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We study the task of learning state representations from potentially high-dimensional observations, with the goal of controlling an unknown partially observable system. We pursue a direct latent model learning approach, where a dynamic model in some latent state space is learned by predicting quantities directly related to planning (e.g., costs) without reconstructing the observations. In particular, we focus on an intuitive cost-driven state representation learning method for solving Linear Quadratic Gaussian (LQG) control, one of the most fundamental partially observable control problems. As our main results, we establish finite-sample guarantees of finding a near-optimal state representation function and a near-optimal controller using the directly learned latent model. To the best of our knowledge, despite various empirical successes, prior to this work it was unclear if such a cost-driven latent model learner enjoys finite-sample guarantees. Our work underscores the value of predicting multi-step costs, an idea that is key to our theory, and notably also an idea that is known to be empirically valuable for learning state representations. 

We study projection-free methods for constrained Riemannian optimization. In particular, we propose a Riemannian Frank-Wolfe (RFW) method that handles constraints directly, in contrast to prior methods that rely on (potentially costly) projections. We analyze non-asymptotic convergence rates ofRFWto an optimum for geodesically convex problems, and to a critical point for nonconvex objectives. We also present a practical setting under whichRFWcan attain a linear convergence rate. As a concrete example, we specializeRFWto the manifold of positive definite matrices and apply it to two tasks: (i) computing the matrix geometric mean (Riemannian centroid); and (ii) computing the Bures-Wasserstein barycenter. Both tasks involve geodesically convex interval constraints, for which we show that the Riemannian “linear” oracle required byRFWadmits a closed form solution; this result may be of independent interest. We complement our theoretical results with an empirical comparison ofRFWagainst state-of-the-art Riemannian optimization methods, and observe thatRFWperforms competitively on the task of computing Riemannian centroids.

 

Standard contrastive learning approaches usually require a large number of negatives for effective unsupervised learning and often exhibit slow convergence. We suspect this behavior is due to the suboptimal selection of negatives used for offering contrast to the positives. We counter this difficulty by taking inspiration from support vector machines (SVMs) to present max-margin contrastive learning (MMCL). Our approach selects negatives as the sparse support vectors obtained via a quadratic optimization problem, and contrastiveness is enforced by maximizing the decision margin. As SVM optimization can be computationally demanding, especially in an end-to-end setting, we present simplifications that alleviate the computational burden. We validate our approach on standard vision benchmark datasets, demonstrating better performance in unsupervised representation learning over state-of-the-art, while having better empirical convergence properties.