Why Linear Recurrent Memory Works in Partially Observable Reinforcement Learning

The family of linear recurrent neural networks has shown strong performance as recurrent memory units in partially observable reinforcement learning. We provide a theoretical justification for their empirical effectiveness by constructing and studying two linear filters: (i) the first exactly reproduces the pre-softmax logits of the belief vector in a hidden Markov model (HMM) under a deterministic transition matrix, thereby serving as a sufficient statistic for optimal policy learning, (ii) the second achieves vanishing state-decoding error under a nearly deterministic transition matrix, thus reducing state ambiguity to near zero. The results extend to action-controlled HMMs, where the corresponding linear filters become time-varying with action-dependent dynamics. We illustrate our main results through numerical experiments and further show that the constructed linear filter serves as a strong feature extractor in a small reinforcement learning game.

Commit to the Bit: Reactive Reinforcement Learning Done Right

Reinforcement learning algorithms are commonly analyzed (and designed) under the Markov assumption. This is unrealistic, as most environments encountered in practice are either partially observable, or require function approximation that restricts the agent to access non-Markovian state features. We consider the problem of learning an optimal reactive policy in a finite environment with deterministic observations (or equivalently, hard state aggregation). We introduce a new algorithm, Committed Q-learning, and prove almost-sure convergence to the optimal reactive policy under an intuitive assumption we call rewire-robustness. This assumption is strictly weaker than the $q_\star$-realizability condition used in prior work. Our algorithm is a variant of classical Q-learning in which the behavior policy commits to a single action upon entering a feature, and only resamples actions when the observed feature changes. A crucial part of our analysis is the introduction of quasi-Markov environments.

Adaptive Inverted-Index Routing for Granular Mixtures-of-Experts

Mixture-of-experts (MoE) models enable scalable transformer architectures by activating only a subset of experts per token. Recent evidence suggests that performance improves with increasingly granular experts, i.e., many small experts instead of a few large ones. However, this regime substantially increases routing cost, which can dominate computation. We introduce adaptive inverted-index routing for MoE (AIR-MoE), an inverted-index-inspired routing architecture based on vector quantization (VQ). In a first stage, AIR-MoE performs coarse shortlisting by assigning tokens to VQ codewords to construct a candidate set of experts. In a second stage, fine scoring computes exact routing scores restricted to this shortlist. This two-stage procedure approximates true top-k routing while avoiding full expert scoring and, in contrast to prior work, imposing no structural constraints on expert parameters. AIR-MoE serves as a drop-in replacement for standard routers and requires no modifications to the model architecture or loss function. We further provide a lower bound on the mass recall achieved by AIR-MoE that yields insights into its inner workings. Empirically, we demonstrate that AIR-MoE achieves improved performance compared to existing routing approaches in granular MoE settings.

Efficient Diffusion Models under Nonconvex Equality and Inequality constraints via Landing

Generative modeling within constrained sets is essential for scientific and engineering applications involving physical, geometric, or safety requirements (e.g., molecular generation, robotics). We present a unified framework for constrained diffusion models on generic nonconvex feasible sets $Σ$ that simultaneously enforces equality and inequality constraints throughout the diffusion process. Our framework incorporates both overdamped and underdamped dynamics for forward and backward sampling. A key algorithmic innovation is a computationally efficient landing mechanism that replaces costly and often ill-defined projections onto $Σ$, ensuring feasibility without iterative Newton solves or projection failures. By leveraging underdamped dynamics, we accelerate mixing toward the prior distribution, effectively alleviating the high simulation costs typically associated with constrained diffusion. Empirically, this approach reduces function evaluations and memory usage during both training and inference while preserving sample quality. On benchmarks featuring equality and mixed constraints, our method achieves comparable sample quality to state-of-the-art baselines while significantly reducing computational cost, providing a practical and scalable solution for diffusion on nonconvex feasible sets.

Zeroth-Order Optimization at the Edge of Stability

Zeroth-order (ZO) methods are widely used when gradients are unavailable or prohibitively expensive, including black-box learning and memory-efficient fine-tuning of large models, yet their optimization dynamics in deep learning remain underexplored. In this work, we provide an explicit step size condition that exactly captures the (mean-square) linear stability of a family of ZO methods based on the standard two-point estimator. Our characterization reveals a sharp contrast with first-order (FO) methods: whereas FO stability is governed solely by the largest Hessian eigenvalue, mean-square stability of ZO methods depends on the entire Hessian spectrum. Since computing the full Hessian spectrum is infeasible in practical neural network training, we further derive tractable stability bounds that depend only on the largest eigenvalue and the Hessian trace. Empirically, we find that full-batch ZO methods operate at the edge of stability: ZO-GD, ZO-GDM, and ZO-Adam consistently stabilize near the predicted stability boundary across a range of deep learning training problems. Our results highlight an implicit regularization effect specific to ZO methods, where large step sizes primarily regularize the Hessian trace, whereas in FO methods they regularize the top eigenvalue.

SALAAD: Sparse And Low-Rank Adaptation via ADMM

Modern large language models are increasingly deployed under compute and memory constraints, making flexible control of model capacity a central challenge. While sparse and low-rank structures naturally trade off capacity and performance, existing approaches often rely on heuristic designs that ignore layer and matrix heterogeneity or require model-specific architectural modifications. We propose SALAAD, a plug-and-play framework applicable to different model architectures that induces sparse and low-rank structures during training. By formulating structured weight learning under an augmented Lagrangian framework and introducing an adaptive controller that dynamically balances the training loss and structural constraints, SALAAD preserves the stability of standard training dynamics while enabling explicit control over the evolution of effective model capacity during training. Experiments across model scales show that SALAAD substantially reduces memory consumption during deployment while achieving performance comparable to ad-hoc methods. Moreover, a single training run yields a continuous spectrum of model capacities, enabling smooth and elastic deployment across diverse memory budgets without the need for retraining.

Teaching Machine Learning Fundamentals with LEGO Robotics

This paper presents the web-based platform Machine Learning with Bricks and an accompanying two-day course designed to teach machine learning concepts to students aged 12 to 17 through programming-free robotics activities. Machine Learning with Bricks is an open source platform and combines interactive visualizations with LEGO robotics to teach three core algorithms: KNN, linear regression, and Q-learning. Students learn by collecting data, training models, and interacting with robots via a web-based interface. Pre- and post-surveys with 14 students demonstrate significant improvements in conceptual understanding of machine learning algorithms, positive shifts in AI perception, high platform usability, and increased motivation for continued learning. This work demonstrates that tangible, visualization-based approaches can make machine learning concepts accessible and engaging for young learners while maintaining technical depth. The platform is freely available at https://learning-and-dynamics.github.io/ml-with-bricks/, with video tutorials guiding students through the experiments at https://youtube.com/playlist?list=PLx1grFu4zAcwfKKJZ1Ux4LwRqaePCOA2J.

A Systems-Theoretic View on the Convergence of Algorithms under Disturbances

Algorithms increasingly operate within complex physical, social, and engineering systems where they are exposed to disturbances, noise, and interconnections with other dynamical systems. This article extends known convergence guarantees of an algorithm operating in isolation (i.e., without disturbances) and systematically derives stability bounds and convergence rates in the presence of such disturbances. By leveraging converse Lyapunov theorems, we derive key inequalities that quantify the impact of disturbances. We further demonstrate how our result can be utilized to assess the effects of disturbances on algorithmic performance in a wide variety of applications, including communication constraints in distributed learning, sensitivity in machine learning generalization, and intentional noise injection for privacy. This underpins the role of our result as a unifying tool for algorithm analysis in the presence of noise, disturbances, and interconnections with other dynamical systems.

A Critical Perspective on Finite Sample Conformal Prediction Theory in Medical Applications

Machine learning (ML) is transforming healthcare, but safe clinical decisions demand reliable uncertainty estimates that standard ML models fail to provide. Conformal prediction (CP) is a popular tool that allows users to turn heuristic uncertainty estimates into uncertainty estimates with statistical guarantees. CP works by converting predictions of a ML model, together with a calibration sample, into prediction sets that are guaranteed to contain the true label with any desired probability. An often cited advantage is that CP theory holds for calibration samples of arbitrary size, suggesting that uncertainty estimates with practically meaningful statistical guarantees can be achieved even if only small calibration sets are available. We question this promise by showing that, although the statistical guarantees hold for calibration sets of arbitrary size, the practical utility of these guarantees does highly depend on the size of the calibration set. This observation is relevant in medical domains because data is often scarce and obtaining large calibration sets is therefore infeasible. We corroborate our critique in an empirical demonstration on a medical image classification task.

PENEX: AdaBoost-Inspired Neural Network Regularization

AdaBoost sequentially fits so-called weak learners to minimize an exponential loss, which penalizes mislabeled data points more severely than other loss functions like cross-entropy. Paradoxically, AdaBoost generalizes well in practice as the number of weak learners grows. In the present work, we introduce Penalized Exponential Loss (PENEX), a new formulation of the multi-class exponential loss that is theoretically grounded and, in contrast to the existing formulation, amenable to optimization via first-order methods. We demonstrate both empirically and theoretically that PENEX implicitly maximizes margins of data points. Also, we show that gradient increments on PENEX implicitly parameterize weak learners in the boosting framework. Across computer vision and language tasks, we show that PENEX exhibits a regularizing effect often better than established methods with similar computational cost. Our results highlight PENEX's potential as an AdaBoost-inspired alternative for effective training and fine-tuning of deep neural networks.