Differentiable decision trees (DDTs) support end-to-end optimisation and explicit decision paths, but dense oblique splits can obscure the role of individual input features. We propose the Euclidean-Concentrated Differentiable Decision Tree (EU-DDT), which regularises the relative absolute feature weights at each internal node using their normalised squared Euclidean distance from the uniform distribution. This encourages dominance by a small number of features while retaining probabilistic soft routing. Across five matched random seeds, EU-DDT reduces the mean effective number of features from 8.94 to 1.54 on Wine and from 45.28 to 9.06 on Digits. Mean test accuracy changes from 100.0% to 93.9% on Wine and from 92.3% to 94.6% on Digits, illustrating the dataset-dependent trade-off between concentration and predictive performance. We also use EU-DDT to parameterise a PPO actor with an MLP critic. Two random-seed runs per environment show a consistent qualitative increase in concentration while preserving effective task performance. In representative CartPole-v1 and MiniPacman runs, EU-DDT reaches 100% final evaluation success, while mean node concentration increases from 0.1272 to 0.7179 and from 0.0437 to 0.7237, respectively. These results support EU regularisation as a mechanism for reducing node-level feature complexity; the reinforcement-learning evidence remains preliminary because only two seeds are evaluated.
We study how boundary geometry affects $L^q$ approximation by shallow ReLU neural networks on convex bodies $K\subset\mathbb R^d$. The target classes are unit balls of weighted variation spaces that can enlarge classical spaces by incorporating domain geometry into constraints on output coefficients. We establish upper and lower bounds for arbitrary convex bodies and determine the optimal polynomial approximation exponents for all planar convex bodies and their products with polytopes, including cubes and circular cylinders. For $C^2$ convex bodies, we determine the optimal exponents in a dimension-dependent range of $q$ extending beyond $2$. When all principal curvatures are positive, the result extends to every finite $q\ge2$. For these classes, the dependence of the optimal exponent on the domain is captured by a single geometric growth exponent associated with small boundary caps. We also identify sharp weight thresholds for retaining the classical approximation exponent and improve previous weights on Euclidean balls while preserving the classical $L^2$ approximation rate. When the target is given as a finite ReLU network, the upper bounds can be realized by a compressed network using only the original neurons, without increasing the weighted coefficient sum. These results show that boundary geometry can change the optimal exponent even when dimension, error norm, and weighting rule are fixed.
AI 速览：将多模态智能体的开放式推理改为从预设自然语言候选集中选择，以提升推理效率并保持任务表现。
Multimodal agents commonly generate free-form reasoning before each action. For small models, limited model capacity can result in lengthy reasoning that provides little useful guidance for action generation while incurring substantial inference cost. To address this challenge, we introduce Selection-based Structured Reasoning (SSR), a framework that reformulates reasoning as selection instead of open-ended generation. SSR represents recurring high-level reasoning as pre-specified, reusable natural-language candidates. At each turn, the model selects from these reasoning candidates based on their likelihoods given the current context, without requiring an auxiliary task head. Using pre-specified reasoning traces enables parallel scoring, where teacher-forced prefilling computes token likelihoods concurrently within and across candidates using a shared context KV cache. We evaluate SSR on seven multimodal search benchmarks using 2B and 4B models. Across multiple reinforcement learning objectives and supervised fine-tuning, SSR delivers significant efficiency gains without sacrificing task performance. SSR achieves an average success rate competitive with leading search agents of the same scale, while reducing per-turn reasoning latency by over 90% and total per-question model inference latency by 28-54%. Project page: https://zfy0314.github.io/ssr-webpage/.