Jiajin Li

Yiping Lu, Jiajin Li, Lexing Ying et al. · stanford

The optimal design of experiments typically involves solving an NP-hard combinatorial optimization problem. In this paper, we aim to develop a globally convergent and practically efficient optimization algorithm. Specifically, we consider a setting where the pre-treatment outcome data is available and the synthetic control estimator is invoked. The average treatment effect is estimated via the difference between the weighted average outcomes of the treated and control units, where the weights are learned from the observed data. {Under this setting, we surprisingly observed that the optimal experimental design problem could be reduced to a so-called \textit{phase synchronization} problem.} We solve this problem via a normalized variant of the generalized power method with spectral initialization. On the theoretical side, we establish the first global optimality guarantee for experiment design when pre-treatment data is sampled from certain data-generating processes. Empirically, we conduct extensive experiments to demonstrate the effectiveness of our method on both the US Bureau of Labor Statistics and the Abadie-Diemond-Hainmueller California Smoking Data. In terms of the root mean square error, our algorithm surpasses the random design by a large margin.

Taoli Zheng, Linglingzhi Zhu, Anthony Man-Cho So et al.

Nonconvex-nonconcave minimax optimization has received intense attention over the last decade due to its broad applications in machine learning. Most existing algorithms rely on one-sided information, such as the convexity (resp. concavity) of the primal (resp. dual) functions, or other specific structures, such as the Polyak-Łojasiewicz (PŁ) and Kurdyka-Łojasiewicz (KŁ) conditions. However, verifying these regularity conditions is challenging in practice. To meet this challenge, we propose a novel universally applicable single-loop algorithm, the doubly smoothed gradient descent ascent method (DS-GDA), which naturally balances the primal and dual updates. That is, DS-GDA with the same hyperparameters is able to uniformly solve nonconvex-concave, convex-nonconcave, and nonconvex-nonconcave problems with one-sided KŁ properties, achieving convergence with $\mathcal{O}(ε^{-4})$ complexity. Sharper (even optimal) iteration complexity can be obtained when the KŁ exponent is known. Specifically, under the one-sided KŁ condition with exponent $θ\in(0,1)$, DS-GDA converges with an iteration complexity of $\mathcal{O}(ε^{-2\max\{2θ,1\}})$. They all match the corresponding best results in the literature. Moreover, we show that DS-GDA is practically applicable to general nonconvex-nonconcave problems even without any regularity conditions, such as the PŁ condition, KŁ condition, or weak Minty variational inequalities condition. For various challenging nonconvex-nonconcave examples in the literature, including ``Forsaken'', ``Bilinearly-coupled minimax'', ``Sixth-order polynomial'', and ``PolarGame'', the proposed DS-GDA can all get rid of limit cycles. To the best of our knowledge, this is the first first-order algorithm to achieve convergence on all of these formidable problems.

Jiajin Li, Linglingzhi Zhu, Anthony Man-Cho So

Nonconvex-nonconcave minimax optimization has gained widespread interest over the last decade. However, most existing works focus on variants of gradient descent-ascent (GDA) algorithms, which are only applicable to smooth nonconvex-concave settings. To address this limitation, we propose a novel algorithm named smoothed proximal linear descent-ascent (smoothed PLDA), which can effectively handle a broad range of structured nonsmooth nonconvex-nonconcave minimax problems. Specifically, we consider the setting where the primal function has a nonsmooth composite structure and the dual function possesses the Kurdyka-Lojasiewicz (KL) property with exponent $θ\in [0,1)$. We introduce a novel convergence analysis framework for smoothed PLDA, the key components of which are our newly developed nonsmooth primal error bound and dual error bound. Using this framework, we show that smoothed PLDA can find both $ε$-game-stationary points and $ε$-optimization-stationary points of the problems of interest in $\mathcal{O}(ε^{-2\max\{2θ,1\}})$ iterations. Furthermore, when $θ\in [0,\frac{1}{2}]$, smoothed PLDA achieves the optimal iteration complexity of $\mathcal{O}(ε^{-2})$. To further demonstrate the effectiveness and wide applicability of our analysis framework, we show that certain max-structured problem possesses the KL property with exponent $θ=0$ under mild assumptions. As a by-product, we establish algorithm-independent quantitative relationships among various stationarity concepts, which may be of independent interest.

Jiajin Li, Sirui Lin, Jose Blanchet et al.

Distributionally robust optimization has been shown to offer a principled way to regularize learning models. In this paper, we find that Tikhonov regularization is distributionally robust in an optimal transport sense (i.e., if an adversary chooses distributions in a suitable optimal transport neighborhood of the empirical measure), provided that suitable martingale constraints are also imposed. Further, we introduce a relaxation of the martingale constraints which not only provides a unified viewpoint to a class of existing robust methods but also leads to new regularization tools. To realize these novel tools, tractable computational algorithms are proposed. As a byproduct, the strong duality theorem proved in this paper can be potentially applied to other problems of independent interest.

Lemin Kong, Jiajin Li, Jianheng Tang et al.

Gromov-Wasserstein (GW) distance is a powerful tool for comparing and aligning probability distributions supported on different metric spaces. Recently, GW has become the main modeling technique for aligning heterogeneous data for a wide range of graph learning tasks. However, the GW distance is known to be highly sensitive to outliers, which can result in large inaccuracies if the outliers are given the same weight as other samples in the objective function. To mitigate this issue, we introduce a new and robust version of the GW distance called RGW. RGW features optimistically perturbed marginal constraints within a Kullback-Leibler divergence-based ambiguity set. To make the benefits of RGW more accessible in practice, we develop a computationally efficient and theoretically provable procedure using Bregman proximal alternating linearized minimization algorithm. Through extensive experimentation, we validate our theoretical results and demonstrate the effectiveness of RGW on real-world graph learning tasks, such as subgraph matching and partial shape correspondence.

Lingxiao Li, Noam Aigerman, Vladimir G. Kim et al.

Finding multiple solutions of non-convex optimization problems is a ubiquitous yet challenging task. Most past algorithms either apply single-solution optimization methods from multiple random initial guesses or search in the vicinity of found solutions using ad hoc heuristics. We present an end-to-end method to learn the proximal operator of a family of training problems so that multiple local minima can be quickly obtained from initial guesses by iterating the learned operator, emulating the proximal-point algorithm that has fast convergence. The learned proximal operator can be further generalized to recover multiple optima for unseen problems at test time, enabling applications such as object detection. The key ingredient in our formulation is a proximal regularization term, which elevates the convexity of our training loss: by applying recent theoretical results, we show that for weakly-convex objectives with Lipschitz gradients, training of the proximal operator converges globally with a practical degree of over-parameterization. We further present an exhaustive benchmark for multi-solution optimization to demonstrate the effectiveness of our method.

Jia Li, Jiajin Li, Yang Liu et al.

In this paper, we consider an inverse problem in graph learning domain -- ``given the graph representations smoothed by Graph Convolutional Network (GCN), how can we reconstruct the input graph signal?" We propose Graph Deconvolutional Network (GDN) and motivate the design of GDN via a combination of inverse filters in spectral domain and de-noising layers in wavelet domain, as the inverse operation results in a high frequency amplifier and may amplify the noise. We demonstrate the effectiveness of the proposed method on several tasks including graph feature imputation and graph structure generation.

Jiajin Li, Caihua Chen, Anthony Man-Cho So

Wasserstein \textbf{D}istributionally \textbf{R}obust \textbf{O}ptimization (DRO) is concerned with finding decisions that perform well on data that are drawn from the worst-case probability distribution within a Wasserstein ball centered at a certain nominal distribution. In recent years, it has been shown that various DRO formulations of learning models admit tractable convex reformulations. However, most existing works propose to solve these convex reformulations by general-purpose solvers, which are not well-suited for tackling large-scale problems. In this paper, we focus on a family of Wasserstein distributionally robust support vector machine (DRSVM) problems and propose two novel epigraphical projection-based incremental algorithms to solve them. The updates in each iteration of these algorithms can be computed in a highly efficient manner. Moreover, we show that the DRSVM problems considered in this paper satisfy a Hölderian growth condition with explicitly determined growth exponents. Consequently, we are able to establish the convergence rates of the proposed incremental algorithms. Our numerical results indicate that the proposed methods are orders of magnitude faster than the state-of-the-art, and the performance gap grows considerably as the problem size increases.

Baoxiang Wang, Shuai Li, Jiajin Li et al.

We analyze the Gambler's problem, a simple reinforcement learning problem where the gambler has the chance to double or lose the bets until the target is reached. This is an early example introduced in the reinforcement learning textbook by Sutton and Barto (2018), where they mention an interesting pattern of the optimal value function with high-frequency components and repeating non-smooth points. It is however without further investigation. We provide the exact formula for the optimal value function for both the discrete and the continuous cases. Though simple as it might seem, the value function is pathological: fractal, self-similar, derivative taking either zero or infinity, and not written as elementary functions. It is in fact one of the generalized Cantor functions, where it holds a complexity that has been uncharted thus far. Our analyses could provide insights into improving value function approximation, gradient-based algorithms, and Q-learning, in real applications and implementations.

Jiajin Li, Sen Huang, Anthony Man-Cho So

Wasserstein distance-based distributionally robust optimization (DRO) has received much attention lately due to its ability to provide a robustness interpretation of various learning models. Moreover, many of the DRO problems that arise in the learning context admits exact convex reformulations and hence can be tackled by off-the-shelf solvers. Nevertheless, the use of such solvers severely limits the applicability of DRO in large-scale learning problems, as they often rely on general purpose interior-point algorithms. On the other hand, there are very few works that attempt to develop fast iterative methods to solve these DRO problems, which typically possess complicated structures. In this paper, we take a first step towards resolving the above difficulty by developing a first-order algorithmic framework for tackling a class of Wasserstein distance-based distributionally robust logistic regression (DRLR) problem. Specifically, we propose a novel linearized proximal ADMM to solve the DRLR problem, whose objective is convex but consists of a smooth term plus two non-separable non-smooth terms. We prove that our method enjoys a sublinear convergence rate. Furthermore, we conduct three different experiments to show its superb performance on both synthetic and real-world datasets. In particular, our method can achieve the same accuracy up to 800+ times faster than the standard off-the-shelf solver.

Jiajin Li, Baoxiang Wang

Policy optimization on high-dimensional continuous control tasks exhibits its difficulty caused by the large variance of the policy gradient estimators. We present the action subspace dependent gradient (ASDG) estimator which incorporates the Rao-Blackwell theorem (RB) and Control Variates (CV) into a unified framework to reduce the variance. To invoke RB, our proposed algorithm (POSA) learns the underlying factorization structure among the action space based on the second-order advantage information. POSA captures the quadratic information explicitly and efficiently by utilizing the wide & deep architecture. Empirical studies show that our proposed approach demonstrates the performance improvements on high-dimensional synthetic settings and OpenAI Gym's MuJoCo continuous control tasks.