These limitations make LSTM models difficult to deploy on specialized hardware requiring real-time processes with inferior hardware resources and power budget.

Homomorphic encryption has recently attracted attention as a key solution to preserve privacy in machine learning applications.

We thank the reviewers for the time they spent evaluating our manuscript and for their valuable comments. We agree that having theoretical guarantees would be a big plus. As for scalability, the bottleneck of our method is the single-linkage algorithm. Similarly to Monath et al. (NeurIPS 2017), our idea consists Given the significant body of additional material, we feel that this topic is best left to a future publication. Line 8,56,70,93: I would suggest a more cautious usage of the word "equivalent".

Large language models (LLMs) experience significant performance degradation when the input exceeds the pretrain-ing context window, primarily due to the out-of-distribution (OOD) behavior of Rotary Position Embedding (RoPE). Recent studies mitigate this problem by remapping OOD positions into the in-distribution range with fixed mapping strategies, ignoring the dynamic relationship between input length and the model's effective context window. To this end, we propose Length-aware M ulti-grained P ositional Encoding (LaMPE), a training-free method that fully utilizes the model's effective context window for adaptive long-context scaling in LLMs. Motivated by the left-skewed frequency distribution of relative positions, LaMPE establishes a dynamic relationship between mapping length and input length through a parametric scaled sigmoid function to adaptively allocate positional capacity across varying input lengths. Meanwhile, LaMPE devises a novel multi-grained attention mechanism that strategically allocates positional resolution across different sequence regions to capture both fine-grained locality and long-range dependencies. Our method can be seamlessly applied to a wide range of RoPE-based LLMs without training. Extensive experiments on three representative LLMs across five mainstream long-context benchmarks demonstrate that LaMPE achieves significant performance improvements compared to existing length extrapolation methods. The code will be released at https://github.com/scar-on/LaMPE.

We point out that neural networks are not black boxes, and their generalization stems from the ability to dynamically map a dataset to the local extrema of the model function. We further prove that the number of local extrema in a neural network is positively correlated with the number of its parameters, and on this basis, we give a new algorithm that is different from the back-propagation algorithm, which we call the extremum-increment algorithm. Some difficult situations, such as gradient vanishing and overfitting, can be reasonably explained and dealt with in this framework.