We study Gaussian regression under squared population L_2 loss in a known m-dimensional subspace of degree-at-most-k functions on the d-dimensional Boolean cube. Random inputs can undersample regions essential for prediction, delaying the parametric rate even when the model is known. For fixed q_0<1/2, 1kq_0d, and sufficiently large fixed A, the worst-subspace sample threshold for minimax error Aσ^2(m+t)/n with confidence 1-e^{-t}, t4, is [ N=(m+t)\{E_{d,k}+O(k^{1/3})\}, E_{d,k}=dΨ(k/d), ] where Ψ(q)=2-H(12-{q(1-q)}) and H is binary entropy with natural logarithms. The upper bound holds for every feasible m; the matching lower bound holds when md{k^{1/3}} or tm. We sharpen the Polyanskiy--Samorodnitsky uncertainty principle in two respects. First, for fixed leakage ρ(0,1), the smallest set carrying a fraction 1-ρ of a nonzero degree-at-most-k polynomial's energy has probability \{-E_{d,k}+O_{ρ,q_0}(k^{1/3})\}. An Airy-kernel construction proves that the remainder cannot be o(k^{1/3}) in general. Second, we construct a subspace of dimension d{k^{1/3}} such that every function in the subspace has at least a fraction 1-ρ of its energy on the same set, whose probability is at most \{-E_{d,k}+C_{ρ,q_0}k^{1/3}\}. For sufficiently large k, this set is a Hamming ball. A striking consequence is an exponential cost of noise: the parametric rate can require (m+t)4^k\{-O(k^{1/3})\} samples, whereas O((m+t)2^k) suffice for noiseless identification. As kwith k/d0, the noisy threshold is (m+t)\{2k+o(k)\}.
Spatial reasoning remains a persistent weakness of vision-language models (VLMs), because RGB inputs do not directly provide geometric evidence. Existing remedies either inject 3D into the model at inference, paying architecture and latency costs, or train with outcome rewards that supervise only the final answer. Spatial errors originate in perception: a misjudged depth or direction can be corrected only by the scene's true geometry, which the 3D-scanned sources of spatial training corpora already provide. We propose GPD (Geometry-Privileged Distillation), which makes geometric evidence the privilege in on-policy self-distillation (OPSD). For each question, depth, semantic, and bird's-eye-view (BEV) cues are rendered as compact text and routed to the teacher alongside the reference answer; a privileged KL, applied only to incorrect trajectories, augments GRPO, and the deployed model remains RGB-only. On the 4B backbone, GPD achieves 57.1 on VSI-Bench and 37.6 average across MindCube, SPARBench, MMSI-Bench, and ViewSpatial, outperforming both GRPO and answer-privileged OPSD across spatial reasoning benchmarks. Ablations confirm the complementarity of 3D and answer privilege, the advantage of question-conditioned routing over full-context injection, and the benefit of restricting distillation to incorrect trajectories.
Understanding a dynamical world calls for more than a latent state that summarizes its observations: the state should also be organized into the factors that stay shared across related observations and the factors that vary between them. For example, a robot pushing a cube to a goal should take the same action when the camera shifts or the lights dim, since nothing in the scene has moved. Existing approaches to this decomposition commonly obtain it through reconstruction, so the latent variables must first explain the entire observational world before their organization can be trusted. Joint embedding predictive architectures (JEPAs) model the latent state directly and never reconstruct, yet no existing result recovers the invariant and variant parts of the state they learn. How to learn the invariant-variant structure of the latent world without paying for its reconstruction therefore remains open. To close this gap, we introduce SplitJEPA, a JEPA that jointly recovers the latent state and its invariant and variant organization directly in representation space, without any reconstruction. We prove that, under stationary Gaussian predictive dynamics and a full-rank variation condition, SplitJEPA identifies the invariant and variant subspaces up to independent block-wise isometries, without introducing an observation decoder. Since the guarantee needs no decoder, the result extends reconstruction-free latent recovery to invariant-variant block identification. Experiments on synthetic nonlinear systems and robotic manipulation tasks support the theoretical results and show their practical value for both robustness and efficiency.
Supervised fine-tuning (SFT) adapts pretrained large language models (LLMs) to downstream tasks, but the required concepts can receive substantially different levels of pretrained support. Frequent concepts are more likely to be well learned, whereas rare concepts may remain weakly represented. We introduce a novel notion named prior barrier to quantify how strongly the pretrained model supports competing concepts over the target concept. We observe that prior barriers follow a long-tail distribution, placing head and tail concepts at different starting points for SFT: head concepts face lower prior barriers, whereas tail concepts require additional instructions to overcome their higher prior barriers. Our theoretical analysis further derives a predictive risk bound for SFT under long-tail prior barriers, explicitly characterizing how the prior barrier and accumulated SFT evidence jointly determine predictive performance. Motivated by this prior barrier-dependent demand, we propose PASS, an adaptive SFT instruction selection method that constructs reference-derived concepts and estimates the distinguishing evidence provided by each instruction, and adaptively allocates the selection budget toward concepts that remain insufficiently covered under the current selection. In this way, PASS jointly considers which instructions can provide useful evidence and where additional supervision is needed under a limited budget. Experiments show that our method consistently outperforms seven state-of-the-art instruction selection methods on four backbone-budget settings. An ablation study further shows that PASS's adaptive allocation consistently improves over uniform allocation.
Adversarial games have driven advances from heuristic search to reinforcement learning, yet learning and adapting strategies from limited samples remain challenging. AI agents offer an alternative by turning game experience into revisions of executable policies. Building on heuristic learning (HL), we formalize Adversarial Heuristic Learning (AHL), a paradigm that uses AI agents as learning engines to refine game policies and supporting software while keeping model weights fixed. We introduce AAArena, a benchmark comprising 12 authentic adversarial games and 1,920 archived human programs, with an evaluation protocol modeled on real-world game competitions. Agents interpret rules, choose opponents, analyze replays, and revise game agents to achieve their highest ranking within fixed match and evaluation budgets. We evaluate {completedmodels} model and harness configurations: Opus5.5 with Claude Code earns 6 gold medals, while no evaluated configuration tops the remaining 6 human ladders. Performance is generally weaker in games with more complex rule specifications. Further experiments show that opponent selection and dense feedback support policy improvement, and that agents learn from both on-policy replays of their own matches and off-policy replays of other players' matches. These results highlight HL's potential in adversarial games and identify persistent challenges in game understanding, strategy implementation, and long-horizon policy development.
We introduce RefineMix, a framework for training discrete diffusion models under severe data scarcity, a common constraint in scientific applications. RefineMix uses out-of-distribution data at selected diffusion times to improve generalization without biasing the sampling distribution. Although this strategy has been explored in continuous diffusion, discrete diffusion presents a distinct challenge: unlike Gaussian noise, masking preserves domain information in surviving tokens, limiting the use of related data at high noise levels. At low noise levels, however, the domains effectively disjoint supports become an advantage, allowing the model to learn from both in-domain and out-of-distribution data without biasing the sampler. We formalize these intuitions and provide a theoretical analysis for the proposed method. Experimentally, across five domain-shift settings, RefineMix matches or outperforms in-domain finetuning and data mixing. For protein sequence generation, finetuning with just 197 in-domain examples nearly doubles the fraction of generated proteins that are simultaneously novel, foldable, and in-family compared to standard finetuning.
While caching key-value (KV) states accelerates Large Language Model (LLM) decoding, this cache can dominate memory usage at long context lengths. One solution is to compress this memory by exploiting inter-layer cache similarities. However, most existing techniques necessitate architectural changes to LLMs and incur substantial overhead. In this work, we propose a symmetry-aware value cache merging strategy that reduces cache memory while avoiding both harmful performance degradation and architectural overhead during decoding. Furthermore, we show that this approach can be exploited alongside existing cache compression techniques, composing with high-ratio quantization or key cache pruning to reach compression ratios that neither method reaches alone, with minimal additional cost. Ultimately, our findings reveal a major source of underutilized capacity in the value cache, offering a simple yet highly effective direction for scaling context windows under memory constraints.
Anatomical continuity provides complementary information for medical image restoration, but its use requires accounting for local anatomy and variations in slice sampling. We introduce ContiLNN, which augments two-dimensional restoration backbones with bidirectional closed-form continuous-time (Bi-CfC) modules for cross-slice modeling while retaining in-plane feature extraction. Slice-index intervals modulate gates determined by local features and hidden states, enabling propagation to respond to sampling variations without numerical ODE integration. Reference-guided consistency aligns first- and second-order cross-slice intensity differences to preserve anatomical variation, while distillation from a frozen backbone helps retain in-plane fidelity. Across five training seeds, ContiLNN improves mean PSNR over Restore-RWKV by 0.1907, 1.0176, and 1.2482 dB for CT denoising, MRI super-resolution, and reduced-count PET restoration, respectively, with lower RMSE in all three tasks. CT results are descriptive for one held-out patient. PET ablations support ordered propagation beyond additional pointwise capacity. Under contiguous training, Bi-CfC achieves higher fidelity than a Bi-GRU with similar parameter counts and arithmetic costs across all tested sampling conditions. Matched seven-slice profiling shows 52.8% lower latency and 57.0% lower peak GPU memory use than Bi-GRU. Mixed-gap training improves sparse and irregular-context performance for both operators, without a uniform ranking across metrics and contexts. Experiments with fewer training patients and a second backbone further support data efficiency and backbone compatibility.
Many tasks in scientific computing and machine learning require the Jacobian or Hessian matrix of a function. Automatic differentiation (AD) computes these derivatives to machine precision, but materializing a dense m n Jacobian requires n forward-mode or m reverse-mode AD passes, one per column or row. For a large class of functions, each output depends on only a few inputs, making the derivative matrix sparse. Automatic sparse differentiation (ASD) exploits this structure in four steps: detection of the input-agnostic sparsity pattern, coloring of a graph to group columns or rows that can share an AD pass, compressed differentiation to compute a compressed derivative matrix with one AD pass per color, and finally decompression into the original sparsity pattern. The number of colors, and hence of AD passes, is often independent of the problem dimension: a banded Jacobian with b contiguous bands, for instance, only ever requires b colors, regardless of its size. asdex offers the first standalone ASD toolkit in the popular JAX ecosystem. With asdex.jacobian and asdex.hessian, it provides sparse drop-in replacements for jax.jacobian and jax.hessian.
Anthropic is making changes to its usage policy for the first time in over a year to reflect new and high-risk cases of misuse …
A reference architecture for securely sharing one Amazon SageMaker HyperPod EKS cluster across multiple teams, using AWS IAM Identity Center for authentication, per-team SageMaker Domains and Kubernetes namespaces for isolation, HyperPod Task Governance for fairness, and namespace-level cost allocation for chargeback.
