Regularized Adaptation: Theory, Algorithms and Applications

Regularized Adaptation: Theory, Algorithms and Applications Xiao Li Electrical Engineering Department University of Washington

Roadmap • Introduction • Theoretical results • A Bayesian fidelity prior for adaptation • Generalization error bounds • Regularized adaptation algorithms • SVM and MLP adaptation • Experiments on vowel and object classification • The application to the Vocal Joystick • Conclusions and future work

Inductive Learning • Given • a set of m samples (xi, yi) ~p(x, y) • a decision function space F: X  {±1} • Goal • learn a decision function that minimizes the expected error • In practice • minimize the empirical error • while applying certain regularization strategy to achieve good generalization performance

Why Is Regularization Helpful? • Learning theory says • Frequentist: Vapnik’s VC bound expresses Φ as a function of the VC dimension of F • Bayesian: the Occam’s Razor bound expresses Φ as a function of the prior probability of f • “Accuracy-regularization” • We want to minimize the empirical error as well as the capacity • Frequentist: support vector machines • Bayesian: Bayesian model selection

Adaptive Learning • Two related yet different distributions • Training • target (test-time) • Given • An unadapted model • Adaptation data (labeled) • Goal • Learn an adapted model that is as close as possible to our desired model • Notes • Assume sufficient training data but limited adaptation data • Training data is not preserved

Scenarios • Customization • Speech recognition: speaker adaptation • Handwriting recognition: writer adaptation • Language processing: domain adaptation • Evolutionary environments • Spam filtering • Incremental/sequential learning • Start from a simple or rough model and refine incrementally

Practical Work on Adaptation • Gaussian mixture models (GMMs) • MAP (Gauvain 94); MLLR (Leggetter 95) • Support vector machines (SVMs) • Boosting-like approach (Matic 93) • Weighted combination of old support vectors and adaptation data (Wu 04) • Multi-layer perceptrons (MLPs) • Shared “internal representation” (Baxter 95, Caruana 97, Stadermann 05) • Linear input network (Neto 95) • Conditional maximum entropy models • Gaussian prior (Chelba 04)

This Work Seeks Answers to … • A unified and principled approach to adaptation • applicable to a variety of classifiers • amenable to variations in the amount of adaptation data • Quantitative relationships between • the generalization error bound (or sample complexity bound) and • the divergence between training and target distributions

Bayesian Fidelity Prior • Adaptation objective • Remp( f ) – empirical error on the adaptation data • Pfid( f )– Bayesian “fidelity prior” • Fidelity prior • How likely a classifier is given a training distribution (rather than a training set – key difference from hierarchical Bayes approaches, e.g. Baxter 97) • Applicable to different classifiers • Relates to the KL-divergence

Generative Models • Generative models p( x, y | f ) • Classification • Posterior • Assume f tr and f ad are the true models generating the training and target distributions respectively, i.e. • Note that this assumption is justifiable if the function space contains the true models and if we use the log likelihood loss standard prior, chosen before training

Fidelity Prior for Generative Models • Key result where β > 0 • Implication • Fidelity prior at the desired model We are more likely to learn our desired model using the fidelity prior than using the standard prior

Instantiations • To compute the fidelity prior • assuming a “uniform” standard prior, this prior is determined by the KL-divergence • In cases the KL-divergence does not have a close form, we use an upper bound instead (hence a lower bound on the prior) • Gaussian models • The fidelity prior is a normal-Wishart distribution • Mixture models • An upper bound on the KL-divergence (using log sum inequality) • Hidden Markov models • An upper bound on the KL-divergence (Silva 06)

Discriminative Models • A unified view of SVMs, MLPs, CRFs and etc. • Affine classifiers in a transformed space f = ( w, b ) • Classification • Conditional likelihood (for binary case)

Discriminative Models (cont.) • Conditional models p( y | x, f ) • Classification • Posterior • Assume f tr and f ad are the true models generating the training and target conditionaldistributions respectively, i.e.

Fidelity Prior for Conditional Models • Again a divergence where β > 0 • What if we do not know ptr(x, y) • We seek an upper bound on the KL-divergence and hence a lower bound on the prior • Key result where

Occam’s Razor Bound for Adaptation • For a countable function space

Bound using standard prior Bounds using divergence prior m

PAC-Bayesian Bounds for Adaptation • For both countable and uncountable function spaces • Choice of prior p( f ) and posterior q( f ) • D( q( f )||p( f ) ) and stochastic error are easily computable • Use pfid( f )or its related forms as prior • Choose q( f )to have the same parametric form • Examples • Gaussian models • Linear classifier

Algorithms Derived from the Fidelity Prior • Generative Models • Relation to MAP adaptation • Conditional Models • Log linear models • We focus on SVMs and MLPs

Regularized SVM Adaptation • Optimization objective • Globally optimal solution • Regularized – fixing old support vectors and their coefficients • Extended regularized – update coefficients of old support vectors as well

Algorithms in Comparison • Unadapted • Retrained • Use adaptation data only • Boosted (Matic 93) • Select adaptation data misclassified by the unadapted model • Combine with old support vectors • Bootstrapped (proposed in thesis) • Train a seed classifier using adaptation data only • Select old support vectors correctly classified by the seed classifier; combine with adaptation data • Combine with adaptation data • Regularizedand Extended regularized

Regularized MLP Adaptation • Optimization objective for a multi-class, two-layer MLP • Wh2o andWh2o – the hidden-to-output and input-to-hidden layer weight matrix respectively • ||W|| is the L2 norm • Remp( f ) – cross-entropy, corresponding to log loss • Locally optimal solution found using back-propogation

Algorithms in Comparison • Unadapted • Retrained • Start from randomly initialized weight and train with weight decay • Linear input network (Neto 95) • Add a linear transformation in the input space • Retrained speaker-independent (Neto 95) • Start from the unadapted; train both layers • Retrained last layer (Baxter 95, Caruana 97, Stadermann 05) • Start from the unadapted; only train the last layer • Retrained first layer (proposed in thesis) • Start from the unadapted; only train the first layer • Regularized • Note that all above (except retrained) can be considered as special cases of regularized

Experimental Paradigm • Goal • To compare adaptation algorithms for a given classifier • Not to compare adaptation algorithms across classifiers • Procedure • Train an unadapted model on training set • Adapt (with supervision) and evaluate via n-fold CV on test set • Select regularization coefficients on the dev set • Corpora • VJ vowel dataset (Kilanski 06) • NORB image dataset (LeCun 04)

VJ Vowel Dataset • Task • 8 Vowel classes • Frame-level classification error rate • Speaker adaptation • Data allocation • Training set – 21 speakers, 420K samples For SVM, we random selected 80K samples for training • Test set – 10 speakers, 200 samples • Dev set – 4 speakers, 80 samples • Features • 182 dimensions – 7 frames of MFCC+delta features

SVM Adaptation • RBF kernel (std=10) optimized for training and fixed for adaptation • Mean and std. dev over 10 speakers; red are significant at p<0.001 level

MLP Adaptation (I) • 50 hidden nodes • Mean and std. dev over 10 speakers

MLP Adaptation (II) • Varying number of vowel classes available in adaptation data

NORB Image Dataset • Task • 5 object classes • Lighting condition adaptation • Data allocation • Training set – 2700 samples • Test set – 2700 samples • Features • 32x32 raw images

SVM Adaptation • RBF kernel (std=500) optimized for training and fixed for adaptation • Mean and std. dev over 6 lighting conditions

MLP Adaptation • 30 hidden node • Mean and std. dev over 6 lighting conditions

Why the Vocal Joystick • Computer interfaces for individuals with motor-impairments • Head tracking • Eye-gaze tracking • Brain-computer interfaces • Expensive and error prone • Speech is a natural solution, but… • Most suitable for discrete commands • Or, requires a more complex syntax

What Is the Vocal Joystick • A voice-based interface • produce real-time, continuous signals to control standard computing devices and robotic arms • Acoustic Parameters • Vowel quality • Loudness • Pitch • Discrete sound identity • VJ mouse

VJ Engine Dynamic Bayesian network Two-Layer MLP … Phoneme HMMs

Adaptation in the VJ • Why adaptation is important User variability, style mismatch and channel mismatch • Adaptation tools • Regularized MLP adaptation for vowel classification • Regularized GMM adaptation for discrete sound recognition

Regularized Adaptation: Theory, Algorithms and Applications