NeurIPS 2023 Papers

We propose a novel formulation for dictionary learning that minimizes the determinant of the dictionary matrix, also known as its volume, subject to the constraint that each row of the sparse coefficient matrix has unit $\ell_1$ norm. The main motivation for the proposed formulation is that it provides global identifiability guarantee of the groundtruth dictionary and sparse coefficient matrices, up to the inherent and inconsequential permutation and scaling ambiguity, if a set of vectors obtained from the coefficient matrix lies inside the $\ell_\infty$ norm ball but contains the $\ell_2$ norm ball in their convex hull. Unlike existing work on identifiability of dictionary learning, our result is global, meaning that a globally optimal solution to our proposed formulation has to be a permuted and rescaled version of the groundtruth factors. Another major improvement in our result is that there is no additional assumption on the dictionary matrix other than it is nonsingular, unlike most other work that require the atoms of the dictionary to be mutually incoherent. We also provide a probabilistic analysis and show that if the sparse coefficient matrix is generated from the widely adopted Bernoulli-Gaussian model, then it is globally identifiable if the sample size is bigger than a constant times $k\log k$, where $k$ is the number atoms in the dictionary, with overwhelming probability. The bound is essentially the same as those local identifiability results, but we show that it is also global. Finally, we propose algorithms to solve the new proposed formulation, specifically one based on the linearized-ADMM with efficient per-iteration updates. The proposed algorithms exhibit surprisingly effective performance in correctly and efficiently recovering the dictionary, as demonstrated in the numerical experiments.

Plug-and-play (PnP) prior is a well-known class of methods for solving imaging inverse problems by computing fixed-points of operators combining physical measurement models and learned image denoisers. While PnP methods have been extensively used for image recovery with known measurement operators, there is little work on PnP for solving blind inverse problems. We address this gap by presenting a new block-coordinate PnP (BC-PnP) method that efficiently solves this joint estimation problem by introducing learned denoisers as priors on both the unknown image and the unknown measurement operator. We present a new convergence theory for BC-PnP compatible with blind inverse problems by considering nonconvex data-fidelity terms and expansive denoisers. Our theory analyzes the convergence of BC-PnP to a stationary point of an implicit function associated with an approximate minimum mean-squared error (MMSE) denoiser. We numerically validate our method on two blind inverse problems: automatic coil sensitivity estimation in magnetic resonance imaging (MRI) and blind image deblurring. Our results show that BC-PnP provides an efficient and principled framework for using denoisers as PnP priors for jointly estimating measurement operators and images.

Functional constrained optimization (FCO) has emerged as a powerful tool for solving various machine learning problems. However, with the rapid increase in applications of neural networks in recent years, it has become apparent that both the objective and constraints often involve nonconvex functions, which poses significant challenges in obtaining high-quality solutions. In this work, we focus on a class of nonconvex FCO problems with nonconvex constraints, where the two optimization variables are nonlinearly coupled in the inequality constraint. Leveraging the primal-dual optimization framework, we propose a smoothed first-order Lagrangian method (SLM) for solving this class of problems. We establish the theoretical convergence guarantees of SLM to the Karush-Kuhn-Tucker (KKT) solutions through quantifying dual error bounds. By establishing connections between this structured FCO and equilibrium-constrained nonconvex problems (also known as bilevel optimization), we apply the proposed SLM to tackle bilevel optimization oriented problems where the lower-level problem is nonconvex. Numerical results obtained from both toy examples and hyper-data cleaning problems demonstrate the superiority of SLM compared to benchmark methods.

Real-world optimization problems often require black-box optimization with observation failure, where we can obtain the objective function value if we succeed, otherwise, we can only obtain a fact of failure. Moreover, this failure region can be complex by several latent constraints, whose number is also unknown. For this problem, we propose a failure-aware Gaussian process upper confidence bound (F-GP-UCB), which only requires a mild assumption for the observation failure that an optimal solution lies on an interior of a feasible region. Furthermore, we show that the number of successful observations grows linearly, by which we provide the first regret upper bounds and the convergence of F-GP-UCB. We demonstrate the effectiveness of F-GP-UCB in several benchmark functions, including the simulation function motivated by material synthesis experiments.

Conditional stochastic optimization has found applications in a wide range of machine learning tasks, such as invariant learning, AUPRC maximization, and meta-learning. As the demand for training models with large-scale distributed data grows in these applications, there is an increasing need for communication-efficient distributed optimization algorithms, such as federated learning algorithms. This paper considers the nonconvex conditional stochastic optimization in federated learning and proposes the first federated conditional stochastic optimization algorithm (FCSG) with a conditional stochastic gradient estimator and a momentum-based algorithm (\emph{i.e.}, FCSG-M). To match the lower bound complexity in the single-machine setting, we design an accelerated algorithm (Acc-FCSG-M) via the variance reduction to achieve the best sample and communication complexity. Compared with the existing optimization analysis for Meta-Learning in FL, federated conditional stochastic optimization considers the sample of tasks. Extensive experimental results on various tasks validate the efficiency of these algorithms.

We initiate the study of stochastic optimization with oblivious noise, broadly generalizing the standard heavy-tailed noise setup. In our setting, in addition to random observation noise, the stochastic gradient may be subject to independent \emph{oblivious noise}, which may not have bounded moments and is not necessarily centered. Specifically, we assume access to a noisy oracle for the stochastic gradient of $f$ at $x$, which returns a vector $\nabla f(\gamma, x) + \xi$, where $\gamma$ is the bounded variance observation noise and $\xi$ is the oblivious noise that is independent of $\gamma$ and $x$. The only assumption we make on the oblivious noise $\xi$ is that $\Pr[\xi = 0] \ge \alpha$, for some $\alpha \in (0, 1)$. In this setting, it is not information-theoretically possible to recover a single solution close to the target when the fraction of inliers $\alpha$ is less than $1/2$. Our main result is an efficient {\em list-decodable} learner that recovers a small list of candidates at least one of which is close to the true solution. On the other hand, if $\alpha = 1-\epsilon$, where $0< \epsilon < 1/2$ is sufficiently small constant, the algorithm recovers a single solution. Along the way, we develop a rejection-sampling-based algorithm to perform noisy location estimation, which may be of independent interest.

In this paper we study the second-order optimality of decentralized stochastic algorithm that escapes saddle point efficiently for nonconvex optimization problems. We propose a new pure gradient-based decentralized stochastic algorithm PEDESTAL with a novel convergence analysis framework to address the technical challenges unique to the decentralized stochastic setting. Our method is the first decentralized stochastic algorithm to achieve second-order optimality with non-asymptotic analysis. We provide theoretical guarantees with the gradient complexity of $\tilde{O} (\epsilon^{-3})$ to find $O(\epsilon, \sqrt{\epsilon})$-second-order stationary point, which matches state-of-the-art results of centralized counterparts or decentralized methods to find first-order stationary point. We also conduct two decentralized tasks in our experiments, a matrix sensing task with synthetic data and a matrix factorization task with a real-world dataset to validate the performance of our method.

The knowledge gradient (KG) algorithm is a popular policy for the best arm identification (BAI) problem. It is built on the simple idea of always choosing the measurement that yields the greatest expected one-step improvement in the estimate of the best mean of the arms. In this research, we show that this policy has limitations, causing the algorithm not asymptotically optimal. We next provide a remedy for it, by following the manner of one-step look ahead of KG, but instead choosing the measurement that yields the greatest one-step improvement in the probability of selecting the best arm. The new policy is called improved knowledge gradient (iKG). iKG can be shown to be asymptotically optimal. In addition, we show that compared to KG, it is easier to extend iKG to variant problems of BAI, with the $\epsilon$-good arm identification and feasible arm identification as two examples. The superior performances of iKG on these problems are further demonstrated using numerical examples.

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.

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 $\mathcal{O}(\epsilon^{-4})$ complexity. Sharper (even optimal) iteration complexity can be obtained when the KŁ exponent is known. Specifically, under the one-sided KŁ condition with exponent $\theta\in(0,1)$, DS-GDA converges with an iteration complexity of $\mathcal{O}(\epsilon^{-2\max\\{2\theta,1\\}})$. 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.

Parallelization is a popular strategy for improving the performance of methods. Optimization methods are no exception: design of efficient parallel optimization methods and tight analysis of their theoretical properties are important research endeavors. While the minimax complexities are well known for sequential optimization methods, the theory of parallel optimization methods is less explored. In this paper, we propose a new protocol that generalizes the classical oracle framework approach. Using this protocol, we establish minimax complexities for parallel optimization methods that have access to an unbiased stochastic gradient oracle with bounded variance. We consider a fixed computation model characterized by each worker requiring a fixed but worker-dependent time to calculate stochastic gradient. We prove lower bounds and develop optimal algorithms that attain them. Our results have surprising consequences for the literature of asynchronous optimization methods.

We consider a non-convex constrained optimization problem, where the objective function is weakly convex and the constraint function is either convex or weakly convex. To solve this problem, we consider the classical switching subgradient method, which is an intuitive and easily implementable first-order method whose oracle complexity was only known for convex problems. This paper provides the first analysis on the oracle complexity of the switching subgradient method for finding a nearly stationary point of non-convex problems. Our results are derived separately for convex and weakly convex constraints. Compared to existing approaches, especially the double-loop methods, the switching gradient method can be applied to non-smooth problems and achieves the same complexity using only a single loop, which saves the effort on tuning the number of inner iterations.

Communication compression is a common technique in distributed optimization that can alleviate communication overhead by transmitting compressed gradients and model parameters. However, compression can introduce information distortion, which slows down convergence and incurs more communication rounds to achieve desired solutions. Given the trade-off between lower per-round communication costs and additional rounds of communication, it is unclear whether communication compression reduces the total communication cost. This paper explores the conditions under which unbiased compression, a widely used form of compression, can reduce the total communication cost, as well as the extent to which it can do so. To this end, we present the first theoretical formulation for characterizing the total communication cost in distributed optimization with unbiased compressors. We demonstrate that unbiased compression alone does not necessarily save the total communication cost, but this outcome can be achieved if the compressors used by all workers are further assumed independent. We establish lower bounds on the communication rounds required by algorithms using independent unbiased compressors to minimize smooth convex functions and show that these lower bounds are tight by refining the analysis for ADIANA. Our results reveal that using independent unbiased compression can reduce the total communication cost by a factor of up to $\Theta(\sqrt{\min\\{n,\kappa\\}})$ when all local smoothness constants are constrained by a common upper bound, where $n$ is the number of workers and $\kappa$ is the condition number of the functions being minimized. These theoretical findings are supported by experimental results.

The mean-field Langevin dynamics (MFLD) is a nonlinear generalization of the Langevin dynamics that incorporates a distribution-dependent drift, and it naturally arises from the optimization of two-layer neural networks via (noisy) gradient descent. Recent works have shown that MFLD globally minimizes an entropy-regularized convex functional in the space of measures. However, all prior analyses assumed the infinite-particle or continuous-time limit, and cannot handle stochastic gradient updates. We provide a general framework to prove a uniform-in-time propagation of chaos for MFLD that takes into account the errors due to finite-particle approximation, time-discretization, and stochastic gradient. To demonstrate the wide applicability of our framework, we establish quantitative convergence rate guarantees to the regularized global optimal solution for $(i)$ a wide range of learning problems such as mean-field neural network and MMD minimization, and $(ii)$ different gradient estimators including SGD and SVRG. Despite the generality of our results, we achieve an improved convergence rate in both the SGD and SVRG settings when specialized to the standard Langevin dynamics.

A recent development in Bayesian optimization is the use of local optimization strategies, which can deliver strong empirical performance on high-dimensional problems compared to traditional global strategies. The "folk wisdom" in the literature is that the focus on local optimization sidesteps the curse of dimensionality; however, little is known concretely about the expected behavior or convergence of Bayesian local optimization routines. We first study the behavior of the local approach, and find that the statistics of individual local solutions of Gaussian process sample paths are surprisingly good compared to what we would expect to recover from global methods. We then present the first rigorous analysis of such a Bayesian local optimization algorithm recently proposed by Müller et al. (2021), and derive convergence rates in both the noisy and noiseless settings.

Classical analysis of convex and non-convex optimization methods often requires the Lipschitz continuity of the gradient, which limits the analysis to functions bounded by quadratics. Recent work relaxed this requirement to a non-uniform smoothness condition with the Hessian norm bounded by an affine function of the gradient norm, and proved convergence in the non-convex setting via gradient clipping, assuming bounded noise. In this paper, we further generalize this non-uniform smoothness condition and develop a simple, yet powerful analysis technique that bounds the gradients along the trajectory, thereby leading to stronger results for both convex and non-convex optimization problems. In particular, we obtain the classical convergence rates for (stochastic) gradient descent and Nesterov's accelerated gradient method in the convex and/or non-convex setting under this general smoothness condition. The new analysis approach does not require gradient clipping and allows heavy-tailed noise with bounded variance in the stochastic setting.

The differential equation (DE) approach for convex optimization, which relates optimization methods to specific continuous DEs with rate-revealing Lyapunov functionals, has gained increasing interest since the seminal paper by Su--Boyd--Candès (2014). However, the approach still lacks a crucial component to make it truly useful: there is no general, consistent way to transition back to discrete optimization methods. Consequently, even if we derive insights from continuous DEs, we still need to perform individualized and tedious calculations for the analysis of each method. This paper aims to bridge this gap by introducing a new concept called ``weak discrete gradient'' (wDG), which consolidates the conditions required for discrete versions of gradients in the DE approach arguments. We then define abstract optimization methods using wDG and provide abstract convergence theories that parallel those in continuous DEs. We demonstrate that many typical optimization methods and their convergence rates can be derived as special cases of this abstract theory. The proposed unified discretization framework for the differential equation approach to convex optimization provides an easy environment for developing new optimization methods and achieving competitive convergence rates with state-of-the-art methods, such as Nesterov's accelerated gradient.

In appropriate frameworks, automatic differentiation is transparent to the user, at the cost of being a significant computational burden when the number of operations is large. For iterative algorithms, implicit differentiation alleviates this issue but requires custom implementation of Jacobian evaluation. In this paper, we study one-step differentiation, also known as Jacobian-free backpropagation, a method as easy as automatic differentiation and as performant as implicit differentiation for fast algorithms (e.g. superlinear optimization methods). We provide a complete theoretical approximation analysis with specific examples (Newton's method, gradient descent) along with its consequences in bilevel optimization. Several numerical examples illustrate the well-foundness of the one-step estimator.

Biases with respect to socially-salient attributes of individuals have been well documented in evaluation processes used in settings such as admissions and hiring. We view such an evaluation process as a transformation of a distribution of the true utility of an individual for a task to an observed distribution and model it as a solution to a loss minimization problem subject to an information constraint. Our model has two parameters that have been identified as factors leading to biases: the resource-information trade-off parameter in the information constraint and the risk-averseness parameter in the loss function. We characterize the distributions that arise from our model and study the effect of the parameters on the observed distribution. The outputs of our model enrich the class of distributions that can be used to capture variation across groups in the observed evaluations. We empirically validate our model by fitting real-world datasets and use it to study the effect of interventions in a downstream selection task. These results contribute to an understanding of the emergence of bias in evaluation processes and provide tools to guide the deployment of interventions to mitigate biases.