Stochastic Gradient Hamiltonian Monte Carlo (SGHMC) is a method that extends Hamiltonian Monte Carlo (HMC) to handle large-scale and streaming data by using stochastic gradient estimates. HMC is efficient for sampling from complex distributions but requires computing the gradient of the potential energy function, which is infeasible for large datasets. SGHMC addresses this by incorporating a friction term into the momentum update, counteracting the effects of noisy gradient estimates and maintaining the desired target distribution as the invariant distribution. This approach allows for efficient exploration of the state space while using stochastic gradients to reduce computational costs. The paper explores the properties of stochastic gradient HMC, showing that naive implementations can lead to poor performance due to the noise introduced by stochastic gradients. By introducing second-order Langevin dynamics with a friction term, SGHMC maintains the desired target distribution as the invariant distribution, enabling efficient sampling. Theoretical analysis and experiments validate the effectiveness of SGHMC, demonstrating its performance in simulated data and real-world applications such as Bayesian neural networks and online Bayesian matrix factorization. SGHMC is compared to other methods like SGLD and standard HMC, showing that it achieves better performance in terms of convergence and accuracy. The method is particularly effective in scenarios with large datasets or streaming data, where traditional HMC methods are not feasible. The computational complexity of SGHMC is similar to that of SGLD, making it a viable option for scalable Bayesian inference. The paper also discusses the connection between SGHMC and stochastic gradient descent with momentum, highlighting the benefits of incorporating friction to counteract noise in the gradient estimates. Overall, SGHMC provides a robust and efficient approach for Bayesian inference in large-scale and online settings.Stochastic Gradient Hamiltonian Monte Carlo (SGHMC) is a method that extends Hamiltonian Monte Carlo (HMC) to handle large-scale and streaming data by using stochastic gradient estimates. HMC is efficient for sampling from complex distributions but requires computing the gradient of the potential energy function, which is infeasible for large datasets. SGHMC addresses this by incorporating a friction term into the momentum update, counteracting the effects of noisy gradient estimates and maintaining the desired target distribution as the invariant distribution. This approach allows for efficient exploration of the state space while using stochastic gradients to reduce computational costs. The paper explores the properties of stochastic gradient HMC, showing that naive implementations can lead to poor performance due to the noise introduced by stochastic gradients. By introducing second-order Langevin dynamics with a friction term, SGHMC maintains the desired target distribution as the invariant distribution, enabling efficient sampling. Theoretical analysis and experiments validate the effectiveness of SGHMC, demonstrating its performance in simulated data and real-world applications such as Bayesian neural networks and online Bayesian matrix factorization. SGHMC is compared to other methods like SGLD and standard HMC, showing that it achieves better performance in terms of convergence and accuracy. The method is particularly effective in scenarios with large datasets or streaming data, where traditional HMC methods are not feasible. The computational complexity of SGHMC is similar to that of SGLD, making it a viable option for scalable Bayesian inference. The paper also discusses the connection between SGHMC and stochastic gradient descent with momentum, highlighting the benefits of incorporating friction to counteract noise in the gradient estimates. Overall, SGHMC provides a robust and efficient approach for Bayesian inference in large-scale and online settings.