When Does Online Imitation Learning Help in LLM Post-Training? The Role of (Non-)Realizability Beyond Horizon

Online imitation learning (IL), particularly on-policy distillation, has emerged as a strong LLM post-training approach, often outperforming offline supervised fine-tuning (SFT). Yet a principled understanding of when and why online interaction helps remains unclear. In this work, we challenge the view that error accumulation is the main source of online IL's advantage, and instead show that the benefits of online interaction depend critically on whether the setting is realizable, i.e., whether the student policy class can represent the expert policy. Under realizability, we empirically find that offline IL already matches expert performance. In contrast, in non-realizable (misspecified) settings, we prove that offline IL encounters an information-theoretic bottleneck even when horizon $H=1$, and propose a structural characterization of misspecification relative to the reward, under which online IL provably achieves high performance despite a large distributional mismatch between the expert and student policies.

Less Data, Faster Training: repeating smaller datasets speeds up learning via sampling biases

This work investigates the ``small-vs-large gap'', where repeating on fewer samples can lead to compute saving during training compared to using a larger dataset. This is observed across algorithmic tasks, architectures and optimizers and cannot be explained using prior theory. We argue that the speedup comes from appropriate layer-wise growth enabled by sampling biases, which is more pronounced when the dataset size is smaller. We provide both theoretical analysis and empirical evidence from various interventions. Our results suggest that using a smaller dataset with more repetitions is not just a fallback strategy under data scarcity, but can be proactively leveraged as a favorable inductive biases for optimization, particularly in reasoning tasks.

The two clocks and the innovation window: When and how generative models learn rules

Generative models trained on finite data face a fundamental tension: their score-matching or next-token objective converges to the empirical training distribution rather than the population distribution we seek to learn. Using rule-valid synthetic tasks, we trace this tension across two training timescales: $τ_{\mathrm{rule}}$, the step at which generations first become rule-valid, and $τ_{\mathrm{mem}}$, the step at which models begin reproducing training samples. Focusing on parity and extending to other binary rules and combinatorial puzzles, we characterize how these two clocks, $τ_{\mathrm{rule}}$ and $τ_{\mathrm{mem}}$, depend on key aspects of the learning setup. Specifically, we show that $τ_{\mathrm{rule}}$ increases with rule complexity and decreases with model capacity, while $τ_{\mathrm{mem}}$ is approximately invariant to the rule and scales nearly linearly with dataset size $N$. We define the \emph{innovation window} as the interval $[τ_{\mathrm{rule}}, τ_{\mathrm{mem}}]$. This window widens with increasing $N$ and narrows with rule complexity, and may vanish entirely when $τ_{\mathrm{rule}} \geq τ_{\mathrm{mem}}$. The same two-clock structure arises in both diffusion (DiT) and autoregressive (GPT) models, with architecture-dependent offsets. Dissecting the learned score of DiT models reveals a corresponding evolution of the optimization landscapes, where rule-valid samples' basins expand substantially around $τ_{\mathrm{rule}}$, while training samples' basins begin to dominate around $τ_{\mathrm{mem}}$. Together, these results yield a unified and predictive account of when and how generative models exhibit genuine innovation.

Progressive distillation induces an implicit curriculum

Knowledge distillation leverages a teacher model to improve the training of a student model. A persistent challenge is that a better teacher does not always yield a better student, to which a common mitigation is to use additional supervision from several ``intermediate'' teachers. One empirically validated variant of this principle is progressive distillation, where the student learns from successive intermediate checkpoints of the teacher. Using sparse parity as a sandbox, we identify an implicit curriculum as one mechanism through which progressive distillation accelerates the student's learning. This curriculum is available only through the intermediate checkpoints but not the final converged one, and imparts both empirical acceleration and a provable sample complexity benefit to the student. We then extend our investigation to Transformers trained on probabilistic context-free grammars (PCFGs) and real-world pre-training datasets (Wikipedia and Books). Through probing the teacher model, we identify an analogous implicit curriculum where the model progressively learns features that capture longer context. Our theoretical and empirical findings on sparse parity, complemented by empirical observations on more complex tasks, highlight the benefit of progressive distillation via implicit curriculum across setups.

TinyGSM: achieving >80% on GSM8k with small language models

Small-scale models offer various computational advantages, and yet to which extent size is critical for problem-solving abilities remains an open question. Specifically for solving grade school math, the smallest model size so far required to break the 80\% barrier on the GSM8K benchmark remains to be 34B. Our work studies how high-quality datasets may be the key for small language models to acquire mathematical reasoning. We introduce \texttt{TinyGSM}, a synthetic dataset of 12.3M grade school math problems paired with Python solutions, generated fully by GPT-3.5. After finetuning on \texttt{TinyGSM}, we find that a duo of a 1.3B generation model and a 1.3B verifier model can achieve 81.5\% accuracy, outperforming existing models that are orders of magnitude larger. This also rivals the performance of the GPT-3.5 ``teacher'' model (77.4\%), from which our model's training data is generated. Our approach is simple and has two key components: 1) the high-quality dataset \texttt{TinyGSM}, 2) the use of a verifier, which selects the final outputs from multiple candidate generations.

Transformers are uninterpretable with myopic methods: a case study with bounded Dyck grammars

Interpretability methods aim to understand the algorithm implemented by a trained model (e.g., a Transofmer) by examining various aspects of the model, such as the weight matrices or the attention patterns. In this work, through a combination of theoretical results and carefully controlled experiments on synthetic data, we take a critical view of methods that exclusively focus on individual parts of the model, rather than consider the network as a whole. We consider a simple synthetic setup of learning a (bounded) Dyck language. Theoretically, we show that the set of models that (exactly or approximately) solve this task satisfy a structural characterization derived from ideas in formal languages (the pumping lemma). We use this characterization to show that the set of optima is qualitatively rich; in particular, the attention pattern of a single layer can be ``nearly randomized'', while preserving the functionality of the network. We also show via extensive experiments that these constructions are not merely a theoretical artifact: even after severely constraining the architecture of the model, vastly different solutions can be reached via standard training. Thus, interpretability claims based on inspecting individual heads or weight matrices in the Transformer can be misleading.

Exposing Attention Glitches with Flip-Flop Language Modeling

Why do large language models sometimes output factual inaccuracies and exhibit erroneous reasoning? The brittleness of these models, particularly when executing long chains of reasoning, currently seems to be an inevitable price to pay for their advanced capabilities of coherently synthesizing knowledge, pragmatics, and abstract thought. Towards making sense of this fundamentally unsolved problem, this work identifies and analyzes the phenomenon of attention glitches, in which the Transformer architecture's inductive biases intermittently fail to capture robust reasoning. To isolate the issue, we introduce flip-flop language modeling (FFLM), a parametric family of synthetic benchmarks designed to probe the extrapolative behavior of neural language models. This simple generative task requires a model to copy binary symbols over long-range dependencies, ignoring the tokens in between. We find that Transformer FFLMs suffer from a long tail of sporadic reasoning errors, some of which we can eliminate using various regularization techniques. Our preliminary mechanistic analyses show why the remaining errors may be very difficult to diagnose and resolve. We hypothesize that attention glitches account for (some of) the closed-domain hallucinations in natural LLMs.

Understanding Augmentation-based Self-Supervised Representation Learning via RKHS Approximation

Good data augmentation is one of the key factors that lead to the empirical success of self-supervised representation learning such as contrastive learning and masked language modeling, yet theoretical understanding of its role in learning good representations remains limited. Recent work has built the connection between self-supervised learning and approximating the top eigenspace of a graph Laplacian operator. Learning a linear probe on top of such features can naturally be connected to RKHS regression. In this work, we use this insight to perform a statistical analysis of augmentation-based pretraining. We start from the isometry property, a key geometric characterization of the target function given by the augmentation. Our first main theorem provides, for an arbitrary encoder, near tight bounds for both the estimation error incurred by fitting the linear probe on top of the encoder, and the approximation error entailed by the fitness of the RKHS the encoder learns. Our second main theorem specifically addresses the case where the encoder extracts the top-d eigenspace of a Monte-Carlo approximation of the underlying kernel with the finite pretraining samples. Our analysis completely disentangles the effects of the model and the augmentation. A key ingredient in our analysis is the augmentation complexity, which we use to quantitatively compare different augmentations and analyze their impact on downstream performance on synthetic and real datasets.

Transformers Learn Shortcuts to Automata

Algorithmic reasoning requires capabilities which are most naturally understood through recurrent models of computation, like the Turing machine. However, Transformer models, while lacking recurrence, are able to perform such reasoning using far fewer layers than the number of reasoning steps. This raises the question: what solutions are these shallow and non-recurrent models finding? We investigate this question in the setting of learning automata, discrete dynamical systems naturally suited to recurrent modeling and expressing algorithmic tasks. Our theoretical results completely characterize shortcut solutions, whereby a shallow Transformer with only $o(T)$ layers can exactly replicate the computation of an automaton on an input sequence of length $T$. By representing automata using the algebraic structure of their underlying transformation semigroups, we obtain $O(\log T)$-depth simulators for all automata and $O(1)$-depth simulators for all automata whose associated groups are solvable. Empirically, we perform synthetic experiments by training Transformers to simulate a wide variety of automata, and show that shortcut solutions can be learned via standard training. We further investigate the brittleness of these solutions and propose potential mitigations.

Masked prediction tasks: a parameter identifiability view

The vast majority of work in self-supervised learning, both theoretical and empirical (though mostly the latter), have largely focused on recovering good features for downstream tasks, with the definition of "good" often being intricately tied to the downstream task itself. This lens is undoubtedly very interesting, but suffers from the problem that there isn't a "canonical" set of downstream tasks to focus on -- in practice, this problem is usually resolved by competing on the benchmark dataset du jour. In this paper, we present an alternative lens: one of parameter identifiability. More precisely, we consider data coming from a parametric probabilistic model, and train a self-supervised learning predictor with a suitably chosen parametric form. Then, we ask whether we can read off the ground truth parameters of the probabilistic model from the optimal predictor. We focus on the widely used self-supervised learning method of predicting masked tokens, which is popular for both natural languages and visual data. While incarnations of this approach have already been successfully used for simpler probabilistic models (e.g. learning fully-observed undirected graphical models), we focus instead on latent-variable models capturing sequential structures -- namely Hidden Markov Models with both discrete and conditionally Gaussian observations. We show that there is a rich landscape of possibilities, out of which some prediction tasks yield identifiability, while others do not. Our results, borne of a theoretical grounding of self-supervised learning, could thus potentially beneficially inform practice. Moreover, we uncover close connections with uniqueness of tensor rank decompositions -- a widely used tool in studying identifiability through the lens of the method of moments.