Concentration

Modern predictive systems expose multiple quantities that are commonly interpreted as measures of confidence. However, these quantities can summarize different aspects of the predictive process. This distinction matters when confidence is used to evaluate reliability or inform downstream oversight and control. We investigate whether different confidence readouts are empirically interchangeable in an autoregressive language model by comparing local confidence, defined from the probability of the greedy-selected answer token, with global confidence, defined from modal-answer frequency under repeated sampling. Across MMLU and ARC Challenge, the two signals are weakly correlated and differ substantially in their association with correctness: global confidence is moderately associated with correctness, whereas local confidence shows little association. We further test whether question-level disagreement between the signals is associated with sampling instability. On ARC, larger local--global confidence gaps are associated with higher answer entropy, more distinct sampled answers, and lower modal-answer concentration. The gap--entropy association persists when disagreement and instability are estimated from disjoint stochastic samples, indicating that it is not explained by shared finite-sample variation. The corresponding relationship is substantially weaker on MMLU, where only 4% of questions exhibit sampling instability. These results show that confidence readouts derived from the same predictive system are not empirically interchangeable and that their disagreement can provide a diagnostic of unstable sampling behavior. Confidence should therefore be treated as an explicitly defined measurement rather than as a single intrinsic scalar property of a model, particularly when it is used to inform downstream evaluation, oversight, or control.

Joint motion forecasts pair each autonomous-vehicle (AV) future with surrounding traffic, but actor-level metrics do not show whether that structure matters to a planner. We study this question with a marginal-preserving product control that removes AV-traffic pairing among the learned modes while retaining the fixed constant-velocity pair and holding trajectories, actor-level marginals before planner conditioning, candidates, the cost terms and weights, and fallback fixed. The intervention also changes candidate-conditioned concentration. Across twelve runs on 1,400 held-out Argoverse 2 scenarios, the intervention changes 3.0% of route-level offline selections at $τ=4$ m. Control-minus-joint recorded-trajectory regret is $-0.026$ and $-0.118$ at the two training sizes; crossed and seed-$t$ intervals span zero. At $τ=1$ m, relative costs change in 87.9% of route evaluations and route-level offline selections in 8.1%. Before concentration matching, descriptive outcome estimates favor the control. Most of this gap disappears along an approximate concentration-matching path; the remaining contrasts are $+0.112$ and $-0.047$, and both crossed intervals span zero. Actor-level forecast metrics remain identical. Pairing-strength and temperature sweeps show that the decision contrast grows with pairing removal and sharper conditioning. The intervention changes planner decisions even though actor-level metrics remain unchanged. The matching analysis, however, cannot separate any recorded-outcome effect of learned-mode AV-traffic pairing from the accompanying change in conditioned concentration.

AI-generated text is flowing back into the training corpora of the next generation of models. Recursive training on it drives model collapse, and recent work extends the setting to many models feeding one another -- but almost always with the market split evenly, while real generative AI is an oligopoly. Concentration raises two worries: fewer, more uniform sources may make collapse faster, and later models may be dragged toward the oligarch's output. We test both in controlled ecosystems: 13 open 1--4B models form natural ecosystems of 3 to 13 players, plus an injected probe that pushes the top share to 90%; each generation, every model's output is mixed into a shared pool by market share and every model is retrained on that pool from clean base weights, for five generations. Yet within the range we test, neither worry materializes; what emerges instead is an invariance. Making the split more unequal barely changes the speed of collapse. Destinations move even less: the share and identity knobs shift five-generation endpoints by only a few percent of the drift common to all arms -- the ecosystems collapse to nearly the same place. An extreme share paired with the strongest injected bias still does not guarantee steering, and the topic shifts it does produce leave only a faint trace on the ruler that measures collapse. What sets the speed is who supplies the pool and how readily those suppliers are carried along: with every share held fixed, swapping the members of a K=3 ecosystem changes five-generation drift by 2.8x; a share-weighted index of each member's susceptibility explains the speed differences across nineteen arms with R^2 = 0.68; and replacing half the pool with human text roughly halves drift without changing its course. Within the tested range, concentration sets neither the destination nor the pace of collapse; the pace follows whose text fills the pool.

Lightweight channel attention mechanisms are widely used in image classification, yet their effectiveness in fine-grained visual recognition (FGVR) remains limited. Most modules summarize each channel by global average pooling (GAP), which captures activation magnitude but ignores spatial concentration, so channels with different spatial distributions but identical means receive the same descriptor. We propose Concentration-Aware Channel Attention (ConCA), which pairs the mean with a shift-invariant negative-input entropy (NegEnt), computed via a softmax over the negated activations, forming a dual descriptor that jointly encodes magnitude and concentration. A depthwise 1-D convolutional multi-layer perceptron (MLP), whose parameter count is linear in the number of channels, maps the pair to a per-channel weight. On six fine-grained benchmarks, ConCA improves over attention-free, SE-Net, and ECA-Net baselines as well as four richer descriptor-based modules under a controlled from-scratch protocol, and it generalizes across eight backbones on iNat2021-mini. These results indicate that the channel descriptor, together with the per-channel gating that maps it to attention weights, is an important but underexplored aspect of lightweight channel attention in FGVR.

We consider for an arbitrary fixed $ρ$ and for each positive integer $n$ a multilayer feedforward artificial neural network with $ρ$ layers, $n$ neurons in the first layer (the input layer) and only one neuron, the output neuron, in the last layer. Very roughly formulated, the main result is that if the distribution of weights of connections from a layer to the next are, for all large $n$, approximated well by a fixed continuous (but otherwise arbitrary) curve which does not depend on $n$, and if the values of the $n$ input neurons are independently and identically distributed with a continuous probability density function, then there is a number $ψ$ such that for all $\varepsilon > 0$ the probability that the value of the output neuron is in $[ψ- \varepsilon, ψ+ \varepsilon]$ tends to 1 as $n$ tends to infinity.

Group Relative Policy Optimization (GRPO) has become a standard reinforcement learning method for post-training language models. Recent work shows that GRPO can reduce the base model's reasoning capacity and underperform it in Pass@k when k is large, indicating reduced coverage of reasoning paths. We find that this reduction is associated with GRPO concentrating on responses that the base model already generates with high probability. We trace this concentration to two mechanisms in the GRPO update. At the response level, high-probability responses dominate the group gradient through repeated occurrence. At the token level, GRPO's importance ratio scales gradients, further reinforcing tokens that become more likely under the current policy. We propose ReCo, a reweighting method that addresses both effects. Response contributions are normalized by their expected occurrence within the rollout group, and the token-level importance ratio is replaced with a variance-based ratio that gives larger update scale to non-saturated decision points where alternative token choices remain plausible. Across Qwen2.5-Math-1.5B/7B and Llama-3.1-8B-Instruct on five mathematical reasoning benchmarks, ReCo improves Pass@k for large values of k and is comparable to GRPO for small values of k.

Shippers are beginning to delegate carrier selection to large language model (LLM) agents. We ask what such delegation does to a freight matching market, and which platform design choices contain it. We carried out agent-based simulations in which fifty shipper agents, built on commercial LLMs from OpenAI (GPT), Anthropic (Claude), and Google (Gemini), procure truckload capacity for thirty days. The market implements the rules of digital freight matching: each load is offered down the shipper's ranked list of carriers (waterfall tendering), carriers have daily capacity limits, spot prices respond to congestion, and carrier ratings accumulate with transactions. We found three risks and one remedy that works. Agents converged at once: for a fixed sampled carrier population, the same carrier was the modal first choice of every model on day one, attracting up to 76% of requests. Because each agent picks from its own randomly drawn list of displayed candidates, the platform controls how many options each shipper sees; concentration rose steeply once lists exceeded about ten carriers, with the onset differing across models. Which carriers ended up dominant varied widely from one sampled market to another, and displaying true quality instead of estimated ratings changed neither the level nor this variability (by design, quality affects only what agents see, never delivery outcomes). Against these risks, disclosing each carrier's remaining daily capacity cut concentration by a third and doubled shipper surplus, while vendor diversification, list-order randomization, and popularity display showed no clearly detectable effect. Platform information design, ahead of model choice or model regulation, is the lever that works.

The mammalian kidney concentrates urine using a mechanism with no analogue in current neural architectures: the countercurrent multiplier. Two anti-parallel flows joined at a hairpin recirculate a weak magnitude-bounded local pump into a large axial gradient achieving a four-fold concentration increase from a single-effect gradient that never exceeds 200 mOsm at any point. We formalize this mechanism as a differentiable sequence operator the Countercurrent Multiplier (CCM) layer and study it as an alternative to residual iterative refinement.

We establish mean-square and concentration bounds for stochastic approximation (SA) with arbitrary norm contractive mappings, under a multiplicative noise model where the noise may scale affinely with the norm of the iterates, and the iterates are potentially unbounded. These settings arise in reinforcement learning, where operators are often contractive in the $\ell_\infty$ norm and the noise scales with the iterates. To address the arbitrary norm, earlier works replace the non-smooth squared norm with a smooth Lyapunov function constructed via the generalized Moreau envelope. For concentration analysis, these works handle multiplicative noise and unbounded iterates through a multi-stage bootstrapping argument that starts from a time-varying worst-case bound and iteratively refines it. We instead present a unified and elementary analysis that yields both bounds. Using an averaged noise sequence and corresponding auxiliary iterates, we obtain a one-step Lyapunov drift inequality for the normed error directly, without smoothing the norm or constructing an envelope. For the mean-square bound, we combine this drift inequality with an induction argument showing that the iterates remain bounded in expectation. For the concentration bound, we develop a probabilistic induction over a sequence of "good" events on which the iterates are controlled, allowing the standard Azuma-Hoeffding bound to be applied. Our approach yields the first sub-Gaussian tailed maximal (all-time) concentration bound for SA under multiplicative noise, by allowing the stepsize to depend logarithmically on the confidence level. Beyond the specific setting considered here, we discuss the generalizability of these proof techniques to other noise models and iterative algorithms.

Sim-to-real policy transfer -- deploying policies trained in simulation in the real world -- is a promising paradigm for scaling robot manipulation without large-scale real-world data. However, transferring simulation-trained policies remains challenging due to discrepancies in contact dynamics -- particularly in contact-rich tasks where subtle differences can alter task outcomes entirely. Because interaction between the manipulated object and the environment is mediated through contact, task success depends on accurately reproducing task-relevant contacts. Accordingly, in manipulation, contact-centric fidelity -- reproducing both the contact event sequence (when, where, and how contacts occur) and the local contact dynamics (how forces and motions evolve at each contact) -- is a necessary condition for task success. Based on this insight, we propose a contact-centric real-to-sim-to-real RL framework that uses task-relevant contact event sequences extracted from real demonstrations as the learning objective. We approximate objects as groups of primitives and optimize their contact geometry in simulation so that the resulting local contact dynamics explain the observed state transitions. The contact event sequence is automatically extracted by replaying the demonstration. This sequence serves as a structured reward signal, guiding the policy toward physically plausible contact regimes validated in reality and preventing exploitation of unrealistic simulator contacts. The signal is obtained automatically, requiring no per-task reward design. Experiments on contact-rich manipulation tasks demonstrate more stable and robust sim-to-real policy transfer compared to unconstrained RL baselines.

Inspired by advances in natural language processing and computer vision, "time-series foundation models" (TSFMs) have recently been introduced with the promise of strong generalization across diverse time-series tasks, including forecasting, classification, and anomaly detection, as well as across domains such as healthcare, climate science, and manufacturing. However, their utility for gas-sensing data remains largely unexplored. To address this gap, this paper systematically evaluates recent TSFMs on electronic nose (E-Nose) data. In particular, we investigate whether embeddings produced by representative TSFMs, including Chronos-2 and MOMENT, provide effective representations for gas identification and concentration prediction. Specifically, we show that fine-tuning is necessary to achieve satisfactory performance on E-Nose data, and fusing TSFM embeddings with representations learned by specialized predictive models can further improve the performance, suggesting both the potential and limitations of current TSFMs for gas-sensing applications.

Discrete diffusion language model (DLM) fine-tuning inherits inexpensive diagnostics from denoising-time confidence monitors, but their PEFT-training meaning is untested. We test top-1 argmax concentration as a collapse warning. Across 816 LoRA/PEFT configurations from three DLM families, the warning fires for every configuration while logs record 0/816 actual collapses at the 200 step horizon, giving zero precision. The cause is pre-equilibrium saturation: top-1 concentration is already high before optimization and quickly becomes insensitive to final training stability. We then evaluate max LoRA gradient norm, a parameter-side signal that samples gradient routing rather than token concentration. On a pooled held-out LLaDA-family split, a train-optimized threshold identifies top-decile final-loss configurations with precision 0.68 and F1=0.79, above the all-positive top-1 baseline even at the lower split-bootstrap confidence bound. Autoregressive controls and cross-family threshold failures bound the result to short-horizon DLM-LoRA inspection rather than a universal collapse detector. Workflow: drop top-1 as a PEFT alarm, log max-gradient early in training, and calibrate thresholds per DLM family before routing runs for inspection.

Large language models now mediate how buyers discover products and services, making the competitive structure of AI-generated recommendations a strategic concern for brands. A basic question has lacked large-scale empirical answers: in a given category, which brand does a model recommend, and how concentrated is that ownership? Across 3,750 responses spanning 50 brands, five industries, and 250 brand-free category queries on three models (GPT-5.2, Google Gemini 3 Flash, and Perplexity sonar-pro), each query repeated five times under a dice-roll stability protocol, we propose three exploratory metrics: the Category Ownership Index (COI), a brand's share of mentions within a category; the Competitive Vacuum Index (CVI), flagging categories with no single leader; and the Displacement Score (DS), quantifying asymmetric substitution between brand pairs. In this sample, recommendation concentration was moderate: the mean Gini coefficient was 0.28 (95% CI [0.16, 0.41]), below the 0.60 power-law threshold we set. Competitive vacuums were rare, appearing in 8.0% of queries, so the models named at least one sampled brand in most cases. Cross-model agreement on the top-recommended brand was 41.6%: a top position on one model did not reliably hold on another. Displacement was industry-dependent, from co-recommendation in consulting (0.4:1) to one-directional substitution up to 4.3:1, with an unweighted mean of 2.4:1 across the five industries. A BERTopic check placed only 4.2% of discovered topic clusters outside the original categories. Within the scope studied, these results sit in tension with a strong winner-takes-all narrative around AI recommendation, and the three metrics offer a candidate, reproducible procedure for competitive-intelligence analysis that future work can validate.

Wall shear stress (WSS) governs near-wall transport dynamics and is a key hemodynamic indicator in cardiovascular flows, yet remains difficult to infer accurately due to the need for precise computation of near-wall velocity gradients. Passive scalar fields, such as concentration or temperature, are advected by the same underlying velocity field and have the potential to uncover hidden flow physics metrics such as WSS. In this work, we demonstrate such reconstruction from spatially limited passive scalar observations using two fundamentally different inverse frameworks: a differentiable physics framework based on discrete adjoint, PDE-constrained optimization, which enforces the governing equations as hard constraints, and physics-informed neural networks (PINNs), which treat them as soft constraints. Benchmark problems include a 2D canonical backward-facing step (2D-BFS) and a 3D patient-specific stenotic coronary artery. For the 2D-BFS case, evaluated under three measurement scenarios (near-wall, far-field, and combined), PINN achieves high accuracy when near-wall data are available but fails when restricted to far-field measurements, whereas the differentiable physics approach recovers accurate WSS across all scenarios. In the 3D patient-specific case, the differentiable physics framework outperforms PINNs, yielding accurate WSS reconstruction. These results establish that measurement location and inverse formulation jointly determine reconstruction fidelity in scalar-based near-wall flow inference. The proposed framework opens a path toward estimation of near-wall hemodynamics from scalar transport data, with broader applicability to fluid flow problems where passive scalars can be observed.

We address the issue of global convergence in stochastic continuous optimization. For that purpose, we formulate the Canonical Evolutionary Strategy (CES) as a controlled mathematical framework to analyze global convergence in evolutionary algorithms via the semiclassical limit of a Schr{ö}dinger-type replicator-mutator equation. We provide a rigorous hierarchy from a discrete individual-based dynamics to a deterministic mean-field limit, demonstrating that global convergence is governed by the principal eigenfunction of the underlying operator. This property, defined as Geometric Selection, naturally prioritizes robust, flat optima over narrow local traps, offering a mathematical justification for the ''survival of the flattest'' phenomenon. Moreover, unlike consensus-driven methods that are prone to premature variance collapse when the global minimizer resides outside the initial support, the replicator-mutator dynamics of CES facilitate intrinsic mass transport. High-dimensional benchmarks (d = 30) confirm this advantage, showing that CES achieves lower residual errors in shifted initialization scenarios where standard consensus-driven and gradient-based methods fail to migrate effectively. By shifting the focus from point-wise consensus to spectral concentration, our framework provides a robust theoretical foundation for global convergence in Evolution Strategies (ES) without the need for additional numerical heuristics.

In this paper, we propose Concentrate and Concentrate (CaC), a coarse-to-fine anomaly reward model based on Vision-Language Models. During inference, it first conducts a global temporal scan to anchor anomalous time windows, then performs fine-grained spatial grounding within the localized interval, and finally derives robust judgments via structured spatiotemporal Chain-of-Thought reasoning. To equip the model with these capabilities, we construct the first large-scale generated video anomaly dataset with per-frame bounding-box annotations, temporal anomaly windows, and fine-grained attribution labels. Building on this dataset, we design a three-stage progressive training paradigm. The model initially learns spatial and temporal anchoring through single- and multi-frame supervised fine-tuning, and then is optimized by a reinforcement learning strategy based on two-turn Group Relative Policy Optimization (GRPO). Beyond conventional accuracy rewards, we introduce Temporal and Spatial IoU rewards to supervise the intermediate localization process, effectively guiding the model toward more grounded and interpretable spatiotemporal reasoning. Extensive experiments demonstrate that CaC can stably concentrate on subtle anomalies, achieving a 25.7% accuracy improvement on fine-grained anomaly benchmarks and, when used as a reward signal, CaC reduces generated-video anomalies by 11.7% while improving overall video quality.

Transformers with self-attention modules as their core components have become an integral architecture in modern large language and foundation models. In this paper, we study the evolution of tokens in deep encoder-only transformers at inference time which is described in the large-token limit by a mean-field continuity equation. Leveraging ideas from the convergence analysis of interacting multi-particle systems, with particles corresponding to tokens, we prove that the token distribution rapidly concentrates onto the push-forward of the initial distribution under a projection map induced by the key, query, and value matrices, and remains metastable for moderate times. Specifically, we show that the Wasserstein distance of the two distributions scales like $\sqrt{{\log(β+1)}/β}\exp(Ct)+\exp(-ct)$ in terms of the temperature parameter $β^{-1}\to 0$ and inference time $t\geq 0$. For the proof, we establish Lyapunov-type estimates for the zero-temperature equation, identify its limit as $t\to\infty$, and employ a stability estimate in Wasserstein space together with a quantitative Laplace principle to couple the two equations. Our result implies that for time scales of order $\logβ$ the token distribution concentrates at the identified limiting distribution. Numerical experiments confirm this and, beyond that, complement our theory by showing that for finite $β$ and large $t$ the dynamics enter a different terminal phase, dominated by the spectrum of the value matrix.

Mechanistic interpretability assumes that circuit analysis becomes harder as models scale. We challenge this assumption by showing that the attention architecture matters more than parameter count. Studying three circuit types across Pythia and Qwen2.5, we find that grouped query attention produces circuits that are far more concentrated and mechanistically stable than standard multi-head attention at comparable scales. The same concentration pattern holds across indirect object identification, induction heads, and factual recall. Within a single architecture family (Qwen2.5), factual recall circuits undergo a discrete phase transition above a critical scale, collapsing to a single bottleneck rather than degrading gradually. These findings suggest that some architectural choices make large models more tractable to study and that interpretability difficulty is not a fixed consequence of model size.

Sequence-level evaluations in autoregressive Large Language Models (LLMs) rely on highly dependent token generation. Establishing tight concentration bounds for these processes remains a challenge due to two fundamental bottlenecks in existing frameworks: (i) classical inequalities typically separate dependency structures from target sensitivities, leading to a scalar collapse that inflates the variance proxy to a suboptimal $\mathcal{O}(N)$ for sparse terminal rewards; (ii) conversely, while certain spatial methods achieve tighter bounds, they lack the strictly causal filtration required by sequential generation, rendering them inapplicable to the autoregressive setting. To resolve both bottlenecks, we establish a sharp McDiarmid-type inequality for dependent sequences, governed strictly by the exact matrix-vector multiplication of the causal dependency resolvent and the target sensitivity vector. This Matrix-Decoupled Concentration (MDC) framework natively recovers optimal constants for Markov chains and exploits directed $d$-separation to yield order-optimal bounds for causal trees. Crucially, by exactly preserving the coordinate-wise sparsity of rewards within a strictly causal framework, MDC mathematically prevents scalar collapse, guaranteeing a dimension-free $\mathcal{O}(1)$ variance proxy and providing a rigorous mathematical justification for the stability of long-context reasoning.

Artificial Jagged Intelligence (AJI) denotes a recurring pattern in which large learning systems exhibit strong local capabilities while remaining weak or brittle in other domains. This paper develops a formal theory of AJI as uneven allocation of optimization pressure. We model training as a finite-budget process that distributes gradient-driven update energy across capability-relevant directions in parameter space. In this model, jagged capability profiles arise from anisotropic objective structure, data geometry, and representational coupling rather than from a single scalar quantity called intelligence. The paper defines capability gain, optimization energy share, and jaggedness, then proves that persistent concentration of cumulative update energy yields lower bounds on dispersion in capability gains. A finite-budget tradeoff theorem shows why prioritizing one capability can impose opportunity costs on others unless positive coupling or shared structure offsets the cost. The analysis also studies redistribution mechanisms, including energy-variance regularization and auxiliary structural objectives, as interventions that reshape the optimization field. The resulting framework links uneven emergence, training architecture, and optimization governance. It predicts that early concentration of update energy should forecast later capability jaggedness; that scaling under a narrow objective need not eliminate anisotropy; and that explicitly funded auxiliary objectives can revive neglected capabilities. AJI is therefore not merely a descriptive label for uneven model behavior, but a testable theory of how finite optimization resources produce concentrated, delayed, and structurally uneven capability formation.

Multi-agent deliberation systems using large language models (LLMs) are increasingly proposed for policy simulation, yet they suffer from artificial consensus: evaluator agents converge on the same option regardless of their assigned value perspectives. We present the AI Council, a three-phase deliberation framework, and conduct 120 deliberations across two policy scenarios to test two interventions. First, architectural heterogeneity (assigning a different 7-9B parameter model to each value perspective) significantly reduces first-choice concentration compared to a homogeneous baseline (child welfare: 70.9% to 46.1%, p < 0.001, r = 0.58; housing: 46.0% to 22.9%, p < 0.001, r = 0.50). This contrasts with accuracy-oriented multi-agent debate, where heterogeneity does not reduce convergence, suggesting model diversity operates differently when no objectively correct answer exists. Second, coherence validation (using a frontier model to assess whether each evaluator's reasoning is grounded in its assigned values) reveals a fidelity-diversity tradeoff: on a scenario with a dominant option, it further reduces concentration (46.1% to 40.8%, p = 0.004), but on a scenario with genuinely competitive options, it increases concentration (22.9% to 26.6%, p = 0.96) by amplifying high-coherence evaluators who cluster on one option. This tradeoff may be a general property of multi-agent systems employing quality weighting. We report negative results from three failed Delphi designs, demonstrate that 8B models exhibit binary rather than graded responses to counter-arguments, and propose the trustworthy tension rate as a diagnostic measure of small-model deliberation capabilities.

Existing single image dehazing methods have demonstrated satisfactory performance on homogeneous thin-haze images; however, they often struggle with non-homogeneous hazy images that exhibit spatially varying haze concentrations and abrupt density transitions across different regions. To address this fundamental limitation, we propose a novel multi-branch deep neural network framework, termed Concentration Partitioning and Image Fusion Network (CPIFNet), which decomposes the challenging non-homogeneous dehazing problem into a set of tractable homogeneous sub-problems. Our key insight is that a single non-homogeneous hazy image can be viewed as a composite of multiple local regions, each exhibiting approximately homogeneous haze characteristics. CPIFNet employs a two-stage architecture consisting of an Image Enhancement Network (IENet) stage and an Image Fusion Network (IFNet) stage. In the first stage, multiple IENet branches are independently trained on homogeneous haze datasets of different concentration levels, producing enhancement models that excel at restoring regions matching their respective haze densities. In the second stage, the IFNet intelligently aggregates the advantageous regions from all enhancement outputs through deep feature stacking and merging, yielding a unified high-quality dehazed result. Furthermore, we introduce a comprehensive loss function incorporating reconstruction, perceptual, structural, and color losses to jointly supervise both stages.