Partial Identification with Noisy Covariates: A Robust Optimization Approach

Causal inference from observational datasets often relies on measuring and adjusting for covariates. In practice, measurements of the covariates can often be noisy and/or biased, or only measurements of their proxies may be available. Directly adjusting for these imperfect measurements of the covariates can lead to biased causal estimates. Moreover, without additional assumptions, the causal effects are not point-identifiable due to the noise in these measurements. To this end, we study the partial identification of causal effects given noisy covariates, under a user-specified assumption on the noise level. The key observation is that we can formulate the identification of the average treatment effects (ATE) as a robust optimization problem. This formulation leads to an efficient robust optimization algorithm that bounds the ATE with noisy covariates. We show that this robust optimization approach can extend a wide range of causal adjustment methods to perform partial identification, including backdoor adjustment, inverse propensity score weighting, double machine learning, and front door adjustment. Across synthetic and real datasets, we find that this approach provides ATE bounds with a higher coverage probability than existing methods.

No-Regret Learning in Partially-Informed Auctions

Auctions with partially-revealed information about items are broadly employed in real-world applications, but the underlying mechanisms have limited theoretical support. In this work, we study a machine learning formulation of these types of mechanisms, presenting algorithms that are no-regret from the buyer's perspective. Specifically, a buyer who wishes to maximize his utility interacts repeatedly with a platform over a series of $T$ rounds. In each round, a new item is drawn from an unknown distribution and the platform publishes a price together with incomplete, "masked" information about the item. The buyer then decides whether to purchase the item. We formalize this problem as an online learning task where the goal is to have low regret with respect to a myopic oracle that has perfect knowledge of the distribution over items and the seller's masking function. When the distribution over items is known to the buyer and the mask is a SimHash function mapping $\mathbb{R}^d$ to $\{0,1\}^{\ell}$, our algorithm has regret $\tilde {\mathcal{O}}((Td\ell)^{\frac{1}{2}})$. In a fully agnostic setting when the mask is an arbitrary function mapping to a set of size $n$, our algorithm has regret $\tilde {\mathcal{O}}(T^{\frac{3}{4}}n^{\frac{1}{2}})$. Finally, when the prices are stochastic, the algorithm has regret $\tilde {\mathcal{O}}((Tn)^{\frac{1}{2}})$.

Conformal prediction for the design problem

In many real-world deployments of machine learning, we use a prediction algorithm to choose what data to test next. For example, in the protein design problem, we have a regression model that predicts some real-valued property of a protein sequence, which we use to propose new sequences believed to exhibit higher property values than observed in the training data. Since validating designed sequences in the wet lab is typically costly, it is important to know how much we can trust the model's predictions. In such settings, however, there is a distinct type of distribution shift between the training and test data: one where the training and test data are statistically dependent, as the latter is chosen based on the former. Consequently, the model's error on the test data -- that is, the designed sequences -- has some non-trivial relationship with its error on the training data. Herein, we introduce a method to quantify predictive uncertainty in such settings. We do so by constructing confidence sets for predictions that account for the dependence between the training and test data. The confidence sets we construct have finite-sample guarantees that hold for any prediction algorithm, even when a trained model chooses the test-time input distribution. As a motivating use case, we demonstrate how our method quantifies uncertainty for the predicted fitness of designed protein using several real data sets.

Improving Generalization via Uncertainty Driven Perturbations

Recently Shah et al., 2020 pointed out the pitfalls of the simplicity bias - the tendency of gradient-based algorithms to learn simple models - which include the model's high sensitivity to small input perturbations, as well as sub-optimal margins. In particular, while Stochastic Gradient Descent yields max-margin boundary on linear models, such guarantee does not extend to non-linear models. To mitigate the simplicity bias, we consider uncertainty-driven perturbations (UDP) of the training data points, obtained iteratively by following the direction that maximizes the model's estimated uncertainty. Unlike loss-driven perturbations, uncertainty-guided perturbations do not cross the decision boundary, allowing for using a larger range of values for the hyperparameter that controls the magnitude of the perturbation. Moreover, as real-world datasets have non-isotropic distances between data points of different classes, the above property is particularly appealing for increasing the margin of the decision boundary, which in turn improves the model's generalization. We show that UDP is guaranteed to achieve the maximum margin decision boundary on linear models and that it notably increases it on challenging simulated datasets. Interestingly, it also achieves competitive loss-based robustness and generalization trade-off on several datasets.

Transferred Q-learning

We consider $Q$-learning with knowledge transfer, using samples from a target reinforcement learning (RL) task as well as source samples from different but related RL tasks. We propose transfer learning algorithms for both batch and online $Q$-learning with offline source studies. The proposed transferred $Q$-learning algorithm contains a novel re-targeting step that enables vertical information-cascading along multiple steps in an RL task, besides the usual horizontal information-gathering as transfer learning (TL) for supervised learning. We establish the first theoretical justifications of TL in RL tasks by showing a faster rate of convergence of the $Q$ function estimation in the offline RL transfer, and a lower regret bound in the offline-to-online RL transfer under certain similarity assumptions. Empirical evidences from both synthetic and real datasets are presented to back up the proposed algorithm and our theoretical results.

Robust Estimation for Nonparametric Families via Generative Adversarial Networks

We provide a general framework for designing Generative Adversarial Networks (GANs) to solve high dimensional robust statistics problems, which aim at estimating unknown parameter of the true distribution given adversarially corrupted samples. Prior work focus on the problem of robust mean and covariance estimation when the true distribution lies in the family of Gaussian distributions or elliptical distributions, and analyze depth or scoring rule based GAN losses for the problem. Our work extend these to robust mean estimation, second moment estimation, and robust linear regression when the true distribution only has bounded Orlicz norms, which includes the broad family of sub-Gaussian, sub-Exponential and bounded moment distributions. We also provide a different set of sufficient conditions for the GAN loss to work: we only require its induced distance function to be a cumulative density function of some light-tailed distribution, which is easily satisfied by neural networks with sigmoid activation. In terms of techniques, our proposed GAN losses can be viewed as a smoothed and generalized Kolmogorov-Smirnov distance, which overcomes the computational intractability of the original Kolmogorov-Smirnov distance used in the prior work.

Reinforcement Learning with Heterogeneous Data: Estimation and Inference

Reinforcement Learning (RL) has the promise of providing data-driven support for decision-making in a wide range of problems in healthcare, education, business, and other domains. Classical RL methods focus on the mean of the total return and, thus, may provide misleading results in the setting of the heterogeneous populations that commonly underlie large-scale datasets. We introduce the K-Heterogeneous Markov Decision Process (K-Hetero MDP) to address sequential decision problems with population heterogeneity. We propose the Auto-Clustered Policy Evaluation (ACPE) for estimating the value of a given policy, and the Auto-Clustered Policy Iteration (ACPI) for estimating the optimal policy in a given policy class. Our auto-clustered algorithms can automatically detect and identify homogeneous sub-populations, while estimating the Q function and the optimal policy for each sub-population. We establish convergence rates and construct confidence intervals for the estimators obtained by the ACPE and ACPI. We present simulations to support our theoretical findings, and we conduct an empirical study on the standard MIMIC-III dataset. The latter analysis shows evidence of value heterogeneity and confirms the advantages of our new method.

Online Active Learning with Dynamic Marginal Gain Thresholding

The blessing of ubiquitous data also comes with a curse: the communication, storage, and labeling of massive, mostly redundant datasets. In our work, we seek to solve the problem at its source, collecting only valuable data and throwing out the rest, via active learning. We propose an online algorithm which, given any stream of data, any assessment of its value, and any formulation of its selection cost, extracts the most valuable subset of the stream up to a constant factor while using minimal memory. Notably, our analysis also holds for the federated setting, in which multiple agents select online from individual data streams without coordination and with potentially very different appraisals of cost. One particularly important use case is selecting and labeling training sets from unlabeled collections of data that maximize the test-time performance of a given classifier. In prediction tasks on ImageNet and MNIST, we show that our selection method outperforms random selection by up to 5-20%.

Nonconvex Stochastic Scaled-Gradient Descent and Generalized Eigenvector Problems

Motivated by the problem of online canonical correlation analysis, we propose the \emph{Stochastic Scaled-Gradient Descent} (SSGD) algorithm for minimizing the expectation of a stochastic function over a generic Riemannian manifold. SSGD generalizes the idea of projected stochastic gradient descent and allows the use of scaled stochastic gradients instead of stochastic gradients. In the special case of a spherical constraint, which arises in generalized eigenvector problems, we establish a nonasymptotic finite-sample bound of $\sqrt{1/T}$, and show that this rate is minimax optimal, up to a polylogarithmic factor of relevant parameters. On the asymptotic side, a novel trajectory-averaging argument allows us to achieve local asymptotic normality with a rate that matches that of Ruppert-Polyak-Juditsky averaging. We bring these ideas together in an application to online canonical correlation analysis, deriving, for the first time in the literature, an optimal one-time-scale algorithm with an explicit rate of local asymptotic convergence to normality. Numerical studies of canonical correlation analysis are also provided for synthetic data.