Multiscale Markov Decision Problems: Compression, Solution, and Transfer Learning

Many problems in sequential decision making and stochastic control often have natural multiscale structure: sub-tasks are assembled together to accomplish complex goals. Systematically inferring and leveraging hierarchical structure, particularly beyond a single level of abstraction, has remained a longstanding challenge. We describe a fast multiscale procedure for repeatedly compressing, or homogenizing, Markov decision processes (MDPs), wherein a hierarchy of sub-problems at different scales is automatically determined. Coarsened MDPs are themselves independent, deterministic MDPs, and may be solved using existing algorithms. The multiscale representation delivered by this procedure decouples sub-tasks from each other and can lead to substantial improvements in convergence rates both locally within sub-problems and globally across sub-problems, yielding significant computational savings. A second fundamental aspect of this work is that these multiscale decompositions yield new transfer opportunities across different problems, where solutions of sub-tasks at different levels of the hierarchy may be amenable to transfer to new problems. Localized transfer of policies and potential operators at arbitrary scales is emphasized. Finally, we demonstrate compression and transfer in a collection of illustrative domains, including examples involving discrete and continuous statespaces.Comment: 86 pages, 15 figure

Synchronization and Noise: A Mechanism for Regularization in Neural Systems

To learn and reason in the presence of uncertainty, the brain must be capable of imposing some form of regularization. Here we suggest, through theoretical and computational arguments, that the combination of noise with synchronization provides a plausible mechanism for regularization in the nervous system. The functional role of regularization is considered in a general context in which coupled computational systems receive inputs corrupted by correlated noise. Noise on the inputs is shown to impose regularization, and when synchronization upstream induces time-varying correlations across noise variables, the degree of regularization can be calibrated over time. The proposed mechanism is explored first in the context of a simple associative learning problem, and then in the context of a hierarchical sensory coding task. The resulting qualitative behavior coincides with experimental data from visual cortex.Comment: 32 pages, 7 figures. under revie

Multiple Resolution Image Classification

Binary image classifiction is a problem that has received much attention in recent years. In this paper we evaluate a selection of popular techniques in an effort to find a feature set/ classifier combination which generalizes well to full resolution image data. We then apply that system to images at one-half through one-sixteenth resolution, and consider the corresponding error rates. In addition, we further observe generalization performance as it depends on the number of training images, and lastly, compare the system's best error rates to that of a human performing an identical classification task given teh same set of test images

Notes on the Shannon Entropy of the Neural Response

In these notes we focus on the concept of Shannon entropy in an attempt to provide a systematic way of assessing the discrimination properties of the neural response, and quantifying the role played by the number of layers and the number of templates

Learning and Invariance in a Family of Hierarchical Kernels

Understanding invariance and discrimination properties of hierarchical models is arguably the key to understanding how and why such models, of which the the mammalian visual system is one instance, can lead to good generalization properties and reduce the sample complexity of a given learning task. In this paper we explore invariance to transformation and the role of layer-wise embeddings within an abstract framework of hierarchical kernels motivated by the visual cortex. Here a novel form of invariance is induced by propagating the effect of locally defined, invariant kernels throughout a hierarchy. We study this notion of invariance empirically. We then present an extension of the abstract hierarchical modeling framework to incorporate layer-wise embeddings, which we demonstrate can lead to improved generalization and scalable algorithms. Finally we analyze experimentally sample complexity properties as a function of architectural parameters

Generalization and Properties of the Neural Response

Hierarchical learning algorithms have enjoyed tremendous growth in recent years, with many new algorithms being proposed and applied to a wide range of applications. However, despite the apparent success of hierarchical algorithms in practice, the theory of hierarchical architectures remains at an early stage. In this paper we study the theoretical properties of hierarchical algorithms from a mathematical perspective. Our work is based on the framework of hierarchical architectures introduced by Smale et al. in the paper "Mathematics of the Neural Response", Foundations of Computational Mathematics, 2010. We propose a generalized definition of the neural response and derived kernel that allows us to integrate some of the existing hierarchical algorithms in practice into our framework. We then use this generalized definition to analyze the theoretical properties of hierarchical architectures. Our analysis focuses on three particular aspects of the hierarchy. First, we show that a wide class of architectures suffers from range compression; essentially, the derived kernel becomes increasingly saturated at each layer. Second, we show that the complexity of a linear architecture is constrained by the complexity of the first layer, and in some cases the architecture collapses into a single-layer linear computation. Finally, we characterize the discrimination and invariance properties of the derived kernel in the case when the input data are one-dimensional strings. We believe that these theoretical results will provide a useful foundation for guiding future developments within the theory of hierarchical algorithms

Mathematics of the Neural Response

We propose a natural image representation, the neural response, motivated by the neuroscience of the visual cortex. The inner product defined by the neural response leads to a similarity measure between functions which we call the derived kernel. Based on a hierarchical architecture, we give a recursive definition of the neural response and associated derived kernel. The derived kernel can be used in a variety of application domains such as classification of images, strings of text and genomics data

Phonetic Classification Using Hierarchical, Feed-forward, Spectro-temporal Patch-based Architectures

A preliminary set of experiments are described in which a biologically-inspired computer vision system (Serre, Wolf et al. 2005; Serre 2006; Serre, Oliva et al. 2006; Serre, Wolf et al. 2006) designed for visual object recognition was applied to the task of phonetic classification. During learning, the systemprocessed 2-D wideband magnitude spectrograms directly as images, producing a set of 2-D spectrotemporal patch dictionaries at different spectro-temporal positions, orientations, scales, and of varying complexity. During testing, features were computed by comparing the stored patches with patches fromnovel spectrograms. Classification was performed using a regularized least squares classifier (Rifkin, Yeo et al. 2003; Rifkin, Schutte et al. 2007) trained on the features computed by the system. On a 20-class TIMIT vowel classification task, the model features achieved a best result of 58.74% error, compared to 48.57% error using state-of-the-art MFCC-based features trained using the same classifier. This suggests that hierarchical, feed-forward, spectro-temporal patch-based architectures may be useful for phoneticanalysis