Feature-Frequency–Adaptive On-line - IA de Investigación especializada en el MIT

Feature-Frequency–Adaptive On-line
Training for Fast and Accurate Natural
Procesamiento del lenguaje

Xu Sun∗
Peking University

Wenjie Li∗∗
Hong Kong Polytechnic University

Houfeng Wang†
Peking University

Qin Lu‡
Hong Kong Polytechnic University

Training speed and accuracy are two major concerns of large-scale natural language processing
sistemas. Typically, we need to make a tradeoff between speed and accuracy. It is trivial to improve
the training speed via sacriﬁcing accuracy or to improve the accuracy via sacriﬁcing speed.
Sin embargo, it is nontrivial to improve the training speed and the accuracy at the same time,
which is the target of this work. To reach this target, we present a new training method, feature-
frequency–adaptive on-line training, for fast and accurate training of natural language process-
ing systems. It is based on the core idea that higher frequency features should have a learning rate
that decays faster. Theoretical analysis shows that the proposed method is convergent with a fast
convergence rate. Experiments are conducted based on well-known benchmark tasks, incluido
named entity recognition, word segmentation, phrase chunking, and sentiment analysis. Estos
tasks consist of three structured classiﬁcation tasks and one non-structured classiﬁcation task,
with binary features and real-valued features, respectivamente. Experimental results demonstrate
that the proposed method is faster and at the same time more accurate than existing methods,
achieving state-of-the-art scores on the tasks with different characteristics.

∗ Key Laboratory of Computational Linguistics (Peking University), Ministry of Education, Beijing, Porcelana,

and School of EECS, Peking University, Beijing, Porcelana. Correo electrónico: xusun@pku.edu.cn.

∗∗ Department of Computing, Hong Kong Polytechnic University, Hung Hom, Kowloon 999077, hong

kong. Correo electrónico: cswjli@comp.polyu.edu.hk.

† Key Laboratory of Computational Linguistics (Peking University), Ministry of Education, Beijing, Porcelana,

and School of EECS, Peking University, Beijing, Porcelana. Correo electrónico: wanghf@pku.edu.cn.

‡ Department of Computing, Hong Kong Polytechnic University, Hung Hom, Kowloon 999077, hong

kong. Correo electrónico: csluqin@comp.polyu.edu.hk.

Envío recibido: 27 December 2012; versión revisada recibida: 30 Puede 2013; accepted for publication:
16 Septiembre 2013.

doi:10.1162/COLI a 00193

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Ligüística computacional

Volumen 40, Número 3

1. Introducción

Training speed is an important concern of natural language processing (NLP) sistemas.
Large-scale NLP systems are computationally expensive. In many real-world applica-
ciones, we further need to optimize high-dimensional model parameters. Por ejemplo,
the state-of-the-art word segmentation system uses more than 40 million features (Sol,
Wang, and Li 2012). The heavy NLP models together with high-dimensional parameters
lead to a challenging problem on model training, which may require week-level training
time even with fast computing machines.

Accuracy is another very important concern of NLP systems. Sin embargo, usually
it is quite difﬁcult to build a system that has fast training speed and at the same time
has high accuracy. Typically we need to make a tradeoff between speed and accuracy,
to trade training speed for higher accuracy or vice versa. En este trabajo, we have tried
to overcome this problem: to improve the training speed and the model accuracy at the
mismo tiempo.

There are two major approaches for parameter training: batch and on-line. Estándar
gradient descent methods are normally batch training methods, in which the gradient
computed by using all training instances is used to update the parameters of the model.
The batch training methods include, Por ejemplo, steepest gradient descent, conjugate
gradient descent (CG), and quasi-Newton methods like limited-memory BFGS (Nocedal
and Wright 1999). The true gradient is usually the sum of the gradients from each
individual training instance. Por lo tanto, batch gradient descent requires the training
method to go through the entire training set before updating parameters. This is why
batch training methods are typically slow.

On-line learning methods can signiﬁcantly accelerate the training speed compared
with batch training methods. A representative on-line training method is the stochastic
gradient descent method (SGD) and its extensions (p.ej., stochastic meta descent) (Bottou
1998; Vishwanathan et al. 2006). The model parameters are updated more frequently
compared with batch training, and fewer passes are needed before convergence. Para
large-scale data sets, on-line training methods can be much faster than batch training
methods.

Sin embargo, we ﬁnd that the existing on-line training methods are still not good
enough for training large-scale NLP systems—probably because those methods are
not well-tailored for NLP systems that have massive features. Primero, the convergence
speed of the existing on-line training methods is not fast enough. Our studies show that
the existing on-line training methods typically require more than 50 training passes
before empirical convergence, which is still slow. For large-scale NLP systems, el
training time per pass is typically long and fast convergence speed is crucial. Segundo,
the accuracy of the existing on-line training methods is not good enough. We want to
further improve the training accuracy. We try to deal with the two challenges at the
mismo tiempo. Our goal is to develop a new training method for faster and at the same time
more accurate natural language processing.

In this article, we present a new on-line training method, adaptive on-line gradient
descent based on feature frequency information (ADF),1 for very accurate and fast
on-line training of NLP systems. Other than the high training accuracy and fast train-
ing speed, we further expect that the proposed training method has good theoretical

1 ADF source code and tools can be obtained from http://klcl.pku.edu.cn/member/sunxu/index.htm.

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Sun et al.

Feature-Frequency–Adaptive On-line Training for Natural Language Processing

propiedades. We want to prove that the proposed method is convergent and has a fast
convergence rate.

In the proposed ADF training method, we use a learning rate vector in the on-line
updating. This learning rate vector is automatically adapted based on feature frequency
information in the training data set. Each model parameter has its own learning rate
adapted on feature frequency information. This proposal is based on the simple intu-
ition that a feature with higher frequency in the training process should have a learning
rate that decays faster. This is because a higher frequency feature is expected to be
well optimized with higher conﬁdence. De este modo, a higher frequency feature is expected to
have a lower learning rate. We systematically formalize this intuition into a theoretically
sound training algorithm, ADF.

The main contributions of this work are as follows:

On the methodology side, we propose a general purpose on-line training
método, ADF. The ADF method is signiﬁcantly more accurate than
existing on-line and batch training methods, and has faster training speed.
Además, theoretical analysis demonstrates that the ADF method is
convergent with a fast convergence rate.

On the application side, for the three well-known tasks, including named
entity recognition, word segmentation, and phrase chunking, the proposed
simple method achieves equal or even better accuracy than the existing
gold-standard systems, which are complicated and use extra resources.

2. Trabajo relacionado

Our main focus is on structured classiﬁcation models with high dimensional features.
For structured classiﬁcation, the conditional random ﬁelds model is widely used. A
illustrate that the proposed method is a general-purpose training method not limited to
a speciﬁc classiﬁcation task or model, we also evaluate the proposal for non-structured
classiﬁcation tasks like binary classiﬁcation. For non-structured classiﬁcation, El máximo-
imum entropy model (Berger, Della Pietra, and Della Pietra 1996; Ratnaparkhi 1996)
is widely used. Aquí, we review the conditional random ﬁelds model and the related
work of on-line training methods.

2.1 Conditional Random Fields

The conditional random ﬁeld (CRF) model is a representative structured classiﬁcation
model and it is well known for its high accuracy in real-world applications. The CRF
model is proposed for structured classiﬁcation by solving “the label bias problem”
(Lafferty, McCallum, and Pereira 2001). Assuming a feature function that maps a pair of
observation sequence xxx and label sequence yyy to a global feature vector fff, the probability
of a label sequence yyy conditioned on the observation sequence xxx is modeled as follows
(Lafferty, McCallum, and Pereira 2001):

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PAG(yyy|xxx, www) =

where www is a parameter vector.

∑

⊤

exp. {www
fff (yyy, xxx)}
′
′
′y
′ exp {www⊤fff (y
y
′
′y
∀y
y

, xxx)}

(1)

565

Ligüística computacional

Volumen 40, Número 3

Given a training set consisting of n labeled sequences, zzzi = (xxxi, yyyi), for i = 1 . . . norte,

parameter estimation is performed by maximizing the objective function,

l(www) =

n∑

yo=1

registro P(yyyi

|xxxi, www) − R(www)

(2)

The ﬁrst term of this equation represents a conditional log-likelihood of training
datos. The second term is a regularizer for reducing overﬁtting. We use an L2 prior,
||www||2
R(www) =
2(cid:27)2 . In what follows, we denote the conditional log-likelihood of each sample
|xxxi, www) as ℓ(zzzi, www). The ﬁnal objective function is as follows:
as log P(yyyi

l(www) =

n∑

yo=1

ℓ(zzzi, www) −

||www||2
2p2

(3)

2.2 On-line Training

The most representative on-line training method is the SGD method (Bottou 1998;
Tsuruoka, Tsujii, and Ananiadou 2009; Sun et al. 2013). The SGD method uses a
randomly selected small subset of the training sample to approximate the gradient of
an objective function. The number of training samples used for this approximation is
called the batch size. By using a smaller batch size, one can update the parameters
more frequently and speed up the convergence. The extreme case is a batch size of 1,
and it gives the maximum frequency of updates, which we adopt in this work. En esto
caso, the model parameters are updated as follows:

wwwt+1 = wwwt + γt

∇

wwwt

stoch(zzzi, wwwt)

(4)

where t is the update counter, γt is the learning rate or so-called decaying rate, y
l
stoch(zzzi, wwwt) is the stochastic loss function based on a training sample zzzi. (More details
of SGD are described in Bottou [1998], Tsuruoka, Tsujii, and Ananiadou [2009], y
Sun et al. [2013].) Following the most recent work of SGD, the exponential decaying
rate works the best for natural language processing tasks, and it is adopted in our
implementation of the SGD (Tsuruoka, Tsujii, and Ananiadou 2009; Sun et al. 2013).

Other well-known on-line training methods include perceptron training (Freund
and Schapire 1999), averaged perceptron training (collins 2002), more recent devel-
opment/extensions of stochastic gradient descent (p.ej., the second-order stochastic
gradient descent training methods like stochastic meta descent) (Vishwanathan et al.
2006; Hsu et al. 2009), etcétera. Sin embargo, the second-order stochastic gradient descent
method requires the computation or approximation of the inverse of the Hessian matrix
of the objective function, which is typically slow, especially for heavily structured classi-
ﬁcation models. Usually the convergence speed based on number of training iterations
is moderately faster, but the time cost per iteration is slower. Thus the overall time cost
is still large.

Compared with the related work on batch and on-line training (Jacobs 1988;
Sperduti and Starita 1993; Dragar, Crammer, and Pereira 2008; Duchi, Hazan, y
Cantante 2010; McMahan and Streeter 2010), our work is fundamentally different. El
proposed ADF training method is based on feature frequency adaptation, and to the best
of our knowledge there is no prior work on direct feature-frequency–adaptive on-line

566

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Sun et al.

Feature-Frequency–Adaptive On-line Training for Natural Language Processing

training. Compared with the conﬁdence-weighted (CW) classiﬁcation method and its
variation AROW (Dragar, Crammer, and Pereira 2008; Crammer, Kulesza, and Dredze
2009), the proposed method is substantially different. While the feature frequency
information is implicitly modeled via a complicated Gaussian distribution framework
in Dredze, Crammer, and Pereira (2008) and Crammer, Kulesza, and Dredze (2009),
the frequency information is explicitly modeled in our proposal via simple learning
rate adaptation. Our proposal is more straightforward in capturing feature frequency
información, and it has no need to use Gaussian distributions and KL divergence,
which are important in the CW and AROW methods. Además, our proposal is a
probabilistic learning method for training probabilistic models such as CRFs, mientras
the CW and AROW methods (Dragar, Crammer, and Pereira 2008; Crammer, Kulesza,
and Dredze 2009) are non-probabilistic learning methods extended from perceptron-
style approaches. De este modo, the framework is different. This work is a substantial extension
of the conference version (Sol, Wang, and Li 2012). Sol, Wang, and Li (2012) focus on
the speciﬁc task of word segmentation, whereas this article focuses on the proposed
training algorithm.

3. Feature-Frequency–Adaptive On-line Learning

In traditional on-line optimization methods such as SGD, no distinction is made for
different parameters in terms of the learning rate, and this may result in slow conver-
gence of the model training. Por ejemplo, in the on-line training process, suppose the
high frequency feature f1 and the low frequency feature f2 are observed in a training
sample and their corresponding parameters w1 and w2 are to be updated via the same
learning rate γt. Suppose the high frequency feature f1 has been updated 100 veces
and the low frequency feature f2 has only been updated once. Entonces, es posible que
the weight w1 is already well optimized and the learning rate γt is too aggressive for
updating w1. Updating the weight w1 with the learning rate γt may make w1 be far
from the well-optimized value, and it will require corrections in the future updates. Este
causes ﬂuctuations in the on-line training and results in slow convergence speed. On
the other hand, it is possible that the weight w2 is poorly optimized and the same learn-
ing rate γt is too conservative for updating w2. This also results in slow convergence
velocidad.

Para resolver este problema, we propose ADF. In spite of the high accuracy and fast
convergence speed, the proposed method is easy to implement. The proposed method
with feature-frequency–adaptive learning rates can be seen as a learning method with
speciﬁc diagonal approximation of the Hessian information based on assumptions of
feature frequency information. In this approximation, the diagonal elements of the
diagonal matrix correspond to the feature-frequency–adaptive learning rates. Accord-
ing to the aforementioned example and analysis, it assumes that a feature with higher
frequency in the training process should have a learning rate that decays faster.

3.1 Algoritmo

In the proposed ADF method, we try to use more reﬁned learning rates than traditional
SGD training. Instead of using a single learning rate (a scalar) for all weights, we extend
the learning rate scalar to a learning rate vector, which has the same dimension as the
weight vector www. The learning rate vector is automatically adapted based on feature

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Ligüística computacional

Volumen 40, Número 3

frequency information. By doing so, each weight has its own learning rate, and we will
show that this can signiﬁcantly improve the convergence speed of on-line learning.

In the ADF learning method, the update formula is:

wwwt+1 = wwwt + γγγt

··· gggt

(5)

The update term gggt is the gradient term of a randomly sampled instance:

gggt = ∇

wwwt

stoch(zzzi, wwwt) = ∇
l

wwwt

ℓ(zzzi, wwwt) −

{

}

||wwwt
||2
2nσ2

Además, γγγt
component-wise (Hadamard) product of two vectors.

+ is a positive vector-valued learning rate and ··· denotes the

∈ R f

The learning rate vector γγγt is automatically adapted based on feature frequency
information in the updating process. Intuitivamente, a feature with higher frequency in the
training process has a learning rate that decays faster. This is because a weight with
higher frequency is expected to be more adequately trained, hence a lower learning
rate is preferable for fast convergence. We assume that a high frequency feature should
have a lower learning rate, and a low frequency feature should have a relatively higher
learning rate in the training process. We systematically formalize this idea into a theoret-
ically sound training algorithm. The proposed method with feature-frequency–adaptive
learning rates can be seen as a learning method with speciﬁc diagonal approximation
of the inverse of the Hessian matrix based on feature frequency information.

Given a window size q (number of samples in a window), we use a vector vvv to record
the feature frequency. The kth entry vvvk corresponds to the frequency of the feature k in
this window. Given a feature k, we use u to record the normalized frequency:

u = vvvk/q

For each feature, an adaptation factor η is calculated based on the normalized frequency
información, como sigue:

η = α − u(α − β)

where α and β are the upper and lower bounds of a scalar, con 0 < β < α < 1. Intu- itively, the upper bound α corresponds to the adaptation factor of the lowest frequency features, and the lower bound β corresponds to the adaptation factor of the highest frequency features. The optimal values of α and β can be tuned based on speciﬁc real- world tasks, for example, via cross-validation on the training data or using held-out data. In practice, via cross-validation on the training data of different tasks, we found that the following setting is sufﬁcient to produce adequate performance for most of the real-world natural language processing tasks: α around 0.995, and β around 0.6. This indicates that the feature frequency information has similar characteristics across many different natural language processing tasks. As we can see, a feature with higher frequency corresponds to a smaller scalar via linear approximation. Finally, the learning rate is updated as follows: γγγk ← ηγγγk 568 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / c o l i / l a r t i c e - p d f / / / / 4 0 3 5 6 3 1 8 0 3 7 9 7 / c o l i _ a _ 0 0 1 9 3 p d . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Sun et al. Feature-Frequency–Adaptive On-line Training for Natural Language Processing ADF learning algorithm stoch(zzzi, www) www ← 0, t ← 0, vvv ← 0, γγγ ← c repeat until convergence . Draw a sample zzzi at random from the data set ZZZ . vvv ← UPDATEFEATUREFREQ(vvv, zzzi) if t > 0 and t mod q = 0
.
. γγγ ← UPDATELEARNRATE(γγγ, vvv)
.
. vvv ← 0
.
. ggg ← ∇
l
www
. www ← www + γγγ ··· ggg
t ← t + 1
.
return www

1: procedure ADF(ZZZ, www, q, C, a, b)
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14: procedure UPDATEFEATUREFREQ(vvv, zzzi)
for k ∈ features used in sample zzzi
15:
. vvvk
16:
return vvv
17:
18:
19: procedure UPDATELEARNRATE(γγγ, vvv)
20:
21:
22:
23:
24:

for k ∈ all features
. u ← vvvk/q
. η ← α − u(α − β)
← ηγγγk
. γγγk
return γγγ

← vvvk + 1

Cifra 1
The proposed ADF on-line learning algorithm. In the algorithm, ZZZ is the training data set; q, C, a,
and β are hyper-parameters; q is an integer representing window size; c is for initializing the
learning rates; and α and β are the upper and lower bounds of a scalar, con 0 < β < α < 1. With this setting, different features correspond to different adaptation factors based on feature frequency information. Our ADF algorithm is summarized in Figure 1. The ADF training method is efﬁcient because the only additional computation (compared with traditional SGD) is the derivation of the learning rates, which is simple and efﬁcient. As we know, the regularization of SGD can perform efﬁciently via the opti- mization based on sparse features (Shalev-Shwartz, Singer, and Srebro 2007). Similarly, the derivation of γγγt can also perform efﬁciently via the optimization based on sparse features. Note that although binary features are common in natural language processing tasks, the ADF algorithm is not limited to binary features and it can be applied to real- valued features. 3.2 Convergence Analysis We want to show that the proposed ADF learning algorithm has good convergence properties. There are two steps in the convergence analysis. First, we show that the ADF update rule is a contraction mapping. Then, we show that the ADF training is asymptotically convergent, and with a fast convergence rate. To simplify the discussion, our convergence analysis is based on the convex loss function of traditional classiﬁcation or regression problems: L(www) = n∑ i=1 ℓ(xxxi, yi, www · fff i) − ||www||2 2σ2 569 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / c o l i / l a r t i c e - p d f / / / / 4 0 3 5 6 3 1 8 0 3 7 9 7 / c o l i _ a _ 0 0 1 9 3 p d . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Computational Linguistics Volume 40, Number 3 where fff i is the feature vector generated from the training sample (xxxi, yi). L(www) is a func- tion in www · fff i, such as 1 − www · fff i)2 for regression or log[1 + exp(−yiwww · fff i)] for binary classiﬁcation. 2 (yi To make convergence analysis of the proposed ADF training algorithm, we need to introduce several mathematical deﬁnitions. First, we introduce Lipschitz continuity: Deﬁnition 1 (Lipschitz continuity) A function F : X → R is Lipschitz continuous with the degree of D if |F(x) − F(y)| ≤ D|x − y| for ∀x, y ∈ X . X can be multi-dimensional space, and |x − y| is the distance between the points x and y. Based on the deﬁnition of Lipschitz continuity, we give the deﬁnition of the Lipschitz constant ||F|| Lip as follows: Deﬁnition 2 (Lipschitz constant) ||F|| Lip := inf{D where |F(x) − F(y)| ≤ D|x − y| for ∀x, y} In other words, the Lipschitz constant ||F|| that makes the function F Lipschitz continuous. Lip is the lower bound of the continuity degree Further, based on the deﬁnition of Lipschitz constant, we give the deﬁnition of contraction mapping as follows: Deﬁnition 3 (Contraction mapping) A function F : X → X is a contraction mapping if its Lipschitz constant is smaller than 1: ||F|| Lip < 1. Then, we can show that the traditional SGD update is a contraction mapping. Lemma 1 (SGD update rule is contraction mapping) −1, Let γ be a ﬁxed low learning rate in SGD updating. If γ ≤ (||x2 i the SGD update rule is a contraction mapping in Euclidean space with Lipschitz con- tinuity degree 1 − γ/σ2. || · ||∇ ′ℓ(xxxi, yi, y ′ ′y y y Lip) )|| ′ The proof can be extended from the related work on convergence analysis of parallel SGD training (Zinkevich et al. 2010). The stochastic training process is a one-following- one dynamic update process. In this dynamic process, if we use the same update rule F, we have wwwt+1 = F(wwwt) and wwwt+2 = F(wwwt+1). It is only necessary to prove that the dynamic update is a contraction mapping restricted by this one-following-one dynamic process. That is, for the proposed ADF update rule, it is only necessary to prove it is a dynamic contraction mapping. We formally deﬁne dynamic contraction mapping as follows. Deﬁnition 4 (Dynamic contraction mapping) Given a function F : X → X , suppose the function is used in a dynamic one-following- ∈ X . Then, the function F is a one process: xt+1 = F(xt) and xt+2 = F(xt+1) for ∀xt | for ∀xt − xt+1 dynamic contraction mapping if ∃D < 1, |xt+2 | ≤ D|xt+1 ∈ X . − xt We can see that a contraction mapping is also a dynamic contraction mapping, but a dynamic contraction mapping is not necessarily a contraction mapping. We ﬁrst show 570 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / c o l i / l a r t i c e - p d f / / / / 4 0 3 5 6 3 1 8 0 3 7 9 7 / c o l i _ a _ 0 0 1 9 3 p d . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Sun et al. Feature-Frequency–Adaptive On-line Training for Natural Language Processing that the ADF update rule with a ﬁxed learning rate vector of different learning rates is a dynamic contraction mapping. Theorem 1 (ADF update rule with ﬁxed learning rates) Let γγγ be a ﬁxed learning rate vector with different learning rates. Let γmax be the max- imum learning rate in the learning rate vector γγγ: γmax := sup{γi where γi ∈ γγγ}. Then −1, the ADF update rule is a dynamic contraction if γmax mapping in Euclidean space with Lipschitz continuity degree 1 − γmax/σ2. The proof is sketched in Section 5. ′ || · ||∇ ′ℓ(xxxi, yi, y ′ ′y y y ≤ (||x2 i Lip) )|| )|| Lip) Further, we need to prove that the ADF update rule with a decaying learning rate vector is a dynamic contraction mapping, because the real ADF algorithm has a || · decaying learning rate vector. In the decaying case, the condition that γmax −1 can be easily achieved, because γγγ continues to decay with an ′ ||∇ ′ℓ(xxxi, yi, y ′ ′y y y exponential decaying rate. Even if the γγγ is initialized with high values of learning rates, after a number of training passes (denoted as T) γγγT is guaranteed to be small enough so −1. Without that γmax := sup{γi where γi losing generality, our convergence analysis starts from the pass T and we take γγγT as γγγ0 in the following analysis. Thus, we can show that the ADF update rule with a decaying learning rate vector is a dynamic contraction mapping: ′ || · ||∇ ′ℓ(xxxi, yi, y ′ ′y y y } and γmax ≤ (||x2 i ≤ (||x2 i ∈ γγγT Lip) )|| Theorem 2 (ADF update rule with decaying learning rates) Let γγγt be a learning rate vector in the ADF learning algorithm, which is decaying over the time t and with different decaying rates based on feature frequency infor- ≤ mation. Let γγγt start from a low enough learning rate vector γγγ0 such that γmax −1, where γmax is the maximum element in γγγ0. Then, the ADF (||x2 i update rule with decaying learning rate vector is a dynamic contraction mapping in Euclidean space with Lipschitz continuity degree 1 − γmax/σ2. The proof is sketched in Section 5. ′ || · ||∇ ′ℓ(xxxi, yi, y ′ ′y y y Lip) )|| Based on the connections between ADF training and contraction mapping, we demonstrate the convergence properties of the ADF training method. First, we prove the convergence of the ADF training. Theorem 3 (ADF convergence) ADF training is asymptotically convergent. The proof is sketched in Section 5. Further, we analyze the convergence rate of the ADF training. When we have the lowest learning rate γγγt+1 = βγγγt, the expectation of the obtained wwwt is as follows (Murata 1998; Hsu et al. 2009): E(wwwt) = www ∗ + t∏ m=1 ∗ (III − γγγ0βmHHH(www ))(www0 ∗ − www ) ∗ where www is the optimal weight vector, and HHH is the Hessian matrix of the objective function. The rate of convergence is governed by the largest eigenvalue of the function CCCt = )). Following Murata (1998) and Hsu et al. (2009), we can derive a bound of rate of convergence, as follows. m=1(III − γγγ0βmHHH(www ∏ t ∗ 571 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / c o l i / l a r t i c e - p d f / / / / 4 0 3 5 6 3 1 8 0 3 7 9 7 / c o l i _ a _ 0 0 1 9 3 p d . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Computational Linguistics Volume 40, Number 3 Theorem 4 (ADF convergence rate) m=1(III − γγγ0βmHHH(www Assume ϕ is the largest eigenvalue of the function CCCt = proposed ADF training, its convergence rate is bounded by ϕ, and we have ∏ t ∗ )). For the ϕ ≤ exp { γγγ0λβ β − 1 } ∗ where λ is the minimum eigenvalue of HHH(www The proof is sketched in Section 5. ). The convergence analysis demonstrates that the proposed method with feature- frequency-adaptive learning rates is convergent and the bound of convergence rate is analyzed. It demonstrates that increasing the values of γγγ0 and β leads to a lower bound of the convergence rate. Because the bound of the convergence rate is just an up-bound rather than the actual convergence rate, we still need to conduct automatic tuning of the hyper-parameters, including γγγ0 and β, for optimal convergence rate in practice. The ADF training method has a fast convergence rate because the feature- frequency-adaptive schema can avoid the ﬂuctuations on updating the weights of high frequency features, and it can avoid the insufﬁcient training on updating the weights of low frequency features. In the following sections, we perform experiments to conﬁrm the fast convergence rate of the proposed method. 4. Evaluation Our main focus is on training heavily structured classiﬁcation models. We evaluate the proposal on three NLP structured classiﬁcation tasks: biomedical named entity recogni- tion (Bio-NER), Chinese word segmentation, and noun phrase (NP) chunking. For the structured classiﬁcation tasks, the ADF training is based on the CRF model (Lafferty, McCallum, and Pereira 2001). Further, to demonstrate that the proposed method is not limited to structured classiﬁcation tasks, we also perform experiments on a non- structured binary classiﬁcation task: sentiment-based text classiﬁcation. For the non- structured classiﬁcation task, the ADF training is based on the maximum entropy model (Berger, Della Pietra, and Della Pietra 1996; Ratnaparkhi 1996). 4.1 Biomedical Named Entity Recognition (Structured Classiﬁcation) The biomedical named entity recognition (Bio-NER) task is from the BIONLP-2004 shared task. The task is to recognize ﬁve kinds of biomedical named entities, including DNA, RNA, protein, cell line, and cell type, on the MEDLINE biomedical text mining corpus (Kim et al. 2004). A typical approach to this problem is to cast it as a sequential labeling task with the BIO encoding. This data set consists of 20,546 training samples (from 2,000 MEDLINE article abstracts, with 472,006 word tokens) and 4,260 test samples. The properties of the data are summarized in Table 1. State-of-the-art systems for this task include Settles (2004), Finkel et al. (2004), Okanohara et al. (2006), Hsu et al. (2009), Sun, Matsuzaki, et al. (2009), and Tsuruoka, Tsujii, and Ananiadou (2009). Following previous studies for this task (Okanohara et al. 2006; Sun, Matsuzaki, et al. 2009), we use word token–based features, part-of-speech (POS) based features, and orthography pattern–based features (preﬁx, uppercase/lowercase, etc.), as listed in Table 2. With the traditional implementation of CRF systems (e.g., the HCRF package), the edges features usually contain only the information of yi−1 and yi, and ignore the 572 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / c o l i / l a r t i c e - p d f / / / / 4 0 3 5 6 3 1 8 0 3 7 9 7 / c o l i _ a _ 0 0 1 9 3 p d . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Sun et al. Feature-Frequency–Adaptive On-line Training for Natural Language Processing Table 1 Summary of the Bio-NER data set. #Abstracts #Sentences #Words Train Test 2,000 404 20,546 (10/abs) 4,260 (11/abs) 472,006 (23/sen) 96,780 (23/sen) Table 2 Feature templates used for the Bio-NER task. wi is the current word token on position i. ti is the POS tag on position i. oi is the orthography mode on position i. yi is the classiﬁcation label on position i. yi−1yi represents label transition. A × B represents a Cartesian product between two sets. } } Word Token–based Features: {wi−2, wi−1, wi, wi+1, wi+2, wi−1wi, wiwi+1 ×{yi, yi−1yi Part-of-Speech (POS)–based Features: {ti−2, ti−1, ti, ti+1, ti+2, ti−2ti−1, ti−1ti, titi+1, ti+1ti+2, ti−2ti−1ti, ti−1titi+1, titi+1ti+2 ×{yi, yi−1yi Orthography Pattern–based Features: {oi−2, oi−1, oi, oi+1, oi+2, oi−2oi−1, oi−1oi, oioi+1, oi+1oi+2 ×{yi, yi−1yi } } } } information of the observation sequence (i.e., xxx). The major reason for this simple real- ization of edge features in traditional CRF implementation is to reduce the dimension of features. To improve the model accuracy, we utilize rich edge features following Sun, Wang, and Li (2012), in which local observation information of xxx is combined in edge features just like the implementation of node features. A detailed introduction to rich edge features can be found in Sun, Wang, and Li (2012). Using the feature templates, we extract a high dimensional feature set, which contains 5.3 × 107 features in total. Following prior studies, the evaluation metric for this task is the balanced F-score deﬁned as 2PR/(P + R), where P is precision and R is recall. 4.2 Chinese Word Segmentation (Structured Classiﬁcation) Chinese word segmentation aims to automatically segment character sequences into word sequences. Chinese word segmentation is important because it is the ﬁrst step for most Chinese language information processing systems. Our experiments are based on the Microsoft Research data provided by The Second International Chinese Word Segmentation Bakeoff. In this data set, there are 8.8 × 104 word-types, 2.4 × 106 word- tokens, 5 × 103 character-types, and 4.1 × 106 character-tokens. State-of-the-art systems for this task include Tseng et al. (2005), Zhang, Kikui, and Sumita (2006), Zhang and Clark (2007), Gao et al. (2007), Sun, Zhang, et al. (2009), Sun (2010), Zhao et al. (2010), and Zhao and Kit (2011). The feature engineering follows previous work on word segmentation (Sun, Wang, and Li 2012). Rich edge features are used. For the classiﬁcation label yi and the label transition yi−1yi on position i, we use the feature templates as follows (Sun, Wang, and Li 2012): r Character unigrams located at positions i − 2, i − 1, i, i + 1, and i + 2. 573 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / c o l i / l a r t i c e - p d f / / / / 4 0 3 5 6 3 1 8 0 3 7 9 7 / c o l i _ a _ 0 0 1 9 3 p d . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Computational Linguistics Volume 40, Number 3 r r r r r r r Character bigrams located at positions i − 2, i − 1, i and i + 1. Whether xj and xj+1 are identical, for j = i − 2, . . . , i + 1. Whether xj and xj+2 are identical, for j = i − 3, . . . , i + 1. The character sequence xj,i if it matches a word w ∈ U, with the constraint i − 6 < j < i. The item xj,i represents the character sequence xj . . . xi. U represents the unigram-dictionary collected from the training data. The character sequence xi,k if it matches a word w ∈ U, with the constraint i < k < i + 6. The word bigram candidate [xj,i−1, xi,k] if it hits a word bigram [wi, wj] ∈ B, and satisﬁes the aforementioned constraints on j and k. B represents the word bigram dictionary collected from the training data. The word bigram candidate [xj,i, xi+1,k] if it hits a word bigram [wi, wj] ∈ B, and satisﬁes the aforementioned constraints on j and k. All feature templates are instantiated with values that occurred in training samples. The extracted feature set is large, and there are 2.4 × 107 features in total. Our evaluation is based on a closed test, and we do not use extra resources. Following prior studies, the evaluation metric for this task is the balanced F-score. 4.3 Phrase Chunking (Structured Classiﬁcation) In the phrase chunking task, the non-recursive cores of noun phrases, called base NPs, are identiﬁed. The phrase chunking data is extracted from the data of the CoNLL-2000 shallow-parsing shared task (Sang and Buchholz 2000). The training set consists of 8,936 sentences, and the test set consists of 2,012 sentences. We use the feature templates based on word n-grams and part-of-speech n-grams, and feature templates are shown in Table 3. Rich edge features are used. Using the feature templates, we extract 4.8 × 105 features in total. State-of-the-art systems for this task include Kudo and Matsumoto (2001), Collins (2002), McDonald, Crammer, and Pereira (2005), Vishwanathan et al. (2006), Sun et al. (2008), and Tsuruoka, Tsujii, and Ananiadou (2009). Following prior studies, the evaluation metric for this task is the balanced F-score. 4.4 Sentiment Classiﬁcation (Non-Structured Classiﬁcation) To demonstrate that the proposed method is not limited to structured classiﬁcation, we select a well-known sentiment classiﬁcation task for evaluating the proposed method on non-structured classiﬁcation. Table 3 Feature templates used for the phrase chunking task. wi, ti, and yi are deﬁned as before. } Word-Token–based Features: {wi−2, wi−1, wi, wi+1, wi+2, wi−1wi, wiwi+1 ×{yi, yi−1yi Part-of-Speech (POS)–based Features: {ti−1, ti, ti+1, ti−2ti−1, ti−1ti, titi+1, ti+1ti+2, ti−2ti−1ti, ti−1titi+1, titi+1ti+2 ×{yi, yi−1yi } } } 574 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / c o l i / l a r t i c e - p d f / / / / 4 0 3 5 6 3 1 8 0 3 7 9 7 / c o l i _ a _ 0 0 1 9 3 p d . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Sun et al. Feature-Frequency–Adaptive On-line Training for Natural Language Processing Generally, sentiment classiﬁcation classiﬁes user review text as a positive or neg- ative opinion. This task (Blitzer, Dredze, and Pereira 2007) consists of four subtasks based on user reviews from Amazon.com. Each subtask is a binary sentiment clas- siﬁcation task based on a speciﬁc topic. We use the maximum entropy model for classiﬁcation. We use the same lexical features as those used in Blitzer, Dredze, and Pereira (2007), and the total number of features is 9.4 × 105. Following prior work, the evaluation metric is binary classiﬁcation accuracy. 4.5 Experimental Setting As for training, we perform gradient descent with the proposed ADF training method. To compare with existing literature, we choose four popular training methods, a rep- resentative batch training method, and three representative on-line training methods. The batch training method is the limited-memory BFGS (LBFGS) method (Nocedal and Wright 1999), which is considered to be one of the best optimizers for log-linear models like CRFs. The on-line training methods include the SGD training method, which we introduced in Section 2.2, the structured perceptron (Perc) training method (Freund and Schapire 1999; Collins 2002), and the averaged perceptron (Avg-Perc) training method (Collins 2002). The structured perceptron method and averaged perceptron method are non-probabilistic training methods that have very fast training speed due to the avoidance of the computation on gradients (Sun, Matsuzaki, and Li 2013). All training methods, including ADF, SGD, Perc, Avg-Perc, and LBFGS, use the same set of features. We also compared the ADF method with the CW method (Dredze, Crammer, and Pereira 2008) and the AROW method (Crammer, Kulesza, and Dredze 2009). The CW and AROW methods are implemented based on the Conﬁdence Weighted Learning Library.2 Because the current implementation of the CW and AROW methods do not utilize rich edge features, we removed the rich edge features in our systems to make more fair comparisons. That is, we removed rich edge features in the CRF-ADF setting, and this simpliﬁed method is denoted as ADF-noRich. The second-order stochastic gradient descent training methods, including the SMD method (Vishwanathan et al. 2006) and the PSA method (Hsu et al. 2009), are not considered in our experiments because we ﬁnd those methods are quite slow when running on our data sets with high dimensional features. We ﬁnd that the settings of q, α, and β in the ADF training method are not sensitive among speciﬁc tasks and can be generally set. We simply set q = n/10 (n is the number of training samples). It means that feature frequency information is updated 10 times per iteration. Via cross-validation only on the training data of different tasks, we ﬁnd that the following setting is sufﬁcient to produce adequate performance for most of the real-world natural language processing tasks: α around 0.995 and β around 0.6. This indicates that the feature frequency information has similar characteristics across many different natural language processing tasks. Thus, we simply use the following setting for all tasks: q = n/10, α = 0.995, and β = 0.6. This leaves c (the initial value of the learning rates) as the only hyper-parameter that requires careful tuning. We perform automatic tuning for c based on the training data via 4-fold cross-validation, testing with c = 0.005, 0.01, 0.05, 0.1, respectively, and the optimal c is chosen based on the best accuracy of cross-validation. Via this automatic 2 http://webee.technion.ac.il/people/koby/code-index.html. 575 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / c o l i / l a r t i c e - p d f / / / / 4 0 3 5 6 3 1 8 0 3 7 9 7 / c o l i _ a _ 0 0 1 9 3 p d . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Computational Linguistics Volume 40, Number 3 tuning, we ﬁnd it is proper to set c = 0.005, 0.1, 0.05, 0.005, for the Bio-NER, word segmentation, phrase chunking, and sentiment classiﬁcation tasks, respectively. To reduce overﬁtting, we use an L2 Gaussian weight prior (Chen and Rosenfeld 1999) for the ADF, LBFGS, and SGD training methods. We vary the σ with different values (e.g., 1.0, 2.0, and 5.0) for 4-fold cross validation on the training data of different tasks, and ﬁnally set σ = 5.0 for all training methods in the Bio-NER task; σ = 5.0 for all training methods in the word segmentation task; σ = 5.0, 1.0, 1.0 for ADF, SGD, and LBFGS in the phrase chunking task; and σ = 1.0 for all training methods in the sentiment classiﬁcation task. Experiments are performed on a computer with an Intel(R) Xeon(R) 2.0-GHz CPU. 4.6 Structured Classiﬁcation Results 4.6.1 Comparisons Based on Empirical Convergence. First, we check the experimental re- sults of different methods on their empirical convergence state. Because the perceptron training method (Perc) does not achieve empirical convergence even with a very large number of training passes, we simply report its results based on a large enough number of training passes (e.g., 200 passes). Experimental results are shown in Table 4. As we can see, the proposed ADF method is more accurate than other training methods, either the on-line ones or the batch one. It is a bit surprising that the ADF method performs even more accurately than the batch training method (LBFGS). We notice that some previous work also found that on-line training methods could have Table 4 Results for the Bio-NER, word segmentation, and phrase chunking tasks. The results and the number of passes are decided based on empirical convergence (with score deviation of adjacent ﬁve passes less than 0.01). For the non-convergent case, we simply report the results based on a large enough number of training passes. As we can see, the ADF method achieves the best accuracy with the fastest convergence speed. Bio-NER Prec Rec F-score Passes Train-Time (sec) LBFGS (batch) SGD (on-line) Perc (on-line) Avg-Perc (on-line) ADF (proposal) 67.69 70.91 65.37 68.76 71.71 70.20 72.69 66.95 72.56 72.80 68.92 71.79 66.15 70.61 72.25 400 91 200 37 35 152,811.34 76,549.21 20,436.69 3,928.01 27,490.24 Segmentation Prec Rec F-score Passes Train-Time (sec) LBFGS (batch) SGD (on-line) Perc (on-line) Avg-Perc (on-line) ADF (proposal) 97.46 97.58 96.99 97.56 97.67 96.86 97.11 96.03 97.05 97.31 97.16 97.34 96.50 97.30 97.49 102 27 200 16 15 13,550.68 6,811.15 8,382.606 716.87 4,260.08 Chunking Prec Rec F-score Passes Train-Time (sec) LBFGS (batch) SGD (on-line) Perc (on-line) Avg-Perc (on-line) ADF (proposal) 94.57 94.48 93.66 94.34 94.66 94.09 94.04 93.31 94.04 94.38 94.33 94.26 93.48 94.19 94.52 105 56 200 12 17 797.04 903.88 543.51 33.45 282.17 576 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / c o l i / l a r t i c e - p d f / / / / 4 0 3 5 6 3 1 8 0 3 7 9 7 / c o l i _ a _ 0 0 1 9 3 p d . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Sun et al. Feature-Frequency–Adaptive On-line Training for Natural Language Processing better performance than batch training methods such as LBFGS (Tsuruoka, Tsujii, and Ananiadou 2009; Schaul, Zhang, and LeCun 2012). The ADF training method can achieve better results probably because the feature-frequency–adaptive training schema can produce more balanced training of features with diversiﬁed frequencies. Traditional SGD training may over-train high frequency features and at the same time may have insufﬁcient training of low frequency features. The ADF training method can avoid such problems. It will be interesting to perform further analysis in future work. We also performed signiﬁcance tests based on t-tests with a signiﬁcance level of 0.05. Signiﬁcance tests demonstrate that the ADF method is signiﬁcantly more accurate than the existing training methods in most of the comparisons, whether on-line or batch. For the Bio-NER task, the differences between ADF and LBFGS, SGD, Perc, and Avg-Perc are signiﬁcant. For the word segmentation task, the differences between ADF and LBFGS, SGD, Perc, and Avg-Perc are signiﬁcant. For the phrase chunking task, the differences between ADF and Perc and Avg-Perc are signiﬁcant; the differences between ADF and LBFGS and SGD are non-signiﬁcant. Moreover, as we can see, the proposed method achieves a convergence state with the least number of training passes, and with the least wall-clock time. In general, the ADF method is about one order of magnitude faster than the LBFGS batch training method and several times faster than the existing on-line training methods. 4.6.2 Comparisons with State-of-the-Art Systems. The three tasks are well-known bench- mark tasks with standard data sets. There is a large amount of published research on those three tasks. We compare the proposed method with the state-of-the-art systems. The comparisons are shown in Table 5. As we can see, our system is competitive with the best systems for the Bio-NER, word segmentation, and NP-chunking tasks. Many of the state-of-the-art systems use extra resources (e.g., linguistic knowledge) or complicated systems (e.g., voting over l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / c o l i / l a r t i c e - p d f / / / / 4 0 3 5 6 3 1 8 0 3 7 9 7 / c o Table 5 Comparing our results with some representative state-of-the-art systems. Bio-NER Method F-score Semi-Markov CRF + global features (Okanohara et al. 2006) (Hsu et al. 2009) CRF + PSA(1) training (Tsuruoka, Tsujii, and Ananiadou 2009) CRF + SGD-L1 training Our Method CRF + ADF training 71.5 69.4 71.6 72.3 Segmentation (Gao et al. 2007) (Sun, Zhang, et al. 2009) (Sun 2010) Our Method Method F-score Semi-Markov CRF Latent-variable CRF Multiple segmenters + voting CRF + ADF training 97.2 97.3 96.9 97.5 Chunking (Kudo and Matsumoto 2001) (Vishwanathan et al. 2006) (Sun et al. 2008) Our Method Method F-score Combination of multiple SVM CRF + SMD training Latent-variable CRF CRF + ADF training 94.2 93.6 94.3 94.5 l i _ a _ 0 0 1 9 3 p d . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 577 Computational Linguistics Volume 40, Number 3 multiple models). Thus, it is impressive that our single model–based system without extra resources achieves good performance. This indicates that the proposed ADF training method can train model parameters with good generality on the test data. 4.6.3 Training Curves. To study the detailed training process and convergence speed, we show the training curves in Figures 2–4. Figure 2 focuses on the comparisons between the ADF method and the existing on-line training methods. As we can see, the ADF method converges faster than other on-line training methods in terms of both training passes and wall-clock time. The ADF method has roughly the same training speed per pass compared with traditional SGD training. Figure 3 (Top Row) focuses on comparing the ADF method with the CW method (Dredze, Crammer, and Pereira 2008) and the AROW method (Crammer, Kulesza, and Dredze 2009). Comparisons are based on similar features. As discussed before, the ADF- noRich method is a simpliﬁed system, with rich edge features removed from the CRF- ADF system. As we can see, the proposed ADF method, whether with or without rich edge features, outperforms the CW and AROW methods. Figure 3 (Bottom Row) focuses on the comparisons with different mini-batch (the training samples in each stochastic update) sizes. Representative results with a mini-batch size of 10 are shown. In general, we ﬁnd larger mini-batch sizes will slow down the convergence speed. Results demon- strate that, compared with the SGD training method, the ADF training method is less sensitive to mini-batch sizes. Figure 4 focuses on the comparisons between the ADF method and the batch training method LBFGS. As we can see, the ADF method converges at least one order l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / c o l i / l a r t i c e - p d f / / / / 4 0 3 5 6 3 1 8 0 3 7 9 7 / c o l i _ a _ 0 0 1 9 3 p d . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Figure 2 Comparisons among the ADF method and other on-line training methods. (Top Row) Comparisons based on training passes. As we can see, the ADF method has the best accuracy and with the fastest convergence speed based on training passes. (Bottom Row) Comparisons based on wall-clock time. 578 0204060801006667686970717273Number of Passes) ADFSGDPerc0204060801009696.296.496.696.89797.297.497.6Segmentation (ADF vs. on-line)Number of Passes) ADFSGDPerc02040608010092.59393.59494.5Chunking (ADF vs. on-line)Number of Passes) ADFSGDPerc0123456x1046667686970717273Training Time (sec)) ADFSGDPerc02,0004,0006,0008,00010,0009696.296.496.696.89797.297.497.6Segmentation (ADF vs. on-line)Training Time (sec)) ADFSGDPerc020040060080092.59393.59494.5Chunking (ADF vs. on-line)Training Time (sec)) ADFSGDPerc Sun et al. Feature-Frequency–Adaptive On-line Training for Natural Language Processing Figure 3 (Top Row) Comparing ADF and ADF-noRich with CW and AROW methods. As we can see, both the ADF and ADF-noRich methods work better than the CW and AROW methods. (Bottom Row) Comparing different methods with mini-batch = 10 in the stochastic learning setting. magnitude faster than the LBFGS training in terms of both training passes and wall- clock time. For the LBFGS training, we need to determine the LBFGS memory parameter m, which controls the number of prior gradients used to approximate the Hessian information. A larger value of m will potentially lead to more accurate estimation of the Hessian information, but at the same time will consume signiﬁcantly more memory. Roughly, the LBFGS training consumes m times more memory than the ADF on-line training method. For most tasks, the default setting of m = 10 is reasonable. We set m = 10 for the word segmentation and phrase chunking tasks, and m = 6 for the Bio-NER task due to the shortage of memory for m > 6 cases in this task.

4.6.4 One-Pass Learning Results. Many real-world data sets can only observe the training
data in one pass. Por ejemplo, some Web-based on-line data streams can only appear
once so that the model parameter learning should be ﬁnished in one-pass learning (ver
Zinkevich et al. 2010). Por eso, it is important to test the performance in the one-pass
learning scenario.

In the one-pass learning scenario, the feature frequency information is computed
“on the ﬂy” during on-line training. As shown in Section 3.1, we only need to have
a real-valued vector vvv to record the cumulative feature frequency information, cual
is updated when observing training instances one by one. Entonces, the learning rate
vector γγγ is updated based on the vvv only and there is no need to observe the training
instances again. This is the same algorithm introduced in Section 3.1 and no change is
required for the one-pass learning scenario. Cifra 5 shows the comparisons between
the ADF method and baselines on one-pass learning. As we can see, the ADF method

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02040608010060626466687072Bio−NER (ADF vs. CW/AROW)Number of PassesF−score (%) ADFADF−noRichCWAROW02040608010092939495969798Segmentation (ADF vs. CW/AROW)Number of PassesF−score (%) ADFADF−noRichCWAROW02040608010092.59393.59494.595Chunking (ADF vs. CW/AROW)Number of PassesF−score (%) ADFADF−noRichCWAROW020406080100686970717273Bio−NER (MiniBatch=10)Number of PassesF−score (%) ADFSGD02040608010096.496.696.89797.297.497.6Segmentation (MiniBatch=10)Number of PassesF−score (%) ADFSGD02040608010093.293.493.693.89494.294.494.6Chunking (MiniBatch=10)Number of PassesF−score (%) ADFSGD

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Volumen 40, Número 3

Cifra 4
Comparisons between the ADF method and the batch training method LBFGS. (Top Row)
Comparisons based on training passes. As we can see, the ADF method converges much faster
than the LBFGS method, and with better accuracy on the convergence state. (Bottom Row)
Comparisons based on wall-clock time.

consistently outperforms the baselines. This also reﬂects the fast convergence speed of
the ADF training method.

4.7 Non-Structured Classiﬁcation Results

In previous experiments, we showed that the proposed method outperforms existing
baselines on structured classiﬁcation. Sin embargo, we want to show that the ADF
method also has good performance on non-structured classiﬁcation. Además, este
task is based on real-valued features instead of binary features.

Cifra 5
Comparisons among different methods based on one-pass learning. As we can see, the ADF
method has the best accuracy on one-pass learning.

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01002003004006667686970717273Bio−NER (ADF vs. lote)Number of PassesF−score (%) ADFLBFGS01002003004009696.296.496.696.89797.297.497.6Segmentation (ADF vs. lote)Number of PassesF−score (%) ADFLBFGS010020030040093.693.89494.294.4Chunking (ADF vs. lote)Number of PassesF−score (%) ADFLBFGS051015x 1046667686970717273Bio−NER (ADF vs. lote)Training Time (segundo)F−score (%) ADFLBFGS012345x 1049696.296.496.696.89797.297.497.6Segmentation (ADF vs. lote)Training Time (segundo)F−score (%) ADFLBFGS01,0002,0003,00092.59393.59494.5Chunking (ADF vs. lote)Training Time (segundo)) ADFLBFGS

Sun et al.

Feature-Frequency–Adaptive On-line Training for Natural Language Processing

Mesa 6
Results on sentiment classiﬁcation (non-structured binary classiﬁcation).

Accuracy

Passes

Train-Time (segundo)

LBFGS (lote)
SGD (on-line)
Perc (on-line)
Avg-Perc (on-line)
ADF (propuesta)

87.00
87.13
84.55
85.04
87.89

86
44
25
46
30

72.20
55.88
5.82
12.22
57.12

Experimental results of different training methods on the convergence state are
mostrado en la tabla 6. As we can see, the proposed method outperforms all of the on-line
and batch baselines in terms of binary classiﬁcation accuracy. Here again we observe
that the ADF and SGD methods outperform the LBFGS baseline.

The training curves are shown in Figure 6. As we can see, the ADF method con-
verges quickly. Because this data set is relatively small and the feature dimension is
much smaller than previous tasks, we ﬁnd the baseline training methods also have
fast convergence speed. The comparisons on one-pass learning are shown in Fig-
ura 7. Just as for the experiments for structured classiﬁcation tasks, the ADF method

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Cifra 6
F-score curves on sentiment classiﬁcation. (Top Row) Comparisons among the ADF method and
on-line training baselines, based on training passes and wall-clock time, respectivamente. (Bottom
Fila) Comparisons between the ADF method and the batch training method LBFGS, Residencia en
training passes and wall-clock time, respectivamente. As we can see, the ADF method outperforms
both the on-line training baselines and the batch training baseline, with better accuracy and
faster convergence speed.

581

0204060801008283848586878889Sentiment (ADF vs. on-line)Number of PassesAccuracy (%) ADFSGDPerc0204060808283848586878889Sentiment (ADF vs. on-line)Training Time (segundo)Accuracy (%) ADFSGDPerc0501001508283848586878889Sentiment (ADF vs. lote)Number of PassesAccuracy (%) ADFLBFGS0501001508283848586878889Sentiment (ADF vs. lote)Training Time (segundo)Accuracy (%) ADFLBFGS

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Volumen 40, Número 3

Cifra 7
One-pass learning results on sentiment classiﬁcation.

outperforms the baseline methods on one-pass learning, with more than 12.7% error
rate reduction.

5. Proofs

This section gives proofs of Theorems 1–4.

Proof of Theorem 1 Following Equation (5), the ADF update rule is F(wwwt) := wwwt+1 =

wwwt + γγγ ··· gggt. For ∀wwwt

∈ X ,

− wwwt+1

(6)

where ai and bi are the ith elements of the vector γγγ and gggt+1, respectivamente. FSGD is the
SGD update rule with the ﬁxed learning rate γmax such that γmax := sup{γi where γi
∈
γγγ}. En otras palabras, for the SGD update rule FSGD, the ﬁxed learning rate γmax is derived
from the ADF update rule. According to Lemma 1, the SGD update rule FSGD is a
contraction mapping in Euclidean space with Lipschitz continuity degree 1 − γmax/σ2,
′
|| · ||∇
′ℓ(xxxi, yi, y
given the condition that γmax
′
′y
y
y

−1. Por eso, it goes to

≤ (||x2
i

Lip)

)||

|FSGD(wwwt+1) − FSGD(wwwt)| ≤ (1 − γmax/σ2)|wwwt+1

− wwwt

(7)

Combining Equations (6) y (7), it goes to

|F(wwwt+1) − F(wwwt)| ≤ (1 − γmax/σ2)|wwwt+1

− wwwt

De este modo, according to the deﬁnition of dynamic contraction mapping, the ADF update rule
is a dynamic contraction mapping in Euclidean space with Lipschitz continuity degree
⊓⊔
1 − γmax/σ2.

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Sun et al.

Feature-Frequency–Adaptive On-line Training for Natural Language Processing

Proof of Theorem 2 As presented in Equation (5), the ADF update rule is F(wwwt) :=

wwwt+1 = wwwt + γγγt

··· gggt. For ∀wwwt

∈ X ,

··· gggt+1

(8)

where ai is the ith element of the vector γγγt+1. bi and FSGD are the same as before.
Similar to the analysis of Theorem 1, the third step of Equation (8) is valid because γmax
is the maximum learning rate at the beginning and all learning rates are decreasing
when t is increasing. The proof can be easily derived following the same steps in the
⊓⊔
proof of Theorem 1. To avoid redundancy, we do not repeat the derivation.

Proof of Theorem 3 Let M be the accumulative change of the ADF weight vector wwwt:

Mt :=

∑

t′=1,2,:::,t

|wwwt′+1

− wwwt′|

To prove the convergence of the ADF, we need to prove the sequence Mt converges as
t → ∞. Following Theorem 2, we have the following formula for the ADF training:

|F(wwwt+1) − F(wwwt)| ≤ (1 − γmax/σ2)|wwwt+1

− wwwt

where γmax is the maximum learning rate at the beginning. Let d0 := |www2
q := 1 − γmax/σ2, then we have:

− www1

| y

Mt =

∑

t′=1,2,:::,t

|wwwt′+1

− wwwt′|

≤ d0 + d0q + d0q2 + · · · + d0qt−1
= d0(1 − qt)/(1 − q)

(9)

When t → ∞, d0(1 − qt)/(1 − q) goes to d0/(1 − q) because q < 1. Hence, we have: Mt ≤ d0/(1 − q) Thus, Mt is upper-bounded. Because we know that Mt is a monotonically increasing function when t → ∞, it follows that Mt converges when t → ∞. This completes the ⊓⊔ proof. 583 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / c o l i / l a r t i c e - p d f / / / / 4 0 3 5 6 3 1 8 0 3 7 9 7 / c o l i _ a _ 0 0 1 9 3 p d . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Computational Linguistics Volume 40, Number 3 Proof of Theorem 4 First, we have eigen(CCCt) = t∏ (1 − γγγ0βmλ) m=1 { ≤ exp − γγγ0λ } βm t∑ m=1 0 ≤ n∏ j=1 (1 − aj) ≤ n∏ j=1 − −aj = e e ∑ n j=1 aj Then, we have This is because 1 − aj t → ∞, we have ≤ e −aj given 0 ≤ aj < 1. Finally, because { eigen(CCCt) ≤ exp − γγγ0λ t∑ } βm { → exp m=1 } −γγγ0λβ 1 − β ∑ t m=1 βm → (cid:12) 1−(cid:12) when This completes the proof. 6. Conclusions ⊓⊔ l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / c o l i / l a r t i c e - p d f / / / / 4 0 3 5 6 3 1 8 0 3 7 9 7 / c o In this work we tried to simultaneously improve the training speed and model accuracy of natural language processing systems. We proposed the ADF on-line training method, based on the core idea that high frequency features should result in a learning rate that decays faster. We demonstrated that the ADF on-line training method is convergent and has good theoretical properties. Based on empirical experiments, we can state the following conclusions. First, the ADF method achieved the major target of this work: faster training speed and higher accuracy at the same time. Second, the ADF method was robust: It had good performance on several structured and non-structured classiﬁ- cation tasks with very different characteristics. Third, the ADF method worked well on both binary features and real-valued features. Fourth, the ADF method outperformed existing methods in a one-pass learning setting. Finally, our method achieved state- of-the-art performance on several well-known benchmark tasks. To the best of our knowledge, our simple method achieved a much better F-score than the existing best reports on the biomedical named entity recognition task. l i _ a _ 0 0 1 9 3 p d . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Acknowledgments This work was supported by the National Natural Science Foundation of China (no. 61300063, no. 61370117), the Doctoral Fund of Ministry of Education of China (no. 20130001120004), a Hong Kong Polytechnic University internal grant (4-ZZD5), a Hong Kong RGC Project (no. PolyU 5230/08E), the National High Technology Research and Development Program of China (863 Program, no. 2012AA011101), and the Major National Social Science Fund of China (no. 12&ZD227). 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