This paper investigates whether sequence models can learn to perform numerical algorithms, e.g. gradient descent, on the fundamental problem of least squares. Our goal is to inherit two properties of standard algorithms from numerical analysis: (1) machine precision, i.e. we want to obtain solutions that are accurate to near floating point error, and (2) numerical generality, i.e. we want them to apply broadly across problem instances. We find that prior approaches using Transformers fail to meet these criteria, and identify limitations present in existing architectures and training procedures. First, we show that softmax Transformers struggle to perform high-precision multiplications, which prevents them from precisely learning numerical algorithms. Second, we identify an alternate class of architectures, comprised entirely of polynomials, that can efficiently represent high-precision gradient descent iterates. Finally, we investigate precision bottlenecks during training and address them via a high-precision training recipe that reduces stochastic gradient noise. Our recipe enables us to train two polynomial architectures, gated convolutions and linear attention, to perform gradient descent iterates on least squares problems. For the first time, we demonstrate the ability to train to near machine precision. Applied iteratively, our models obtain 100,000x lower MSE than standard Transformers trained end-to-end and they incur a 10,000x smaller generalization gap on out-of-distribution problems. We make progress towards end-to-end learning of numerical algorithms for least squares.

There has been a surge in the use of large language models (LLM) conversational agents to generate responses based on long-term history from multiple sessions. However, existing long-term open-domain dialogue datasets lack complex, real-world personalization and fail to capture implicit reasoning-where relevant information is embedded in subtle, syntactic, or semantically distant connections rather than explicit statements. In such cases, traditional retrieval methods fail to capture relevant context, and long-context modeling also becomes inefficient due to numerous complicated persona-related details. To address this gap, we introduce ImplexConv, a large-scale long-term dataset with 2,500 examples, each containing approximately 100 conversation sessions, designed to study implicit reasoning in personalized dialogues. Additionally, we propose TaciTree, a novel hierarchical tree framework that structures conversation history into multiple levels of summarization. Instead of brute-force searching all data, TaciTree enables an efficient, level-based retrieval process where models refine their search by progressively selecting relevant details. Our experiments demonstrate that TaciTree significantly improves the ability of LLMs to reason over long-term conversations with implicit contextual dependencies.

Conversational search utilizes muli-turn natural language contexts to retrieve relevant passages. Existing conversational dense retrieval models mostly view a conversation as a fixed sequence of questions and responses, overlooking the severe data sparsity problem -- that is, users can perform a conversation in various ways, and these alternate conversations are unrecorded. Consequently, they often struggle to generalize to diverse conversations in real-world scenarios. In this work, we propose a framework for generalizing Conversational dense retrieval via LLM-cognition data Augmentation (ConvAug). ConvAug first generates multi-level augmented conversations to capture the diverse nature of conversational contexts. Inspired by human cognition, we devise a cognition-aware process to mitigate the generation of false positives, false negatives, and hallucinations. Moreover, we develop a difficulty-adaptive sample filter that selects challenging samples for complex conversations, thereby giving the model a larger learning space. A contrastive learning objective is then employed to train a better conversational context encoder. Extensive experiments conducted on four public datasets, under both normal and zero-shot settings, demonstrate the effectiveness, generalizability, and applicability of ConvAug.