question, introducing the first sample-efficient algorithm for LMDPs without *any additional structural assumptions*. Our result builds off a new perspective on the role off-policy evaluation guarantees and coverage coefficient in LMDPs, a perspective, which has been overlooked in the context of exploration in partially observed environments. Specifically, we establish a novel off-policy evaluation lemma and introduce a new coverage coefficient for LMDPs. Then, we show how these can be used to derive near-optimal guarantees of an optimistic exploration algorithm. These results, we believe, can be valuable for a wide range of interactive learning problems beyond the LMDP class, and especially, for partially observed environments.

We consider solving the low-rank matrix sensing problem with the Factorized Gradient Descent (FGD) method when the specified rank is larger than the true rank. We refer to this as over-parameterized matrix sensing. If the ground truth signal $\mathbf{X}^* \in \mathbb{R}^{d \times d}$ is of rank $r$, but we try to recover it using $\mathbf{F} \mathbf{F}^\top$ where $\mathbf{F} \in \mathbb{R}^{d \times k}$ and $k>r$, the existing statistical analysis either no longer holds or produces a vacuous statistical error upper bound (infinity) due to the flat local curvature of the loss function around the global maxima. By decomposing the factorized matrix $\mathbf{F}$ into separate column spaces to capture the impact of using $k > r$, we show that $\left\| {\mathbf{F}_t \mathbf{F}_t - \mathbf{X}^*} \right\|_F^2$ converges sub-linearly to a statistical error of $\tilde{\mathcal{O}} (k d \sigma^2/n)$ after $\tilde{\mathcal{O}}(\frac{\sigma_{r}}{\sigma}\sqrt{\frac{n}{d}})$ iterations, where $\mathbf{F}_t$ is the output of FGD after $t$ iterations, $\sigma^2$ is the variance of the observation noise, $\sigma_{r}$ is the $r$-th largest eigenvalue of $\mathbf{X}^*$, and $n$ is the number of samples. With a precise characterization of the convergence behavior and the statistical error, our results, therefore, offer a comprehensive picture of the statistical and computational complexity if we solve the over-parameterized matrix sensing problem with FGD.

Recent results established that EM enjoys global convergence for Gaussian Mixture Models. For Mixed Linear Regression, however, only local convergence results have been established, and those only for the high signal-to-noise ratio (SNR) regime. In this work, we completely characterize the global optimality of EM: we show that starting from any randomly initialized point, the EM algorithm converges to the true parameter${\beta }^{*}$at the minimax statistical rates under all SNR regimes. Toward this goal, we first show the global convergence of the EM algorithm at the population level. Then we provide a complete characterization of statistical and computational behaviors of EM under all SNR regimes with finite samples. In particular: (i) When the SNR is sufficiently large, the EM updates converge to the true parameter$ {\beta }^{*}$at the standard parametric convergence rate$O((d/n)^{1/2})$after$O(\log (n/d))$iterations. (ii) In the regime where the SNR is above$O((d/n)^{1/4})$and below some constant, the EM iterates converge to a$O({\mathrm { SNR}}^{-1} (d/n)^{1/2})$neighborhood of the true parameter, when the number of iterations is of the order$O({\mathrm { SNR}}^{-2} \log (n/d))$. (iii) In the low SNR regime where the SNR is below$O((d/n)^{1/4})$, we show that EM converges to a$O((d/n)^{1/4})$neighborhood of the true parameters, after$O((n/d)^{1/2})$iterations. By providing tight convergence guarantees of the EM algorithm in middle-to-low SNR regimes, we reveal that in low SNR, EM changes rate, matching the$n^{-1/4}$rate of the MLE, a behavior that previous work had been unable to show.

Modern machine learning models deployed in the wild can encounter both covariate and semantic shifts, giving rise to the problems of out-of-distribution (OOD) generalization and OOD detection respectively. While both problems have received significant research attention lately, they have been pursued independently. This may not be surprising, since the two tasks have seemingly conflicting goals. This paper provides a new unified approach that is capable of simultaneously generalizing to covariate shifts while robustly detecting semantic shifts. We propose a margin-based learning framework that exploits freely available unlabeled data in the wild that captures the environmental test-time OOD distributions under both covariate and semantic shifts. We show both empirically and theoretically that the proposed margin constraint is the key to achieving both OOD generalization and detection. Extensive experiments show the superiority of our framework, outperforming competitive baselines that specialize in either OOD generalization or OOD detection. Code is publicly available at https://github.com/deeplearning-wisc/scone.

We consider episodic reinforcement learning in reward-mixing Markov decision processes (RMMDPs): at the beginning of every episode nature randomly picks a latent reward model among $M$ candidates and an agent interacts with the MDP throughout the episode for $H$ time steps. Our goal is to learn a near-optimal policy that nearly maximizes the $H$ time-step cumulative rewards in such a model. Prior work established an upper bound for RMMDPs with $M=2$. In this work, we resolve several open questions for the general RMMDP setting. We consider an arbitrary $M\ge2$ and provide a sample-efficient algorithm–$EM^2$–that outputs an $\epsilon$-optimal policy using $O \left(\epsilon^{-2} \cdot S^d A^d \cdot \text{poly}(H, Z)^d \right)$ episodes, where $S, A$ are the number of states and actions respectively, $H$ is the time-horizon, $Z$ is the support size of reward distributions and $d=O(\min(M,H))$. We also provide a $(SA)^{\Omega(\sqrt{M})} / \epsilon^{2}$ lower bound, supporting that super-polynomial sample complexity in $M$ is necessary.

We consider stochastic unconstrained bilevel optimization problems when only the first-order gradient oracles are available. While numerous optimization methods have been proposed for tackling bilevel problems, existing methods either tend to require possibly expensive calculations regarding Hessians of lower-level objectives, or lack rigorous finite-time performance guarantees. In this work, we propose a Fully First-order Stochastic Approximation (F2SA) method, and study its non-asymptotic convergence properties. Specifically, we show that F2SA converges to an $\epsilon$-stationary solution of the bilevel problem after $\epsilon^{-7/2}, \epsilon^{-5/2}$, and $\epsilon^{-3/2}$ iterations (each iteration using $O(1)$ samples) when stochastic noises are in both level objectives, only in the upper-level objective, and not present (deterministic settings), respectively. We further show that if we employ momentum-assisted gradient estimators, the iteration complexities can be improved to $\epsilon^{-5/2}, \epsilon^{-4/2}$, and $\epsilon^{-3/2}$, respectively. We demonstrate even superior practical performance of the proposed method over existing second-order based approaches on MNIST data-hypercleaning experiments.

Motivated by online recommendation systems, we propose the problem of finding the optimal policy in multitask contextual bandits when a small fraction $\alpha < 1/2$ of tasks (users) are arbitrary and adversarial. The remaining fraction of good users share the same instance of contextual bandits with $S$ contexts and $A$ actions (items). Naturally, whether a user is good or adversarial is not known in advance. The goal is to robustly learn the policy that maximizes rewards for good users with as few user interactions as possible. Without adversarial users, established results in collaborative filtering show that $O(1/\epsilon^2)$ per-user interactions suffice to learn a good policy, precisely because information can be shared across users. This parallelization gain is fundamentally altered by the presence of adversarial users: unless there are super-polynomial number of users, we show a lower bound of $\tilde{\Omega}(\min(S,A) \cdot \alpha^2 / \epsilon^2)$ per-user interactions to learn an $\epsilon$-optimal policy for the good users. We then show we can achieve an $\tilde{O}(\min(S,A)\cdot \alpha/\epsilon^2)$ upper-bound, by employing efficient robust mean estimators for both uni-variate and high-dimensional random variables. We also show that this can be improved depending on the distributions of contexts.

We consider a multi-armed bandit problem with $M$ latent contexts, where an agent interacts with the environment for an episode of $H$ time steps. Depending on the length of the episode, the learner may not be able to estimate accurately the latent context. The resulting partial observation of the environment makes the learning task significantly more challenging. Without any additional structural assumptions, existing techniques to tackle partially observed settings imply the decision maker can learn a near-optimal policy with $O(A)^H$ episodes, but do not promise more. In this work, we show that learning with {\em polynomial} samples in $A$ is possible. We achieve this by using techniques from experiment design. Then, through a method-of-moments approach, we design a procedure that provably learns a near-optimal policy with $O(\poly(A) + \poly(M,H)^{\min(M,H)})$ interactions. In practice, we show that we can formulate the moment-matching via maximum likelihood estimation. In our experiments, this significantly outperforms the worst-case guarantees, as well as existing practical methods.

We study the convergence rates of the EM algorithm for learning two-component mixed linear regression under all regimes of signal-to-noise ratio (SNR). We resolve a long-standing question that many recent results have attempted to tackle: we completely characterize the convergence behavior of EM, and show that the EM algorithm achieves minimax optimal sample complexity under all SNR regimes. In particular, when the SNR is sufficiently large, the EM updates converge to the true parameter $\theta^{*}$ at the standard parametric convergence rate $\calo((d/n)^{1/2})$ after $\calo(\log(n/d))$ iterations. In the regime where the SNR is above $\calo((d/n)^{1/4})$ and below some constant, the EM iterates converge to a $\calo({\rm SNR}^{-1} (d/n)^{1/2})$ neighborhood of the true parameter, when the number of iterations is of the order $\calo({\rm SNR}^{-2} \log(n/d))$. In the low SNR regime where the SNR is below $\calo((d/n)^{1/4})$, we show that EM converges to a $\calo((d/n)^{1/4})$ neighborhood of the true parameters, after $\calo((n/d)^{1/2})$ iterations. Notably, these results are achieved under mild conditions of either random initialization or an efficiently computable local initialization. By providing tight convergence guarantees of the EM algorithm in middle-to-low SNR regimes, we fill the remaining gap in the literature, and significantly, reveal that in low SNR, EM changes rate, matching the $n^{-1/4}$ rate of the MLE, a behavior that previous work had been unable to show.

Learning a near optimal policy in a partially observable system remains an elusive challenge in contemporary reinforcement learning. In this work, we consider episodic reinforcement learning in a reward-mixing Markov decision process (MDP). There, a reward function is drawn from one of $M$ possible reward models at the beginning of every episode, but the identity of the chosen reward model is not revealed to the agent. Hence, the latent state space, for which the dynamics are Markovian, is not given to the agent. We study the problem of learning a near optimal policy for two reward-mixing MDPs. Unlike existing approaches that rely on strong assumptions on the dynamics, we make no assumptions and study the problem in full generality. Indeed, with no further assumptions, even for two switching reward-models, the problem requires several new ideas beyond existing algorithmic and analysis techniques for efficient exploration. We provide the first polynomial-time algorithm that finds an $\epsilon$-optimal policy after exploring $\tilde{O}(poly(H,\epsilon^{-1}) \cdot S^2 A^2)$ episodes, where $H$ is time-horizon and $S, A$ are the number of states and actions respectively. This is the first efficient algorithm that does not require any assumptions in partially observed environments where the observation space is smaller than the latent state space.

In this work, we consider the regret minimization problem for reinforcement learning in latent Markov Decision Processes (LMDP). In an LMDP, an MDP is randomly drawn from a set of $M$ possible MDPs at the beginning of the interaction, but the identity of the chosen MDP is not revealed to the agent. We first show that a general instance of LMDPs requires at least $\Omega((SA)^M)$ episodes to even approximate the optimal policy. Then, we consider sufficient assumptions under which learning good policies requires polynomial number of episodes. We show that the key link is a notion of separation between the MDP system dynamics. With sufficient separation, we provide an efficient algorithm with local guarantee, {\it i.e.,} providing a sublinear regret guarantee when we are given a good initialization. Finally, if we are given standard statistical sufficiency assumptions common in the Predictive State Representation (PSR) literature (e.g., \cite{boots2011online}) and a reachability assumption, we show that the need for initialization can be removed.

We study the convergence of the Expectation-Maximization (EM) algorithm for mixtures of linear regressions with an arbitrary number $k$ of components. We show that as long as signal-to-noise ratio (SNR) is $\tilde{\Omega}(k)$, well-initialized EM converges to the true regression parameters. Previous results for $k \geq 3$ have only established local convergence for the noiseless setting, i.e., where SNR is infinitely large. Our results enlarge the scope to the environment with noises, and notably, we establish a statistical error rate that is independent of the norm (or pairwise distance) of the regression parameters. In particular, our results imply exact recovery as $\sigma \rightarrow 0$, in contrast to most previous local convergence results for EM, where the statistical error scaled with the norm of parameters. Standard moment-method approaches may be applied to guarantee we are in the region where our local convergence guarantees apply.

We consider the problem of spherical Gaussian Mixture models with $k \geq 3$ components when the components are well separated. A fundamental previous result established that separation of $\Omega(\sqrt{\log k})$ is necessary and sufficient for identifiability of the parameters with \textit{polynomial} sample complexity (Regev and Vijayaraghavan, 2017). In the same context, we show that $\tilde{O} (kd/\epsilon^2)$ samples suffice for any $\epsilon \lesssim 1/k$, closing the gap from polynomial to linear, and thus giving the first optimal sample upper bound for the parameter estimation of well-separated Gaussian mixtures. We accomplish this by proving a new result for the Expectation-Maximization (EM) algorithm: we show that EM converges locally, under separation $\Omega(\sqrt{\log k})$. The previous best-known guarantee required $\Omega(\sqrt{k})$ separation (Yan, et al., 2017). Unlike prior work, our results do not assume or use prior knowledge of the (potentially different) mixing weights or variances of the Gaussian components. Furthermore, our results show that the finite-sample error of EM does not depend on non-universal quantities such as pairwise distances between means of Gaussian components.

The Expectation-Maximization algorithm is perhaps the most broadly used algorithm for inference of latent variable problems. A theoretical understanding of its performance, however, largely remains lacking. Recent results established that EM enjoys global convergence for Gaussian Mixture Models. For Mixed Linear Regression, however, only local convergence results have been established, and those only for the high SNR regime. We show here that EM converges for mixed linear regression with two components (it is known that it may fail to converge for three or more), and moreover that this convergence holds for random initialization. Our analysis reveals that EM exhibits very different behavior in Mixed Linear Regression from its behavior in Gaussian Mixture Models, and hence our proofs require the development of several new ideas.