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Large Margin Hidden Markov Models for Automatic Speech Recognition

Fei Sha Computer Science Division University of California Berkeley, CA 94720-1776 feisha@cs.berkeley.edu

Lawrence K. Saul Department of Computer Science and Engineering University of California (San Diego) La Jolla, CA 92093-0404 saul@cs.ucsd.edu

Abstract

We study the problem of parameter estimation in continuous density hidden Markov models (CD-HMMs) for automatic speech recognition (ASR). As in support vector machines, we propose a learning algorithm based on the goal of margin maximization. Unlike earlier work on max-margin Markov networks, our approach is speci?cally geared to the modeling of real-valued observations (such as acoustic feature vectors) using Gaussian mixture models. Unlike previous discriminative frameworks for ASR, such as maximum mutual information and minimum classi?cation error, our framework leads to a convex optimization, without any spurious local minima. The objective function for large margin training of CD-HMMs is de?ned over a parameter space of positive semide?nite matrices. Its optimization can be performed ef?ciently with simple gradient-based methods that scale well to large problems. We obtain competitive results for phonetic recognition on the TIMIT speech corpus.

1 Introduction

As a result of many years of widespread use, continuous density hidden Markov models (CDHMMs) are very well matched to current front and back ends for automatic speech recognition (ASR) [21]. Typical front ends compute real-valued feature vectors from the short-time power spectra of speech signals. The distributions of these acoustic feature vectors are modeled by Gaussian mixture models (GMMs), which in turn appear as observation models in CD-HMMs. Viterbi decoding is used to solve the problem of sequential classi?cation in ASR—namely, the mapping of sequences of acoustic feature vectors to sequences of phonemes and/or words, which are modeled by state transitions in CD-HMMs. The simplest method for parameter estimation in CD-HMMs is the Expectation-Maximization (EM) algorithm. The EM algorithm is based on maximizing the joint likelihood of observed feature vectors and label sequences. It is widely used due to its simplicity and scalability to large data sets, which are common in ASR. A weakness of this approach, however, is that the model parameters of CDHMMs are not optimized for sequential classi?cation: in general, maximizing the joint likelihood does not minimize the phoneme or word error rates, which are more relevant metrics for ASR. Noting this weakness, many researchers in ASR have studied alternative frameworks for parameter estimation based on conditional maximum likelihood [11], minimum classi?cation error [4] and maximum mutual information [20]. The learning algorithms in these frameworks optimize discriminative criteria that more closely track actual error rates, as opposed to the EM algorithm for maximum likelihood estimation. These algorithms do not enjoy the simple update rules and relatively fast convergence of EM, but carefully and skillfully implemented, they lead to lower error rates [13, 20].

Recently, in a new approach to discriminative acoustic modeling, we proposed the use of “large margin GMMs” for multiway classi?cation [15]. Inspired by support vector machines (SVMs), the learning algorithm in large margin GMMs is designed to maximize the distance between labeled examples and the decision boundaries that separate different classes [19]. Under mild assumptions, the required optimization is convex, without any spurious local minima. In contrast to SVMs, however, large margin GMMs are very naturally suited to problems in multiway (as opposed to binary) classi?cation; also, they do not require the kernel trick for nonlinear decision boundaries. We showed how to train large margin GMMs as segment-based phonetic classi?ers, yielding signi?cantly lower error rates than maximum likelihood GMMs [15]. The integrated large margin training of GMMs and transition probabilities in CD-HMMs, however, was left as an open problem. We address that problem in this paper, showing how to train large margin CD-HMMs in the more general setting of sequential (as opposed to multiway) classi?cation. In this setting, the GMMs appear as acoustic models whose likelihoods are integrated over time by Viterbi decoding. Experimentally, we ?nd that large margin training of HMMs for sequential classi?cation leads to signi?cant improvement beyond the frame-based and segment-based discriminative training in [15]. Our framework for large margin training of CD-HMMs builds on ideas from many previous studies in machine learning and ASR. It has similar motivation as recent frameworks for sequential classi?cation in the machine learning community [1, 6, 17], but differs in its focus on the real-valued acoustic feature representations used in ASR. It has similar motivation as other discriminative paradigms in ASR [3, 4, 5, 11, 13, 20], but differs in its goal of margin maximization and its formulation of the learning problem as a convex optimization over positive semide?nite matrices. The recent margin-based approach of [10] is closest in terms of its goals, but entirely different in its mechanics; moreover, its learning is limited to the mean parameters in GMMs.

2 Large margin GMMs for multiway classi?cation

Before developing large margin HMMs for ASR, we brie?y review large margin GMMs for multiway classi?cation [15]. The problem of multiway classi?cation is to map inputs x ∈ ?d to labels y ∈ {1, 2, . . . , C}, where C is the number of classes. Large margin GMMs are trained from a set of labeled examples {(xn , yn )}N . They have many parallels to SVMs, including the goal of margin n=1 maximization and the use of a convex surrogate to the zero-one loss [19]. Unlike SVMs, where classes are modeled by half-spaces, in large margin GMMs the classes are modeled by collections of ellipsoids. For this reason, they are more naturally suited to problems in multiway as opposed to binary classi?cation. Sections 2.1–2.3 review the basic framework for large margin GMMs: ?rst, the simplest setting in which each class is modeled by a single ellipsoid; second, the formulation of the learning problem as a convex optimization; third, the general setting in which each class is modeled by two or more ellipsoids. Section 2.4 presents results on handwritten digit recognition. 2.1 Parameterization of the decision rule The simplest large margin GMMs model each class by a single ellipsoid in the input space. The ellipsoid for class c is parameterized by a centroid vector ?c ∈ ?d and a positive semide?nite matrix Ψc ∈ ?d×d that determines its orientation. Also associated with each class is a nonnegative scalar offset θc ≥ 0. The decision rule labels an example x ∈ ?d by the class whose centroid yields the smallest Mahalanobis distance: (1) y = argmin (x??c )T Ψc (x??c ) + θc .

c

The decision rule in eq. (1) is merely an alternative way of parameterizing the maximum a posterior (MAP) label in traditional GMMs with mean vectors ?c , covariance matrices Ψ?1 , and prior class c probabilities pc , given by y = argminc { pc N (?c , Ψ?1 ) }. c The argument on the right hand side of the decision rule in eq. (1) is nonlinear in the ellipsoid parameters ?c and Ψc . As shown in [15], however, a useful reparameterization yields a simpler expression. For each class c, the reparameterization collects the parameters {?c , Φc , θc } in a single enlarged matrix Φc ∈ ?(d+1)×(d+1) : Φc = Ψc ??T Ψc c ?Ψc ?c ?T Ψc ?c + θc c . (2)

Note that Φc is positive semide?nite. Furthermore, if Φc is strictly positive de?nite, the parameters {?c , Ψc , θc } can be uniquely recovered from Φc . With this reparameterization, the decision rule in eq. (1) simpli?es to: x y = argmin z T Φc z . (3) where z = 1 c The argument on the right hand side of the decision rule in eq. (3) is linear in the parameters Φc . In what follows, we will adopt the representation in eq. (3), implicitly constructing the “augmented” vector z for each input vector x. Note that eq. (3) still yields nonlinear (piecewise quadratic) decision boundaries in the vector z. 2.2 Margin maximization Analogous to learning in SVMs, we ?nd the parameters {Φc } that minimize the empirical risk on the training data—i.e., parameters that not only classify the training data correctly, but also place the decision boundaries as far away as possible. The margin of a labeled example is de?ned as its distance to the nearest decision boundary. If possible, each labeled example is constrained to lie at least one unit distance away from the decision boundary to each competing class: ?c = yn , z T (Φc ? Φyn ) z n ≥ 1. n (4)

Fig. 1 illustrates this idea. Note that in the “realizable” setting where these constraints can be simultaneously satis?ed, they do not uniquely determine the parameters {Φc }, which can be scaled to yield arbitrarily large margins. Therefore, as in SVMs, we propose a convex optimization that selects the “smallest” parameters that satisfy the large margin constraints in eq. (4). In this case, the optimization is an instance of semide?nite programming [18]: min c trace(Ψc ) s.t. 1 + z T (Φyn ? Φc )z n ≤ 0, n Φc ? 0, c = 1, 2, . . . , C ?c = yn , n = 1, 2, . . . , N (5)

Note that the trace of the matrix Ψc appears in the above objective function, as opposed to the trace of the matrix Φc , as de?ned in eq. (2); minimizing the former imposes the scale regularization only on the inverse covariance matrices of the GMM, while the latter would improperly regularize the mean vectors as well. The constraints Φc ? 0 restrict the matrices to be positive semide?nite. The objective function must be modi?ed for training data that lead to infeasible constraints in eq. (5). As in SVMs, we introduce nonnegative slack variables ξnc to monitor the amount by which the margin constraints in eq. (4) are violated [15]. The objective function in this setting balances the margin violations versus the scale regularization: min c trace(Ψc ) nc ξnc + γ s.t. 1 + z T (Φyn ? Φc )z n ≤ ξnc , n ξnc ≥ 0, ?c = yn , n = 1, 2, . . . , N Φc ? 0, c = 1, 2, . . . , C (6)

where the balancing hyperparameter γ > 0 is set by some form of cross-validation. This optimization is also an instance of semide?nite programming. 2.3 Softmax margin maximization for multiple mixture components Lastly we review the extension to mixture modeling where each class is represented by multiple ellipsoids [15]. Let Φcm denote the matrix for the mth ellipsoid (or mixture component) in class c. We imagine that each example xn has not only a class label yn , but also a mixture component label mn . Such labels are not provided a priori in the training data, but we can generate “proxy” labels by ?tting GMMs to the examples in each class by maximum likelihood estimation, then for each example, computing the mixture component with the highest posterior probability. In the setting where each class is represented by multiple ellipsoids, the goal of learning is to ensure that each example is closer to its “target” ellipsoid than the ellipsoids from all other classes. Speci?cally, for a labeled example (xn , yn , mn ), the constraint in eq. (4) is replaced by the M constraints: ?c = yn , ?m, z T (Φcm ? Φyn mn )z n ≥ 1, n (7)

Figure 1: Decision boundary in a large margin GMM: labeled examples lie at least one unit of distance away.

where M is the number of mixture components (assumed, for simplicity, to be the same for each class). We fold these multiple constraints into a single one by appealing to the “softmax” inequality: minm am ≥ ? log m e?am . Speci?cally, using the inequality to derive a lower bound on minm z T Φcm z n , we replace the M constraints in eq. (7) by the stricter constraint: n ?c = yn , ? log

m

We will use a similar technique in section 3 to handle the exponentially many constraints that arise in sequential classi?cation. Note that the inequality in eq. (8) implies the inequality of eq. (7) but not vice versa. Also, though nonlinear in the matrices {Φcm }, the constraint in eq. (8) is still convex. The objective function in eq. (6) extends straightforwardly to this setting. It balances a regularizing term that sums over ellipsoids versus a penalty term that sums over slack variables, one for each constraint in eq. (8). The optimization is given by: min cm trace(Ψcm ) nc ξnc + γ T T s.t. 1 + z n Φyn mn z n + log m e?zn Φcm zn ≤ ξnc , ξnc ≥ 0, ?c = yn , n = 1, 2, . . . , N Φcm ? 0, c = 1, 2, . . . , C, m = 1, 2, . . . , M

This optimization is not an instance of semide?nite programming, but it is convex. We discuss how to perform the optimization ef?ciently for large data sets in appendix A. 2.4 Handwritten digit recognition We trained large margin GMMs for multiway classi?cation of MNIST handwritten digits [8]. The MNIST data set has 60000 training examples and 10000 test examples. Table 1 shows that the large margin GMMs yielded signi?cantly lower test error rates than GMMs trained by maximum likelihood estimation. Our best results are comparable to the best SVM results (1.0-1.4%) on deskewed images [8] that do not make use of prior knowledge. For our best model, with four mixture components per digit class, the core training optimization over all training examples took ?ve minutes on a PC. (Multiple runs of this optimization on smaller validation sets, however, were also required to set two hyperparameters: the regularizer for model complexity, and the termination criterion for early stopping.)

3 Large margin HMMs for sequential classi?cation

In this section, we extend the framework in the previous section from multiway classi?cation to sequential classi?cation. Particularly, we have in mind the application to ASR, where GMMs are used to parameterize the emission densities of CD-HMMs. Strictly speaking, the GMMs in our framework cannot be interpreted as emission densities because their parameters are not constrained to represent normalized distributions. Such an interpretation, however, is not necessary for their use as discriminative models. In sequential classi?cation by CD-HMMs, the goal is to infer the correct hidden state sequence y = [y1 , y2 , . . . , yT ] given the observation sequence X = [x1 , x2 , . . . , xT ]. In the application to ASR, the hidden states correspond to phoneme labels, and the observations are

n

i

g

r

a m y r a d n u o b n o i s i c e d

mixture 1 2 4 8

EM 4.2% 3.4% 3.0% 3.3%

margin 1.4% 1.4% 1.2% 1.5%

Table 1: Test error rates on MNIST digit recognition: maximum likelihood versus large margin GMMs.

e?zn Φcm zn ? z T Φyn mn z n ≥ 1. n

T

(8)

(9)

acoustic feature vectors. Note that if an observation sequence has length T and each label can belong to C classes, then the number of incorrect state sequences grows as O(C T ). This combinatorial explosion presents the main challenge for large margin methods in sequential classi?cation: how to separate the correct hidden state sequence from the exponentially large number of incorrect ones. The section is organized as follows. Section 3.1 explains the way that margins are computed for sequential classi?cation. Section 3.2 describes our algorithm for large margin training of CD-HMMs. Details are given only for the simple case where the observations in each hidden state are modeled by a single ellipsoid. The extension to multiple mixture components closely follows the approach in section 2.3 and can be found in [14, 16]. Margin-based learning of transition probabilities is likewise straightforward but omitted for brevity. Both these extensions were implemented, however, for the experiments on phonetic recognition in section 3.3. 3.1 Margin constraints for sequential classi?cation We start by de?ning a discriminant function over state (label) sequences of the CD-HMM. Let a(i, j) denote the transition probabilities of the CD-HMM, and let Φs denote the ellipsoid parameters of state s. The discriminant function D(X, s) computes the score of the state sequence s = [s1 , s2 , . . . , sT ] on an observation sequence X = [x1 , x2 , . . . , xT ] as:

T

D(X, s) =

t

log a(st?1 , st ) ?

t=1

z T Φst z t . t

(10)

This score has the same form as the log-probability log P (X, s) in a CD-HMM with Gaussian emission densities. The ?rst term accumulates the log-transition probabilities along the state sequence, while the second term accumulates “acoustic scores” computed as the Mahalanobis distances to each state’s centroid. In the setting where each state is modeled by multiple mixture components, the acoustic scores from individual Mahalanobis distances are replaced with “softmax” distances of T the form log M e?zt Φst m zt , as described in section 2.3 and [14, 16]. m=1 We introduce margin constraints in terms of the above discriminant function. Let H(s, y) denote the Hamming distance (i.e., the number of mismatched labels) between an arbitrary state sequence s and the target state sequence y. Earlier, in section 2 on multiway classi?cation, we constrained each labeled example to lie at least one unit distance from the decision boundary to each competing class; see eq. (4). Here, by extension, we constrain the score of each target sequence to exceed that of each competing sequence by an amount equal to or greater than the Hamming distance: ?s = y, D(X, y) ? D(X, s) ≥ H(s, y) (11) Intuitively, eq. (11) requires that the (log-likelihood) gap between the score of an incorrect sequence s and the target sequence y should grow in proportion to the number of individual label errors. The appropriateness of such proportional constraints for sequential classi?cation was ?rst noted by [17]. 3.2 Softmax margin maximization for sequential classi?cation The challenge of large margin sequence classi?cation lies in the exponentially large number of constraints, one for each incorrect sequence s, embodied by eq. (11). We will use the same softmax inequality, previously introduced in section 2.3, to fold these multiple constraints into one, thus considerably simplifying the optimization required for parameter estimation. We ?rst rewrite the constraint in eq. (11) as: ?D(X, y) + max{H(s, y) + D(X, s)} ≤ 0

s=y

(12)

We obtain a more manageable constraint by substituting a softmax upper bound for the max term and requiring that the inequality still hold: ?D(X, y) + log

s=y

eH(s,y)+D(X,s) ≤ 0

(13)

Note that eq. (13) implies eq. (12) but not vice versa. As in the setting for multiway classi?cation, the objective function for sequential classi?cation balances two terms: one regularizing the scale of

the GMM parameters, the other penalizing margin violations. Denoting the training sequences by {X n , y n }N and the slack variables (one for each training sequence) by ξn ≥ 0, we obtain the n=1 following convex optimization: min cm trace(Ψcm ) n ξn + γ s.t. ?D(X n , y n ) + log s=yn eH(s,yn )+D(X n ,s) ≤ ξn , ξn ≥ 0, n = 1, 2, . . . , N Φcm ? 0, c = 1, 2, . . . , C, m = 1, 2, . . . , M

(14)

It is worth emphasizing several crucial differences between this optimization and previous ones [4, 11, 20] for discriminative training of CD-HMMs for ASR. First, the softmax large margin constraint in eq. (13) is a differentiable function of the model parameters, as opposed to the “hard” maximum in eq. (12) and the number of classi?cation errors in the MCE training criteria [4]. The constraint and its gradients with respect to GMM parameters Φcm and transition parameters a(·, ·) can be computed ef?ciently using dynamic programming, by a variant of the standard forward-backward procedure in HMMs [14]. Second, due to the reparameterization in eq. (2), the discriminant function D(X n , y n ) and the softmax function are convex in the model parameters. Therefore, the optimization eq. (14) can be cast as convex optimization, avoiding spurious local minima [14]. Third, the optimization not only increases the log-likelihood gap between correct and incorrect state sequences, but also drives the gap to grow in proportion to the number of individually incorrect labels (which we believe leads to more robust generalization). Finally, compared to the large margin framework in [17], the softmax handling of exponentially large number of margin constraints makes it possible to train on larger data sets. We discuss how to perform the optimization ef?ciently in appendix A. 3.3 Phoneme recognition We used the TIMIT speech corpus [7, 9, 12] to perform experiments in phonetic recognition. We followed standard practices in preparing the training, development, and test data. Our signal processing front-end computed 39-dimensional acoustic feature vectors from 13 mel-frequency cepstral coef?cients and their ?rst and second temporal derivatives. In total, the training utterances gave rise to roughly 1.2 million frames, all of which were used in training. We trained baseline maximum likelihood recognizers and two different types of large margin recognizers. The large margin recognizers in the ?rst group were “low-cost” discriminative CD-HMMs whose GMMs were merely trained for frame-based classi?cation. In particular, these GMMs were estimated by solving the optimization in eq. (8), then substituted into ?rst-order CD-HMMs for sequence decoding. The large margin recognizers in the second group were fully trained for sequential classi?cation. In particular, their CD-HMMs were estimated by solving the optimization in eq. (14), generalized to multiple mixture components and adaptive transition parameters [14, 16]. In all the recognizers, the acoustic feature vectors were labeled by 48 phonetic classes, each represented by one state in a ?rst-order CD-HMM. For each recognizer, we compared the phonetic state sequences obtained by Viterbi decoding to the “ground-truth” phonetic transcriptions provided by the TIMIT corpus. For the purpose of computing error rates, we followed standard conventions in mapping the 48 phonetic state labels down to 39 broader phone categories. We computed two different types of phone error rates, one based on Hamming distance, the other based on edit distance. The former was computed simply from the percentage of mismatches at the level of individual frames. The latter was computed by aligning the Viterbi and ground truth transcriptions using dynamic programming [9] and summing the substitution, deletion, and insertion error rates from the alignment process. The “frame-based” phone error rate computed from Hamming distances is more closely tracked by our objective function for large margin training, while the “string-based” phone error rate computed from edit distances provides a more relevant metric for ASR. Tables 2 and 3 show the results of our experiments. For both types of error rates, and across all model sizes, the best performance was consistently obtained by large margin CD-HMMs trained for sequential classi?cation. Moreover, among the two different types of large margin recognizers, utterance-based training generally yielded signi?cant improvement over frame-based training. Discriminative learning of CD-HMMs is an active research area in ASR. Two types of algorithms have been widely used: maximum mutual information (MMI) [20] and minimum classi?cation er-

mixture (per state) 1 2 4 8

baseline (EM) 45% 45% 42% 41%

margin (frame) 37% 36% 35% 34%

margin (utterance) 30% 29% 28% 27%

mixture (per state) 1 2 4 8

baseline (EM) 40.1% 36.5% 34.7% 32.7%

margin (frame) 36.3% 33.5% 32.6% 31.0%

margin (utterance) 31.2% 30.8% 29.8% 28.2%

Table 2: Frame-based phone error rates, from Hamming distance, of different recognizers. See text for details.

Table 3: String-based phone error rates, from edit distance, of different recognizers. See text for details.

ror [4]. In [16], we compare the large margin training proposed in this paper to both MMI and MCE systems for phoneme recognition trained on the exact same acoustic features. There we ?nd that the large margin approach leads to lower error rates, owing perhaps to the absence of local minima in the objective function and/or the use of margin constraints based on Hamming distances.

4 Discussion

Discriminative learning of sequential models is an active area of research in both ASR [10, 13, 20] and machine learning [1, 6, 17]. This paper makes contributions to lines of work in both communities. First, in distinction to previous work in ASR, we have proposed a convex, margin-based cost function that penalizes incorrect decodings in proportion to their Hamming distance from the desired transcription. The use of the Hamming distance in this context is a crucial insight from the work of [17] in the machine learning community, and it differs profoundly from merely penalizing the log-likelihood gap between incorrect and correct transcriptions, as commonly done in ASR. Second, in distinction to previous work in machine learning, we have proposed a framework for sequential classi?cation that naturally integrates with the infrastructure of modern speech recognizers. Using the softmax function, we have also proposed a novel way to monitor the exponentially many margin constraints that arise in sequential classi?cation. For real-valued observation sequences, we have shown how to train large margin HMMs via convex optimizations over their parameter space of positive semide?nite matrices. Finally, we have demonstrated that these learning algorithms lead to improved sequential classi?cation on data sets with over one million training examples (i.e., phonetically labeled frames of speech). In ongoing work, we are applying our approach to large vocabulary ASR and other tasks such as speaker identi?cation and visual object recognition.

A

Solver

The optimizations in eqs. (5), (6), (9) and (14) are convex: speci?cally, in terms of the matrices that parameterize large margin GMMs and HMMs, the objective functions are linear, while the constraints de?ne convex sets. Despite being convex, however, these optimizations cannot be managed by off-the-shelf numerical optimization solvers or generic interior point methods for problems as large as the ones in this paper. We devised our own special-purpose solver for these purposes. For simplicity, we describe our solver for the optimization of eq. (6), noting that it is easily extended to eqs. (9) and (14). To begin, we eliminate the slack variables and rewrite the objective function in terms of the hinge loss function: hinge(z) = max(0, z). This yields the objective function: L=

n,c=yn

hinge 1 + z T (Φyn ?Φc )z n + γ n

c

trace(Ψc ),

(15)

which is convex in terms of the positive semide?nite matrices Φc . We minimize L using a projected subgradient method [2], taking steps along the subgradient of L, then projecting the matrices {Φc } back onto the set of positive semide?nite matrices after each update. This method is guaranteed to converge to the global minimum, though it typically converges very slowly. For faster convergence, we precede this method with an unconstrained conjugate gradient optimization in the square-root matrices {?c }, where Φc = ?c ?T . The latter optimization is not convex, but in practice it rapidly c converges to an excellent starting point for the projected subgradient method.

Acknowledgment

This work was supported by the National Science Foundation under grant Number 0238323. We thank F. Pereira, K. Crammer, and S. Roweis for useful discussions and correspondence. Part of this work was conducted while both authors were af?liated with the University of Pennsylvania.

References

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- Hidden Markov Models for Automatic Speech Recognition
- Large margin hidden markov models for speech recognition
- Hidden Markov models for speech recognition
- Hidden Model Sequence Models for Automatic Speech Recognition
- The application of hidden markov models in speech recognition
- Gain adapted hidden markov models for recognition of clean and noisy speech
- Continuous speech recognition using multilayer perceptrons with hidden Markov models
- a tutorial on hidden markov models and selected applications in speech recognition
- On the application of hidden markov models for enhancing noisy speech
- Factor analysed hidden Markov models for speech recognition

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