We consider high dimensional dynamic multi-product pricing with an evolving but low-dimensional linear demand model. Assuming the temporal variation in cross-elasticities exhibits low-rank structure based on fixed (latent) features of the products, we show that the revenue maximization problem reduces to an online bandit convex optimization with side information given by the observed demands. We design dynamic pricing algorithms whose revenue approaches that of the best fixed price vector in hindsight, at a rate that only depends on the intrinsic rank of the demand model and not the number of products. Our approach applies a bandit convex optimization algorithm in a projected low-dimensional space spanned by the latent product features, while simultaneously learning this span via online singular value decomposition of a carefully-crafted matrix containing the observed demands.

There has been growing interest in how economists can import machine learning tools designed for prediction to accelerate and automate the model selection process, while still retaining desirable inference properties for causal parameters. Focusing on partially linear models, we extend the Double ML framework to allow for (1) a number of treatments that may grow with the sample size and (2) the analysis of panel data under sequentially exogenous errors. Our low-dimensional treatment (LD) regime directly extends the work in [Chernozhukov et al., 2016], by showing that the coefficients from a second stage, ordinary least squares estimator attain root-n convergence and desired coverage even if the dimensionality of treatment is allowed to grow. In a high-dimensional sparse (HDS) regime, we show that second stage LASSO and debiased LASSO have asymptotic properties equivalent to oracle estimators with no upstream error. We argue that these advances make Double ML methods a desirable alternative for practitioners estimating short-term demand elasticities in non-contractual settings.

Estimators computed from adaptively collected data do not behave like their non-adaptive brethren. Rather, the sequential dependence of the collection policy can lead to severe distributional biases that persist even in the infinite data limit. We develop a general method decorrelation procedure -- W-decorrelation -- for transforming the bias of adaptive linear regression estimators into variance. The method uses only coarse-grained information about the data collection policy and does not need access to propensity scores or exact knowledge of the policy. We bound the finite-sample bias and variance of the W-estimator and develop asymptotically correct confidence intervals based on a novel martingale central limit theorem. We then demonstrate the empirical benefits of the generic W-decorrelation procedure in two different adaptive data settings: the multi-armed bandits and autoregressive time series models.

We are in the middle of a remarkable rise in the use and capability of artificial intelligence. Much of this growth has been fueled by the success of deep learning architectures: models that map from observables to outputs via multiple layers of latent representations. These deep learning algorithms are effective tools for unstructured prediction, and they can be combined in AI systems to solve complex automated reasoning problems. This paper provides a recipe for combining ML algorithms to solve for causal effects in the presence of instrumental variables -- sources of treatment randomization that are conditionally independent from the response. We show that a flexible IV specification resolves into two prediction tasks that can be solved with deep neural nets: a first-stage network for treatment prediction and a second-stage network whose loss function involves integration over the conditional treatment distribution. This Deep IV framework imposes some specific structure on the stochastic gradient descent routine used for training, but it is general enough that we can take advantage of off-the-shelf ML capabilities and avoid extensive algorithm customization. We outline how to obtain out-of-sample causal validation in order to avoid over-fit. We also introduce schemes for both Bayesian and frequentist inference: the former via a novel adaptation of dropout training, and the latter via a data splitting routine.