Ning Zhao

Lifu Wang, Bo Shen, Ning Zhao et al.

The residual network is now one of the most effective structures in deep learning, which utilizes the skip connections to ``guarantee" the performance will not get worse. However, the non-convexity of the neural network makes it unclear whether the skip connections do provably improve the learning ability since the nonlinearity may create many local minima. In some previous works \cite{freeman2016topology}, it is shown that despite the non-convexity, the loss landscape of the two-layer ReLU network has good properties when the number $m$ of hidden nodes is very large. In this paper, we follow this line to study the topology (sub-level sets) of the loss landscape of deep ReLU neural networks with a skip connection and theoretically prove that the skip connection network inherits the good properties of the two-layer network and skip connections can help to control the connectedness of the sub-level sets, such that any local minima worse than the global minima of some two-layer ReLU network will be very ``shallow". The ``depth" of these local minima are at most $O(m^{(η-1)/n})$, where $n$ is the input dimension, $η<1$. This provides a theoretical explanation for the effectiveness of the skip connection in deep learning.

Lifu Wang, Bo Shen, Ning Zhao

In machine learning, asynchronous parallel stochastic gradient descent (APSGD) is broadly used to speed up the training process through multi-workers. Meanwhile, the time delay of stale gradients in asynchronous algorithms is generally proportional to the total number of workers, which brings additional deviation from the accurate gradient due to using delayed gradients. This may have a negative influence on the convergence of the algorithm. One may ask: How many workers can we use at most to achieve a good convergence and the linear speedup? In this paper, we consider the second-order convergence of asynchronous algorithms in non-convex optimization. We investigate the behaviors of APSGD with consistent read near strictly saddle points and provide a theoretical guarantee that if the total number of workers is bounded by $\widetilde{O}(K^{1/3}M^{-1/3})$ ($K$ is the total steps and $M$ is the mini-batch size), APSGD will converge to good stationary points ($||\nabla f(x)||\leq ε, \nabla^2 f(x)\succeq -\sqrtε\bm{I}, ε^2\leq O(\sqrt{\frac{1}{MK}}) $) and the linear speedup is achieved. Our works give the first theoretical guarantee on the second-order convergence for asynchronous algorithms. The technique we provide can be generalized to analyze other types of asynchronous algorithms to understand the behaviors of asynchronous algorithms in distributed asynchronous parallel training.