  Federated learning (FL), as a distributed machine learning approach, has
drawn a great amount of attention in recent years. FL shows an inherent
advantage in privacy preservation, since users' raw data are processed locally.
However, it relies on a centralized server to perform model aggregation.
Therefore, FL is vulnerable to server malfunctions and external attacks. In
this paper, we propose a novel framework by integrating blockchain into FL,
namely, blockchain assisted decentralized federated learning (BLADE-FL), to
enhance the security of FL. The proposed BLADE-FL has a good performance in
terms of privacy preservation, tamper resistance, and effective cooperation of
learning. However, it gives rise to a new problem of training deficiency,
caused by lazy clients who plagiarize others' trained models and add artificial
noises to conceal their cheating behaviors. To be specific, we first develop a
convergence bound of the loss function with the presence of lazy clients and
prove that it is convex with respect to the total number of generated blocks
$K$. Then, we solve the convex problem by optimizing $K$ to minimize the loss
function. Furthermore, we discover the relationship between the optimal $K$,
the number of lazy clients, and the power of artificial noises used by lazy
clients. We conduct extensive experiments to evaluate the performance of the
proposed framework using the MNIST and Fashion-MNIST datasets. Our analytical
results are shown to be consistent with the experimental results. In addition,
the derived optimal $K$ achieves the minimum value of loss function, and in
turn the optimal accuracy performance.
