Giulia Livieri

Matthieu Darcy, Boumediene Hamzi, Giulia Livieri et al.

We consider the problem of learning Stochastic Differential Equations of the form $dX_t = f(X_t)dt+σ(X_t)dW_t $ from one sample trajectory. This problem is more challenging than learning deterministic dynamical systems because one sample trajectory only provides indirect information on the unknown functions $f$, $σ$, and stochastic process $dW_t$ representing the drift, the diffusion, and the stochastic forcing terms, respectively. We propose a method that combines Computational Graph Completion and data adapted kernels learned via a new variant of cross validation. Our approach can be decomposed as follows: (1) Represent the time-increment map $X_t \rightarrow X_{t+dt}$ as a Computational Graph in which $f$, $σ$ and $dW_t$ appear as unknown functions and random variables. (2) Complete the graph (approximate unknown functions and random variables) via Maximum a Posteriori Estimation (given the data) with Gaussian Process (GP) priors on the unknown functions. (3) Learn the covariance functions (kernels) of the GP priors from data with randomized cross-validation. Numerical experiments illustrate the efficacy, robustness, and scope of our method.

Luca Galimberti, Anastasis Kratsios, Giulia Livieri · eth-zurich

Several non-linear operators in stochastic analysis, such as solution maps to stochastic differential equations, depend on a temporal structure which is not leveraged by contemporary neural operators designed to approximate general maps between Banach space. This paper therefore proposes an operator learning solution to this open problem by introducing a deep learning model-design framework that takes suitable infinite-dimensional linear metric spaces, e.g. Banach spaces, as inputs and returns a universal \textit{sequential} deep learning model adapted to these linear geometries specialized for the approximation of operators encoding a temporal structure. We call these models \textit{Causal Neural Operators}. Our main result states that the models produced by our framework can uniformly approximate on compact sets and across arbitrarily finite-time horizons Hölder or smooth trace class operators, which causally map sequences between given linear metric spaces. Our analysis uncovers new quantitative relationships on the latent state-space dimension of Causal Neural Operators, which even have new implications for (classical) finite-dimensional Recurrent Neural Networks. In addition, our guarantees for recurrent neural networks are tighter than the available results inherited from feedforward neural networks when approximating dynamical systems between finite-dimensional spaces.

Shijin Gong, Erhan Xu, Kai Ye et al.

Reinforcement learning with verifiable rewards has become a standard recipe for improving the reasoning abilities of large language models. Existing algorithms face a tradeoff between computational efficiency and sample efficiency in value estimation and policy learning. We introduce BASIS, a critic-free post-training algorithm designed to address this tradeoff. At each online training step, BASIS samples only one rollout per prompt, but leverages rich information across prompts in the entire batch to improve value function estimation. Our experiments demonstrate that BASIS reduces MSE in value function estimation by 69% compared to REINFORCE++, a representative single-rollout baseline, and achieves lower MSE with one rollout than group mean estimators with 8 rollouts. This improvement in value estimation translates to better policy optimization: using substantially less training time, BASIS achieves performance close to multi-rollout GRPO-type baselines and often outperforms single-rollout REINFORCE-type baselines.

Pingfan Su, Kai Ye, Shijin Gong et al.

Recent advances in large language models (LLMs) have made it increasingly difficult to distinguish human-written text from AI-generated content. Many existing detectors train supervised neural classifiers that achieve strong in-distribution performance but are often opaque and can degrade substantially under distribution shift. We present READER, a reasoning-enhanced AI text detector that outputs both a human/AI label and a structured rationale describing the evidence for its decision. A key component of our approach is READ, a curated supervision set of rationales and verdicts. We fine-tune an LLM on READ to build READER, which reasons before detecting at inference time. Despite having only 1.5B parameters, READER consistently outperforms existing detectors as well as prompted, high-capacity LLM baselines (GPT-5.2, Gemini-3-Pro, and DeepSeek-V3.2), which are 100 to 1000 times larger in scale.

Nils Detering, Luca Galimberti, Anastasis Kratsios et al. · eth-zurich

Since their introduction by Kipf and Welling in $2017$, a primary use of graph convolutional networks is transductive node classification, where missing labels are inferred within a single observed graph and its feature matrix. Despite the widespread use of the network model, the statistical foundations of transductive learning remain limited, as standard inference frameworks typically rely on multiple independent samples rather than a single graph. In this work, we address these gaps by developing new concentration-of-measure tools that leverage the geometric regularities of large graphs via low-dimensional metric embeddings. The emergent regularities are captured using a random graph model; however, the methods remain applicable to deterministic graphs once observed. We establish two principal learning results. The first concerns arbitrary deterministic $k$-vertex graphs, and the second addresses random graphs that share key geometric properties with an Erdős-Rényi graph $\mathbf{G}=\mathbf{G}(k,p)$ in the regime $p \in \mathcal{O}((\log (k)/k)^{1/2})$. The first result serves as the basis for and illuminates the second. We then extend these results to the graph convolutional network setting, where additional challenges arise. Lastly, our learning guarantees remain informative even with a few labelled nodes $N$ and achieve the optimal nonparametric rate $\mathcal{O}(N^{-1/2})$ as $N$ grows.

Anastasis Kratsios, Giulia Livieri, A. Martina Neuman

We study the statistical behaviour of reasoning probes in a stylized model of looped reasoning, given by Boolean circuits whose computational graph is a perfect $ν$-ary tree ($ν\ge 2$) and whose output is appended to the input and fed back iteratively for subsequent computation rounds. A reasoning probe has access to a sampled subset of internal computation nodes, possibly without covering the entire graph, and seeks to infer which $ν$-ary Boolean gate is executed at each queried node, representing uncertainty via a probability distribution over a fixed collection of $\mathtt{m}$ admissible $ν$-ary gates. This partial observability induces a generalization problem, which we analyze in a realizable, transductive setting. We show that, when the reasoning probe is parameterized by a graph convolutional network (GCN)-based hypothesis class and queries $N$ nodes, the worst-case generalization error attains the optimal rate $\mathcal{O}(\sqrt{\log(2/δ)}/\sqrt{N})$ with probability at least $1-δ$, for $δ\in (0,1)$. Our analysis combines snowflake metric embedding techniques with tools from statistical optimal transport. A key insight is that this optimal rate is achievable independently of graph size, owing to the existence of a low-distortion one-dimensional snowflake embedding of the induced graph metric. As a consequence, our results provide a sharp characterization of how structural properties of the computational graph govern the statistical efficiency of reasoning under partial access.