  Estimating causal effects from observational data is a central problem in
many domains. A general approach is to balance covariates with weights such
that the distribution of the data mimics randomization. We present generalized
balancing weights, Neural Balancing Weights (NBW), to estimate the causal
effects of an arbitrary mixture of discrete and continuous interventions. The
weights were obtained through direct estimation of the density ratio between
the source and balanced distributions by optimizing the variational
representation of $f$-divergence. For this, we selected $\alpha$-divergence as
it presents efficient optimization because it has an estimator whose sample
complexity is independent of its ground truth value and unbiased mini-batch
gradients; moreover, it is advantageous for the vanishing-gradient problem. In
addition, we provide the following two methods for estimating the balancing
weights: improving the generalization performance of the balancing weights and
checking the balance of the distribution changed by the weights. Finally, we
discuss the sample size requirements for the weights as a general problem of a
curse of dimensionality when balancing multidimensional data. Our study
provides a basic approach for estimating the balancing weights of
multidimensional data using variational $f$-divergences.
