Personalized decision policies are increasingly central in areas such as healthcare [Bertsimas et al., 2017], education[Mandeletal.,2014], andpublicpolicy[Kubeetal.,2019], wheretailoringactions to individual characteristics can improve outcomes. In many of these settings, however, actively experimenting with new policies to generate "online data" is expensive, risky, or infeasible, which motivates methods that can evaluate and optimize policies using pre-existing "offline data." A variety of work studies semiparametric efficient estimation of the value of a fixed policy from offline data [Chernozhukov et al., 2018, Dud ık et al., 2011, Jiang and Li, 2016, Kallus and Uehara, 2020, 2022, Kallus et al., 2022, Scharfstein et al., 1999]. And, a variety of work considers selecting the policy that optimizes such estimates over policies in a given class [Athey and Wager, 2021, Chernozhukov et al., 2019, Foster and Syrgkanis, 2023, Kallus, 2021, Zhang et al., 2013, Zhou et al., 2023], which generally yields rates the scale with policy class complexity, e.g., OP(N 1/2) for VC classes. Luedtke and Chambaz [2020] get regret acceleration to oP(N 1/2) by leveraging an equicontinuity argument.

The natural gradient method is widely used in statistical optimization, but its standard formulation assumes a Euclidean parameter space. This paper proposes an inversion-free stochastic natural gradient method for probability distributions whose parameters lie on a Riemannian manifold. The manifold setting offers several advantages: one can implicitly enforce parameter constraints such as positive definiteness and orthogonality, ensure parameters are identifiable, or guarantee regularity properties of the objective like geodesic convexity. Building on an intrinsic formulation of the Fisher information matrix (FIM) on a manifold, our method maintains an online approximation of the inverse FIM, which is efficiently updated at quadratic cost using score vectors sampled at successive iterates. In the Riemannian setting, these score vectors belong to different tangent spaces and must be combined using transport operations. We prove almost-sure convergence rates of $O(\log{s}/s^α)$ for the squared distance to the minimizer when the step size exponent $α>2/3$. We also establish almost-sure rates for the approximate FIM, which now accumulates transport-based errors. A limited-memory variant of the algorithm with sub-quadratic storage complexity is proposed. Finally, we demonstrate the effectiveness of our method relative to its Euclidean counterparts on variational Bayes with Gaussian approximations and normalizing flows.

However, their utility is limited by their entanglement with respect to different visual attributes.

Throughout this paper, all convexity-related concepts are defined in the Euclidean space.

Our method overcomes this limitation by sampling in a single function evaluation. The key innovation is to optimize a neural network via maximum likelihood on the manifold, possible by adapting the free-form flow framework to Riemannian manifolds.

The Wasserstein distance readily applies to rank-reduced databutlacksaffine-invariance.

Theoretical error bounds are provided when the tangent spaces of the manifold satisfy certain incoherence conditions.

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