Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1812.00139

Number of Connected Components in a Graph: Estimation via Counting Patterns

Ashish Khetan·Harshay Shah·Sewoong Oh
View original source ↗Read source document ↗Export metadata ↓

Abstract

Due to the limited resources and the scale of the graphs in modern datasets, we often get to observe a sampled subgraph of a larger original graph of interest, whether it is the worldwide web that has been crawled or social connections that have been surveyed. Inferring a global property of the original graph from such a sampled subgraph is of a fundamental interest. In this work, we focus on estimating the number of connected components. It is a challenging problem and, for general graphs, little is known about the connection between the observed subgraph and the number of connected components of the original graph. In order to make this connection, we propose a highly redundant and large-dimensional representation of the subgraph, which at first glance seems counter-intuitive. A subgraph is represented by the counts of patterns, known as network motifs. This representation is crucial in introducing a novel estimator for the number of connected components for general graphs, under the knowledge of the spectral gap of the original graph. The connection is made precise via the Schatten $k$-norms of the graph Laplacian and the spectral representation of the number of connected components. We provide a guarantee on the resulting mean squared error that characterizes the bias variance tradeoff. Experiments on synthetic and real-world graphs suggest that we improve upon competing algorithms for graphs with spectral gaps bounded away from zero.

Explore related subjects

cs.LGstat.ML

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ashish Khetan, Harshay Shah, Sewoong Oh. 2018-12-01. Number of Connected Components in a Graph: Estimation via Counting Patterns. https://arxiv.org/abs/1812.00139

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↗