Causal effect estimation seeks to determine the impact of an intervention from observational data. However, the existing causal inference literature primarily addresses treatment effects on frequently occurring events. But what if we are interested in estimating the effects of a policy intervention whose benefits, while potentially important, can only be observed and measured in rare yet impactful events, such as extreme climate events? The standard causal inference methodology is not designed for this type of inference since the events of interest may be scarce in the observed data and some degree of extrapolation is necessary. Extreme Value Theory (EVT) provides methodologies for analyzing statistical phenomena in such extreme regimes. We introduce a novel framework for assessing treatment effects in extreme data to capture the causal effect at the occurrence of rare events of interest. In particular, we employ the theory of multivariate regular variation to model extremities. We develop a consistent estimator for extreme treatment effects and present a rigorous non-asymptotic analysis of its performance. We illustrate the performance of our estimator using both synthetic and semi-synthetic data.

Recent progress in Neural Causal Models (NCMs) showcased how identification and partial identification of causal effects can be automatically carried out via training of neural generative models that respect the constraints encoded in a given causal graph [Xia et al. 2022, Balazadeh et al. 2022]. However, formal consistency of these methods has only been proven for the case of discrete variables or only for linear causal models. In this work, we prove the consistency of partial identification via NCMs in a general setting with both continuous and categorical variables. Further, our results highlight the impact of the design of the underlying neural network architecture in terms of depth and connectivity as well as the importance of applying Lipschitz regularization in the training phase. In particular, we provide a counterexample showing that without Lipschitz regularization this method may not be asymptotically consistent. Our results are enabled by new results on the approximability of Structural Causal Models (SCMs) via neural generative models, together with an analysis of the sample complexity of the resulting architectures and how that translates into an error in the constrained optimization problem that defines the partial identification bounds.

Gradient dominance property is a condition weaker than strong convexity, yet sufficiently ensures global convergence even in non-convex optimization. This property finds wide applications in machine learning, reinforcement learning (RL), and operations management. In this paper, we propose the stochastic homogeneous second-order descent method (SHSODM) for stochastic functions enjoying gradient dominance property based on a recently proposed homogenization approach. Theoretically, we provide its sample complexity analysis, and further present an enhanced result by incorporating variance reduction techniques. Our findings show that SHSODM matches the best-known sample complexity achieved by other second-order methods for gradient-dominated stochastic optimization but without cubic regularization. Empirically, since the homogenization approach only relies on solving extremal eigenvector problem at each iteration instead of Newton-type system, our methods gain the advantage of cheaper computational cost and robustness in ill-conditioned problems. Numerical experiments on several RL tasks demonstrate the better performance of SHSODM compared to other off-the-shelf methods.

SOLNP \(+\) is a derivative-free solver for constrained nonlinear optimization. It starts from SOLve Nonlinear Programming (SOLNP) proposed in 1989 by Ye. The main ideas are to use finite difference to approximate the gradient of the objective function and constraints, and use augmented Lagrangian method and sequential quadratic programming to deal with nonlinear constraints. We incorporate the techniques of implicit filtering, a new restart mechanism, and a modern quadratic programming solver into this new version with an ANSI C implementation. The algorithm exhibits a great advantage in running time and robustness under noise compared with the old version implemented in MATLAB. The numerical experiments show that SOLNP \(+\) is comparable with two widely used solvers, COBYLA and NOMAD. SOLNP \(+\) is available at https://github.com/COPT-Public/SOLNP_plus .

We study episodic two-player zero-sum Markov games (MGs) in the offline setting, where the goal is to find an approximate Nash equilibrium (NE) policy pair based on a dataset collected a priori. When the dataset does not have uniform coverage over all policy pairs, finding an approximate NE involves challenges in three aspects: (i) distributional shift between the behavior policy and the optimal policy, (ii) function approximation to handle large state space, and (iii) minimax optimization for equilibrium solving. We propose a pessimism-based algorithm, dubbed as pessimistic minimax value iteration (PMVI), which overcomes the distributional shift by constructing pessimistic estimates of the value functions for both players and outputs a policy pair by solving a correlated coarse equilibrium based on the two value functions. Furthermore, we establish a data-dependent upper bound on the suboptimality which recovers a sublinear rate without the assumption on uniform coverage of the dataset. We also prove an information-theoretical lower bound, which shows our upper bound is nearly minimax optimal, which suggests that the data-dependent term is intrinsic. Our theoretical results also highlight a notion of “relative uncertainty”, which characterizes the necessary and sufficient condition for achieving sample efficiency in offline MGs. To the best of our knowledge, we provide the first nearly minimax optimal result for offline MGs with function approximation.