Atieh Bastan, Xiaoling He, A. Pavan, N. V. Vinodchandran
原文摘要
We study Binary Tokenization: given a corpus of binary strings over $\{0,1\}$ and a vocabulary budget $K$, select at most $K$ substrings (called vocabulary) to minimize the total number of tokens needed to represent the corpus. The primitive symbols $0$ and $1$ are always available, and each document must be segmented separately. Letting $n$ denote the total corpus length and $M$ for the maximum document length, we prove that, for every fixed $\varepsilon\in(0,1)$, Binary Tokenization problem does not have polynomial-time $n^{1-\varepsilon}$-approximation algorithms unless $\mathsf{P}=\mathsf{NP}$. To complement the hardness result, we give a polynomial-time $(γ,γ)$-bicriteria algorithm, with $γ= \sqrt{M}\log n$: The algorithm outputs a vocabulary of size at most $γK$ with number of tokens at most $γ{\rm OPT}_K$, where ${\rm OPT}_K$ is the optimal number of tokens under the original budget $K$. We also study a storage-budget variant, in which the sum of the selected substrings' lengths is bounded by $B$. For $2\le B<n$, we obtain an $(O(\log n\log B),O(\log n\log B))$-bicriteria approximation algorithm.
Chain-of-thought (CoT) reasoning has emerged as a powerful paradigm for extending the computational capabilities of transformer-based models. A growing body of recent work has begun to uncover the fine-grained computational power of transformers augmented with CoT, as well as their connections to classical complexity theory. In this work, we advance this line of research by establishing the first hierarchy theorem for CoT-based computation. Our main result shows that for any reasonable resource bound $r(n) \geq n$ and any $\varepsilon > 0$, \[ \mathrm{CoT}(r(n)) \subsetneq \mathrm{CoT}(r(n)^{1+\varepsilon}), \] where $\mathrm{CoT}(r(n))$ denotes the class of decision problems solvable by transformers using $O(r(n))$ CoT reasoning steps on inputs of length $n$. Our result shows that increasing the number of CoT reasoning steps provably yields strictly greater computational power, establishing CoT steps as a fundamental resource in transformer-based computation. Unlike classical hierarchy theorems, which are typically proved via diagonalization, our approach leverages known coarse-grained simulations between CoT-augmented transformers and Turing machines in both directions. Building on these connections, we adapt the classical padding technique from complexity theory to this setting to obtain a tight hierarchy. A key technical contribution of our work is the implementation of padding within the transformer-based CoT model.
Michael Chen, Atieh Bastan, A. Pavan, N. V. Vinodchandran
原文摘要
We study the sample complexity of list replicable algorithms for estimating the biases of $d$ coins. An algorithm is $(k,\varepsilon,δ)$-list replicable if for every bias vector $\mathbf{p}\in[0,1]^d$, there is a set of outputs $L_{\mathbf{p}}\subseteq \mathbb{R}^d$ of at most $k$ elements such that every $\mathbf{y}\in L_{\mathbf{p}}$ satisfies $\lVert\mathbf{y}-\mathbf{p}\rVert_\infty\leq\varepsilon$, and the algorithm output is in $L_{\mathbf{p}}$ with probability at least $1-δ$. Prior work gave a $(d+1,\varepsilon,δ)$-list replicable algorithm with sample complexity $\widetilde{O}\left(\frac{d^3}{\varepsilon^2}\right)$, but no nontrivial sample complexity lower bound was known. We establish the first non-trivial sample complexity lower bound for this problem and give an improved algorithm that matches it up to polylogarithmic factors. Together, these results determine the optimal sample complexity for list size $d+1$ to be $\widetildeΘ\left(\frac{d^2}{\varepsilon^2}\right)$, showing that the quadratic dependence on dimension is inherent up to polylogarithmic factors. More generally, for every $k\in[d+1,2^d]$, we establish matching upper and lower bounds of $\widetildeΘ\left(\frac{d^2}{\varepsilon^2\log k}\right)$ on the total sample complexity.