Towards Understanding the Universality of Transformers for Next-Token Prediction
- URL: http://arxiv.org/abs/2410.03011v1
- Date: Thu, 3 Oct 2024 21:42:21 GMT
- Title: Towards Understanding the Universality of Transformers for Next-Token Prediction
- Authors: Michael E. Sander, Gabriel Peyré,
- Abstract summary: Causal Transformers are trained to predict the next token for a given context.
We take a step towards understanding this phenomenon by studying the approximation ability of Transformers for next-token prediction.
- Score: 20.300660057193017
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: Causal Transformers are trained to predict the next token for a given context. While it is widely accepted that self-attention is crucial for encoding the causal structure of sequences, the precise underlying mechanism behind this in-context autoregressive learning ability remains unclear. In this paper, we take a step towards understanding this phenomenon by studying the approximation ability of Transformers for next-token prediction. Specifically, we explore the capacity of causal Transformers to predict the next token $x_{t+1}$ given an autoregressive sequence $(x_1, \dots, x_t)$ as a prompt, where $ x_{t+1} = f(x_t) $, and $ f $ is a context-dependent function that varies with each sequence. On the theoretical side, we focus on specific instances, namely when $ f $ is linear or when $ (x_t)_{t \geq 1} $ is periodic. We explicitly construct a Transformer (with linear, exponential, or softmax attention) that learns the mapping $f$ in-context through a causal kernel descent method. The causal kernel descent method we propose provably estimates $x_{t+1} $ based solely on past and current observations $ (x_1, \dots, x_t) $, with connections to the Kaczmarz algorithm in Hilbert spaces. We present experimental results that validate our theoretical findings and suggest their applicability to more general mappings $f$.
Related papers
- Attention with Trained Embeddings Provably Selects Important Tokens [73.77633297039097]
Token embeddings play a crucial role in language modeling but, despite this practical relevance, their theoretical understanding remains limited.<n>Our paper addresses the gap by characterizing the structure of embeddings obtained via gradient descent.<n>Experiments on real-world datasets (IMDB, Yelp) exhibit a phenomenology close to that unveiled by our theory.
arXiv Detail & Related papers (2025-05-22T21:00:09Z) - Computational-Statistical Tradeoffs at the Next-Token Prediction Barrier: Autoregressive and Imitation Learning under Misspecification [50.717692060500696]
Next-token prediction with the logarithmic loss is a cornerstone of autoregressive sequence modeling.
Next-token prediction can be made robust so as to achieve $C=tilde O(H)$, representing moderate error amplification.
No computationally efficient algorithm can achieve sub-polynomial approximation factor $C=e(log H)1-Omega(1)$.
arXiv Detail & Related papers (2025-02-18T02:52:00Z) - IT$^3$: Idempotent Test-Time Training [95.78053599609044]
This paper introduces Idempotent Test-Time Training (IT$3$), a novel approach to addressing the challenge of distribution shift.
IT$3$ is based on the universal property of idempotence.
We demonstrate the versatility of our approach across various tasks, including corrupted image classification.
arXiv Detail & Related papers (2024-10-05T15:39:51Z) - Revisiting Step-Size Assumptions in Stochastic Approximation [1.3654846342364308]
It is shown for the first time that this assumption is not required for convergence and finer results.<n>Rates of convergence are obtained for the standard algorithm and for estimates obtained via the averaging technique of Polyak and Ruppert.<n>Results from numerical experiments illustrate that $beta_theta$ may be large due to a combination of multiplicative noise and Markovian memory.
arXiv Detail & Related papers (2024-05-28T05:11:05Z) - Transformer In-Context Learning for Categorical Data [51.23121284812406]
We extend research on understanding Transformers through the lens of in-context learning with functional data by considering categorical outcomes, nonlinear underlying models, and nonlinear attention.
We present what is believed to be the first real-world demonstration of this few-shot-learning methodology, using the ImageNet dataset.
arXiv Detail & Related papers (2024-05-27T15:03:21Z) - Conv-Basis: A New Paradigm for Efficient Attention Inference and Gradient Computation in Transformers [16.046186753149]
Self-attention mechanism is the key to the success of transformers in recent Large Language Models (LLMs)
We leverage the convolution-like structure of attention matrices to develop an efficient approximation method for attention using convolution matrices.
We hope our new paradigm for accelerating attention computation in transformer models can help their application to longer contexts.
arXiv Detail & Related papers (2024-05-08T17:11:38Z) - Mechanics of Next Token Prediction with Self-Attention [41.82477691012942]
Transformer-based language models are trained on large datasets to predict the next token given an input sequence.
We show that training self-attention with gradient descent learns an automaton which generates the next token in two distinct steps.
We hope that these findings shed light on how self-attention processes sequential data and pave the path toward demystifying more complex architectures.
arXiv Detail & Related papers (2024-03-12T21:15:38Z) - How do Transformers perform In-Context Autoregressive Learning? [76.18489638049545]
We train a Transformer model on a simple next token prediction task.
We show how a trained Transformer predicts the next token by first learning $W$ in-context, then applying a prediction mapping.
arXiv Detail & Related papers (2024-02-08T16:24:44Z) - How to Capture Higher-order Correlations? Generalizing Matrix Softmax
Attention to Kronecker Computation [12.853829771559916]
We study a generalization of attention which captures triple-wise correlations.
This generalization is able to solve problems about detecting triple-wise connections that were shown to be impossible for transformers.
We show that our construction, algorithms, and lower bounds naturally generalize to higher-order tensors and correlations.
arXiv Detail & Related papers (2023-10-06T07:42:39Z) - A Unified Framework for Uniform Signal Recovery in Nonlinear Generative
Compressed Sensing [68.80803866919123]
Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $mathbfx*$ rather than for all $mathbfx*$ simultaneously.
Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples.
We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy.
arXiv Detail & Related papers (2023-09-25T17:54:19Z) - Misspecified Phase Retrieval with Generative Priors [15.134280834597865]
We estimate an $n$-dimensional signal $mathbfx$ from $m$ i.i.d.realizations of the single index model $y.
We show that both steps enjoy a statistical rate of order $sqrt(klog L)cdot (log m)/m$ under suitable conditions.
arXiv Detail & Related papers (2022-10-11T16:04:11Z) - Approximate Function Evaluation via Multi-Armed Bandits [51.146684847667125]
We study the problem of estimating the value of a known smooth function $f$ at an unknown point $boldsymbolmu in mathbbRn$, where each component $mu_i$ can be sampled via a noisy oracle.
We design an instance-adaptive algorithm that learns to sample according to the importance of each coordinate, and with probability at least $1-delta$ returns an $epsilon$ accurate estimate of $f(boldsymbolmu)$.
arXiv Detail & Related papers (2022-03-18T18:50:52Z) - Linear Time Sinkhorn Divergences using Positive Features [51.50788603386766]
Solving optimal transport with an entropic regularization requires computing a $ntimes n$ kernel matrix that is repeatedly applied to a vector.
We propose to use instead ground costs of the form $c(x,y)=-logdotpvarphi(x)varphi(y)$ where $varphi$ is a map from the ground space onto the positive orthant $RRr_+$, with $rll n$.
arXiv Detail & Related papers (2020-06-12T10:21:40Z) - $O(n)$ Connections are Expressive Enough: Universal Approximability of
Sparse Transformers [71.31712741938837]
We show that sparse Transformers with only $O(n)$ connections per attention layer can approximate the same function class as the dense model with $n2$ connections.
We also present experiments comparing different patterns/levels of sparsity on standard NLP tasks.
arXiv Detail & Related papers (2020-06-08T18:30:12Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.