The in-context learning capabilities of modern language models have motivated a deeper mathematical understanding of sequence models. A line of recent work has shown that linear attention models can emulate projected gradient descent iterations to implicitly learn the task vector from the data provided in the context window. In this work, we consider a novel setting where the global task distribution can be partitioned into a union of conditional task distributions. We then examine the use of task-specific prompts and prediction heads for learning the prior information associated with the conditional task distribution using a one-layer attention model. Our results on loss landscape show that task-specific prompts facilitate a covariance-mean decoupling where prompt-tuning explains the conditional mean of the distribution whereas the variance is learned/explained through in-context learning. Incorporating task-specific head further aids this process by entirely decoupling estimation of mean and variance components. This covariance-mean perspective similarly explains how jointly training prompt and attention weights can provably help over fine-tuning after pretraining.

Recent research shows that in-context learning (ICL) can be effective even when demonstrations have missing or incorrect labels. To shed light on this capability, we examine a canonical setting where the demonstrations are drawn according to a binary Gaussian mixture model (GMM) and a certain fraction of the demonstrations have missing labels. We provide a comprehensive theoretical study to show that: (1) The loss landscape of one-layer linear attention models recover the optimal fully-supervised estimator but completely fail to exploit unlabeled data; (2) In contrast, multilayer or looped transformers can effectively leverage unlabeled data by implicitly constructing estimators of the form $\sum_{i\ge 0} a_i (X^\top X)^iX^\top y$ with $X$ and $y$ denoting features and partially-observed labels (with missing entries set to zero). We characterize the class of polynomials that can be expressed as a function of depth and draw connections to Expectation Maximization, an iterative pseudo-labeling algorithm commonly used in semi-supervised learning. Importantly, the leading polynomial power is exponential in depth, so mild amount of depth/looping suffices. As an application of theory, we propose looping off-the-shelf tabular foundation models to enhance their semi-supervision capabilities. Extensive evaluations on real-world datasets show that our method significantly improves the semisupervised tabular learning performance over the standard single pass inference.

Small language models (SLMs) struggle to learn complex reasoning behaviors, especially when high-quality traces are scarce or difficult to learn from. A typical approach for training such models combines a supervised fine-tuning (SFT) stage, often to distill reasoning capabilities from a larger model, followed by a reinforcement learning (RL) stage such as Group Relative Policy Optimization (GRPO). In this paper, we investigate the fundamental limitations of this SFT + RL paradigm and propose methods to overcome them. Using a toy student-expert model over Markov chains, we demonstrate that the SFT + RL strategy can fail completely when (1) the expert's traces are too difficult for the small model to express, or (2) the small model's initialization achieves exponentially sparse rewards as task complexity grows. To address these, we introduce BREAD, a GRPO variant that bridges SFT and RL via partial expert guidance and branch rollouts. When self-generated traces fail, BREAD adaptively inserts short expert prefixes/hints, allowing the small model to complete the rest of the reasoning path, and ensuring that each update includes at least one successful trace. This mechanism both densifies the reward signal and induces a natural learning curriculum. BREAD requires fewer than 40\% of ground-truth traces, consistently outperforming standard GRPO while speeding up the training by about 3$\times$. Importantly, we find that BREAD helps the model solve problems that are otherwise unsolvable by the SFT + RL strategy, highlighting how branch rollouts and expert guidance can aid SLM reasoning.

Froth velocity is one of the key control indices affecting the performance of froth flotation. In the flotation production process, the froth velocity is usually set based on manual experience, which is subjective and arbitrary, which is highly subjective. Moreover, the relationship between the flotation process characteristics and froth velocity is complex, making it difficult to optimize the froth velocity setting. Therefore, an intelligent optimization setting method for froth velocity that integrates multi-source flotation operation information is proposed. In this study, we first analyze the relationship between different operational conditions and the flotation speed, establishing a model for the flotation speed. Subsequently, we employ case-based reasoning (CBR), combined with a flotation speed evaluation and learning model, to obtain the set values for the flotation speed. Finally, an intelligent compensation prediction model based on extreme trees regression (ETR) is utilized to derive the final predicted flotation speed values. Throughout these steps, a flotation production index prediction model is applied to forecast various flotation production indices under the predicted flotation speed. Experiments utilizing historical process data from a flotation production experimental platform demonstrated that the froth velocity settings derived from this method resulted in reduced fluctuation in production indicators and a 12% increase in the comprehensive qualification rate of flotation process indicators. Additionally, the concentrate output rose by 13.9%. This approach significantly enhances flotation production conditions and key performance indicators, confirming its practical application value.

Transformer-based language models are trained on large datasets to predict the next token given an input sequence. Despite this simple training objective, they have led to revolutionary advances in natural language processing. Underlying this success is the self-attention mechanism. In this work, we ask: What does a single self-attention layer learn from next-token prediction? We show that training self-attention with gradient descent learns an automaton which generates the next token in two distinct steps: (1) Hard retrieval: Given input sequence, self-attention precisely selects the high-priority input tokens associated with the last input token. (2) Soft composition: It then creates a convex combination of the high-priority tokens from which the next token can be sampled. Under suitable conditions, we rigorously characterize these mechanics through a directed graph over tokens extracted from the training data. We prove that gradient descent implicitly discovers the strongly-connected components (SCC) of this graph and self-attention learns to retrieve the tokens that belong to the highest-priority SCC available in the context window. Our theory relies on decomposing the model weights into a directional component and a finite component that correspond to hard retrieval and soft composition steps respectively. This also formalizes a related implicit bias formula conjectured in [Tarzanagh et al. 2023]. We hope that these findings shed light on how self-attention processes sequential data and pave the path toward demystifying more complex architectures.

Modern language models rely on the transformer architecture and attention mechanism to perform language understanding and text generation. In this work, we study learning a 1-layer self-attention model from a set of prompts and the associated outputs sampled from the model. We first establish a formal link between the self-attention mechanism and Markov models under suitable conditions: Inputting a prompt to the self-attention model samples the output token according to a *context-conditioned Markov chain* (CCMC). *CCMC* is obtained by weighing the transition matrix of a standard Markov chain according to the sufficient statistics of the prompt/context. Building on this formalism, we develop identifiability/coverage conditions for the data distribution that guarantee consistent estimation of the latent model under a teacher-student setting and establish sample complexity guarantees under IID data. Finally, we study the problem of learning from a single output trajectory generated in response to an initial prompt. We characterize a *winner-takes-all* phenomenon where the generative process of self-attention evolves to sampling from a small set of *winner tokens* that dominate the context window. This provides a mathematical explanation to the tendency of modern LLMs to generate repetitive text.

Recent research has shown that Transformers with linear attention are capable of in-context learning (ICL) by implementing a linear estimator through gradient descent steps. However, the existing results on the optimization landscape apply under stylized settings where task and feature vectors are assumed to be IID and the attention weights are fully parameterized. In this work, we develop a stronger characterization of the optimization and generalization landscape of ICL through contributions on architectures, low-rank parameterization, and correlated designs: (1) We study the landscape of 1-layer linear attention and 1-layer H3, a state-space model. Under a suitable correlated design assumption, we prove that both implement 1-step preconditioned gradient descent. We show that thanks to its native convolution filters, H3 also has the advantage of implementing sample weighting and outperforming linear attention in suitable settings. (2) By studying correlated designs, we provide new risk bounds for retrieval augmented generation (RAG) and task-feature alignment which reveal how ICL sample complexity benefits from distributional alignment. (3) We derive the optimal risk for low-rank parameterized attention weights in terms of covariance spectrum. Through this, we also shed light on how LoRA can adapt to a new distribution by capturing the shift between task covariances. Experimental results corroborate our theoretical findings. Overall, this work explores the optimization and risk landscape of ICL in practically meaningful settings and contributes to a more thorough understanding of its mechanics.

Constructing useful representations across a large number of tasks is a key requirement for sample-efficient intelligent systems. A traditional idea in multitask learning (MTL) is building a shared representation across tasks which can then be adapted to new tasks by tuning last layers. A desirable refinement of using a shared one-fits-all representation is to construct task-specific representations. To this end, recent PathNet/muNet architectures represent individual tasks as pathways within a larger supernet. The subnetworks induced by pathways can be viewed as task-specific representations that are composition of modules within supernet's computation graph. This work explores the pathways proposal from the lens of statistical learning: We first develop novel generalization bounds for empirical risk minimization problems learning multiple tasks over multiple paths (Multipath MTL). In conjunction, we formalize the benefits of resulting multipath representation when adapting to new downstream tasks. Our bounds are expressed in terms of Gaussian complexity, lead to tangible guarantees for the class of linear representations, and provide novel insights into the quality and benefits of a multipath representation. When computation graph is a tree, Multipath MTL hierarchically clusters the tasks and builds cluster-specific representations. We provide further discussion and experiments for hierarchical MTL and rigorously identify the conditions under which Multipath MTL is provably superior to traditional MTL approaches with shallow supernets.

The growing interest in complex decision-making and language modeling problems highlights the importance of sample-efficient learning over very long horizons. This work takes a step in this direction by investigating contextual linear bandits where the current reward depends on at most s prior actions and contexts (not necessarily consecutive), up to a time horizon of h. In order to avoid polynomial dependence on h, we propose new algorithms that leverage sparsity to discover the dependence pattern and arm parameters jointly. We consider both the data-poor (T= h) regimes and derive respective regret upper bounds O(d square-root(sT) +min(q, T) and O( square-root(sdT) ), with sparsity s, feature dimension d, total time horizon T, and q that is adaptive to the reward dependence pattern. Complementing upper bounds, we also show that learning over a single trajectory brings inherent challenges: While the dependence pattern and arm parameters form a rank-1 matrix, circulant matrices are not isometric over rank-1 manifolds and sample complexity indeed benefits from the sparse reward dependence structure. Our results necessitate a new analysis to address long-range temporal dependencies across data and avoid polynomial dependence on the reward horizon h. Specifically, we utilize connections to the restricted isometry property of circulant matrices formed by dependent sub-Gaussian vectors and establish new guarantees that are also of independent interest.

In the context of multitask learning (MTL), representation learning is often accomplished through a feature-extractor ϕ that is shared across all tasks. This way, intuitively, the statistical cost of learning ϕ is collaboratively split across all tasks which enables sample efficiency. In this work, we consider a novel fairness scenario where T tasks can be split into majority and minority groups of sizes Tmajand Tminrespectively: The group assignments are unknown during MTL and Tmin/Tmajratio corresponds to the imbalance level of the problem. We further assume that these groups admit r0, r1-dimensional linear representations which are orthogonal to each other, thus, they would not benefit each other during MTL. Our main finding is that misspecification disproportionately hurts the minority tasks and over-parameterization is key to ensuring fairness of MTL representations. Specifically, we prove that, when we fit a R = r0dimensional misspecified representation, MTL model achieves small task-averaged risk however it has vanishing explanatory power on minority tasks. Conversely, when we fit a R = r0+ r1dimensional well-specified representation, MTL model achieves small risks on both majority and minority tasks which are on par with the oracle baseline of training each group individually with the hindsight knowledge of assignments. Finally, we provide experimental results which are consistent with our theoretical findings.

In-context learning (ICL) is a type of prompting where a transformer model operates on a sequence of (input, output) examples and performs inference on-the-fly. In this work, we formalize in-context learning as an algorithm learning problem where a transformer model implicitly constructs a hypothesis function at inference-time. We first explore the statistical aspects of this abstraction through the lens of multitask learning: We obtain generalization bounds for ICL when the input prompt is (1) a sequence of i.i.d. (input, label) pairs or (2) a trajectory arising from a dynamical system. The crux of our analysis is relating the excess risk to the stability of the algorithm implemented by the transformer. We characterize when transformer/attention architecture provably obeys the stability condition and also provide empirical verification. For generalization on unseen tasks, we identify an inductive bias phenomenon in which the transfer learning risk is governed by the task complexity and the number of MTL tasks in a highly predictable manner. Finally, we provide numerical evaluations that (1) demonstrate transformers can indeed implement near-optimal algorithms on classical regression problems with i.i.d. and dynamic data, (2) provide insights on stability, and (3) verify our theoretical predictions.

Chain-of-thought (CoT) is a method that enables language models to handle complex reasoning tasks by decomposing them into simpler steps. Despite its success, the underlying mechanics of CoT are not yet fully understood. In an attempt to shed light on this, our study investigates the impact of CoT on the ability of transformers to in-context learn a simple to study, yet general family of compositional functions: multi-layer perceptrons (MLPs). In this setting, we find that the success of CoT can be attributed to breaking down in-context learning of a compositional function into two distinct phases: focusing on and filtering data related to each step of the composition and in-context learning the single-step composition function. Through both experimental and theoretical evidence, we demonstrate how CoT significantly reduces the sample complexity of in-context learning (ICL) and facilitates the learning of complex functions that non-CoT methods struggle with. Furthermore, we illustrate how transformers can transition from vanilla in-context learning to mastering a compositional function with CoT by simply incorporating additional layers that perform the necessary data-filtering for CoT via the attention mechanism. In addition to these test-time benefits, we show CoT helps accelerate pretraining by learning shortcuts to represent complex functions and filtering plays an important role in this process. These findings collectively provide insights into the mechanics of CoT, inviting further investigation of its role in complex reasoning tasks.

Attention mechanism is a central component of the transformer architecture which led to the phenomenal success of large language models. However, the theoretical principles underlying the attention mechanism are poorly understood, especially its nonconvex optimization dynamics. In this work, we explore the seminal softmax-attention model $f(X)=\langle Xv, \texttt{softmax}(XWp)\rangle$, where $X$ is the token sequence and $(v,W,p)$ are trainable parameters. We prove that running gradient descent on $p$, or equivalently $W$, converges in direction to a max-margin solution that separates *locally-optimal* tokens from non-optimal ones. This clearly formalizes attention as an optimal token selection mechanism. Remarkably, our results are applicable to general data and precisely characterize *optimality* of tokens in terms of the value embeddings $Xv$ and problem geometry. We also provide a broader regularization path analysis that establishes the margin maximizing nature of attention even for nonlinear prediction heads. When optimizing $v$ and $p$ simultaneously with logistic loss, we identify conditions under which the regularization paths directionally converge to their respective hard-margin SVM solutions where $v$ separates the input features based on their labels. Interestingly, the SVM formulation of $p$ is influenced by the support vector geometry of $v$. Finally, we verify our theoretical findings via numerical experiments and provide insights.

Deep networks are typically trained with many more parameters than the size of the training dataset. Recent empirical evidence indicates that the practice of overparameterization not only benefits training large models, but also assists – perhaps counterintuitively – building lightweight models. Specifically, it suggests that overparameterization benefits model pruning / sparsification. This paper sheds light on these empirical findings by theoretically characterizing the high-dimensional asymptotics of model pruning in the overparameterized regime. The theory presented addresses the following core question: ``should one train a small model from the beginning, or first train a large model and then prune?''. We analytically identify regimes in which, even if the location of the most informative features is known, we are better off fitting a large model and then pruning rather than simply training with the known informative features. This leads to a new double descent in the training of sparse models: growing the original model, while preserving the target sparsity, improves the test accuracy as one moves beyond the overparameterization threshold. Our analysis further reveals the benefit of retraining by relating it to feature correlations. We find that the above phenomena are already present in linear and random-features models. Our technical approach advances the toolset of high-dimensional analysis and precisely characterizes the asymptotic distribution of over-parameterized least-squares. The intuition gained by analytically studying simpler models is numerically verified on neural networks.