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This paper studies the problem of computing a linear approximation of quadratic Wasserstein distance $W_2$. In particular, we compute an approximation of the negative homogeneous weighted Sobolev norm whose connection to Wasserstein distance follows from a classic linearization of a general Monge-Amp\'ere equation. Our contribution is threefold. First, we provide expository material on this classic linearization of Wasserstein distance including a quantitative error estimate. econd, we reduce the computational problem to solving a elliptic boundary value problem involving the Witten Laplacian, which is a Schr\"odinger operator of the form $H = -\Delta + V$, and describe an associated embedding. Third, for the case of probability distributions on the unit square $[0,1]^2$ represented by $n \times n$ arrays we present a fast code demonstrating our approach. Several numerical examples are presented.

Massive sized survival datasets are becoming increasingly prevalent with the development of the healthcare industry. Such datasets pose computational challenges unprecedented in traditional survival analysis use-cases. A popular way for coping with massive datasets is downsampling them to a more manageable size, such that the computational resources can be afforded by the researcher. Cox proportional hazards regression has remained one of the most popular statistical models for the analysis of survival data to-date. This work addresses the settings of right censored and possibly left truncated data with rare events, such that the observed failure times constitute only a small portion of the overall sample. We propose Cox regression subsampling-based estimators that approximate their full-data partial-likelihood-based counterparts, by assigning optimal sampling probabilities to censored observations, and including all observed failures in the analysis. Asymptotic properties of the proposed estimators are established under suitable regularity conditions, and simulation studies are carried out to evaluate the finite sample performance of the estimators. We further apply our procedure on UK-biobank colorectal cancer genetic and environmental risk factors.

Optimum parameter estimation methods require knowledge of a parametric probability density that statistically describes the available observations. In this work we examine Bayesian and non-Bayesian parameter estimation problems under a data-driven formulation where the necessary parametric probability density is replaced by available data. We present various data-driven versions that either result in neural network approximations of the optimum estimators or in well defined optimization problems that can be solved numerically. In particular, for the data-driven equivalent of non-Bayesian estimation we end up with optimization problems similar to the ones encountered for the design of generative networks.

We study a syntax for specifying quantitative "assertions" - functions mapping program states to numbers - for probabilistic program verification. We prove that our syntax is expressive in the following sense: Given any probabilistic program $C$, if a function $f$ is expressible in our syntax, then the function mapping each initial state $\sigma$ to the expected value of $f$ evaluated in the final states reached after termination of $C$ on $\sigma$ (also called the weakest preexpectation $\textit{wp} [C](f)$) is also expressible in our syntax. As a consequence, we obtain a relatively complete verification system for reasoning about expected values and probabilities in the sense of Cook: Apart from proving a single inequality between two functions given by syntactic expressions in our language, given $f$, $g$, and $C$, we can check whether $g \preceq \textit{wp} [C] (f)$.

Second-order optimizers are thought to hold the potential to speed up neural network training, but due to the enormous size of the curvature matrix, they typically require approximations to be computationally tractable. The most successful family of approximations are Kronecker-Factored, block-diagonal curvature estimates (KFAC). Here, we combine tools from prior work to evaluate exact second-order updates with careful ablations to establish a surprising result: Due to its approximations, KFAC is not closely related to second-order updates, and in particular, it significantly outperforms true second-order updates. This challenges widely held believes and immediately raises the question why KFAC performs so well. We answer this question by showing that KFAC approximates a first-order algorithm, which performs gradient descent on neurons rather than weights. Finally, we show that this optimizer often improves over KFAC in terms of computational cost and data-efficiency.

The conventional approach to data-driven inversion framework is based on Gaussian statistics that presents serious difficulties, especially in the presence of outliers in the measurements. In this work, we present maximum likelihood estimators associated with generalized Gaussian distributions in the context of R\'enyi, Tsallis and Kaniadakis statistics. In this regard, we analytically analyse the outlier-resistance of each proposal through the so-called influence function. In this way, we formulate inverse problems by constructing objective functions linked to the maximum likelihood estimators. To demonstrate the robustness of the generalized methodologies, we consider an important geophysical inverse problem with high noisy data with spikes. The results reveal that the best data inversion performance occurs when the entropic index from each generalized statistic is associated with objective functions proportional to the inverse of the error amplitude. We argue that in such a limit the three approaches are resistant to outliers and are also equivalent, which suggests a lower computational cost for the inversion process due to the reduction of numerical simulations to be performed and the fast convergence of the optimization process.

We derive and analyze a symmetric interior penalty discontinuous Galerkin scheme for the approximation of the second-order form of the radiative transfer equation in slab geometry. Using appropriate trace lemmas, the analysis can be carried out as for more standard elliptic problems. Supporting examples show the accuracy and stability of the method also numerically. For discretization, we employ quadtree-like grids, which allow for local refinement in phase-space, and we show exemplary that adaptive methods can efficiently approximate discontinuous solutions.

This paper studies the accelerated gradient descent for general nonconvex problems under the gradient Lipschitz and Hessian Lipschitz assumptions. We establish that a simple restarted accelerated gradient descent (AGD) finds an $\epsilon$-approximate first-order stationary point in $O(\epsilon^{-7/4})$ gradient computations with simple proofs. Our complexity does not hide any polylogarithmic factors, and thus it improves over the state-of-the-art one by the $O(\log\frac{1}{\epsilon})$ factor. Our simple algorithm only consists of Nesterov's classical AGD and a restart mechanism, and it does not need the negative curvature exploitation or the optimization of regularized surrogate functions. Technically, our simple proof does not invoke the analysis for the strongly convex AGD, which is crucial to remove the $O(\log\frac{1}{\epsilon})$ factor.

Obtaining first-order regret bounds -- regret bounds scaling not as the worst-case but with some measure of the performance of the optimal policy on a given instance -- is a core question in sequential decision-making. While such bounds exist in many settings, they have proven elusive in reinforcement learning with large state spaces. In this work we address this gap, and show that it is possible to obtain regret scaling as $\mathcal{O}(\sqrt{V_1^\star K})$ in reinforcement learning with large state spaces, namely the linear MDP setting. Here $V_1^\star$ is the value of the optimal policy and $K$ is the number of episodes. We demonstrate that existing techniques based on least squares estimation are insufficient to obtain this result, and instead develop a novel robust self-normalized concentration bound based on the robust Catoni mean estimator, which may be of independent interest.

We study the problem of learning in the stochastic shortest path (SSP) setting, where an agent seeks to minimize the expected cost accumulated before reaching a goal state. We design a novel model-based algorithm EB-SSP that carefully skews the empirical transitions and perturbs the empirical costs with an exploration bonus to guarantee both optimism and convergence of the associated value iteration scheme. We prove that EB-SSP achieves the minimax regret rate $\widetilde{O}(B_{\star} \sqrt{S A K})$, where $K$ is the number of episodes, $S$ is the number of states, $A$ is the number of actions and $B_{\star}$ bounds the expected cumulative cost of the optimal policy from any state, thus closing the gap with the lower bound. Interestingly, EB-SSP obtains this result while being parameter-free, i.e., it does not require any prior knowledge of $B_{\star}$, nor of $T_{\star}$ which bounds the expected time-to-goal of the optimal policy from any state. Furthermore, we illustrate various cases (e.g., positive costs, or general costs when an order-accurate estimate of $T_{\star}$ is available) where the regret only contains a logarithmic dependence on $T_{\star}$, thus yielding the first horizon-free regret bound beyond the finite-horizon MDP setting.