Can Gradient Descent Provably Learn Linear Dynamic Systems?

We study the learning ability of linear recurrent neural networks with gradient descent. We prove the first theoretical guarantee on linear RNNs with Gradient Descent to learn any stable linear dynamic system. We show that despite the non-convexity of the optimization loss if the width of the RNN is large enough (and the required width in hidden layers does not rely on the length of the input sequence), a linear RNN can provably learn any stable linear dynamic system with the sample and time complexity polynomial in $\frac{1}{1-\rho_C}$ where $\rho_C$ is roughly the spectral radius of the stable system. Our results provide the first theoretical guarantee to learn a linear RNN and demonstrate how can the recurrent structure help to learn a dynamic system.

On the Provable Generalization of Recurrent Neural Networks

Recurrent Neural Network (RNN) is a fundamental structure in deep learning. Recently, some works study the training process of over-parameterized neural networks, and show that over-parameterized networks can learn functions in some notable concept classes with a provable generalization error bound. In this paper, we analyze the training and generalization for RNNs with random initialization, and provide the following improvements over recent works: 1) For a RNN with input sequence $x=(X_1,X_2,...,X_L)$, previous works study to learn functions that are summation of $f(\beta^T_lX_l)$ and require normalized conditions that $||X_l||\leq\epsilon$ with some very small $\epsilon$ depending on the complexity of $f$. In this paper, using detailed analysis about the neural tangent kernel matrix, we prove a generalization error bound to learn such functions without normalized conditions and show that some notable concept classes are learnable with the numbers of iterations and samples scaling almost-polynomially in the input length $L$. 2) Moreover, we prove a novel result to learn N-variables functions of input sequence with the form $f(\beta^T[X_{l_1},...,X_{l_N}])$, which do not belong to the ``additive'' concept class, i,e., the summation of function $f(X_l)$. And we show that when either $N$ or $l_0=\max(l_1,..,l_N)-\min(l_1,..,l_N)$ is small, $f(\beta^T[X_{l_1},...,X_{l_N}])$ will be learnable with the number iterations and samples scaling almost-polynomially in the input length $L$.

Is the Skip Connection Provable to Reform the Neural Network Loss Landscape?

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 $O(m^{(\eta-1)/n})$, where $n$ is the input dimension, $\eta<1$. This provides a theoretical explanation for the effectiveness of the skip connection in deep learning.

On the Convergence of Perturbed Distributed Asynchronous Stochastic Gradient Descent to Second Order Stationary Points in Non-convex Optimization

In this paper, the second order convergence of non-convex optimization in the asynchronous stochastic gradient descent (ASGD) algorithm is studied systematically. We investigate the behavior of ASGD near and away from saddle points. Different from the general stochastic gradient descent(SGD), we show that ASGD might return back even if it has escaped the saddle points, yet after staying near a strict saddle point for a long enough time ($O(T)$), ASGD will finally go away from strict saddle points. An inequality is given to describe the process of ASGD to escape saddle points. Using a novel Razumikhin-Lyapunov method, we show the exponential instability of the perturbed gradient dynamics near the strict saddle points and give a more detailed estimation about how the time delay parameter $T$ influences the speed to escape. In particular, we consider the optimization of smooth nonconvex functions, and propose a perturbed asynchronous stochastic gradient descent algorithm with guarantee of convergence to second order stationary points with high probability in $O(1/\epsilon^4)$ iterations. To the best of our knowledge, this is the first work on the second order convergence of asynchronous algorithm.