Recent studies show a close connection between neural networks (NN) and kernel methods. However, most of these analyses (e.g., NTK) focus on the influence of (infinite) width instead of the depth of NN models. There remains a gap between theory and practical network designs that benefit from the depth. This paper first proposes a novel kernel family named Neural Optimization Kernel (NOK). Our kernel is defined as the inner product between two $T$-step updated functionals in RKHS w.r.t. a regularized optimization problem. Theoretically, we proved the monotonic descent property of our update rule for both convex and non-convex problems, and a $O(1/T)$ convergence rate of our updates for convex problems. Moreover, we propose a data-dependent structured approximation of our NOK, which builds the connection between training deep NNs and kernel methods associated with NOK. The resultant computational graph is a ResNet-type finite width NN. Our structured approximation preserved the monotonic descent property and $O(1/T)$ convergence rate. Namely, a $T$-layer NN performs $T$-step monotonic descent updates. Notably, we show our $T$-layered structured NN with ReLU maintains a $O(1/T)$ convergence rate w.r.t. a convex regularized problem, which explains the success of ReLU on training deep NN from a NN architecture optimization perspective. For the unsupervised learning and the shared parameter case, we show the equivalence of training structured NN with GD and performing functional gradient descent in RKHS associated with a fixed (data-dependent) NOK at an infinity-width regime. For finite NOKs, we prove generalization bounds. Remarkably, we show that overparameterized deep NN (NOK) can increase the expressive power to reduce empirical risk and reduce the generalization bound at the same time. Extensive experiments verify the robustness of our structured NOK blocks.
翻译:最近的研究表明神经网络( NN) 和内核方法之间有着密切的联系。 然而, 这些分析( 例如 NTK ) 大多侧重于( 无限) 宽度的影响, 而不是 NN 模型的深度。 理论与实际网络设计之间仍然存在差距, 从而受益于深度 。 此论文首先提出一个新的内核家族名为 Neal Optimization Kernel( NOK ) 。 我们的内核定义为 RHS w.r. t. 中两个步骤更新的内核功能之间的内产。 由此产生的内核计算图是一个常规化的优化问题。 从理论上, 我们的更新规则对于 convex 和非convex 问题的影响, 以及我们对于 convex 问题更新的 $( 1/ T) 的内核内核内核内核内核内核内核内核内核内核内核内化的内核内核内核内核内核内化。 我们的内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内 内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内