Appendix382 AAdditional Experiments3383 A.1 Experiments on the ETT datasets384 In the main body, we present a comparison of the benchmark methods on the ETTm2 dataset. In this385 section, we extend our analysis to the remaining three ETT datasets, namely ETTh1, ETTh2, and386 ETTm1, as summarized in Table 7. Our experimental results reveal that Basisformer outperforms all387 other methods in terms of MSE and MAE. In all experiments, lower MSE values indicate better model performance, and we present the best results in boldface. Experimental results with longer length input setting391 Throughout our research, we maintain consistency in our experimental settings by fixing the input392 length to be 96(with a reduced input length of 36for the illness dataset), instead of using a longer393 length.

Since the proposal of transformers (Vaswani et al., 2017), these models have been limited to bounded input lengths, because of their need to attend to every token in the input. In this work, we propose Unlimiformer: a general approach that wraps any existing pretrained encoder-decoder transformer, and offloads the cross-attention computation to a single k-nearest-neighbor (kNN) index, while the returned kNN distances are the attention dot-product scores. This kNN index can be kept on either the GPU or CPU memory and queried in sub-linear time; this way, we can index practically unlimited input sequences, while every attention head in every decoder layer retrieves its top-k keys, instead of attending to every key. We evaluate Unlimiformer on several long-document and book-summarization benchmarks, showing that it can process even 500k token-long inputs from the BookSum dataset, without any input truncation at test time. We demonstrate that Unlimiformer improves pretrained models such as BART (Lewis et al., 2020a) and Longformer (Beltagy et al., 2020) by extending them to unlimited inputs without additional learned weights and without modifying their code. Our code and models are publicly available, and support LLaMA-2 as well2.

Time series forecasting is crucial for applications across multiple domains and various scenarios. Although Transformers have dramatically advanced the landscape of forecasting, their effectiveness remains debated.

In this paper, we intent to handle this emerging issue.

Recurrent Neural Networks (RNNs) are among the most popular models in sequential data analysis. Yet, in the foundational PAC learning language, what concept class can it learn? Moreover, how can the same recurrent unit simultaneously learn functions from different input tokens to different output tokens, without affecting each other?

Recurrent Neural Network (RNN) is a fundamental structure in deep learning. Recently, some works study the training process of over-parameterized neural networks, and show that over-parameterized networks can learn functions in some notable concept classes with a provable generalization error bound. In this paper, we analyze the training and generalization for RNNs with random initialization, and provide the following improvements over recent works:(1) For a RNN with input sequence $x=(X_1,X_2,...,X_L)$, previous works study to learn functions that are summation of $f(\beta^T_lX_l)$ and require normalized conditions that $||X_l||\leq\epsilon$ with some very small $\epsilon$ depending on the complexity of $f$. In this paper, using detailed analysis about the neural tangent kernel matrix, we prove a generalization error bound to learn such functions without normalized conditions and show that some notable concept classes are learnable with the numbers of iterations and samples scaling almost-polynomially in the input length $L$.(2) Moreover, we prove a novel result to learn N-variables functions of input sequence with the form $f(\beta^T[X_{l_1},...,X_{l_N}])$, which do not belong to the ``additive'' concept class, i,e., the summation of function $f(X_l)$. And we show that when either $N$ or $l_0=\max(l_1,..,l_N)-\min(l_1,..,l_N)$ is small, $f(\beta^T[X_{l_1},...,X_{l_N}])$ will be learnable with the number iterations and samples scaling almost-polynomially in the input length $L$.