Open research questions in Stochastic Gradient Optimization Techniques

What the literature leaves open

• We give a near-linear-time algorithm which requires (cid:101)𝑂(𝑑) samples under mild conditions (𝜀𝜅 ≲ 1), with error 𝑂(𝜎 𝜀𝜅) for Gaussian distributions, improving upon prior works, and answering an open problem by [JLST21]– improved error guarantees for fast algorithms do not require an increased sample complexity.

  On efficient robust regression with subquadratic samples · 2026

• 21 7 Conclusions and open problems This work confirms the widespread belief that a random orthonormal matrix is the optimal embedding for randomized matrix approximations.

  Sharp analysis of sketched least squares and randomized low-rank approximation · 2026

• We hope our work encourages further study of matrix-aware optimization beyond LLM pretraining alongside the safety practices already established for VLA and RLVR systems.

  Rethinking Muon Beyond Pretraining: Spectral Failures and High-Pass Remedies for VLA and RLVR · 2026

• We established directional late-phase implicit bias results for a class of mirror descent algorithms in homogeneous neural networks. Our analysis is based on a novel balance equation derived via the Fenchel-Young identity. In addition, we show that the hyperparameter λ > 0 plays a critical role in whether the implicit bias induced by mirror flow is realized in practice, as large values of λ can substantially slow convergence to the corresponding max-margin solution. Together, these results show how mirror geometry steers optimization toward distinct max-margin solutions and induces either sparse or dense feature learning depending on the chosen geometry. In particular, the L1 max-margin promotes sparsity by selecting a small subset of active parameters. In contrast, the Lp max-margin for p ≥ 2 encourages dense but homogeneous weight distributions, where weights concentrate around similar magnitudes. While the former aligns naturally with sparse feature selection, the latter may benefit quantization, as more uniform weight scales facilitate mapping parameters to discrete levels. Overall, our results suggest that mirror geometry provides a unified mechanism for steering models toward either sparse or quantization-friendly representations. Finally, this raises several questions beyond the scope of the present work, including extending Theorem 4.7 to the α < 2 regime (discussed in Appendix G.4), as well as understanding the effects of finite learning rate, early stopping, and the generalization properties of these max-margin solutions. 1Here λ is layerwise rescaled correcting for the width of each layer as detailed in Appendix J. 9 :HLJKWPDJQLWXGH)UHTXHQF\+\SHUEROLF*'p=106SDUVLW\9DOLGDWLRQ$FFXUDF\+\SHUEROLF*'p=10:HLJKWPDJQLWXGH)UHTXHQF\=0.1*'=1e5 Acknowledgments This work was supported in part by the DFG project 464109215 within the Priority Programme SPP 2298 “Theoretical Foundations of Deep Learning”. TJ has been supported by funding from the European Research Council (ERC) under the Horizon Europe Framework Programme (HORIZON) for proposal number 101116395 SPARSE-ML. GM has been supported in part by the DARPA AIQ grant HR00112520014, NSF grants DMS-2522495, DMS-2145630, CCF-2212520, and the BMFTR in DAAD project 57616814 (SECAI).

  Implicit Bias of Mirror Flow in Homogeneous Neural Networks: Sparse and Dense Feature Learning · 2026

• References We have presented a dynamic view of the phenomenon of memorization reported in the training of diffusion models, drawing upon some analysis of the related phe- nomenon of ‘collapse’ in machine learning. This also leads to some qualitative feel for the various factors af- fecting its occurrence and intensity, and throws up some interesting directions for further research. The work is grounded in some recent developments in stochastic ap- proximation with constant step size due to Azizian et al. [2024] and flags its importance in the analysis of machine learning algorithms. Going forward, one important problem is to estimate the error in the global minima of the Freidlin-Wentzell potential V for the constant stepsize stochastic approx- imation, and those of the function Γ′(·) := Γ(λ(·), ·) be- ing minimized. Γ′ would be the Freidlin-Wentzell poten- tial for the Langevin dynamics and ipso facto, an ap- proximation for the Freidlin-Wentzell potential for its discretization, viz., the Langevin algorithm. Γ′ has an explicit control theoretic expression given by Γ′(x) = inf

  Adynamical systems view of training generativemodels and the memorization phenomenon · 2026

• Further testing of the CHONKNORIS method on various PDE and inverse problem solution maps. Exploration of the application of FONKNORIS to other types of differential equations.

  Operator learning at machine precision · 2026 · DOI

• The lack of a reusable model that can learn operators to machine precision. Prior methods, such as Deep Operator Nets and Fourier Neural Operators, have limitations. The need for a method that can generalize to unseen equations without retraining.

  Operator learning at machine precision · 2026 · DOI

• The paper identifies the challenge of assuming constant model parameters over prolonged periods of time. It also notes the challenge of ensuring stability and contractivity in mean squared error for existing methods. The paper highlights the need to handle non-log-concave observation densities and non-Lipschitz gradients.

  Implicit score-driven filters for time-varying parameter models · 2026 · DOI

• Investigating the properties of the ISD filter for non-log-concave observation densities. Comparing the ISD filter with other filtering methods in various settings. Extending the ISD filter to more complex models and applications.

  Implicit score-driven filters for time-varying parameter models · 2026 · DOI

• However, these approaches typically require additional stabilization procedures whose numerical robustness and complexity are not fully understood in general.

  Spherical Harmonic Optimal Transport: Application to Climate Models Comparisons · 2026

• Approximating the input-output behavior of a multivariable black-box function from limited data is challenging when blind to the importance of its inputs and their interactions.

  A Weighted Kernel Method for Approximation that Adapts to Learned Multivariable Structure · 2026

• However, the generalization performance of asynchronous delayed SGD, which is an essential metric for assessing machine learning algorithms, has rarely been explored.

  Toward Understanding the Generalizability of Delayed Stochastic Gradient Descent · 2025 · DOI

• Bridging the gap between theory and practice in non-convex optimization. Developing interpretable machine learning models. Integrating analytical modelling with computational techniques to shape the next generation of intelligent systems.

  Mathematical Foundations of Machine Learning Optimizing Algorithms through Analytical Modelling · 2026 · DOI

• The gap between theoretical guarantees and practical strategies for algorithm design. The challenge of optimizing non-convex loss surfaces in deep learning.

  Mathematical Foundations of Machine Learning Optimizing Algorithms through Analytical Modelling · 2026 · DOI

• The paper identifies the need for understanding the mathematical foundations of AI and ML. The paper highlights the importance of linear algebra, calculus, probability theory, and optimization in AI and ML.

  The Impact of Mathematics in Artificial Intelligence and Machine Learning · 2026 · DOI

• The vanishing gradient phenomenon near local optima. The shrinkage behavior of gradient-based optimizers. The lack of independence of the reconstruction loss between adjacent iterations.

  A Theoretical and Experimental Exploration in Permutation Randomization on Nonsmooth Nonconvex Optimization · 2026 · DOI

• Applying the STAM algorithm to other practical problems. Exploring other applications of the STAM algorithm.

  A Stochastic Three-Block Alternating Minimization Algorithm and its Application to Quantized Deep Neural Networks · 2026 · DOI

• To compare the proposed method to other second-order methods. To evaluate the proposed method on other datasets and applications. To develop more efficient and scalable curvature-informed methods for neural network training.

  A Damped Hessian-Free Newton--Conjugate Gradient Method for Weighted Multiclass Neural Classification · 2026 · DOI

• There is a need for curvature-informed methods for weighted multiclass neural classification. First-order methods such as SGD, momentum-based variants, RMSProp, and Adam can be sensitive to ill-conditioning, local curvature variation, and unfavorable geometry of the objective.

  A Damped Hessian-Free Newton--Conjugate Gradient Method for Weighted Multiclass Neural Classification · 2026 · DOI

• The presence of noise in the function and gradient evaluations. The need for a novel reduction test to control the impact of noise. The requirement for the algorithm to be stable and effective in obtaining the approximate optimal value.

  A trust region gradient sampling method for noisy nonsmooth convex optimization · 2026 · DOI

• The lack of effective algorithms for nonsmooth unconstrained optimization problems in noisy environments. The need for a novel reduction test to control the impact of noise on the algorithm's performance.

  A trust region gradient sampling method for noisy nonsmooth convex optimization · 2026 · DOI

• The lack of a comparable theory for transformer architectures on low-dimensional manifolds. The need for a non-asymptotic approximation result with an intrinsic-dimensional rate and an ambient-dimension-independent multiplicative constant.

  Dimension-Independent Approximations on Low-Dimensional Manifolds Using Transformers · 2026 · DOI

• Experimental validation of the theoretical analysis. Application of the Law of Large Numbers to other stochastic algorithms. Analysis of the behavior of SGD in non-convex optimization problems.

  Stochastic Gradient Descent and the Law of Large Numbers: A Probabilistic Analysis of Convergence · 2026 · DOI

• The lack of a probabilistic analysis of the convergence of SGD. The need for a theoretical foundation for the effectiveness of SGD. The limited understanding of the behavior of SGD in large-scale machine learning problems.

  Stochastic Gradient Descent and the Law of Large Numbers: A Probabilistic Analysis of Convergence · 2026 · DOI

• To develop a taxonomy of distributional distances according to the computational primitives they induce. To understand when a particular alignment task is the right one for a given application.

  The tractability landscape of diffusion alignment: regularization, rewards, and computational primitives · 2026

Most-cited papers in Stochastic Gradient Optimization Techniques

• The Gaussian hare and the Laplacian tortoise: computability of squared-error versus absolute-error estimators · Statistical Science · 1997 · 403 citations
• Likelihood ratio gradient estimation for stochastic systems · Communications of the ACM · 1990 · 223 citations
• Federated Variance-Reduced Stochastic Gradient Descent With Robustness to Byzantine Attacks · IEEE Transactions on Signal Processing · 2020 · 195 citations
• A Novel Framework for the Analysis and Design of Heterogeneous Federated Learning · IEEE Transactions on Signal Processing · 2021 · 128 citations
• Convergence properties of the expected improvement algorithm with fixed mean and covariance functions · Journal of Statistical Planning and Inference · 2010 · 127 citations
• OR Forum—An Algorithmic Approach to Linear Regression · Operations Research · 2015 · 96 citations
• Logistic Regression: From Art to Science · Statistical Science · 2017 · 78 citations
• An Improved Convergence Analysis for Decentralized Online Stochastic Non-Convex Optimization · IEEE Transactions on Signal Processing · 2021 · 73 citations
• Variance-Reduced Decentralized Stochastic Optimization With Accelerated Convergence · IEEE Transactions on Signal Processing · 2020 · 72 citations
• Time complexity of iterative-deepening-A∗ · Artificial Intelligence · 2001 · 61 citations

Most recent work

• Neural Network Optimization Reimagined: Decoupled Techniques for Scratch and Fine-Tuning · IEEE Transactions on Pattern Analysis and Machine Intelligence · 2026
• What Happens in Successful Optimizations? A Survey of 2018–2024 Literature · Journal of Medicinal Chemistry · 2026
• High-Probability Minimax Lower Bounds · Statistical Science · 2026
• Parallel Collaborative ADMM Privacy Computing and Adaptive GPU Acceleration for Distributed Edge Networks · IEEE Transactions on Mobile Computing · 2026
• Communication-Efficient Stochastic Distributed Learning · IEEE Transactions on Automatic Control · 2026
• Structured First-Layer Initialization Pre-Training Techniques to Accelerate Training Process Based on $\varepsilon$-Rank · Communications in Computational Physics · 2026
• Exact worst-case convergence rates of gradient descent: a complete analysis for all constant stepsizes over nonconvex and convex functions · Mathematical Programming · 2026
• Optimal rates for estimating the covariance kernel from synchronously sampled functional data · Bernoulli · 2026
• On the randomized Euler scheme for stochastic differential equations with integral-form drift · Journal of Computational and Applied Mathematics · 2026
• Poisson noise removal via a non-convex log total variation regularization · Applied Mathematical Modelling · 2026

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