Zhiyuan Liu

Junhao Chen, Shengding Hu, Zhiyuan Liu et al.

Large Language Models (LLMs) exhibit various emergent abilities. Among these abilities, some might reveal the internal working mechanisms of models. In this paper, we uncover a novel emergent capability in models: the intrinsic ability to perform extended sequences of calculations without relying on chain-of-thought step-by-step solutions. Remarkably, the most advanced models can directly output the results of two-digit number additions with lengths extending up to 15 addends. We hypothesize that the model emerges Implicit Discrete State Representations (IDSRs) within its hidden states and performs symbolic calculations internally. To test this hypothesis, we design a sequence of experiments that look into the hidden states. Specifically, we first confirm that IDSRs exist. Then, we provide interesting observations about the formation of IDSRs from layer, digit, and sequence perspectives. Finally, we confirm that models indeed use IDSRs to produce the final answers. However, we also discover that these state representations are far from lossless in current open-sourced models, leading to inaccuracies in their final performance. Our work presents a novel exploration of LLMs' symbolic calculation abilities and the underlying mechanisms.

Zhuoyang Wu, Xinze Li, Zhenghao Liu et al.

Large Language Models (LLMs) have exhibited strong reasoning capabilities and achieved remarkable performance in mathematical problem-solving tasks. Recently, distilling reasoning ability from long-form Chains-of-Thought (CoTs) has emerged as a promising approach for enhancing Small Language Models (SLMs). Existing studies typically treat SLMs as student models and use long-form CoTs as supervision signals for Supervised Fine-Tuning (SFT) to transfer reasoning ability. However, such long-form CoT teachers are usually unaware of the student model's capacity, which limits the effective utilization of the provided reasoning traces. To overcome this limitation, we propose errOr-aware self-ReflectION (ORION), a framework that refines teacher CoTs through an Error-Aware Reflection process. ORION enables the student model to construct more tailored teacher CoTs by refining teacher CoTs and incorporating its own reasoning errors. Experiments on multiple mathematical reasoning benchmarks demonstrate that ORION consistently improves performance by more than 2% over all baselines. Further analysis reveals that the CoTs constructed by ORION exhibit higher coherence and logical consistency, thereby serving as more effective supervision signals for SFT. All codes are available at https://github.com/NEUIR/ORION.git.

Zhiyuan Liu, Huazheng Wang, Fan Shen et al.

We study incentivized exploration for the multi-armed bandit (MAB) problem where the players receive compensation for exploring arms other than the greedy choice and may provide biased feedback on reward. We seek to understand the impact of this drifted reward feedback by analyzing the performance of three instantiations of the incentivized MAB algorithm: UCB, $\varepsilon$-Greedy, and Thompson Sampling. Our results show that they all achieve $\mathcal{O}(\log T)$ regret and compensation under the drifted reward, and are therefore effective in incentivizing exploration. Numerical examples are provided to complement the theoretical analysis.

Zhiyuan Liu, Guohui Ding, Lijun Chen et al.

We propose an alternating optimization algorithm to the nonconvex Koopman operator learning problem for nonlinear dynamic systems. We show that the proposed algorithm will converge to a critical point with rate $O(1/T)$ and $O(\frac{1}{\log T})$ for the constant and diminishing learning rates, respectively, under some mild conditions. To cope with the high dimensional nonlinear dynamical systems, we present the first-ever distributed Koopman operator learning algorithm. We show that the distributed Koopman operator learning has the same convergence properties as the centralized Koopman operator learning, in the absence of optimal tracker, so long as the basis functions satisfy a set of state-based decomposition conditions. Numerical experiments are provided to complement our theoretical results.

Zhiyuan Liu, Soumya Kundu, Lijun Chen et al.

In this paper we propose a new Koopman operator approach to the decomposition of nonlinear dynamical systems using Koopman Gramians. We introduce the notion of an input-Koopman operator, and show how input-Koopman operators can be used to cast a nonlinear system into the classical state-space form, and identify conditions under which input and state observable functions are well separated. We then extend an existing method of dynamic mode decomposition for learning Koopman operators from data known as deep dynamic mode decomposition to systems with controls or disturbances. We illustrate the accuracy of the method in learning an input-state separable Koopman operator for an example system, even when the underlying system exhibits mixed state-input terms. We next introduce a nonlinear decomposition algorithm, based on Koopman Gramians, that maximizes internal subsystem observability and disturbance rejection from unwanted noise from other subsystems. We derive a relaxation based on Koopman Gramians and multi-way partitioning for the resulting NP-hard decomposition problem. We lastly illustrate the proposed algorithm with the swing dynamics for an IEEE 39-bus system.