This paper proposes near-optimal algorithms for the pure-exploration linear bandit problem in the fixed confidence and fixed budget settings. Leveraging ideas from the theory of suprema of empirical processes, we provide an algorithm whose sample complexity scales with the geometry of the instance and avoids an explicit union bound over the number of arms. Unlike previous approaches which sample based on minimizing a worst-case variance (e.g. G-optimal design), we define an experimental design objective based on the Gaussian-width of the underlying arm set. We provide a novel lower bound in terms of this objective that highlights its fundamental role in the sample complexity. The sample complexity of our fixed confidence algorithm matches this lower bound, and in addition is computationally efficient for combinatorial classes, e.g. shortest-path, matchings and matroids, where the arm sets can be exponentially large in the dimension. Finally, we propose the first algorithm for linear bandits in the the fixed budget setting. Its guarantee matches our lower bound up to logarithmic factors.

Having a perfect model to compute the optimal policy is often infeasible in reinforcement learning. It is important in high-stakes domains to quantify and manage risk induced by model uncertainties. Entropic risk measure is an exponential utility-based convex risk measure that satisfies many reasonable properties. In this paper, we propose an entropic risk constrained policy gradient and actor-critic algorithms that are risk-averse to the model uncertainty. We demonstrate the usefulness of our algorithms on several problem domains.

Inverse reinforcement learning (IRL) aims to estimate the reward function of optimizing agents by observing their response (estimates or actions). This paper considers IRL when noisy estimates of the gradient of a reward function generated by multiple stochastic gradient agents are observed. We present a generalized Langevin dynamics algorithm to estimate the reward function $R(\theta)$; specifically, the resulting Langevin algorithm asymptotically generates samples from the distribution proportional to $\exp(R(\theta))$. The proposed IRL algorithms use kernel-based passive learning schemes. We also construct multi-kernel passive Langevin algorithms for IRL which are suitable for high dimensional data. The performance of the proposed IRL algorithms are illustrated on examples in adaptive Bayesian learning, logistic regression (high dimensional problem) and constrained Markov decision processes. We prove weak convergence of the proposed IRL algorithms using martingale averaging methods. We also analyze the tracking performance of the IRL algorithms in non-stationary environments where the utility function $R(\theta)$ jump changes over time as a slow Markov chain.

Combining the predictions of collections of neural networks often outperforms the best single network. Such ensembles are typically trained independently, and their superior `wisdom of the crowd' originates from the differences between networks. Collective foraging and decision making in socially interacting animal groups is often improved or even optimal thanks to local information sharing between conspecifics. We therefore present a model for co-learning by ensembles of interacting neural networks that aim to maximize their own performance but also their functional relations to other networks. We show that ensembles of interacting networks outperform independent ones, and that optimal ensemble performance is reached when the coupling between networks increases diversity and degrades the performance of individual networks. Thus, even without a global goal for the ensemble, optimal collective behavior emerges from local interactions between networks. We show the scaling of optimal coupling strength with ensemble size, and that networks in these ensembles specialize functionally and become more `confident' in their assessments. Moreover, optimal co-learning networks differ structurally, relying on sparser activity, a wider range of synaptic weights, and higher firing rates - compared to independently trained networks. Finally, we explore interactions-based co-learning as a framework for expanding and boosting ensembles.

We study the problem of efficient exact partitioning of the hypergraphs generated by high-order planted models. A high-order planted model assumes some underlying cluster structures, and simulates high-order interactions by placing hyperedges among nodes. Example models include the disjoint hypercliques, the densest subhypergraphs, and the hypergraph stochastic block models. We show that exact partitioning of high-order planted models (a NP-hard problem in general) is achievable through solving a computationally efficient convex optimization problem with a tensor nuclear norm constraint. Our analysis provides the conditions for our approach to succeed on recovering the true underlying cluster structures, with high probability.

Reliably transferring treatment policies learned in one clinical environment to another is currently limited by challenges related to domain shift. In this paper we address off-policy learning for sequential decision making under domain shift -- a scenario susceptible to catastrophic overconfidence -- which is highly relevant to a high-stakes clinical settings where the target domain may also be data-scarce. We propose a two-fold counterfactual regularization procedure to improve off-policy learning, addressing domain shift and data scarcity. First, we utilize an informative prior derived from a data-rich source environment to indirectly improve drawing counterfactual example observations. Then, these samples are then used to learn a policy for the target domain, regularized by the source policy through KL-divergence. In simulated sepsis treatment, our counterfactual policy transfer procedure significantly improves the performance of a learned treatment policy.

The slow convergence rate and pathological curvature issues of first-order gradient methods for training deep neural networks, initiated an ongoing effort for developing faster $\mathit{second}$-$\mathit{order}$ optimization algorithms beyond SGD, without compromising the generalization error. Despite their remarkable convergence rate ($\mathit{independent}$ of the training batch size $n$), second-order algorithms incur a daunting slowdown in the $\mathit{cost}$ $\mathit{per}$ $\mathit{iteration}$ (inverting the Hessian matrix of the loss function), which renders them impractical. Very recently, this computational overhead was mitigated by the works of [ZMG19, CGH+19], yielding an $O(Mn^2)$-time second-order algorithm for training overparametrized neural networks with $M$ parameters. We show how to speed up the algorithm of [CGH+19], achieving an $\tilde{O}(Mn)$-time backpropagation algorithm for training (mildly overparametrized) ReLU networks, which is near-linear in the dimension ($Mn$) of the full gradient (Jacobian) matrix. The centerpiece of our algorithm is to reformulate the Gauss-Newton iteration as an $\ell_2$-regression problem, and then use a Fast-JL type dimension reduction to $\mathit{precondition} $ the underlying Gram matrix in time independent of $M$, allowing to find a sufficiently good approximate solution via $\mathit{first}$-$\mathit{order}$ conjugate gradient. Our result provides a proof-of-concept that advanced machinery from randomized linear algebra-which led to recent breakthroughs in $\mathit{convex}$ $\mathit{optimization}$ (ERM, LPs, Regression)-can be carried over to the realm of deep learning as well.

We consider the classic problem of $(\epsilon,\delta)$-PAC learning a best arm where the goal is to identify with confidence $1-\delta$ an arm whose mean is an $\epsilon$-approximation to that of the highest mean arm in a multi-armed bandit setting. This problem is one of the most fundamental problems in statistics and learning theory, yet somewhat surprisingly its worst-case sample complexity is not well understood. In this paper, we propose a new approach for $(\epsilon,\delta)$-PAC learning a best arm. This approach leads to an algorithm whose sample complexity converges to \emph{exactly} the optimal sample complexity of $(\epsilon,\delta)$-learning the mean of $n$ arms separately and we complement this result with a conditional matching lower bound. More specifically:

We present a sparse representation of model uncertainty for Deep Neural Networks (DNNs) where the parameter posterior is approximated with an inverse formulation of the Multivariate Normal Distribution (MND), also known as the information form. The key insight of our work is that the information matrix, i.e. the inverse of the covariance matrix tends to be sparse in its spectrum. Therefore, dimensionality reduction techniques such as low rank approximations (LRA) can be effectively exploited. To achieve this, we develop a novel sparsification algorithm and derive a cost-effective analytical sampler. As a result, we show that the information form can be scalably applied to represent model uncertainty in DNNs. Our exhaustive theoretical analysis and empirical evaluations on various benchmarks show the competitiveness of our approach over the current methods.

Prior Networks are a class of models which yield interpretable measures of uncertainty and have been shown to outperform state-of-the-art ensemble approaches on a range of tasks. However, Prior Networks have so far been developed only for classification tasks. The properties of Regression Prior Networks are demonstrated on synthetic data, selected UCI datasets, and two monocular depth estimation tasks. They yield performance competitive with ensemble approaches. However, in order to improve the safety of AI systems (Amodei et al., 2016) and avoid costly mistakes in high-risk applications, such as self-driving cars, it is desirable for models to yield estimates of uncertainty in their predictions. Ensemble methods are known to yield both improved predictive performance and robust uncertainty estimates (Gal & Ghahramani, 2016; Lakshminarayanan et al., 2017; Maddox et al., 2019). Importantly, ensemble approaches allow interpretable measures of uncertainty to be derived via a mathematically consistent probabilistic framework. Specifically, the overall total uncertainty can be decomposed into data uncertainty, or uncertainty due to inherent noise in the data, and knowledge uncertainty, which is due to the model having limited uncertainty of the test data (Malinin, 2019). Uncertainty estimates derived from ensembles have been applied to the detection of misclassifications, out-of-domain inputs and adversarial attack detection (Carlini & Wagner, 2017; Smith & Gal, 2018), and active learning (Kirsch et al., 2019). Unfortunately, ensemble methods may be computationally expensive to train and are always expensive during inference.