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Nicolas Flammarion

Publications and source records attributed to Nicolas Flammarion.

At least 19 recordsLinked to original sources
arXiv2026

Controlling Sharpening in LLM Post-Training

LLM post-training via reinforcement learning has shown consistent improvements in accuracy at the cost of reducing behavioral coverage - a phenomenon known as sharpening that can stall progress on difficult tasks. Repeated sampling objectives such as $\mathrm{pass@k}$ and $\mathrm{max@k}$ are known to mitigate sharpening, but the mechanism behind it remains poorly understood. Moreover, existing approaches for $\mathrm{@k}$ optimization use dedicated algorithms, and no general framework exists for adapting established policy optimization methods from $\mathrm{@}1$ to $\mathrm{@k}$. We derive the optimal policies for KL-regularized $\mathrm{pass@k}$ and $\mathrm{max@k}$ objectives in closed form. Our explicit solutions reveal that $\mathrm{@k}$ optimization naturally redistributes learning pressure based on prompt difficulty, directing it toward harder problems. Crucially, we show that standard $\mathrm{@}1$ objectives achieve these exact same optimal policies through prompt-adaptive regularization or reward transformation. Ultimately, this yields an optimizer-independent explanation for why $\mathrm{@k}$ objectives reduce sharpening on easy tasks while prioritizing success on harder ones. Building on these insights, we introduce $\mathrm{@k}$ Adaptive Pressure for Policy Optimization (KAPPO). Following a one-time reference calibration, KAPPO seamlessly adapts existing $\mathrm{@}1$ methods to $\mathrm{@k}$ via prompt-adaptive regularization or reward transformation, without otherwise altering their training procedures. Experiments with several RL methods on mathematical reasoning and code generation demonstrate that KAPPO provides direct control over sharpening. Crucially, we demonstrate that $\mathrm{@k}$ pressure can be fully realized with single completion training, matching the performance of multi-completion $\mathrm{@k}$ optimizers at a fraction of the cost.

cs.LG↗
arXiv2026

Learning Distributions from Samples and Qualities

We study parameter estimation when each sample from a continuous parametric distribution is accompanied by a binary quality label governed by the same unknown parameter. By imposing hard feasibility constraints, deterministic labels can accelerate estimation from the classical $n^{-1/2}$ rate of smooth quality-aware M-estimators to $n^{-1}$. Combining PAC guarantees with local error-to-parameter conversions, we establish this rate for both identifiable and finitely ambiguous quality classifiers, with likelihood information resolving the latter ambiguity. When the quality model is known only up to an additive constant, we construct threshold-free pairwise preferences from oppositely labeled samples and retain the accelerated rate despite the unknown threshold. We further show that pointwise qualities and pairwise preferences are incomparable: either can be strictly more informative than the other. Our assumptions hold for Gaussian, Laplace, and exponential families under thresholded log-likelihood rewards. Finally, a minimax lower bound establishes the optimal $n^{-1}$ dependence on sample size, up to logarithmic and dimension-dependent factors, while experiments confirm the benefits of constraint-based estimation.

stat.ML↗
arXiv2026

Two-Timescale Fine-tuning Provably Learns New Features for Two-Layer ReLU Networks

Fine-tuning pre-trained models on specialized tasks with scarce data is central to modern deep learning. Despite its empirical success, theoretical understanding of fine-tuning remains limited. We introduce a Gaussian multi-index setting to study fine-tuning from pre-trained weights, where the teacher network has $m+1$ features, $m$ of which are learned during pre-training and one of which must be learned during fine-tuning. For two-layer ReLU networks, we show that two-timescale training, i.e., updating the outer weights infinitely faster than the hidden ones, learns the new task-specific feature while preserving the pre-trained ones in the model representation. Moreover, only $\mathcal{O}(d)$ fine-tuning samples are required for this recovery, independently of the number of pre-trained features. In contrast, with random initialization, the same number of samples is insufficient to recover the target parameters. Our results therefore demonstrate that pre-training can induce an implicit bias with a clear statistical advantage over random initialization, enabling feature learning from scarce fine-tuning data.

stat.ML↗
arXiv2026

Statistical Benefits of Fine-Tuning from Pretrained Initialization in Diagonal Linear Networks

Adapting pretrained models to downstream tasks with limited data has become a central paradigm in modern deep learning. Yet, despite its widespread practical success, how fine-tuning leverages information from pretraining remains poorly understood theoretically. We study fine-tuning from pretrained weights through the lens of sparse linear regression and two-layer diagonal linear networks. In our setting, pretraining provides information through the support (and signs) of the initialization predictor, which may contain coordinates relevant to the downstream task. We show how pretrained information reshapes the implicit bias and training dynamics, and can thereby reduce the sample complexity of recovering the target parameters and support. In particular, for a clean initialization with correctly inherited signs, we show that the required sample size is comparable to that of a weighted Lasso estimator that explicitly exploits the pretrained support through a suitably chosen regularizer. Our results thus show how information encoded in pretrained weights can be implicitly exploited by gradient-based fine-tuning, reducing the amount of data needed to recover a downstream task.

stat.ML↗
arXiv2026

FuseLIP: Multimodal Embeddings via Early Fusion of Discrete Tokens

Contrastive language-image pre-training aligns features of text-image pairs in a common latent space via distinct encoders for each modality. While this approach achieves impressive performance in several zero-shot tasks, it cannot natively handle multimodal inputs, i.e., encoding image and text into a single feature vector. As a remedy, it is common practice to use additional modules to merge the features extracted by unimodal encoders. In this work, we present FuseLIP, a new architecture for multimodal embedding. Leveraging recent progress in discrete image tokenizers, we propose to use a single transformer model operating on a unified vocabulary of text and image tokens. This early fusion approach allows the different modalities to interact at each depth of encoding and obtain richer representations compared to common late fusion. We collect new datasets for multimodal pre-training and evaluation, designing challenging tasks for multimodal encoders. We show that FuseLIP outperforms late fusion approaches in several multimodal and unimodal embedding tasks.

cs.CV↗
arXiv2026

(How) Learning Rates Regulate Catastrophic Overtraining

Supervised fine-tuning (SFT) is a common first stage of LLM post-training, teaching the model to follow instructions and shaping its behavior as a helpful assistant. At the same time, SFT may harm the fundamental capabilities of an LLM, particularly after long pretraining: a phenomenon known as catastrophic overtraining (Springer et al., 2025). To understand overtraining, we first investigate catastrophic forgetting in finetuning through the lens of implicit regularization of the learning rate. For models trained to the same SFT loss, we identify how the learning rate mediates optimization: finetuning with large and small steps converges to qualitatively different models. Next, we link forgetting to overtraining: learning rate decay increases the sharpness of the pretrained model, which in turn exacerbates catastrophic forgetting during SFT, leading to overtraining. Our findings paint a picture of the overtraining mechanism in LLMs and broadly contribute to the understanding of the interplay between optimization dynamics during pretraining and finetuning.

cs.LG↗
arXiv2026

Incremental Learning of Sparse Attention Patterns in Transformers

This paper studies simple transformers trained on a high-order Markov chain, where the model must incorporate information from multiple past positions, each with different statistical importance. We show that transformers learn the task incrementally, with each stage corresponding to learning how to copy information from a subset of positions via a sparse attention pattern. Notably, the learning dynamics transition from a competitive phase, where all heads focus on the statistically most important positions, to a cooperative phase, where different heads specialize in different patterns. We model these dynamics with simplified differential equations and prove stage-wise convergence of the resulting system. Functionally, these stages correspond to a sequence of increasingly expressive misspecified models, with the full model class reached only at the end. Overall, we give a theoretical account of how structured attention patterns and head specialization emerge in stages without an explicit curriculum, with implications for generalization in sequential tasks.

cs.LG↗
arXiv2026

Induction Heads Interpolate N-Grams

Induction heads are attention circuits believed to underlie in-context learning in transformers, yet a precise characterization of the estimators they implement remains elusive. We study transformers trained on order-$k$ Markov chains and identify two complementary smoothing mechanisms. First, at finite attention-weight scale, the circuit implements a soft context-matching estimator: it aggregates contributions from exact and partial context matches, weighted exponentially by their overlap, and induces a data-dependent interpolation across context orders analogous to Jelinek-Mercer smoothing. Second, a beginning-of-sequence (BOS) token induces additive pseudo-counts, recovering Dirichlet-style smoothing. We construct a disentangled transformer implementing both mechanisms and show that trained transformers recover the predicted attention patterns. Across settings where pseudo-count smoothing is optimal or lower-order contexts provide structured evidence, trained transformers match or outperform classical count-based baselines. Our results bridge mechanistic interpretability of induction heads with classical statistical smoothing, revealing that transformers learn to regularize in-context estimation rather than simply count.

cs.LG↗
arXiv2026

Space-Efficient Language Generation in the Limit

We initiate a resource-aware theory of \textit{language generation in the limit} under the minimal constraint of space efficiency. In our framework, a learner observes an adversarial positive stream from a target language $K$ and must eventually output a hallucination-free hypothesis language $L \subseteq K$ while omitting at most $Δ$ strings of $K$. We focus on $\mathcal{C}_{s,k}$, the collection of languages recognized by DFAs with at most $s$ states over an alphabet of size $k$, as the natural hypothesis class for memory-bounded learners. In the exponential-space regime, we prove that a learner can exactly identify the target $K$. Under a stricter memory budget, we characterize the strongest possible generation guarantees. In particular, we present a streaming algorithm using $\mathrm{poly}(s,k)$ space that converges to a hypothesis with generation gap $Δ= O(k^{2s-2})$. Moreover, the learned hypothesis captures every string in $K$ of length at least $2s-1$. We complement this result with a near-matching lower bound through a reduction from a standard communication complexity problem. Specifically, achieving generation gap $Δ\le k^{(1-\varepsilon)s}$ requires $k^{Ω(\varepsilon s)}$ memory. Together, these results reveal a sharp transition between polynomial-space generation and exponential-space exact identification.

cs.DS↗
arXiv2026

Learning What to Forget: Improving LLM Unlearning via Learned Token-Level Importance

Machine unlearning aims to remove targeted knowledge from a trained model while preserving its general capabilities. For autoregressive language models, not all tokens in a forget sample are equally relevant to forgetting. Existing approaches either ignore this heterogeneity or rely on auxiliary models, heuristics, or external annotations to estimate each token's relevance for forgetting. We instead characterize it through the interaction with the retain objective: a token is forget-specific to the extent that minimizing the forget loss on that token does not conflict with retain optimality. We formalize this perspective as a joint optimization problem over the model parameters and the token weights and show that, under a natural separation condition, the resulting objective recovers the oracle forget-specific token support. Motivated by this formulation, we introduce Alternating Token-Weighted Unlearning (ATWU), a lightweight framework that jointly learns token forget-specificity and model parameters during unlearning using a simple linear scorer over the hidden states, without external token level supervision. Across TOFU and RWKU, ATWU achieves state of the art forget-retain trade-offs, outperforming sample-level methods, probability-based token weighting heuristics, and auxiliary-model-based approaches. Moreover, the learned scores align substantially better with ground truth forget-specific spans, indicating that ATWU identifies semantically meaningful token level forgetting signals. Overall, our results suggest that retain conflict provides an effective criterion for identifying what language models should forget, enabling unsupervised learning of token level forget-specificity directly from model representations with minimal computational overhead.

cs.LG↗
arXiv2026

Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs

The training of neural networks by gradient descent methods is a cornerstone of the deep learning revolution. Yet, despite some recent progress, a complete theory explaining its success is still missing. This article presents, for orthogonal input vectors, a precise description of the gradient flow dynamics of training one-hidden layer ReLU neural networks for the mean squared error at small initialisation. In this setting, despite non-convexity, we show that the gradient flow converges to zero loss and characterise its implicit bias towards minimum variation norm. Furthermore, some interesting phenomena are highlighted: a quantitative description of the initial alignment phenomenon and a proof that the process follows a specific saddle to saddle dynamics.

stat.ML↗
arXiv2026

Transformers Learn Latent Mixture Models In-Context via Mirror Descent

Sequence modelling requires determining which past tokens are causally relevant from the context and their importance: a process inherent to the attention layers in transformers, yet whose underlying learned mechanisms remain poorly understood. In this work, we formalize the task of estimating token importance as an in-context learning problem by introducing a framework based on Mixture of Transition Distributions, where a latent variable determines the influence of past tokens on the next. The distribution over this latent variable is parameterized by unobserved mixture weights that transformers must learn in-context. We demonstrate that transformers can implement Mirror Descent to learn these weights from the context. Specifically, we give an explicit construction of a three-layer transformer that exactly implements one step of Mirror Descent and prove that the resulting estimator is a first-order approximation of the Bayes-optimal predictor. Corroborating our construction and its learnability via gradient descent, we empirically show that transformers trained from scratch learn solutions consistent with our theory: their predictive distributions, attention patterns, and learned transition matrix closely match the construction, while deeper models achieve performance comparable to multi-step Mirror Descent.

cs.LG↗
arXiv2026

Chain-of-Frames: Advancing Video Understanding in Multimodal LLMs via Frame-Aware Reasoning

Recent work has shown that eliciting Large Language Models (LLMs) to generate reasoning traces in natural language before answering the user's request can significantly improve their performance across tasks. This approach has been extended to multimodal LLMs, where the models can produce chains-of-thoughts (CoT) about the content of input images and videos. For video inputs, prior works use complex multi-step pipelines that extract and include relevant frames from videos in the CoT, or produce simpler single-stage reasoning traces at the expense of poor temporal grounding. Here, we propose the first video LLMs with single-stage reasoning that includes explicit references to relevant frames, thereby reducing temporal inconsistencies in the reasoning process. Our approach is simple, unified, and self-contained, employing a single-stage inference to handle complex video understanding tasks without relying on auxiliary modules for frame selection or caption generation. For this, we first create COF-DATA, a large dataset of diverse questions, answers, and corresponding frame-grounded reasoning traces from both natural and synthetic videos, spanning various topics and tasks. Our models, obtained fine-tuning video LLMs on this chain-of-frames (CoF) data, generate reasoning traces that accurately identify key frames to answer given questions. In turn, this consistently improves performance across multiple video understanding benchmarks. Surprisingly, we find that synthetic data alone, despite being out-of-distribution with respect to these real-world benchmarks, provides a significant boost in model accuracy. Code is available at https://github.com/SaraGhazanfari/CoF.

cs.CV↗
arXiv2026

Gradient Flow Polarizes Softmax Outputs towards Low-Entropy Solutions

Understanding the intricate non-convex training dynamics of softmax-based models is crucial for explaining the empirical success of transformers. In this article, we analyze the gradient flow dynamics of the value-softmax model, defined as ${L}(\mathbf{V} σ(\mathbf{a}))$, where $\mathbf{V}$ and $\mathbf{a}$ are a learnable value matrix and attention vector, respectively. As the matrix times softmax vector parameterization constitutes the core building block of self-attention, our analysis provides direct insight into transformer's training dynamics. We reveal that gradient flow on this structure inherently drives the optimization toward solutions characterized by low-entropy outputs. We demonstrate the universality of this polarizing effect across various objectives, including logistic and square loss. Furthermore, we discuss the practical implications of these theoretical results, offering a formal mechanism for empirical phenomena such as attention sinks and massive activations.

cs.LG↗
arXiv2026

On the Adversarial Robustness of Discrete Image Tokenizers

Discrete image tokenizers encode visual inputs as sequences of tokens from a finite vocabulary and are gaining popularity in multimodal systems, including encoder-only, encoder-decoder, and decoder-only models. However, unlike CLIP encoders, their vulnerability to adversarial attacks has not been explored. Ours being the first work studying this topic, we first formulate attacks that aim to perturb the features extracted by discrete tokenizers, and thus change the extracted tokens. These attacks are computationally efficient, application-agnostic, and effective across classification, multimodal retrieval, and captioning tasks. Second, to defend against this vulnerability, inspired by recent work on robust CLIP encoders, we fine-tune popular tokenizers with unsupervised adversarial training, keeping all other components frozen. While unsupervised and task-agnostic, our approach significantly improves robustness to both unsupervised and end-to-end supervised attacks and generalizes well to unseen tasks and data. Unlike supervised adversarial training, our approach can leverage unlabeled images, making it more versatile. Overall, our work highlights the critical role of tokenizer robustness in downstream tasks and presents an important step in the development of safe multimodal foundation models.

cs.CV↗
arXiv2026

On the Out-of-Distribution Generalization of Reasoning in Multimodal LLMs for Simple Visual Planning Tasks

Integrating reasoning in large language models and large vision-language models has recently led to significant improvement of their capabilities. However, the generalization of reasoning models is still vaguely defined and poorly understood. In this work, we present an evaluation framework to rigorously examine how well chain-of-thought (CoT) approaches generalize on a simple planning task. Specifically, we consider a grid-based navigation task in which a model is provided with a map and must output a sequence of moves that guides a player from a start position to a goal while avoiding obstacles. The versatility of the task and its data allows us to fine-tune model variants using different input representations (visual and textual) and CoT reasoning strategies, and systematically evaluate them under both in-distribution (ID) and out-of-distribution (OOD) test conditions. Our experiments show that, while CoT reasoning improves in-distribution generalization across all representations, out-of-distribution generalization (e.g., to larger maps) remains very limited in most cases when controlling for trivial matches with the ID data. Surprisingly, we find that reasoning traces which combine multiple text formats yield the best (and non-trivial) OOD generalization. Finally, purely text-based models consistently outperform those utilizing image-based inputs, including a recently proposed approach relying on latent space reasoning.

cs.LG↗
arXiv2026

HalluHard: A Hard Multi-Turn Hallucination Benchmark

Large language models (LLMs) still produce plausible-sounding but ungrounded factual claims, a problem that worsens in multi-turn dialogue as context grows and early errors cascade. We introduce $\textbf{HalluHard}$, a challenging multi-turn hallucination benchmark with 950 seed questions spanning four high-stakes domains: legal cases, research questions, medical guidelines, and coding. We operationalize groundedness by requiring inline citations for factual assertions. To support reliable evaluation in open-ended settings, we propose a judging pipeline that iteratively retrieves evidence via web search. It can fetch, filter, and parse full-text sources (including PDFs) to assess whether cited material actually supports the generated content. Across a diverse set of frontier proprietary and open-weight models, hallucinations remain substantial even with web search ($\approx 30\%$ for the strongest configuration, Opus-4.5 with web search), with content-grounding errors persisting at high rates. Finally, we show that hallucination behavior is shaped by model capacity, turn position, effective reasoning, and the type of knowledge required.

cs.AI↗
arXiv2025

Exact Learning of Arithmetic with Differentiable Agents

We explore the possibility of exact algorithmic learning with gradient-based methods and introduce a differentiable framework capable of strong length generalization on arithmetic tasks. Our approach centers on Differentiable Finite-State Transducers (DFSTs), a Turing-complete model family that avoids the pitfalls of prior architectures by enabling constant-precision, constant-time generation, and end-to-end log-parallel differentiable training. Leveraging policy-trajectory observations from expert agents, we train DFSTs to perform binary and decimal addition and multiplication. Remarkably, models trained on tiny datasets generalize without error to inputs thousands of times longer than the training examples. These results show that training differentiable agents on structured intermediate supervision could pave the way towards exact gradient-based learning of algorithmic skills. Code available at \href{https://github.com/dngfra/differentiable-exact-algorithmic-learner.git}{https://github.com/dngfra/differentiable-exact-algorithmic-learner.git}.

cs.LG↗