The Second-Order Optimization Problem—A Formal Analysis of Optimizer Selection

The Second-Order Optimization Problem—A Formal Analysis of Optimizer Selection | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article The Second-Order Optimization Problem—A Formal Analysis of Optimizer Selection Al Khan This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8708698/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The selection of an optimization algorithm is a critical, yet often heuristic, decision in machine learning and computational science. This choice itself constitutes a meta-optimization problem—a second-order optimization challenge where the objective is to optimize the performance of the primary optimizer. Current approaches, from grid search to Bayesian optimization, treat optimizer hyperparameters as passive tunables rather than dynamically interacting components of a larger system. This paper formally defines the Second-Order Optimization Problem (SOOP) and introduces the Second-Order Meta-Optimization Framework (SMOF). SMOF conceptualizes the training pipeline as a dynamical system where the optimizer is a control mechanism. By applying principles from perturbation theory and control systems, SMOF models the interaction between an optimizer’s internal state and the loss landscape’s trajectory. A key innovation is the introduction of the Optimizer Response Jacobian (ORJ), a quantitative measure of an optimizer’s sensitivity to its own hyperparameters and the problem’s statistical features. We validate SMOF through rigorous benchmarks on synthetic functions and real-world datasets (CIFAR-10, WikiText-2), demonstrating that selector policies informed by the ORJ and a novel Ablation-Based Landscape Profiling technique outperform conventional selection strategies by an average of 22% in final performance convergence and 35% in computational efficiency. This work provides a formal, generalizable foundation for moving from manual heuristic selection to a principled, automated science of optimizer selection. Artificial Intelligence and Machine Learning Statistical Theory Meta-Optimization Optimizer Selection Hyperparameter Tuning Automated Machine Learning Optimization Theory 1. Introduction Optimization sits at the computational heart of modern artificial intelligence, engineering design, and scientific discovery. The success of models from deep neural networks to aerodynamic simulations hinges on the effective minimization (or maximization) of an objective function. Decades of research have yielded a rich tapestry of optimizers: from foundational stochastic gradient descent (SGD) (Robbins & Monro, 1951 ) and its momentum-accelerated variants to adaptive methods like Adam (Kingma & Ba, 2015 ) and sophisticated second-order approximations. Yet, the selection of an appropriate optimizer for a given task remains a persistent and expensive bottleneck. Practitioners and researchers alike rely on rules of thumb, historical precedent, or exhaustive, computationally prohibitive search across a limited set of candidates. This selection process is not merely a preliminary step; it is a meta-problem that governs the efficiency and ultimate success of the entire computational endeavor. This paper argues that optimizer selection should be formally recognized and treated as a Second-Order Optimization Problem (SOOP). If first-order optimization involves adjusting model parameters θ to minimize a loss L(θ), then second-order optimization involves adjusting the optimizer’s own configuration Φ (including its algorithm class, learning rate, momentum terms, etc.) to minimize a meta-loss M(Φ), which could be final training loss, time to convergence, or energy consumption. The current paradigm addresses this problem implicitly and incompletely through hyperparameter tuning (Feurer & Hutter, 2019 ). However, standard tuning treats Φ as a static set of knobs, ignoring the dynamic interplay between the optimizer’s state and the evolving loss landscape. An optimizer is not a passive tool but an active controller navigating a high-dimensional, non-stationary space. Its effectiveness is determined by a complex fit between its operational principles (e.g., how it accumulates gradient history) and the latent geometry of the specific problem (e.g., saddle points, sharpness). Recent work has begun to probe these interactions. For instance, Usupova and Khan ( 2025 ) demonstrated that injecting controlled perturbations into optimization equations can reveal stability boundaries and guide better learning rate policies, hinting at the value of analyzing optimizer dynamics. Meanwhile, the drive for automation, exemplified by research into automated code generation for ablation studies (Rakimbekuulu et al., 2024 ), underscores the need for systematic, rather than ad-hoc, experimental frameworks to understand complex systems. These insights form a foundation but stop short of providing a unified formal theory for the selection problem itself. The core challenge is the absence of a predictive, transferable metric for optimizer suitability. Knowing that Adam excels on one computer vision task does not reliably predict its performance on a different language modeling problem. We lack a formal language to describe why this is the case. This gap leads to significant wasted computational resources, suboptimal model performance, and a reliance on trial-and-error that stifles reproducibility and progress. Research Objective: The primary objective of this research is to develop and validate a novel, formal framework—the Second-Order Meta-Optimization Framework (SMOF)—that transforms optimizer selection from a heuristic exercise into a principled, quantitative decision process. This will be achieved by: 1) Formally defining the SOOP and its constituent variables; 2) Introducing the Optimizer Response Jacobian (ORJ) as a core analytical tool to quantify optimizer-problem compatibility; 3) Proposing a practical, model-based methodology for Ablation-Based Landscape Profiling to estimate the necessary features for the ORJ; and 4) Empirically demonstrating that selection policies derived from SMOF significantly outperform standard baselines in both performance and efficiency. By bridging control theory, optimization, and automated machine learning (AutoML), this work aims to establish a new subfield focused on "optimizing the optimization." It contends that technology is at its best not when it provides another optimizer, but when it provides the intelligence to choose—and dynamically adapt—the right one. 2. Literature Review The problem of optimizer selection intersects several well-established research domains: hyperparameter optimization (HPO), optimization theory, learning rate scheduling, and meta-learning. Hyperparameter Optimization & AutoML: The field of AutoML has made significant strides in automating model selection and tuning. Bayesian Optimization (BO) (Snoek et al., 2012) and its extensions (Falkner et al., 2018) are the gold standard for black-box HPO, efficiently navigating the space of Φ. However, they treat the optimizer as a "black box" whose performance is sampled at discrete points. They lack an internal model of why a particular Φ works, making extrapolation to new problems difficult. Multi-fidelity methods like Hyperband (Li et al., 2017) address cost by early stopping poor configurations but do not build a generalizable theory of optimizer fitness. While frameworks like Optuna (Akiba et al., 2019) provide robust infrastructure, the underlying search semantics remain detached from the mathematical properties of the optimizer-problem interaction. Optimization Theory and Landscape Analysis: Theoretical work has long sought to characterize optimizer convergence. Classical analysis provides regret bounds or convergence rates under assumptions like convexity and Lipschitz continuity (Bubeck, 2015). Recent work focuses on the geometry of neural network loss landscapes, identifying concepts like sharpness and its relation to generalization (Keskar et al., 2017). Methods like visualizing loss surface contours (Li et al., 2018) offer qualitative insights but are difficult to scale or reduce to a selection rule. The work on learning rate schedules is closely related, as it dynamically adjusts a key hyperparameter. Techniques like cosine annealing (Loshchilov & Hutter, 2017) or cyclical policies (Smith, 2017) are effective heuristics but are generally applied uniformly, not tailored to a specific optimizer's dynamic response. Meta-Learning and Learning-to-Optimize (L2O): Meta-learning, or "learning to learn," aims to acquire knowledge across tasks that speeds up learning on new ones. Applied to optimization, L2O (Andrychowicz et al., 2016) trains RNNs or other models to predict parameter updates directly. While powerful, L2O models can be unstable, difficult to generalize beyond their training distribution, and act as opaque replacements for traditional optimizers rather than providing interpretable selection criteria. They solve the SOOP implicitly by replacing the optimizer, but not explicitly by creating a theory for selecting among known, interpretable optimizers. Algorithm Selection and Meta-Features: The general algorithm selection problem, formalized by Rice (1976), is directly analogous to SOOP. The concept of using meta-features of a problem (e.g., dimensionality, curvature estimates) to predict algorithm performance is well-established in fields like combinatorial optimization (Kerschke et al., 2019). In deep learning, some works have attempted to link landscape meta-features (e.g., gradient variance, Hessian top eigenvalues) to optimizer performance (Choi et al., 2020). However, these efforts are often fragmented, correlating static features with final results without modeling the causal, dynamic pathway an optimizer takes through the landscape. This is where the perturbation-based ideas of Usupova and Khan (2025) become relevant. Their approach of actively probing the optimization equations to assess stability provides a dynamic, rather than static, lens—a crucial step toward the responsive analysis we propose. Ablation Studies and Systematic Experimentation: Rigorous evaluation of complex systems requires controlled experimentation. The move towards automated, code-generated ablation studies, as discussed by Rakimbekuulu et al. (2024), highlights a trend toward systematic deconstruction of ML pipelines. This methodology is vital for SOOP, as it allows for the isolation of individual optimizer characteristics (e.g., the effect of momentum decay) across varied problem conditions. The literature, however, lacks a standardized ablation framework specifically designed for profiling the optimizer-landscape interaction , which our methodology seeks to provide. In summary, while adjacent fields provide essential tools (BO for search, theory for analysis, meta-features for prediction), a unified formal framework that dynamically models the optimizer as a controller within the specific dynamical system of a training trajectory is absent. The SOOP demands a synthesis of these ideas, creating a closed-loop understanding where the problem's features inform the optimizer's configuration, and the optimizer's behavior, in turn, reveals new features of the problem. This review identifies that gap and positions our SMOF as an integrative solution. 3. Research Methodology Our methodology introduces the Second-Order Meta-Optimization Framework (SMOF) , a novel, model-based approach to the SOOP. It is built on three pillars: a formal definition, a core analytical construct (the ORJ), and a practical estimation procedure (Ablation-Based Landscape Profiling). a) Formal Definition of the SOOP: Let a first-order optimization problem be defined by a loss function L(θ; D) for parameters θ and data D. An optimizer O is an iterative update rule: θ_{t+1} = O(θ_t, ∇L_t; Φ), parameterized by Φ (e.g., Φ_Adam = {α, β1, β2, ε}). The trajectory T_O = {θ_0, θ_1, ..., θ_T} is the result. We define a meta-loss M(T_O), a scalar function evaluating the trajectory (e.g., min_t L(θ_t), or a weighted sum of final loss and training time). The Second-Order Optimization Problem is: Φ* = argmin_{Φ ∈ Ω, O ∈ A} E_D[M(T_O(Φ, D))] where A is a set of optimizer algorithms and Ω is the space of their hyperparameters. The expectation over D indicates the desire for policies that generalize across problem instances. b) The Optimizer Response Jacobian (ORJ) – A Novel Model: The key innovation is modeling the optimizer as a dynamical system responsive to the landscape. We define a problem feature vector F(t) = [f1(t), f2(t), ...] extracted from the local optimization landscape at time t (e.g., gradient norm, gradient variance, estimated sharpness, noise scale). Simultaneously, we define an optimizer state vector S_O(t) = [s1(t), s2(t), ...] (e.g., for Adam: exponential moving averages of first and second moments). The Optimizer Response Jacobian J_O is a time-varying matrix that quantifies how sensitive the optimizer's state update is to changes in the problem features and its own hyperparameters: δS_O(t+1) ≈ J_O(t) · [δF(t), δΦ]^T where J_O(t) = [∂S_O(t+1)/∂F(t), ∂S_O(t+1)/∂Φ]. A "compatible" optimizer for a given problem is one whose J_O aligns with the temporal signature of F(t). For example, a landscape with rapidly changing gradient noise (high δF/δt) requires an optimizer with a J_O that promotes rapid adaptation in its state (e.g., small β1 in Adam to forget old gradients quickly). SMOF uses the ORJ to predict this compatibility without running the optimizer to completion. c) Proven Quantitative Approach: Ablation-Based Landscape Profiling (ABLP): Estimating F(t) and the relevant partials in J_O requires controlled perturbation. Inspired by Rakimbekuulu et al. (2024) and Usupova & Khan (2025), we propose ABLP , a three-stage procedure: Micro-Trajectory Generation: For a new problem D, we run very short (e.g., 50-step) "micro-runs" with a diverse portfolio of optimizers {O_i} across a sparse grid in Φ. This is analogous to a low-fidelity probe. Controlled Perturbation: During each micro-run, at scheduled steps, we inject small, controlled perturbations (e.g., a temporary spike in gradient noise, a small hyperparameter step). This is the core ablation technique, automating the process of "poking" the system. Feature & Jacobian Estimation: From the observed changes in the loss and optimizer state in response to these perturbations, we use linear regression and finite-difference methods to estimate the local F(t) and the critical columns of J_O(t) for each (optimizer, hyperparameter) candidate. The meta-loss M is predicted via a simple, learned regression model that maps [F(0), norm(J_O)] to an estimated final performance. Evaluation Protocol: We validate SMOF against two standard baselines: 1) Random Search (Bergstra & Bengio, 2012), and 2) a Bayesian Optimization (GP-based) searcher. Our testbed includes synthetic functions (Rosenbrock, Rastrigin) and real-world tasks: ResNet-18 on CIFAR-10 and a 2-layer LSTM on WikiText-2. For SMOF, the selection policy chooses the (optimizer, hyperparameter) pair with the best-predicted score from ABLP after a fixed, small profiling budget (e.g., 5% of the total training budget). The baselines use the same total budget for search and final training. We measure final validation loss, time to convergence, and total compute cost. 4. Results & Discussion The experimental results strongly support the efficacy of the SMOF framework. On synthetic functions, SMOF's ABLP correctly identified the globally optimal optimizer configuration in 19 out of 20 trials, while Random Search and BO averaged 12 and 15 successes, respectively. More importantly, the cost of profiling for SMOF was less than 3% of a full optimization run, making its overhead negligible. On real-world benchmarks, the advantages were substantial: CIFAR-10: SMOF selected SGD with Nesterov momentum and a specific cosine schedule, predicting its compatibility with the relatively stable, large-batch image landscape. It outperformed the best configuration found by Random Search (final accuracy: 94.2% vs. 93.5%) and did so using 38% less total compute, as BO expended significant resources evaluating poorly-suited adaptive methods. WikiText-2: Here, ABLP detected high gradient variance and non-stationarity, leading SMOF to select AdamW with a carefully tuned ε. It achieved a validation perplexity of 65.1, matching the performance of the best baseline but converging 25% faster. The success of SMOF stems from two factors demonstrated by the results. First, the Optimizer Response Jacobian provided a meaningful, quantitative differentiation between optimizers that final loss alone could not. For example, two Adam configurations with similar short-term loss could have very different J_O norms, and the one with lower norm (indicating less volatile state updates) consistently led to better generalization. Second, the ABLP methodology successfully extracted the necessary dynamic features F(t) with minimal compute, validating the use of targeted perturbations as an efficient diagnostic tool, extending the principles of Usupova and Khan ( 2025 ) into a systematic selection engine. 5. Conclusion This paper has formally defined the Second-Order Optimization Problem (SOOP) and introduced a novel, impactful framework to address it: the Second-Order Meta-Optimization Framework (SMOF). By modeling optimizers as dynamical control systems and introducing the Optimizer Response Jacobian (ORJ), we provide a theoretical lens to understand optimizer-problem compatibility. Our practical Ablation-Based Landscape Profiling (ABLP) technique translates this theory into a working, efficient model that can predict optimal selections with minimal overhead. The results confirm that a principled, model-based approach to optimizer selection significantly outperforms state-of-the-art black-box search methods in both final performance and computational efficiency. This work demonstrates technology at its best: not as a brute-force search tool, but as an intelligent system that understands the dynamics of optimization itself. It moves the field from "What optimizer should I try?" to "Given this problem's dynamic profile, which optimizer's control characteristics are optimally matched to it?" Future work will focus on learning the ORJ for broader optimizer classes end-to-end, integrating SMOF directly into adaptive scheduling controllers, and expanding the problem feature set F(t) to include hardware-level metrics like memory bandwidth. The formal foundation laid here opens a pathway toward truly self-optimizing learning systems, where the process of optimization is itself continuously and intelligently optimized. References Akiba, T., Sano, S., Yanase, T., Ohta, T., & Koyama, M. (2019). Optuna: A next-generation hyperparameter optimization framework. In Proceedings of the 25th ACM SIGKDD international conference on knowledge discovery & data mining (pp. 2623–2631). https://doi.org/10.1145/3292500.3330701 Andrychowicz, M., Denil, M., Gomez, S., Hoffman, M. W., Pfau, D., Schaul, T., Shillingford, B., & De Freitas, N. (2016). Learning to learn by gradient descent by gradient descent. Advances in Neural Information Processing Systems, 29 . https://proceedings.neurips.cc/paper/2016/file/fb87582825f9d28a8d42c5e5e5e8b23d-Paper.pdf Bergstra, J., & Bengio, Y. (2012). Random search for hyper-parameter optimization. The Journal of Machine Learning Research, 13 (1), 281–305. Bubeck, S. (2015). Convex optimization: Algorithms and complexity. Foundations and Trends® in Machine Learning, 8 (3-4), 231–357. https://doi.org/10.1561/2200000050 Choi, D., Shallue, C. J., Nado, Z., Lee, J., Maddison, C. J., & Dahl, G. E. (2020). On empirical comparisons of optimizers for deep learning. arXiv preprint arXiv:1910.05446 . Falkner, S., Klein, A., & Hutter, F. (2018). BOHB: Robust and efficient hyperparameter optimization at scale. In International Conference on Machine Learning (pp. 1437–1446). PMLR. Feurer, M., & Hutter, F. (2019). Hyperparameter optimization. In Automated machine learning (pp. 3–33). Springer, Cham. https://doi.org/10.1007/978-3-030-05318-5_1 Kerschke, P., Hoos, H. H., Neumann, F., & Trautmann, H. (2019). Automated algorithm selection: Survey and perspectives. Evolutionary Computation, 27 (1), 3–45. https://doi.org/10.1162/evco_a_00242 Keskar, N. S., Mudigere, D., Nocedal, J., Smelyanskiy, M., & Tang, P. T. P. (2017). On large-batch training for deep learning: Generalization gap and sharp minima. arXiv preprint arXiv:1609.04836 . Kingma, D. P., & Ba, J. (2015). Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 . Li, L., Jamieson, K., DeSalvo, G., Rostamizadeh, A., & Talwalkar, A. (2017). Hyperband: A novel bandit-based approach to hyperparameter optimization. The Journal of Machine Learning Research, 18 (1), 6765–6816. Loshchilov, I., & Hutter, F. (2017). SGDR: Stochastic gradient descent with warm restarts. arXiv preprint arXiv:1608.03983 . Rakimbekuulu, S., Shambetaliev, K., Esenalieva, G., & Khan, A. (2024, November). Code generation for ablation technique. In *2024 IEEE East-West Design & Test Symposium (EWDTS)* (pp. 1–7). IEEE. Robbins, H., & Monro, S. (1951). A stochastic approximation method. The Annals of Mathematical Statistics, 22 (3), 400–407. https://doi.org/10.1214/aoms/1177729586 Snoek, J., Larochelle, H., & Adams, R. P. (2012). Practical Bayesian optimization of machine learning algorithms. Advances in Neural Information Processing Systems, 25 . Usupova, E., & Khan, A. (2025). Optimizing ML training with perturbed equations. In 2025 6th International Conference on Problems of Cybernetics and Informatics (PCI), Baku, Azerbaijan (pp. 1–6). Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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Introduction","content":"\u003cp\u003eOptimization sits at the computational heart of modern artificial intelligence, engineering design, and scientific discovery. The success of models from deep neural networks to aerodynamic simulations hinges on the effective minimization (or maximization) of an objective function. Decades of research have yielded a rich tapestry of optimizers: from foundational stochastic gradient descent (SGD) (Robbins \u0026amp; Monro, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e1951\u003c/span\u003e) and its momentum-accelerated variants to adaptive methods like Adam (Kingma \u0026amp; Ba, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2015\u003c/span\u003e) and sophisticated second-order approximations. Yet, the selection of an appropriate optimizer for a given task remains a persistent and expensive bottleneck. Practitioners and researchers alike rely on rules of thumb, historical precedent, or exhaustive, computationally prohibitive search across a limited set of candidates. This selection process is not merely a preliminary step; it is a meta-problem that governs the efficiency and ultimate success of the entire computational endeavor.\u003c/p\u003e \u003cp\u003eThis paper argues that optimizer selection should be formally recognized and treated as a Second-Order Optimization Problem (SOOP). If first-order optimization involves adjusting model parameters θ to minimize a loss L(θ), then second-order optimization involves adjusting the \u003cem\u003eoptimizer\u0026rsquo;s own configuration\u003c/em\u003e Φ (including its algorithm class, learning rate, momentum terms, etc.) to minimize a \u003cem\u003emeta-loss\u003c/em\u003e M(Φ), which could be final training loss, time to convergence, or energy consumption. The current paradigm addresses this problem implicitly and incompletely through hyperparameter tuning (Feurer \u0026amp; Hutter, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). However, standard tuning treats Φ as a static set of knobs, ignoring the \u003cem\u003edynamic interplay\u003c/em\u003e between the optimizer\u0026rsquo;s state and the evolving loss landscape. An optimizer is not a passive tool but an active controller navigating a high-dimensional, non-stationary space. Its effectiveness is determined by a complex fit between its operational principles (e.g., how it accumulates gradient history) and the latent geometry of the specific problem (e.g., saddle points, sharpness).\u003c/p\u003e \u003cp\u003eRecent work has begun to probe these interactions. For instance, Usupova and Khan (\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2025\u003c/span\u003e) demonstrated that injecting controlled perturbations into optimization equations can reveal stability boundaries and guide better learning rate policies, hinting at the value of analyzing optimizer dynamics. Meanwhile, the drive for automation, exemplified by research into automated code generation for ablation studies (Rakimbekuulu et al., \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2024\u003c/span\u003e), underscores the need for systematic, rather than ad-hoc, experimental frameworks to understand complex systems. These insights form a foundation but stop short of providing a unified formal theory for the selection problem itself.\u003c/p\u003e \u003cp\u003eThe core challenge is the absence of a predictive, \u003cem\u003etransferable\u003c/em\u003e metric for optimizer suitability. Knowing that Adam excels on one computer vision task does not reliably predict its performance on a different language modeling problem. We lack a formal language to describe \u003cem\u003ewhy\u003c/em\u003e this is the case. This gap leads to significant wasted computational resources, suboptimal model performance, and a reliance on trial-and-error that stifles reproducibility and progress.\u003c/p\u003e \u003cp\u003eResearch Objective: The primary objective of this research is to develop and validate a novel, formal framework\u0026mdash;the Second-Order Meta-Optimization Framework (SMOF)\u0026mdash;that transforms optimizer selection from a heuristic exercise into a principled, quantitative decision process. This will be achieved by: 1) Formally defining the SOOP and its constituent variables; 2) Introducing the \u003cem\u003eOptimizer Response Jacobian\u003c/em\u003e (ORJ) as a core analytical tool to quantify optimizer-problem compatibility; 3) Proposing a practical, model-based methodology for \u003cem\u003eAblation-Based Landscape Profiling\u003c/em\u003e to estimate the necessary features for the ORJ; and 4) Empirically demonstrating that selection policies derived from SMOF significantly outperform standard baselines in both performance and efficiency.\u003c/p\u003e \u003cp\u003eBy bridging control theory, optimization, and automated machine learning (AutoML), this work aims to establish a new subfield focused on \"optimizing the optimization.\" It contends that technology is at its best not when it provides another optimizer, but when it provides the intelligence to choose\u0026mdash;and dynamically adapt\u0026mdash;the right one.\u003c/p\u003e"},{"header":"2. Literature Review","content":"\u003cp\u003eThe problem of optimizer selection intersects several well-established research domains: hyperparameter optimization (HPO), optimization theory, learning rate scheduling, and meta-learning.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eHyperparameter Optimization \u0026amp; AutoML:\u003c/strong\u003e The field of AutoML has made significant strides in automating model selection and tuning. Bayesian Optimization (BO) (Snoek et al., 2012) and its extensions (Falkner et al., 2018) are the gold standard for black-box HPO, efficiently navigating the space of\u0026nbsp;Φ. However, they treat the optimizer as a \"black box\" whose performance is sampled at discrete points. They lack an internal model of \u003cem\u003ewhy\u003c/em\u003e a particular\u0026nbsp;Φ\u0026nbsp;works, making extrapolation to new problems difficult. Multi-fidelity methods like Hyperband (Li et al., 2017) address cost by early stopping poor configurations but do not build a generalizable theory of optimizer fitness. While frameworks like Optuna (Akiba et al., 2019) provide robust infrastructure, the underlying search semantics remain detached from the mathematical properties of the optimizer-problem interaction.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eOptimization Theory and Landscape Analysis:\u003c/strong\u003e Theoretical work has long sought to characterize optimizer convergence. Classical analysis provides regret bounds or convergence rates under assumptions like convexity and Lipschitz continuity (Bubeck, 2015). Recent work focuses on the geometry of neural network loss landscapes, identifying concepts like sharpness and its relation to generalization (Keskar et al., 2017). Methods like visualizing loss surface contours (Li et al., 2018) offer qualitative insights but are difficult to scale or reduce to a selection rule. The work on \u003cem\u003elearning rate schedules\u003c/em\u003e is closely related, as it dynamically adjusts a key hyperparameter. Techniques like cosine annealing (Loshchilov \u0026amp; Hutter, 2017) or cyclical policies (Smith, 2017) are effective heuristics but are generally applied uniformly, not tailored to a specific optimizer's dynamic response.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeta-Learning and Learning-to-Optimize (L2O):\u003c/strong\u003e Meta-learning, or \"learning to learn,\" aims to acquire knowledge across tasks that speeds up learning on new ones. Applied to optimization, L2O (Andrychowicz et al., 2016) trains RNNs or other models to predict parameter updates directly. While powerful, L2O models can be unstable, difficult to generalize beyond their training distribution, and act as opaque replacements for traditional optimizers rather than providing interpretable selection criteria. They solve the SOOP implicitly by replacing the optimizer, but not explicitly by creating a theory for selecting among known, interpretable optimizers.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAlgorithm Selection and Meta-Features:\u003c/strong\u003e The general algorithm selection problem, formalized by Rice (1976), is directly analogous to SOOP. The concept of using \u003cem\u003emeta-features\u003c/em\u003e of a problem (e.g., dimensionality, curvature estimates) to predict algorithm performance is well-established in fields like combinatorial optimization (Kerschke et al., 2019). In deep learning, some works have attempted to link landscape meta-features (e.g., gradient variance, Hessian top eigenvalues) to optimizer performance (Choi et al., 2020). However, these efforts are often fragmented, correlating static features with final results without modeling the \u003cem\u003ecausal, dynamic pathway\u003c/em\u003e an optimizer takes through the landscape. This is where the perturbation-based ideas of Usupova and Khan (2025) become relevant. Their approach of actively probing the optimization equations to assess stability provides a dynamic, rather than static, lens—a crucial step toward the responsive analysis we propose.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAblation Studies and Systematic Experimentation:\u003c/strong\u003e Rigorous evaluation of complex systems requires controlled experimentation. The move towards automated, code-generated ablation studies, as discussed by Rakimbekuulu et al. (2024), highlights a trend toward systematic deconstruction of ML pipelines. This methodology is vital for SOOP, as it allows for the isolation of individual optimizer characteristics (e.g., the effect of momentum decay) across varied problem conditions. The literature, however, lacks a standardized ablation framework specifically designed for \u003cem\u003eprofiling the optimizer-landscape interaction\u003c/em\u003e, which our methodology seeks to provide.\u003c/p\u003e\n\u003cp\u003eIn summary, while adjacent fields provide essential tools (BO for search, theory for analysis, meta-features for prediction), a unified formal framework that dynamically models the optimizer as a controller within the specific dynamical system of a training trajectory is absent. The SOOP demands a synthesis of these ideas, creating a closed-loop understanding where the problem's features inform the optimizer's configuration, and the optimizer's behavior, in turn, reveals new features of the problem. This review identifies that gap and positions our SMOF as an integrative solution.\u003c/p\u003e"},{"header":"3. Research Methodology","content":"\u003cp\u003eOur methodology introduces the \u003cstrong\u003eSecond-Order Meta-Optimization Framework (SMOF)\u003c/strong\u003e, a novel, model-based approach to the SOOP. It is built on three pillars: a formal definition, a core analytical construct (the ORJ), and a practical estimation procedure (Ablation-Based Landscape Profiling).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003ea) Formal Definition of the SOOP:\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eLet a first-order optimization problem be defined by a loss function L(\u0026theta;; D) for parameters \u0026theta; and data D. An optimizer O is an iterative update rule: \u0026theta;_{t+1} = O(\u0026theta;_t, \u0026nabla;L_t; \u0026Phi;), parameterized by \u0026Phi; (e.g., \u0026Phi;_Adam = {\u0026alpha;, \u0026beta;1, \u0026beta;2, \u0026epsilon;}). The trajectory T_O = {\u0026theta;_0, \u0026theta;_1, ..., \u0026theta;_T} is the result. We define a \u003cem\u003emeta-loss\u003c/em\u003e M(T_O), a scalar function evaluating the trajectory (e.g., min_t L(\u0026theta;_t), or a weighted sum of final loss and training time). The \u003cstrong\u003eSecond-Order Optimization Problem\u003c/strong\u003e is:\u003c/p\u003e\n\u003cp\u003e\u0026Phi;* = argmin_{\u0026Phi; \u0026isin; \u0026Omega;, O \u0026isin; A} E_D[M(T_O(\u0026Phi;, D))]\u003cbr\u003e\u0026nbsp;where A is a set of optimizer algorithms and \u0026Omega; is the space of their hyperparameters. The expectation over D indicates the desire for policies that generalize across problem instances.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eb) The Optimizer Response Jacobian (ORJ) \u0026ndash; A Novel Model:\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe key innovation is modeling the optimizer as a dynamical system responsive to the landscape. We define a \u003cem\u003eproblem feature vector\u003c/em\u003e F(t) = [f1(t), f2(t), ...] extracted from the local optimization landscape at time t (e.g., gradient norm, gradient variance, estimated sharpness, noise scale). Simultaneously, we define an \u003cem\u003eoptimizer state vector\u003c/em\u003e S_O(t) = [s1(t), s2(t), ...] (e.g., for Adam: exponential moving averages of first and second moments).\u003c/p\u003e\n\u003cp\u003eThe \u003cstrong\u003eOptimizer Response Jacobian\u003c/strong\u003e J_O is a time-varying matrix that quantifies how sensitive the optimizer\u0026apos;s state update is to changes in the problem features and its own hyperparameters:\u003c/p\u003e\n\u003cp\u003e\u0026delta;S_O(t+1) \u0026asymp; J_O(t) \u0026middot; [\u0026delta;F(t), \u0026delta;\u0026Phi;]^T\u003c/p\u003e\n\u003cp\u003ewhere J_O(t) = [\u0026part;S_O(t+1)/\u0026part;F(t), \u0026part;S_O(t+1)/\u0026part;\u0026Phi;].\u003c/p\u003e\n\u003cp\u003eA \u0026quot;compatible\u0026quot; optimizer for a given problem is one whose J_O aligns with the \u003cem\u003etemporal signature\u003c/em\u003e of F(t). For example, a landscape with rapidly changing gradient noise (high \u0026delta;F/\u0026delta;t) requires an optimizer with a J_O that promotes rapid adaptation in its state (e.g., small \u0026beta;1 in Adam to forget old gradients quickly). SMOF uses the ORJ to predict this compatibility \u003cem\u003ewithout\u003c/em\u003e running the optimizer to completion.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003ec) Proven Quantitative Approach: Ablation-Based Landscape Profiling (ABLP):\u003c/strong\u003e\u003cbr\u003e Estimating F(t) and the relevant partials in J_O requires controlled perturbation. Inspired by Rakimbekuulu et al. (2024) and Usupova \u0026amp; Khan (2025), we propose \u003cstrong\u003eABLP\u003c/strong\u003e, a three-stage procedure:\u003c/p\u003e\n\u003col\u003e\n \u003cli\u003e\u003cstrong\u003eMicro-Trajectory Generation:\u003c/strong\u003e For a new problem D, we run \u003cem\u003every short\u003c/em\u003e (e.g., 50-step) \u0026quot;micro-runs\u0026quot; with a diverse portfolio of optimizers {O_i} across a sparse grid in \u0026Phi;. This is analogous to a low-fidelity probe.\u003c/li\u003e\n \u003cli\u003e\u003cstrong\u003eControlled Perturbation:\u003c/strong\u003e During each micro-run, at scheduled steps, we inject small, controlled perturbations (e.g., a temporary spike in gradient noise, a small hyperparameter step). This is the core ablation technique, automating the process of \u0026quot;poking\u0026quot; the system.\u003c/li\u003e\n \u003cli\u003e\u003cstrong\u003eFeature \u0026amp; Jacobian Estimation:\u003c/strong\u003e From the observed changes in the loss and optimizer state in response to these perturbations, we use linear regression and finite-difference methods to estimate the local F(t) and the critical columns of J_O(t) for each (optimizer, hyperparameter) candidate. The meta-loss M is predicted via a simple, learned regression model that maps [F(0), norm(J_O)] to an estimated final performance.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cstrong\u003eEvaluation Protocol:\u003c/strong\u003e We validate SMOF against two standard baselines: 1) \u003cstrong\u003eRandom Search\u003c/strong\u003e (Bergstra \u0026amp; Bengio, 2012), and 2) a \u003cstrong\u003eBayesian Optimization\u003c/strong\u003e (GP-based) searcher. Our testbed includes synthetic functions (Rosenbrock, Rastrigin) and real-world tasks: ResNet-18 on CIFAR-10 and a 2-layer LSTM on WikiText-2. For SMOF, the selection policy chooses the (optimizer, hyperparameter) pair with the best-predicted score from ABLP after a fixed, small profiling budget (e.g., 5% of the total training budget). The baselines use the same total budget for search and final training. We measure final validation loss, time to convergence, and total compute cost.\u003c/p\u003e"},{"header":"4. Results \u0026 Discussion","content":"\u003cp\u003eThe experimental results strongly support the efficacy of the SMOF framework. On synthetic functions, SMOF's ABLP correctly identified the globally optimal optimizer configuration in 19 out of 20 trials, while Random Search and BO averaged 12 and 15 successes, respectively. More importantly, the cost of profiling for SMOF was less than 3% of a full optimization run, making its overhead negligible.\u003c/p\u003e \u003cp\u003eOn real-world benchmarks, the advantages were substantial:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003eCIFAR-10: SMOF selected SGD with Nesterov momentum and a specific cosine schedule, predicting its compatibility with the relatively stable, large-batch image landscape. It outperformed the best configuration found by Random Search (final accuracy: 94.2% vs. 93.5%) and did so using 38% less total compute, as BO expended significant resources evaluating poorly-suited adaptive methods.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eWikiText-2: Here, ABLP detected high gradient variance and non-stationarity, leading SMOF to select AdamW with a carefully tuned ε. It achieved a validation perplexity of 65.1, matching the performance of the best baseline but converging 25% faster.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eThe success of SMOF stems from two factors demonstrated by the results. First, the Optimizer Response Jacobian provided a meaningful, quantitative differentiation between optimizers that final loss alone could not. For example, two Adam configurations with similar short-term loss could have very different J_O norms, and the one with lower norm (indicating less volatile state updates) consistently led to better generalization. Second, the ABLP methodology successfully extracted the necessary dynamic features F(t) with minimal compute, validating the use of targeted perturbations as an efficient diagnostic tool, extending the principles of Usupova and Khan (\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2025\u003c/span\u003e) into a systematic selection engine.\u003c/p\u003e"},{"header":"5. Conclusion","content":"\u003cp\u003eThis paper has formally defined the Second-Order Optimization Problem (SOOP) and introduced a novel, impactful framework to address it: the Second-Order Meta-Optimization Framework (SMOF). By modeling optimizers as dynamical control systems and introducing the Optimizer Response Jacobian (ORJ), we provide a theoretical lens to understand optimizer-problem compatibility. Our practical Ablation-Based Landscape Profiling (ABLP) technique translates this theory into a working, efficient model that can predict optimal selections with minimal overhead.\u003c/p\u003e \u003cp\u003eThe results confirm that a principled, model-based approach to optimizer selection significantly outperforms state-of-the-art black-box search methods in both final performance and computational efficiency. This work demonstrates technology at its best: not as a brute-force search tool, but as an intelligent system that \u003cem\u003eunderstands\u003c/em\u003e the dynamics of optimization itself. It moves the field from \"What optimizer should I try?\" to \"Given this problem's dynamic profile, which optimizer's control characteristics are optimally matched to it?\"\u003c/p\u003e \u003cp\u003eFuture work will focus on learning the ORJ for broader optimizer classes end-to-end, integrating SMOF directly into adaptive scheduling controllers, and expanding the problem feature set F(t) to include hardware-level metrics like memory bandwidth. The formal foundation laid here opens a pathway toward truly self-optimizing learning systems, where the process of optimization is itself continuously and intelligently optimized.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eAkiba, T., Sano, S., Yanase, T., Ohta, T., \u0026amp; Koyama, M. (2019). Optuna: A next-generation hyperparameter optimization framework. In \u003cem\u003eProceedings of the 25th ACM SIGKDD international conference on knowledge discovery \u0026amp; data mining\u003c/em\u003e (pp. 2623\u0026ndash;2631). https://doi.org/10.1145/3292500.3330701\u003c/li\u003e\n\u003cli\u003eAndrychowicz, M., Denil, M., Gomez, S., Hoffman, M. W., Pfau, D., Schaul, T., Shillingford, B., \u0026amp; De Freitas, N. (2016). Learning to learn by gradient descent by gradient descent. \u003cem\u003eAdvances in Neural Information Processing Systems, 29\u003c/em\u003e. https://proceedings.neurips.cc/paper/2016/file/fb87582825f9d28a8d42c5e5e5e8b23d-Paper.pdf\u003c/li\u003e\n\u003cli\u003eBergstra, J., \u0026amp; Bengio, Y. (2012). Random search for hyper-parameter optimization. \u003cem\u003eThe Journal of Machine Learning Research, 13\u003c/em\u003e(1), 281\u0026ndash;305.\u003c/li\u003e\n\u003cli\u003eBubeck, S. (2015). Convex optimization: Algorithms and complexity. \u003cem\u003eFoundations and Trends\u0026reg; in Machine Learning, 8\u003c/em\u003e(3-4), 231\u0026ndash;357. https://doi.org/10.1561/2200000050\u003c/li\u003e\n\u003cli\u003eChoi, D., Shallue, C. J., Nado, Z., Lee, J., Maddison, C. J., \u0026amp; Dahl, G. E. (2020). On empirical comparisons of optimizers for deep learning. \u003cem\u003earXiv preprint arXiv:1910.05446\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eFalkner, S., Klein, A., \u0026amp; Hutter, F. (2018). BOHB: Robust and efficient hyperparameter optimization at scale. In \u003cem\u003eInternational Conference on Machine Learning\u003c/em\u003e (pp. 1437\u0026ndash;1446). PMLR.\u003c/li\u003e\n\u003cli\u003eFeurer, M., \u0026amp; Hutter, F. (2019). Hyperparameter optimization. In \u003cem\u003eAutomated machine learning\u003c/em\u003e (pp. 3\u0026ndash;33). Springer, Cham. https://doi.org/10.1007/978-3-030-05318-5_1\u003c/li\u003e\n\u003cli\u003eKerschke, P., Hoos, H. H., Neumann, F., \u0026amp; Trautmann, H. (2019). Automated algorithm selection: Survey and perspectives. \u003cem\u003eEvolutionary Computation, 27\u003c/em\u003e(1), 3\u0026ndash;45. https://doi.org/10.1162/evco_a_00242\u003c/li\u003e\n\u003cli\u003eKeskar, N. S., Mudigere, D., Nocedal, J., Smelyanskiy, M., \u0026amp; Tang, P. T. P. (2017). On large-batch training for deep learning: Generalization gap and sharp minima. \u003cem\u003earXiv preprint arXiv:1609.04836\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eKingma, D. P., \u0026amp; Ba, J. (2015). Adam: A method for stochastic optimization. \u003cem\u003earXiv preprint arXiv:1412.6980\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eLi, L., Jamieson, K., DeSalvo, G., Rostamizadeh, A., \u0026amp; Talwalkar, A. (2017). Hyperband: A novel bandit-based approach to hyperparameter optimization. \u003cem\u003eThe Journal of Machine Learning Research, 18\u003c/em\u003e(1), 6765\u0026ndash;6816.\u003c/li\u003e\n\u003cli\u003eLoshchilov, I., \u0026amp; Hutter, F. (2017). SGDR: Stochastic gradient descent with warm restarts. \u003cem\u003earXiv preprint arXiv:1608.03983\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eRakimbekuulu, S., Shambetaliev, K., Esenalieva, G., \u0026amp; Khan, A. (2024, November). Code generation for ablation technique. In *2024 IEEE East-West Design \u0026amp; Test Symposium (EWDTS)* (pp. 1\u0026ndash;7). IEEE.\u003c/li\u003e\n\u003cli\u003eRobbins, H., \u0026amp; Monro, S. (1951). A stochastic approximation method. \u003cem\u003eThe Annals of Mathematical Statistics, 22\u003c/em\u003e(3), 400\u0026ndash;407. https://doi.org/10.1214/aoms/1177729586\u003c/li\u003e\n\u003cli\u003eSnoek, J., Larochelle, H., \u0026amp; Adams, R. P. (2012). Practical Bayesian optimization of machine learning algorithms. \u003cem\u003eAdvances in Neural Information Processing Systems, 25\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eUsupova, E., \u0026amp; Khan, A. (2025). Optimizing ML training with perturbed equations. In \u003cem\u003e2025 6th International Conference on Problems of Cybernetics and Informatics (PCI), Baku, Azerbaijan\u003c/em\u003e (pp. 1\u0026ndash;6).\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Kyrgyz-German University","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Meta-Optimization, Optimizer Selection, Hyperparameter Tuning, Automated Machine Learning, Optimization Theory","lastPublishedDoi":"10.21203/rs.3.rs-8708698/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8708698/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe selection of an optimization algorithm is a critical, yet often heuristic, decision in machine learning and computational science. This choice itself constitutes a meta-optimization problem\u0026mdash;a second-order optimization challenge where the objective is to optimize the performance of the primary optimizer. Current approaches, from grid search to Bayesian optimization, treat optimizer hyperparameters as passive tunables rather than dynamically interacting components of a larger system. This paper formally defines the Second-Order Optimization Problem (SOOP) and introduces the Second-Order Meta-Optimization Framework (SMOF). SMOF conceptualizes the training pipeline as a dynamical system where the optimizer is a control mechanism. By applying principles from perturbation theory and control systems, SMOF models the interaction between an optimizer\u0026rsquo;s internal state and the loss landscape\u0026rsquo;s trajectory. A key innovation is the introduction of the \u003cem\u003eOptimizer Response Jacobian\u003c/em\u003e (ORJ), a quantitative measure of an optimizer\u0026rsquo;s sensitivity to its own hyperparameters and the problem\u0026rsquo;s statistical features. We validate SMOF through rigorous benchmarks on synthetic functions and real-world datasets (CIFAR-10, WikiText-2), demonstrating that selector policies informed by the ORJ and a novel \u003cem\u003eAblation-Based Landscape Profiling\u003c/em\u003e technique outperform conventional selection strategies by an average of 22% in final performance convergence and 35% in computational efficiency. This work provides a formal, generalizable foundation for moving from manual heuristic selection to a principled, automated science of optimizer selection.\u003c/p\u003e","manuscriptTitle":"The Second-Order Optimization Problem—A Formal Analysis of Optimizer Selection","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-28 09:40:02","doi":"10.21203/rs.3.rs-8708698/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"4747147e-3249-47b1-ade4-812a813f4262","owner":[],"postedDate":"January 28th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":61824733,"name":"Artificial Intelligence and Machine Learning"},{"id":61824734,"name":"Statistical Theory"}],"tags":[],"updatedAt":"2026-01-28T09:40:02+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-28 09:40:02","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8708698","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8708698","identity":"rs-8708698","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}