Many practical applications of online reinforcement learning require the satisfaction of safety constraints while learning about the unknown environment. In this work, we study Linear Quadratic Regulator (LQR) learning with unknown dynamics, but with the additional constraint that the position must stay within a safe region for the entire trajectory with high probability. Unlike in previous works, we allow for both bounded and unbounded noise distributions and study stronger baselines of nonlinear controllers that are better suited for constrained problems than linear controllers. Due to these complications, we focus on 1-dimensional state- and action- spaces, however we also discuss how we expect the high-level takeaways can generalize to higher dimensions. Our primary contribution is the first $\tilde{O}_T(\sqrt{T})$-regret bound for constrained LQR learning, which we show relative to a specific baseline of non-linear controllers. We then prove that, for any non-linear baseline satisfying natural assumptions, $\tilde{O}_T(\sqrt{T})$-regret is possible when the noise distribution has sufficiently large support and $\tilde{O}_T(T^{2/3})$-regret is possible for any subgaussian noise distribution. An overarching theme of our results is that enforcing safety provides "free exploration" that compensates for the added cost of uncertainty in safety constrained control, resulting in the same regret rate as in the unconstrained problem.

Shampoo, a second-order optimization algorithm which uses a Kronecker product preconditioner, has recently garnered increasing attention from the machine learning community. The preconditioner used by Shampoo can be viewed either as an approximation of the Gauss--Newton component of the Hessian or the covariance matrix of the gradients maintained by Adagrad. We provide an explicit and novel connection between the $\textit{optimal}$ Kronecker product approximation of these matrices and the approximation made by Shampoo. Our connection highlights a subtle but common misconception about Shampoo's approximation. In particular, the $\textit{square}$ of the approximation used by the Shampoo optimizer is equivalent to a single step of the power iteration algorithm for computing the aforementioned optimal Kronecker product approximation. Across a variety of datasets and architectures we empirically demonstrate that this is close to the optimal Kronecker product approximation. Additionally, for the Hessian approximation viewpoint, we empirically study the impact of various practical tricks to make Shampoo more computationally efficient (such as using the batch gradient and the empirical Fisher) on the quality of Hessian approximation.

When resources are scarce, an allocation policy is needed to decide who receives a resource. This problem occurs, for instance, when allocating scarce medical resources and is often solved using modern ML methods. This paper introduces methods to evaluate index-based allocation policies -- that allocate a fixed number of resources to those who need them the most -- by using data from a randomized control trial. Such policies create dependencies between agents, which render the assumptions behind standard statistical tests invalid and limit the effectiveness of estimators. Addressing these challenges, we translate and extend recent ideas from the statistics literature to present an efficient estimator and methods for computing asymptotically correct confidence intervals. This enables us to effectively draw valid statistical conclusions, a critical gap in previous work. Our extensive experiments validate our methodology in practical settings, while also showcasing its statistical power. We conclude by proposing and empirically verifying extensions of our methodology that enable us to reevaluate a past randomized control trial to evaluate different ML allocation policies in the context of a mHealth program, drawing previously invisible conclusions.

In these settings, such as communicable disease management (Tuldrà et al., the underlying transition dynamics are often unknown 1999; Killian et al., 2019), prenatal and infant care (Hegde a priori, requiring online reinforcement & Doshi, 2016; Ope, 2020; Bashingwa et al., 2021), and learning (RL). However, existing methods in online cancer prevention (Wells et al., 2011; Lee et al., 2019), beneficiaries RL for RMABs cannot incorporate properties may at any time enter an adhering (e.g., following often present in real-world public health applications, their treatment regimen) or non-adhering (e.g., missing a such as contextual information and treatment) state. As adherence is often vital for ensuring non-stationarity. We present Bayesian Learning certain health outcomes, programs may allocate resources for Contextual RMABs (BCoR), an online or interventions to patients at risk of drop-out from the program RL approach for RMABs that novelly combines due to continued non-adherence. We can model this techniques in Bayesian modeling with Thompson problem as an RMAB by representing each beneficiary as an sampling to flexibly model a wide range of arm, their adherence status as the state of the corresponding complex RMAB settings, such as contextual and MDP, and the allocation of an intervention as the action.

Online reinforcement learning and other adaptive sampling algorithms are increasingly used in digital intervention experiments to optimize treatment delivery for users over time. In this work, we focus on longitudinal user data collected by a large class of adaptive sampling algorithms that are designed to optimize treatment decisions online using accruing data from multiple users. Combining or "pooling" data across users allows adaptive sampling algorithms to potentially learn faster. However, by pooling, these algorithms induce dependence between the sampled user data trajectories; we show that this can cause standard variance estimators for i.i.d. data to underestimate the true variance of common estimators on this data type. We develop novel methods to perform a variety of statistical analyses on such adaptively sampled data via Z-estimation. Specifically, we introduce the \textit{adaptive} sandwich variance estimator, a corrected sandwich estimator that leads to consistent variance estimates under adaptive sampling. Additionally, to prove our results we develop novel theoretical tools for empirical processes on non-i.i.d., adaptively sampled longitudinal data which may be of independent interest. This work is motivated by our efforts in designing experiments in which online reinforcement learning algorithms optimize treatment decisions, yet statistical inference is essential for conducting analyses after experiments conclude.

Conjoint analysis is a popular experimental design used to measure multidimensional preferences. Researchers examine how varying a factor of interest, while controlling for other relevant factors, influences decision-making. Currently, there exist two methodological approaches to analyzing data from a conjoint experiment. The first focuses on estimating the average marginal effects of each factor while averaging over the other factors. Although this allows for straightforward design-based estimation, the results critically depend on the distribution of other factors and how interaction effects are aggregated. An alternative model-based approach can compute various quantities of interest, but requires researchers to correctly specify the model, a challenging task for conjoint analysis with many factors and possible interactions. In addition, a commonly used logistic regression has poor statistical properties even with a moderate number of factors when incorporating interactions. We propose a new hypothesis testing approach based on the conditional randomization test to answer the most fundamental question of conjoint analysis: Does a factor of interest matter in any way given the other factors? Our methodology is solely based on the randomization of factors, and hence is free from assumptions. Yet, it allows researchers to use any test statistic, including those based on complex machine learning algorithms. As a result, we are able to combine the strengths of the existing design-based and model-based approaches. We illustrate the proposed methodology through conjoint analysis of immigration preferences and political candidate evaluation. We also extend the proposed approach to test for regularity assumptions commonly used in conjoint analysis.

In this work, we analyze an efficient sampling-based algorithm for general-purpose reachability analysis, which remains a notoriously challenging problem with applications ranging from neural network verification to safety analysis of dynamical systems. By sampling inputs, evaluating their images in the true reachable set, and taking their ɛ-padded convex hull as a set estimator, this algorithm applies to general problem settings and is simple to implement. Our main contribution is the derivation of asymptotic and finite-sample accuracy guarantees using random set theory. This analysis informs algorithmic design to obtain an ɛ-close reachable set approximation with high probability, provides insights into which reachability problems are most challenging, and motivates safety-critical applications of the technique. On a neural network verification task, we show that this approach is more accurate and significantly faster than prior work. Informed by our analysis, we also design a robust model predictive controller that we demonstrate in hardware experiments. Keywords: reachability analysis, random set theory, robust control, neural network verification.

Reward functions are at the heart of every reinforcement learning (RL) algorithm. In robotic grasping, rewards are often complex and manually engineered functions that do not rely on well-justified physical models from grasp analysis. This work demonstrates that analytic grasp stability metrics constitute powerful optimization objectives for RL algorithms that refine grasps on a three-fingered hand using only tactile and joint position information. We outperform a binary-reward baseline by 42.9% and find that a combination of geometric and force-agnostic grasp stability metrics yields the highest average success rates of 95.4% for cuboids, 93.1% for cylinders, and 62.3% for spheres across wrist position errors between 0 and 7 centimeters and rotational errors between 0 and 14 degrees. In a second experiment, we show that grasp refinement algorithms trained with contact feedback (contact positions, normals, and forces) perform up to 6.6% better than a baseline that receives no tactile information.

This work develops central limit theorems for cross-validation and consistent estimators of its asymptotic variance under weak stability conditions on the learning algorithm. Together, these results provide practical, asymptotically-exact confidence intervals for $k$-fold test error and valid, powerful hypothesis tests of whether one learning algorithm has smaller $k$-fold test error than another. These results are also the first of their kind for the popular choice of leave-one-out cross-validation. In our real-data experiments with diverse learning algorithms, the resulting intervals and tests outperform the most popular alternative methods from the literature.

As bandit algorithms are increasingly utilized in scientific studies, there is an associated increasing need for reliable inference methods based on the resulting adaptively-collected data. In this work, we develop methods for inference regarding the treatment effect on data collected in batches using a bandit algorithm. We focus on the setting in which the total number of batches is fixed and develop approximate inference methods based on the asymptotic distribution as the size of the batches goes to infinity. We first prove that the ordinary least squares estimator (OLS), which is asymptotically normal on independently sampled data, is not asymptotically normal on data collected using standard bandit algorithms when the treatment effect is zero. This asymptotic non-normality result implies that the naive assumption that the OLS estimator is approximately normal can lead to Type-1 error inflation and confidence intervals with below-nominal coverage probabilities. Second, we introduce the Batched OLS estimator (BOLS) that we prove is asymptotically normal---even in the zero treatment effect case---on data collected from both multi-arm and contextual bandits. Moreover, BOLS is robust to changes in the baseline reward and can be used for obtaining simultaneous confidence intervals for the treatment effect from all batches in non-stationary bandits. We demonstrate in simulations that BOLS can be used reliably for hypothesis testing and obtaining a confidence interval for the treatment effect, even in small sample settings.