Large Margin Classification for Moving Targets

Studies in Applied Mathematics, 2010

Gaussians are important tools for learning from data of large dimensions. The variance of a Gaussian kernel is a measurement of the frequency range of function components or features retrieved by learning algorithms induced by the Gaussian. The learning ability and approximation power increase when the variance of the Gaussian decreases. Thus, it is natural to use Gaussians with decreasing variances for online algorithms when samples are imposed one by one. In this paper, we consider fully online classification algorithms associated with a general loss function and varying Gaussians which are closely related to regularization schemes in reproducing kernel Hilbert spaces. Learning rates are derived in terms of the smoothness of a target function associated with the probability measure controlling sampling and the loss function. A critical estimate is given for the norm of the difference of regularized target functions as the variance of the Gaussian changes. Concrete learning rates are presented for the online learning algorithm with the least square loss function.

We present a family of margin based online learning algorithms for various prediction tasks. In particular we derive and analyze algorithms for binary and multiclass categorization, regression, uniclass prediction and sequence prediction. The update steps of our different algorithms are all based on analytical solutions to simple constrained optimization problems. This unified view allows us to prove worst-case loss bounds for the different algorithms and for the various decision problems based on a single lemma. Our bounds on the cumulative loss of the algorithms are relative to the smallest loss that can be attained by any fixed hypothesis, and as such are applicable to both realizable and unrealizable settings. We demonstrate some of the merits of the proposed algorithms in a series of experiments with synthetic and real data sets.

2018

Making use of predictions is a crucial, but under-explored, area of sequential decision problems with limited information. While in practice most online algorithms rely on predictions to make real time decisions, in theory their performance is only analyzed in simplified models of prediction noise, either adversarial or i.i.d. The goal of this thesis is to bridge this divide between theory and practice: to study online algorithm under more practical predictions models, gain better understanding about the value of prediction, and design online algorithms that make the best use of predictions. This thesis makes three main contributions. First, we propose a stochastic prediction error model that generalizes prior models in the learning and stochastic control communities, incorporates correlation among prediction errors, and captures the fact that predictions improve as time passes. Using this general prediction model, we prove that Averaging Fixed Horizon Control (AFHC) can simultaneou...

2012

We prove logarithmic regret bounds that depend on the loss L * T of the competitor rather than on the number T of time steps. In the general online convex optimization setting, our bounds hold for any smooth and exp-concave loss (such as the square loss or the logistic loss). This bridges the gap between the O(ln T ) regret exhibited by expconcave losses and the O( L * T ) regret exhibited by smooth losses. We also show that these bounds are tight for specific losses, thus they cannot be improved in general. For online regression with square loss, our analysis can be used to derive a sparse randomized variant of the online Newton step, whose expected number of updates scales with the algorithm's loss. For online classification, we prove the first logarithmic mistake bounds that do not rely on prior knowledge of a bound on the competitor's norm.

2009

Online learning algorithms have impressive convergence properties when it comes to risk minimization and convex games on very large problems. However, they are inherently sequential in their design which prevents them from taking advantage of modern multi-core architectures. In this paper we prove that online learning with delayed updates converges well, thereby facilitating parallel online learning.

2008

Any change in the classification problem in the course of online classification is termed changing environments. Examples of changing environments include change in the underlying data distribution, change in the class definition, adding or removing a feature. The two general strategies for handling changing environments are (i) constant update of the classifier and (ii) re-training of the classifier after change detection. The former strategy is useful with gradual changes while the latter is useful with abrupt changes. If the type of changes is not known in advance, a combination of the two strategies may be advantageous. We propose a classifier ensemble using Winnow. For the constant-update strategy we used the nearest neighbour with a fixed size window and two methods with a learning rate: the online perceptron and an online version of the linear discriminant classifier (LDC). For the detect-and-retrain strategy we used the nearest neighbour classifier and the online LDC. Experiments were carried out on 28 data sets and 3 different scenarios: no change, gradual change and abrupt change. The results indicate that the combination works better than each strategy on its own.

2007

We study the rates of growth of the regret in online convex optimization. First, we show that a simple extension of the algorithm of Hazan et al eliminates the need for a priori knowledge of the lower bound on the second derivatives of the observed functions. We then provide an algorithm, Adaptive Online Gradient Descent, which interpolates between the results of Zinkevich for linear functions and of Hazan et al for strongly convex functions, achieving intermediate rates between T and log T . Furthermore, we show strong optimality of the algorithm. Finally, we provide an extension of our results to general norms.

Knowledge Science, Engineering and Management, 2010

Online classification learners operating under concept drift can be subject to latency in examples arriving at the training base. A discussion of latency and the related notion of example filtering leads to the development of an example life cycle for online learning (OLLC). Latency in a data stream is modelled in a new Example Life-cycle Integrated Simulation Environment (ELISE). In a series of experiments, the online learner algorithm CD3 is evaluated under several drift and latency scenarios. Results show that systems subject to large random latencies can, when drift occurs, suffer substantial deterioration in classification rate with slow recovery.

2013

In this work, we analyze the generalization ability of distributed online learning algorithms under stationary and non-stationary environments. We derive bounds for the excess-risk attained by each node in a connected network of learners and study the performance advantage that diffusion strategies have over individual non-cooperative processing. We conduct extensive simulations to illustrate the results.

Information Sciences, 2019

We are living in a world progressively driven by data. Besides mining big data which cannot be altogether stored in the main memory as required by traditional offline methods, the problem of learning rare data that can only be collected over time is also very prevalent. Consequently, there is a need of online methods which can handle arriving data and offer the same accuracy as offline methods. In this paper, we introduce a new lossless online Bayesianbased classifier which uses the arriving data in a 1-by-1 manner and discards each data right after use. The lossless property of our proposed method guarantees that it can reach the same prediction model as its offline counterpart regardless of the incremental training order. Experimental results demonstrate its superior performance over many well-known state-of-theart methods in the literature.