In Section 3.2, we proposed the crossdistillation (XD) learning scheme. The distillation objective in Eq (10) is the inner decorrelation minimization between embeddings z and [ z]. In addition to the correlation-based distillation loss, we also investigate the negative logarithm(e.g, To avoid the unbalanced loss magnitude, the distillation loss is introduced as the regularization term controlled by the penalty level γ: L = LSACL(zA,zB)+γLCD (1) LCD = ( [ zA]logzA + [ zB]logzB)/2 (2) We empirically observe that the negative logarithm-based distillation loss failed to outperform the proposed cross-distillation loss LCD with inner-decorrelation minimization. As shown in the ImageNet-100 results below: Method Encoder # of Params (M) Linear Eval Acc.

Liu et al. [2018] first showed that stationary importance sampling methods can be viewed as Rao-Blackwellization of IS estimator, and claimed that the expectation of the likelihood-ratios conditioned on state and action is equal to the distribution ratio, as stated in Property 1. For completeness, we present a proof of Property 1. Recall that d This gives us the expression " This additional marginalization step over time allows us to consider time-independent distribution ratios. Then, using the law of total expectation, we can write the expectation of the second sum in (4) as: " Assumption 1. Plugging in the final expression from (5) back into (4) gives us " Note that in the infinite horizon setting where L!1and for finite n, (6) becomes " Similarly, by generalizing this pattern it can be observed that on unrolling n times, we will get, 1 " 0 X For all experiments, we utilize the domains and algorithm implementations from Caltech OPE Benchmarking Suite (COBS) library by Voloshin et al. [2019]. We include a brief description of each of these domains below, and a full description of each can be found in the work by Voloshin et al. [2019]. Graph Environment The Graph environment is a two-chain environment with 2L states and 2 actions.

The IDs of the 10 datasets used in this work, as well as the number of examples and features, are provided in Table 1 in the main manuscript. All of the datasets correspond to binary classification problems, with varying degrees of class imbalance. While the prediction is always performed in the logarithmic domain, when evaluating the models we transform both the labels and the model predictions back into their original domain. The loss function used for training and evaluation is the standard root mean-squared error (sklearn.metrics.mean_squared_error). We download the raw data programmatically using the Kaggle API, which produces the filetrain.tsv.