On Convex Duality in Linear Inverse Problems

A Convex Duality Framework for GANs

Generative adversarial network (GAN) is a minimax game between a generator mimicking the true model and a discriminator distinguishing the samples produced by the generator from the real training samples. Given an unconstrained discriminator able to approximate any function, this game reduces to finding the generative model minimizing a divergence measure, e.g. the Jensen-Shannon (JS) divergence, to the data distribution. However, in practice the discriminator is constrained to be in a smaller class $\mathcal{F}$ such as neural nets. Then, a natural question is how the divergence minimization …

Parallel Deep Neural Networks Have Zero Duality Gap

Training deep neural networks is a challenging non-convex optimization problem. Recent work has proven that the strong duality holds (which means zero duality gap) for regularized finite-width two-layer ReLU networks and consequently provided an equivalent convex training problem. However, extending this result to deeper networks remains to be an open problem. In this paper, we prove that the duality gap for deeper linear networks with vector outputs is non-zero. In contrast, we show that the zero duality gap can be obtained by stacking standard deep networks in parallel, which we call a paral…

Revealing the Structure of Deep Neural Networks via Convex Duality

We study regularized deep neural networks (DNNs) and introduce a convex analytic framework to characterize the structure of the hidden layers. We show that a set of optimal hidden layer weights for a norm regularized DNN training problem can be explicitly found as the extreme points of a convex set. For the special case of deep linear networks, we prove that each optimal weight matrix aligns with the previous layers via duality. More importantly, we apply the same characterization to deep ReLU networks with whitened data and prove the same weight alignment holds. As a corollary, we also prove …

Matrix Completion and Related Problems via Strong Duality

This work studies the strong duality of non-convex matrix factorization problems: we show that under certain dual conditions, these problems and its dual have the same optimum. This has been well understood for convex optimization, but little was known for non-convex problems. We propose a novel analytical framework and show that under certain dual conditions, the optimal solution of the matrix factorization program is the same as its bi-dual and thus the global optimality of the non-convex program can be achieved by solving its bi-dual which is convex. These dual conditions are satisfied by a…

Convex Geometry and Duality of Over-parameterized Neural Networks

We develop a convex analytic approach to analyze finite width two-layer ReLU networks. We first prove that an optimal solution to the regularized training problem can be characterized as extreme points of a convex set, where simple solutions are encouraged via its convex geometrical properties. We then leverage this characterization to show that an optimal set of parameters yield linear spline interpolation for regression problems involving one dimensional or rank-one data. We also characterize the classification decision regions in terms of a kernel matrix and minimum $\ell_1$-norm solutions.…

Convex duality for stochastic shortest path problems in known and unknown environments

This paper studies Stochastic Shortest Path (SSP) problems in known and unknown environments from the perspective of convex optimisation. It first recalls results in the known parameter case, and develops understanding through different proofs. It then focuses on the unknown parameter case, where it studies extended value iteration (EVI) operators. This includes the existing operators used in Rosenberg et al. [26] and Tarbouriech et al. [31] based on the l-1 norm and supremum norm, as well as defining EVI operators corresponding to other norms and divergences, such as the KL-divergence. This p…

Deep Neural Networks with Multi-Branch Architectures Are Less Non-Convex

Several recently proposed architectures of neural networks such as ResNeXt, Inception, Xception, SqueezeNet and Wide ResNet are based on the designing idea of having multiple branches and have demonstrated improved performance in many applications. We show that one cause for such success is due to the fact that the multi-branch architecture is less non-convex in terms of duality gap. The duality gap measures the degree of intrinsic non-convexity of an optimization problem: smaller gap in relative value implies lower degree of intrinsic non-convexity. The challenge is to quantitatively measure …

Reinforcement Learning via Fenchel-Rockafellar Duality

We review basic concepts of convex duality, focusing on the very general and supremely useful Fenchel-Rockafellar duality. We summarize how this duality may be applied to a variety of reinforcement learning (RL) settings, including policy evaluation or optimization, online or offline learning, and discounted or undiscounted rewards. The derivations yield a number of intriguing results, including the ability to perform policy evaluation and on-policy policy gradient with behavior-agnostic offline data and methods to learn a policy via max-likelihood optimization. Although many of these results …

通过凸对偶探索正则化神经网络的损失景观

Exploring the loss landscape of regularized neural networks via convex duality

本文通过将正则化神经网络转化为等价凸问题并利用对偶理论，系统刻画了其损失景观的驻点结构、最优解连通性与非唯一性，并证明相关结果可推广至向量值与三层网络架构。

基于共轭对偶的广义凸性与光滑性：深度神经网络的优化理论

Generalized Convexity and Smoothness via Conjugate Duality: Optimization Theory for Deep Neural Networks

该研究通过Legendre函数与凸共轭构建DNN训练的统一优化框架，引入广义凸性与光滑性，推导广义梯度类优化器的收敛速率，定量分析训练影响因素，实验验证理论与训练动态的一致性。