Approximation Bounds for Transformer Networks with Application to Regression

We explore the approximation capabilities of Transformer networks for H\"older and Sobolev functions, and apply these results to address nonparametric regression estimation with dependent observations. First, we establish novel upper bounds for standard Transformer networks approximating sequence-to-sequence mappings whose component functions are H\"older continuous with smoothness index $\gamma \in (0,1]$. To achieve an approximation error $\varepsilon$ under the $L^p$-norm for $p \in [1, \infty]$, it suffices to use a fixed-depth Transformer network whose total number of parameters scales as $\varepsilon^{-d_x n / \gamma}$. This result not only extends existing findings to include the case $p = \infty$, but also matches the best known upper bounds on number of parameters previously obtained for fixed-depth FNNs and RNNs. Similar bounds are also derived for Sobolev functions. Second, we derive explicit convergence rates for the nonparametric regression problem under various $\beta$-mixing data assumptions, which allow the dependence between observations to weaken over time. Our bounds on the sample complexity impose no constraints on weight magnitudes. Lastly, we propose a novel proof strategy to establish approximation bounds, inspired by the Kolmogorov-Arnold representation theorem. We show that if the self-attention layer in a Transformer can perform column averaging, the network can approximate sequence-to-sequence H\"older functions, offering new insights into the interpretability of self-attention mechanisms.

Approximation Bounds for Recurrent Neural Networks with Application to Regression

We study the approximation capacity of deep ReLU recurrent neural networks (RNNs) and explore the convergence properties of nonparametric least squares regression using RNNs. We derive upper bounds on the approximation error of RNNs for H\"older smooth functions, in the sense that the output at each time step of an RNN can approximate a H\"older function that depends only on past and current information, termed a past-dependent function. This allows a carefully constructed RNN to simultaneously approximate a sequence of past-dependent H\"older functions. We apply these approximation results to derive non-asymptotic upper bounds for the prediction error of the empirical risk minimizer in regression problem. Our error bounds achieve minimax optimal rate under both exponentially $\beta$-mixing and i.i.d. data assumptions, improving upon existing ones. Our results provide statistical guarantees on the performance of RNNs.

Convergence Analysis of Flow Matching in Latent Space with Transformers

We present theoretical convergence guarantees for ODE-based generative models, specifically flow matching. We use a pre-trained autoencoder network to map high-dimensional original inputs to a low-dimensional latent space, where a transformer network is trained to predict the velocity field of the transformation from a standard normal distribution to the target latent distribution. Our error analysis demonstrates the effectiveness of this approach, showing that the distribution of samples generated via estimated ODE flow converges to the target distribution in the Wasserstein-2 distance under mild and practical assumptions. Furthermore, we show that arbitrary smooth functions can be effectively approximated by transformer networks with Lipschitz continuity, which may be of independent interest.