This paper explores decentralized learning in a graph-based setting, where data is distributed across nodes. We investigate a decentralized SGD algorithm that utilizes a random walk to update a global model based on local data. Our focus is on designing the transition probability matrix to speed up convergence. While importance sampling can enhance centralized learning, its decentralized counterpart, using the Metropolis-Hastings (MH) algorithm, can lead to the entrapment problem, where the random walk becomes stuck at certain nodes, slowing convergence. To address this, we propose the Metropolis-Hastings with L\'evy Jumps (MHLJ) algorithm, which incorporates random perturbations (jumps) to overcome entrapment. We theoretically establish the convergence rate and error gap of MHLJ and validate our findings through numerical experiments.

Effective hyper-parameter tuning is essential to guarantee the performance that neural networks have come to be known for. In this work, a principled approach to choosing the learning rate is proposed for shallow feedforward neural networks. We associate the learning rate with the gradient Lipschitz constant of the objective to be minimized while training. An upper bound on the mentioned constant is derived and a search algorithm, which always results in non-divergent traces, is proposed to exploit the derived bound. It is shown through simulations that the proposed search method significantly outperforms the existing tuning methods such as Tree Parzen Estimators (TPE). The proposed method is applied to two different existing applications, namely, channel estimation in a wireless communication system and prediction of the exchange currency rates, and it is shown to pick better learning rates than the existing methods using the same or lesser compute power.