Over the past decade there has been considerable interest in spectral algorithms for learning Predictive State Representations (PSRs). Spectral algorithms have appealing theoretical guarantees; however, the resulting models do not always perform well on inference tasks in practice. One reason for this behavior is the mismatch between the intended task (accurate filtering or prediction) and the loss function being optimized by the algorithm (estimation error in model parameters). A natural idea is to improve performance by refining PSRs using an algorithm such as EM. Unfortunately it is not obvious how to apply apply an EM style algorithm in the context of PSRs as the Log Likelihood is not well defined for all PSRs. We show that it is possible to overcome this problem using ideas from Predictive State Inference Machines. We combine spectral algorithms for PSRs as a consistent and efficient initialization with PSIM-style updates to refine the resulting model parameters. By combining these two ideas we develop Inference Gradients, a simple, fast, and robust method for practical learning of PSRs. Inference Gradients performs gradient descent in the PSR parameter space to optimize an inference-based loss function like PSIM. Because Inference Gradients uses a spectral initialization we get the same consistency benefits as PSRs. We show that Inference Gradients outperforms both PSRs and PSIMs on real and synthetic data sets.
@cite_8 propose a two pass algorithm for learning a dynamical system model, where they first learn a PSR using a spectral algorithm, then subsequently convert the PSR into a valid HMM using an exterior point method. Their approach is different from ours as they produce an HMM as the final model rather than a PSR. Additionally, their algorithm focuses on model parameter optimization rather than optimal performance on inference tasks.
@cite_8
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Probabilistic latent-variable models are a fundamental tool in statistics and machine learning. Despite their widespread use, identifying the parameters of basic latent variable models continues to be an extremely challenging problem. Traditional maximum likelihood-based learning algorithms find valid parameters, but suffer from high computational cost, slow convergence, and local optima. In contrast, recently developed spectral algorithms are computationally efficient and provide strong statistical guarantees, but are not guaranteed to find valid parameters. In this work, we introduce a two-stage learning algorithm for latent variable models. We first use a spectral method of moments algorithm to find a solution that is close to the optimal solution but not necessarily in the valid set of model parameters. We then incrementally refine the solution via an exterior point method until a local optima that is arbitrarily near the valid set of parameters is found. We perform several experiments on synthetic and real-world data and show that our approach is more accurate than previous work, especially when training data is limited.