thanks to their highly strong nonlinear modeling capabilities. Among neural networks, especially recurrent neural networks (RNNs) are used to model time series and temporal data due to their inherent memory storing the past information. However, since simple RNNs lack control structures, the norm of gradient may grow or decay in a fast manner during training . Therefore, simple RNNs are insufficient to capture time dependencies . To circumvent this issue, a novel RNN architecture with control structures, i.e., the LSTM network, is introduced . In this study, we consider online nonlinear regression with LSTM-based networks due to their superior performance in capturing long-term dependencies.
For LSTM-based networks, there exists a wide range of online training methods to learn network parameters [6, 3, 7]. Among them, the first-order gradient-based methods  are widely preferred due to their efficiency. However, the first-order techniques, in general, provide poorer performance compared to the second-order techniques . As a second-order technique, the extended Kalman filter (EKF) learning algorithm has often been favored in terms of its accuracy and speed of convergence [7, 6]. However, the EKF learning algorithm has a quadratic computational requirement in the parameter size, which is usually prohibitive for practical applications due to the large number of parameters in LSTMs. To reduce the computational requirement of EKF, the independent EKF (IEKF) algorithm has been introduced in .111We note that IEKF is firstly introduced in  with the name of the Node Level Extended Kalman Filter Algorithm (NEKA). However, in the following studies [9, 7], NEKA is renamed as IEKF to distinguish the algorithm from the Node Decoupled Extended Kalman Filter algorithm . In this study, we prefer to use IEKF to avoid confusion. The main motivation of IEKF is the observation that during the EKF-based neural network training, the correlation between the weights belonging to different neural nodes is usually much lower than the correlation between the weights in the same neural node [6, 10]. Based on this observation, in IEKF, each neural node is assumed as an independent subsystem, and seperate EKF learning algorithms are used to learn the weights in different nodes. By this method, the computational requirement of the learning procedure is reduced by the number of neural nodes in the network while avoiding considerable performance reduction . We note that since practical LSTM models usually consist of neural nodes, the computational saving with IEKF leads to a considerable run-time reduction in the LSTM training.
Although IEKF provides comparable performance to the second-order methods in a considerably smaller run-time, it is more vulnerable to the divergence problems (compared to the EKF and stochastic gradient descent (SGD) algorithms) due to its treatment of each node as an independent subsystem . In this study, to provide both efficient and robust LSTM-based online learning procedure, we introduce an IEKF-based training algorithm with a theoretical convergence guarantee. To the best of our knowledge, our paper is the first study that provides a theoretical convergence guarantee for an IEKF-based algorithm in the neural network literature. We note that by using IEKF, we introduce a highly efficient counterpart of the state-of-the-art online learning algorithms [11, 12, 13], especially for LSTM-based online learning, where IEKF provides considerable improvements in training time without significant performance reduction .
2 Model and Problem Description
We222All vectors are column vectors and denoted by boldface lower case letters. Matrices are represented by boldface capital letters. I is the identity matrix, whose dimensions are understood from the context.
is the identity matrix, whose dimensions are understood from the context.and denote the Euclidean norm and trace operators. Given two vectors x and y, is their vertical concatenation. We use bracket notation to denote the set of the first positive integers, i.e., . define the online regression problem as follows: We sequentially receive ,
, and input vectors,,
such that our goal is to estimatebased on our current and past observations .333We assume for notational simplicity; however, our derivations hold for any bounded desired data sequence after shifting and scaling in magnitude. Given our estimate , which can only be a function of and , we suffer the loss
. The aim is to optimize the network with respect to the loss function. In this study, we particularly work with the squared error, i.e., . However, our work can be extended to a wide range of cost functions (including the cross-entropy) using the analysis in [15, Section 3].
In this paper, we study online regression with LSTM-based networks. As the LSTM cell, we use the most widely used LSTM model, where the activation functions are set to the hyperbolic tangent function and the peep-hole connections are eliminated. As the network model, we use a single hidden layer based on the LSTM structure, and an output layer with the hyperbolic tangent function. Hence, the network equations are:
Here, denotes the element-wise multiplication, is the state vector, is the input vector, and is the output vector, and is our final estimation. Furthermore, , and
are the input, forget and output gates respectively. The sigmoid functionand the hyperbolic tangent function applies point wise to the vector elements. The weight matrices are and . We note that although we do not explicitly write the bias terms, they can be included in (1)-(7) by augmenting the input vector with a constant dimension.
3 Independent Extended Kalman Filter
In this section, we derive the IEKF update rules for LSTM-based online learning. To convert the LSTM training into a state estimation problem, we model the desired signal as an autoregressive process that is realized by the LSTM network in (1)-(7), which we describe with the following dynamical system:444For notational convenience, we group all the LSTM parameters, i.e., , , , and , into a vector , where . We also use to denote the input sequence up to time , i.e., .
Here, we represent the optimal LSTM weights that realize the incoming data stream with a vector , which is modeled as a stationary process. As detailed in Fig. 1, we use to represent the unfolded version of our network model in (1)-(7) over all the time steps up to the current time step , where all forward passes are parametrized by . Here, the dependence of on is due to the increased length of the recursion at each time step.
In the IEKF framework, we apply different (and independent) EKF learning algorithms to the state-space model in (8)-(9) to learn the weights in different nodes. To describe the learning rule, let us denote the LSTM nodes with the first different integers, i.e., , and use to index the nodes, i.e., . Let us also use to denote our estimation for the optimal weights in node at time step . Then, we perform the weight updates in IEKF with the following:
Here, is our prediction for , which is calculated by performing the forward LSTM propagation in (1)-(7) with the estimated weights, i.e., for all . is the state covariance matrix, is the Jacobian matrix, and is the Kalman gain matrix corresponding to the LSTM weights in node . The noise covariance matrices and are artifically introduced to the algorithm to enhance the training performance . In order to efficiently implement the algorithm, we use diagonal matrices for the artificial noise terms, i.e., and , where for all .
We recall that the EKF algorithm has quadratic complexity in the number of the network weights , i.e., . On the other hand, the computational complexity of the IEKF learning algorithm is due to the update rules in (11) and (13). Since in practical LSTM models, the dimension of the state vectors, i.e., , is usually selected between and , the reduction in the computational requirement with IEKF leads to considerable run-time savings in LSTM-based online learning with respect to the EKF algorithm. However, as noted earlier, IEKF is vulnerable to the divergence problems since it assumes each node as an independent system . Therefore, to provide an efficient and robust learning algorithm, we use the IEKF framework and introduce an IEKF-based LSTM training algorithm with a convergence guarantee in the following section.
4 Algorithm Development
In this section, we introduce an IEKF-based training algorithm with a theoretical convergence guarantee. For the analysis in the following, we write the error dynamics of the independent EKF structures. To this end, we first write the Taylor series expansion of around :
where is the Jacobian matrix of evaluated at , and is the non-linear term in the expansion. Note that and . For notational simplicity, we introduce two shorthand notations: , and . Then, the error dynamics of the EKF learning algorithm applied to node can be written as:
where we consider the effect of partitioning the weights as additional non-linearity for node .
Since our network model only consists of smooth functions, is also smooth, which means the norm of is bounded by a scalar value for all the nodes throughout the training, i.e., . In the following subsection, we use to guarantee convergence by our algorithm.
4.1 Main Algorithm
In this subsection, we present the main result of this paper, i.e., Algorithm 1. In Algorithm 1, we take as the input. We initialize the state covariance matrix of each independent EKF as , where . In each time step, we first generate a prediction , then receive the desired data , and suffer the loss . We perform the parameter updates only if the loss is bigger than , i.e., . If so, we calculate the Jacobian matrix , measurement noise level , and the Kalman gain matrix for each in lines 7-9 of Algorithm 1. We update the weights and state covariance matrix of the weights belonging to node in lines 10 and 11.
In the following theorem, we state the theoretical guarantees of Algorithm 1.
If stays bounded during training, Algorithm 1 guarantees the following statements:
The LSTM weights stay bounded during training.
The loss sequence converges to the interval in the deterministic sense.
See the Appendix. ∎
Due to the Kalman gain matrix formulation (line 9 in Algorithm 1), is always smaller than or equal to for each node , i.e., for all . Since , and the artificial process noise level is a user-dependent parameter, the condition in Theorem 1 can be satisfied by the user by selecting sufficiently small .
In this section, we illustrate the performance of our algorithm on two real-life data sets: elevators , and the Alcoa price stock dataset . To demonstrate the performance improvements of our algorithm, we compare it with two widely used LSTM training methods, i.e., the EKF learning algorithm , and the stochastic gradient descent algorithm . In the following, we use Alg1 to denote Algorithm 1, EKF for the EKF learning algorithm, and SGD for the stochastic gradient descent algorithm. We run each experiment times and provide the mean performances.
We first consider the elevators data set, which is obtained from the procedure that is related to controlling an F16 aircraft . Here, our aim is to predict the scalar variable that expresses the actions of the aircraft, i.e., . For this data set, we use -dimensional input vectors of the dataset with an additional bias dimension, i.e., , where . To get small loss values with relatively lower run-time, we use -dimensional state vectors in LSTM, i.e., . For Alg1, we choose as , the artificial process noise level as for all , and the initial state covariance matrices as for all . In EKF, we use the measurement noise level for all , linearly decreasing process noise level sequence annealed from to , and set the initial state covariance matrix as . For SGD, we choose the learning rate as . In Table 1, we provide the mean squared error and run-time of the compared algorithms for this setup. Here, we observe that the mean squared errors of Alg1 and EKF are very close to each other and significantly smaller than the mean squared error of SGD. However, since we use the IEKF framework to develop our algorithm, Alg1 achieves the resulting error performance with times smaller run-time compared to the EKF learning algorithm.
In our second simulation, we use the Alcoa stock price dataset , which contains the daily stock price values of Alcoa Inc. between the years -. Our goal is to predict the opening, closing, highest, lowest and adjacent lowest values of the next day’s stock price by using the observed prices, i.e., . As the input vector, we use the opening, closing, highest, lowest and adjacent lowest stock price values of the current day with an additional bias dimension, where , and . We set the dimension of the state vectors , i.e., . For Alg1, we choose as , and set the initial state covariance matrices as for all . In EKF, we chose the measurement noise level as for all , and set the initial state covariance matrix as . In both Alg1 and EKF, we choose the process noise level as linearly decreasing from to . For SGD, we choose the learning rate as . In Table 1, we provide the mean squared error and run-time of the compared algorithms for this experiment. Similar to the previous experiments, here, we observe that the resulting errors of Alg1 and EKF are very close to each other and lower than the error of SGD. We also observe that Alg1 provides this performance in times shorter training time than EKF, and in a comparable run-time with SGD. We add that Alg1 did not have any divergence problem in both experiments (in a total of simulations), which is parallel with our theoretical results in the previous section.
6 Concluding Remarks
We studied online nonlinear regression with long short term memory (LSTM) based networks. For this problem, we introduced a highly efficient and robust IEKF-based training algorithm with a theoretical convergence guarantee. In the simulations, we demonstrate that our algorithm achieves significant performance improvements with respect to the conventional LSTM training methods [3, 6]. We particularly show that our algorithm provides superior error performance compared to SGD and very similar error performance (with times smaller run-time) compared to EKF.
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Before the proof, we present several propositions that will be used to prove the theoretical guarantees of Algorithm 1.
Algorithm 1 guarantees the following statements:
For each node , the difference between the locally optimal weights and LSTM weights is governed with the following dynamical equation:
which can also be written as
For each node , and exist and they are always positive definite as such
As a result of the previous two statements,
holds for each node .
Proof of Lemma 1.
By (11), for all ,
(26) (27) (28) (29)
By noting that , and using (29) as the induction hypothesis, we can show that exists and , for all . Since , has the same properties, which leads to (21). Also, (20) can be reached by taking the inverse of both sides in (26).
Now, we can prove Theorem 1.
Proof of Thorem 1.
In the following, we use the second method of Lyapunov to prove the statements in the theorem. Let us fix an arbitrary node , and choose the Lyapunov function as
Let us say that . Since we update , and only when , for , . Therefore in the following, we only consider the time steps, where we perform the weight update, i.e, .
To begin with, we write the open formula of :
For the sake of notational simplicity, we introduce , where . Then, we write (36) as
We write the last term in (37) as
|By using the 3rd statement in Lemma 1|
We add to (42), and group the terms as