In multi-source learning with discrete labels, distributional heterogeneity across domains poses a central challenge to developing predictive models that transfer reliably to unseen domains. We study multi-source unsupervised domain adaptation, where labeled data are drawn from multiple source domains and only unlabeled data from a target domain. To address potential distribution shifts, we propose a novel Conditional Group Distributionally Robust Optimization (CG-DRO) framework that learns a classifier by minimizing the worst-case cross-entropy loss over the convex combinations of the conditional outcome distributions from the sources. To solve the resulting minimax problem, we develop an efficient Mirror Prox algorithm, where we employ a double machine learning procedure to estimate the risk function. This ensures that the errors of the machine learning estimators for the nuisance models enter only at higher-order rates, thereby preserving statistical efficiency under covariate shift. We establish fast statistical convergence rates for the estimator by constructing two surrogate minimax optimization problems that serve as theoretical bridges. A distinguishing challenge for CG-DRO is the emergence of nonstandard asymptotics: the empirical estimator may fail to converge to a standard limiting distribution due to boundary effects and system instability. To address this, we introduce a perturbation-based inference procedure that enables uniformly valid inference, including confidence interval construction and hypothesis testing.

Solutions of optimization problems, including policy optimization in reinforcement learning, typically rely upon some variant of gradient descent. There has been much recent work in the machine learning, control, and optimization communities applying the Polyak-Łojasiewicz Inequality (PLI) to such problems in order to establish an exponential rate of convergence (a.k.a. ``linear convergence'' in the local-iteration language of numerical analysis) of loss functions to their minima under the gradient flow. Often, as is the case of policy iteration for the continuous-time LQR problem, this rate vanishes for large initial conditions, resulting in a mixed globally linear / locally exponential behavior. This is in sharp contrast with the discrete-time LQR problem, where there is global exponential convergence. That gap between CT and DT behaviors motivates the search for various generalized PLI-like conditions, and this talk will address that topic. Moreover, these generalizations are key to understanding the transient and asymptotic effects of errors in the estimation of the gradient, errors which might arise from adversarial attacks, wrong evaluation by an oracle, early stopping of a simulation, inaccurate and very approximate digital twins, stochastic computations (algorithm ``reproducibility''), or learning by sampling from limited data. We describe an ``input to state stability'' (ISS) analysis of this issue. The second part discusses convergence and PLI-like properties of ``linear feedforward neural networks'' in feedback control. Much of the work described here was done in collaboration with Arthur Castello B. de Oliveira, Leilei Cui, Zhong-Ping Jiang, and Milad Siami.

We study online optimization methods for zero-sum games, a fundamental problem in adversarial learning in machine learning, economics, and many other domains. Traditional methods approximate Nash equilibria (NE) using either regret-based methods (time-average convergence) or contraction-map-based methods (last-iterate convergence). We propose a new method based on Hamiltonian dynamics in physics and prove that it can characterize the set of NE in a finite (linear) number of iterations of alternating gradient descent in the unbounded setting, modulo degeneracy, a first in online optimization. Unlike standard methods for computing NE, our proposed approach can be parallelized and works with arbitrary learning rates, both firsts in algorithmic game theory. Experimentally, we support our results by showing our approach drastically outperforms standard methods.

Training deep neural networks for computer vision is inherently challenging due to issues like unstable gradients, local minima, and pervasive noisy data [48]. These challenges are magnified in anomalous environments where data distributions deviate from the norm, critically impairing the optimization process. Such instability hinders the model's ability to learn robust representations and significantly affects its generalization to unseen data. The choice of optimizer is central to alleviating these issues, as it governs both convergence speed and stability during training. Over the decades, various optimizers have been proposed to tackle different facets of this optimization challenge. Early work on Stochastic Gradient Descent (SGD) [30, 33] laid the foundation for iterative gradient-based methods by employing a simple yet effective parameter update scheme. AdaGrad [4] introduced per-parameter learning rate adjustments to better handle sparse gradients, while Adam [13] fused momentum-based updates with adaptive learning rates, accelerating convergence. Subsequently, Adam variants such as AdamW [23], AdaBelief [52], and Adan [21] have sought to address limitations in Adam's adaptive mechanism and enhance robustness in complex, non-convex landscapes.

Gradient descent (GD) on logistic regression has many fascinating properties. When the dataset is linearly separable, it is known that the iterates converge in direction to the maximum-margin separator regardless of how large the step size is. In the non-separable case, however, it has been shown that GD can exhibit a cycling behaviour even when the step sizes is still below the stability threshold $2/λ$, where $λ$ is the largest eigenvalue of the Hessian at the solution. This short paper explores whether restricting the data to have equal magnitude is a sufficient condition for global convergence, under any step size below the stability threshold. We prove that this is true in a one dimensional space, but in higher dimensions cycling behaviour can still occur. We hope to inspire further studies on quantifying how common these cycles are in realistic datasets, as well as finding sufficient conditions to guarantee global convergence with large step sizes.

--Federated learning is a machine learning approach that enables multiple devices (i.e., agents) to train a shared model cooperatively without exchanging raw data. This technique keeps data localized on user devices, ensuring privacy and security, while each agent trains the model on their own data and only shares model updates. The communication overhead is a significant challenge due to the frequent exchange of model updates between the agents and the central server . In this paper, we propose a communication-efficient federated learning scheme that utilizes low-rank approximation of neural network gradients and quantization to significantly reduce the network load of the decentralized learning process with minimal impact on the model's accuracy. I. INTRODUCTION As artificial intelligence and machine learning evolve, new computational paradigms are emerging to address the increasing demand for privacy, efficiency, and scalability.

Achieving fast and stable off-policy learning in deep reinforcement learning (RL) is challenging. Most existing methods rely on semi-gradient temporal-difference (TD) methods for their simplicity and efficiency, but are consequently susceptible to divergence. While more principled approaches like Gradient TD (GTD) methods have strong convergence guarantees, they have rarely been used in deep RL. Recent work introduced the Generalized Projected Bellman Error ($\GPBE$), enabling GTD methods to work efficiently with nonlinear function approximation. However, this work is only limited to one-step methods, which are slow at credit assignment and require a large number of samples. In this paper, we extend the $\GPBE$ objective to support multistep credit assignment based on the $λ$-return and derive three gradient-based methods that optimize this new objective. We provide both a forward-view formulation compatible with experience replay and a backward-view formulation compatible with streaming algorithms. Finally, we evaluate the proposed algorithms and show that they outperform both PPO and StreamQ in MuJoCo and MinAtar environments, respectively. Code available at https://github.com/esraaelelimy/gtd\_algos

Catastrophic forgetting is the primary challenge that hinders continual learning, which refers to a neural network ability to sequentially learn multiple tasks while retaining previously acquired knowledge. Elastic Weight Consolidation, a regularization-based approach inspired by synaptic consolidation in biological neural systems, has been used to overcome this problem. In this study prior research is replicated and extended by evaluating EWC in supervised learning settings using the PermutedMNIST and RotatedMNIST benchmarks. Through systematic comparisons with L2 regularization and stochastic gradient descent (SGD) without regularization, we analyze how different approaches balance knowledge retention and adaptability. Our results confirm what was shown in previous research, showing that EWC significantly reduces forgetting compared to naive training while slightly compromising learning efficiency on new tasks. Moreover, we investigate the impact of dropout regularization and varying hyperparameters, offering insights into the generalization of EWC across diverse learning scenarios. These results underscore EWC's potential as a viable solution for lifelong learning in neural networks.

Despite its wide range of applications across various domains, the optimization foundations of deep matrix factorization (DMF) remain largely open. In this work, we aim to fill this gap by conducting a comprehensive study of the loss landscape of the regularized DMF problem. Toward this goal, we first provide a closed-form characterization of all critical points of the problem. Building on this, we establish precise conditions under which a critical point is a local minimizer, a global minimizer, a strict saddle point, or a non-strict saddle point. Leveraging these results, we derive a necessary and sufficient condition under which every critical point is either a local minimizer or a strict saddle point. This provides insights into why gradient-based methods almost always converge to a local minimizer of the regularized DMF problem. Finally, we conduct numerical experiments to visualize its loss landscape to support our theory.

In this work, we analyze the optimization behaviour of common private learning optimization algorithms under heavy-tail class imbalanced distribution. We show that, in a stylized model, optimizing with Gradient Descent with differential privacy (DP-GD) suffers when learning low-frequency classes, whereas optimization algorithms that estimate second-order information do not. In particular, DP-AdamBC that removes the DP bias from estimating loss curvature is a crucial component to avoid the ill-condition caused by heavy-tail class imbalance, and empirically fits the data better with $\approx8\%$ and $\approx5\%$ increase in training accuracy when learning the least frequent classes on both controlled experiments and real data respectively.