The diversity of time series applications and scarcity of domain-specific data highlight the need for time-series models with strong few-shot learning capabilities. In this work, we propose a novel training scheme and a transformer-based architecture, collectively referred to as TimePFN, for multivariate time-series (MTS) forecasting. TimePFN is based on the concept of Prior-data Fitted Networks (PFN), which aims to approximate Bayesian inference. Our approach consists of (1) generating synthetic MTS data through diverse Gaussian process kernels and the linear coregionalization method, and (2) a novel MTS architecture capable of utilizing both temporal and cross-channel dependencies across all input patches. We evaluate TimePFN on several benchmark datasets and demonstrate that it outperforms the existing state-of-the-art models for MTS forecasting in both zero-shot and few-shot settings. Notably, fine-tuning TimePFN with as few as 500 data points nearly matches full dataset training error, and even 50 data points yield competitive results. We also find that TimePFN exhibits strong univariate forecasting performance, attesting to its generalization ability. Overall, this work unlocks the power of synthetic data priors for MTS forecasting and facilitates strong zero- and few-shot forecasting performance.

A growing number of machine learning scenarios rely on knowledge distillation where one uses the output of a surrogate model as labels to supervise the training of a target model. In this work, we provide a sharp characterization of this process for ridgeless, high-dimensional regression, under two settings: *(i)* model shift, where the surrogate model is arbitrary, and *(ii)* distribution shift, where the surrogate model is the solution of empirical risk minimization with out-of-distribution data. In both cases, we characterize the precise risk of the target model through non-asymptotic bounds in terms of sample size and data distribution under mild conditions. As a consequence, we identify the form of the optimal surrogate model, which reveals the benefits and limitations of discarding weak features in a data-dependent fashion. In the context of weak-to-strong (W2S) generalization, this has the interpretation that *(i)* W2S training, with the surrogate as the weak model, can provably outperform training with strong labels under the same data budget, but *(ii)* it is unable to improve the data scaling law. We validate our results on numerical experiments both on ridgeless regression and on neural network architectures.

Test-time training (TTT) methods explicitly update the weights of a model to adapt to the specific test instance, and they have found success in a variety of settings, including most recently language modeling and reasoning. To demystify this success, we investigate a gradient-based TTT algorithm for in-context learning, where we train a transformer model on the in-context demonstrations provided in the test prompt. Specifically, we provide a comprehensive theoretical characterization of linear transformers when the update rule is a single gradient step. Our theory (i) delineates the role of alignment between pretraining distribution and target task, (ii) demystifies how TTT can alleviate distribution shift, and (iii) quantifies the sample complexity of TTT including how it can significantly reduce the eventual sample size required for in-context learning. As our empirical contribution, we study the benefits of TTT for TabPFN, a tabular foundation model. In line with our theory, we demonstrate that TTT significantly reduces the required sample size for tabular classification (3 to 5 times fewer) unlocking substantial inference efficiency with a negligible training cost.

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.

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.

This paper is eligible for the Jack Keil Wolf ISIT Student Paper Award. We study a communication system in which a source node is to send status updates over a channel to a remote destination. The update packets will be received error-free, however the channel imposes independent and identically distributed transmission delay on each transmission. The destination node will utilize the upcoming update packets at certain instants that are referred to as query instants. The Query Age of Information (QAoI) at the destination is defined to be the time-average Age of Information (AoI) measured at query instants. We define the problem where the receiver decides when to send requests to "pull" data from the source, to minimize the QAoI, as the pull-or-wait (PoW) problem. We contrast the solution of the PoW problem with that of the update-or-wait (UoW) problem, where the source decides when to generate updates to minimize the time-average AoI. We show that when the query instants occur according to a Poisson process, the solution of the PoW problem is equivalent to that of the UoW problem; however, under periodic query arrivals, the optimal QAoI is always less than or equal to the time average AoI under the same power constraint. We believe this motivates using QAoI as an objective instead of plain AoI in many applications requiring time-sensitive updates.