Inventory control using discrete-time models is a wellstudied problem, where orders of items to hold in stock must anticipate future demand [1, 2]. By defining the costs of insufficient stocks, it is possible to find cost-minimizing policies using dynamic programming [3, 4, 5]. In practice, however, maintaining a certain service level of an inventory control system is a greater priority than cost minimization [6, 7]. Under certain restrictive assumptions on the demand process - such as memoryless and identically distributed demand - there are explicit formulations of the duality between service levels and costs [8].

We present an efficient parameter-free approach for statistical learning from corrupted training sets. We identify corrupted and non-corrupted samples using latent Bernoulli variables, and therefore formulate the robust learning problem as maximization of the likelihood where latent variables are marginalized out. The resulting optimization problem is solved via variational inference using an efficient Expectation-Maximization based method. The proposed approach improves over the state-of-the-art by automatically inferring the corruption level and identifying outliers, while adding minimal computational overhead. We demonstrate our robust learning method on a wide variety of machine learning tasks including online learning and deep learning where it exhibits ability to adapt to different levels of noise and attain high prediction accuracy.

Randomized trials are widely considered as the gold standard for evaluating the effects of decision policies. Trial data is, however, drawn from a population which may differ from the intended target population and this raises a problem of external validity (aka. generalizability). In this paper we seek to use trial data to draw valid inferences about the outcome of a policy on the target population. Additional covariate data from the target population is used to model the sampling of individuals in the trial study. We develop a method that yields certifiably valid trial-based policy evaluations under any specified range of model miscalibrations. The method is nonparametric and the validity is assured even with finite samples. The certified policy evaluations are illustrated using both simulated and real data.

Assessment of model fitness is a key part of machine learning. The standard paradigm of model evaluation is analysis of the average loss over future data. This is often explicit in model fitting, where we select models that minimize the average loss over training data as a surrogate, but comes with limited theoretical guarantees. In this paper, we consider the problem of characterizing a batch of out-of-sample losses of a model using a calibration data set. We provide finite-sample limits on the out-of-sample losses that are statistically valid under quite general conditions and propose a diagonistic tool that is simple to compute and interpret. Several numerical experiments are presented to show how the proposed method quantifies the impact of distribution shifts, aids the analysis of regression, and enables model selection as well as hyperparameter tuning.

State-of-the-art machine learning models can be vulnerable to very small input perturbations that are adversarially constructed. Adversarial training is an effective approach to defend against it. Formulated as a min-max problem, it searches for the best solution when the training data were corrupted by the worst-case attacks. Linear models are among the simple models where vulnerabilities can be observed and are the focus of our study. In this case, adversarial training leads to a convex optimization problem which can be formulated as the minimization of a finite sum. We provide a comparative analysis between the solution of adversarial training in linear regression and other regularization methods. Our main findings are that: (A) Adversarial training yields the minimum-norm interpolating solution in the overparameterized regime (more parameters than data), as long as the maximum disturbance radius is smaller than a threshold. And, conversely, the minimum-norm interpolator is the solution to adversarial training with a given radius. (B) Adversarial training can be equivalent to parameter shrinking methods (ridge regression and Lasso). This happens in the underparametrized region, for an appropriate choice of adversarial radius and zero-mean symmetrically distributed covariates. (C) For $\ell_\infty$-adversarial training -- as in square-root Lasso -- the choice of adversarial radius for optimal bounds does not depend on the additive noise variance. We confirm our theoretical findings with numerical examples.

We consider the problem of evaluating the performance of a decision policy using past observational data. The outcome of a policy is measured in terms of a loss (aka. disutility or negative reward) and the main problem is making valid inferences about its out-of-sample loss when the past data was observed under a different and possibly unknown policy. Using a sample-splitting method, we show that it is possible to draw such inferences with finite-sample coverage guarantees about the entire loss distribution, rather than just its mean. Importantly, the method takes into account model misspecifications of the past policy - including unmeasured confounding. The evaluation method can be used to certify the performance of a policy using observational data under a specified range of credible model assumptions.

We consider the problem of finding tuned regularized parameter estimators for linear models. We start by showing that three known optimal linear estimators belong to a wider class of estimators that can be formulated as a solution to a weighted and constrained minimization problem. The optimal weights, however, are typically unknown in many applications. This begs the question, how should we choose the weights using only the data? We propose using the covariance fitting SPICE-methodology to obtain data-adaptive weights and show that the resulting class of estimators yields tuned versions of known regularized estimators - such as ridge regression, LASSO, and regularized least absolute deviation. These theoretical results unify several important estimators under a common umbrella. The resulting tuned estimators are also shown to be practically relevant by means of a number of numerical examples.

The paper considers the problem of multi-objective decision support when outcomes are uncertain. We extend the concept of Pareto-efficient decisions to take into account the uncertainty of decision outcomes across varying contexts. This enables quantifying trade-offs between decisions in terms of tail outcomes that are relevant in safety-critical applications. We propose a method for learning efficient decisions with statistical confidence, building on results from the conformal prediction literature. The method adapts to weak or nonexistent context covariate overlap and its statistical guarantees are evaluated using both synthetic and real data.

We consider the problem of learning from training data obtained in different contexts, where the test data is subject to distributional shifts. We develop a distributionally robust method that focuses on excess risks and achieves a more appropriate trade-off between performance and robustness than the conventional and overly conservative minimax approach. The proposed method is computationally feasible and provides statistical guarantees. We demonstrate its performance using both real and synthetic data.

Conventional methods in causal effect inference typically rely on specifying a valid set of adjustment variables. When this set is unknown or misspecified, inferences will be erroneous. We propose a method for inferring average causal effects when the adjustment set is unknown. When the data-generating process belongs to the class of acyclical linear structural equation models, we prove that the method yields asymptotically valid confidence intervals. Our results build upon a smooth characterization of linear acyclic directed graphs. We verify the capability of the method to produce valid confidence intervals for average causal effects using synthetic data, even when the appropriate adjustment sets are unknown.