In modern optimization methods used in deep learning, each update depends on the history of previous iterations, often referred to as memory, and this dependence decays fast as the iterates go further into the past. For example, gradient descent with momentum has exponentially decaying memory through exponentially averaged past gradients. We introduce a general technique for identifying a memoryless algorithm that approximates an optimization algorithm with memory. It is obtained by replacing all past iterates in the update by the current one, and then adding a correction term arising from memory (also a function of the current iterate). This correction term can be interpreted as a perturbation of the loss, and the nature of this perturbation can inform how memory implicitly (anti-)regularizes the optimization dynamics. As an application of our theory, we find that Lion does not have the kind of implicit anti-regularization induced by memory that AdamW does, providing a theory-based explanation for Lion's better generalization performance recently documented. Empirical evaluations confirm our theoretical findings.

Tree Search (TS) is crucial to some of the most influential successes in reinforcement learning. Here, we tackle two major challenges with TS that limit its usability: distribution shift and scalability. We first discover and analyze a counter-intuitive phenomenon: action selection through TS and a pre-trained value function often leads to lower performance compared to the original pre-trained agent, even when having access to the exact state and reward in future steps. We show this is due to a distribution shift to areas where value estimates are highly inaccurate and analyze this effect using Extreme Value theory. To overcome this problem, we introduce a novel off-policy correction term that accounts for the mismatch between the pre-trained value and its corresponding TS policy by penalizing under-sampled trajectories.

Throughout this paper, all convexity-related concepts are defined in the Euclidean space.

Neural Information Processing Systems http://nips.cc/

Prediction sets provide a means of quantifying the uncertainty in predictive tasks. Using held out calibration data, conformal prediction and risk control can produce prediction sets that exhibit statistically valid error control in a computationally efficient manner. However, in the standard formulations, the error is only controlled on average over many possible calibration datasets of fixed size. In this paper, we extend the control to remain valid with high probability over a cumulatively growing calibration dataset at any time point. We derive such guarantees using quantile-based arguments and illustrate the applicability of the proposed framework to settings involving distribution shift. We further establish a matching lower bound and show that our guarantees are asymptotically tight. Finally, we demonstrate the practical performance of our methods through both simulations and real-world numerical examples.

The complexity of a neural network can be reduced by promoting sparse interconnection structures.

Despite significant work on low-bit quantization-aware training (QAT), there is still an accuracy gap between such techniques and native training. To address this, we introduce CAGE (Curvature-Aware Gradient Estimation), a new QAT method that augments the straight-through estimator (STE) gradient with a curvature-aware correction designed to counteract the loss increase induced by quantization. CAGE is derived from a multi-objective view of QAT that balances loss minimization with the quantization constraints, yielding a principled correction term that depends on local curvature information. On the theoretical side, we introduce the notion of Pareto-optimal solutions for quantized optimization, and establish that CAGE yields strong convergence guarantees in the smooth non-convex setting. In terms of implementation, our approach is optimizer-agnostic, but we provide a highly-efficient implementation that leverages Adam statistics. CAGE significantly improves upon the prior state-of-the-art methods in terms of accuracy, for similar computational cost: for QAT fine-tuning, it halves the compression accuracy loss relative to the prior best method, while for QAT pre-training of Llama models, its accuracy for 3-bit weights-and-activations (W3A3) matches the accuracy achieved at 4-bits (W4A4) with the prior best method. The official implementation can be found over https://github.com/IST-DASLab/CAGE .

Federated optimization is a constrained form of distributed optimization that enables training a global model without directly sharing client data. Although existing algorithms can guarantee convergence in theory and often achieve stable training in practice, the reasons behind performance degradation under data heterogeneity remain unclear. To address this gap, the main contribution of this paper is to provide a theoretical perspective that explains why such degradation occurs. We introduce the assumption that heterogeneous client data lead to distinct local optima, and show that this assumption implies two key consequences: 1) the distance among clients' local optima raises the lower bound of the global objective, making perfect fitting of all client data impossible; and 2) in the final training stage, the global model oscillates within a region instead of converging to a single optimum, limiting its ability to fully fit the data. These results provide a principled explanation for performance degradation in non-iid settings, which we further validate through experiments across multiple tasks and neural network architectures. The framework used in this paper is open-sourced at: https://github.com/NPCLEI/fedtorch.

Throughout this paper, all convexity-related concepts are defined in the Euclidean space.

Abstract-- Accurate estimation of the relative attitude and angular velocity between two rigid bodies is fundamental in aerospace applications such as spacecraft rendezvous and docking. In these scenarios, a chaser vehicle must determine the orientation and angular velocity of a target object using onboard sensors. This work addresses the challenge of designing an Equivariant Filter (EqF) that can reliably estimate both the relative attitude and the target's angular velocity using noisy observations of two known, non-collinear vectors fixed in the target frame. T o derive the EqF, a symmetry for the system is proposed and an equivariant lift onto the symmetry group is calculated. Observability and convergence properties are analyzed. Simulations demonstrate the filter's performance, with Monte Carlo runs yielding statistically significant results. The impact of low-rate measurements is also examined and a strategy to mitigate this effect is proposed. I. INTRODUCTION In the past decade, there has been a growing interest in the development and validation of On-Orbit Servicing (OOS) and Active Debris Removal (ADR) technologies [1], [2], [3], driven by the ever-increasing number of satellites deployed each year. In OOS and ADR missions, the chaser or servicer spacecraft will need to approach the target spacecraft and synchronize its motion before performing the planned operations. As such, the chaser must estimate the relative attitude and angular velocity of the target, as well as the relative position and linear velocity.