Estimating the density of a distribution from samples is a fundamental problem in statistics. In many practical settings, the Wasserstein distance is an appropriate error metric for density estimation. For example, when estimating population densities in a geographic region, a small Wasserstein distance means that the estimate is able to capture roughly where the population mass is. In this work we study differentially private density estimation in the Wasserstein distance. We design and analyze instance-optimal algorithms for this problem that can adapt to easy instances.

Then, we intend to calculate the constraint part. The algorithm for Licence method for single-target interventiion scenario is shown in Algorithm 1. The details of experimental baselines are demonstrated as follows. AIT [11] is an active learning method that utilize f-score to select intervention queries. REAL fidelity means the model always choose the highest fidelity to conduct experiments.

For example, to study the causal relations between the drugs and diseases, one can conduct clinical tests via selectively administering the medicines to the patients.

Scientific modeling applications often require estimating a distribution of parameters consistent with a dataset of observations--an inference task also known as source distribution estimation. This problem can be ill-posed, however, since many different source distributions might produce the same distribution of data-consistent simulations. To make a principled choice among many equally valid sources, we propose an approach which targets the maximum entropy distribution, i.e., prioritizes retaining as much uncertainty as possible.

V ov95, CBL06] and online convex optimization [Haz16, Ora19] have been developed. Until the labels of all examples of X have been predicted: The learning algorithm picks a point x X and makes a prediction z R about its label.

Bilevel optimization has recently regained interest owing to its applications in emerging machine learning fields such as hyperparameter optimization, meta-learning, and reinforcement learning. Recent results have shown that simple alternating (implicit) gradient-based algorithms can match the convergence rate of single-level gradient descent (GD) when addressing bilevel problems with a strongly convex lower-level objective. However, it remains unclear whether this result can be generalized to bilevel problems beyond this basic setting.