Jian Li

Xunyu Zhu, Jian Li, Can Ma et al.

Large Language Models (LLMs) have demonstrated exceptional proficiency in mathematical reasoning tasks due to their extensive parameter counts and training on vast datasets. Despite these capabilities, deploying LLMs is hindered by their computational demands. Distilling LLM mathematical reasoning into Smaller Language Models (SLMs) has emerged as a solution to this challenge, although these smaller models often suffer from errors in calculation and semantic understanding. Prior work has proposed Program-of-Thought Distillation (PoTD) to avoid calculation error. To further address semantic understanding errors, we propose Key-Point-Driven Mathematical Reasoning Distillation (KPDD). KPDD enhances the reasoning performance of SLMs by breaking down the problem-solving process into three stages: Core Question Extraction, Problem-Solving Information Extraction, and Step-by-Step Solution. This method is further divided into KPDD-CoT, which generates Chain-of-Thought rationales, and KPDD-PoT, which creates Program-of-Thought rationales. The experiment results show that KPDD-CoT significantly improves reasoning abilities, while KPDD-PoT achieves state-of-the-art performance in mathematical reasoning tasks. Our approach effectively mitigates misunderstanding errors, advancing the deployment of efficient and capable SLMs.

Guojun Xiong, Jian Li, Rahul Singh

We study a finite-horizon restless multi-armed bandit problem with multiple actions, dubbed R(MA)^2B. The state of each arm evolves according to a controlled Markov decision process (MDP), and the reward of pulling an arm depends on both the current state of the corresponding MDP and the action taken. The goal is to sequentially choose actions for arms so as to maximize the expected value of the cumulative rewards collected. Since finding the optimal policy is typically intractable, we propose a computationally appealing index policy which we call Occupancy-Measured-Reward Index Policy. Our policy is well-defined even if the underlying MDPs are not indexable. We prove that it is asymptotically optimal when the activation budget and number of arms are scaled up, while keeping their ratio as a constant. For the case when the system parameters are unknown, we develop a learning algorithm. Our learning algorithm uses the principle of optimism in the face of uncertainty and further uses a generative model in order to fully exploit the structure of Occupancy-Measured-Reward Index Policy. We call it the R(MA)^2B-UCB algorithm. As compared with the existing algorithms, R(MA)^2B-UCB performs close to an offline optimum policy, and also achieves a sub-linear regret with a low computational complexity. Experimental results show that R(MA)^2B-UCB outperforms the existing algorithms in both regret and run time.

Lijie Chen, Jian Li, Mingda Qiao

In the Best-$k$-Arm problem, we are given $n$ stochastic bandit arms, each associated with an unknown reward distribution. We are required to identify the $k$ arms with the largest means by taking as few samples as possible. In this paper, we make progress towards a complete characterization of the instance-wise sample complexity bounds for the Best-$k$-Arm problem. On the lower bound side, we obtain a novel complexity term to measure the sample complexity that every Best-$k$-Arm instance requires. This is derived by an interesting and nontrivial reduction from the Best-$1$-Arm problem. We also provide an elimination-based algorithm that matches the instance-wise lower bound within doubly-logarithmic factors. The sample complexity of our algorithm strictly dominates the state-of-the-art for Best-$k$-Arm (module constant factors).

Lijie Chen, Jian Li, Mingda Qiao

In the classical best arm identification (Best-$1$-Arm) problem, we are given $n$ stochastic bandit arms, each associated with a reward distribution with an unknown mean. We would like to identify the arm with the largest mean with probability at least $1-δ$, using as few samples as possible. Understanding the sample complexity of Best-$1$-Arm has attracted significant attention since the last decade. However, the exact sample complexity of the problem is still unknown. Recently, Chen and Li made the gap-entropy conjecture concerning the instance sample complexity of Best-$1$-Arm. Given an instance $I$, let $μ_{[i]}$ be the $i$th largest mean and $Δ_{[i]}=μ_{[1]}-μ_{[i]}$ be the corresponding gap. $H(I)=\sum_{i=2}^nΔ_{[i]}^{-2}$ is the complexity of the instance. The gap-entropy conjecture states that $Ω\left(H(I)\cdot\left(\lnδ^{-1}+\mathsf{Ent}(I)\right)\right)$ is an instance lower bound, where $\mathsf{Ent}(I)$ is an entropy-like term determined by the gaps, and there is a $δ$-correct algorithm for Best-$1$-Arm with sample complexity $O\left(H(I)\cdot\left(\lnδ^{-1}+\mathsf{Ent}(I)\right)+Δ_{[2]}^{-2}\ln\lnΔ_{[2]}^{-1}\right)$. If the conjecture is true, we would have a complete understanding of the instance-wise sample complexity of Best-$1$-Arm. We make significant progress towards the resolution of the gap-entropy conjecture. For the upper bound, we provide a highly nontrivial algorithm which requires \[O\left(H(I)\cdot\left(\lnδ^{-1} +\mathsf{Ent}(I)\right)+Δ_{[2]}^{-2}\ln\lnΔ_{[2]}^{-1}\mathrm{polylog}(n,δ^{-1})\right)\] samples in expectation. For the lower bound, we show that for any Gaussian Best-$1$-Arm instance with gaps of the form $2^{-k}$, any $δ$-correct monotone algorithm requires $Ω\left(H(I)\cdot\left(\lnδ^{-1} + \mathsf{Ent}(I)\right)\right)$ samples in expectation.

Lijie Chen, Jian Li

We study the best arm identification (BEST-1-ARM) problem, which is defined as follows. We are given $n$ stochastic bandit arms. The $i$th arm has a reward distribution $D_i$ with an unknown mean $μ_{i}$. Upon each play of the $i$th arm, we can get a reward, sampled i.i.d. from $D_i$. We would like to identify the arm with the largest mean with probability at least $1-δ$, using as few samples as possible. We provide a nontrivial algorithm for BEST-1-ARM, which improves upon several prior upper bounds on the same problem. We also study an important special case where there are only two arms, which we call the sign problem. We provide a new lower bound of sign, simplifying and significantly extending a classical result by Farrell in 1964, with a completely new proof. Using the new lower bound for sign, we obtain the first lower bound for BEST-1-ARM that goes beyond the classic Mannor-Tsitsiklis lower bound, by an interesting reduction from Sign to BEST-1-ARM. We propose an interesting conjecture concerning the optimal sample complexity of BEST-1-ARM from the perspective of instance-wise optimality.