AAAI 2026 论文列表

AAAI Conference on Artificial Intelligence

录用类型

4,436篇论文

按官方录用论文列表统计,可能与会议公布的数字略有出入。

  • 其他 / 未标注
    4,436100.0%

Top 10 主题(点击筛选)

暂无主题数据。

全部 207 个研究方向的趋势

浏览全部论文

Provably Data-Driven Projection Method for Quadratic Programming

Anh Tuan Nguyen, Viet Anh Nguyen

展开摘要

Projection methods aim to reduce the dimensionality of the optimization instance, thereby improving the scalability of high-dimensional problems. Recently, Sakaue and Oki (2024) proposed a data-driven approach for linear programs (LPs), where the projection matrix is learned from observed problem instances drawn from an application-specific distribution of problems. We analyze the generalization guarantee for the data-driven projection matrix learning for convex quadratic programs (QPs). Unlike in LPs, the optimal solutions of convex QPs are not confined to the vertices of the feasible polyhedron, and this complicates the analysis of the optimal value function. To overcome this challenge, we demonstrate that the solutions of convex QPs can be localized within a feasible region corresponding to a special active set, utilizing Caratheodory's theorem. Building on such observation, we propose the unrolled active set method, which models the computation of the optimal value as a Goldberg-Jerrum algorithm with bounded complexities, thereby establishing learning guarantees. We then extend our analysis further to other settings, including learning to match the optimal solution and an input-aware setting, where we learn to map QP problem instances to projection matrices.

Technical
Hexaïssa: Standing on Giants’ Shoulders—Routing the Best Chess Engines with Mixture-of-Experts and Latent Reward Learning

Bach Ngo, Nguyen Hoang Khoi Do

展开摘要

We present Hexaïssa, a novel framework for adaptive chess engine routing that formulates expert selection as a Mixture-of-Experts (MoE) problem. Hexaïssa learns a gating policy that dynamically selects among heterogeneous state-of-the-art engines—such as Stockfish, LCZero, and Obsidian—based on the tactical and strategic complexity of each board state. This adaptivity enables stronger play and more efficient computation than any fixed engine or static configuration. However, training such a gating policy is fundamentally challenging due to sparse optimization signals and long-horizon credit assignment in chess games. To address these challenges, we introduce a score-based inverse reinforcement learning (IRL) method that models expert engine trajectories as samples from a latent distribution over optimal behaviors. By recovering the Stein score function of this distribution via stochastic differential equations (SDEs), we infer dense, per-move reward signals consistent with potential-based IRL. These latent rewards allow efficient training of the gating network without requiring additional environment interaction or human supervision. Empirical results on standard chess benchmarks demonstrate that Hexaïssa significantly outperforms individual engines, conventional MoE models, and IRL baselines.

Technical
Context-Sensitive Abstractions for Reinforcement Learning with Parameterized Actions

Rashmeet Kaur Nayyar, Naman Shah, Siddharth Srivastava

展开摘要

Real-world sequential decision-making often involves parameterized action spaces that require both, decisions regarding discrete actions and decisions about continuous action parameters governing how an action is executed. Existing approaches exhibit severe limitations in this setting---planning methods demand hand-crafted action models, and standard reinforcement learning (RL) algorithms are designed for either discrete or continuous actions but not both, and the few RL methods that handle parameterized actions typically rely on domain-specific engineering and fail to exploit the latent structure of these spaces. This paper extends the scope of RL algorithms to long-horizon, sparse-reward settings with parameterized actions by enabling agents to autonomously learn both state and action abstractions online. We introduce algorithms that progressively refine these abstractions during learning, increasing fine-grained detail in the critical regions of the state–action space where greater resolution improves performance. Across several continuous-state, parameterized-action domains, our abstraction-driven approach enables TD(λ) to achieve markedly higher sample efficiency than state-of-the-art baselines.

Technical
GP-MoLFormer-Sim: Test Time Molecular Optimization Through Contextual Similarity Guidance

Jiri Navratil, Jarret Ross, Payel Das, Youssef Mroueh, Samuel C Hoffman, Vijil Chenthamarakshan, Brian Belgodere

展开摘要

The ability to design molecules while preserving similarity to a target molecule and/or property is crucial for various applications in drug discovery, chemical design, and biology. We introduce in this paper an efficient training-free method for navigating and sampling from the molecular space with a generative Chemical Language Model (CLM), while using the molecular similarity to the target as a guide. Our method leverages the contextual representations learned from the CLM itself to estimate the molecular similarity, which is then used to adjust the autoregressive sampling strategy of the CLM. At each step of the decoding process, the method tracks the distance of the current generations from the target and updates the logits to encourage the preservation of similarity in generations. We implement the method using a recently proposed ~47M parameter SMILES-based CLM, GP-MoLFormer, and therefore refer to the method as GP-MoLFormer-Sim, which enables a test-time update of the deep generative policy to reflect the contextual similarity to a set of guide molecules. The method is further integrated into a genetic algorithm (GA) and tested on a set of standard molecular optimization benchmarks involving property optimization, molecular rediscovery, and structure-based drug design. Results show that, GP-MoLFormer-Sim, combined with GA (GP-MoLFormer-Sim+GA) outperforms existing training-free baseline methods, when the oracle remains black-box. The findings in this work are a step forward in understanding and guiding the generative mechanisms of CLMs.

Technical
Policy Search, Retrieval, and Composition via Task Similarity in Collaborative Agentic Systems

Saptarshi Nath, Christos Peridis, Eseoghene Benjamin, Xinran Liu, Soheil Kolouri, Peter Kinnell, Zexin Li, Cong Liu, Shirin Dora, Andrea Soltoggio

展开摘要

Agentic AI aims to create systems that set their own goals, adapt proactively to change, and refine behavior through continuous experience. Recent advances suggest that, when facing multiple and unforeseen tasks, agents could benefit from sharing machine-learned knowledge and reusing policies that have already been fully or partially learned by other agents. However, how to query, select, and retrieve policies from a pool of agents, and how to integrate such policies remains a largely unexplored area. This study explores how an agent decides what knowledge to select, from whom, and when and how to integrate it in its own policy in order to accelerate its own learning. The proposed algorithm, Modular Sharing and Composition in Collective Learning (MOSAIC), improves learning in agentic collectives by combining (1) knowledge selection using performance signals and cosine similarity on Wasserstein task embeddings, (2) modular and transferable neural representations via masks, and (3) policy integration, composition and fine-tuning. MOSAIC outperforms isolated learners and global sharing approaches in both learning speed and overall performance, and in some cases solves tasks that isolated agents cannot. The results also demonstrate that selective, goal-driven reuse leads to less susceptibility to task interference. We also observe the emergence of self-organization, where agents solving simpler tasks accelerate the learning of harder ones through shared knowledge.

Technical
Parameter-free Optimal Rates for Nonlinear Semi-Norm Contractions with Applications to Q-Learning

Ankur Naskar, Gugan Thoppe, Vijay Gupta

展开摘要

Algorithms for solving nonlinear fixed-point equations---such as average-reward Q-learning and TD-learning---often involve semi-norm contractions. Achieving parameter-free optimal convergence rates for these methods via Polyak–Ruppert averaging has remained elusive, largely due to the non-monotonicity of such semi-norms. We close this gap by (i.) recasting the averaged error as a linear recursion involving a nonlinear perturbation, and (ii.) taming the nonlinearity by coupling the semi-norm's contraction with the monotonicity of a suitably induced norm. Our main result yields the first parameter-free ~O(1/√t) optimal rates for Q-learning in both average-reward and exponentially discounted settings, where t denotes the iteration index. The result applies within a broad framework that accommodates both synchronous and asynchronous updates, single-agent and distributed deployments, and data streams obtained from either simulators or along Markovian trajectories.

Technical
RankList – a Listwise Preference Learning Framework for Predicting Subjective Preferences

Abinay Reddy Naini, Fernando Diaz, Carlos Busso

展开摘要

Preference learning has gained significant attention in tasks involving subjective human judgments, such as speech emotion recognition (SER) and image aesthetic assessment. While pairwise frameworks such as RankNet offer robust modeling of relative preferences, they are inherently limited to local comparisons and struggle to capture global ranking consistency. To address these limitations, we propose RankList, a novel listwise preference learning framework that generalizes RankNet to structured list-level supervision. Our formulation explicitly models local and non-local ranking constraints within a probabilistic framework. The paper introduces a log-sum-exp approximation to improve training efficiency. We further extend RankList with skip-wise comparisons, enabling progressive exposure to complex list structures and enhancing global ranking fidelity. Extensive experiments demonstrate the superiority of our method across diverse modalities. On benchmark SER datasets (MSP-Podcast, IEMOCAP, BIIC Podcast), RankList achieves consistent improvements in Kendall's Tau and ranking accuracy compared to standard listwise baselines. We also validate our approach on aesthetic image ranking using the Artistic Image Aesthetics dataset, highlighting its broad applicability. Through ablation and cross-domain studies, we show that RankList not only improves in-domain ranking but also generalizes better across datasets. Our framework offers a unified, extensible approach for modeling ordered preferences in subjective learning scenarios.

Technical
From Decision Trees to Boolean Logic: A Fast and Unified SHAP Algorithm

Alexander Nadel, Ron Wettenstein

展开摘要

SHapley Additive exPlanations (SHAP) is a key tool for interpreting decision tree ensembles by assigning contribution values to features. It is widely used in finance, advertising, medicine, and other domains. Two main approaches to SHAP calculation exist: Path-Dependent SHAP, which leverages the tree structure for efficiency, and Background SHAP, which uses a background dataset to estimate feature distributions. We introduce Woodelf, a SHAP algorithm that integrates decision trees, game theory, and Boolean logic into a unified framework. For each consumer, Woodelf constructs a pseudo-Boolean formula that captures their feature values, the structure of the decision tree ensemble, and the entire background dataset. It then leverages this representation to compute Background SHAP in linear time. Woodelf can also compute Path-Dependent SHAP, Shapley interaction values, Banzhaf values, and Banzhaf interaction values. Woodelf is designed to run efficiently on CPU and GPU hardware alike. Available via the Python package woodelf, it is implemented using NumPy, SciPy, and CuPy without relying on custom C++ or CUDA code. This design enables fast performance and seamless integration into existing frameworks, supporting large-scale computation of SHAP and other game-theoretic values in practice. For example, on a dataset with 3,000,000 rows, 5,000,000 background samples, and 127 features, Woodelf computed all Background Shapley values in 162 seconds on CPU and 16 seconds on GPU—compared to 44 minutes required by the best method on any hardware platform, representing 16x and 165x speedups, respectively.

Technical
Unlocking Efficient Vehicle Dynamics Modeling via Analytic World Models

Asen Nachkov, Danda Pani Paudel, Jan-Nico Zaech, Davide Scaramuzza, Luc Van Gool

展开摘要

Differentiable simulators represent an environment’s dynamics as a differentiable function. Within robotics and autonomous driving, this property is used in Analytic Policy Gradients (APG), which relies on backpropagating through the dynamics to train accurate policies for diverse tasks. Here we show that differentiable simulation also has an important role in world modeling, where it can impart predictive, prescriptive, and counterfactual capabilities to an agent. Specifically, we design three novel task setups in which the differentiable dynamics are combined within an end-to-end computation graph not with a policy, but a state predictor. This allows us to learn relative odometry, optimal planners, and optimal inverse states. We collectively call these predictors Analytic World Models (AWMs) and demonstrate how differentiable simulation enables their efficient, end-to-end learning. In autonomous driving scenarios, they have broad applicability and can augment an agent’s decision-making beyond reactive control.

Technical
CoRe-Fed: Bridging Collaborative and Representation Fairness via Federated Embedding Distillation

Noorain Mukhtiar, Adnan Mahmood, Quan Z. Sheng

展开摘要

With the proliferation of distributed data sources, Federated Learning (FL) has emerged as a key approach to enable collaborative intelligence through decentralized model training while preserving data privacy. However, conventional FL algorithms often suffer from performance disparities across clients caused by heterogeneous data distributions and unequal participation, which leads to unfair outcomes. Specifically, we focus on two core fairness challenges, i.e., representation bias, arising from misaligned client representations, and collaborative bias, stemming from inequitable contribution during aggregation, both of which degrade model performance and generalizability. To mitigate these disparities, we propose CoRe-Fed, a unified optimization framework that bridges collaborative and representation fairness via embedding-level regularization and fairness-aware aggregation. Initially, an alignment-driven mechanism promotes semantic consistency between local and global embeddings to reduce representational divergence. Subsequently, a dynamic reward-penalty-based aggregation strategy adjusts each client’s weight based on participation history and embedding alignment to ensure contribution-aware aggregation. Extensive experiments across diverse models and datasets demonstrate that CoRe-Fed improves both fairness and model performance over state-of-the-art baseline algorithms.

Technical
Inference Offloading for Cost-Sensitive Binary Classification at the Edge

Vishnu Narayanan Moothedath, Umang Agarwal, Umeshraja N, James Richard Gross, Jaya Prakash Champati, Sharayu Moharir

展开摘要

We investigate a binary classification problem in an edge intelligence system where false negatives are more costly than false positives. The system features a compact, locally deployed model, supplemented by a larger, remote model that is accessible via the network, albeit at an offloading cost. For each sample, our system first uses the locally deployed model for inference. Based on the output of the local model, the sample may be offloaded to the remote model. This work aims to understand the fundamental trade-off between classification accuracy and the offloading costs within such a hierarchical inference (HI) system. To optimise this system, we propose an online learning framework that continuously adapts a pair of thresholds on the local model's confidence scores. These thresholds determine the prediction of the local model and whether a sample is classified locally or offloaded to the remote model. We present a closed-form solution for the setting where the local model is calibrated. For the more general case of uncalibrated models, we introduce H2T2, an online two-threshold hierarchical inference policy, and prove it achieves sublinear regret. H2T2 is model-agnostic, requires no training, and learns during the inference phase using limited feedback. Simulations on real-world datasets show that H2T2 consistently outperforms naive and single-threshold HI policies, sometimes even surpassing single-threshold offline optima. The policy also demonstrates robustness to distribution shifts and adapts effectively to mismatched classifiers.

Technical
Expandable and Differentiable Dual Memories with Orthogonal Regularization for Exemplar-free Continual Learning

Hyung-Jun Moon, Sung-Bae Cho

展开摘要

Continual learning methods used to force neural networks to process sequential tasks in isolation, preventing them from leveraging useful inter-task relationships and causing them to repeatedly relearn similar features or overly differentiate them. To address this problem, we propose a fully differentiable, exemplar-free expandable method composed of two complementary memories: One learns common features that can be used across all tasks, and the other combines the shared features to learn discriminative characteristics unique to each sample. Both memories are differentiable so that the network can autonomously learn latent representations for each sample. For each task, the memory adjustment module adaptively prunes critical slots and minimally expands capacity to accommodate new concepts, and orthogonal regularization enforces geometric separation between preserved and newly learned memory components to prevent interference. Experiments on CIFAR-10, CIFAR-100, and Tiny-ImageNet show that the proposed method outperforms 14 state-of-the-art methods for class-incremental learning, achieving final accuracies of 55.13%, 37.24%, and 30.11%, respectively. Additional analysis confirms that, through effective integration and utilization of knowledge, the proposed method can increase average performance across sequential tasks, and it produces feature extraction results closest to the upper bound, thus establishing a new milestone in continual learning.

Technical
Bant: Byzantine Antidote via Trial Function and Trust Scores

Gleb Molodtsov, Daniil Medyakov, Sergey Skorik, Nikolas Khachaturov, Shahane Tigranyan, Vladimir Aletov, Aram Avetisyan, Martin Takac, Aleksandr Beznosikov

展开摘要

Recent advancements in machine learning have improved performance while also increasing computational demands. While federated and distributed setups address these issues, their structures remain vulnerable to malicious influences. In this paper, we address a specific threat: Byzantine attacks, wherein compromised clients inject adversarial updates to derail global convergence. We combine the concept of trust scores with trial function methodology to dynamically filter outliers. Our methods address the critical limitations of previous approaches, allowing operation even when Byzantine nodes are in the majority. Moreover, our algorithms adapt to widely used scaled methods such as Adam and RMSProp, as well as practical scenarios, including local training and partial participation. We validate the robustness of our methods by conducting extensive experiments on both public datasets and private ECG data collected from medical institutions. Furthermore, we provide a broad theoretical analysis of our algorithms and their extensions to the aforementioned practical setups. The convergence guaranties of our methods are comparable to those of classical algorithms developed without Byzantine interference.

Technical
Neuro-Symbolic Federated Learning over Heterogeneous Data-Views: A Structured Approach to Distributive EHR Modelling

Soheila Molaei, Bahareh Fatemi, Anshul Thakur, Andrew Soltan, Fazle Rabbi, Andreas L. Opdahl, Kim Branson, Patrick Schwab, Danielle Belgrave, David A Clifton

展开摘要

Federated learning (FL) enables privacy-preserving model training across distributed Electronic Health Records (EHRs), but its deployment remains limited by data-view heterogeneity, where institutions maintain incompatible local schemas. Most existing methods address this by enforcing flat, aligned data views, which require extensive cross-site preprocessing and manual harmonisation that often discards client-specific features, or by projecting inputs into a shared latent space, which sacrifices interpretability. We propose a modelling shift from conventional FL with vectorised inputs to a symbolic, relation-centric framework, where each client organises its EHR data as a structured, type-aware relational graph. This enables client-specific inference without requiring schema alignment and supports FL across heterogeneous data views. To model over these symbolic structures, we introduce an architecture that combines relation-aware message passing with a learnable feature relevance mechanism, jointly enabling accurate local predictions and client-specific interpretability while supporting parameter sharing across clients. Beyond strong performance on three real-world EHR datasets exhibiting data-view heterogeneity, we further show that our framework supports multimodal FL under modality-level heterogeneity. Using MC-MED, a publicly available multimodal emergency department dataset, we demonstrate that our method accommodates clients with partially missing modalities, highlighting its robustness and scalability in real-world clinical settings.

Technical
Exact Shapley Attributions in Quadratic-time for FANOVA Gaussian Processes

Majid Mohammadi, Krikamol Muandet, Ilaria Tiddi, Annette Ten Teije, Siu Lun Chau

展开摘要

Shapley values are widely recognized as a principled method for attributing importance to input features in machine learning. However, the exact computation of Shapley values scales exponentially with the number of features, severely limiting the practical application of this powerful approach. The challenge is further compounded when the predictive model is probabilistic---as in Gaussian processes (GPs)---where the outputs are random variables rather than point estimates, necessitating additional computational effort in modeling higher-order moments. In this work, we demonstrate that for an important class of GPs known as FANOVA GP, which explicitly models all main effects and interactions, exact Shapley attributions for both local and global explanations can be computed in *quadratic* time. For *local, instance-wise explanations*, we define a stochastic cooperative game over function components and compute the *exact stochastic Shapley value* in quadratic time only, capturing both the expected contribution and uncertainty. For *global explanations*, we introduce a deterministic, variance-based value function and compute exact Shapley values that quantify each feature’s contribution to the model’s overall sensitivity. Our methods leverage a closed-form (stochastic) Möbiusrepresentation of the FANOVA decomposition and introduce recursive algorithms, inspired by Newton's identities, to efficiently compute the mean and variance of Shapley values. Our work enhances the utility of explainable AI, as demonstrated by empirical studies, by providing more scalable, axiomatically sound, and uncertainty-aware explanations for predictions generated by structured probabilistic models.

Technical
Neural Architecture and Hyperparameter Selection Through Meta-Learning on Time Series

Erfan Moeini, Christopher Vox, Marie Anastacio, Wadie Skaf, Mitra Baratchi, Holger H. Hoos

展开摘要

Active research in time series classification and forecasting has led to the development of a wide range of machine learning models. For practitioners, the selection of a suitable model among these, along with their hyperparameters, remains a challenging task. While automated machine learning offers approaches for automatic selection of models for a given task, the practical efficacy of these methods is often limited, due to the computational complexity of searching over a large design space and the high dimensionality of time series datasets that poses additional challenges on generalisation quality. To fill this gap, we propose a meta-learning framework that transfers past knowledge from previous searches to recommend an architecture and its hyperparameters; specifically, this framework utilises a joint representation of deep neural architectures and time series datasets, and predicts the performance of neural architectures along with their hyperparameters on time series datasets. Our computational experiments reveal that the configurations proposed by our meta-learned surrogate achieve a performance gain of up to 34% on 4 out of the 8 forecasting datasets we considered and up to 60% on 36 out of 73 of our classification datasets, whilst reducing the computational cost to 10% of that required by the hyperparameter optimisation method HEBO to tune the architectures, showcasing the effectiveness of meta-learning in the time series domain.

Technical
Explaining Temporal Graph Neural Network via Quantum-Inspired Evolutionary Algorithm

Masahiro Mitani, Yuya Sasaki

展开摘要

Temporal Graph Neural Network (TGNN) explanation has attracted increasing attention due to its applicability in dynamic scenarios such as recommendation systems. However, existing explanation methods for TGNNs face two key limitations: (1) computational inefficiency and (2) a restricted focus on either factual or counterfactual explanations, but not both. In this paper, we propose QIEA-TGX, an efficient and unified explanation algorithm based on a quantum-inspired evolutionary algorithm. QIEA-TGX effectively generates explanatory subgraphs that significantly influence TGNN predictions, without requiring additional model training or extensive inference. Experimental results on real-world datasets demonstrate that QIEA-TGX improves explanation fidelity by up to 31% while reducing computation time by up to 92% compared to state-of-the-art baselines.

Technical
T4NMTD: Transition-Centric Reinforcement Learning for Non-Markovian Task Decomposition

Ruixuan Miao, Xu Lu, Cong Tian, Bin Yu, Zhenhua Duan

展开摘要

Non-Markovian Tasks (NMTs) are distinguished by their dependence on long-term memory and state-dependent dynamics, setting them apart from the traditional Markovian models typically employed in Reinforcement Learning (RL). NMTs not only suffer from reward sparseness but also rely on historical information, making their resolution considerably more challenging. In this paper, we propose a novel RL framework T4NMTD (Transition-centric framework for NMT Decomposition), designed specifically for learning NMTs which are specified by temporal logic. The core of T4NMTD is a task decomposition mechanism along with a parallel training approach for NMTs. An NMT is first decomposed as basic units based on the transitions of the automata which are derived from temporal logic formulae. The units are then modularized into sub-tasks according to their semantic similarity under logical interpretation. The training strategy of T4NMTD adopts a dual-level structure: the high-level learns to shape the boundaries and coordinate arrangement of the sub-tasks from a global perspective, while the low-level learns those sub-tasks in parallel. In addition, we invent a dynamic policy intervention scheme to mitigate the policy myopic issue during parallel training. A comprehensive evaluation is conducted on benchmark problems with respect to various metrics. The experimental results demonstrate that T4NMTD effectively addresses NMTs, achieving significant performance improvements compared with related studies.

Technical
Learning Branching Policies for MILPs with Proximal Policy Optimization

Abdelouahed Ben Mhamed, Assia Kamal Idrissi, Amal Seghrouchni

展开摘要

Branch-and-Bound (B&B) is the dominant exact solution method for Mixed Integer Linear Programs (MILP), yet its exponential time complexity poses significant challenges for large-scale instances. The growing capabilities of machine learning have spurred efforts to improve B&B by learning data-driven branching policies. However, most existing approaches rely on Imitation Learning (IL), which tends to overfit to expert demonstrations and struggles to generalize to structurally diverse or unseen instances. In this work, we propose Tree-Gate Proximal Policy Optimization (TGPPO), a novel framework that employs Proximal Policy Optimization (PPO), a Reinforcement Learning (RL) algorithm, to train a branching policy aimed at improving generalization across heterogeneous MILP instances. Our approach builds on a parameterized state space representation that dynamically captures the evolving context of the search tree. Empirical evaluations show that TGPPO often outperforms existing learning-based policies in terms of reducing the number of nodes explored and improving p-Primal-Dual Integrals (PDI), particularly in out-of-distribution instances. These results highlight the potential of RL to develop robust and adaptable branching strategies for MILP solvers.

Technical
Online Linear Regression with Paid Stochastic Features

Nadav Merlis, Kyoungseok Jang, Nicolo Cesa-Bianchi

展开摘要

We study an online linear regression setting in which the observed feature vectors are corrupted by noise and the learner can pay to reduce the noise level. In practice, this may happen for several reasons: for example, because features can be measured more accurately using more expensive equipment, or because data providers can be incentivized to release less private features. Assuming feature vectors are drawn i.i.d. from a fixed but unknown distribution, we measure the learner's regret against the linear predictor minimizing a notion of loss that combines the prediction error and payment. We first study the case in which the mapping between payments and noise covariance is known and prove order-optimal regret bounds in the interaction length (up to log-factors). We then derive order-optimal bounds also when the noise covariance is unknown and prove that the regret rate is worse than the case of known covariances. Our analysis leverages matrix martingale concentration, showing that the empirical loss uniformly converges to the expected one for all payments and linear predictors.

Technical