P. S. Sastry

Deep Patel, P. S. Sastry

Deep Neural Networks, often owing to the overparameterization, are shown to be capable of exactly memorizing even randomly labelled data. Empirical studies have also shown that none of the standard regularization techniques mitigate such overfitting. We investigate whether the choice of the loss function can affect this memorization. We empirically show, with benchmark data sets MNIST and CIFAR-10, that a symmetric loss function, as opposed to either cross-entropy or squared error loss, results in significant improvement in the ability of the network to resist such overfitting. We then provide a formal definition for robustness to memorization and provide a theoretical explanation as to why the symmetric losses provide this robustness. Our results clearly bring out the role loss functions alone can play in this phenomenon of memorization.

Deep Patel, P. S. Sastry

Deep Neural Networks (DNNs) have been shown to be susceptible to memorization or overfitting in the presence of noisily-labelled data. For the problem of robust learning under such noisy data, several algorithms have been proposed. A prominent class of algorithms rely on sample selection strategies wherein, essentially, a fraction of samples with loss values below a certain threshold are selected for training. These algorithms are sensitive to such thresholds, and it is difficult to fix or learn these thresholds. Often, these algorithms also require information such as label noise rates which are typically unavailable in practice. In this paper, we propose an adaptive sample selection strategy that relies only on batch statistics of a given mini-batch to provide robustness against label noise. The algorithm does not have any additional hyperparameters for sample selection, does not need any information on noise rates and does not need access to separate data with clean labels. We empirically demonstrate the effectiveness of our algorithm on benchmark datasets.

Vidyadhar Upadhya, P S Sastry

The Gaussian-Bernoulli restricted Boltzmann machine (GB-RBM) is a useful generative model that captures meaningful features from the given $n$-dimensional continuous data. The difficulties associated with learning GB-RBM are reported extensively in earlier studies. They indicate that the training of the GB-RBM using the current standard algorithms, namely, contrastive divergence (CD) and persistent contrastive divergence (PCD), needs a carefully chosen small learning rate to avoid divergence which, in turn, results in slow learning. In this work, we alleviate such difficulties by showing that the negative log-likelihood for a GB-RBM can be expressed as a difference of convex functions if we keep the variance of the conditional distribution of visible units (given hidden unit states) and the biases of the visible units, constant. Using this, we propose a stochastic {\em difference of convex functions} (DC) programming (S-DCP) algorithm for learning the GB-RBM. We present extensive empirical studies on several benchmark datasets to validate the performance of this S-DCP algorithm. It is seen that S-DCP is better than the CD and PCD algorithms in terms of speed of learning and the quality of the generative model learnt.

Kulin Shah, P. S. Sastry, Naresh Manwani

In this paper, we propose a novel mixture of expert architecture for learning polyhedral classifiers. We learn the parameters of the classifierusing an expectation maximization algorithm. Wederive the generalization bounds of the proposedapproach. Through an extensive simulation study, we show that the proposed method performs comparably to other state-of-the-art approaches.

Soumyajit Mitra, P S Sastry

In this paper we address the problem of discovering a small set of frequent serial episodes from sequential data so as to adequately characterize or summarize the data. We discuss an algorithm based on the Minimum Description Length (MDL) principle and the algorithm is a slight modification of an earlier method, called CSC-2. We present a novel generative model for sequence data containing prominent pairs of serial episodes and, using this, provide some statistical justification for the algorithm. We believe this is the first instance of such a statistical justification for an MDL based algorithm for summarizing event sequence data. We then present a novel application of this data mining algorithm in text classification. By considering text documents as temporal sequences of words, the data mining algorithm can find a set of characteristic episodes for all the training data as a whole. The words that are part of these characteristic episodes could then be considered the only relevant words for the dictionary thus resulting in a considerably reduced feature vector dimension. We show, through simulation experiments using benchmark data sets, that the discovered frequent episodes can be used to achieve more than four-fold reduction in dictionary size without losing any classification accuracy.

Vidyadhar Upadhya, P. S. Sastry

Learning RBMs using standard algorithms such as CD(k) involves gradient descent on the negative log-likelihood. One of the terms in the gradient, which involves expectation w.r.t. the model distribution, is intractable and is obtained through an MCMC estimate. In this work we show that the Hessian of the log-likelihood can be written in terms of covariances of hidden and visible units and hence, all elements of the Hessian can also be estimated using the same MCMC samples with small extra computational costs. Since inverting the Hessian may be computationally expensive, we propose an algorithm that uses inverse of the diagonal approximation of the Hessian, instead. This essentially results in parameter-specific adaptive learning rates for the gradient descent process and improves the efficiency of learning RBMs compared to the standard methods. Specifically we show that using the inverse of diagonal approximation of Hessian in the stochastic DC (difference of convex functions) program approach results in very efficient learning of RBMs.

Aritra Ghosh, Himanshu Kumar, P. S. Sastry

In many applications of classifier learning, training data suffers from label noise. Deep networks are learned using huge training data where the problem of noisy labels is particularly relevant. The current techniques proposed for learning deep networks under label noise focus on modifying the network architecture and on algorithms for estimating true labels from noisy labels. An alternate approach would be to look for loss functions that are inherently noise-tolerant. For binary classification there exist theoretical results on loss functions that are robust to label noise. In this paper, we provide some sufficient conditions on a loss function so that risk minimization under that loss function would be inherently tolerant to label noise for multiclass classification problems. These results generalize the existing results on noise-tolerant loss functions for binary classification. We study some of the widely used loss functions in deep networks and show that the loss function based on mean absolute value of error is inherently robust to label noise. Thus standard back propagation is enough to learn the true classifier even under label noise. Through experiments, we illustrate the robustness of risk minimization with such loss functions for learning neural networks.

Suresh Kirthi Kumaraswamy, PS Sastry, KR Ramakrishnan

Convolutional Neural Networks (CNNs) are very effective for many pattern recognition tasks. However, training deep CNNs needs extensive computation and large training data. In this paper we propose Bank of Filter-Trees (BFT) as a trans- fer learning mechanism for improving efficiency of learning CNNs. A filter-tree corresponding to a filter in k^{th} convolu- tional layer of a CNN is a subnetwork consisting of the filter along with all its connections to filters in all preceding layers. An ensemble of such filter-trees created from the k^{th} layers of many CNNs learnt on different but related tasks, forms the BFT. To learn a new CNN, we sample from the BFT to select a set of filter trees. This fixes the target net up to the k th layer and only the remaining network would be learnt using train- ing data of new task. Through simulations we demonstrate the effectiveness of this idea of BFT. This method constitutes a novel transfer learning technique where transfer is at a sub- network level; transfer can be effected from multiple source networks; and, with no finetuning of the transferred weights, the performance achieved is on par with networks that are trained from scratch.

Vidyadhar Upadhya, P. S. Sastry

By exploiting the property that the RBM log-likelihood function is the difference of convex functions, we formulate a stochastic variant of the difference of convex functions (DC) programming to minimize the negative log-likelihood. Interestingly, the traditional contrastive divergence algorithm is a special case of the above formulation and the hyperparameters of the two algorithms can be chosen such that the amount of computation per mini-batch is identical. We show that for a given computational budget the proposed algorithm almost always reaches a higher log-likelihood more rapidly, compared to the standard contrastive divergence algorithm. Further, we modify this algorithm to use the centered gradients and show that it is more efficient and effective compared to the standard centered gradient algorithm on benchmark datasets.

Aritra Ghosh, Naresh Manwani, P. S. Sastry

In most practical problems of classifier learning, the training data suffers from the label noise. Hence, it is important to understand how robust is a learning algorithm to such label noise. This paper presents some theoretical analysis to show that many popular decision tree algorithms are robust to symmetric label noise under large sample size. We also present some sample complexity results which provide some bounds on the sample size for the robustness to hold with a high probability. Through extensive simulations we illustrate this robustness.

Vidyadhar Upadhya, P. S. Sastry

The Restricted Boltzmann Machines (RBM) can be used either as classifiers or as generative models. The quality of the generative RBM is measured through the average log-likelihood on test data. Due to the high computational complexity of evaluating the partition function, exact calculation of test log-likelihood is very difficult. In recent years some estimation methods are suggested for approximate computation of test log-likelihood. In this paper we present an empirical comparison of the main estimation methods, namely, the AIS algorithm for estimating the partition function, the CSL method for directly estimating the log-likelihood, and the RAISE algorithm that combines these two ideas. We use the MNIST data set to learn the RBM and then compare these methods for estimating the test log-likelihood.

Aritra Ghosh, Naresh Manwani, P. S. Sastry

In many applications, the training data, from which one needs to learn a classifier, is corrupted with label noise. Many standard algorithms such as SVM perform poorly in presence of label noise. In this paper we investigate the robustness of risk minimization to label noise. We prove a sufficient condition on a loss function for the risk minimization under that loss to be tolerant to uniform label noise. We show that the $0-1$ loss, sigmoid loss, ramp loss and probit loss satisfy this condition though none of the standard convex loss functions satisfy it. We also prove that, by choosing a sufficiently large value of a parameter in the loss function, the sigmoid loss, ramp loss and probit loss can be made tolerant to non-uniform label noise also if we can assume the classes to be separable under noise-free data distribution. Through extensive empirical studies, we show that risk minimization under the $0-1$ loss, the sigmoid loss and the ramp loss has much better robustness to label noise when compared to the SVM algorithm.