Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.08911

Instance-Optimal Matrix Multiplicative Weight Update and Its Quantum Applications

Weiyuan Gong·Tongyang Li·Xinzhao Wang·Zhiyu Zhang
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Matrix Multiplicative Weight Update (MMWU) is a seminal online learning algorithm with numerous applications. Applied to the matrix version of the Learning from Expert Advice (LEA) problem on the $d$-dimensional spectraplex, it is well known that MMWU achieves the minimax-optimal regret bound of $O(\sqrt{T\log d})$, where $T$ is the time horizon. In this paper, we present an improved algorithm achieving the instance-optimal regret bound of $O(\sqrt{T\cdot S(X||d^{-1}I_d)})$, where $X$ is the comparator in the regret, $I_d$ is the identity matrix, and $S(\cdot||\cdot)$ denotes the quantum relative entropy. Furthermore, our algorithm has the same computational complexity as MMWU, indicating that the improvement in the regret bound is ``free''. Technically, we first develop a general potential-based framework for matrix LEA, with MMWU being its special case induced by the standard exponential potential. Then, the crux of our analysis is a new ``one-sided'' Jensen's trace inequality built on a Laplace transform technique, which allows the application of general potential functions beyond exponential to matrix LEA. Our algorithm is finally induced by an optimal potential function from the vector LEA problem, based on the imaginary error function. Complementing the above, we provide a memory lower bound for matrix LEA, and explore the applications of our algorithm in quantum learning theory. We show that it outperforms the state of the art for learning quantum states corrupted by depolarization noise, random quantum states, and Gibbs states. In addition, applying our algorithm to linearized convex losses enables predicting nonlinear quantum properties, such as purity, quantum virtual cooling, and Rényi-$2$ correlation.

Explore related subjects

cs.LGcs.AIcs.DSquant-phstat.ML

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Weiyuan Gong, Tongyang Li, Xinzhao Wang, Zhiyu Zhang. 2025-09-10. Instance-Optimal Matrix Multiplicative Weight Update and Its Quantum Applications. https://arxiv.org/abs/2509.08911

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rollout Total Correlation for Deep Reinforcement Learning

Learning task-relevant representations is crucial for reinforcement learning. Recent approaches aim to learn such representations by improving the temporal consistency in the observed transitions. However, they only consider individual transitions and can fail to achieve long-term consistency. Instead, we argue that capturing aspects of the state that correlate with other states and actions of the trajectory---even more distant in the future---could further help in extracting task-relevant information. Hence, in this paper we investigate how to learn representations by maximizing the rollout total correlation, the correlation among all learned representations and actions within the trajectories produced by the agent. For improving rollout total correlation, we propose to combine two complementary lower bounds based on a generative and a discriminative model, combined with a simple and effective technique of chunk-wise mini-batching. Furthermore, we propose an intrinsic reward based on the learned representation for better exploration. Experimental evaluations on a set of challenging image-based simulated control tasks show that our method achieves better sample efficiency, and robustness to both white noise and natural video backgrounds compared to leading baselines.

cs.LG↗
arXiv2026

Reward Shaping to Mitigate Reward Hacking in RLHF

Reinforcement learning from human feedback (RLHF) is widely used to align large language models (LLMs) with human preferences. However, RLHF remains vulnerable to \emph{reward hacking}, whereby a policy exploits imperfections in the reward function instead of learning the intended behavior, thereby undermining alignment. Although reward shaping can stabilize RLHF training and partially mitigate reward hacking, shaping methods and their underlying design principles have not been systematically investigated. To address this gap, we conduct a comprehensive study of prevalent reward-shaping techniques. Our analysis identifies two key design principles: (1) the reinforcement-learning reward should be bounded, and (2) it should grow rapidly at first and then gradually saturate. Motivated by these principles, we propose Preference as Reward (PAR), a novel method that uses the latent preferences encoded in the reward model as the reinforcement-learning signal. We further show that PAR possesses two variance-reduction properties that stabilize RLHF training and substantially widen the practical window for early stopping. Our evaluation consists of two parts. First, we compare PAR with several other reward-shaping strategies using Proximal Policy Optimization (PPO) as the reinforcement-learning algorithm and Gemma2-2B as the base model. Second, we compare PAR with the vanilla baseline (i.e., unshaped reward) across four base models and four reinforcement-learning algorithms. In the first set of experiments, PAR consistently outperforms other reward-shaping methods and also reflects high data efficiency and robustness. The second set of experiments shows that PAR is particularly effective for actor-critic RL algorithms when value estimates become unstable and demonstrates its effectiveness across different base models. The code is available at https://github.com/PorUna-byte/PAR.

cs.LG↗
arXiv2026

Trajectory Entropy Reinforcement Learning for Robust Robot Motor Skill Learning

Simplicity is a critical inductive bias for designing data-driven controllers, especially when robustness is important. Despite the impressive results of deep reinforcement learning in complex control tasks, it is prone to capturing intricate and spurious correlations between observations and actions, leading to failure under slight perturbations to the environment. To tackle this problem, in this work we introduce a novel inductive bias towards simple policies in reinforcement learning. The simplicity inductive bias is introduced by minimizing the entropy of entire action trajectories, corresponding to the number of bits required to describe information in action trajectories after the agent observes state trajectories. Our reinforcement learning agent, Trajectory Entropy Reinforcement Learning, is optimized to minimize the trajectory entropy while maximizing rewards. We show that the trajectory entropy can be effectively estimated by learning a variational parameterized action prediction model, and use the prediction model to construct an information-regularized reward function. Furthermore, we construct a practical algorithm that enables the joint optimization of models, including the policy and the prediction model. Experimental evaluations on several high-dimensional locomotion tasks show that our learned policies produce more cyclical and consistent action trajectories, and achieve superior performance, and robustness to noise and dynamic changes than the state-of-the-art.

cs.LG↗