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arXiv2026

Training nGPT

The normalized Transformer (nGPT) realizes hyperspherical representation learning by constraining model parameter vectors and activation vectors to the unit hypersphere. In this paper, we describe a practical training recipe for nGPT and evaluate it on modern hybrid Mamba-2--Transformer Mixture-of-Experts (MoE) models. The recipe introduces Logit Gradient Preconditioning, Logarithmic Learning Rate Decay, GatedAdamW, angular update control, and optional exploration mechanisms. Compared with an unnormalized model of the same hybrid MoE architecture trained with AdamW, the 30B-total-parameter nGPT model reaches the same validation loss using approximately half as many training tokens. The recipe scales across the models considered, which contain up to 30B total parameters.

cs.LG↗
arXiv2026

Spruce: Scalable Private Outsourced Retrieval Using Compact Embeddings

Retrieval-Augmented Generation (RAG) has made dense retrieval over large document collections a standard building block. Organizations increasingly outsource vector indexes to untrusted clouds, exposing proprietary corpora and user queries. Cryptographic protection is challenging because each query searches corpus-scale state, causing computation, correlated randomness, and communication to grow with the corpus. At million-document scale, a naive secure implementation takes minutes and about 90 GB of communication per query. Even recent optimized systems require 10--22 seconds. We propose Spruce (Scalable Private Outsourced Retrieval Using Compact Embeddings), which co-designs representations with the cryptographic protocol. Spruce learns compact binary codes that preserve candidates for full-precision reranking, replacing corpus-wide embedding scoring with efficient Hamming-distance computation under two-server multi-party computation (MPC). A corpus-calibrated fixed-radius protocol avoids multi-round candidate selection while preserving retrieval quality. Spruce also provides private cluster pruning, which trades minor quality loss for substantially less computation, and a one-core owner-operated dealer that removes cloud OT preprocessing bottlenecks. Across four corpora containing 383K--5.42M documents, Spruce preserves the original search quality with median candidate sets of only 382--1,952. At 10 Gbps inter-server bandwidth, full scans take 0.21--2.97 seconds, $4.8$--$6.7\times$ faster than the closest measured prior work. Private pruning takes 0.06--1.09 seconds, achieves $13.1$--$22.9\times$ speedups, and retains $93.9\%$--$97.3\%$ of full-float NDCG. On the largest corpus, pruning and the dealer jointly improve sustained throughput by $31.5\times$ at 1 Gbps per link.

cs.CR↗
arXiv2026

Exact Recovery Thresholds for Weighted Data Selection in Vector-Valued Linear Regression

We resolve the threshold part of Question 4 of the COLT 2025 open problem "Data Selection for Regression Tasks" of Hanneke, Moran, Shlimovich and Yehudayoff. We study vector-valued linear regression with square loss $\ell_{(x,y)}(W)=\lVert Wx-y\rVert_2^2$, where $x\in\mathbb{R}^d$ and $y\in\mathbb{R}^m$. The learner returns the minimum-Frobenius-norm empirical risk minimizer. We prove that the minimal budget of weighted examples for recovering the full-data loss on every finite dataset is exactly $n^{\star}(d,m)=(m+1)d$. We determine the weighted selection profile $F_{\mathrm{weighted}}(d,m,n)$ at the near-threshold budget: $F_{\mathrm{weighted}}(d,m,(m+1)d-1)=1+1/(dm^2)$. We recover the known spanning-budget value $F_{\mathrm{weighted}}(d,m,d)=d+1$ for every $m$, and $F_{\mathrm{weighted}}(d,m,n)=\infty$ for $n<d$. For the smallest open intermediate cell $(d,m)=(2,2)$ we prove $F_{\mathrm{weighted}}(2,2,3)\in[13/8,15/8]$ and $F_{\mathrm{weighted}}(2,2,4)\in[5/4,3/2]$. We reduce the conjectured exact values $13/8$ and $5/4$ to a finite moment problem on the circle with at most seven atoms and assemble structural evidence for it. The upper bounds use a fixed-basis conic compression lemma, a determinant--facet rigidity theorem for maximal certificates, and sharp sparsification lemmas for zero-mean weighted point systems. These tools may be of independent interest. We also exhibit an explicit six-point integer dataset with $d=m=2$ on which no weighted selection of $2d$ points recovers the optimal loss. Thus the scalar sufficient budget $2d$ does not extend to vector-valued outputs. Our new regression-profile results for $m\ge2$ extend the scalar theory for $m=1$.

cs.LG↗
arXiv2026

Fractal dimension predicts quantum kernel collapse in angle-encoded data

Angle-encoded quantum kernels on tabular data collapse when the feature map is wider than the intrinsic dimension of the data. We propose the correlation fractal dimension D2 as an a priori qubit budget: encode D2 coordinates chosen by FD-ASE instead of the PCA-95% width or all E attributes. On nine data sets and a statevector simulator (n= 32), a one-layer ZZ fidelity kernel at q=D2 stays geometrically alive while the same kernel at the PCA-95% width has already collapsed. The budget is map-dependent: product-state and IQP maps overshoot it; a second ZZ layer undershoots it. Packed dense-angle and re-uploading encodings still live at the fractal q, but not when PCA-95% features are stacked onto those qubits. Shrinking the angle bandwidth moves the ZZ knee later; stretching it kills the kernel earlier. On IBM Quantum (ibm_fez, 256 shots, n=8) the one-layer ZZ kernel at the fractal width matches the exact kernel (MAE 0.021); past that width both hardware and simulator have collapsed. The ceiling is a property of the map-data pair at a stated bandwidth, not of the classical table alone.

quant-ph↗
arXiv2026

DCCQ: From Ordered Bernoulli Levels to Critical-Line Geometry: Integer Quantization, Bernoulli Residual Phase, and Prime-Power Spectra

We present the discrete--complex complement quotient (DCCQ) framework through the ordered Bernoulli-word kernel $f(p,n,k)=p^k(1-p)^{n-k}$ and its inverse-integer level sets. The binary level $2^{-n}$ selects $p=1/2$ as the unique real split-independent anchor. Imposing the additional continuation rule $1-z=\overline z$ gives the conjugation-symmetric line $z=1/2+iu$. On the central normalized branch, the inverse-level split $κ=k/n$ also has real part $1/2$. The quadratic coordinate $Q(z)=1/4+u^2$ identifies complementary conjugate points, has minimum $1/4$, and has an explicit two-sign inverse; restricting its values to integers defines an arithmetic quantization. For supplied positive critical-line zero ordinates $γ_k$, the induced level has the exact decomposition \[ L_k=\frac14+γ_k^2=N_k+δ_k,\qquad δ_k=\widetilde B_1\!\left(γ_k^2+\frac34\right), \] where $N_k=\lfloor L_k+1/2\rfloor$ and $\widetilde B_1(x)=\{x\}-1/2$. The integer--phase pair $(N_k,e^{2πiδ_k})$ retains the level and the positive ordinate. Prime-exponent coordinates describe the factorization of the integer component, while conditional Fourier moments formulate an open question about the residual phases. The finite numerical diagnostics are descriptive. A distinct complex exponent $s$ gives an inverse-level representation of the supplied term $m^{-s}$; direct substitution cancels the auxiliary coordinates. These terms assemble into the classical Dirichlet series and Euler product for $\Re s>1$. The framework organizes exact coordinate identities, classical zeta connections, and numerical diagnostics; it establishes neither conditional equidistribution nor a zero-location theorem.

cs.LG↗
arXiv2026

Suan: Rectifying Direct Preference Safety Alignment in Large Language Models

Integrating robust safety guardrails into Large Language Models (LLMs) is essential for delivering helpful yet harmless responses. While proprietary systems exhibit reliable safety controls, their underlying methodologies and trade-offs remain largely undisclosed. Achieving comparable security in open-weight models remains a persistent challenge, as post-trained variants frequently suffer from over-refusal and degraded general quality. To overcome these drawbacks, we introduce Suan, a novel preference optimization algorithm. Unlike existing methods, we formulate the optimization objective directly at the gradient level, bypassing the standard variational derivation. As a result, we obtain more interpretable and robust training dynamics. Extensive evaluations across a diverse suite of competitive baselines and benchmarks demonstrate that Suan achieves superior safety alignment while fully preserving response utility.

cs.LG↗
arXiv2026

Intelligence per Watt: Measuring Intelligence Efficiency of Local AI

Large language model (LLM) queries are predominantly processed by frontier models in centralized cloud infrastructure. Demand growth strains this paradigm faster than providers can scale. Two advances create an opportunity to rethink it: small, local LMs (<=20B active parameters) now achieve competitive performance to frontier models on many tasks, and local accelerators (e.g., Apple M4 Max) can host these models at interactive latencies. This raises the question: can local inference viably redistribute demand from centralized infrastructure? This requires measuring both whether local LMs can accurately answer real-world queries and whether they can do so efficiently on power-constrained devices (e.g., laptops). We propose intelligence per watt (IPW), task accuracy per unit of power, as a unified metric for the capability and efficiency of local inference across model-accelerator configurations. We evaluate 20+ state-of-the-art local LMs, 8 hardware accelerators (local and cloud), and 1M real-world single-turn chat and reasoning queries. For each query, we measure accuracy (local LM win rate against frontier models), energy, latency, and power. We find three key results. First, local LMs successfully answer 88.7% of these queries, with accuracy varying by domain. Second, longitudinal analysis from 2023-2025 shows IPW improved 5.3x, driven by both algorithmic and accelerator advances, with locally-serviceable query coverage rising from 23.2% to 71.3%. Third, local accelerators achieve at least 1.4x lower IPW than cloud accelerators running identical models, revealing significant headroom for local accelerator optimization. These findings demonstrate that local inference can meaningfully redistribute demand from centralized infrastructure for a substantial subset of queries, with IPW serving as the critical metric for tracking this transition.

cs.DC↗
arXiv2026

Universal Approximation of Nonlinear Operators and Their Derivatives

We show that Universal Approximation (UA) of nonlinear operators and their derivatives via Operator Learning (OL) architectures fails in ${C^k_F}$ (Fréchet) compact-open topologies and in Fréchet--Sobolev norms (i.e. under operator norms). We solve this obstruction by restoring UA in natural weaker topologies: $C^k_B$ (Bastiani) compact-open topologies and (novel) weighted Bastiani--Sobolev spaces for general finite input measures. In full Banach-space generality, these are the first complete generalizations of the corresponding influential classical results in [Hornik, 1991] to infinite-dimensional spaces and OL. Based on our UATs, we formulate Bastiani--Sobolev training in DIOL. These results launch Derivative-Informed Operator Learning (DIOL) (i.e. learning nonlinear operators and their derivatives) on general Banach spaces. We parameterize nonlinear operators via Encoder-Decoder Architectures, classical OL architectures available in general Banach spaces; these include DeepONets, Deep-H-ONets, and PCA-Nets, which our UATs cover. A key mathematical result is that our new weighted Bastiani--Sobolev spaces generalize classical Gaussian (Malliavin) Sobolev spaces on Banach spaces. Open frontiers where DIOL and our UATs find applications are: high-order accuracy in OL; fast constrained optimization in Banach spaces (e.g. optimal control of PDEs, inverse problems) via Learn-Then-Optimize; numerical methods for infinite-dimensional PDEs (e.g. HJB PDEs on Banach spaces from infinite-dimensional optimal control via Optimize-Then-Learn, such as optimal control of PDEs, SPDEs, path-dependent systems, partially observed systems, mean-field control).

cs.LG↗
arXiv2026

Disciplined Bilevel Programming

Bilevel optimization provides a natural modeling language for hierarchical decision problems. However, applying existing numerical solvers usually requires substantial manual analysis and reformulation. In this paper, we introduce disciplined bilevel programming (DBLP), a symbolic framework that allows users to specify and solve optimistic bilevel problems in a high-level, human-readable way that is close to the mathematical formulation. For problems with a disciplined nonlinear upper problem and a convex lower problem satisfying the disciplined parameterized programming rules, DBLP automatically canonicalizes the lower problem into conic form and constructs an equivalent single-level reformulation using the conic Karush-Kuhn-Tucker conditions. We relax the resulting complementarity constraint and use a gap continuation procedure to approximately solve a sequence of smooth nonlinear problems. We implement DBLP in the open-source Python package BLVPY, an extension of CVXPY for bilevel programming. We demonstrate the modeling and solution capabilities of BLVPY on a range of bilevel optimization problems from several application domains. The proposed framework and implementation allow users to specify and solve bilevel optimization problems within a few lines of code, without prior expertise in bilevel modeling and numerical optimization.

math.OC↗
arXiv2026

SMILE: Bridging Continuous Optimization and Discrete Symbolic Recovery

Symbolic regression (SR) discovers closed-form mathematical expressions from data, offering interpretability beyond black-box models. Existing methods suffer from slow convergence in combinatorial search spaces and lack mechanisms to exploit compositional structure in the data. We introduce SMILE (Sine, Multiplication, Identity, Logarithm, Exponential), a hybrid framework that unifies continuous gradient-based optimization with discrete symbolic recovery through three stages: structural analysis of the data to identify the compositional hierarchy of the target expression, continuous optimization to learn parameters of a network that encodes the target expression using interpretable activations, and symbolic recovery through structured pruning, coefficient optimization, and rounding. This final stage distills the learned network into a compact expression with exact symbolic constants. We evaluate SMILE on SRBench across ground-truth and black-box datasets, with ablation studies validating each component. SMILE achieves the highest symbolic solution rate at the largest noise levels, demonstrating strong robustness where competing methods degrade substantially. It consistently lies on the Pareto front of accuracy versus complexity, recovering significantly simpler expressions in a fraction of the time required by the competing methods.

cs.LG↗
arXiv2026

GLASS: Graph-Language Alignment with Spherical Scoring for Transferable Graph-Level Anomaly Detection

We introduce GLASS, a framework for graph-level anomaly detection (GLAD) that achieves robust cross-domain transferability through graph-language alignment on the unit hypersphere. GLASS builds a unified representation space by aligning a structure-aware graph encoder with an instruction-aware text embedding via a multi-slice soft cosine objective. Our framework serializes local, global, and semantic graph properties into a compact Graph Descriptor Prompt (GraphDP), creating a text bridge that enables domain-agnostic anomaly scoring. By enforcing multi-scale consistency through Matryoshka representation slices, the model captures anomalous deviations at multiple levels of granularity. We formulate anomaly detection as density estimation on the aligned hypersphere and introduce Spherical Multi-Modal Scoring (SMS), which instantiates von Mises-Fisher kernel density estimators in both graph and text embedding spaces. This probabilistic formulation recovers angular 1-nearest-neighbor scoring in the high-concentration limit, motivates the practical mean k-nearest-neighbor scorer, and provides a principled fusion of structural and semantic anomaly signals. The shared text embedding space further serves as a cross-domain bridge: by encoding a target domain's GraphDP without target-domain training data, GLASS performs zero-shot anomaly detection, and with only a handful of normal examples, few-shot adaptation via reference-set calibration. For privacy-sensitive deployment, we extend reference-set calibration with a bounded joint graph-text kernel summary that provides graph-record differential privacy while keeping the encoders fixed independently of the private target references. Across twelve benchmarks and three meta-domains, GLASS obtains the best average AUROC and rank compared with recent advanced GLAD baselines and enables effective cross-domain transfer.

cs.LG↗
arXiv2026

Tensor-Train Weak SINDy: Identifying High-Dimensional Nonlinear Dynamics

Weak Sparse Identification of Nonlinear Dynamics (WSINDy) provides a noise-robust approach for learning dynamical systems from data without requiring numerical differentiation. However, for high-dimensional systems, tensor-product libraries of candidate functions grow exponentially with the state dimension, making standard WSINDy expensive in both computation and memory. The Multidimensional Approximation of Nonlinear Dynamics (MANDy) addresses this scaling through a tensor-train (TT) representation of the candidate library, but does not provide a mechanism for sparse model selection. Here, we combine these approaches to develop TT-WSINDy, which performs the weak-form transformation, regression, and sparsification in TT format. We show that the TT formulation recovers the corresponding WSINDy regression problem and derive polynomial time and memory complexity bounds for the tensor-train sparsification procedure. Numerical experiments demonstrate robustness to measurement noise and computational savings for high-dimensional systems.

cs.LG↗
arXiv2026

Supraglacial Lake Fate Is Knowable Long Before the Season Ends

A supraglacial lake on the Greenland Ice Sheet ends its melt season in one of four ways: it drains rapidly through a hydrofracture, drains slowly across the surface, refreezes in place, or is buried by late-season snowfall. Which one occurs decides whether the meltwater reaches the ice bed. Satellite classifiers recover the outcome accurately but only after the season closes, and how much of a season each outcome actually requires has never been measured. We measure it directly: holding the representation and the classifier fixed, we truncate the input at 14 cutoffs from May 1 to December 31, retrain at each, and record the earliest cutoff at which each outcome's per-class F1 reaches a fixed target. The outcomes resolve in a consistent order, two of them months early: rapid drainage by July 15 and slow drainage by August 1, respectively 92 and 75 days ahead of the earliest date a full-season pipeline can be computed at all, with buried and refreeze following at 44 and 30 days. Five further learners, from a majority-class floor and 54 summary statistics to a trigger-based early classifier, leave the ordering largely intact: the three that produce a per-class trajectory reproduce it in five of six cases despite end-of-season accuracies differing by up to 18 percentage points, and it survives leave-one-basin-out evaluation, though not a move to machine-labeled lakes in an unseen season. Every feature we compute at day t reads only days up to t, at a cost of at most 1.3 percentage points. A monitoring system should therefore not have one release date: rapid drainage can be flagged on July 15, three months before a full-season pipeline can be computed at all.

cs.LG↗
arXiv2026

RegionFM: Interpretable Region-Based Brain MRI Classification Using Foundation Model Embeddings

Foundation models provide powerful representations for brain MRI analysis, but their predictions remain difficult to interpret in anatomically meaningful terms. Clinical assessment of brain MRI is commonly organized around anatomically defined structures and regional abnormalities, whereas conventional explanation methods typically produce voxel- or patch-level importance maps that do not explicitly quantify the contributions of individual brain regions. To address this mismatch, we propose RegionFM, an interpretable framework that integrates anatomical segmentation with brain MRI foundation-model embeddings. RegionFM first divides each MRI scan into anatomical regions and constructs a separate MRI volume for each region. A frozen foundation model then encodes each region into an embedding, and a region-additive logistic model combines these embeddings such that every anatomical region contributes an explicit scalar term to the final prediction. This formulation supports both subject-level and cohort-level analyses of regional contributions. We evaluate RegionFM on cognitive-impairment classification using embeddings from multiple pretrained brain MRI foundation models. The results show that RegionFM maintains performance comparable to less interpretable fine-tuning approaches while providing anatomically grounded explanations. Randomized embedding ablations yield near-chance performance, indicating that the predictions rely on meaningful structure captured by the foundation-model embeddings rather than simple feature statistics. Overall, RegionFM better aligns model explanations with anatomy-based clinical reasoning while maintaining competitive predictive performance.

cs.CV↗
arXiv2026

Dec-BFTRL: Squre-Root Regret for Decentralized Online Upper-Linearizable Optimization under Separation Access with Application to Continuous Submodular Maximization

We study decentralized online optimization of upper-linearizable payoffs over an action set under efficient separation access, with applications to online continuous diminishing-return (DR) submodular maximization. We propose Decentralized Barrier Follow-the-Regularized-Leader (Dec-BFTRL), and evaluate each agent's played action against the average of all local objectives. Each agent maps an internal iterate to a feasible action through an approximate gauge projection, communicates only a cumulative surrogate-gradient dual state, and invokes the local HybridNewton procedure to approximately minimize its post-communication BFTRL potential. For every agent, we achieve expected network-aggregate regret of $\widetilde O(\sqrt{T})$. Over $T$ rounds, each agent uses $T$ neighbor-mixing steps and $\widetilde O(T)$ separation-oracle calls. We give wrapper instantiations covering four up-concave or DR-submodular maximization problems.

math.OC↗
arXiv2026

CoER: Defending against Adaptive Indirect Prompt Injection via Adversarial Co-Evolution and Refinement

Language-model agents are vulnerable to indirect prompt injection (IPI) during tool use: adversarial instructions hidden in untrusted tool outputs can covertly redirect legitimate task execution. Existing work often trains and evaluates defenses against fixed attacks that do not adapt to the defender's behavior, so the resulting defenses may struggle against adaptive attacks in real-world settings. We argue that a strong defense against adaptive IPI must adapt during training to a continually evolving attacker. Building on this insight, we propose CoER, a verifier-grounded co-evolution and refinement framework that models interleaved tool calls and adaptive injections within a task as a general-sum Markov game: the defender advances the task through successive tool calls, while the attacker can inject multiple times within the same task and adapt subsequent attacks to the defender's responses and prior execution traces. After initializing the attacker from successful trajectories, bilateral adversarial reinforcement learning (BA-RL) retains historical policies from both roles as opponent populations and mixes current and historical opponents, extending training beyond the latest matchup. Attackers from these populations are then reused to challenge teacher agents, and only demonstrations verified for both safety and task completion are used to fine-tune the co-evolved defender. Across seven domains and three evaluation seeds, CoER reduces adaptive attack success from 41.3% to 0.2% and raises safe task completion from 39.6% to 76.2%; external benchmarks also show improved attack resistance. Further experiments validate the effectiveness of bilateral historical-opponent mixing and population-guided refinement. Attacker analyses show that co-evolution strengthens attack capabilities and that the trained attacker uses execution feedback to adapt subsequent injections.

cs.LG↗
arXiv2026

I Don't Miss You, but I Do: Self-Explanation Faithfulness of Modality Missingness in Vision-Language Models

Vision-language models (VLMs) are increasingly used in settings where some input modalities may be unavailable, yet we know little about whether they can faithfully explain how such missing information affects their own predictions. We introduce an interventional protocol for evaluating self-explanations of modality dynamics: models state what each modality alone would support, whether restoring a missing modality would change their answer, and whether the available evidence is sufficient; we then execute the corresponding intervention and compare these claims with realized behavior. We evaluate ten VLMs spanning open-weight and proprietary models across four tasks covering mixed, redundant, and unique modality regimes. We find a systematic tendency to overstate the sufficiency of available evidence. Models substantially underestimate the effect of restoring missing modalities: executed change exceeds predicted change in 78 of 80 model-task-condition settings, with task-level median executed change rates reaching 70.1\% while median predicted rates remain at most 9.6\%. Insufficiency claims have low recall, leaving many cases in which behavior changes despite a stated claim of sufficiency. Retrospective self-explanations show the same tendency, over-crediting single-input sufficiency in mixed regimes and interchangeability in redundant ones. Together, these results show that VLMs systematically mischaracterize how their predictions depend on available and missing evidence, motivating executable interventions as a behavioral test of multimodal self-explanations.

cs.LG↗
arXiv2026

Equivariance Breaks the Learning Rate

Equivariant networks are commonly trained with Adam, yet recent work reports that matrix structured optimizers such as Muon can perform better, with the reasons for these gains only partly understood. We identify one source of this difference inside equivariant layers. An equivariant layer learns one channel mixing matrix $W_l$ per degree $l$, which we call an irrep block, and shares it across the $2l+1$ components, giving the expanded map $W_l \otimes I_{2l+1}$. This sharing sums gradient contributions across components and can produce different update scales under SGD. Adam's entrywise normalization reduces sensitivity to gradient scale, but neither optimizer directly controls the effective step size of each block. A single learning rate can therefore produce different effective step sizes across blocks. Muon instead controls the effective step size by approximately equalizing the singular values of each momentum matrix. We normalize each irrep block update by a single scalar, preserving its singular value ratios while letting the learning rate control its size. We implement this with spectral normalization or a simpler root-mean-square normalization. We evaluate spectral normalization in a controlled $\mathrm{SO}(3)$-equivariant model with a matched non-equivariant model. In this setting, the step size mismatch grows with width in the equivariant model but not in the non-equivariant model. We evaluate both variants across molecular force prediction on the rMD17 and MD22 datasets, QM9 molecular property prediction, and charged particle dynamics. Across these applications, block normalization generally improves Adam and closes part of its gap to Muon. These results highlight an overlooked interaction between equivariant architectures and their optimizers. Studying and designing the two together may help explain and address training difficulties often attributed to equivariance itself.

cs.LG↗