arXiv:2610.12541v1 Announce Type: new Abstract: Boundary-value problems for the Laplace equation develop singular behaviour near corners, edges and junctions that smooth approximators resolve poorly. We combine holomorphic neural networks with singular functions derived from the local geometry and boundary conditions. The representation is harmonic by construction, so no interior residual is needed: the network fits the smooth remainder from boundary data alone, and the enrichment terms capture the singular behaviour. Singular exponents are trainable, so the construction also applies when the local expansion is not known in advance. We extend the approach to three dimensions using independent networks evaluated along complex null directions. The enriched model is the most accurate tested method on every problem. Its margin is several orders of magnitude where the enrichment captures the singularity exactly, and a factor of about 1.7-2.1 over the next-best method on a CdZnTe radiation detector with many singular sites, in both two and three dimensions.
arXiv:2610.12520v1 Announce Type: new Abstract: The goal of circuit discovery is to decompose a monolithic neural network into tiny modular subnetworks which are interpretable in the sense that each does an identifiable subtask. We develop a practical method for composing arbitrarily many circuits into a single model. We propose a greedy, density-scaled composition algorithm that overcomes limitations of previous attempts to compose circuits by naive union, then generalize it to $N$ circuits via CircuitTree, a new data structure built on a $\Theta(N\log N)$ tree-compose schedule. We compose circuits pruned from HuBERT for classifying individual phonemes into a syllable classifier, a capability none of the 39 phoneme circuits has individually. Chaining CircuitTrees into a hierarchy-of-circuits pipeline, we transcribe open syllables at 64.8\% accuracy, within 1.2 points of a 66.0\% ceiling that shares no circuitry at all, with 99.5\% of circuit components shared.
arXiv:2610.12496v1 Announce Type: new Abstract: Sequence models with a fixed-size state avoid the quadratic cost of attention, but face a design choice. An input-dependent state transition gives content-dependent memory at the cost of FFT-based training, while a fixed transition keeps the O(T log T) convolutional path but reads memory the same way for every input. ResonatorLM, a recent model of the second kind, reads a bank of damped complex oscillators through a fixed learned phase per head. We make that readout phase a function of the current token, which adds input dependence to the read only, keeps the convolution exact, and costs 0.2 percent more parameters. An analysis of the untrained bank shows that phase does encode elapsed time, but an ordinary token falls below the background within 2-8 steps and no choice of damping extends linear reconstruction beyond about 16 tokens. In 6M-parameter character-level models on WikiText-2, the input-dependent phase matches the original at the training length and removes its failure beyond it: trained on 256 tokens and tested on 1024, perplexity falls from 12.81 to 3.76. We trace the failure to long-memory heads that accumulate more history than training exposed them to. The learned phase tracks the current character rather than distance and does not improve retrieval.
arXiv:2610.12495v1 Announce Type: new Abstract: Small language models trained on a single local machine leave little room for architectural trial and error. Kairos combines bidirectional gated-DeltaNet linear attention and sliding-window attention over shared projections (LiZAttention2), block-wise Attention-Residual aggregation, a sparse Mixture-of-Experts (MoE) feed-forward layer and a multi-scale byte-level codec, trained with a uniform-noise denoising objective over one byte space shared by text, image, audio, video, lidar and control streams; this report asks whether the combination optimizes at all before asking whether it is useful. We give code-derived algorithms for each component, including a cross-session recurrent-state memory gate, then run convergence sanity checks: overfittinga single 440-byte sentence at 1--15M parameters, text only, with three seeds for codec and attention choices and single-run sweeps for seven further factors. The two attention variants tested at 3\,000 steps (LiZAttention2 and window attention alone) reach a minimum loss near $10^{-3}$. After 600 steps, the mean minimum loss ranges from 0.018 (patch codec) to 0.354 (depthwise convolutional codec) with an identical ranking on three seeds. An apparent advantage of shared over separate attention projections closes with more steps or more diverse data and reverses when the projections are frozen: it is a convergence-speed effect, not a capacity effect. These checks suggest that the implementation optimizes; they say nothing about generalization, multimodal training or comparison with existing models, which a planned pretraining run is meant to address. The Discussion sizes a 200M-total/25M-active target analytically; none of its throughput figures is measured. Code is available at https://github.com/fabienfrfr/Kairos . Python package at https://pypi.org/project/kairos-fm/ .
Curriculum learning typically organizes training samples from easy to hard. In GRPO, medium-difficulty samples are attractive for online selection because they are more likely to yield informative group-relative learning signals, and prioritizing them can improve training efficiency. However, it remains unclear whether persistently selecting such samples also leads to the strongest final model performance, or whether training can benefit from progressing toward harder samples. Meanwhile, sample difficulty changes as the policy evolves, so fixed difficulty estimates can become misaligned with the current policy. We therefore re-estimate sample difficulty under the current policy and, inspired by the easy-to-hard principle of curriculum learning, design and evaluate a staged curriculum that progressively shifts the active difficulty band toward harder samples. Experiments on GSM8K show that persistent medium-difficulty selection does not achieve the strongest observed performance. Across three random seeds, progressing to a moderately harder difficulty band improves test accuracy to $0.6591 \pm 0.0077$, compared with $0.6449 \pm 0.0024$ for the fixed-medium curriculum. However, further progression to a harder difficulty band reduces accuracy to $0.6405 \pm 0.0053$. These results show that staged difficulty progression can outperform persistent medium-difficulty selection, but its benefit is non-monotonic. Effective GRPO curricula can therefore benefit from progressing beyond medium difficulty, while avoiding excessive progression toward overly difficult samples.}
Standard Policy Gradient (PG) algorithms suffer from high estimation variance because they rely on Monte Carlo updates. While off-policy variance reduction mitigates this problem, establishing optimal sample complexity currently requires either restrictive assumptions, such as uniformly bounded importance weights, or impractical second-order operations. In this work, we introduce Loopless Variance-Reduced PG (LVR-PG), a unified theoretical framework that abstracts a class of first-order estimators that integrate recursive updates with defensive importance sampling, including known estimators such as PAGE-PG and STORM-PG. We prove that this algorithmic class achieves $\tilde{\mathcal{O}}(ε^{-2})$ last-iterate global convergence under weak gradient domination for general parametric policies. We establish this guarantee using purely first-order updates, with minimal assumptions and without gradient preconditioners. By extending state-of-the-art lower bounds to continuous parametric policies, we establish an $Ω(ε^{-2})$ sample complexity limit, confirming our framework is minimax-optimal up to logarithmic factors and horizon terms.
Discrete graph diffusion models corrupt graph structure through independent edge updates with a Markovian noise model. Combined with the exchangeability requirement on the output distribution, the terminal prior is necessarily an Erdős--Rényi (E-R) random graph. This has a fundamental consequence for unconditional graph generation without node features: in the large graph limit, any GNN denoiser applied to an E-R graph approximately outputs an E-R graph, making effective denoising difficult. In practice, generated graphs fail to recover meaningful node heterogeneity and higher-order structure. Previous models introduced specialized node features to compensate, framing this as an expressivity problem, while the role of the prior remained unexamined. We instead address the root cause by introducing the Latent Sociability Prior (LSP) and a join} diffusion process that co-evolves latent node sociability variables and graph structure, removing edge independence while preserving exchangeability. Unlike existing latent graph diffusion methods, the sociability mechanism requires no complex encoder: its sole role is to guide edge updates and preserve node heterogeneity throughout the trajectory. We validate our approach on multiple graph generation datasets and show that the proposed framework more accurately captures graph distributions than baseline methods without node features.