We present GradientDICE for estimating the density ratio between the state distribution of the target policy and the sampling distribution in off-policy reinforcement learning. GradientDICE fixes several problems of GenDICE (Zhang et al., 2020), the state-of-the-art for estimating such density ratios. Namely, the optimization problem in GenDICE is not a convex-concave saddle-point problem once nonlinearity in optimization variable parameterization is introduced to ensure positivity, so any primal-dual algorithm is not guaranteed to converge or find the desired solution. However, such nonlinearity is essential to ensure the consistency of GenDICE even with a tabular representation. This is a fundamental contradiction, resulting from GenDICEs original formulation of the optimization problem. In GradientDICE, we optimize a different objective from GenDICE by using the Perron-Frobenius theorem and eliminating GenDICEs use of divergence. Consequently, nonlinearity in parameterization is not necessary for GradientDICE, which is provably convergent under linear function approximation.
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