The Singular Value Decomposition is a key operation in many machine learning methods. Its computational cost, however, makes it unscalable and impractical for the massive-sized datasets becoming common in applications. We present a new method, QUIC-SVD, for fast approximation of the full SVD with automatic sample size minimization and empirical relative error control. Previous Monte Carlo approaches have not addressed the full SVD nor benefited from the efficiency of automatic, empirically-driven sample sizing. Our empirical tests show speedups of several orders of magnitude over exact SVD. Such scalability should enable QUIC-SVD to meet the needs of a wide array of methods and applications.

Machine learning contains many computational bottlenecks in the form of nested summations over datasets. Kernel estimators and other methods are burdened by these expensive computations. Exact evaluation is typically O(n2 ) or higher, which severely limits application to large datasets. We present a multi-stage stratified Monte Carlo method for approximating such summations with probabilistic relative error control. The essential idea is fast approximation by sampling in trees. This method differs from many previous scalability techniques (such as standard multi-tree methods) in that its error is stochastic, but we derive conditions for error control and demonstrate that they work. Further, we give a theoretical sample complexity for the method that is independent of dataset size, and show that this appears to hold in experiments, where speedups reach as high as 1014 , many orders of magnitude beyond the previous state of the art.

We present a solution for inferring hidden state from sensorimotor experience when the environment takes the form of a POMDP with deterministic transition and observation functions. Such environments can appear to be arbitrarily complex and non-deterministic on the surface, but are actually deterministic with respect to the unobserved underlying state. We show that there always exists a finite history-based representation that fully captures the unobserved world state, allowing for perfect prediction of action effects. This representation takes the form of a looping prediction suffix tree (PST). We derive a sound and complete algorithm for learning a looping PST from a sufficient sample of sensorimotor experience. We also give empirical illustrations of the advantages conferred by this approach, and characterize the approximations to the looping PST that are made by existing algorithms such as Variable Length Markov Models, Utile Suffix Memory and Causal State Splitting Reconstruction.

Schema learning is a way to discover probabilistic, constructivist, pre- dictive action models (schemas) from experience. It includes meth- ods for finding and using hidden state to make predictions more accu- rate. We extend the original schema mechanism [1] to handle arbitrary discrete-valued sensors, improve the original learning criteria to handle POMDP domains, and better maintain hidden state by using schema pre- dictions. These extensions show large improvement over the original schema mechanism in several rewardless POMDPs, and achieve very low prediction error in a difficult speech modeling task. Further, we compare extended schema learning to the recently introduced predictive state rep- resentations [2], and find their predictions of next-step action effects to be approximately equal in accuracy. This work lays the foundation for a schema-based system of integrated learning and planning.