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In-Context Deep Learning via Transformer Models

Weimin Wu, Maojiang Su, Jerry Yao-Chieh Hu, Zhao Song, Han Liu

TL;DR

This work tackles whether a transformer can feasibly simulate the training of a deep neural network via in-context learning (ICL). By constructing explicit ReLU- and Softmax-based transformer architectures, it shows how a transformer can perform multiple gradient-descent steps on an $N$-layer NN in-context, with rigorous approximation and convergence guarantees. The key contributions include a $(2N+4)L$-layer ReLU transformer that emulates $L$ GD steps and extends to varying input/output dimensions, plus a 4L-layer Softmax transformer with universal-approximation support for similar capabilities, all supported by detailed gradient-decomposition analyses and error bounds. Empirical results on synthetic data demonstrate that ICL can match direct training performance for 3-, 4-, and 6-layer networks, highlighting the potential of foundation-model in-context learning to perform deep learning tasks without explicit parameter updates.

Abstract

We investigate the transformer's capability to simulate the training process of deep models via in-context learning (ICL), i.e., in-context deep learning. Our key contribution is providing a positive example of using a transformer to train a deep neural network by gradient descent in an implicit fashion via ICL. Specifically, we provide an explicit construction of a $(2N+4)L$-layer transformer capable of simulating $L$ gradient descent steps of an $N$-layer ReLU network through ICL. We also give the theoretical guarantees for the approximation within any given error and the convergence of the ICL gradient descent. Additionally, we extend our analysis to the more practical setting using Softmax-based transformers. We validate our findings on synthetic datasets for 3-layer, 4-layer, and 6-layer neural networks. The results show that ICL performance matches that of direct training.

In-Context Deep Learning via Transformer Models

TL;DR

This work tackles whether a transformer can feasibly simulate the training of a deep neural network via in-context learning (ICL). By constructing explicit ReLU- and Softmax-based transformer architectures, it shows how a transformer can perform multiple gradient-descent steps on an -layer NN in-context, with rigorous approximation and convergence guarantees. The key contributions include a -layer ReLU transformer that emulates GD steps and extends to varying input/output dimensions, plus a 4L-layer Softmax transformer with universal-approximation support for similar capabilities, all supported by detailed gradient-decomposition analyses and error bounds. Empirical results on synthetic data demonstrate that ICL can match direct training performance for 3-, 4-, and 6-layer networks, highlighting the potential of foundation-model in-context learning to perform deep learning tasks without explicit parameter updates.

Abstract

We investigate the transformer's capability to simulate the training process of deep models via in-context learning (ICL), i.e., in-context deep learning. Our key contribution is providing a positive example of using a transformer to train a deep neural network by gradient descent in an implicit fashion via ICL. Specifically, we provide an explicit construction of a -layer transformer capable of simulating gradient descent steps of an -layer ReLU network through ICL. We also give the theoretical guarantees for the approximation within any given error and the convergence of the ICL gradient descent. Additionally, we extend our analysis to the more practical setting using Softmax-based transformers. We validate our findings on synthetic datasets for 3-layer, 4-layer, and 6-layer neural networks. The results show that ICL performance matches that of direct training.

Paper Structure

This paper contains 48 sections, 26 theorems, 193 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: In-Context Learning and In-Context Gradient Descent
  3. In-Context Gradient Descent on -Layer Neural Networks
  4. Problem Setup: ICGD for -Layer Neural Networks
  5. Explicit Gradient Descent of -Layer Neural Networks
  6. Transformers Approximate Gradient Descent of -Layer Neural Networks In-Context
  7. In-Context Deep Learning with Softmax Transformers
  8. Numerical Studies
  9. Conclusion
  10. Related Work, Broader Impact, Further Discussion and Limitations
  11. Related Work
  12. In-Context Learning.
  13. In-Context Gradient Descent on Deep Models wangcontextpanigrahi2023trainable.
  14. Broader Impact
  15. Further Discussion
  16. ...and 33 more sections

Key Result

Lemma 1

Fix any $B_v, \eta > 0$. Suppose loss function $\mathcal{L}_n(w)$ on $n$ data points $\{(x_i,y_i)\}_{i \in [n]}$ follows eqn:loss. Suppose closed domain $\mathcal{W}$ and projection function ${\rm Proj}_{\mathcal{W}}(w)$ follows eqn:domain_w_m. Let $A_i(j), r'_i(j), R_i(j), V_j$ be as defined in def where $A_i(j)$ denote the derivative of $\ell(p_i(N), y_i)$ with respect to the parameters in the $

Figures (7)

  • Figure 1: One Step In-Context Gradient Descent (ICGD) with $(2N+4)$-layer Transformer. This illustration presents the backpropagation process within an ICGD in a transformer model with $2N+4$ layers. It simulates a single gradient descent step for an $N$-layer neural network, trained with loss $\mathcal{L}_n$ and datasets $\{(x_i, y_i)\}_{i \in [n]}$. The term $p_i(j)$ denotes the output after the $j$-th layer for input $x_i$. The terms $r'_i(j)$, $u(p_i(N), y_i)$, and $s_i(j)$ are intermediate gradient terms of gradient $\nabla \mathcal{L}_n(w)$ from the chain rule. The expression ${\rm Proj}_{\mathcal{W}}(w - \eta \nabla \mathcal{L}_n(w))$ shows one gradient descent step. Here, $\eta$ is the learning rate, and $\mathcal{W}$ denotes the bounded domain for the $N$-layer NN parameters $w$.
  • Figure 2: Performance of ICL in ReLU-Transformer and Softmax-Transformer: ICL learns 6-layer NN and achieves R-squared values comparable to those from training with prompt samples.
  • Figure 3: Performance of ICL in ReLU-Transformer: ICL learns 3-layer, 4-layer, and 6-layer NN and achieves R-squared values comparable to those from training with prompt samples. The results also show the ICL performance declines as the testing distribution diverges from the pretraining one.
  • Figure 4: Performance of ICL in $\mathop{\rm{Softmax}}$-Transformer: ICL learns 3-layer, 4-layer, and 6-layer NN and achieves R-squared values comparable to those from training with prompt samples. The results also show the ICL performance declines as the testing distribution diverges from the pretraining one. Note that performance decreases when the prompt length exceeds the pretraining length (i.e., 50), a well-known issue dai2019transformeranil2022exploring. We believe this is due to the absolute positional encodings in GPT-2, as noted in zhang2023trained
  • Figure 5: Performance of ICL Across Various $N$-layer Network Parameter Distributions for the ReLU-Transformer: ICL learns 4-layer NN and achieves R-squared values comparable to those from training with prompt samples, even when the parameter distribution in the $N$-layer network during testing diverges from that in the pretraining phase ($N(0, I)$).
  • ...and 2 more figures

Theorems & Definitions (65)

  • Definition 1: $N$-Layer Neural Network
  • Remark 1: Prediction Function for $j$-th layer on $i$-th Data: $p_i(j)$
  • Remark 2: Why Bounded Domain $\mathcal{W}$?
  • Definition 2: Abbreviations
  • Lemma 1: Decomposition of One Gradient Descent Step
  • proof : Proof Sketch
  • Definition 3: Definition of intermediate terms
  • Definition 4: Approximability by Sum of ReLUs, Definition 12 of bai2023transformers
  • Lemma 2: Approximate $p_i(j)$
  • proof
  • ...and 55 more