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This paper examines learning the optimal filtering policy, known as theKalman gain, for a linear system with unknown noise covariance matrices usingnoisy output data. The learning problem is formulated as a stochastic policyoptimization problem, aiming to minimize the output prediction error. Thisformulation provides a direct bridge between data-driven optimal control and,its dual, optimal filtering. Our contributions are twofold. Firstly, we conducta thorough convergence analysis of the stochastic gradient descent algorithm,adopted for the filtering problem, accounting for biased gradients andstability constraints. Secondly, we carefully leverage a combination of toolsfrom linear system theory and high-dimensional statistics to derivebias-variance error bounds that scale logarithmically with problem dimension,and, in contrast to subspace methods, the length of output trajectories onlyaffects the bias term.

Data-driven Optimal Filtering for Linear Systems with Unknown Noise Covariances