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

Interactive Segmentation in Multimodal Medical Imagery Using a Bayesian Transductive Learning Approach

Labeled training data in the medical domain is rare and expensive to obtain. The lack of labeled multimodal medical image data is a major obstacle for devising learning-based interactive segmentation tools. Transductive learning (TL) or semi-supervised learning (SSL) offers a workaround by leveraging unlabeled and labeled data to infer labels for the test set given a small portion of label information. In this paper we propose a novel algorithm for interactive segmentation using transductive learning and inference in conditional mixture nave Bayes models (T-CMNB) with spatial regularization constraints. T-CMNB is an extension of the transductive nave Bayes algorithm [1, 20]. The multimodal Gaussian mixture assumption on the class-conditional likelihood and spatial regularization constraints allow us to explain more complex distributions required for spatial classification in multimodal imagery. To simplify the estimation we reduce the parameter space by assuming nave conditional independence between the feature space and the class label. The nave conditional independence assumption allows efficient inference of marginal and conditional distributions for large scale learning and inference [19]. We evaluate the proposed algorithm on multimodal MRI brain imagery using ROC statistics and provide preliminary results. The algorithm shows promising segmentation performance with a sensitivity and specificity of 90.37% and 99.74% respectively and compares competitively to alternative interactive segmentation schemes

On-line learning in radial basis functions networks

An analytic investigation of the average case learning and generalization properties of Radial Basis Function Networks (RBFs) is presented, utilising on-line gradient descent as the learning rule. The analytic method employed allows both the calculation of generalization error and the examination of the internal dynamics of the network. The generalization error and internal dynamics are then used to examine the role of the learning rate and the specialization of the hidden units, which gives insight into decreasing the time required for training. The realizable and over-realizable cases are studied in detail; the phase of learning in which the hidden units are unspecialized (symmetric phase) and the phase in which asymptotic convergence occurs are analyzed, and their typical properties found. Finally, simulations are performed which strongly confirm the analytic results

A Nonparametric Approach to Pricing Options Learning Networks

For practitioners of equity markets, option pricing is a major challenge during high volatility periods and Black-Scholes formula for option pricing is not the proper tool for very deep out-of-the-money options. The Black-Scholes pricing errors are larger in the deeper out-of-the money options relative to the near the-money options, and it's mispricing worsens with increased volatility. Experts opinion is that the Black-Scholes model is not the proper pricing tool in high volatility situations especially for very deep out-of-the-money options. They also argue that prior to the 1987 crash, volatilities were symmetric around zero moneyness, with in-the-money and out-of-the money having higher implied volatilities than at-the-money options. However, after the crash, the call option implied volatilities were decreasing monotonically as the call went deeper into out-of-the-money, while the put option implied volatilities were decreasing monotonically as the put went deeper into in-the-money. Since these findings cannot be explained by the Black-Scholes model and its variations, researchers searched for improved option pricing models. Feedforward networks provide more accurate pricing estimates for the deeper out-of-the money options and handles pricing during high volatility with considerably lower errors for out-of-the-money call and put options. This could be invaluable information for practitioners as option pricing is a major challenge during high volatility periods. In this article a nonparametric method for estimating S&P 100 index option prices using artificial neural networks is presented. To show the value of artificial neural network pricing formulas, Black-Scholes option prices are compared with the network prices against market prices. To illustrate the practical relevance of the network pricing approach, it is applied to the pricing of S&P 100 index options from April 4, 2014 to April 9, 2014. On the five days data while Black-Scholes formula prices have a mean $10.17 error for puts, and$1.98 for calls, while neural network’s error is less than $5 for puts, and$1 for calls

Bias/Variance is not the same as Approximation/Estimation

We study the relation between two classical results: the bias-variance decomposition, and the approximation-estimation decomposition. Both are important conceptual tools in Machine Learning, helping us describe the nature of model fitting. It is commonly stated that they are “closely related”, or “similar in spirit”. However, sometimes it is said they are equivalent. In fact they are different, but have subtle connections cutting across learning theory, classical statistics, and information geometry, that (very surprisingly) have not been previously observed. We present several results for losses expressible as a Bregman divergence: a broad family with a known bias-variance decomposition. Discussion and future directions are presented for more general losses, including the 0/1 classification loss

A Nonparametric Approach to Pricing and Hedging Derivative Securities Via Learning Networks

We propose a nonparametric method for estimating the pricing formula of a derivative asset using learning networks. Although not a substitute for the more traditional arbitrage-based pricing formulas, network pricing formulas may be more accurate and computationally more efficient alternatives when the underlying asset's price dynamics are unknown, or when the pricing equation associated with no-arbitrage condition cannot be solved analytically. To assess the potential value of network pricing formulas, we simulate Black-Scholes option prices and show that learning networks can recover the Black-Scholes formula from a two-year training set of daily options prices, and that the resulting network formula can be used successfully to both price and delta-hedge options out-of-sample. For comparison, we estimate models using four popular methods: ordinary least squares, radial basis function networks, multilayer perceptron networks, and projection pursuit. To illustrate the practical relevance of our network pricing approach, we apply it to the pricing and delta-hedging of S&P 500 futures options from 1987 to 1991.

Foundational principles for large scale inference: Illustrations through correlation mining

When can reliable inference be drawn in the "Big Data" context? This paper presents a framework for answering this fundamental question in the context of correlation mining, with implications for general large scale inference. In large scale data applications like genomics, connectomics, and eco-informatics the dataset is often variable-rich but sample-starved: a regime where the number $n$ of acquired samples (statistical replicates) is far fewer than the number $p$ of observed variables (genes, neurons, voxels, or chemical constituents). Much of recent work has focused on understanding the computational complexity of proposed methods for "Big Data." Sample complexity however has received relatively less attention, especially in the setting when the sample size $n$ is fixed, and the dimension $p$ grows without bound. To address this gap, we develop a unified statistical framework that explicitly quantifies the sample complexity of various inferential tasks. Sampling regimes can be divided into several categories: 1) the classical asymptotic regime where the variable dimension is fixed and the sample size goes to infinity; 2) the mixed asymptotic regime where both variable dimension and sample size go to infinity at comparable rates; 3) the purely high dimensional asymptotic regime where the variable dimension goes to infinity and the sample size is fixed. Each regime has its niche but only the latter regime applies to exa-scale data dimension. We illustrate this high dimensional framework for the problem of correlation mining, where it is the matrix of pairwise and partial correlations among the variables that are of interest. We demonstrate various regimes of correlation mining based on the unifying perspective of high dimensional learning rates and sample complexity for different structured covariance models and different inference tasks

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

Feedforward networks are a class of regression techniques that can be used to learn to perform some task from a set of examples. The question of generalization of network performance from a finite training set to unseen data is clearly of crucial importance. In this article we first show that the generalization error can be decomposed in two terms: the approximation error, due to the insufficient representational capacity of a finite sized network, and the estimation error, due to insufficient information about the target function because of the finite number of samples. We then consider the problem of approximating functions belonging to certain Sobolev spaces with Gaussian Radial Basis Functions. Using the above mentioned decomposition we bound the generalization error in terms of the number of basis functions and number of examples. While the bound that we derive is specific for Radial Basis Functions, a number of observations deriving from it apply to any approximation t..