Symbol Description Vt Set of observable variables at time t V0:TUnion of observable variables at time t= 0,...,T Xt Manipulative variables at time t Yt Target variable at time t P(Xt) Power set of Xt Mt Set of MIS sets at time t Xs,ts-th intervention set at time t In this section we give the proof for Theorem 1 in the main text. This means that W includes those variables that are parents of Yt but are nor target at previous time steps nor intervened variables. In the following proof the values of IV0:t 1, XPYs,t, IPY0:t 1 and W are denoted by i, xPY, iPY and w respectively. Finally, fYY and fNYYare the functions in the SCM for Yt (see Assumptions (1) in the main text). Eq. (2) follows from the Eq. Finally, noticing that p(yPTt |I0:t 1) is the distribution targeted when optimizing the objective function at previous time steps one can obtain Eq. (6). The derivations above show how the objective function at time t is given by the expected value of the output of the functional relationship fNYYwhere the expectation is taken with respect to the variables that are not intervened on. This expectation is then shifted to account for the interventions implemented in the system at previous time steps that are affecting the target variable through fYY .

Increasingly, copious amounts of consumer data are used to learn high-revenue mechanisms via machine learning. Existing research on mechanism design via machine learning assumes that there is a distribution over the buyers' values for the items for sale and that the learning algorithm's input is a training set sampled from this distribution. This setup makes the strong assumption that no noise is introduced during data collection. In order to help place mechanism design via machine learning on firm foundations, we investigate the extent to which this learning process is robust to noise. Optimizing revenue using noisy data is challenging because revenue functions are extremely volatile: an infinitesimal change in the buyers' values can cause a steep drop in revenue. Nonetheless, we provide guarantees when arbitrarily correlated noise is added to the training set; we only require that the noise has bounded magnitude or is sub-Gaussian. We conclude with an application of our guarantees to multi-task mechanism design, where there are multiple distributions over buyers' values and the goal is to learn a high-revenue mechanism per distribution. To our knowledge, we are the first to study mechanism design via machine learning with noisy data as well as multi-task mechanism design.

Increasingly, copious amounts of consumer data are used to learn high-revenue mechanisms via machine learning. Existing research on mechanism design via machine learning assumes that there is a distribution over the buyers' values for the items for sale and that the learning algorithm's input is a training set sampled from this distribution. This setup makes the strong assumption that no noise is introduced during data collection. In order to help place mechanism design via machine learning on firm foundations, we investigate the extent to which this learning process is robust to noise. Optimizing revenue using noisy data is challenging because revenue functions are extremely volatile: an infinitesimal change in the buyers' values can cause a steep drop in revenue. Nonetheless, we provide guarantees when arbitrarily correlated noise is added to the training set; we only require that the noise has bounded magnitude or is sub-Gaussian. We conclude with an application of our guarantees to multi-task mechanism design, where there are multiple distributions over buyers' values and the goal is to learn a high-revenue mechanism per distribution. To our knowledge, we are the first to study mechanism design via machine learning with noisy data as well as multi-task mechanism design.

For each image's list of candidate objects, we heuristically downsample to a set of "most interesting" regions by: 1) selecting the at-most k " 4 largest/most central people; 2) keeping the most central/large objects; 3) over-sampling rarer objects according to prior frequency of detection in the LVIS vocabulary; 4) limiting the number of objects of a single type per-image; and 5) downsampling overlapping region proposals to encourage broader coverage of the pixel area of the image.