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Byte level language models avoid tokenizer dependence and provide universal coverage, but processing every byte at full resolution is expensive. We introduce DeepMem, a causal multiscale architecture that separates semantic compression from exact byte preservation. A learned invertible subband transform constructs a hierarchy of shorter sequences, while a lossless carrier and explicit detail sidebands preserve byte identity. Fast weight memory blocks operate along the encoder, with iterative shared state computation at the compressed bottleneck, while an attention based super resolution decoder restores byte resolution representations. DeepMem achieves stronger language modeling performance than a matched byte level Transformer on FineWeb-Edu and PG19, and obtains the best macro average performance across the evaluated FLORES languages. It also improves PIQA accuracy while remaining competitive on code modeling, multiple choice reasoning, and byte level corruption robustness. Bottleneck refinement requires approximately $7{,}600\times$ fewer counted FLOPs than equivalent full resolution reasoning, while maintaining competitive evaluation throughput.

Predicting social network churn before it becomes visible in aggregate metrics remains an open challenge, particularly in enterprise settings where heterogeneous usage patterns, sparse data, and deployment constraints preclude standard supervised approaches. We formalize this problem as finite-step engagement projection over contributor-role Markov chains, with a stationary-distribution interpretation for ergodic chains in the long-run limit. For each engagement feature type within a network (e.g., Q&A, communities, articles, etc.), users are classified monthly into four behavioral states and a transition matrix is estimated from six months of history. Six-step propagation yields a label-free projection of the contributor mix; for ergodic chains, the Perron-Frobenius theorem establishes convergence under repeated propagation. Comparing projected future engagement to current engagement produces the Health Ratio, a single score requiring no trained model, no labels, and no infrastructure beyond SQL. Validated across seven rolling prediction windows on 11,500+ tenant-level networks, the Health Ratio achieves mean AUC 0.917 with a 156 times churn-rate gradient across risk tiers, outperforming all single-signal baselines and the growth-adjusted variant previously deployed in production. An ablation study shows that six-step propagation achieves similar churn discrimination to the eigenvector-based score. Deployed on Microsoft Engage since October 2025, the framework powers customer health monitoring and has enabled proactive intervention on at-risk networks before decline becomes visible in traditional metrics.

Task instructions from users for agents are inherently underspecified, especially regarding safety boundaries--creating behavioral risks. To study these risks, we introduce BARC-Bench, a collection of 110 tasks across nine simulated workspace environments where benign requests leave openings for unsafe actions. In this benchmark, we evaluate agent behavior against safety boundaries established by workspace records, implicit environmental context, and unstated social norms. For example, an agent asked to schedule off-boarding discussions must arrange the meeting without prematurely revealing an employee's resignation through a shared calendar invitation. Evaluating four frontier model configurations across five attempts per task reveals that models are not consistently safe: the share of tasks completed safely on all five attempts ranges from 9.1% to 72.7%, with even the strongest configuration producing a safety violation on 27.3% of tasks. These results highlight an asymmetric operational risk: while models frequently complete complex workflows, their inability to consistently respect unstated boundaries presents a substantial barrier to safe autonomous delegation.

We study the degree needed for sum-of-squares (SoS) proofs of several basic principles of real linear algebra. For any $n\times n$ matrix $A$ of rank greater than $r$, the Weak Rank Principle states that the equations $XY^\top=A$ with $n\times r$ matrix variables $X, Y$ have no solution. We show that this principle requires degree exactly $2r+2$ to prove. On the other hand, for $A=I_n$, setting $X = Y$ gives the Tight Frame Principle, which has a degree-$4$ proof. For $n\times n$ matrices, proving that a right inverse is also a left inverse requires degree exactly $2n+2$, while proving that orthonormal rows imply orthonormal columns again has a degree-4 proof. Using our lower bounds, we also derive related SoS degree lower bounds for polynomial systems encoding sign rank, matrix rigidity, small-width homogeneous non-commutative algebraic branching programs, and tensor CP rank. Lastly, we show that quantitative bounds on the Frobenius norms of the factors in the Weak Rank Principle can lower the degree required to refute $XY^\top=I_n$, but sufficiently large bounds still preserve the exact degree $2r+2$ from the unbounded case, giving a quantitative version of the Archimedean property arising in classical results on SoS proofs.

Vision-language-action (VLA) models increasingly pair a pre-trained vision-language model (VLM) with a flow-matching expert that conditions its action predictions on the VLM's key-value representations. Those representations must carry information useful for control, yet the VLM was never trained for the action expert's flow-matching objective. Two signals can address this: gradients backpropagated from the expert's loss (coupling), or VLM supervision to predict discrete tokens encoding the same target actions. Prominent training recipes forgo the first, blocking expert-to-VLM gradients (insulation) to protect the VLM's existing capabilities, while also supervising it on vision-language data and action tokens. Because vision-language co-training also aims to preserve those capabilities, the additional protection provided by insulation is unclear. At the same time, insulation removes a direct signal about which features continuous action prediction requires. On the other hand, action-token supervision offers an alternative way to learn action-relevant representations, which may explain why insulation succeeds in practice. We disentangle these factors in autonomous driving and manipulation by independently varying action-token supervision and whether the expert's loss updates the VLM. In open-loop driving, insulation is competitive with coupling when action tokens are supervised; without them, coupling reduces trajectory-prediction error by roughly 25%. When both components are trained jointly from the outset, coupling increases closed-loop driving route progress by approximately 17% with action-token supervision and 40% without it, and similarly improves robot manipulation performance under both settings. These findings show that the value of gradient insulation depends on the VLM's action supervision and the evaluation setting, challenging insulation as a default VLA training recipe.

A computational path records the steps used to establish an equality. Topologies on computational equality witnesses can depend on the chosen generators and execution schedule, even when their algebraic and homotopical meanings agree. We study continuous presentations of finite computational paths whose declared rewrites are realized by endpoint-fixed homotopies. We compare a topology retaining the full signed trace with an observable topology recording endpoints, length, and exact path realization. Continuous finite-word interpretations that are inverse up to scoped rewriting preserve trace-sensitive quotient arrows and their final composable-pair spaces. Observable quotients behave differently. Over a point with a nonempty primitive alphabet, we classify their topology: it records length parity when all named relations preserve parity, and is otherwise indiscrete. Adjoining a conservative abbreviation can therefore turn a discrete two-element quotient into an indiscrete space, even though ordinary multiplication remains continuous. A circle presentation also gives nonhomeomorphic quotients under binary and equal-slot timing. Geometric completeness alone does not identify the trace-sensitive and observable topologies. Multiplication is continuous on the quotient of composable representatives. We characterize when this topology agrees with the ordinary pullback topology and give an open-projection criterion. Universal presentations recover classical fundamental-groupoid topology. Normal forms, primitive interchange, and a Cayley covering prove completeness for circle, torus, product, and rose presentations. A literal-word correspondence relates the semantics to the reflexivity, symmetry, and transitivity fragment of equality derivations. Selected results are formalized in Lean, with the realization models and proof coverage stated explicitly.

Diagonal state space models process long sequences at linear cost, yet the same transition is applied at every step of a rollout, so an unconstrained recurrence holds no quantity fixed and the state decays or grows without bound. Current designs constrain one quantity, and preserving one does not guarantee preservation of the others, so a model can hold its norm to machine precision while momentum drifts by hundreds of percent. We claim that the conserved quantities of a diagonal recurrence are layered, that each layer requires a strictly stronger condition on the operator, and that a unit-modulus spectrum reaches only the outermost. Acting on the claim, we enforce the conditions required for every layer by construction. The generator of the Kronecker-Cayley transition (KC) is a graph Laplacian over object subspaces, $\mathbf{H}=\mathbf{L}\otimes I_m$, advanced by a Cayley transform, and the step stays diagonal at $O(N^3+N^2m)$ cost. The conserved set is exactly a $2m^2$-dimensional family of quadratic forms, held at every parameter value, so the quantities and the dynamics the construction reaches are fixed before training. On four conservation benchmarks, KC holds total mass at or near the reference floor while every diagonal baseline loses it by orders of magnitude. We further compare against mass correction and find that two correctors fail once the underlying prediction diverges, which shows that assembly rather than correction is necessary for a guarantee that holds across the rollout. A free modal basis trained under a conservation penalty conserves only by freezing its spectrum, whereas the assembled operator conserves with every mode but one free to rotate. Beyond these benchmarks, KC holds the linear momentum of three $N$-body systems and, at the operator level, the conserved quantity in molecular dynamics, chemical kinetics, pharmacokinetics and power-grid swing dynamics.

We establish an $Ω(d)$ lower bound on the competitive ratio of convex body chasing in $\mathbb{R}^d$ with movement costs measured in any norm. Previously, the best known lower bound valid across all norms was $Ω(d^{1/6})$, established by Papazov (2023); stronger lower bounds were known for $\ell_p$ norms, where Bubeck, Klartag, Lee, Li, and Sellke (2019) had established a lower bound of $Ω(\max\{\sqrt{d}, d^{1-1/p}\})$. In particular, a linear lower bound was known only for the $\ell_\infty$ norm, realized by the standard construction of chasing faces of a hypercube. Our lower bound strictly improves on the prior universal and $\ell_p$ lower bounds for $p < \infty$ and shows that the functional Steiner point algorithm of Sellke (2023) - which has competitive ratio $O(d)$ - is asymptotically optimal for every norm. Our approach is based on the recursive construction of Bubeck, Coester, and Rabani (2023) for metrical service systems, which gives a lower bound of $Ω(r^2)$ for requests of size $O(r)$. We show that requests in their construction, each comprising a set of discrete points in a graph metric, can be mapped to the vertices of convex requests in a $(2r+1)$-dimensional $\ell_p$ vector space while approximately preserving movement costs up to a factor of $O(r)$; the general result then follows via Milman's (1985) quotient-of-a-subspace theorem.

Single-cell RNA foundation models (scFMs) are increasingly used for in-silico biological discovery, yet predictive performance alone does not reveal whether their decisions rely on biologically meaningful signals or model-specific shortcuts. We systematically investigate post-hoc attribution of downstream predictions across two tasks: cell-type annotation using pretrained Cell2Sentence and scGPT models, and perturbation prediction using scGPT+STATE in a controlled setting. For cell annotation, we use attribution to assess whether predictions rely on established marker genes and how their relevance varies with model architecture, expression rank, and the surrounding cellular context. For perturbation prediction, we compare attribution patterns with known regulatory ground-truth in a synthetic setting to assess what they reveal about the model's learned computations. Across these tasks, attribution reveals biological feature use, contextual interactions, contributions from different input pathways, learning progression, associations with prediction quality, and potential pitfalls that are not fully captured by standard performance metrics. Experiments with complete and limited regulatory knowledge further demonstrate that conclusions drawn from attribution depend strongly on the choice of biological reference, baseline, and null model. Together, these findings highlight the value of attribution as a flexible post-hoc diagnostic for understanding model behavior, identifying potential issues, and informing the development of scFMs.

We study whether the equilibrium of a drop on a chemically patterned wall, and the capillary forces that the pattern exerts on it, can be learned from the free energy alone, without solution data. We introduce DropFENO (Droplet Free-Energy Neural Operator), in which Fourier (FNO), tensor-factorised Fourier (TFNO) and wavelet (WNO) neural operators are trained in two and three dimensions on the diffuse-interface free energy of the drop, with its volume and centroid constraints imposed as penalties. The predictions are tested with the virtual-work (Hellmann--Feynman) theorem: the force exerted by the pattern on a held drop is a wall integral of the equilibrium field which, for a drop straddling a wettability step, reduces identically to the contact-line law $F=σΔ\cosθ\,L$. DropFENO reproduces this law and the energy landscape of a drop crossing a step, and reproduces the pinning of the receding contact line less accurately. The residual error lies in the soft modes of the capillary energy (small displacements of the interface and the contact line). This explains why the energy is predicted accurately while contact-line positions are not, and why on striped walls the operators return Cassie-like states instead of the edge-pinned states whose anisotropic apparent angles the reference minimiser shares with lattice Boltzmann simulations. On held-out patterned walls the median energy of the predictions lies $\NdEmedRandFnoTwo$ of the free-drop energy above the best known minimum (FNO, two dimensions), at about one hundredth of the cost of a per-instance minimisation, and a refinement by a classical solver started from the prediction returns the reference state in almost all cases.

Alternating online mirror descent (AltOMD) is a simple learning rule for zero-sum games, which improves upon simultaneous OMD numerically and theoretically. Previous work obtained $O(T^{1/3})$ and $O(T^{1/5})$ regret bounds for AltOMD, by leveraging modified-energy-function-based analyses. However, settling whether AltOMD achieves near-constant regret has remained an open question. In this paper we resolve the question of whether AltOMD achieves near-constant regret in the affirmative. In particular, we show an $O(log(T))$ regret bound on the sum of the players' external regrets for any two-player zero-sum bilinear game with compact convex strategy sets, under mild regularity assumptions on the regularizers. We verify that our assumptions hold for the negative entropy regularizer, thereby obtaining logarithmic regret for the alternating multiplicative weights updates algorithm. The regret bound also yields an $O(log(T)/T)$ duality gap bound for the averaged strategies.

Implicit generative models based on deep neural networks have revolutionized many scientific fields, from weather research to language generation. They successfully model complex random objects arising in real-world problems. In this work, we propose a sample-efficient training mechanism for these models. Our approach leverages regularized inference with shrinkage priors and generalized Bayes. In the large $p$ (parameter dimension) and small $n$ (sample size) setting, sparsity-inducing priors have been successfully used for linear regression when $p \gg n$. However, their application to implicit generative models remains challenging due to the intractability of the likelihood function. To address this, we adapt a scoring rule posterior based on a strictly proper scoring rule, following the generalized Bayes framework. We then propose a block SGMCMC within Gibbs sampling mechanism. This enables efficient exploration of high-dimensional parameter spaces by identifying a sparse representation of neural implicit generative models. We demonstrate the performance of our proposed method through experiments on $p \gg n$ linear regression and an application of neural generative models.

Importance sampling is a building block of modern statistical computing, but the weighted estimate is prone to high or infinite variance. Moreover, its empirical standard error is often unreliable, leaving importance sampling without practical uncertainty quantification. Motivated by multilevel regression and poststratification (MRP) in survey sampling, we propose post-stratified importance sampling (PIS) for estimating target expectations from proposal draws. PIS reformulates arbitrarily high-dimensional weighting into two one-dimensional probabilistic modeling problems: regression for the conditional mean given the importance ratio and marginal density estimation for the importance ratio. Different modeling choices yield a family of PIS estimators, encompassing ordinary importance sampling as a special case. We develop a model-based implementation that combines one-dimensional Gaussian process regression for the conditional mean and score-informed heavy-tailed density estimation for the marginal law, using both the observed weights and the available density evaluations. We study the oracle gains of PIS and establish consistency under general and model-specific conditions. Simulations and applications to cross-validation demonstrate improvements in estimation accuracy and uncertainty quantification.

As a current trend in Artificial Intelligence (AI), large foundation models are increasingly employed as the core of AI services. However, even after training, serving such models at scale remains a challenging task due to their heavy resource footprints, particularly in terms of GPU memory. While recent works revealed unique characteristics of systems serving foundation models that distinguish them from traditional distributed computing systems, there is still a lack of fundamental understanding of the underlying system management problems. This work aims at addressing this gap by extracting a novel problem of "server chain composition" via block placement and cache allocation for serving chainstructured jobs with large memory footprints, which models a fundamental problem in serving large foundation models through pipeline parallelism. After showing the NP-hardness of the optimal solution, the focus is turned to developing scalable algorithms with guaranteed performance under state-of-the-art load balancing. Application of the proposed solution to a distributed large language model (LLM) serving system shows significant reduction of response times compared to state-of-the-art solutions.

A defining feature of the linear-probing hash table is its spatial locality: at load factor $1 - \varepsilon$, the hash table can be implemented so that each operation scans a single contiguous region of expected size $O(1/\varepsilon)$ (this is known as the \emph{scan cost}). This guarantee, however, relies on all items having unit size. In this paper, we ask whether it is possible to achieve a similar guarantee, but in a hash table that stores items of arbitrary lengths $|x| \ge 1$. The main result of the paper is a new hash table called \emph{Catalan hashing}, which supports amortized expected scan cost $O(|x|/\varepsilon)$ for positive queries, $O(1/\varepsilon)$ for negative queries, and $O(|x|\varepsilon^{-1}\log^{1+δ}|x|)$ for insertions/deletions, for any constant $δ> 0$. As a corollary, we obtain the same bounds in the cache-oblivious model, with a scan cost of $S$ translating to $1 + O(S/B)$ I/Os. Along the way, we introduce several high-level techniques that may be of independent interest; this includes a novel way of using erasure codes in order to perform write optimization within a hash table, and a connection to scheduling that allows us to place elements in a way which avoids clustering at high load factors.

Looped transformers increase effective depth by repeatedly applying a shared set of transformer layers, offering a parameter-efficient way to scale computation. However, standard residual connections aggregate information across recurrent depth into a single stream, providing limited control over which intermediate representations from preceding layers should be reused. In this paper, we introduce ARLoop, a looped transformer architecture based on attention residuals, where the input of each layer is selectively aggregated through attention over model depth. Distinct from standard transformers, we find that looped transformers do not require access to the full residual history: sliding-window depth attention matches or outperforms full-history attention at larger recurrent depths. Across controlled experiments with multiple parameter scales (140M and 370M) and recurrent depths (R = 1 to 5), ARLoop consistently outperforms prior residual designs for looped transformers. More importantly, we further scale ARLoop to 0.6B and 1.2B parameters under a realistic LLM pretraining recipe, demonstrating that the gains transfer to realistic pretraining, with particularly strong improvements on math and coding tasks. Our 1.2B model, pretrained on only 0.5T tokens with three recurrences, outperforms Huginn-3.5B (the only fully-open looped LLM as of our release date) across all evaluated tasks and is competitive with similar-sized non-looped models such as Qwen3.5-2B despite being trained on substantially less data.

Fault-tolerant quantum computers run millions of logical cycles to perform quantum error correction (QEC). As logical operations span entire logical cycles, the logical cycle time (LCT) dictates program runtime. Qubit measurements, critical to extract parity information for QEC, are the slowest operations and account for 43-86% of LCT. Measurement latencies impact fidelity: fast measurements lower our ability to correctly classify quantum states, whereas slow measurements increase the likelihood of decoherence. Hardware providers thus calibrate measurement latencies that maximize readout fidelity per qubit and use them to schedule all measurements during program execution. However, calibrated latencies are sub-optimal for logical performance because logical cycles span multiple QEC rounds, during which data qubits are not measured. This increases LCT and logical error rates (LERs) due to decoherence at the logical level. We propose DynaMite, a lightweight real-time measurement latency selection policy built on two key insights. First, logical cycles must use a measurement latency that is fast enough to reduce both LCT and LER for the QEC code, instead of the calibrated latency that only maximizes readout fidelity. Second, not all logical cycles are equal; many do not fully utilize the error budget of the QEC code, i.e., the number of failed parity checks it can tolerate without compromising LER. These cycles could use even faster measurements but do not, because latencies are chosen statically, whereas error information is only available at runtime. DynaMite exploits the fact that a logical cycle comprises multiple QEC rounds and selects the fastest latency for an upcoming round based on the error budget left and rounds remaining. Our evaluations show that DynaMite reduces LCT by 1.69x on average and up to 2.29x, and reduces LER to 0.46x on average and up to 0.13x of baseline.

The goal of Sign Language Production (SLP) is to translate natural language text into corresponding sign language movements performed by a virtual avatar. The quality of the generated motion is typically evaluated by a motion-space Frechet distance (FID) and back-translation BLEU score on benchmarks such as How2Sign. The existing approaches can improve both metrics substantially while the underlying generator fails to represent the input sentence, a failure we call conditional collapse. In this work we propose to evaluate the generated motion at three independent levels: (tau1) initial-pose conditioning, (tau2) output diversity, and (tau3) target faithfulness. We compute these as pairwise-distance ratios using latent representations of a frozen motion autoencoder (MoAE), judged against each dataset's own GT-vs-GT reference. We evaluate diffusion and regression SLP models on How2Sign, including a re-implemented Neural Sign Actors, and show that tau3 faithfulness is never attained, while FID varies by nearly two orders of magnitude and is uncorrelated with faithfulness. We then apply the same models and diagnostic to Phoenix-2014T, CSL-Daily, and gloss-level ASL3DWord and show that faithfulness varies proportionally to how densely the training sentences cover the test conditions (conditioning coverage) and not corpus size. The smaller but densely covered Phoenix-2014T and ASL3DWord datasets generate faithful sign motion, while CSL-Daily follows the input text yet falls short of sign language faithfulness. Using the largest, open-domain How2Sign dataset, the models collapse at every scale. We introduce two measurements of the text data that rank the corpora in the order of how densely the training sentences fill the embedding space, and how close each test sentence lies to its nearest training sentence.

Deep models for radiological diagnosis are typically trained to predict disease labels directly from images, which limits them to diseases seen during training and makes their decisions hard to interpret in radiologists' terms. Radiology, however, proceeds from visual findings to disease through published guidelines that make this decision rule explicit, interpretable, and reproducible. We introduce diagnosis by composition (DbC), a representation learning framework that encodes this findings-first logic. DbC learns a vocabulary of radiological findings by aligning image features with the text embeddings of the findings, each finding represented as a dense map over the scan. A disease is defined by a set of required findings derived from its guidelines, and its diagnosis map is a fixed, parameter-free composition of the corresponding finding maps. Because the finding-to-disease mapping is derived from guidelines rather than learned, DbC is interpretable and auditable by design, can extend to new diseases by specifying their findings without retraining, and can diagnose unseen diseases by composing their findings. On multi-phase liver CT, DbC detects and localizes unseen diseases, rare diseases from a different institution, and an external cohort significantly better than all comparators, including models that align images with disease names. DbC also improves detection and localization in the fully supervised setting and remains superior across training-set sizes. Although demonstrated on liver CT, DbC applies wherever radiological guidelines define diseases through findings.

TCBench is a benchmark for evaluating global, short to medium-range (1-5 days) forecasts of tropical cyclone (TC) track and intensity. To allow a fair and model-agnostic comparison, TCBench builds on the IBTrACS observational dataset and formulates TC forecasting as predicting the time evolution of an existing tropical system conditioned on its initial position and intensity. TCBench includes state-of-the-art physics-based (TIGGE) and Artificial Intelligence Weather Prediction (AIWP) models (AIFS, Pangu-Weather, FourCastNetV2, GenCast, FNV3P2). If not readily available, TC tracks are consistently derived from model outputs using the TempestExtremes library. TCBench provides deterministic and probabilistic storm-following metrics. On 2023 test cases, AIWP models skillfully forecast TC tracks, while realistic intensity forecasts require additional steps such as post-processing or task-specific training. Designed for accessibility, TCBench helps AI practitioners tackle domain-relevant TC challenges and equips tropical meteorologists with data-driven tools and workflows to improve prediction and TC process understanding. By lowering barriers to reproducible, process-aware evaluation of extreme events, TCBench aims to democratize data-driven TC forecasting.

Let $μ$ be a probability distribution on a spin system with $n$ sites and possibly infinite spin spaces. We prove local-to-global spectral-gap bounds for block Glauber dynamics. Given a symmetric, entrywise nonnegative matrix $A$ with zero diagonal and $λ_{\max}(A)<1$ encoding local spectral bounds for a spin system, we derive global spectral-gap bounds for block dynamics on its marginal distributions. These bounds are sharp: the Gaussian system with covariance $(I-A)^{-1}$ attains equality for every choice of disjoint blocks. Our proof extends the Bochner method through an abstract inequality for orthogonal projections on Hilbert spaces. As an application, we prove that Glauber dynamics for the $O(N)$ model, with $N\geq 2$, arbitrary external fields, and couplings of either sign, has spectral gap at least $(1-\|J_{\mathrm{abs}}\|_{\mathrm{op}}/N)/n$, where $J$ is the interaction matrix and $J_{\mathrm{abs}}=(|J_{ij}|)$. We also obtain spectral bounds for swap walks and local spectral expansion of partite simplicial complexes. For complexes formed from disjoint groups of ranks in random flags of subspaces over $\mathbb F_q$, we obtain a one-sided local spectral expansion bound of $2\sqrt q/(q+1)$. This bound is independent of the number and sizes of each group, and is nontrivial for every prime power $q$. Finally, for random forests $F$ of any finite graph with arbitrary positive edge weights, we prove that \[ \Pr(e,f\in F)\le \frac{4}{3} \Pr(e\in F)\Pr(f\in F) \] for distinct edges $e,f$, improving the factor 2 supplied by general log-concavity bounds for matroid independent sets. The argument combines our extended Bochner method with the analysis of an auxiliary real stable polynomial.

A policy can earn its performance by where it locates on average or by reacting to information. For any Markov policy $π$ in a finite-horizon MDP, let $ν$ be its open-loop marginal, which plays $π$'s average action distribution at each step. We show that $J(π)-J(ν)$ equals $\sum_t tr Cov(Q^ν_t,π_t)$ under endogenous dynamics. Audits based on immediate costs alone can have the wrong sign. Writing the covariance as a joint-versus-product gap gives the bound $|J(π)-J(ν)|\le\sum_t span_t\sqrt{I(s_t;a_t)/2}$, with a sharp constant. Hence, an MDP's adaptivity gap is bounded by the information that its optimal policy uses. Estimating $J(π)-J(ν)$ from $π$'s own bandit-feedback logs takes $Θ(\barχ^2 \log(1/η)/δ^2)$ rounds, where $\barχ^2$ is an average $χ^2$-divergence. At the same time, deterministic non-constant policies cannot be audited at all. We also provide an importance-sampling auditor for episodic MDPs.

End-to-end digital twins can be constructed by fitting separate stage models and connecting them after inference. We investigate the Frequentist prediction properties of a commonly used Monte Carlo simulation scheme that forwards complete predictive draws through such a chain when measured stage outputs contain independent normal errors, but the latent stage means are deterministic. Analytical two-stage examples show persistent asymptotic interval and coverage inflation and, with quadratic propagation, mean bias dependent on the parameters to be estimated. Simulations of four- and eight-stage linear-quadratic univariate chains with 7, 15 or 30 training runs demonstrate pronounced input-dependent distortions. A diagnostic using error-free latent training predictors shows that the effect is mainly not attributable to errors in variables, but to the MCSim approach even showing underestimation of coverage. We propose a model-based bootstrap: an initial chain forwards fitted means, bootstrap chains are refitted on fixed-center bootstrap data while forwarding their mean prediction, and a target-stage future error enters a studentized prediction root. Across the investigated scenarios, relative mean bias is substantially reduced, and coverage is closer to the nominal 90%. At sample size 30, the eight-stage bootstrap has mean width ratios of 1.16-1.42 relative to the known-parameter normal reference, compared with 4.11-4.97 for observed-to-observed MCSim. These results support improved endpoint prediction in the tested setting. Further investigations of the bootstrap scheme in the supplement for a bivariate 6-stage process show similar satisfactory results.

A general mathematical model of experience formation is considered, in which the learning level is defined as the probability that the next value of the uncertainty factor has already been mastered and not forgotten. Within this model the shape of the learning curve is derived, and it turns out to be determined primarily by the distribution of situation occurrence: a uniform distribution yields an exponential curve, whereas a power-law distribution yields a power-law curve whose exponent equals the ratio of the distribution exponent to that same exponent increased by one. Both mechanisms generating power-law curves, established earlier in the analysis of debugging, assembly and repair activities, are shown to have been independently rediscovered in deep learning theory as the "frequency" and "spectral" mechanisms of scaling laws for artificial neural networks, while the exponential curves of machine learning (contraction operators, strongly convex optimization, stationary reinforcement) are generated by the same constancy of the contraction coefficient that was revealed half a century earlier in models of radioactive decay, the first-order lag element, and others. A solution is given for the problem of choosing a training distribution that differs from the deployment distribution. A methodological consequence is discussed: three distinct mechanisms generate curves indistinguishable in shape, so an observed learning curve does not allow one to judge the "internal" structure of the learning system.

We introduce PLANTEUR, a benchmark for crop-type mapping from satellite images in the French overseas territories. Building on the PASTIS design, PLANTEUR provides georeferenced and aligned Sentinel-2, Sentinel-1, and Landsat-8 sequences with parcel-derived labels spanning six tropical regions and an 8-class label space. We define both a standard test split and a leave-one-island-out protocol to measure cross-territory generalization, and evaluate fully supervised models, linear probes, and fine-tuned foundation models. The benchmark code and data are open source and available on GitHub and Hugging Face.

Let $A_1,\ldots,A_n$ be independent $d \times d$ real symmetric Gaussian random matrices, and consider the linear operator $A(x) = n^{-1/2}\sum_{i=1}^n x_iA_i$, $x\in \mathbb R^n$. We construct an iterative algorithm in the Approximate Message Passing family which iterates over $A$ and its adjoint $A^*$, and establish a state evolution result which characterizes its behavior in the limit $d\to \infty, 2n/d^2 \to α$ in terms of a correlated Gaussian-semicircular process in a free probability space, in the sense of strong convergence of operators. We then apply this iteration to the random matrix discrepancy problem which asks for a binary vector $x \in \{-1,+1\}^n$ such that $A(x)$ has a small operator norm. Our algorithm achieves an operator norm $2σ(α)$, for a standard deviation $σ(α)<1$ for all $α>0$, and $σ(α) = O(1/\sqrt{α\log α})$ for large $α$. This resolves the algorithmic question of Kunisky-Zhang (2023) and Maillard (2025).

Fairness evaluation of clinical prediction models for binary outcomes often involves fairness metrics that compare the area under the receiver operating characteristic curve (AUC, or c-statistic) across subgroups. However, the AUC has been shown to be a non-collapsible metric, meaning that its subgroup specific and overall values are not directly comparable as often implicitly assumed. This raises important questions about how existing AUC-based fairness metrics should be used and interpreted. We examine eight AUC-based fairness metrics proposed in the literature, including: within-group AUC parity; cross-group AUC (xAUC) parity; gap-to-overall AUC; equity-scaled AUC; minimum-based metrics; $\mathrm{xAUC}^1$; $\mathrm{xAUC}^0$; and $\mathrm{xAUC}^1$ or $\mathrm{xAUC}^0$ parity. We characterise their estimands, relate them to the decomposition of overall AUC (into within- and cross-group components), and critically appraise them. We then illustrate their behaviour using simulations based on the EuroSCORE II, a clinical prediction model for in-hospital mortality. We recommend reporting an AUC matrix, containing the overall AUC and all within- and cross-group AUC components, alongside population characteristics. When fairness metrics are needed, we recommend reporting the within-group AUC parity, with cross-group AUC parity providing supplementary information but requiring caution due to its potential conflict with calibration. Metrics involving $\mathrm{xAUC}^1$ and $\mathrm{xAUC}^0$ have merit, summarising within- and cross-group AUCs into single values, but their interpretation is non-trivial, and they should be used with caution. Gap-to-overall AUC, equity-scaled AUC, and minimum-based metrics compare quantities that should not be compared and it is unclear how to operationalise them, making them not fit for purpose and generally not recommended.

Colonel Blotto is a fundamental resource-allocation game in which two players divide their budgets across battlefields, and the higher allocation on each battlefield wins its value. The problem of computing an equilibrium efficiently has remained unresolved for its classical continuous version with arbitrary budgets and with values that may vary across battlefields but are the same for both players. We give a fully polynomial-time approximation scheme that computes an $\varepsilon$-Nash equilibrium in time polynomial in the input length and $1/\varepsilon$, and nearly linear in the input length for fixed $\varepsilon$. For sufficiently balanced budgets and battlefield values, we also compute exact equilibria in linear time.