On the Relationship Between Generalization Error, Hypothesis Complexity, and Sample Complexity for Radial Basis Functions

In this paper, we bound the generalization error of a class of Radial Basis Function networks, for certain well defined function learning tasks, in terms of the number of parameters and number of examples. We show that the total generalization error is partly due to the insufficient representational capacity of the network (because of its finite size) and partly due to insufficient information about the target function (because of finite number of samples). We make several observations about generalization error which are valid irrespective of the approximation scheme. Our result also sheds light on ways to choose an appropriate network architecture for a particular problem.

Sharing visual features for multiclass and multiview object detection

We consider the problem of detecting a large number of different classes of objects in cluttered scenes. Traditional approaches require applying a battery of different classifiers to the image, at multiple locations and scales. This can be slow and can require a lot of training data, since each classifier requires the computation of many different image features. In particular, for independently trained detectors, the (run-time) computational complexity, and the (training-time) sample complexity, scales linearly with the number of classes to be detected. It seems unlikely that such an approach will scale up to allow recognition of hundreds or thousands of objects. We present a multi-class boosting procedure (joint boosting) that reduces the computational and sample complexity, by finding common features that can be shared across the classes (and/or views). The detectors for each class are trained jointly, rather than independently. For a given performance level, the total number of features required, and therefore the computational cost, is observed to scale approximately logarithmically with the number of classes. The features selected jointly are closer to edges and generic features typical of many natural structures instead of finding specific object parts. Those generic features generalize better and reduce considerably the computational cost of an algorithm for multi-class object detection.

Reinforcement Learning by Policy Search

One objective of artificial intelligence is to model the behavior of an intelligent agent interacting with its environment. The environment's transformations can be modeled as a Markov chain, whose state is partially observable to the agent and affected by its actions; such processes are known as partially observable Markov decision processes (POMDPs). While the environment's dynamics are assumed to obey certain rules, the agent does not know them and must learn. In this dissertation we focus on the agent's adaptation as captured by the reinforcement learning framework. This means learning a policy---a mapping of observations into actions---based on feedback from the environment. The learning can be viewed as browsing a set of policies while evaluating them by trial through interaction with the environment. The set of policies is constrained by the architecture of the agent's controller. POMDPs require a controller to have a memory. We investigate controllers with memory, including controllers with external memory, finite state controllers and distributed controllers for multi-agent systems. For these various controllers we work out the details of the algorithms which learn by ascending the gradient of expected cumulative reinforcement. Building on statistical learning theory and experiment design theory, a policy evaluation algorithm is developed for the case of experience re-use. We address the question of sufficient experience for uniform convergence of policy evaluation and obtain sample complexity bounds for various estimators. Finally, we demonstrate the performance of the proposed algorithms on several domains, the most complex of which is simulated adaptive packet routing in a telecommunication network.

Sequential Optimal Recovery: A Paradigm for Active Learning

In most classical frameworks for learning from examples, it is assumed that examples are randomly drawn and presented to the learner. In this paper, we consider the possibility of a more active learner who is allowed to choose his/her own examples. Our investigations are carried out in a function approximation setting. In particular, using arguments from optimal recovery (Micchelli and Rivlin, 1976), we develop an adaptive sampling strategy (equivalent to adaptive approximation) for arbitrary approximation schemes. We provide a general formulation of the problem and show how it can be regarded as sequential optimal recovery. We demonstrate the application of this general formulation to two special cases of functions on the real line 1) monotonically increasing functions and 2) functions with bounded derivative. An extensive investigation of the sample complexity of approximating these functions is conducted yielding both theoretical and empirical results on test functions. Our theoretical results (stated insPAC-style), along with the simulations demonstrate the superiority of our active scheme over both passive learning as well as classical optimal recovery. The analysis of active function approximation is conducted in a worst-case setting, in contrast with other Bayesian paradigms obtained from optimal design (Mackay, 1992).

A Formulation for Active Learning with Applications to Object Detection

We discuss a formulation for active example selection for function learning problems. This formulation is obtained by adapting Fedorov's optimal experiment design to the learning problem. We specifically show how to analytically derive example selection algorithms for certain well defined function classes. We then explore the behavior and sample complexity of such active learning algorithms. Finally, we view object detection as a special case of function learning and show how our formulation reduces to a useful heuristic to choose examples to reduce the generalization error.

The Effect of Indexing on the Complexity of Object Recognition

Many current recognition systems use constrained search to locate objects in cluttered environments. Previous formal analysis has shown that the expected amount of search is quadratic in the number of model and data features, if all the data is known to come from a sinlge object, but is exponential when spurious data is included. If one can group the data into subsets likely to have come from a single object, then terminating the search once a "good enough" interpretation is found reduces the expected search to cubic. Without successful grouping, terminated search is still exponential. These results apply to finding instances of a known object in the data. In this paper, we turn to the problem of selecting models from a library, and examine the combinatorics of determining that a candidate object is not present in the data. We show that the expected search is again exponential, implying that naﶥ approaches to indexing are likely to carry an expensive overhead, since an exponential amount of work is needed to week out each of the incorrect models. The analytic results are shown to be in agreement with empirical data for cluttered object recognition.

The Informational Complexity of Learning from Examples

This thesis attempts to quantify the amount of information needed to learn certain tasks. The tasks chosen vary from learning functions in a Sobolev space using radial basis function networks to learning grammars in the principles and parameters framework of modern linguistic theory. These problems are analyzed from the perspective of computational learning theory and certain unifying perspectives emerge.

Complexity of Human Language Comprehension

The goal of this article is to reveal the computational structure of modern principle-and-parameter (Chomskian) linguistic theories: what computational problems do these informal theories pose, and what is the underlying structure of those computations? To do this, I analyze the computational complexity of human language comprehension: what linguistic representation is assigned to a given sound? This problem is factored into smaller, interrelated (but independently statable) problems. For example, in order to understand a given sound, the listener must assign a phonetic form to the sound; determine the morphemes that compose the words in the sound; and calculate the linguistic antecedent of every pronoun in the utterance. I prove that these and other subproblems are all NP-hard, and that language comprehension is itself PSPACE-hard.

The Complexity of Safety Stock Placement in General-Network Supply Chains

We consider the optimization problem of safety stock placement in a supply chain, as formulated in [1]. We prove that this problem is NP-Hard for supply chains modeled as general acyclic networks. Thus, we do not expect to find a polynomial-time algorithm for safety stock placement for a general-network supply chain.