Dachao Lin

Dachao Lin, Yuze Han, Haishan Ye et al.

We study finite-sum distributed optimization problems involving a master node and $n-1$ local nodes under the popular $δ$-similarity and $μ$-strong convexity conditions. We propose two new algorithms, SVRS and AccSVRS, motivated by previous works. The non-accelerated SVRS method combines the techniques of gradient sliding and variance reduction and achieves a better communication complexity of $\tilde{\mathcal{O}}(n {+} \sqrt{n}δ/μ)$ compared to existing non-accelerated algorithms. Applying the framework proposed in Katyusha X, we also develop a directly accelerated version named AccSVRS with the $\tilde{\mathcal{O}}(n {+} n^{3/4}\sqrt{δ/μ})$ communication complexity. In contrast to existing results, our complexity bounds are entirely smoothness-free and exhibit superiority in ill-conditioned cases. Furthermore, we establish a nearly matched lower bound to verify the tightness of our AccSVRS method.

Dachao Lin, Zhihua Zhang

In this short note, we give the convergence analysis of the policy in the recent famous policy mirror descent (PMD). We mainly consider the unregularized setting following [11] with generalized Bregman divergence. The difference is that we directly give the convergence rates of policy under generalized Bregman divergence. Our results are inspired by the convergence of value function in previous works and are an extension study of policy mirror descent. Though some results have already appeared in previous work, we further discover a large body of Bregman divergences could give finite-step convergence to an optimal policy, such as the classical Euclidean distance.

Kun Chen, Dachao Lin, Zhihua Zhang

In this paper, we follow Eftekhari's work to give a non-local convergence analysis of deep linear networks. Specifically, we consider optimizing deep linear networks which have a layer with one neuron under quadratic loss. We describe the convergent point of trajectories with arbitrary starting point under gradient flow, including the paths which converge to one of the saddle points or the original point. We also show specific convergence rates of trajectories that converge to the global minimizer by stages. To achieve these results, this paper mainly extends the machinery in Eftekhari's work to provably identify the rank-stable set and the global minimizer convergent set. We also give specific examples to show the necessity of our definitions. Crucially, as far as we know, our results appear to be the first to give a non-local global analysis of linear neural networks from arbitrary initialized points, rather than the lazy training regime which has dominated the literature of neural networks, and restricted benign initialization in Eftekhari's work. We also note that extending our results to general linear networks without one hidden neuron assumption remains a challenging open problem.

Dachao Lin, Zhihua Zhang

We consider the fundamental problem of learning linear predictors (i.e., separable datasets with zero margin) using neural networks with gradient flow or gradient descent. Under the assumption of spherically symmetric data distribution, we show directional convergence guarantees with exact convergence rate for two-layer non-linear networks with only two hidden nodes, and (deep) linear networks. Moreover, our discovery is built on dynamic from the initialization without both initial loss and perfect classification constraint in contrast to previous works. We also point out and study the challenges in further strengthening and generalizing our results.

Guangzeng Xie, Hao Jin, Dachao Lin et al.

We propose \textit{Meta-Regularization}, a novel approach for the adaptive choice of the learning rate in first-order gradient descent methods. Our approach modifies the objective function by adding a regularization term on the learning rate, and casts the joint updating process of parameters and learning rates into a maxmin problem. Given any regularization term, our approach facilitates the generation of practical algorithms. When \textit{Meta-Regularization} takes the $\varphi$-divergence as a regularizer, the resulting algorithms exhibit comparable theoretical convergence performance with other first-order gradient-based algorithms. Furthermore, we theoretically prove that some well-designed regularizers can improve the convergence performance under the strong-convexity condition of the objective function. Numerical experiments on benchmark problems demonstrate the effectiveness of algorithms derived from some common $\varphi$-divergence in full batch as well as online learning settings.

Dachao Lin, Ruoyu Sun, Zhihua Zhang

Sparse neural networks have received increasing interest due to their small size compared to dense networks. Nevertheless, most existing works on neural network theory have focused on dense neural networks, and the understanding of sparse networks is very limited. In this paper, we study the loss landscape of one-hidden-layer sparse networks. First, we consider sparse networks with a dense final layer. We show that linear networks can have no spurious valleys under special sparse structures, and non-linear networks could also admit no spurious valleys under a wide final layer. Second, we discover that spurious valleys and spurious minima can exist for wide sparse networks with a sparse final layer. This is different from wide dense networks which do not have spurious valleys under mild assumptions.

Dachao Lin, Peiqin Sun, Guangzeng Xie et al.

Quantized Neural Networks (QNNs) use low bit-width fixed-point numbers for representing weight parameters and activations, and are often used in real-world applications due to their saving of computation resources and reproducibility of results. Batch Normalization (BN) poses a challenge for QNNs for requiring floating points in reciprocal operations, and previous QNNs either require computing BN at high precision or revise BN to some variants in heuristic ways. In this work, we propose a novel method to quantize BN by converting an affine transformation of two floating points to a fixed-point operation with shared quantized scale, which is friendly for hardware acceleration and model deployment. We confirm that our method maintains same outputs through rigorous theoretical analysis and numerical analysis. Accuracy and efficiency of our quantization method are verified by experiments at layer level on CIFAR and ImageNet datasets. We also believe that our method is potentially useful in other problems involving quantization.

Mo Zhou, Tianyi Liu, Yan Li et al.

Numerous empirical evidence has corroborated that the noise plays a crucial rule in effective and efficient training of neural networks. The theory behind, however, is still largely unknown. This paper studies this fundamental problem through training a simple two-layer convolutional neural network model. Although training such a network requires solving a nonconvex optimization problem with a spurious local optimum and a global optimum, we prove that perturbed gradient descent and perturbed mini-batch stochastic gradient algorithms in conjunction with noise annealing is guaranteed to converge to a global optimum in polynomial time with arbitrary initialization. This implies that the noise enables the algorithm to efficiently escape from the spurious local optimum. Numerical experiments are provided to support our theory.

Hao Jin, Dachao Lin, Zhihua Zhang

Stochastic variance-reduced gradient (SVRG) is a classical optimization method. Although it is theoretically proved to have better convergence performance than stochastic gradient descent (SGD), the generalization performance of SVRG remains open. In this paper we investigate the effects of some training techniques, mini-batching and learning rate decay, on the generalization performance of SVRG, and verify the generalization performance of Batch-SVRG (B-SVRG). In terms of the relationship between optimization and generalization, we believe that the average norm of gradients on each training sample as well as the norm of average gradient indicate how flat the landscape is and how well the model generalizes. Based on empirical observations of such metrics, we perform a sign switch on B-SVRG and derive a practical algorithm, BatchPlus-SVRG (BP-SVRG), which is numerically shown to enjoy better generalization performance than B-SVRG, even SGD in some scenarios of deep neural networks.