Online Learning From Finite Training Sets and

Neural Information Processing Systems, 1995

We consider the problem of on-line gradient descent learning for general two-layer neural networks. An analytic solution is presented and used to investigate the role of the learning rate in controlling the evolution and convergence of the learning process.

IEEE Transactions on Information Theory, 2004

In this paper, we study the generalization properties of online learning based stochastic methods for supervised learning problems where the loss function is dependent on more than one training sample (e.g., metric learning, ranking). We present a generic decoupling technique that enables us to provide Rademacher complexity-based generalization error bounds. Our bounds are in general tighter than those obtained by Wang et al. (2012) for the same problem. Using our decoupling technique, we are further able to obtain fast convergence rates for strongly convex pairwise loss functions. We are also able to analyze a class of memory efficient online learning algorithms for pairwise learning problems that use only a bounded subset of past training samples to update the hypothesis at each step. Finally, in order to complement our generalization bounds, we propose a novel memory efficient online learning algorithm for higher order learning problems with bounded regret guarantees.

In this paper, a training data selection method for multilayer neural networks (MLNNs) in online training is proposed. Purpose of the reduction in training data is reducing the computation complexity of the training and saving the memory to store the data without losing generalization performance. This method uses a pairing method, which selects the nearest neighbor data by finding the nearest data in the different classes. The network is trained by the selected data. Since the selected data located along data class boundary, the trained network can guarantee generalization performance. Efficiency of this method for the online training is evaluated by computer simulation

2013

In this paper, we study the generalization properties of online learning based stochastic methods for supervised learning problems where the loss function is dependent on more than one training sample (e.g., metric learning, ranking). We present a generic decoupling technique that enables us to provide Rademacher complexity-based generalization error bounds. Our bounds are in general tighter than those obtained by Wang et al (COLT 2012) for the same problem. Using our decoupling technique, we are further able to obtain fast convergence rates for strongly convex pairwise loss functions. We are also able to analyze a class of memory efficient online learning algorithms for pairwise learning problems that use only a bounded subset of past training samples to update the hypothesis at each step. Finally, in order to complement our generalization bounds, we propose a novel memory efficient online learning algorithm for higher order learning problems with bounded regret guarantees.

2010

We propose a general framework to online learning for classification problems with time-varying potential functions in the adversarial setting. This framework allows to design and prove relative mistake bounds for any generic loss function. The mistake bounds can be specialized for the hinge loss, allowing to recover and improve the bounds of known online classification algorithms. By optimizing the general bound we derive a new online classification algorithm, called NAROW, that hybridly uses adaptive-and fixed-second order information. We analyze the properties of the algorithm and illustrate its performance using synthetic dataset.

Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, 2016

We consider the fundamental problem of prediction with expert advice where the experts are "optimizable": there is a black-box optimization oracle that can be used to compute, in constant time, the leading expert in retrospect at any point in time. In this setting, we give a novel online algorithm that attains vanishing regret with respect to N experts in total r Op ? N q computation time. We also give a lower bound showing that this running time cannot be improved (up to log factors) in the oracle model, thereby exhibiting a quadratic speedup as compared to the standard, oracle-free setting where the required time for vanishing regret is r ΘpN q. These results demonstrate an exponential gap between the power of optimization in online learning and its power in statistical learning: in the latter, an optimization oracle-i.e., an efficient empirical risk minimizer-allows to learn a finite hypothesis class of size N in time Oplog N q. We also study the implications of our results to learning in repeated zero-sum games, in a setting where the players have access to oracles that compute, in constant time, their bestresponse to any mixed strategy of their opponent. We show that the runtime required for approximating the minimax value of the game in this setting is r Θp ? N q, yielding again a quadratic improvement upon the oracle-free setting, where r ΘpN q is known to be tight.

The Journal of Machine Learning Research, 2007

The experimental investigation on the efficient learning of highly non-linear problems by online training, using ordinary feed forward neural networks and stochastic gradient descent on the errors computed by back-propagation, gives evidence that the most crucial factors for efficient training are the hidden units' differentiation, the attenuation of the hidden units' interference and the selective attention on the parts of the problems where the approximation error remains high. In this report, we present global and local selective attention techniques and a new hybrid activation function that enables the hidden units to acquire individual receptive fields which may be global or local depending on the problem's local complexities. The presented techniques enable very efficient training on complex classification problems with embedded subproblems.

Physical Review E, 2003

The dynamical and stationary properties of on-line learning from finite training sets are analyzed by using the cavity method. For large input dimensions, we derive equations for the macroscopic parameters, namely, the student-teacher correlation, the student-student autocorrelation and the learning force fluctuation. This enables us to provide analytical solutions to Adaline learning as a benchmark. Theoretical predictions of training errors in transient and stationary states are obtained by a Monte Carlo sampling procedure. Generalization and training errors are found to agree with simulations. The physical origin of the critical learning rate is presented. Comparison with batch learning is discussed throughout the paper.

Journal of Physics A: Mathematical and General, 1997

We present a method for determining the globally optimal on-line learning rule for a soft committee machine under a statistical mechanics framework. This rule maximizes the total reduction in generalization error over the whole learning process. A simple example demonstrates that the locally optimal rule, which maximizes the rate of decrease in generalization error, may perform poorly in comparison.

Neural Information Processing Systems, 1998

O(ws(s log d+log(dqh/ s))) and O(ws((h/ s) log q) +log(dqh/ s)) are upper bounds for the VC-dimension of a set of neural networks of units with piecewise polynomial activation functions, where s is the depth of the network, h is the number of hidden units, w is the number of adjustable parameters, q is the maximum of the number of polynomial segments of the activation function, and d is the maximum degree of the polynomials; also n(wslog(dqh/s)) is a lower bound for the VC-dimension of such a network set, which are tight for the cases s = 8(h) and s is constant. For the special case q = 1, the VC-dimension is 8(ws log d).