Sequential Learning Framework

Updated 10 January 2026

Sequential learning frameworks are iterative methods that update models using prior information and adaptive data acquisition strategies.
They integrate techniques such as Bayesian optimization, active sampling, and reinforcement learning to efficiently guide model refinement.
These frameworks balance exploration and exploitation, enabling robust decision-making and domain generalization across various scientific and engineering applications.

Sequential learning encompasses a methodology and associated algorithmic paradigms in which information is acquired and models are incrementally improved through a sequence of adaptively chosen steps, each using the results of prior learning to inform future data collection or inference. Such frameworks are fundamental in domains where data acquisition is costly, the optimization landscape is non-stationary or multi-objective, or models must generalize across a sequence of related environments. Contemporary sequential learning frameworks operationalize this principle in active Bayesian optimization, adaptive decision-making, data-efficient classification, and model generalization across domains, leveraging advances in surrogate modeling, acquisition function design, online complexity measures, reinforcement learning, and multi-agent architectures.

1. The Sequential Learning Paradigm: Definitions and General Principles

The core idea in sequential learning frameworks is to perform model updates, data selection, or parameter estimation iteratively, often with a bounded data-collection or compute budget. Sequential steps are not independent; each incorporates information from the accumulation of prior actions, observations, or model states. This process may be driven by explicit criteria such as information gain, uncertainty or surprise, identification of Pareto-optimal fronts, or minimization of excess risk.

Distinguishing features include:

Active data acquisition: Selecting the next data point(s) to maximize an objective, e.g., information-theoretic utility, acquisition function value, or uncertainty.
Adaptive model refinement: Iterative update of surrogates or parameter estimates based on new data, often under sampling constraints.
Non-i.i.d. data handling: Frameworks are often designed for non-i.i.d. settings, including adversarial, non-stationary, or sequentially dependent environments.
Multi-objective or multi-task extension: Many frameworks support optimization or inference in the presence of multiple, potentially conflicting objectives.

2. Framework Architectures: Core Algorithms and Workflow

Several distinct theoretical and practical frameworks instantiate sequential learning; the following exemplify major approaches:

Bayesian Optimization-based Sequential Learning

MOBOSL integrates Gaussian process (GP) surrogates, multi-objective acquisition (qParEGO), and an active loop. The workflow consists of initialization (initial data selection, GP surrogate fitting), batch acquisition via scalarization and qEI maximization, experimental evaluation, GP retraining, and Pareto-front update/stopping by hypervolume threshold. The surrogate posterior is updated sequentially, and acquisition is optimized in the simplex of scalarization weights (Khosravi et al., 2023).

Active Sampling with Confidence and Space-Filling

SL-RF+ for defect classification adopts a random forest as the base learner with two distinct, sequentially applied sampling methods: Least Confidence Sampling (LCS), which queries regions of maximal posterior uncertainty, and Sobol sequence–based synthetic sampling, which explores the parameter space quasi-uniformly before matching synthetic points to candidate real samples for label requests (Raihan et al., 2024).

Sequential Decision Making in Reinforcement and Optimal Control

Unified RL/control frameworks define optimal policies over a sequence, often with resource constraints; e.g., BCRLSP applies offline batch-constrained Q-learning for targeted promotion, combining value estimation with real-time linear programming to enforce budget constraints at each decision epoch (Chen et al., 2022).
The general sequential decision model (Powell) frames all such processes as optimization over policies, admitting four universal classes: Policy Function Approximations, Cost Function Approximations, Value Function Approximations, and Direct Lookahead Approximations. Each policy class can be sequentially updated as new state, reward, or constraint observations arrive (Powell, 2019).

Sequential Learning for Domain Generalization

Sequential MLDG addresses sequential domain shifts by simulating a sequence of tasks (domains), updating model parameters via chained meta-learning steps and backpropagating losses not only to the next but across the entire subsequent sequence of domains. This imposes gradient alignment among domains, enhancing representation invariance (Li et al., 2020).

Sequential Local Learning in Latent Graphical Models

Frameworks for latent graphical models employ a combination of graph-theoretic marginalization and conditioning operations to sequentially recover the parameters of latent-variable models via local black-box solvers (e.g., tensor decomposition on well-posed subgraphs) and label-consistency checks (Park et al., 2017).

3. Mathematical Foundations and Algorithmic Details

Sequential learning frameworks are formalized via probabilistic, optimization, and information-theoretic models:

Surrogate Modeling: For Bayesian optimization, a GP surrogate for each objective is updated as new pairs $(x_i, y_i)$ are evaluated. Posterior updates, acquisition calculation (e.g., expected improvement or surprise metrics), and candidate selection comprise the main loop (Khosravi et al., 2023, Raihan et al., 2024).
Acquisition and Query Policies: Multi-objective acquisition uses scalarizations ( $\lambda$ -weighted Tchebycheff), and batch extensions (qParEGO) jointly optimize acquisition for a batch of candidates. Strategies such as LCS select points $x$ for which $\max_c P(y_c|x)$ is minimal, guiding the learning to sharp boundaries or poorly understood domains (Raihan et al., 2024).
Online Predictive Complexity: The minimax regret of sequential prediction games is governed by sequential Rademacher complexities, sequential fat-shattering, and covering numbers. These characterize necessary and sufficient conditions for online learnability and directly inform the achievable regret bounds and sample complexity in sequential learning (Rakhlin et al., 2010).
Adaptive Sample Sizing and Drift Estimation: When solving a series of drifting stochastic optimization or estimation tasks, sample sizes are adaptively determined based on online estimates of the change in the optimizer (e.g., via empirical risk differences or gradient-based drift estimators), ensuring that excess risk remains below a specified target (Bu et al., 2018, Wilson et al., 2019).

4. Evaluation Metrics and Performance Guarantees

Quantitative evaluation of sequential learning frameworks involves multifaceted metrics:

Pareto-Front Discovery and Data Efficiency: The adjusted proportional hypervolume (APHV) metric combines Pareto-front quality (PHV) and data efficiency ( $K$ ). $\mathrm{APHV} = \alpha(1-K) + \beta\,\mathrm{PHV}$ explicitly trades off the fraction of data used against the quality of the approximate Pareto set (Khosravi et al., 2023).
Classification Metrics: SL-RF+ yields accuracy, precision, recall, and F1 through sequentially expanding training sets, with reduced labeling cost, particularly improving performance for underrepresented classes (Raihan et al., 2024).
Excess Risk: Near-optimal excess risk bounds of the form $O(\Delta + 1/\sqrt{K_t})$ are established for drifting sequential estimation with adaptive sample size selection (Bu et al., 2018).
Empirical Superiority: MOBOSL matches or exceeds prior passive approaches in manufacturing optimization, recovering $100\%$ of the Pareto front with $60$– $70\%$ fewer experiments (Khosravi et al., 2023). SL-RF+ surpasses traditional RF classifiers using $40\%$ fewer labeled samples (Raihan et al., 2024). BCRLSP achieves higher customer retention at the same or lower cost compared to both SL and batch baselines (Chen et al., 2022).

5. Applications Across Scientific and Engineering Domains

Sequential learning frameworks have found application in:

Advanced Manufacturing: Accelerated Pareto-front discovery in multi-objective materials optimization and process parameter selection in additive manufacturing — reducing experimental cost and time (Khosravi et al., 2023, Raihan et al., 2024, Raihan et al., 2024).
Cyber-Manufacturing and Defect Detection: Iterative improvement of surrogate models for melt pool geometry prediction and defect classification, combining active querying with synthetic data generation (CTGAN) to address data scarcity (Raihan et al., 2024, Raihan et al., 2024).
Business Decision-Making: Budget-constrained personalized promotion targeting via reinforcement learning and linear programming (Chen et al., 2022).
Domain Adaptation: Enhanced domain generalization for visual recognition through chained meta-learning steps and explicit backpropagation across domain permutations (Li et al., 2020).
Latent Graphical Model Inference: Scalable identification of high-dimensional, loopy latent GMs through recursive, graph-structured operator sequences (Park et al., 2017).

6. Limitations, Challenges, and Open Directions

Despite broad applicability, several limitations persist:

Compute and Data Constraints: Many frameworks depend on surrogate retraining or sample-size optimization at every iteration; scalability in high dimensions or very large data pools remains challenging (Khosravi et al., 2023, Raihan et al., 2024).
Non-Stationarity and Model Drift: The frameworks assume bounded drift or slow-fluctuating environments; abrupt or adversarial changes require change-point detection, reinitialization, or robust alternatives (Bu et al., 2018, Wilson et al., 2019).
Permutational Complexity: In domain generalization, full enumeration of domain orderings is infeasible; random sampling and first-order approximations serve as proxies but may not recover global optima (Li et al., 2020).
Heuristic Design Choices: The "closest-point-in-dataset" selection or Sobol-based sampling has no domain-specific optimality guarantee and may require further tuning or principled exploration-exploitation balance (Khosravi et al., 2023, Raihan et al., 2024).

Emerging research directions include automated acquisition design, integration with cost-aware or physics-informed models, hybrid continuous-discrete optimization, and extension to settings where the data-generating process is non-stationary or only partially observable.

7. Theoretical Foundations and Generalizations

Sequential learning frameworks represent an intersection of statistical learning theory, stochastic optimization, and reinforcement learning. Online learnability is characterized by sequential complexities (Rademacher, fat-shattering) that govern achievable minimax regret. The models unify settings as diverse as sequential experimental design, adaptive data acquisition, multi-objective optimization, and lifelong meta-learning. Many successful architectures are modular, allowing swapping of surrogate models, acquisition functions, or adaptive routines, and can be ported across domains, including engineering design, medical diagnosis, and scientific discovery (Khosravi et al., 2023, Raihan et al., 2024, Powell, 2019, Rakhlin et al., 2010).

These frameworks underpin principled, resource-efficient learning under sequential, adaptive conditions, with an emphasis on robust performance, data efficiency, and rapid adaptation to changing objectives or environments.