We present Position Interpolation (PI) that extends the context window sizes of RoPE-based pretrained LLMs such as LLaMA models to up to 32768 with minimal fine-tuning (within 1000 steps), while demonstrating strong empirical results on various tasks that require long context, including passkey retrieval, language modeling, and long document summarization from LLaMA 7B to 65B. Meanwhile, the extended model by Position Interpolation preserve quality relatively well on tasks within its original context window. To achieve this goal, Position Interpolation linearly down-scales the input position indices to match the original context window size, rather than extrapolating beyond the trained context length which may lead to catastrophically high attention scores that completely ruin the self-attention mechanism. Our theoretical study shows that the upper bound of interpolation is at least $\sim 600 \times$ smaller than that of extrapolation, further demonstrating its stability. Models extended via Position Interpolation retain its original architecture and can reuse most pre-existing optimization and infrastructure.

We study the {\em combinatorial pure exploration (CPE)} problem in the stochastic multi-armed bandit setting, where a learner explores a set of arms with the objective of identifying the optimal member of a \emph{decision class}, which is a collection of subsets of arms with certain combinatorial structures such as size-$K$ subsets, matchings, spanning trees or paths, etc. The CPE problem represents a rich class of pure exploration tasks which covers not only many existing models but also novel cases where the object of interest has a non-trivial combinatorial structure. In this paper, we provide a series of results for the general CPE problem. We present general learning algorithms which work for all decision classes that admit offline maximization oracles in both fixed confidence and fixed budget settings. We prove problem-dependent upper bounds of our algorithms.

We study the {\em combinatorial pure exploration (CPE)} problem in the stochastic multi-armed bandit setting, where a learner explores a set of arms with the objective of identifying the optimal member of a \emph{decision class}, which is a collection of subsets of arms with certain combinatorial structures such as size-$K$ subsets, matchings, spanning trees or paths, etc. The CPE problem represents a rich class of pure exploration tasks which covers not only many existing models but also novel cases where the object of interest has a non-trivial combinatorial structure. In this paper, we provide a series of results for the general CPE problem. We present general learning algorithms which work for all decision classes that admit offline maximization oracles in both fixed confidence and fixed budget settings. We prove problem-dependent upper bounds of our algorithms. Our analysis exploits the combinatorial structures of the decision classes and introduces a new analytic tool. We also establish a general problem-dependent lower bound for the CPE problem. Our results show that the proposed algorithms achieve the optimal sample complexity (within logarithmic factors) for many decision classes. In addition, applying our results back to the problems of top-$K$ arms identification and multiple bandit best arms identification, we recover the best available upper bounds up to constant factors and partially resolve a conjecture on the lower bounds.

Tensor completion from incomplete observations is a problem of significant practical interest. However, it is unlikely that there exists an efficient algorithm with provable guarantee to recover a general tensor from a limited number of observations. In this paper, we study the recovery algorithm for pairwise interaction tensors, which has recently gained considerable attention for modeling multiple attribute data due to its simplicity and effectiveness. Specifically, in the absence of noise, we show that one can exactly recover a pairwise interaction tensor by solving a constrained convex program which minimizes the weighted sum of nuclear norms of matrices from $O(nr\log^2(n))$ observations. For the noisy cases, we also prove error bounds for a constrained convex program for recovering the tensors. Our experiments on the synthetic dataset demonstrate that the recovery performance of our algorithm agrees well with the theory. In addition, we apply our algorithm on a temporal collaborative filtering task and obtain state-of-the-art results.