Rajarshi Mukherjee

Kinjal Basu, Rajarshi Mukherjee

In a very recent work, Basu and Owen (2015) propose the use of scrambled geometric nets in numerical integration when the domain is a product of $s$ arbitrary spaces of dimension $d$ having a certain partitioning constraint. It was shown that for a class of smooth functions, the integral estimate has variance $O( n^{-1 -2/d} (\log n)^{s-1})$ for scrambled geometric nets, compared to $O(n^{-1})$ for ordinary Monte Carlo. The main idea of this paper is to develop on the work by Loh (2003), to show that the scrambled geometric net estimate has an asymptotic normal distribution for certain smooth functions defined on products of suitable subsets of $\mathbb{R}^d$.

Rebekka Burkholz, Nilanjana Laha, Rajarshi Mukherjee et al.

The lottery ticket hypothesis conjectures the existence of sparse subnetworks of large randomly initialized deep neural networks that can be successfully trained in isolation. Recent work has experimentally observed that some of these tickets can be practically reused across a variety of tasks, hinting at some form of universality. We formalize this concept and theoretically prove that not only do such universal tickets exist but they also do not require further training. Our proofs introduce a couple of technical innovations related to pruning for strong lottery tickets, including extensions of subset sum results and a strategy to leverage higher amounts of depth. Our explicit sparse constructions of universal function families might be of independent interest, as they highlight representational benefits induced by univariate convolutional architectures.

Aaron Sonabend-W, Nilanjana Laha, Ashwin N. Ananthakrishnan et al.

Reinforcement learning (RL) has shown great success in estimating sequential treatment strategies which take into account patient heterogeneity. However, health-outcome information, which is used as the reward for reinforcement learning methods, is often not well coded but rather embedded in clinical notes. Extracting precise outcome information is a resource intensive task, so most of the available well-annotated cohorts are small. To address this issue, we propose a semi-supervised learning (SSL) approach that efficiently leverages a small sized labeled data with true outcome observed, and a large unlabeled data with outcome surrogates. In particular, we propose a semi-supervised, efficient approach to Q-learning and doubly robust off policy value estimation. Generalizing SSL to sequential treatment regimes brings interesting challenges: 1) Feature distribution for Q-learning is unknown as it includes previous outcomes. 2) The surrogate variables we leverage in the modified SSL framework are predictive of the outcome but not informative to the optimal policy or value function. We provide theoretical results for our Q-function and value function estimators to understand to what degree efficiency can be gained from SSL. Our method is at least as efficient as the supervised approach, and moreover safe as it robust to mis-specification of the imputation models.

Lin Liu, Rajarshi Mukherjee, James M. Robins

For many causal effect parameters of interest, doubly robust machine learning (DRML) estimators $\hatψ_{1}$ are the state-of-the-art, incorporating the good prediction performance of machine learning; the decreased bias of doubly robust estimators; and the analytic tractability and bias reduction of sample splitting with cross fitting. Nonetheless, even in the absence of confounding by unmeasured factors, the nominal $(1 - α)$ Wald confidence interval $\hatψ_{1} \pm z_{α/ 2} \widehat{\mathsf{se}} [\hatψ_{1}]$ may still undercover even in large samples, because the bias of $\hatψ_{1}$ may be of the same or even larger order than its standard error of order $n^{-1/2}$. In this paper, we introduce essentially assumption-free tests that (i) can falsify the null hypothesis that the bias of $\hatψ_{1}$ is of smaller order than its standard error, (ii) can provide an upper confidence bound on the true coverage of the Wald interval, and (iii) are valid under the null under no smoothness/sparsity assumptions on the nuisance parameters. The tests, which we refer to as \underline{A}ssumption \underline{F}ree \underline{E}mpirical \underline{C}overage \underline{T}ests (AFECTs), are based on a U-statistic that estimates part of the bias of $\hatψ_{1}$.

Yanjun Han, Jiantao Jiao, Rajarshi Mukherjee

We provide a complete picture of asymptotically minimax estimation of $L_r$-norms (for any $r\ge 1$) of the mean in Gaussian white noise model over Nikolskii-Besov spaces. In this regard, we complement the work of Lepski, Nemirovski and Spokoiny (1999), who considered the cases of $r=1$ (with poly-logarithmic gap between upper and lower bounds) and $r$ even (with asymptotically sharp upper and lower bounds) over Hölder spaces. We additionally consider the case of asymptotically adaptive minimax estimation and demonstrate a difference between even and non-even $r$ in terms of an investigator's ability to produce asymptotically adaptive minimax estimators without paying a penalty.