Gradient_Descent
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A Mechanistic Model for Collective Motion from Sensorimotor Regularities
2026-05-15 18:17:41Vito Mengers, Bao Duc Cao, Oliver BrockAbstract
Collective behavior in animals has long been modeled through self-propelled particle models, which reproduce striking group-level phenomena through abstract interaction forces. Yet these models are fundamentally descriptive: they leave open the question of how collective behavior is actually produced. Recent empirical work makes this gap concrete: locusts do not align with neighbors, sensory and cognitive mechanisms mediate interaction instead. A mechanistic model must therefore operate at the sensorimotor level, grounded in what individual organisms can actually perceive, estimate, and physically execute. We present such a model based on a modeling framework from robotics, extended here to collective motion. Each agent perceives neighbors through bearing and apparent-size cues within a limited field of view, maintains uncertain internal state estimates, and selects actions through gradient descent on a desired social distance -- without any prescribed interaction forces. This simple model produces diverse collective behaviors including polarized motion, milling, ring formations, and subgroup fragmentation. A global sensitivity analysis shows that behavioral transitions are governed by sensorimotor parameters corresponding to measurable biological quantities: field of view geometry, sensory noise, turning agility, and memory. Collective behavior can therefore be understood as the emergent outcome of interacting sensorimotor regularities, and differences across species as the emergent outcome of differences in embodiment and environment.
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URL
https://arxiv.org/abs/2605.16522
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XOResNet: Exclusive-OR Meta-Residuals Facilitate Deep Spiking Neural Networks Learning
Abstract
Spiking neural networks (SNNs) hold promise for demonstrating superior learning and representation capabilities in deep models. Given the tremendous success of ResNet in deep learning, it would naturally follow to train deep SNNs with residual learning. However, existing residual structures for constructing deep SNNs still present challenges of spike redundancy or information loss, as well as redundant learning. In the present study, we first aim to address issues of relative spike redundancy in identity mapping and information loss in non-identity mapping. To this end, we propose an OR-ADD (OA) shortcut connection to merge output spikes/currents from two branches in the residual structure. Furthermore, to mitigate redundant learning in the backbone branch of the residual structure, we introduce the concept of XOR meta-residuals, i.e., selecting pre-learning residuals using the Exclusive-OR (XOR) operation for the backbone branch. Finally, by integrating the OA shortcut and XOR meta-residuals, we devise the XOR residual block and further construct XOResNet with varying depths based on this block. Extensive experiments on four datasets, Fashion-MNIST, CIFAR-10, CIFAR-100, and miniImageNet, show that the proposed XOResNet outperforms existing state-of-the-art deep SNNs optimized via gradient descent. These results validate the effectiveness of our OA shortcut and XOR meta-residual components in overcoming fundamental limitations of residual learning in SNNs, providing new architectural insights for building high-performance neuromorphic systems.
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URL
https://arxiv.org/abs/2605.30362
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Emergent Sparsity in Frozen Random CNN Feature Extractors for Deep Reinforcement Learning
2026-05-15 00:01:05Scott M. NortonAbstract
We report a striking phenomenon: deep reinforcement learning agents trained with frozen, randomly initialized CNN feature extractors spontaneously develop extremely sparse fully-connected representations, without any sparsity-inducing objective. In the first fully-connected layer (FC1, $3{,}136 \to 64$), agents compress task-relevant information through as few as 1-3 neurons out of 64 for deterministic Pong (5-11 for stochastic Pong), while trainable CNNs activate 55-64 neurons under matched conditions. We establish four principal findings. First, FC1 sparsity scales with task complexity: 1-11 for Pong, 19-26 for Breakout, and $\sim$42 for Space Invaders. Width-scaling confirms this reflects task structure rather than a fixed capacity fraction. Second, within-game scaling emerges: three identical Pong seeds produce 5, 7, and 11 active neurons. The 5-neuron seed plateaus at $+14$ reward, while the others reach expert performance ($+18.4$, $+18.7$), suggesting the random projection's usable dimensionality bounds achievable performance. Third, ablation confirms necessity: removing these active neurons crashes performance across two PPO implementations and four games. Fourth, the information bottleneck commits early: a sweep shows the active set locks by 15-30M steps, while reward turns positive 35-105M steps later. A complementary finding in Breakout shows frozen and trainable CNNs reach competitive rewards via structurally different bottlenecks: frozen agents use 17-25 active neurons (participation ratio $\sim$10-14), while trainable agents use 51 (participation ratio $\sim$3.6). Finally, wherever input dimensionality dwarfs intrinsic task dimensionality, gradient descent on a frozen random projection may reveal the effective rank of the underlying problem without explicit sparsity machinery.
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URL
https://arxiv.org/abs/2607.26059
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Critic-Driven Voronoi-Quantization for Distilling Deep RL Policies to Explainable Models
2026-05-14 14:38:56Senne Deproost, Denis Steckelmacher, Ann Now\'eAbstract
Despite many successful attempts at explaining Deep Reinforcement Learning policies using distillation, it remains difficult to balance the performance-interpretability trade-off and select a fitting surrogate model. In addition to this, traditional distillation only minimizes the distance between the behavior of the original and the surrogate policy while other RL-specific components such as action value are disregarded. To solve this, we introduce a new model-agnostic method called Critic-Driven Voronoi State Partitioning, which partitions a black box control policy into regions where a simple class of model can be optimized using gradient descent. By exploiting the critic value network of the original policy, we iteratively introduce new subpolicies in regions with insufficient value, standing in for a measure of policy complexity. The partitioning, a Voronoi quantizer, uses nearest neighbor lookups to assign a linear function to each point in the state space resulting in a cell-like diagram. We validate our approach on several well known benchmarks and proof that this distillation approaches the original policy using a reasonable sized set of linear functions.
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URL
https://arxiv.org/abs/2605.14897
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Heuristic Pathologies and Further Variance Reduction via Uncertainty Propagation in the AIVAT Family of Techniques
Abstract
How should an agent's performance in a multiagent environment be evaluated when there is a limited sample size or a high cost of running a trial? The AIVAT family of variance reduction techniques was proposed to address this challenge by introducing unbiased low-variance estimators of agents' expected payoffs. An important component of AIVAT is a heuristic value function that discriminates between potentially low- and high-value counterfactual histories. A notable gap in the literature is that there is little to no constraint or guideline on how the heuristic value function should be chosen or how uncertainty in its output should be handled. In our first contribution, we parameterize the heuristic value function to highlight AIVAT's potential vulnerabilities: a) the sample variance can be set pathologically low by directly applying gradient descent on the sample variance, and b) one can p-hack to draw a desired statistical conclusion via gradient descent/ascent on the test statistic. The main takeaway is that the heuristic value function should be fixed prior to observing the evaluation data! In our second contribution, we show how the heuristic uncertainty can be propagated to quantify the uncertainty of AIVAT estimates. It is then possible to further reduce the variance using inverse-variance weighted averaging, but AIVAT's unbiasedness guarantee may have to be sacrificed. In our experiments, we use a dataset of 10,000 poker hands to demonstrate our heuristic pathology and uncertainty results, with the latter yielding a 43.0% reduction in the number of samples (poker hands) needed to draw statistical conclusions.
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URL
https://arxiv.org/abs/2605.14261
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Spectral Flattening Is All Muon Needs: How Orthogonalization Controls Learning Rate and Convergence
2026-05-13 06:54:01Tien-Phat Nguyen, Truong Nguyen, Minh-Phuc Truong, Tuc Nguyen, James Bailey, Trung LeAbstract
Muon orthogonalizes the momentum buffer before each update, replacing its singular values with ones via Newton-Schulz iterations. This simple change lets Muon tolerate far larger learning rates and converge faster than other optimizers, but why? We show that the mechanism is spectral flattening, and develop two results around it. First, we prove that Muon's maximal stable step size scales with the average singular value of the gradient rather than the largest, which bottlenecks standard gradient descent. Second, we recast Muon as a preconditioned gradient method and show, under a Kronecker-factored curvature model, that it improves the effective convergence factor, with the improvement controlled by the spectrum of the gradient covariance. Extensive experiments validate both results: Muon remains stable at learning rates that cause SGD to diverge within the first few iterations, and reaches accuracy milestones several epochs earlier even at identical step sizes. Taken together, our results offer a principled, geometric explanation for Muon's empirical success.
Abstract (translated)
URL
https://arxiv.org/abs/2605.13079
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FeatCal: Feature Calibration for Post-Merging Models
2026-05-13 05:35:03Yanggan Gu, Shuo Cai, Zihao Wang, Wenjun Wang, Yuanyi Wang, Pengkai Wang, Sirui Huang, Su Lu, Jianmin Wu, Hongxia YangAbstract
Model merging combines task experts into one model and avoids joint training, retraining, or deploying many expert models, but the merged model often still underperforms task experts. We study this performance gap through feature drift, the difference between features produced by the merged model and by the expert on the same input. Our theory decomposes this drift into upstream propagation and local mismatch, tracks how it propagates and combines through later layers in forward order, and links final feature drift to output drift. This view motivates FeatCal, which uses a small calibration set to calibrate the merged model weights layer by layer in forward order, reducing feature drift while staying close to merged weights and preserving the benefits of model merging. FeatCal uses an efficient closed-form solution to update model weights, with no gradient descent, iterative optimization, or extra modules. On the main CLIP and GLUE benchmarks, FeatCal beats Surgery and ProbSurgery, the closest post-merging calibration baselines: 85.5% vs. 77.0%/78.8% on CLIP-ViT-B/32 Task Arithmetic (TA) and 85.2% vs. 83.7%/82.2% on FLAN-T5-base GLUE. On CLIP-ViT-B/32, 8 examples per task reach 82.9%, and 256 examples per task take 53 seconds, about 4x faster than both baselines, showing better sample efficiency and lower calibration cost.
Abstract (translated)
URL
https://arxiv.org/abs/2605.13030
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Geometric Factual Recall in Transformers
2026-05-12 17:22:22Shauli Ravfogel, Gilad Yehudai, Joan Bruna, Alberto BiettiAbstract
How do transformer language models memorize factual associations? A common view casts internal weight matrices as associative memories over pairs of embeddings, requiring parameter counts that scale linearly with the number of facts. We develop a theoretical and empirical account of an alternative, \emph{geometric} form of memorization in which learned embeddings encode relational structure directly, and the MLP plays a qualitatively different role. In a controlled setting where a single-layer transformer must memorize random bijections from subjects to a shared attribute set, we prove that a logarithmic embedding dimension suffices: subject embeddings encode \emph{linear superpositions} of their associated attribute vectors, and a small MLP acts as a relation-conditioned selector that extracts the relevant attribute via ReLU gating, and not as an associative key-value mapping. We extend these results to the multi-hop setting -- chains of relational queries such as ``Who is the mother of the wife of $x$?'' -- providing constructions with and without chain-of-thought that exhibit a provable capacity-depth tradeoff, complemented by a matching information-theoretic lower bound. Empirically, gradient descent discovers solutions with precisely the predicted structure. Once trained, the MLP transfers zero-shot to entirely new bijections when subject embeddings are appropriately re-initialized, revealing that it has learned a generic selection mechanism rather than memorized any particular set of facts.
Abstract (translated)
URL
https://arxiv.org/abs/2605.12426
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Emergence of Frontier Superposition: M'obius attractor and Cascade Supervision
Abstract
Superposition allows Transformers to reason in depth, carrying an entire reasoning frontier in parallel through a bounded-depth forward pass instead of unrolling serial chain-of-thought tokens. While Zhu et al. (2025) hand-crafted an equal-weight breadth-first frontier in a single residual stream for graph reachability, it remained open whether gradient descent could ever find this target amidst permutation-symmetric saddles. We close this gap on Reachability-by-Superposition over ErdÅs-Rényi graphs by isolating architectural and supervisional contributions. Architecturally, we identify a Möbius attractor: under $S_n$-symmetry in the tree regime, layerwise dynamics reduce to a 1D Möbius map whose zero set is a codimension-one manifold of global optima containing the equal-weight superposition state. On the supervision side, we identify Cascade Supervision: a loss class whose backward pass simultaneously delivers (A) selectivity bootstrap, (B) gradient persistence across depth, and (C) per-step discrimination (e.g., \mathcal{L}_{sup} and \mathcal{L}_{node}). End-to-end supervision fails condition (B) and is provably insufficient: internal gradients at layer c decay as (np)^{-(D-c-2)/2} in the graph fan-out and stall before the manifold is reached. Our thesis: Möbius attractor + Cascade Supervision = emergence of superposition reasoning. The parameter-free decay law predicts a final-step cosine of 0.35 vs. 0.71 (end-to-end vs. cascade) at depth D=3; experiments confirm 0.37 vs. 0.69, matching within 0.02 at every step.
Abstract (translated)
URL
https://arxiv.org/abs/2605.18820
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Distributed Pose Graph Optimization via Continuous Riemannian Dynamics
2026-05-11 20:24:16Jaeho Shin, Maani Ghaffari, Yulun TianAbstract
We present a framework for distributed Pose Graph Optimization (PGO) by formulating the problem as a second-order continuous-time dynamical system evolving on Lie groups. By modeling pose variables as massive particles subject to damping, the equilibrium points of the resulting Riemannian dynamics coincide with first-order critical points of the original PGO problem. Using the governing damped Euler--Poincaré equations and a semi-implicit geometric integrator, we design an optimization algorithm that generalizes existing algorithms such as Riemannian gradient descent and Gauss--Newton. In multi-robot settings, we present a fully distributed and parallel method based on block-diagonal mass and damping matrices, where each robot solves an ordinary differential equation for its own poses with minimal communication overhead. Moreover, modeling both state and velocity enables principled neighbor prediction that significantly improves convergence under delayed communication. Theoretically, we present an analysis and establish sufficient condition that ensures energy dissipation under the employed geometric discretization scheme. Experiments on benchmark PGO datasets demonstrate that the proposed solver achieves superior performance compared to state-of-the-art distributed baselines in both synchronous and asynchronous regimes.
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URL
https://arxiv.org/abs/2605.11210
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DP-LAC: Lightweight Adaptive Clipping for Differentially Private Federated Fine-tuning of Language Models
2026-05-11 09:32:21Haaris Mehmood, Jie Xu, Karthikeyan Saravanan, Rogier Van Dalen, Mete OzayAbstract
Federated learning (FL) enables the collaborative training of large-scale language models (LLMs) across edge devices while keeping user data on-device. However, FL still exposes sensitive information through client-provided gradients. Differentially private stochastic gradient descent (DP-SGD) mitigates this risk by clipping each client's contribution to a threshold $C$ and adding noise proportional to $C$. Existing adaptive clipping techniques dynamically adjust $C$ but demand tedious hyperparameter tuning, which can erode the privacy budget. In this paper, we introduce DP-LAC, a method that first estimates an initial clipping threshold within an order of magnitude of the optimum using private histogram estimation, and then adapts this threshold during training without consuming additional privacy budget or introducing new hyperparameters. Empirical results show that DP-LAC outperforms both state-of-the-art adaptive clipping methods and vanilla DP-SGD, achieving an average accuracy gain of $6.6\%$.
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URL
https://arxiv.org/abs/2605.10272
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EvoPref: Multi-Objective Evolutionary Optimization Discovers Diverse LLM Alignments Beyond Gradient Descent
2026-05-10 21:50:04Dongxin Guo, Jikun Wu, Siu Ming YiuAbstract
Gradient-based preference optimization methods for large language model (LLM) alignment suffer from preference collapse, converging to narrow behavioral modes while neglecting preference diversity. We introduce EvoPref, a multi-objective evolutionary algorithm that maintains populations of Low-Rank Adaptation (LoRA) adapters optimized across helpfulness, harmlessness, and honesty objectives using Non-dominated Sorting Genetic Algorithm II (NSGA-II) selection with archive-based diversity preservation. Our primary contribution is demonstrating that population-based methods discover substantially more diverse alignments than gradient descent. On standard benchmarks, EvoPref improves preference coverage by 18% (median 82.5% vs. 70.0% for ORPO, $p<0.001$, Wilcoxon, $n=30$) and reduces collapse rates by 47% (11.0% vs. 20.6%, $p<0.001$), while achieving competitive alignment quality (median 75.5% RewardBench vs. 75.0% for ORPO, $p<0.05$). We provide theoretical motivation extending recent multi-objective evolutionary algorithm (MOEA) runtime analysis (Dang et al., 2025) suggesting why archive-based methods escape collapse more effectively than single-trajectory optimization. Comprehensive comparisons against MOEA/D, SMS-EMOA, CMA-ES, and gradient baselines (DPO, IPO, KTO, ORPO) with rigorous statistical testing (Friedman with Holm correction, Vargha-Delaney effect sizes, median with IQR) confirm that multi-objective selection with diversity preservation is essential. This work establishes evolutionary optimization as a principled paradigm for diverse LLM alignment.
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URL
https://arxiv.org/abs/2605.09777
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Dual-Path Hyperprior Informed Deep Unfolding Network for Image Compressive Sensing
2026-05-10 14:28:37Tianyi Lu, Wenxue Cui, Shaohui LiuAbstract
Recent Deep Unfolding Networks (DUNs) have significantly advanced Compressive Sensing (CS) by integrating iterative optimization with deep networks. However, existing DUNs still suffer from two challenges: 1) Reliance on a single measurement stream, which limits effective information interaction across distinct measurement subsets. 2) Uniform processing of all image regions, which overlooks varying reconstruction difficulties induced by diverse textures. To address these limitations, a novel Dual-Path Hyperprior Informed Deep Unfolding Network (DPH-DUN) is proposed, which partitions measurements into double subsets to enable hyperprior-guided reconstruction via a dual-path architecture. In the Deep Hyperprior Learning branch, a series of lightweight neural modules are designed to efficiently generate hyperprior knowledge of different domains, enabling collaborative guidance for the CS reconstruction. In the Hyperprior Informed Reconstruction branch, a deep unfolding framework with hyperprior guidance is constructed to iteratively refine reconstruction. Specifically, i) in the gradient descent step, a Hyperprior Informed Step Size Generation network is designed to dynamically generate spatially varying step maps, enabling adaptive fine-grained gradient updates. ii) In the proximal mapping step, two well-designed hyperprior informed attention mechanisms are introduced to dynamically focus on challenging regions via gradient-based hard and soft attentions, facilitating CS reconstruction accuracy. Extensive experiments demonstrate that the proposed DPH-DUN outperforms existing CS methods.
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URL
https://arxiv.org/abs/2605.09566
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Outlier-Robust Diffusion Solvers for Inverse Problems
2026-05-10 11:13:23Yang Zheng, Jiahua Liu, Tongyao Pang, Wen Li, Zhaoqiang LiuAbstract
Methods based on diffusion models (DMs) for solving inverse problems (IPs) have recently achieved remarkable performance. However, DM-based methods typically struggle against outliers, which are common in real-world measurements. In this work, to tackle IPs with outliers, we first refine the measurement via explicit noise estimation to mitigate the effect of noise. Subsequently, we formulate an iteratively reweighted least squares objective based on the Huber loss to address the outliers. We propose a method utilizing gradient descent to approximately solve the corresponding optimization problem for the robust objective. To avoid delicate tuning of the learning rate required by the gradient descent method, we further employ the conjugate gradient method with an efficient strategy for updating. Extensive experiments on multiple image datasets for linear and nonlinear tasks under various conditions demonstrate that our proposed methods exhibit robustness to outliers and outperform recent DM-based methods in most cases.
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URL
https://arxiv.org/abs/2605.09477
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Control Your View: High-Resolution Global Semantic Manipulation in Learned Image Compression
2026-05-09 06:25:20Jiaming Liang, Chi-Man Pun, Weisi Lin, Greta Seng Peng MokAbstract
Learned image compression (LIC) integrates deep neural networks (DNNs) to map high-dimensional images into compact latent representations, reducing redundancy and achieving superior rate-distortion (RD) performance in benign settings. Unfortunately, due to inherent vulnerabilities in DNNs, LIC systems are susceptible to adversarial perturbations that lead to downstream deterioration, compression rate degradation, untargeted distortion, and both local semantic manipulation (LSM) and low-resolution ($3\times28\times28$) global semantic manipulation (GSM). However, high-resolution GSM remains unexplored due to its intractability. Notably, the existing project gradient descent (PGD) method achieves near-perfect white-box attacks for classification, segmentation, and other tasks, yet fails to generalize to high-resolution GSM. Our theoretical and empirical analyses reveal that well-performing GSM drives adversarial examples from the Identity Region to the Amplification Region through the Lazying-Oscillating-Refining stages. General $\ell_{\infty}$-bounded attacks fail on high-resolution GSM because their step-size schedules cannot accommodate both the Oscillating and Refining stages. Based on this, we propose the Periodic Geometric Decay schedule that enables $\ell_{\infty}$-bounded high-resolution GSM. To verify our approach, we integrate it with PGD, yielding a minimal variant, PGD$^{2}$-GSM. Extensive experiments on the Kodak $(3\times768\times512)$ demonstrate that our PGD$^{2}$-GSM is the first to stably achieve high-resolution GSM, thereby exposing a novel threat to LIC systems. Code is available at this https URL.
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URL
https://arxiv.org/abs/2605.08727
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Spectral Dynamics in Deep Networks: Feature Learning, Outlier Escape, and Learning Rate Transfer
2026-05-08 15:28:01Clarissa Lauditi, Cengiz Pehlevan, Blake BordelonAbstract
We study the evolution of hidden-weight spectra in wide neural networks trained by (stochastic) gradient descent. We develop a two-level dynamical mean-field theory (DMFT) that jointly tracks bulk and outlier spectral dynamics for spiked ensembles whose spike directions remain statistically dependent on the random bulk. We apply this framework to two settings: (1) infinite-width nonlinear networks in mean-field/$\mu$P scaling and (2) deep linear networks in the proportional high-dimensional limit, where width, input dimension, and sample size diverge with fixed ratios. Our theory predicts how outliers evolve with training time, width, output scale, and initialization variance. In deep linear networks, $\mu$P yields width-consistent outlier dynamics and hyperparameter transfer, including width-stable growth of the leading NTK mode toward the edge of stability (EoS). In contrast, NTK parameterization exhibits strongly width-dependent outlier dynamics, despite converging to a stable large-width limit. We show that this bulk+outlier picture is descriptive of simple tasks with small output channels, but that tasks involving large numbers of outputs (ImageNet classification or GPT language modeling) are better described by a restructuring of the spectral bulk. We develop a toy model with extensive output channels that recapitulates this phenomenon and show that edge of the spectrum still converges for sufficiently wide networks.
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URL
https://arxiv.org/abs/2605.07870
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Approximation-Free Differentiable Oblique Decision Trees
2026-05-08 15:04:04Subrat Prasad Panda, Blaise Genest, Arvind EaswaranAbstract
Decision Trees (DTs) are widely used in safety-critical domains such as medical diagnosis, valued for their interpretability and effectiveness on tabular data. However, training accurate oblique DTs is challenging due to complex optimization landscapes and overfitting risks, particularly in regression. Recent advances have introduced differentiable formulations that enable gradient-based training and joint optimization of decision boundaries and leaf regressors. Yet, existing approaches typically rely on approximations, either through probabilistic softening of boundaries (soft DTs) or quantized gradients such as the Straight-Through Estimator (STE). To overcome these limitations, we propose DTSemNet, a novel, semantically equivalent, and invertible representation of hard oblique DTs as neural networks. DTSemNet enables end-to-end training with standard gradient descent, eliminating the need for approximations in both classification and regression. While classification aligns naturally with this formulation, regression remains challenging due to the joint optimization of internal nodes and leaf regressors. To address this, we analyze the limitations of STE and introduce an annealed Top-k method that provides accurate gradient signals without approximation. Extensive experiments on classification and regression benchmarks show that DTSemNet-trained oblique DTs outperform state-of-the-art differentiable DTs. Furthermore, we demonstrate that DTSemNet can serve as programmatic DT policies in reinforcement learning environments, thereby broadening their applicability.
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URL
https://arxiv.org/abs/2605.07837
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Decentralized Time-Varying Optimization for Streaming Data via Temporal Weighting
2026-05-07 21:40:42Muhammad Faraz Ul Abrar, Nicol\`o Michelusi, Erik G. LarssonAbstract
Classical optimization theory largely focuses on fixed objective functions, whereas many modern learning systems operate in dynamic environments where data arrive sequentially and decisions must be updated continuously. In this work, we study optimization with streaming data over a distributed network of agents. We adopt a structured, weight-based formulation that explicitly captures the streaming-data origin of the time-varying objective: at each time step, every agent receives a new sample, and the network seeks to track the minimizer of a temporally weighted objective formed from all samples observed across the network so far. We focus on decentralized gradient descent (DGD) with a limited communication/computation budget, where at each time step, only a limited number of DGD iterations can be performed before the objective changes again. For strongly convex and smooth losses, we analyze the tracking error with respect to the time-varying minimizer through a fixed-point theory lens. Our analysis reveals that the tracking error decomposes into a fixed-point tracking term and a bias term induced by data heterogeneity across agents. We specialize the analysis to two natural weighting strategies: uniform weights, which treat all samples equally, and exponentially discounted weights, which geometrically decay the influence of older data. Under uniform weighting, DGD tracks the fixed-point at a rate $\mathcal{O}(1/t)$, whereas discounted weighting yields a non-vanishing fixed-point tracking floor controlled by the discount factor. In both cases, decentralization induces an additional non-zero bias floor under a constant step size. We validate our theoretical findings through numerical simulations.
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URL
https://arxiv.org/abs/2605.06971
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A Rod Flow Model for Adam at the Edge of Stability
Abstract
Cohen et al. (arXiv:2207.14484) observed that adaptive gradient methods such as Adam operate at the edge of stability. While there has been significant work on continuous-time modeling of gradient descent at the edge of stability, extending these models to momentum methods remains underdeveloped. In the gradient descent setting, Regis et al. (arXiv:2602.01480) introduced rod flow, which models consecutive iterates as an extended one-dimensional object -- a "rod." Here we extend rod flow to Adam by working in the joint phase space of parameters and first moment $(w, m)$ and treating the second moment $\nu$ as a smooth auxiliary variable. We also develop rod flows for heavy ball momentum, Nesterov momentum, and scalar and per-component versions of RMSProp, Adam, and NAdam. For all eight optimizers, we empirically evaluate rod flow on representative machine learning architectures, where it tracks the discrete iterates through the edge-of-stability regime significantly more accurately than the corresponding stable flow.
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URL
https://arxiv.org/abs/2605.06821
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The Weight Gram Matrix Captures Sequential Feature Linearization in Deep Networks
2026-05-07 13:35:11Taehun Cha, Daniel Beaglehole, Adityanarayanan Radhakrishnan, Donghun LeeAbstract
Understanding how deep neural networks learn representations remains a central challenge in machine learning theory. In this work, we propose a feature-centric framework for analyzing neural network training by relating weight updates to feature evolution. We introduce a simple identity, the Feature Learning Equation, which identifies the weight Gram matrix as the key object capturing feature dynamics. This enables us to interpret gradient descent as implicitly inducing a hypothetical evolution of features, whose covariance structure - termed the Virtual Covariance - characterizes how representations evolve during training. Building on this perspective, we introduce Target Linearity, a measure quantifying the linear alignment between features and targets. By analyzing the training and layer-wise dynamics, we show that deep networks learn to sequentially transform representations toward target-linear structure. This linearization perspective provides a unified interpretation of several empirical phenomena, including Neural Collapse and linear interpolation in generative models.
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URL
https://arxiv.org/abs/2605.06258
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