In-Context Occam's Razor: How Transformers Prefer Simpler Hypotheses on the Fly

In-context learning (ICL) enables transformers to adapt to new tasks through contextual examples without parameter updates. While existing research has typically studied ICL in fixed-complexity environments, practical language models encounter tasks spanning diverse complexity levels. This paper investigates how transformers navigate hierarchical task structures where higher-complexity categories can perfectly represent any pattern generated by simpler ones. We design well-controlled testbeds based on Markov chains and linear regression that reveal transformers not only identify the appropriate complexity level for each task but also accurately infer the corresponding parameters--even when the in-context examples are compatible with multiple complexity hypotheses. Notably, when presented with data generated by simpler processes, transformers consistently favor the least complex sufficient explanation. We theoretically explain this behavior through a Bayesian framework, demonstrating that transformers effectively implement an in-context Bayesian Occam's razor by balancing model fit against complexity penalties. We further ablate on the roles of model size, training mixture distribution, inference context length, and architecture. Finally, we validate this Occam's razor-like inductive bias on a pretrained GPT-4 model with Boolean-function tasks as case study, suggesting it may be inherent to transformers trained on diverse task distributions.

On the Effect of Negative Gradient in Group Relative Deep Reinforcement Optimization

Reinforcement learning (RL) has become popular in enhancing the reasoning capabilities of large language models (LLMs), with Group Relative Policy Optimization (GRPO) emerging as a widely used algorithm in recent systems. Despite GRPO's widespread adoption, we identify a previously unrecognized phenomenon we term Lazy Likelihood Displacement (LLD), wherein the likelihood of correct responses marginally increases or even decreases during training. This behavior mirrors a recently discovered misalignment issue in Direct Preference Optimization (DPO), attributed to the influence of negative gradients. We provide a theoretical analysis of GRPO's learning dynamic, identifying the source of LLD as the naive penalization of all tokens in incorrect responses with the same strength. To address this, we develop a method called NTHR, which downweights penalties on tokens contributing to the LLD. Unlike prior DPO-based approaches, NTHR takes advantage of GRPO's group-based structure, using correct responses as anchors to identify influential tokens. Experiments on math reasoning benchmarks demonstrate that NTHR effectively mitigates LLD, yielding consistent performance gains across models ranging from 0.5B to 3B parameters.

On the Geometry of Semantics in Next-token Prediction

Modern language models demonstrate a remarkable ability to capture linguistic meaning despite being trained solely through next-token prediction (NTP). We investigate how this conceptually simple training objective leads models to extract and encode latent semantic and grammatical concepts. Our analysis reveals that NTP optimization implicitly guides models to encode concepts via singular value decomposition (SVD) factors of a centered data-sparsity matrix that captures next-word co-occurrence patterns. While the model never explicitly constructs this matrix, learned word and context embeddings effectively factor it to capture linguistic structure. We find that the most important SVD factors are learned first during training, motivating the use of spectral clustering of embeddings to identify human-interpretable semantics, including both classical k-means and a new orthant-based method directly motivated by our interpretation of concepts. Overall, our work bridges distributional semantics, neural collapse geometry, and neural network training dynamics, providing insights into how NTP's implicit biases shape the emergence of meaning representations in language models.

Implicit Bias of SignGD and Adam on Multiclass Separable Data

In the optimization of overparameterized models, different gradient-based methods can achieve zero training error yet converge to distinctly different solutions inducing different generalization properties. While a decade of research on implicit optimization bias has illuminated this phenomenon in various settings, even the foundational case of linear classification with separable data still has important open questions. We resolve a fundamental gap by characterizing the implicit bias of both Adam and Sign Gradient Descent in multi-class cross-entropy minimization: we prove that their iterates converge to solutions that maximize the margin with respect to the classifier matrix's max-norm and characterize the rate of convergence. We extend our results to general p-norm normalized steepest descent algorithms and to other multi-class losses.

Leveraging Online Olympiad-Level Math Problems for LLMs Training and Contamination-Resistant Evaluation

Advances in Large Language Models (LLMs) have sparked interest in their ability to solve Olympiad-level math problems. However, the training and evaluation of these models are constrained by the limited size and quality of available datasets, as creating large-scale data for such advanced problems requires extensive effort from human experts. In addition, current benchmarks are prone to contamination, leading to unreliable evaluations. In this paper, we present an automated pipeline that leverages the rich resources of the Art of Problem Solving (AoPS) forum, which predominantly features Olympiad-level problems and community-driven solutions. Using open-source LLMs, we develop a method to extract question-answer pairs from the forum, resulting in AoPS-Instruct, a dataset of more than 600,000 high-quality QA pairs. Our experiments demonstrate that fine-tuning LLMs on AoPS-Instruct improves their reasoning abilities across various benchmarks. Moreover, we build an automatic pipeline that introduces LiveAoPSBench, an evolving evaluation set with timestamps, derived from the latest forum data, providing a contamination-resistant benchmark for assessing LLM performance. Notably, we observe a significant decline in LLM performance over time, suggesting their success on older examples may stem from pre-training exposure rather than true reasoning ability. Our work presents a scalable approach to creating and maintaining large-scale, high-quality datasets for advanced math reasoning, offering valuable insights into the capabilities and limitations of LLMs in this domain. Our benchmark and code is available at https://github.com/DSL-Lab/aops

Sharper Guarantees for Learning Neural Network Classifiers with Gradient Methods

In this paper, we study the data-dependent convergence and generalization behavior of gradient methods for neural networks with smooth activation. Our first result is a novel bound on the excess risk of deep networks trained by the logistic loss, via an alogirthmic stability analysis. Compared to previous works, our results improve upon the shortcomings of the well-established Rademacher complexity-based bounds. Importantly, the bounds we derive in this paper are tighter, hold even for neural networks of small width, do not scale unfavorably with width, are algorithm-dependent, and consequently capture the role of initialization on the sample complexity of gradient descent for deep nets. Specialized to noiseless data separable with margin $\gamma$ by neural tangent kernel (NTK) features of a network of width $\Omega(\poly(\log(n)))$, we show the test-error rate to be $e^{O(L)}/{\gamma^2 n}$, where $n$ is the training set size and $L$ denotes the number of hidden layers. This is an improvement in the test loss bound compared to previous works while maintaining the poly-logarithmic width conditions. We further investigate excess risk bounds for deep nets trained with noisy data, establishing that under a polynomial condition on the network width, gradient descent can achieve the optimal excess risk. Finally, we show that a large step-size significantly improves upon the NTK regime's results in classifying the XOR distribution. In particular, we show for a one-hidden-layer neural network of constant width $m$ with quadratic activation and standard Gaussian initialization that mini-batch SGD with linear sample complexity and with a large step-size $\eta=m$ reaches the perfect test accuracy after only $\ceil{\log(d)}$ iterations, where $d$ is the data dimension.

DARE the Extreme: Revisiting Delta-Parameter Pruning For Fine-Tuned Models

Storing open-source fine-tuned models separately introduces redundancy and increases response times in applications utilizing multiple models. Delta-parameter pruning (DPP), particularly the random drop and rescale (DARE) method proposed by Yu et al., addresses this by pruning the majority of delta parameters--the differences between fine-tuned and pre-trained model weights--while typically maintaining minimal performance loss. However, DARE fails when either the pruning rate or the magnitude of the delta parameters is large. We highlight two key reasons for this failure: (1) an excessively large rescaling factor as pruning rates increase, and (2) high mean and variance in the delta parameters. To push DARE's limits, we introduce DAREx (DARE the eXtreme), which features two algorithmic improvements: (1) DAREx-q, a rescaling factor modification that significantly boosts performance at high pruning rates (e.g., >30 % on COLA and SST2 for encoder models, with even greater gains in decoder models), and (2) DAREx-L2, which combines DARE with AdamR, an in-training method that applies appropriate delta regularization before DPP. We also demonstrate that DAREx-q can be seamlessly combined with vanilla parameter-efficient fine-tuning techniques like LoRA and can facilitate structural DPP. Additionally, we revisit the application of importance-based pruning techniques within DPP, demonstrating that they outperform random-based methods when delta parameters are large. Through this comprehensive study, we develop a pipeline for selecting the most appropriate DPP method under various practical scenarios.

Implicit Geometry of Next-token Prediction: From Language Sparsity Patterns to Model Representations

Next-token prediction (NTP) over large text corpora has become the go-to paradigm to train large language models. Yet, it remains unclear how NTP influences the mapping of linguistic patterns to geometric properties of the resulting model representations. We frame training of large language models as soft-label classification over sparse probabilistic label vectors, coupled with an analytical approximation that allows unrestricted generation of context embeddings. This approach links NTP training to rank-constrained, nuclear-norm regularized optimization in the logit domain, offering a framework for analyzing the geometry of word and context embeddings. In large embedding spaces, we find that NTP implicitly favors learning logits with a sparse plus low-rank structure. While the sparse component captures the co-occurrence frequency of context-word pairs, the orthogonal low-rank component, which becomes dominant as training progresses, depends solely on the sparsity pattern of the co-occurrence matrix. Consequently, when projected onto an appropriate subspace, representations of contexts that are followed by the same set of next-tokens collapse, a phenomenon we term subspace-collapse. We validate our findings on synthetic and small-scale real language datasets. Finally, we outline potential research directions aimed at deepening the understanding of NTP's influence on the learning of linguistic patterns and regularities.

Upper and lower memory capacity bounds of transformers for next-token prediction

Given a sequence of tokens, such as words, the task of next-token prediction is to predict the next-token conditional probability distribution. Decoder-only transformers have become effective models for this task, but their properties are still not fully understood. In particular, the largest number of distinct context sequences that a decoder-only transformer can interpolate next-token distributions for has not been established. To fill this gap, we prove upper and lower bounds on this number, which are equal up to a multiplicative constant. We prove these bounds in the general setting where next-token distributions can be arbitrary as well as the empirical setting where they are calculated from a finite number of document sequences. Our lower bounds are for one-layer transformers and our proofs highlight an important injectivity property satisfied by self-attention. Furthermore, we provide numerical evidence that the minimal number of parameters for memorization is sufficient for being able to train the model to the entropy lower bound.

Supervised Contrastive Representation Learning: Landscape Analysis with Unconstrained Features

Recent findings reveal that over-parameterized deep neural networks, trained beyond zero training-error, exhibit a distinctive structural pattern at the final layer, termed as Neural-collapse (NC). These results indicate that the final hidden-layer outputs in such networks display minimal within-class variations over the training set. While existing research extensively investigates this phenomenon under cross-entropy loss, there are fewer studies focusing on its contrastive counterpart, supervised contrastive (SC) loss. Through the lens of NC, this paper employs an analytical approach to study the solutions derived from optimizing the SC loss. We adopt the unconstrained features model (UFM) as a representative proxy for unveiling NC-related phenomena in sufficiently over-parameterized deep networks. We show that, despite the non-convexity of SC loss minimization, all local minima are global minima. Furthermore, the minimizer is unique (up to a rotation). We prove our results by formalizing a tight convex relaxation of the UFM. Finally, through this convex formulation, we delve deeper into characterizing the properties of global solutions under label-imbalanced training data.