We study the trade-off between expectation and tail risk for regret distribution in the stochastic multi-armed bandit problem. We fully characterize the interplay among three desired properties for policy design: worst-case optimality, instance-dependent consistency, and light-tailed risk. We show how the order of expected regret exactly affects the decaying rate of the regret tail probability for both the worst-case and instance-dependent scenario. A novel policy is proposed to characterize the optimal regret tail probability for any regret threshold. Concretely, for any given $\alpha\in[1/2, 1)$ and $\beta\in[0, \alpha]$, our policy achieves a worst-case expected regret of $\tilde O(T^\alpha)$ (we call it $\alpha$-optimal) and an instance-dependent expected regret of $\tilde O(T^\beta)$ (we call it $\beta$-consistent), while enjoys a probability of incurring an $\tilde O(T^\delta)$ regret ($\delta\geq\alpha$ in the worst-case scenario and $\delta\geq\beta$ in the instance-dependent scenario) that decays exponentially with a polynomial $T$ term. Such decaying rate is proved to be best achievable. Moreover, we discover an intrinsic gap of the optimal tail rate under the instance-dependent scenario between whether the time horizon $T$ is known a priori or not. Interestingly, when it comes to the worst-case scenario, this gap disappears. Finally, we extend our proposed policy design to (1) a stochastic multi-armed bandit setting with non-stationary baseline rewards, and (2) a stochastic linear bandit setting. Our results reveal insights on the trade-off between regret expectation and regret tail risk for both worst-case and instance-dependent scenarios, indicating that more sub-optimality and inconsistency leave space for more light-tailed risk of incurring a large regret, and that knowing the planning horizon in advance can make a difference on alleviating tail risks.
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Distributed stochastic optimization has drawn great attention recently due to its effectiveness in solving large-scale machine learning problems. Though numerous algorithms have been proposed and successfully applied to general practical problems, their theoretical guarantees mainly rely on certain boundedness conditions on the stochastic gradients, varying from uniform boundedness to the relaxed growth condition. In addition, how to characterize the data heterogeneity among the agents and its impacts on the algorithmic performance remains challenging. In light of such motivations, we revisit the classical Federated Averaging (FedAvg) algorithm for solving the distributed stochastic optimization problem and establish the convergence results under only a mild variance condition on the stochastic gradients for smooth nonconvex objective functions. Almost sure convergence to a stationary point is also established under the condition. Moreover, we discuss a more informative measurement for data heterogeneity as well as its implications.
In this paper, we study the problem of optimal data collection for policy evaluation in linear bandits. In policy evaluation, we are given a target policy and asked to estimate the expected reward it will obtain when executed in a multi-armed bandit environment. Our work is the first work that focuses on such optimal data collection strategy for policy evaluation involving heteroscedastic reward noise in the linear bandit setting. We first formulate an optimal design for weighted least squares estimates in the heteroscedastic linear bandit setting that reduces the MSE of the value of the target policy. We then use this formulation to derive the optimal allocation of samples per action during data collection. We then introduce a novel algorithm SPEED (Structured Policy Evaluation Experimental Design) that tracks the optimal design and derive its regret with respect to the optimal design. Finally, we empirically validate that SPEED leads to policy evaluation with mean squared error comparable to the oracle strategy and significantly lower than simply running the target policy.
This paper considers a stochastic control framework, in which the residual model uncertainty of the dynamical system is learned using a Gaussian Process (GP). In the proposed formulation, the residual model uncertainty consists of a nonlinear function and state-dependent noise. The proposed formulation uses a posterior-GP to approximate the residual model uncertainty and a prior-GP to account for state-dependent noise. The two GPs are interdependent and are thus learned jointly using an iterative algorithm. Theoretical properties of the iterative algorithm are established. Advantages of the proposed state-dependent formulation include (i) faster convergence of the GP estimate to the unknown function as the GP learns which data samples are more trustworthy and (ii) an accurate estimate of state-dependent noise, which can, e.g., be useful for a controller or decision-maker to determine the uncertainty of an action. Simulation studies highlight these two advantages.
This paper studies delayed stochastic algorithms for weakly convex optimization in a distributed network with workers connected to a master node. More specifically, we consider a structured stochastic weakly convex objective function which is the composition of a convex function and a smooth nonconvex function. Recently, Xu et al. 2022 showed that an inertial stochastic subgradient method converges at a rate of $\mathcal{O}(\tau/\sqrt{K})$, which suffers a significant penalty from the maximum information delay $\tau$. To alleviate this issue, we propose a new delayed stochastic prox-linear ($\texttt{DSPL}$) method in which the master performs the proximal update of the parameters and the workers only need to linearly approximate the inner smooth function. Somewhat surprisingly, we show that the delays only affect the high order term in the complexity rate and hence, are negligible after a certain number of $\texttt{DSPL}$ iterations. Moreover, to further improve the empirical performance, we propose a delayed extrapolated prox-linear ($\texttt{DSEPL}$) method which employs Polyak-type momentum to speed up the algorithm convergence. Building on the tools for analyzing $\texttt{DSPL}$, we also develop improved analysis of delayed stochastic subgradient method ($\texttt{DSGD}$). In particular, for general weakly convex problems, we show that convergence of $\texttt{DSGD}$ only depends on the expected delay.
This paper introduces the application of the weak Galerkin (WG) finite element method to solve the Steklov eigenvalue problem, focusing on obtaining lower bounds of the eigenvalues. The noncomforming finite element space of the weak Galerkin finite element method is the key to obtain lower bounds of the eigenvalues. The arbitary high order lower bound estimates are given and the guaranteed lower bounds of the eigenvalues are also discussed. Numerical results demonstrate the accuracy and lower bound property of the numerical scheme.
While distributional reinforcement learning (RL) has demonstrated empirical success, the question of when and why it is beneficial has remained unanswered. In this work, we provide one explanation for the benefits of distributional RL through the lens of small-loss bounds, which scale with the instance-dependent optimal cost. If the optimal cost is small, our bounds are stronger than those from non-distributional approaches. As warmup, we show that learning the cost distribution leads to small-loss regret bounds in contextual bandits (CB), and we find that distributional CB empirically outperforms the state-of-the-art on three challenging tasks. For online RL, we propose a distributional version-space algorithm that constructs confidence sets using maximum likelihood estimation, and we prove that it achieves small-loss regret in the tabular MDPs and enjoys small-loss PAC bounds in latent variable models. Building on similar insights, we propose a distributional offline RL algorithm based on the pessimism principle and prove that it enjoys small-loss PAC bounds, which exhibit a novel robustness property. For both online and offline RL, our results provide the first theoretical benefits of learning distributions even when we only need the mean for making decisions.
I study a stochastic multi-arm bandit problem where rewards are subject to adversarial corruption. I propose a novel attack strategy that manipulates a learner employing the UCB algorithm into pulling some non-optimal target arm $T - o(T)$ times with a cumulative cost that scales as $\widehat{O}(\sqrt{\log T})$, where $T$ is the number of rounds. I also prove the first lower bound on the cumulative attack cost. The lower bound matches the upper bound up to $O(\log \log T)$ factors, showing the proposed attack strategy to be near optimal.
Bayesian optimization has attracted huge attention from diverse research areas in science and engineering, since it is capable of finding a global optimum of an expensive-to-evaluate black-box function efficiently. In general, a probabilistic regression model, e.g., Gaussian processes, random forests, and Bayesian neural networks, is widely used as a surrogate function to model an explicit distribution over function evaluations given an input to estimate and a training dataset. Beyond the probabilistic regression-based Bayesian optimization, density ratio estimation-based Bayesian optimization has been suggested in order to estimate a density ratio of the groups relatively close and relatively far to a global optimum. Developing this line of research further, a supervised classifier can be employed to estimate a class probability for the two groups instead of a density ratio. However, the supervised classifiers used in this strategy tend to be overconfident for a global solution candidate. To solve this overconfidence problem, we propose density ratio estimation-based Bayesian optimization with semi-supervised learning. Finally, we demonstrate the experimental results of our methods and several baseline methods in two distinct scenarios with unlabeled point sampling and a fixed-size pool.
Actor-critic (AC) methods are widely used in reinforcement learning (RL) and benefit from the flexibility of using any policy gradient method as the actor and value-based method as the critic. The critic is usually trained by minimizing the TD error, an objective that is potentially decorrelated with the true goal of achieving a high reward with the actor. We address this mismatch by designing a joint objective for training the actor and critic in a decision-aware fashion. We use the proposed objective to design a generic, AC algorithm that can easily handle any function approximation. We explicitly characterize the conditions under which the resulting algorithm guarantees monotonic policy improvement, regardless of the choice of the policy and critic parameterization. Instantiating the generic algorithm results in an actor that involves maximizing a sequence of surrogate functions (similar to TRPO, PPO) and a critic that involves minimizing a closely connected objective. Using simple bandit examples, we provably establish the benefit of the proposed critic objective over the standard squared error. Finally, we empirically demonstrate the benefit of our decision-aware actor-critic framework on simple RL problems.
It is typically understood that the training of modern neural networks is a process of fitting the probability distribution of desired output. However, recent paradoxical observations in a number of language generation tasks let one wonder if this canonical probability-based explanation can really account for the empirical success of deep learning. To resolve this issue, we propose an alternative utility-based explanation to the standard supervised learning procedure in deep learning. The basic idea is to interpret the learned neural network not as a probability model but as an ordinal utility function that encodes the preference revealed in training data. In this perspective, training of the neural network corresponds to a utility learning process. Specifically, we show that for all neural networks with softmax outputs, the SGD learning dynamic of maximum likelihood estimation (MLE) can be seen as an iteration process that optimizes the neural network toward an optimal utility function. This utility-based interpretation can explain several otherwise-paradoxical observations about the neural networks thus trained. Moreover, our utility-based theory also entails an equation that can transform the learned utility values back to a new kind of probability estimation with which probability-compatible decision rules enjoy dramatic (double-digits) performance improvements. These evidences collectively reveal a phenomenon of utility-probability duality in terms of what modern neural networks are (truly) modeling: We thought they are one thing (probabilities), until the unexplainable showed up; changing mindset and treating them as another thing (utility values) largely reconcile the theory, despite remaining subtleties regarding its original (probabilistic) identity.
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