#01Oct 16, 2026
math.ST
Exact Tail Laws for Nonlinear SGD with Infinite-Variance Gradients
Xiaoli Li, Qianqian Lei, Ziyang Wei and 1 more
With infinite-variance gradient noise, SGD can have infinite expected excess objective even when most iterates are close to the optimum. We characterize its accuracy through exact first-order tail probabilities for the last iterate of nonlinear, constant-stepsize SGD. For regularly varying innovations of index $t\in(1,2)$, the leading measure transports one large gradient shock through the local Hessian flow. This is the same measure that governs the central stable Ornstein--Uhlenbeck limit; we prove its relative accuracy at shrinking tolerances above the central scale, with state-dependent noise and a growing observation horizon. The admissible window is nonempty for every available gradient moment $p>1$. The main technical step is a moment bound at the central fluctuation scale and a lower-moment control of the nonlinear error. The result yields explicit projection and excess-objective tails, extreme quantiles, and a factor $b^{1-t}$ for fixed batch size $b$. For affine population gradients, a sharper comparison gives the same law at fixed tolerances, including random sample Hessians. Coupled simulations in dimensions one to one hundred examine relative calibration, objective quantiles, batch effects, and the error due to random curvature.
#02Oct 16, 2026
eess.SP
ECG-SDI: Single-lead Electrocardiogram Enables Continuous Sleep Depth Index Profiling and Scalable Digital Biomarkers for Sleep Health
Donglin Xie, Qingshuo Zhao, Songchi Zhou and 5 more
Conventional sleep staging provides a coarse-grained representation that obscures within-stage depth variation. We developed ECG-SDI, a single-lead electrocardiography (ECG) framework for continuous sleep-depth profiling and digital biomarker extraction. Using 12,392 overnight polysomnographic recordings, we trained and validated a two-stage deep learning model supervised by conventional sleep-stage labels. Correlations with ordered stages excluding rapid eye movement (REM) sleep were 0.828 in the pooled internal test set and 0.848 in the external Cleveland Family Study; corresponding REM classification AUROCs were 0.954 and 0.977. Overnight trajectories captured within-stage variation. Biomarker-based phenotyping identified a disturbed subtype associated with poorer subjective sleep quality and higher all-cause mortality in the Sleep Heart Health Study (adjusted hazard ratio, 1.20; 95% CI, 1.05-1.38). Among 507 participants with paired recordings before and after continuous positive airway pressure (CPAP), normalized sleep-depth area increased by 6.1% and shallow-sleep burden below 0.3 decreased by 16.0%. Exploratory analyses linked reductions in approximate entropy and depth variability to reduced oxygen desaturation and increased sleep efficiency, respectively (Spearman rho = 0.24 and 0.12); both associations remained positive after covariate adjustment. Mean ECG-SDI within N2 increased by 0.017 (95% CI, 0.004-0.030). A wearable proof-of-concept illustrated the acquisition-to-report workflow. These findings support ECG-based sleep phenotyping and further evaluation of CPAP response monitoring at home.
#03Oct 16, 2026
stat.ML
Task-aware Kernel Stein certificates for Rényi differential privacy
Leoni Carla Wirth, Tara Trautwein, Gesine Reinert
Rènyi differential privacy, based on Rènyi divergence, has become a notion of choice for privacy guarantees. However, outside the Gaussian setting, Rènyi divergence can be difficult to compute. In particular, Rènyi differential privacy guarantees that exploit task-specific information, as in Rènyi Pufferfish privacy, can be challenging. Here we use kernel Stein discrepancies based on escort distributions to provide certificates for Rènyi differential privacy. By designing the kernel to focus on the relevant privacy directions, we also derive task-aware privacy certificates. We provide exact identities for Gaussian and Laplace mechanisms and bounds for exponential mechanisms, with separate task-aware guarantees. A real data set example illustrates how the protected comparison set and the released subspace jointly determine the resulting certificates.
#04Oct 16, 2026
cs.LG
Tool Recommendation for Multi-Step Agents: Finding Every Tool a Task Needs
Sam Partee, River Stevens, Thierry Damiba and 1 more
A multi-step agent task fails when a single tool it needs is missing, yet the tools an agent may need are too many to list: the roughly 8,200 tools on the platform we study have definitions that together exceed the context window of any model we evaluate. We present Tool Recommendation, a deployed system that gives an agent all of them, hundreds of times more than a typical MCP server exposes, through a fixed interface of three tools. It runs at the gateway between agent and tools, executing every call under the end user's own credentials and checking it against the user's consent for each service and the tools the agent may use. We evaluate it on both a public retrieval benchmark and billed end-to-end session cost. On ToolRet, Tool Recommendation places every tool a query needs in the top ten 57% more often than BM25, a lexical ranking that commercial tool search offers, and compares favorably with published systems up to nine times its size. In paired multi-step workflows, its first recommendation covered every required tool in 93% of Codex sessions, and against a fixed list of 100 tools it cut input tokens by 17% with Codex and 40% with Claude Code and the total bill in all 14 runs, by up to 40%, with no change in task completion.
#05Oct 16, 2026
stat.ML
Mesoscopic learning dynamics of extensive-width neural networks
Shu Zhou, Jean Barbier, Sebastian Goldt
Understanding how stochastic gradient descent learns internal representations in wide neural networks trained on rich target functions is a central problem in machine learning. We study this question in two-layer neural networks learning target functions represented by networks of the same architecture, with arbitrary nonlinearity, and with network width and input dimension growing proportionally. In this regime, standard approaches for describing the learning dynamics require tracking a diverging number of overlaps, which measure correlations between learned and target features, leaving no obvious description in terms of a few order parameters. Here, we develop a mesoscopic description of learning by characterizing the distribution of overlaps. In the early stages of learning, we find that a Gaussian description of overlap statistics reduces the learning dynamics to a few collective variables that track learning of the target function's linear and quadratic components. We then estimate the timescale of specialization, when Gaussian closure breaks down and individual learned features begin to strongly correlate with specific target features. We find that specialization in extensive-rank networks is accelerated via a quadratic shielding mechanism, whereby learning the linear and quadratic components of the target function filters gradient fluctuations. We finally describe how the activation function determines the pattern of specialization and characterize the subsequent refinement of learned features. Together, these results show how collective learning gives way to individual feature discovery and clarify when learning in large neural networks admits a simple statistical description.