NeurIPS 2026 Papers

Mixture-of-experts (MoE) is a common approach for increasing parameter capacity, but applying MoE to state space model (SSM) token mixers can multiply the cost of the recurrent state update. We study how to introduce expert specialization into selective SSMs while preserving computational efficiency. We show that MoE--SSM can refer to two designs: (1) MoE over separated SSMs, which maintains multiple state trajectories and thus scales compute with the number of experts; and (2) MoE-parameterized SSM, which mixes experts in parameter space, maintains a single state trajectory, and evaluates the recurrence once. Our method, Switch Mamba (Swimba), follows the second design by routing over expert-produced SSM streams. Theoretically, we establish well-definedness and stability for MoE-parameterized SSMs and characterize the relationship between the two designs. Empirically, we evaluate Swimba on standard benchmark tasks and measure real-time throughput and latency. Under matched FLOPs, Swimba achieves slightly better average performance than the baseline, with a small slowdown in real-time latency and throughput. Overall, these results suggest that parameter-space MoE can increase SSM capacity while keeping the dominant recurrence cost fixed.

Discrete diffusion models are increasingly competitive for language modeling, yet it remains unclear how their denoising objectives organize learning. Although these objectives target the full data distribution, we show that the exact reverse process induces a hierarchy between coarse *support* information and finer *frequency* information. For uniform and absorbing (a.k.a. masking) diffusion, we prove that, in the small-noise regime of the final denoising steps, each single-token reverse edit decomposes into a leading scale, determined by whether it moves toward the data *support* (e.g., grammatically valid sentences), and a finer coefficient, determining *relative probabilities* within the same scale. Thus, recovering validity structure only requires learning the correct order of magnitude of reverse probabilities, whereas recovering data frequencies requires coefficient-level estimation. The separation is mechanism-dependent: uniform diffusion exhibits a trichotomy into validity-improving, validity-preserving, and validity-worsening edits, while absorbing diffusion places its leading-order mass on validity-improving moves. Experiments on a masked language diffusion model and synthetic regular-language tasks support these predictions: support-localization emerges earlier than within-support frequency ranking, and the contrast between uniform and absorbing diffusion matches the predicted rate separation. Together, our results suggest that discrete diffusion models learn data support before data frequencies.

Model distillation can transfer not only intended capabilities but also hidden traits from the teacher. A teacher biased by a system prompt can generate semantically clean training data, such as numeric sequences, that still causes a downstream student to inherit the hidden preference, a phenomenon known as $\textit{subliminal learning}$. Prior work establishes this phenomenon, but the transfer mechanism remains unclear, making targeted mitigation difficult. We propose and validate $\textbf{trait-direction drift}$ as the mechanism underlying subliminal learning: $\textbf{(1)}$ on the teacher side, biased teacher samples retain a measurable preference gap toward the target trait even when their semantic content is unrelated to that trait; $\textbf{(2)}$ on the student side, under a low-rank logit-linear approximation, samples with larger preference gaps induce stronger trait-aligned updates during supervised fine-tuning, and these updates accumulate over training. Guided by this mechanism, we propose $\textbf{probe-space corridor}$, a regularizer that constrains drift along a calibrated trait readout during distillation. The method substantially reduces hidden-trait transfer while preserving task performance: for example, it lowers malicious-response transfer from 29.55% to 6.47% with low main-task accuracy cost, and consistently suppresses animal-preference transfer across the main Qwen-7B preference settings. These results provide both a validated mechanistic explanation of subliminal learning and a targeted recipe for more controllable distillation.

Looped Transformers --- which repeatedly apply a shared transformer block --- are an architecturally natural fit for variable-length algorithmic tasks. Although they can exhibit strong length generalization beyond the length of training sequences, this behavior is brittle, yielding high out-of-distribution (OOD) variance, even across well-performing in-distribution solutions. We trace this variance to the spurious correlation in simple algorithmic tasks between sequence length and number of loops. Introducing stochasticity into the number of loops during training sharply reduces OOD variance and stabilizes predictions across inference-time loop counts. To improve upon heuristic randomization schemes, we further analyze RL-Halting as a learned stochastic schedule and find that it generally improves the accuracy--stability trade-off. Across binary addition, Dyck-1, Unique Set, and Copy, learned stochastic stopping often improves this trade-off but can also stabilize a suboptimal computation. Our work suggests that “when to stop” should be treated as a training-time design choice, not merely an inference-time computation-allocation rule.

Online platforms connect users with relevant products and services using ads. A key challenge is that a user's search query often leaves their true intent ambiguous. Typically, platforms passively predict relevance based on available signals and in some cases offer query refinements. The shift from traditional search to conversational AI provides a new approach. When a user's query is ambiguous, a Large Language Model (LLM) can proactively offer several clarifying follow-up prompts. In this paper we consider the following: what if some of these follow-up prompts can be ``sponsored,'' i.e., selected for their advertising potential. How should these ``suggestion slots'' be allocated? And, how does this new mechanism interact with the traditional ad auction that might follow? This paper introduces a formal model for designing and analyzing these interactive platforms. We use this model to investigate a critical engineering choice: whether it is better to jointly optimize the user interaction and the final ad auction, or to decouple them into separate mechanisms for the suggestion slots and another for the subsequent ad slot. We show that the VCG mechanism can be adopted to jointly optimize the sponsored suggestion and the ads that follow; while this mechanism is more complex, it achieves outcomes that are efficient and truthful. On the other hand, we prove that the simple-to-implement modular approach suffers from strategic inefficiency: its Price of Anarchy is unbounded, for various natural choices for the two mechanisms. While we show that this inefficiency can be mitigated through revenue redistribution, we argue that the resulting mechanism sacrifices the practical simplicity that motivated the modular design in the first place.

Neural avalanches and turbulence-like brain dynamics are usually measured after neural activity has been coarse-grained into continuous fields, leaving open which spike-level mechanisms can generate these macroscopic observables. We build a spatial leaky integrate-and-fire spiking neural network and explicitly transform spikes into activity-density and Hilbert phase fields. A delayed, distance-dependent, passive-dendrite regime reproducibly generates turbulence-like field structure, including high-amplitude phase-defect tracks, structure-function scaling, and power-law-favored continuous-field avalanche tails. Controls show that firing rate, vortex count, and avalanche tail fits are individually insufficient: destroying delayed propagation, dendritic filtering, or spike timing suppresses key field signatures, while random-space nulls can retain many mathematical phase defects but lose spatial scaling and synaptic-delay transfer. Biologically motivated gates further shape the regime: PV-like perisomatic inhibition clamps phase dynamics, SST-like dendritic inhibition and projection-specific inhibitory facilitation preserve long-lived tracks, and a rate-constrained projection-STP regime maintains similar dynamics at 12.25 Hz across 12 seeds. These results use SNNs as a mechanistic testbed for spike-derived brain turbulence observables and show why phase defects and avalanche-like statistics must be interpreted jointly with spatial scaling and spike-level propagation.

Active view selection for 3D Gaussian Splatting (3DGS) requires information-theoretic grounding, computational efficiency, and strong reconstruction quality, yet existing methods satisfy these only partially. Information-theoretic approaches quantify gain over the full parameter space via costly Fisher computation under a Laplace approximation; faster alternatives sacrifice the grounding they started from. We propose SpheriBED, a Bayesian experimental design framework that defines the information objective over per-Gaussian spherical appearance—the quantity each observation fundamentally measures—rather than the full parameter vector, yielding exact closed-form posteriors without Laplace approximation. Scoring a candidate view requires only a single forward rasterization pass with no backward pass or per-candidate optimization. Extensive experiments across benchmarks demonstrate that SpheriBED consistently improves active reconstruction performance and provides more reliable uncertainty quantification compared to state-of-the-art methods.

This paper studies the role of sinks and diagonal patterns as attention switch and anti-oversmoothing mechanisms. We analyze geometric conditions under which sinks can be represented, showing a necessary alignment between the embedding of the sink and all other embeddings. Next, we refine the current understanding of the role of sinks in oversmoothing prevention: we specify the conditions under which dense attention provably smooths more than sparse attention, and empirically verify that such conditions are often satisfied in practice. We further prove an equivalence between sinks and hard attention switch, in which the output of the attention is identically 0. Finally, we relax the hard attention switch by allowing token self-communication: we provide a quantitative comparison of the costs of representing sinks vs.\ diagonal patterns, showing why sinks are favored in pretrained transformers. The introduction and analysis of diagonal patterns and the generalization of the attention switch close the gap between what oversmoothing prevention requires and what sinks provide, while also establishing when and why attention layers act like MLPs if token communication is not necessary.

We study equilibrium learning in discretized Bayesian games, focusing on first-price auctions as a testbed for gradient-based learning dynamics under private information. While the symmetric independent private values model admits a unique Bayesian correlated equilibrium (BCE) in the continuous limit, standard convergence guarantees for gradient-based learners do not apply: classical no-regret theory predicts convergence only to Bayesian coarse correlated equilibria (BCCE), which can remain large and far from the competitive equilibrium even under fine discretizations. To bridge this gap, we introduce the notion of Bayesian semicoarse correlated equilibrium (BSCCE), an equilibrium concept that refines BCCE by restricting deviations to probability-preserving transformations consistent with projected gradient dynamics. We provide a linear programming characterization of BSCCE with polynomially many variables and constraints and establish the inclusion hierarchy $BCE \subseteq BSCCE \subseteq BCCE$. Our main finding is that the BSCCE set contracts in many cases as the discretization of the Bayesian game is refined. Even with priors for which BCCE fails to concentrate, the diameter of the BSCCE polytope shrinks toward zero, indicating convergence to the unique continuous-game Bayes-Nash equilibrium. These results provide a principled explanation for the observed robustness of gradient-based learning in discretized Bayesian games, showing that these dynamics select a smaller equilibrium set than standard no-regret theory predicts.

U-Net--style architectures are widely adopted for modeling physical systems, as their multiscale structure enables efficient processing of high-resolution data while reflecting the hierarchical structure of many physical phenomena. The most successful U-Net-based neural physics simulators often combine convolutions, Fourier layers, or specialized transformer mechanisms such as windowed or axial attention; however, these structured components can limit adaptability across different spatial and spatiotemporal dimensions. In this work, we revisit transformer-based U-Nets with the goal of simplifying attention-based neural physics simulators without sacrificing multiscale efficiency or predictive accuracy. We introduce \texttt{UFlex}, a simple attention-based U-Net whose blocks are obtained through only minimal modifications to standard self-attention, allowing the same architecture to be applied across one-, two-, three-dimensional, and spatiotemporal regular-grid problems without major design changes. Evaluated on seven challenging benchmarks (four 2D and three 3D), \texttt{UFlex} scales to resolutions of up to $512 \times 512$ in 2D and $256 \times 128 \times 256$ in 3D, while reducing training memory and accelerating training compared to state-of-the-art transformer baselines, all while achieving state-of-the-art predictive accuracy.

Large language models (LLMs) have emerged as powerful tools for automated heuristic design (AHD), enabling iterative generation and refinement of heuristics. However, the dominant paradigm embeds LLMs as narrow, fixed components, such as crossover or mutation, within heavily hand‑engineered evolutionary frameworks. We argue this misapprehends LLMs. It treats them as specialized tools rather than general reasoners, constrains them to low‑level operations, and underutilizes their autonomy. Moreover, the extensive human priors in these frameworks violate the bitter lesson principle that general methods scaling with computation surpass hand‑crafted solutions. This raises a key question: *which AHD framework designs best convert stronger LLM capabilities into better heuristics?* To address this, we propose metrics for AHD framework handcraftedness (AHI) and intelligence conversion efficiency (ICE). Evaluating ten LLMs across three challenging optimization problems, we obtain a notable finding that frameworks with fewer human priors consistently yield higher ICE. Based on this finding, we propose SimpleEvol, which removes nearly all structural constraints and allows the LLM to operate autonomously. SimpleEvol consistently achieves the highest ICE, often by a large margin. Our results challenge the trend toward complex AHD pipelines and point to a simpler and more model‑centric alternative, suggesting that reducing human priors is a more effective strategy to scale up with model intelligence.

Verifiable differential privacy enables public auditing of privacy-preserving aggregate computations, addressing the trust deficit in standard implementations. However, existing verifiable counting approaches that distribute trust across multiple provers often require each prover to contribute independent noise. This duplicates the noise added to the aggregate and substantially degrades utility, making accurate private counting difficult in distributed settings. We introduce a shared-noise paradigm for verifiable differentially private counting. Instead of duplicating noise across provers, our approach generates a single noise sample in additive-shared form among mutually distrustful provers, while Pedersen commitments let a public verifier audit consistency without learning the noise. This yields shared-noise, verifier-auditable instantiations of two standard noise families: the Shared-Noise Verifiable Binomial Mechanism (SN-VBM) and the Shared-Noise Verifiable Laplace Mechanism (SN-VLM). We prove that both mechanisms satisfy differential privacy and public verifiability, including completeness, soundness against malicious provers, and zero knowledge against a malicious verifier colluding with corrupted provers. Among the two instantiations, SN-VLM is the more scalable option: it avoids the quadratic growth in shared randomness required by the binomial construction under strict privacy budgets. Experiments on preprocessing load, randomness-generation time, and noise error show that SN-VLM substantially reduces cryptographic overhead while improving empirical accuracy in the evaluated privacy regimes.

We study the problem of $d$-dimensional covariance matrix release under \textit{pure} differential privacy with error measured in Schatten-$p$ norms, where $p\in [1,\infty].$ We identify two meaningful sample-size regimes with qualitatively different behavior, and give a single efficient algorithm together with matching information-theoretic lower bounds throughout. In the \emph{large-sample} regime $n \gtrsim d^{2}/\varepsilon$, we show that the simple $K$-norm mechanism analyzed by \cite{d2026purely} is simultaneously optimal for all Schatten norms $p\in[1,\infty]$, yielding sharp rates from nuclear to spectral norm. In the \emph{moderate-sample} regime $d/\varepsilon \lesssim n \lesssim d^{2}/\varepsilon$, the problem remains non-trivially solvable, but the $K$-norm mechanism becomes suboptimal. We then design a single improved, efficient, $\varepsilon$-differentially private estimator based on the perturb-and-project framework; this estimator is optimal in all regimes and simultaneously for all Schatten norms. Prior to this work, improvements over the $K$-norm mechanism in this regime were only known for Frobenius loss \cite{nikolov2023private}. A key feature of the moderate-sample regime is an inherent dependency of the error on the nuclear norm of the input, first observed by \cite{dong2022differentially} and further investigated by \cite{d2026purely}. Our mechanism captures this dependence uniformly over $p$, and our lower bounds show it to be unavoidable, establishing optimal rates in all regimes, including the correct dependence on $\|\Sigma\|_*$. %The implications of our approach go beyond covariance release: we obtain optimal guarantees for privately releasing arbitrary matrices under nuclear-norm adjacency, and give partial extensions to higher-order moment estimation under pure differential privacy. These extensions highlight the perturb-and-project framework as a flexible tool for private high-dimensional data release, even beyond second moments. Our results extend and are optimal for the more general problem of privately releasing arbitrary matrices under nuclear-norm adjacency. Finally, our lower-bound techniques further generalize to higher-order tensors, establishing similar limitations for higher-order moment estimation under pure differential privacy.

Protein sequence optimization under tight oracle budgets requires methods that explore vast combinatorial spaces while making each evaluation informative. Existing reinforcement learning and off-policy generative approaches often degrade under surrogate noise, and position-agnostic mutation proposals risk disrupting functionally critical residues. We introduce SILO, a trajectory-level self-improvement imitation framework for oracle-budgeted protein design. SILO uses a hierarchical edit policy that decomposes each mutation into a position choice followed by a residue choice. In each active-learning round, the policy samples candidate trajectories via incremental stochastic beam search without replacement (SBS), and a UCB-based proxy ensemble, combined with an alanine-scan fitness score (AFS), selects candidates with functionally relevant edits for in silico oracle evaluation. The policy is then updated by next-action cross-entropy imitation on the round’s best oracle-labeled trajectories, avoiding value-function estimation. Across eight reproduced protein fitness landscapes and five strong baselines from prior work, SILO achieves the highest maximum and top-100 mean fitness on 8 of 8 landscapes within our evaluations, often exhibiting faster early-stage improvement. In low-data and noisy-proxy stress tests on two landscapes per setting, SILO remains competitive or best when several baselines degrade. Ablations show that SBS with AFS account for much of the gains, with iterative imitation providing additional improvement.

The Forward-Forward (FF) algorithm trains networks layer-by-layer using a local "goodness function," yet the principles governing what makes an effective goodness function remain largely unexplored. We systematically explore the goodness-function design space and identify a unifying principle: the goodness function must be sensitive to the shape of neural activity, not its total energy. This principle is motivated by the observation that deep network activations follow heavy-tailed distributions and that discriminative information is often concentrated in peak activities. We propose two complementary families: selective functions (top-k, entmax-weighted energy) that measure only peak activity, and shape-sensitive functions (excess kurtosis / "burstiness" and higher-order moments) that reward heavy-tailed distributions via scale-invariant statistics. Combined with separate label-feature forwarding (FFCL), controlled experiments across 13 goodness functions, 5 activations, 6 datasets, and three continuous sweeps each tracing a characteristic inverted-U yield 89.0% on Fashion-MNIST and 98.2±0.1% on MNIST (4x2000)—a +32.6pp gain over SoS—with consistent improvements across all benchmarks (+72pp USPS, +52pp SVHN). The scale-invariant nature of burstiness makes it particularly robust to magnitude shifts across layers and datasets. Code is available at [https://anonymous.4open.science/r/ff-selectivity-shape](https://anonymous.4open.science/r/ff-selectivity-shape).

Selective state-space models such as Mamba replace fixed linear state transitions by input-dependent transitions, yielding recurrent sequence models that retain linear-time inference while adapting their memory dynamics to the current token. We show that even a simple form of this selective memory update has universal approximation power. Specifically, for every compact input domain, finite horizon, and continuous causal sequence-to-sequence map, a clocked diagonal selective state-space scan with strictly contractive gates and a linear-in-state readout uniformly approximates the target map. The proof is constructive: input-dependent diagonal propagation generates tensor-product partition features over prefixes, and the linear-in-state readout combines these features with sampled target values. Thus the high-order interactions required for universality are produced inside the selective scan itself, rather than by attention, nonlinear hidden-state dynamics, or a universal multilayer perceptron applied after a sequence encoder. These results identify selectivity as a mathematically sufficient mechanism for expressive linear-time state-space sequence modeling. Selectivity lets the model decide, token by token, which parts of the past should be preserved, faded, or combined with the present. The universal approximation theorem shows that this adaptive use of memory is sufficient to reproduce any well-behaved causal sequence rule over a fixed horizon.

Tokenizer-free language models eliminate the tokenizer step of the language modeling pipeline by operating directly on bytes; patch-based variants further aggregate contiguous byte spans into patches for efficiency. However, the average patch size chosen at the model design stage governs a tight trade-off: larger patches reduce compute and KV-cache footprint, but degrade modeling quality. We trace this trade-off to patch lag: until a patch is fully observed, byte predictions within it must rely on a stale representation from the previous patch to preserve causality; this lag widens as patches grow larger. We introduce Scratchpad Patching (SP), which inserts transient scratchpads inside each patch to aggregate the bytes seen so far and refresh patch-level context for subsequent predictions. SP triggers scratchpads using next-byte prediction entropy, selectively allocating compute to information-dense regions and enabling post-hoc adjustment of inference-time compute. Across experiments on natural language and code, SP improves model quality at the same patch size; for example, even at $16$ bytes per patch, SP-augmented models match or closely approach the byte-level baseline on downstream evaluations while using a $16\times$ smaller KV cache over patches and $3$–$4\times$ less inference compute.

Unstructured pruning methods such as Wanda preserve model quality but produce irregular sparsity with no practical compression, while structured methods enable efficient inference at the cost of degraded performance. We ask whether post-hoc linear transformations can convert unstructured sparsity into structured patterns, combining both benefits. Across a broad class of fixed linear transformations—including rotations, decompositions, and permutations—we find no increase in structured sparsity (Δ = 0.0%) in any OPT feed-forward layer. Transformations that yield structure do so identically on dense matrices, independent of pruning. We attribute this to three factors: rank preservation, weak inter-row mask correlation, and spectral dispersion. Together, these prevent post-hoc transformations from inducing structure, suggesting that compressible sparsity must be introduced during pruning rather than recovered afterward.

Probabilities of causation (PoCs), such as the probability of necessity and sufficiency (PNS), are important tools for decision making but are generally not point identifiable. Existing work has derived bounds for these quantities using combinations of experimental and observational data. However, there is very limited research on sample size analysis, namely, how many experimental and observational samples are required to achieve a desired margin of error. In this paper, we propose a sample-size design framework for PoC bounds that can be expressed as finite minima or maxima of smooth functions of experimental and observational probabilities — a representation that covers binary PNS, PN, and PS, as well as representative multi-valued PoC families recently shown to subsume the discrete PoC bounds in the current literature. Our framework gives endpoint-specific sample-size formulas that account for the joint covariance structure of the probability estimates: a distribution-free rule for affine bounds, and a pilot-based plug-in rule with theoretical safety guarantees for ratio-type bounds. Simulations show that the proposed rules achieve the target precision and coverage, recover a much smaller size requirement for binary PNS than the existing calculation, and provide sample-size designs for other discrete PoCs where no prior general sample-size formula is available.