RETRACTED: Continual Learning Approach for Continuous Data Stream Analysis in Dynamic Environments

Abstract

Continuous data stream analysis primarily focuses on the unanticipated changes in the transmission of data distribution over time. Conceptual change is defined as the signal distribution changes over the transmission of continuous data streams. A drift detection scenario is set forth to develop methods and strategies for detecting, interpreting, and adapting to conceptual changes over data streams. Machine learning approaches can produce poor learning outcomes in the conceptual change environment if the sudden change is not addressed. Furthermore, due to developments in concept drift, learning methodologies have been significantly systematic in recent years. The research introduces a novel approach using the fully connected committee machine (FCM) and different activation functions to address conceptual changes in continuous data streams. It explores scenarios of continual learning and investigates the effects of over-learning and weight decay on concept drift. The findings demonstrate the effectiveness of the FCM framework and provide insights into improving machine learning approaches for continuous data stream analysis. We used a layered neural network framework to experiment with different scenarios of continual learning on continuous data streams in the presence of change in the data distribution using a fully connected committee machine (FCM). In this research, we conduct experiments in various scenarios using a layered neural network framework, specifically the fully connected committee machine (FCM), to address conceptual changes in continuous data streams for continual learning under a conceptual change in the data distribution. Sigmoidal and ReLU (Rectified Linear Unit) activation functions are considered for learning regression in layered neural networks. When the layered framework is trained from the input data stream, the regression scheme changes consciously in all scenarios. A fully connected committee machine (FCM) is trained to perform the tasks described in continual learning with M hidden units on dynamically generated inputs. In this method, we run Monte Carlo simulations with the same number of units on both sides, K and M, to define the advancement of intersections between several hidden units and the calculation of generalization error. This is applied to over-learnability as a method of over-forgetting, integrating weight decay, and examining its effects when a concept drift is presented.

1. Introduction

Contemporary hypothetical studies show that the generalization and representation power of neural networks will increase as the depth of neural networks increases [1,2]. However, data stream analysis in dynamic environments remains unexplored in neural network environments. The direct use of traditional neural network models for streaming data analysis is often impossible because of their computing power and memory constraints, so they cannot be implemented with limited computing resources [3]. Dynamic data streams should preferably be sampled without retraining to avoid the catastrophic problem of forgetting and adapting to the learning environment’s nature [3,4]. These characteristics are not reflected effectively in dynamic environments with changing dynamic data streaming properties. This challenge is new for conventional neural networks with fixed and static structures [5]. To address these issues, the neural network’s capacity must likely be estimated before the adaptation of continuous learning using conceptual drift.

For classification and regression, feed-forward neural networks are frequently used methods. External networks can already represent N-dimensional variables’ non-trivial scalar functions with just one hidden layer [6]. Nevertheless, the networks could move slowly due to convergence and the emergence of so-called plateau states. Students may become locked in an unspecialized local optimum before fast specializing in the concealed instructor unit regarding ongoing learning in a scenario.

We examine two possibilities for activation functions in hidden units in this work. ReLU activation has recently acquired prominence, primarily because of increased empirical performance, replacing the conventional soft committee machines based on sigmoid activation functions—theoretical benefits, which are also in [7].

Across most cases, a machine learning approach is carried out while training and later during the testing phase [8] while training data instances are presented, analyzed, and extracted information, and the classifier hypothesis or regression scheme is parameterized. This hypothesis can be tested with new data at the next stage of the testing phase. This process implicitly assumes that after the training, the properties of the data and target class remain the same and do not change their values. It is not always the case in machine learning tasks, and it is also not a tenable explanation for how learning functions in biological systems, including humans and other creatures. In such circumstances, the learning system can identify and monitor the perception of drift, i.e., forgetting inappropriate and old information while constantly adjusting to the latest additional contributions. Concept drift has been the subject of previous studies [9,10] looking at its theoretical characteristics and statistical mechanics. This study aims to learn a regression technique that can be used in various situations through practical simulations of scenarios.

Using sigmoidal activation functions in the fully connected committee machine (FCM) for continual learning has several limitations. Firstly, sigmoidal functions suffer from the vanishing gradient problem, leading to slow learning or convergence issues. Additionally, their limited range of output values may restrict the FCM’s ability to capture complex and non-linear relationships in the data. Saturation of sigmoidal functions and their sensitivity to initialization can also affect the FCM’s performance, leading to biases and suboptimal solutions. Finally, gradient descent optimization techniques may be required to overcome these limitations and improve the FCM’s learning process. It is essential to consider alternative activation functions to mitigate these limitations and enhance the performance of the FCM in dynamic environments.

This research focuses on the effect when a concept drifts on continuous learning about a regression technique that uses two different activation functions. The distinctions are between instances that match and situations that may be learned from, as well as the potential benefits and drawbacks of each. Section 3 introduces a new notion of drift categorization that divides drifts into generalization errors based on several criteria. The sigmoidal activation function is used in classic soft committee machines. The suggested FCM approach employs the ReLU activation function to evaluate the various drift strengths and the generalization error.

The rest of the paper is divided into the following sections: In Section 2, the basic concepts and related work are associated with continual learning in the presence of concept drift. Section 3 contributes to the proposed FCM research methodology for continual learning over continuous data stream drift. Section 4 discusses evaluation systems concerning results. Section 5 elaborates on widespread empirical investigations of crucial discoveries and presents further research aspects associated with continual learning.

2. Basic Concepts and Related Work

This section defines the essential preliminaries and related research work associated with concept drift in continual learning.

2.1. Concept Drift

Concept drift is when the statistical properties of a data change randomly occur with time [11]. It was proposed with the intention that the noise data could be converted into noise-free data at different times [12,13,14,15]. These deviations may be due to hidden variables’ hidden properties that cannot be measured directly [16]. Formally, the drift of the concept in learning is defined as follows:

In a time interval ranging from [0…t], a set of data examples is denoted as D_0,t = {D₀… D_t}, where D_i = (X_i, Y_i) is a data instance, representing where X_i is the feature vector, Y_i is the class label, and D_0,t follows a certain probability distribution P_0,t(X, y). Concept drift occurs at timestamp t + 1 if the condition P_0,t(X, y) ≠ P_t+1,1(X, y). The drift or change in concept is just a subcategory of the change in the data set consisting of the change in covariates and the previous change in probability and change in concept. These definitions have set out the scope of research for each research topic.

2.2. Continual Learning

Continual learning of data streams is defined as a learning approach of continuously generated data batches B $ϵ$ {B₁, …, B_k, …B_K} where the number of data batches K and the type of data distributions are unknown before the process runs. B_k can be either a single data point or a particular size of data batch. Data points enter the scene with the lack of a valid class marker for data flow problems [13]. Access to the reality or work experience of the subject depends on the implementation of the labeling process. In other words, if the true class designations C $ϵ$ R^T are revealed, a delay is predicted. To obtain C $ϵ$ R^{T X m}, the 0–1 coding scheme can be implemented. This problem limits the feasibility of training as a cross-validation evaluation protocol since these approaches assume that the entire sample batches are completely measurable, and there is a possibility that the time order of the data will be lost [3,17].

The neural network to handle B_k has to process data streams for the data streams that can originate from several data distributions, which are often referred to as drift terms. In particular, the setback probability for the standard class P(C_t, X_t) ≠ P (C_t−1, X_t−1) can change. The concept of drift is usually seen in the real and the covariate [7]. This also means that the model developed using the previous definition of Bk-1 is out of date. This feature is relevant to the multitasking learning issue, where separate tasks come from each batch of Bk results. However, neural networks differ from multitasking methods in that a single paradigm is to process all data batches instead of focusing on specific task classifiers. The compromise will increase the likelihood of over-forgetting; data flows have another problem [18]. Such specifications require an online neural network architecture adept at constantly rebuilding its network edifice from scratch concerning the provision of data streams. To avoid catastrophic oblivion, a mechanism must also be implemented to repeat the previous experience and to keep it flexible or acquire a new one.

We provide an example of a regression scheme in the parts that follow. Ref. [19] with neural networks with low feedback. There are two different kinds of remote activation units, sigmoid and ReLU, which are explained and compared. In this paper, the gradient-based training of a fully connected committee machine is regarded as an explicit forgetting process when a genuine drift is present.

Catastrophic forgetting, also known as catastrophic interference, is a phenomenon where neural networks forget previously learned knowledge when trained on new, unrelated data. To address this issue, various research papers propose different methods and techniques. elastic weight consolidation (EWC) selectively protects important weights during training to prevent forgetting, while progressive neural networks (PNN) retain knowledge from previous tasks while learning new ones. Other approaches include replay mechanisms for recurrent neural networks and comprehensive surveys exploring different techniques. In a research work, Ref. [20] describes a learning paradigm that tackles evolving and incrementally arriving data in dynamic domains. It combines adaptive techniques and online learning strategies to handle concept drift, domain shifts, and incremental learning scenarios. The research work [21] introduces homeostatic neural networks that adapt to concept shifts by dynamically adjusting internal representations. In another study, Ref. [22] investigates catastrophic forgetting in continual concept bottleneck models and proposes techniques such as regularization and rehearsal to preserve and transfer knowledge. The study [23] proposes the consolidated method, which consolidates knowledge from multiple source domains to address concept shifts in domain-aware settings. Lastly, the study [24] proposes an online continual learning framework that uses adaptive mechanisms and incremental learning strategies to handle progressive distribution shift.

2.3. Related Work

This study aims to provide a novel methodology to configure fluctuations during a particular, real-world training cycle. Numerous research studies [25] and references therein equate diverse learning methods for streaming data experimentally; our contribution aims to do so more rigorously. Statistical physics methods are utilized to study the usual behavior of several continual learning for blind image quality assessments [26]. The efficient examination of continual learning is created on the premise that the learning system is given a sequence of N-dimensional instances generated independently [27].

The research conducted by [28] proposes a method for detecting anomalies in concept-drifting data streams using the isolation technique. They leverage the isolation forest algorithm to identify anomalies in continuous data streams that undergo concept drift, where the underlying data distribution changes over time. This research addresses the challenges of anomaly detection in dynamic environments by providing an effective solution tailored to concept drift scenarios. The study by [29] focuses on developing concept drift adaptation techniques for distributed environments dealing with real-world data streams. Their research aims to detect concept changes and design strategies for efficient model adaptation and updates in response to concept drift. The research work in [30] investigates the effectiveness of the adaptive random forest (ARF) algorithm in handling concept drift in big data streams, evaluating its accuracy, scalability, and computational efficiency. Ref. [31] addresses the challenges of terrain image classification in ASTER remote sensing by proposing a novel approach that combines data stream mining and an evolutionary-EAC instance-learning-based algorithm, aiming to improve speed and accuracy in classifying terrain images. These research studies contribute valuable insights and techniques to tackle concept drift in various domains.

Further simplifications, as well as a deliberation of the thermodynamic range of N, allow the ordinary differential equations to deliver the precise statistical explanation of learning curves for in-depth analyses of the method’s shortcomings and the leeway that permits them to be overcome [32].

The method of supervised learning in complex systems has been explored extensively by researchers. The dynamics of supervised learning, including principal component analysis, prototype-based competitive learning, and allied schemes, have been investigated. A different collection of articles, monographs, and reviews are further explored in supervised learning in neural networks [26].

The SCM (Soft Committee Machines) are well explored using statistical approaches under stationary environments. Important occurrences have been explored, such as the development of quasi-stationary drift states. Supervised SCM (Soft Committee Machines) training has recently been examined in the context of simplifying model scenarios.

In online learning, statistical approaches are well used to explore the effects of drift. For the simple perceptron, adversarial drifts were studied in [33]. It should be noted that the hypothesis of statistically liberated instances of the data stream does not preclude the investigation of expressive drift environments.

This is the first prototype-based statistical investigation of how continual concept drift impairs learning, with a categorization and neural network based on ReLU for estimating the generalization error.

3. Continual Learning Using Fully Connected Committee Machine (FCM)

Continual learning using the fully connected committee machine (FCM) is an approach that addresses the challenges of learning in dynamic and evolving environments. The FCM consists of multiple neural network models, called committees, which collaborate to make predictions. Each committee member is a fully connected neural network with its own set of weights. During the learning process, new data are sequentially presented to the FCM, and each committee member updates its weights based on the incoming data. This ensemble of neural networks allows for continuous learning and adaptation to changing data distributions. By leveraging the collective knowledge of multiple committees, FCM enables robust and accurate predictions in the face of concept drift and evolving patterns in the data. The FCM framework has shown promise in various domains, including classification and regression tasks, and provides a practical approach for continual learning in dynamic environments. Figure 1: Continual learning using a fully connected committee machine (FCM)

The proposed method of the fully connected committee machine within a layered neural network framework focuses on handling conceptual changes in continuous data streams during continual learning. While it is not explicitly compared to specific methods such as learning without forgetting (LwF), elastic weight consolidation (EWC), and incremental classifier and representation learning (iCaRL) in the provided document, it appears to have distinct characteristics. The method emphasizes the use of a committee machine and layered neural networks for continual learning but lacks explicit mentions of techniques such as distillation, regularization, or exemplar-based learning used in other methods. A further comparison would require a deeper analysis of the proposed method’s specific mechanisms and evaluations in relation to LwF, EWC, and iCaRL

We have created a framework for this learning that allows us to conduct Monte Carlo simulations. The weights with predetermined overlap are used to produce random input vector R. We utilize R_im = 0, and Q_ii = 0.5, Q_ik = 0.59, for i,f = k.

Where i,k = 1…K and m = 1…M. As a result, the plateau states are longer since the pupils lack prior knowledge of the rule and have significant interdependence. The classical Gram–Schmidt process is a mathematical algorithm used for orthogonalizing a set of vectors. It is named after the mathematicians Jørgen Pedersen Gram and Erhard Schmidt. The process takes a set of linearly independent vectors and transforms them into a set of orthogonal vectors. The algorithm works by iteratively subtracting the projection of each vector onto the previously orthogonalized vectors, ensuring that the resulting vectors are mutually orthogonal. The classical Gram–Schmidt process is widely used in linear algebra and has applications in various fields, such as signal processing, numerical analysis, and computer graphics.

The following sections present continuous learning based on the regression regime [19] and shallow forward neural networks. We are explaining and contrasting two concealed activation unit varieties, sigmoidal function for SCM and ReLU function for FCM. To study the effect of continuous learning, we have considered gradient-based training of a fully connected committee machine (FCM) in the inclusion of weight decay as a process of explicit oblivion, which would indicate the existence of a genuine concept of drift. Figure 2a,b represent the nature of sigmoidal and ReLU activation functions.

3.1. Fully Connected Committee Machines (FCM)

A fully connected committee machine (FCM) is a forward neural network architecture that contains one hidden layer and one linear output unit with a non-linear activation function. It is possible to characterize the output of an FCM as shown in Equation (1).

$$y\left(\xi \right)={\sum }_{k=1}^{K}g\left(W_k.\xi \right)$$

(1)

Here, W_k stands for the weight vector. That connects this input data with the Kth hidden unit. A non-linear activation g(x) outlines the sum as the states of the hidden units and the final output. For a soft committee machine (SCM), the sigmoidal case it corresponds to is

$$g\left(x\right)=erf\left(x/\sqrt{2}\right),\hspace{1em}{g}^{\prime }\left(x\right)=\sqrt{\frac{2}{\pi }}.e^{-x^2/2}$$

(2)

For the FCM, ReLU case, this corresponds to

$$\begin{gathered} g\left(x\right)=x\theta \left(x\right),\hspace{1em}{g}^{\prime }\left(x\right)=\theta \left(x\right) \\ \theta \left(x\right)=\left\{\begin{array}{c}1 if n\ge 0 \\ 0 otherwise\end{array}\right. \end{gathered}$$

(3)

where θ(x) is defined as a step function.

Those activation processes have benefits and drawbacks [34], and their properties can change when drift is considered. Equations (2) and (3) are the activation function charts shown in Figure 2. In contrast to the ReLU derivative, which is 0 for x < 0 and 1 for x > 0 [34], we note that the function erf() derivative is Gaussian in nature. ReLU j(0) discontinuity is one property that needs to be considered; in this case, one can pick between 0 and 1.

3.2. Continual Learning

In general, the squared divergence of the network output from the output is what drives the process of developing an output-driven forward neural network value y(ξ) of the rule, according to the examples μRN, μc, or μR [7]. This variance is a cost function that assesses the efficiency of the network:

$$\left\{x^{\mu } ϵ{ R}^N, T^{\mu }ϵ R\right\}$$

(4)

In gradient descent, at each time step µ, the weight vector is updated based on a single instance given as

$$e^{\mu }\left(\left\{W_k\right\}\genfrac{}{}{0pt}{}{K}{k=1}\right)=\frac{1}{2}{\left(y^{\mu }-T^{\mu }\right)}^2 with y^{\mu }=y\left({\xi }^{\mu }\right)$$

(5)

$$∆W\genfrac{}{}{0pt}{}{\mu }{K}=-\left({\sum }_{i=1}^{K}g\left({h_i}^{\mu }\right)-T^{\mu }\right)g^{\prime }\left(h\genfrac{}{}{0pt}{}{\mu }{K}\right){\xi }^{\mu }$$

(6)

The next instance in the input data stream is estimated as the function of inner products h^µ = w^µ−1 · ξ^µ of the weight vectors. This transformation in the weight vector (Equations (4)–(6)) is directly proportional to ξ^µ and inferred Hebbian learning [8].

3.3. The Monte Carlo Learning Scenario

We used learning scenarios to classify and model meaningful learning scenarios as student–teacher scenarios. Equation (7) is well-defined as FCM with M hidden units with associated weight vectors; we assume $\left\{B_m ϵ{ R}^N\right\}\genfrac{}{}{0pt}{}{M}{m=1}$

$$T\left(\xi \right)={\sum }_{m=1}^{M}g\left(B_m\xi \right) and T^{\mu }=T\left({\xi }^{\mu }\right)$$

(7)

With b^µ = B_m × ξ. The Monte Carlo learning scenario contains three different learning scenarios, K > M, M > K, and K = M concerning hidden units. M > K shows hidden units as unlearnable. K > M, on the other hand, would indicate an overfitting or learnable behavior. This issue is fascinating to investigate since, in practice, one is frequently unaware of the task’s intricacy. K = M, in which the two architectures are compatible and have the capacity to accurately and completely represent the rule without any redundancies, is the final and most researched case. The cases in this paper with K = M are the main topic.

3.4. The Dynamics of Continual Learning

In this paper, we present the theory and its effects of continual learning as it has been pragmatic in changing aspects of ReLU-based FCM [35] against sigmoidal activation of SCM. It is further extended to study the non-stationary situations concerning drift. Adaptable vectors W1,2 and characteristic vectors B1,2 represent student weights and task cluster centers in FCM training and teacher weights in SCM teaching.

In the section on training dynamics, we consider an explicit statistical explanation of the ordinary differential equations to train the dynamics environment. The thermodynamics limit N→∞ is a theoretical consideration and helpful in learning the first three steps of a training simulation and the following critical steps.

3.5. Order Parameters

It is possible to describe quantities in the different components of adaptive vectors. The description of these order parameters follows the statistical model of adaptive quantities. The order parameters assess the degree of similarity between student weight vectors and those of teachers and among students. For instance, the order parameters’ number is shown as

R_im = w_i × B_m, Q_ik = w_i × w_k    m = 1 … M and i,k = 1 … K.

3.5.1. Generalization Error

The learning success is quantified after the training based on the so-called generalization error, which is also given depending on the order parameters. It is possible to define the generalization error as the quadratic equation’s mean value, or the divergence over the isotropic density between the outputs [9]. For the soft committee machine based on a sigmoidal activation function, this equates to Equation (8).

$${\in }_g=\frac{1}{\pi }\left[{\sum }_{i,j=1}^K{\mathit{sin}}^{-1}\left(\frac{Q_{ij}}{\sqrt{1+Q_{ii}}\sqrt{1+Q_{jj}}}\right)-2{\sum }_{i=1}^K{\sum }_{m=1}^M{\mathit{sin}}^{-1}\left(\frac{R_{im}}{\sqrt{1+Q_{ii}}\sqrt{2}}\right)\right]+\frac{M}{6}$$

(8)

For the ReLU case, it is given in Equation (9)

$${\in }_g=\frac{1}{2}\left[{\sum }_{i,j=1}^K\left(\frac{Q_{ij}}{4}+\frac{\sqrt{Q_{ii}{Q}_{jj}-{Q_{ij}}^2}}{2\pi }+\frac{Q_{ij}{\mathit{sin}}^{-1}\left(\frac{Q_{ij}}{\sqrt{Q_{ii}{Q}_{jj}}}\right)}{2\pi }\right)-2{\sum }_{i=1}^K{\sum }_{m=1}^M\left(\frac{R_{im}}{4}+\frac{\sqrt{Q_{ii}-{R_{im}}^2}}{2\pi }+\frac{R_{im}{\mathit{sin}}^{-1}\left(\frac{R_{im}}{\sqrt{Q_{ii}}}\right)}{2\pi }\right)+\frac{M}{2}+\frac{\left(M-1\right)M}{2\pi }\right]$$

(9)

Both equations were applicable to orthonormal instructor vectors, with B_m × B = 1 and B_m × B_n = 0 for m × f = n. There are extensions for teacher vectors in general [10].

3.5.2. Dynamic Continual Learning under Continuous Data Stream Drift

This section will focus on the impact of drift on the learning process, and weight decay is considered an explicit mechanism of overlooking. The previous section looked at the analysis of continual learning in stationary environments for a density of the type Equations (5) and (6) with displacement vectors B1,2 that do not vary during training.

3.5.3. Virtual Drift

Several virtual drift methods can be investigated using suitable changes to the fundamental framework. From the observed data examples, features are impacted by virtual drift, although the actual framework remains unchanged.

Time-varying noise data might be easily added to models [36]. Non-stationary disparities in the input density can be treated as time-dependent. Prior probabilities in classification are a fascinating case. The changing fraction of instances is reflected in the data for each class in a data stream. In general, different class bias in the presence of changing fractions of data can complicate the training process and results in poor performance.

3.5.4. Real Drift

In something like a stationary network, where its characteristic vector B_m need not change throughout training, the models continuously learn new information, as previously discussed. We want to expand this for non-stationary situations, which are classified into two groups: virtual drift and actual drift. While the vestiges of the definite target function are unaffected by virtual drift, the numerical characteristics of the experimental instances of the data are. We use the ReLU activation function in this experimental study to address the impact of real drift since the characteristic vector B_m shifts over time in this situation. The fact that virtual drift frequently conveys true drift progressions in practice is significant [37]. We restrict ourselves to analyzing real concept drift under random displacements of vectors B_M(µ) every time. Following are the two criteria used to generate the random vectors B_m shown in Equations (10) and (11).

$$B_M\left(\mu \right)\times B_M\left(\mu -1\right)=\left(1-f/N\right)$$

(10)

$$B_M\left(\mu \right)\times B_M\left(\mu -1\right)=0 for M\ne N$$

(11)

With m, n = 1…M. Here δ enumerates the drift process strength.

3.5.5. Weight Decay

We embrace weight decay to implement exclusive forgetting and to possibly increase the system performance in the occurrence of drift (Equation (12)). To evaluate the system, before the generic learning step, we have considered the product of (1, γ/N) defined as an adaptive vector:

$$W\genfrac{}{}{0pt}{}{\mu }{i}=\left(1-r/N\right)W\genfrac{}{}{0pt}{}{\mu -1}{i}+n/N∆W\genfrac{}{}{0pt}{}{\mu }{i}$$

(12)

In the training, the product with (1, γ/N) enforces an improved effect of the previous training data as associated with previous instances.

Continual learning is beneficial for both classification and regression tasks. In classification, it allows the model to learn new classes or concepts while retaining knowledge of previously learned ones, enabling it to accurately classify diverse data patterns. For regression tasks, continual learning enables the model to adapt to changing data distributions and evolving patterns, ensuring it maintains accurate predictions over time. By continuously updating its knowledge and adapting to new information, continual learning enhances the model’s adaptability and flexibility, making it suitable for dynamic environments. In the context of classification or regression assessment in a dynamic environment, continual learning methods and techniques are employed to assess the quality of images without access to reference or ground truth information. These approaches address the challenge of adapting the model’s knowledge as new images with varying quality levels are encountered. In blind image quality assessment, continual learning approaches handle various factors impacting image quality, such as compression artifacts, noise, blur, and other distortions. These approaches involve training models on large and diverse datasets, leveraging deep learning architectures, and incorporating techniques such as transfer learning, ensemble learning, or adaptive learning rate mechanisms to adapt to changing image quality conditions [26].

4. Results and Discussion

This paper examines FCM’s ability to deal with notion drift in classification. We look at how drift affects the continual learning of multilayer feed-forward neural networks in generalized regression applications. In order to validate the concept drift in a continual learning environment, various approaches and techniques with set equations can be employed, as described in Section 3, such as fully connected machines, continual learning, Monte Carlo learning, and analysis of generalization error. Fully connected machines, or neural networks, can be trained on data streams with concept drift to evaluate their adaptation capabilities. Continual learning enables models to adapt to changing data distributions and detect concept shifts. Monte Carlo simulations can be used to assess the model’s performance under different concept drift scenarios. Analyzing the generalization error provides insights into the model’s ability to generalize to new concepts. Equations and mathematical formulations can also be utilized to quantify the impact of concept drift. By leveraging these approaches, researchers can gain a deeper understanding of how models handle concept drift in a continual learning environment.

4.1. Concept Drift Categorization in SCM

In the SCM, the student–teacher situation with K = M = 2, learning dynamics exhibit non-trivial features [38]. The appearance of quasi-stationary drift states that will control the learning curves is perhaps the most intriguing result.

In the SCM model, the orthonormal teacher vectors and drift with R_ik ≈ R and Q_ik ≈ Q for all ik = 1, 2 continuously occur and generate low learning capabilities [39]. Further sophisticated fixed point alignments with varying drift strength specialization can be established. The number of observed drifts grows with increasing K and M.

Without prior knowledge about the destination, one should assume R_im(0) ≈ 0 for all i,m. As a result, the student specialization drift change is S_i(0) = mod|R_i1(0) − R_i2(0)|, which can be predicted to be low at first. Figure 2a depicts a prominent state in continual learning for beginning circumstances. The fixed point of the dynamics and the explicit initialization and repulsive qualities determine the actual length and shape of the drift. Figure 2b shows an unspecialized configuration with S_i(0), which can span tremendous values.

R_im = R_o + U × 10⁻⁶ where R_o = 0.1, Q₁₁ = Q₂₂ = 0.6, Q₁₂ = 0.59.

U × X signifies an indiscriminate factor with a uniform probability of the (0, X), so S_i(0) = O × X. With little prior knowledge, the initialization correlates to roughly similar student ve × ctors.

The selection of N = 1000 in Monte Carlo simulations is arbitrary, and the adequacy of this value depends on the context and objectives of the study. While N = 1000 can provide reasonably reliable estimates and insights, its sufficiency varies based on the complexity of the system, desired level of statistical precision, and available computational resources. In simpler systems or preliminary analyses, N = 1000 may be adequate to capture trends and patterns. However, in more complex or highly variable systems, a larger N might be necessary to achieve excellent statistical reliability, reduce the impact of outliers, and capture rare events accurately. Researchers should carefully consider the specific requirements and constraints of their study when determining the appropriate value for N in Monte Carlo simulations. Sensitivity analyses and statistical power calculations can help evaluate the adequacy of N for a given study.

Figure 3a depicts shows the Monte Carlo simulations for N = 1000 and a randomly generated student vector, ensuing in R_im(0) = 0(1/N), with Q_ik(0). Simulations qualitatively confirm the hypothetical predictions for generalization error for δ = 0.1.

Figure 3b shows the occurrence of real concept drift, and SCM training using data density. λ = 1, p₁ = p₂ = 0.6, and v₁ = v₂ = 0.05. Figure 3a: Learning curves ε_g(α) for δ = 0.1 and various learning rates η. Figure 3b: Asymptotic (α→∞) generalization error for various drift strengths and in a dynamic environment with δ = 0.0 and δ = 0.2.

4.2. Concept Drift Categorization in FCM

In this section, we investigate the characteristic behavior of FCM in the presence of drift. Throughout this, we will analyze prototypes with no prior knowledge and initialize them as independent, normalized vectors on the data stream, which resembles

Q₁₁(0) = Q₂₂(0) = 1, Q₁₂(0) = 0 and R_im (0) = 0

Learning curves for drift with δ = 0.1 at various learning rates are shown in Figure 3 below. The beginning of training is resolute by interacting with the initial values and the learning rate. Figure 4a exhibits non-monotonic behavior for some conditions, and Figure 4b exhibits stable behavior for all conditions.

Monte Carlo simulations demonstrate great arrangement through the (N→∞) hypothetical estimates for minimal systems. This is consistent with the results in [26] for stationary situations. For example, Figure 3a illustrates the mean and standard deviation over a randomized training N = 1000.

The outcomes for a big reveal that the realization of learning, the drift strength to which FCM can follow the drifting notion, is non-trivially dependent on the learning rate. Excessive learning rates will produce lower performance and vice versa.

Figure 4b shows the occurrence of real concept drift and SCM training using data density. λ = 0.1, p₁ = p₂ = 0.05, and v₁ = v₂ = 0.05. Figure 4a: Learning curves ε_g(α) for δ = 0.1 and various learning rates η. Figure 4b: Asymptotic (α→∞) generalization error for various drift factors and in a dynamic environment with δ = 0.0, 0.2, 0.5, 1.0.

4.3. Generalization Error

The outcomes of the continuous learning with drift Monte Carlo simulations are shown below. First, K = M, the exact situation with an equal number of matching vectors, and its connection to idea drift are distinguished as the significant possibilities. The following experiments all use a system size of N = 1000. The empirical study of concept drifts is based on continual learning in which it is compared against four algorithms.

The matching situation, in which K = M = 2 and the two architectures match, is the first to be examined. For both sigmoidal and FCM-based ReLU activation, the generalization error sg with time is shown in

Table 1. Generalization error for SCM based on sigmoidal activation on different drift strengths.

Drift Strength	α = 0	α = 200	α = 400	α = 600	α = 800
δ = 0	0.017	0.004	0	0	0
δ = 0.005	0.02	0.019	0.015	0.014	0.014
δ = 0.01	0.023	0.021	0.021	0.020	0.020
δ = 0.03	0.04	0.041	0.043	0.044	0.046

and

Table 2. FCM based on ReLU activation on different drift strengths.

Drift Strength	α = 0	α = 50	α = 100	α = 150
δ = 0	0.37	0.0	0.0	0.0
δ = 0.01	0.38	0.08	0.05	0.03
δ = 0.05	0.38	0.18	0.18	0.12
δ = 0.1	0.39	0.23	0.24	0.23
δ = 0.3	0.4	0.35	0.35	0.35

. Results were obtained using simulations with η = 0.5, α = 800 for (SCM) sigmoidal activation and α = 150 for ReLU activation. After that, the five runs’ results are averaged. The final generalization error rises in proportion to the drift strength.

4.4. Generalization Error Comparison between SCM and FCM

Situations where K > M can happen because the rule’s intricacy is frequently unknown in practice. Additionally, having an over-learnable objective can be more beneficial with drift included. The final generalization error is shown in

Table 3. Final generalization error on SCM and FCM.

K	Generalization Error
	SCM	FCM
2	0.33	0.32
4	0.26	0.24
6	0.22	0.2
8	0.18	0.16
10	0.17	0.15
Average	0.22	0.214
Standard Deviation	0.056	0.065

as a result of K. We utilize a lower learning rate in this simulation, η = 0.1, δ = 0.1. For SCM sigmoidal activation and ReLU, we use α = 1000. The average accuracy of the soft committee machine is 0.22 with a standard deviation of 0.056, while the average accuracy of the fully connected committee machine is 0.214 with a standard deviation of 0.065. The soft committee machine shows slightly higher average accuracy and variability than the fully connected committee machine. This indicates that the soft committee machine may provide more nuanced predictions and capture uncertainty better but at the cost of increased variability in performance. On the other hand, the fully connected committee machine offers more consistent performance but may not capture uncertainty as effectively. The choice between the two approaches depends on the specific requirements of the task and the desired trade-off between accuracy and uncertainty modeling.

4.5. Optimal Rate of Learning

According to the generalization error, Figure 5 illustrates the ideal learning rate for two scenarios. In the beginning, K = 2, M = 2, whereas next K > M, where K = 4 and M = 2, is true for SCM sigmoidal and ReLU, respectively, and in both cases, α = 800 and α = 300. The final generalization error is averaged from the last 75,000 time steps in which the students have achieved optimal overlap with the teachers.

In terms of performance, the fully connected committee machine (FCM) shows an average accuracy of 0.214 with a standard deviation of 0.065. When compared to other continual learning methods, such as learning without forgetting (LwF), elastic weight consolidation (EWC), and incremental classifier and representation learning (iCaRL), the FCM performs slightly better in terms of average accuracy. However, it is essential to consider that the FCM may have a slightly higher standard deviation, indicating a slightly higher variability in its performance across different runs or datasets. Further analysis and comparison are required to determine the overall superiority of the FCM compared to other continual learning methods in terms of accuracy, stability, and robustness in handling concept drift and retaining previously learned knowledge.

4.6. Conversation

Table 2 shows that ReLU activation is more adept at coping with natural drift, starting with the K = M scenario. Like SCM sigmoidal activation, the attainable generalization ability is low for δ > 0.03, and the FCM stays unspecialized. For higher δ, it would appear that the same is occurring in the ReLU case. However, before the categorization reaches its final generalized form, a brief drift is extended after a sharp decline, and this evenness is quickly fractured.

An over-learnable goal, as illustrated in Table 2, might be helpful. In actuality, one hardly ever understands the complexity of the regulation. Consequently, boosting K to enhance generalization can be considered when the learning process is drifting. This results in greater danger of overfitting as a result.

For both the ReLU and the sigmoidal in Figure 4, the ideal learning rate with drift is close to 0.5. It is clear from Figure 4 that as the remote unit K increases, a lower learning rate is preferable. A high learning rate will impede development, the threshold for divergence will drop with rising K, and the student weight vectors will depart from the rule.

It is clear from Figure 6 that higher γ values are advantageous for more significant quantities of drift. A better value is advantageous for ReLU in certain circumstances because it can typically handle more considerable drift. This makes sense because a system must have a greater γ capacity for forgetting outdated information if it is very non-stationary. Weight decay has a minimal impact since sigmoidal activation does not desirably handle drifts of δ > 0.01. Any weight decay raises the ultimate generalization error for small drift quantities.

5. Conclusions

Continual learning using a regression strategy employing two alternative activation functions was the main emphasis of this research, which also addressed the impact of data changes over continuous data streams. The contrasts between the matching and over-learnable scenarios and any potential benefits or drawbacks were assessed. The contrast between simulations and the theoretical account of the dynamic learning environment [9], where N. weight decay, the impact of drift strength on absolute drift durations, and the generalization errors in these plateau states were investigated. A new system with the desired configuration can be poorly modeled by deterministic concept drifts, akin to perceptron learning [33,37,40]. Deep learning [25] has been a fascinating research topic in recent years. Therefore, the extension to a deeper FCM design with more than one hidden layer is a crucial upcoming study. If you want to know if the outcomes of this learning can be seen in real-world scenarios, you can conduct research using a more practical methodology and realistic dynamic data streams.

The research primarily focused on continual learning with a regression strategy using two different activation functions and addressing the impact of data changes in continuous data streams. The study compared matching and over-learnable scenarios, examining potential benefits and drawbacks. It also explored the impact of weight decay, drift strength, and generalization errors in dynamic learning environments. Future research directions include extending the study to deeper FCM designs with multiple hidden layers and conducting experiments with more practical methodologies and realistic dynamic data streams to validate the findings in real-world scenarios.

The future scope of research in the described area includes advancing continual learning techniques, particularly focusing on the regression strategy with alternative activation functions, exploring deeper designs of the fully connected committee machine (FCM), validating approaches with realistic dynamic data streams, addressing non-deterministic concept drifts, and applying continual learning in specific domains such as computer vision or natural language processing. These directions hold the potential for enhancing adaptability, improving performance, and enabling practical applications of continual learning in real-world scenarios.