This paper presents ``randomized SINDy", a sequential machine learning algorithm designed for dynamic data that has a time-dependent structure. It employs a probabilistic approach, with its PAC learning property rigorously proven through the mathematical theory of functional analysis. The algorithm dynamically predicts using a learned probability distribution of predictors, updating weights via gradient descent and a proximal algorithm to maintain a valid probability density. Inspired by SINDy (Brunton et al. 2016), it incorporates feature augmentation and Tikhonov regularization. For multivariate normal weights, the proximal step is omitted to focus on parameter estimation. The algorithm's effectiveness is demonstrated through experimental results in regression and binary classification using real-world data.

In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic) gradient descent, and related topics into dynamical statements. We also tackle three concrete challenges. First, we consider the process of information propagation through a neural network, i.e., we study the input-output map for different architectures. We explain the universal embedding property for augmented neural ODEs representing arbitrary functions of given regularity, the classification of multilayer perceptrons and neural ODEs in terms of suitable function classes, and the memory-dependence in neural delay equations. Second, we consider the training aspect of neural networks dynamically. We describe a dynamical systems perspective on gradient descent and study stability for overdetermined problems. We then extend this analysis to the overparameterized setting and describe the edge of stability phenomenon, also in the context of possible explanations for implicit bias. For stochastic gradient descent, we present stability results for the overparameterized setting via Lyapunov exponents of interpolation solutions. Third, we explain several results regarding mean-field limits of neural networks. We describe a result that extends existing techniques to heterogeneous neural networks involving graph limits via digraph measures. This shows how large classes of neural networks naturally fall within the framework of Kuramoto-type models on graphs and their large-graph limits. Finally, we point out that similar strategies to use dynamics to study explainable and reliable AI can also be applied to settings such as generative models or fundamental issues in gradient training methods, such as backpropagation or vanishing/exploding gradients.

Stochastic gradient-based descent (SGD), have long been central to training large language models (LLMs). However, their effectiveness is increasingly being questioned, particularly in large-scale applications where empirical evidence suggests potential performance limitations. In response, this paper proposes a stochastic conjugate subgradient method together with adaptive sampling tailored specifically for training LLMs. The method not only achieves faster convergence per iteration but also demonstrates improved scalability compared to traditional SGD techniques. It leverages sample complexity analysis to adaptively choose the sample size, employs a stochastic conjugate subgradient approach to determine search directions and utilizing an AdamW-like algorithm to adaptively adjust step sizes. This approach preserves the key advantages of first-order methods while effectively addressing the nonconvexity and non-smoothness inherent in LLMs training. Additionally, we provide a detailed analysis of the advantage of the algorithm. Experimental results show that the proposed method not only maintains, but in many cases surpasses, the scalability of traditional SGD techniques, significantly enhancing both the speed and accuracy of the optimization process.

The inevitable presence of data heterogeneity has made federated learning very challenging. There are numerous methods to deal with this issue, such as local regularization, better model fusion techniques, and data sharing. Though effective, they lack a deep understanding of how data heterogeneity can affect the global decision boundary. In this paper, we bridge this gap by performing an experimental analysis of the learned decision boundary using a toy example. Our observations are surprising: (1) we find that the existing methods suffer from forgetting and clients forget the global decision boundary and only learn the perfect local one, and (2) this happens regardless of the initial weights, and clients forget the global decision boundary even starting from pre-trained optimal weights. In this paper, we present FedProj, a federated learning framework that robustly learns the global decision boundary and avoids its forgetting during local training. To achieve better ensemble knowledge fusion, we design a novel server-side ensemble knowledge transfer loss to further calibrate the learned global decision boundary. To alleviate the issue of learned global decision boundary forgetting, we further propose leveraging an episodic memory of average ensemble logits on a public unlabeled dataset to regulate the gradient updates at each step of local training. Experimental results demonstrate that FedProj outperforms state-of-the-art methods by a large margin.

This work assesses both empirically and theoretically, using the performance estimation methodology, how robust different first-order optimization methods are when subject to relative inexactness in their gradient computations. Relative inexactness occurs, for example, when compressing the gradient using fewer bits of information, which happens when dealing with large-scale problems on GPUs. Three major families of methods are analyzed: constant step gradient descent, long-step methods, and accelerated methods. The latter two are first shown to be theoretically not robust to inexactness. Then, a semi-heuristic shortening factor is introduced to improve their theoretical guarantees. All methods are subsequently tested on a concrete inexact problem, with two different types of relative inexactness, and it is observed that both accelerated methods are much more robust than expected, and that the shortening factor significantly helps the long-step methods. In the end, all shortened methods appear to be promising, even in this inexact setting.

Mean-field characterizations of first-order iterative algorithms -- including Approximate Message Passing (AMP), stochastic and proximal gradient descent, and Langevin diffusions -- have enabled a precise understanding of learning dynamics in many statistical applications. For algorithms whose non-linearities have a coordinate-separable form, it is known that such characterizations enjoy a degree of universality with respect to the underlying data distribution. However, mean-field characterizations of non-separable algorithm dynamics have largely remained restricted to i.i.d. Gaussian or rotationally-invariant data. In this work, we initiate a study of universality for non-separable AMP algorithms. We identify a general condition for AMP with polynomial non-linearities, in terms of a Bounded Composition Property (BCP) for their representing tensors, to admit a state evolution that holds universally for matrices with non-Gaussian entries. We then formalize a condition of BCP-approximability for Lipschitz AMP algorithms to enjoy a similar universal guarantee. We demonstrate that many common classes of non-separable non-linearities are BCP-approximable, including local denoisers, spectral denoisers for generic signals, and compositions of separable functions with generic linear maps, implying the universality of state evolution for AMP algorithms employing these non-linearities.

In this paper, we revisit stochastic gradient descent (SGD) with AdaGrad-type preconditioning. Our contributions are twofold. First, we develop a unified convergence analysis of SGD with adaptive preconditioning under anisotropic or matrix smoothness and noise assumptions. This allows us to recover state-of-the-art convergence results for several popular adaptive gradient methods, including AdaGrad-Norm, AdaGrad, and ASGO/One-sided Shampoo. In addition, we establish the fundamental connection between two recently proposed algorithms, Scion and DASGO, and provide the first theoretical guarantees for the latter. Second, we show that the convergence of methods like AdaGrad and DASGO can be provably accelerated beyond the best-known rates using Nesterov momentum. Consequently, we obtain the first theoretical justification that AdaGrad-type algorithms can simultaneously benefit from both diagonal preconditioning and momentum, which may provide an ultimate explanation for the practical efficiency of Adam.

We provide evidence that orthogonalizing gradients during training improves model calibration without sacrificing accuracy. On CIFAR-10 with 10\% labeled data, $\perp$Grad matches SGD in accuracy but yields consistently improved calibration metrics such as lower test loss, reduced softmax overconfidence, and higher predictive entropy. These benefits persist under input corruption (CIFAR-10C) and extended training, where $\perp$Grad models degrade more gracefully than SGD-trained counterparts. $\perp$Grad is optimizer-agnostic, incurs minimal overhead, and works well with post-hoc calibration techniques like temperature scaling. Theoretically, we prove convergence of a simplified version of $\perp$Grad under mild assumptions and characterize its stationary points in positive homogeneous networks: $\perp$Grad converges to solutions where further loss reduction requires confidence scaling rather than decision boundary improvement. Code for this paper can be found at: https://github.com/evanshedges2/orthograd\_improves\_calibration.

We consider centralized distributed optimization in the classical federated learning setup, where $n$ workers jointly find an $\varepsilon$-stationary point of an $L$-smooth, $d$-dimensional nonconvex function $f$, having access only to unbiased stochastic gradients with variance $σ^2$. Each worker requires at most $h$ seconds to compute a stochastic gradient, and the communication times from the server to the workers and from the workers to the server are $τ_{s}$ and $τ_{w}$ seconds per coordinate, respectively. One of the main motivations for distributed optimization is to achieve scalability with respect to $n$. For instance, it is well known that the distributed version of SGD has a variance-dependent runtime term $\frac{h σ^2 L Δ}{n \varepsilon^2},$ which improves with the number of workers $n,$ where $Δ= f(x^0) - f^*,$ and $x^0 \in R^d$ is the starting point. Similarly, using unbiased sparsification compressors, it is possible to reduce both the variance-dependent runtime term and the communication runtime term. However, once we account for the communication from the server to the workers $τ_{s}$, we prove that it becomes infeasible to design a method using unbiased random sparsification compressors that scales both the server-side communication runtime term $τ_{s} d \frac{L Δ}{\varepsilon}$ and the variance-dependent runtime term $\frac{h σ^2 L Δ}{\varepsilon^2},$ better than poly-logarithmically in $n$, even in the homogeneous (i.i.d.) case, where all workers access the same distribution. To establish this result, we construct a new "worst-case" function and develop a new lower bound framework that reduces the analysis to the concentration of a random sum, for which we prove a concentration bound. These results reveal fundamental limitations in scaling distributed optimization, even under the homogeneous assumption.

There are two primary ways of incorporating new information into a language model (LM): changing its prompt or changing its parameters, e.g. via fine-tuning. Parameter updates incur no long-term storage cost for model changes. However, for many model updates, prompting is significantly more effective: prompted models can generalize robustly from single examples and draw logical inferences that do not occur under standard fine-tuning. Can models be modified so that fine-tuning does emulate prompting? This paper describes a method for meta-training LMs such that gradient updates emulate the effects of conditioning on new information. Our approach uses tools from gradient-based meta-learning but uses an LM's own prompted predictions as targets, eliminating the need for ground-truth labels. Subsequent gradient descent training recovers some (and occasionally all) of prompted model performance -- showing improvement on the ``reversal curse'' tasks, and answering questions about text passages after a single gradient update. These results suggest that, with appropriate initialization, gradient descent can be surprisingly expressive. Our results suggest new avenues for long-context modeling and offer insight into the generalization capabilities of gradient-based learning.