Prediction tasks related to congestion are targeted at improving the level of service of the transportation network. With increasing access to larger datasets of higher resolution, the relevance of deep learning in such prediction tasks, is increasing. Several comprehensive survey papers in recent years have summarised the deep learning applications in the transportation domain. However, the system dynamics of the transportation network vary greatly between the non-congested state and the congested state -- thereby necessitating the need for a clear understanding of the challenges specific to congestion prediction. In this survey, we present the current state of deep learning applications in the tasks related to detection, prediction and propagation of congestion. Recurrent and non-recurrent congestion are discussed separately. Our survey leads us to uncover inherent challenges and gaps in the current state of research. Finally, we present some suggestions for future research directions as answers to the identified challenges.

This paper studies fixed step-size stochastic approximation (SA) schemes, including stochastic gradient schemes, in a Riemannian framework. It is motivated by several applications, where geodesics can be computed explicitly, and their use accelerates crude Euclidean methods. A fixed step-size scheme defines a family of time-homogeneous Markov chains, parametrized by the step-size. Here, using this formulation, non-asymptotic performance bounds are derived, under Lyapunov conditions. Then, for any step-size, the corresponding Markov chain is proved to admit a unique stationary distribution, and to be geometrically ergodic. This result gives rise to a family of stationary distributions indexed by the step-size, which is further shown to converge to a Dirac measure, concentrated at the solution of the problem at hand, as the step-size goes to 0. Finally, the asymptotic rate of this convergence is established, through an asymptotic expansion of the bias, and a central limit theorem.

In this paper, we proposed a new technique, {\em variance controlled stochastic gradient} (VCSG), to improve the performance of the stochastic variance reduced gradient (SVRG) algorithm. To avoid over-reducing the variance of gradient by SVRG, a hyper-parameter $\lambda$ is introduced in VCSG that is able to control the reduced variance of SVRG. Theory shows that the optimization method can converge by using an unbiased gradient estimator, but in practice, biased gradient estimation can allow more efficient convergence to the vicinity since an unbiased approach is computationally more expensive. $\lambda$ also has the effect of balancing the trade-off between unbiased and biased estimations. Secondly, to minimize the number of full gradient calculations in SVRG, a variance-bounded batch is introduced to reduce the number of gradient calculations required in each iteration. For smooth non-convex functions, the proposed algorithm converges to an approximate first-order stationary point (i.e. $\mathbb{E}\|\nabla{f}(x)\|^{2}\leq\epsilon$) within $\mathcal{O}(min\{1/\epsilon^{3/2},n^{1/4}/\epsilon\})$ number of stochastic gradient evaluations, which improves the leading gradient complexity of stochastic gradient-based method SCS $(\mathcal{O}(min\{1/\epsilon^{5/3},n^{2/3}/\epsilon\})$. It is shown theoretically and experimentally that VCSG can be deployed to improve convergence.