On the Mechanism and Dynamics of Modular Addition: Fourier Features, Lottery Ticket, and Grokking

We present a comprehensive analysis of how two-layer neural networks learn features to solve the modular addition task. Our work provides a full mechanistic interpretation of the learned model and a theoretical explanation of its training dynamics. While prior work has identified that individual neurons learn single-frequency Fourier features and phase alignment, it does not fully explain how these features combine into a global solution. We bridge this gap by formalizing a diversification condition that emerges during training when overparametrized, consisting of two parts: phase symmetry and frequency diversification. We prove that these properties allow the network to collectively approximate a flawed indicator function on the correct logic for the modular addition task. While individual neurons produce noisy signals, the phase symmetry enables a majority-voting scheme that cancels out noise, allowing the network to robustly identify the correct sum. Furthermore, we explain the emergence of these features under random initialization via a lottery ticket mechanism. Our gradient flow analysis proves that frequencies compete within each neuron, with the "winner" determined by its initial spectral magnitude and phase alignment. From a technical standpoint, we provide a rigorous characterization of the layer-wise phase coupling dynamics and formalize the competitive landscape using the ODE comparison lemma. Finally, we use these insights to demystify grokking, characterizing it as a three-stage process involving memorization followed by two generalization phases, driven by the competition between loss minimization and weight decay.

Kelix Technical Report

Autoregressive large language models (LLMs) scale well by expressing diverse tasks as sequences of discrete natural-language tokens and training with next-token prediction, which unifies comprehension and generation under self-supervision. Extending this paradigm to multimodal data requires a shared, discrete representation across modalities. However, most vision-language models (VLMs) still rely on a hybrid interface: discrete text tokens paired with continuous Vision Transformer (ViT) features. Because supervision is largely text-driven, these models are often biased toward understanding and cannot fully leverage large-scale self-supervised learning on non-text data. Recent work has explored discrete visual tokenization to enable fully autoregressive multimodal modeling, showing promising progress toward unified understanding and generation. Yet existing discrete vision tokens frequently lose information due to limited code capacity, resulting in noticeably weaker understanding than continuous-feature VLMs. We present Kelix, a fully discrete autoregressive unified model that closes the understanding gap between discrete and continuous visual representations.

Learning in Context, Guided by Choice: A Reward-Free Paradigm for Reinforcement Learning with Transformers

In-context reinforcement learning (ICRL) leverages the in-context learning capabilities of transformer models (TMs) to efficiently generalize to unseen sequential decision-making tasks without parameter updates. However, existing ICRL methods rely on explicit reward signals during pretraining, which limits their applicability when rewards are ambiguous, hard to specify, or costly to obtain. To overcome this limitation, we propose a new learning paradigm, In-Context Preference-based Reinforcement Learning (ICPRL), in which both pretraining and deployment rely solely on preference feedback, eliminating the need for reward supervision. We study two variants that differ in the granularity of feedback: Immediate Preference-based RL (I-PRL) with per-step preferences, and Trajectory Preference-based RL (T-PRL) with trajectory-level comparisons. We first show that supervised pretraining, a standard approach in ICRL, remains effective under preference-only context datasets, demonstrating the feasibility of in-context reinforcement learning using only preference signals. To further improve data efficiency, we introduce alternative preference-native frameworks for I-PRL and T-PRL that directly optimize TM policies from preference data without requiring reward signals nor optimal action labels.Experiments on dueling bandits, navigation, and continuous control tasks demonstrate that ICPRL enables strong in-context generalization to unseen tasks, achieving performance comparable to ICRL methods trained with full reward supervision.

ConsisDrive: Identity-Preserving Driving World Models for Video Generation by Instance Mask

Autonomous driving relies on robust models trained on large-scale, high-quality multi-view driving videos. Although world models provide a cost-effective solution for generating realistic driving data, they often suffer from identity drift, where the same object changes its appearance or category across frames due to the absence of instance-level temporal constraints. We introduce ConsisDrive, an identity-preserving driving world model designed to enforce temporal consistency at the instance level. Our framework incorporates two key components: (1) Instance-Masked Attention, which applies instance identity masks and trajectory masks within attention blocks to ensure that visual tokens interact only with their corresponding instance features across spatial and temporal dimensions, thereby preserving object identity consistency; and (2) Instance-Masked Loss, which adaptively emphasizes foreground regions with probabilistic instance masking, reducing background noise while maintaining overall scene fidelity. By integrating these mechanisms, ConsisDrive achieves state-of-the-art driving video generation quality and demonstrates significant improvements in downstream autonomous driving tasks on the nuScenes dataset. Our project page is https://shanpoyang654.github.io/ConsisDrive/page.html.

InstaDrive: Instance-Aware Driving World Models for Realistic and Consistent Video Generation

Autonomous driving relies on robust models trained on high-quality, large-scale multi-view driving videos. While world models offer a cost-effective solution for generating realistic driving videos, they struggle to maintain instance-level temporal consistency and spatial geometric fidelity. To address these challenges, we propose InstaDrive, a novel framework that enhances driving video realism through two key advancements: (1) Instance Flow Guider, which extracts and propagates instance features across frames to enforce temporal consistency, preserving instance identity over time. (2) Spatial Geometric Aligner, which improves spatial reasoning, ensures precise instance positioning, and explicitly models occlusion hierarchies. By incorporating these instance-aware mechanisms, InstaDrive achieves state-of-the-art video generation quality and enhances downstream autonomous driving tasks on the nuScenes dataset. Additionally, we utilize CARLA's autopilot to procedurally and stochastically simulate rare but safety-critical driving scenarios across diverse maps and regions, enabling rigorous safety evaluation for autonomous systems. Our project page is https://shanpoyang654.github.io/InstaDrive/page.html.

Llama-3.1-FoundationAI-SecurityLLM-Reasoning-8B Technical Report

We present Foundation-Sec-8B-Reasoning, the first open-source native reasoning model for cybersecurity. Built upon our previously released Foundation-Sec-8B base model (derived from Llama-3.1-8B-Base), the model is trained through a two-stage process combining supervised fine-tuning (SFT) and reinforcement learning from verifiable rewards (RLVR). Our training leverages proprietary reasoning data spanning cybersecurity analysis, instruction-following, and mathematical reasoning. Evaluation across 10 cybersecurity benchmarks and 10 general-purpose benchmarks demonstrates performance competitive with significantly larger models on cybersecurity tasks while maintaining strong general capabilities. The model shows effective generalization on multi-hop reasoning tasks and strong safety performance when deployed with appropriate system prompts and guardrails. This work demonstrates that domain-specialized reasoning models can achieve strong performance on specialized tasks while maintaining broad general capabilities. We release the model publicly at https://huggingface.co/fdtn-ai/Foundation-Sec-8B-Reasoning.

In-Context Reinforcement Learning From Suboptimal Historical Data

Transformer models have achieved remarkable empirical successes, largely due to their in-context learning capabilities. Inspired by this, we explore training an autoregressive transformer for in-context reinforcement learning (ICRL). In this setting, we initially train a transformer on an offline dataset consisting of trajectories collected from various RL tasks, and then fix and use this transformer to create an action policy for new RL tasks. Notably, we consider the setting where the offline dataset contains trajectories sampled from suboptimal behavioral policies. In this case, standard autoregressive training corresponds to imitation learning and results in suboptimal performance. To address this, we propose the Decision Importance Transformer(DIT) framework, which emulates the actor-critic algorithm in an in-context manner. In particular, we first train a transformer-based value function that estimates the advantage functions of the behavior policies that collected the suboptimal trajectories. Then we train a transformer-based policy via a weighted maximum likelihood estimation loss, where the weights are constructed based on the trained value function to steer the suboptimal policies to the optimal ones. We conduct extensive experiments to test the performance of DIT on both bandit and Markov Decision Process problems. Our results show that DIT achieves superior performance, particularly when the offline dataset contains suboptimal historical data.

Decoding Rewards in Competitive Games: Inverse Game Theory with Entropy Regularization

Estimating the unknown reward functions driving agents' behaviors is of central interest in inverse reinforcement learning and game theory. To tackle this problem, we develop a unified framework for reward function recovery in two-player zero-sum matrix games and Markov games with entropy regularization, where we aim to reconstruct the underlying reward functions given observed players' strategies and actions. This task is challenging due to the inherent ambiguity of inverse problems, the non-uniqueness of feasible rewards, and limited observational data coverage. To address these challenges, we establish the reward function's identifiability using the quantal response equilibrium (QRE) under linear assumptions. Building upon this theoretical foundation, we propose a novel algorithm to learn reward functions from observed actions. Our algorithm works in both static and dynamic settings and is adaptable to incorporate different methods, such as Maximum Likelihood Estimation (MLE). We provide strong theoretical guarantees for the reliability and sample efficiency of our algorithm. Further, we conduct extensive numerical studies to demonstrate the practical effectiveness of the proposed framework, offering new insights into decision-making in competitive environments.

Demystifying the Slash Pattern in Attention: The Role of RoPE

Large Language Models (LLMs) often exhibit slash attention patterns, where attention scores concentrate along the $Δ$-th sub-diagonal for some offset $Δ$. These patterns play a key role in passing information across tokens. But why do they emerge? In this paper, we demystify the emergence of these Slash-Dominant Heads (SDHs) from both empirical and theoretical perspectives. First, by analyzing open-source LLMs, we find that SDHs are intrinsic to models and generalize to out-of-distribution prompts. To explain the intrinsic emergence, we analyze the queries, keys, and Rotary Position Embedding (RoPE), which jointly determine attention scores. Our empirical analysis reveals two characteristic conditions of SDHs: (1) Queries and keys are almost rank-one, and (2) RoPE is dominated by medium- and high-frequency components. Under these conditions, queries and keys are nearly identical across tokens, and interactions between medium- and high-frequency components of RoPE give rise to SDHs. Beyond empirical evidence, we theoretically show that these conditions are sufficient to ensure the emergence of SDHs by formalizing them as our modeling assumptions. Particularly, we analyze the training dynamics of a shallow Transformer equipped with RoPE under these conditions, and prove that models trained via gradient descent exhibit SDHs. The SDHs generalize to out-of-distribution prompts.

Unlocking Out-of-Distribution Generalization in Transformers via Recursive Latent Space Reasoning

Systematic, compositional generalization beyond the training distribution remains a core challenge in machine learning -- and a critical bottleneck for the emergent reasoning abilities of modern language models. This work investigates out-of-distribution (OOD) generalization in Transformer networks using a GSM8K-style modular arithmetic on computational graphs task as a testbed. We introduce and explore a set of four architectural mechanisms aimed at enhancing OOD generalization: (i) input-adaptive recurrence; (ii) algorithmic supervision; (iii) anchored latent representations via a discrete bottleneck; and (iv) an explicit error-correction mechanism. Collectively, these mechanisms yield an architectural approach for native and scalable latent space reasoning in Transformer networks with robust algorithmic generalization capabilities. We complement these empirical results with a detailed mechanistic interpretability analysis that reveals how these mechanisms give rise to robust OOD generalization abilities.