论文检索

Optimization/Non-Convex

Haixiang Zhang, Ying Chen, Javad Lavaei

In this work, we consider the problem of matrix sensing over graphs (MSoG). As a general case of matrix completion and matrix sensing problems, the MSoG problem has not been analyzed in the literature and the existing results cannot be directly applied to the MSoG problem. This work provides the first theoretical results on the optimization landscape of the MSoG problem. More specifically, we propose a new condition, named the $\Omega$-RIP condition, to characterize the optimization complexity of the problem. In addition, with an improved regularizer of the incoherence, we prove that the strict saddle property holds for the MSoG problem with high probability under the incoherence condition and the $\Omega$-RIP condition, which guarantees the polynomial-time global convergence of saddle-avoiding methods. Compared with state-of-the-art results, the bounds in this work are tight up to a constant. Besides the theoretical guarantees, we numerically illustrate the close relation between the $\Omega$-RIP condition and the optimization complexity.

Optimization/Non-Convex

Xixi Jia, Hailin Wang, Jiangjun Peng, Xiangchu Feng, Deyu Meng

Low-rank matrix factorization (LRMF) is a canonical problem in non-convex optimization, the objective function to be minimized is non-convex and even non-smooth, which makes the global convergence guarantee of gradient-based algorithm quite challenging. Recent work made a breakthrough on proving that standard gradient descent converges to the $\varepsilon$-global minima after $O( \frac{d \kappa^2}{\tau^2} {\rm ln} \frac{d \sigma_d}{\tau} + \frac{d \kappa^2}{\tau^2} {\rm ln} \frac{\sigma_d}{\varepsilon})$ iterations from small initialization with a very small learning rate (both are related to the small constant $\tau$). While the dependence of the convergence on the \textit{condition number} $\kappa$ and \textit{small learning rate} makes it not practical especially for ill-conditioned LRMF problem.In this paper, we show that precondition helps in accelerating the convergence and prove that the scaled gradient descent (ScaledGD) and its variant, alternating scaled gradient descent (AltScaledGD) converge to an $\varepsilon$-global minima after $O( {\rm ln} \frac{d}{\delta} + {\rm ln} \frac{d}{\varepsilon})$ iterations from general random initialization. Meanwhile, for small initialization as in gradient descent, both ScaledGD and AltScaledGD converge to $\varepsilon$-global minima after only $O({\rm ln} \frac{d}{\varepsilon})$ iterations. Furthermore, we prove that as a proximity to the alternating minimization, AltScaledGD converges faster than ScaledGD, its global convergence does not rely on small learning rate and small initialization, which certificates the advantages of AltScaledGD in LRMF.

Optimization/Global Optimization

KAZU GHALAMKARI, Mahito Sugiyama, Yoshinobu Kawahara

We present an alternative approach to decompose non-negative tensors, called many-body approximation. Traditional decomposition methods assume low-rankness in the representation, resulting in difficulties in global optimization and target rank selection. We avoid these problems by energy-based modeling of tensors, where a tensor and its mode correspond to a probability distribution and a random variable, respectively. Our model can be globally optimized in terms of the KL divergence minimization by taking the interaction between variables (that is, modes), into account that can be tuned more intuitively than ranks. Furthermore, we visualize interactions between modes as tensor networks and reveal a nontrivial relationship between many-body approximation and low-rank approximation. We demonstrate the effectiveness of our approach in tensor completion and approximation.

Theory/Optimization

Junyi Li, Feihu Huang, Heng Huang

Bilevel Optimization has witnessed notable progress recently with new emerging efficient algorithms. However, its application in the Federated Learning setting remains relatively underexplored, and the impact of Federated Learning's inherent challenges on the convergence of bilevel algorithms remain obscure.In this work, we investigate Federated Bilevel Optimization problems and propose a communication-efficient algorithm, named FedBiOAcc. The algorithm leverages an efficient estimation of the hyper-gradient in the distributed setting and utilizes the momentum-based variance-reduction acceleration. Remarkably, FedBiOAcc achieves a communication complexity $O(\epsilon^{-1})$, a sample complexity $O(\epsilon^{-1.5})$ and the linear speed up with respect to the number of clients. We also analyze a special case of the Federated Bilevel Optimization problems, where lower level problems are locally managed by clients. We prove that FedBiOAcc-Local, a modified version of FedBiOAcc, converges at the same rate for this type of problems. Finally, we validate the proposed algorithms through two real-world tasks: Federated Data-cleaning and Federated Hyper-representation Learning. Empirical results show superior performance of our algorithms.

Theory/Everything Else

Jingzhou Hu, Kejun Huang

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.

Reinforcement Learning/Batch Offline

Ting Li, Chengchun Shi, Jianing Wang, Fan Zhou, hongtu zhu

A/B testing is critical for modern technological companies to evaluate the effectiveness of newly developed products against standard baselines. This paper studies optimal designs that aim to maximize the amount of information obtained from online experiments to estimate treatment effects accurately. We propose three optimal allocation strategies in a dynamic setting where treatments are sequentially assigned over time. These strategies are designed to minimize the variance of the treatment effect estimator when data follow a non Markov decision process or a (time-varying) Markov decision process. We further develop estimation procedures based on existing off-policy evaluation (OPE) methods and conduct extensive experiments in various environments to demonstrate the effectiveness of the proposed methodologies. In theory, we prove the optimality of the proposed treatment allocation design and establish upper bounds for the mean squared errors of the resulting treatment effect estimators.

Applications/Computer Vision

Vishal Asnani, Abhinav Kumar, Suya You, Xiaoming Liu

Previous research in $2D$ object detection focuses on various tasks, including detecting objects in generic and camouflaged images. These works are regarded as passive works for object detection as they take the input image as is. However, convergence to global minima is not guaranteed to be optimal in neural networks; therefore, we argue that the trained weights in the object detector are not optimal. To rectify this problem, we propose a wrapper based on proactive schemes, PrObeD, which enhances the performance of these object detectors by learning a signal. PrObeD consists of an encoder-decoder architecture, where the encoder network generates an image-dependent signal termed templates to encrypt the input images, and the decoder recovers this template from the encrypted images. We propose that learning the optimum template results in an object detector with an improved detection performance. The template acts as a mask to the input images to highlight semantics useful for the object detector. Finetuning the object detector with these encrypted images enhances the detection performance for both generic and camouflaged. Our experiments on MS-COCO, CAMO, COD$10$K, and NC$4$K datasets show improvement over different detectors after applying PrObeD. Our models/codes are available at https://github.com/vishal3477/Proactive-Object-Detection.

Optimization/Control and Optimization

Xinyi Chen, Elad Hazan

Choosing the optimal hyperparameters, including learning rate and momentum, for specific optimization instances is a significant yet non-convex challenge. This makes conventional iterative techniques such as hypergradient descent \cite{baydin2017online} insufficient in obtaining global optimality guarantees.We consider the more general task of meta-optimization -- online learning of the best optimization algorithm given problem instances, and introduce a novel approach based on control theory. We show how meta-optimization can be formulated as an optimal control problem, departing from existing literature that use stability-based methods to study optimization. Our approach leverages convex relaxation techniques in the recently-proposed nonstochastic control framework to overcome the challenge of nonconvexity, and obtains regret guarantees vs. the best offline solution. This guarantees that in meta-optimization, we can learn a method that attains convergence comparable to that of the best optimization method in hindsight from a class of methods.

Optimization/Non-Convex

Ziye Ma, Javad Lavaei, Somayeh Sojoudi

Gradient descent (GD) is crucial for generalization in machine learning models, as it induces implicit regularization, promoting compact representations. In this work, we examine the role of GD in inducing implicit regularization for tensor optimization, particularly within the context of the lifted matrix sensing framework. This framework has been recently proposed to address the non-convex matrix sensing problem by transforming spurious solutions into strict saddles when optimizing over symmetric, rank-1 tensors. We show that, with sufficiently small initialization scale, GD applied to this lifted problem results in approximate rank-1 tensors and critical points with escape directions. Our findings underscore the significance of the tensor parametrization of matrix sensing, in combination with first-order methods, in achieving global optimality in such problems.

Optimization

Kaiqi Jiang, Dhruv Malik, Yuanzhi Li

Adaptive optimization methods are well known to achieve superior convergence relative to vanilla gradient methods. The traditional viewpoint in optimization, particularly in convex optimization, explains this improved performance by arguing that, unlike vanilla gradient schemes, adaptive algorithms mimic the behavior of a second-order method by adapting to the *global* geometry of the loss function. We argue that in the context of neural network optimization, this traditional viewpoint is insufficient. Instead, we advocate for a *local* trajectory analysis. For iterate trajectories produced by running a generic optimization algorithm OPT, we introduce $R^{\text{OPT}}\_{\text{med}}$, a statistic that is analogous to the condition number of the loss Hessian evaluated at the iterates. Through extensive experiments on language models where adaptive algorithms converge faster than vanilla gradient methods like SGD, we show that adaptive methods such as Adam bias the trajectories towards regions where $R^{\text{Adam}}_{\text{med}}$ is small, where one might expect faster optimization. By contrast, SGD (with momentum) biases the trajectories towards regions where $R^{\text{SGD}}\_{\text{med}}$ is comparatively large. We complement these empirical observations with a theoretical result that provably demonstrates this phenomenon in the simplified setting of a two-layer linear network. We view our findings as evidence for the need of a new explanation of the success of adaptive methods, one that is different than the conventional wisdom.

Deep Learning/Theory

Rajat Vadiraj Dwaraknath, Tolga Ergen, Mert Pilanci

Recently, theoretical analyses of deep neural networks have broadly focused on two directions: 1) Providing insight into neural network training by SGD in the limit of infinite hidden-layer width and infinitesimally small learning rate (also known as gradient flow) via the Neural Tangent Kernel (NTK), and 2) Globally optimizing the regularized training objective via cone-constrained convex reformulations of ReLU networks. The latter research direction also yielded an alternative formulation of the ReLU network, called a gated ReLU network, that is globally optimizable via efficient unconstrained convex programs. In this work, we interpret the convex program for this gated ReLU network as a Multiple Kernel Learning (MKL) model with a weighted data masking feature map and establish a connection to the NTK. Specifically, we show that for a particular choice of mask weights that do not depend on the learning targets, this kernel is equivalent to the NTK of the gated ReLU network on the training data. A consequence of this lack of dependence on the targets is that the NTK cannot perform better than the optimal MKL kernel on the training set. By using iterative reweighting, we improve the weights induced by the NTK to obtain the optimal MKL kernel which is equivalent to the solution of the exact convex reformulation of the gated ReLU network. We also provide several numerical simulations corroborating our theory. Additionally, we provide an analysis of the prediction error of the resulting optimal kernel via consistency results for the group lasso.

Probabilistic Methods/Gaussian Processes

Kaiwen Wu, Kyurae Kim, Roman Garnett, Jacob Gardner

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.

Optimization/Global Optimization

Gaspard Beugnot, Julien Mairal, Alessandro Rudi

We present a novel approach to non-convex optimization with certificates, which handles smooth functions on the hypercube or on the torus. Unlike traditional methods that rely on algebraic properties, our algorithm exploits the regularity of the target function intrinsic in the decay of its Fourier spectrum. By defining a tractable family of models, we allow {\em at the same time} to obtain precise certificates and to leverage the advanced and powerful computational techniques developed to optimize neural networks. In this way the scalability of our approach is naturally enhanced by parallel computing with GPUs. Our approach, when applied to the case of polynomials of moderate dimensions but with thousands of coefficients, outperforms the state-of-the-art optimization methods with certificates, as the ones based on Lasserre's hierarchy, addressing problems intractable for the competitors.

Social Aspects/Privacy-preserving Statistics and Machine Learning

Daogao Liu, Arun Ganesh, Sewoong Oh, Abhradeep Guha Thakurta

We reconsider the challenge of non-convex optimization under differential privacy constraint. Building upon the previous variance-reduced algorithm SpiderBoost, we propose a novel framework that employs two types of gradient oracles: one that estimates the gradient at a single point and a more cost-effective option that calculates the gradient difference between two points. Our framework can ensure continuous accuracy of gradient estimations and subsequently enhances the rates of identifying second-order stationary points.Additionally, we consider a more challenging task by attempting to locate the global minima of a non-convex objective via the exponential mechanism without almost any assumptions. Our preliminary results suggest that the regularized exponential mechanism can effectively emulate previous empirical and population risk bounds, negating the need for smoothness assumptions for algorithms with polynomial running time. Furthermore, with running time factors excluded, the exponential mechanism demonstrates promising population risk bound performance, and we provide a nearly matching lower bound.

Reinforcement Learning/Batch Offline

Xiangsen Wang, Haoran Xu, Yinan Zheng, Xianyuan Zhan

Offline reinforcement learning (RL) has received considerable attention in recent years due to its attractive capability of learning policies from offline datasets without environmental interactions. Despite some success in the single-agent setting, offline multi-agent RL (MARL) remains to be a challenge. The large joint state-action space and the coupled multi-agent behaviors pose extra complexities for offline policy optimization. Most existing offline MARL studies simply apply offline data-related regularizations on individual agents, without fully considering the multi-agent system at the global level. In this work, we present OMIGA, a new offline multi-agent RL algorithm with implicit global-to-local value regularization. OMIGA provides a principled framework to convert global-level value regularization into equivalent implicit local value regularizations and simultaneously enables in-sample learning, thus elegantly bridging multi-agent value decomposition and policy learning with offline regularizations. Based on comprehensive experiments on the offline multi-agent MuJoCo and StarCraft II micro-management tasks, we show that OMIGA achieves superior performance over the state-of-the-art offline MARL methods in almost all tasks.

Deep Learning/Robustness

Wanxing Chang, Ye Shi, Jingya Wang

Learning with noisy labels (LNL) poses a significant challenge in training a well-generalized model while avoiding overfitting to corrupted labels.Recent advances have achieved impressive performance by identifying clean labels and correcting corrupted labels for training.However, the current approaches rely heavily on the model’s predictions and evaluate each sample independently without considering either the global or local structure of the sample distribution.These limitations typically result in a suboptimal solution for the identification and correction processes, which eventually leads to models overfitting to incorrect labels.In this paper, we propose a novel optimal transport (OT) formulation, called Curriculum and Structure-aware Optimal Transport (CSOT). CSOT concurrently considers the inter- and intra-distribution structure of the samples to construct a robust denoising and relabeling allocator.During the training process, the allocator incrementally assigns reliable labels to a fraction of the samples with the highest confidence. These labels have both global discriminability and local coherence.Notably, CSOT is a new OT formulation with a nonconvex objective function and curriculum constraints, so it is not directly compatible with classical OT solvers. Here, we develop a lightspeed computational method that involves a scaling iteration within a generalized conditional gradient framework to solve CSOT efficiently.Extensive experiments demonstrate the superiority of our method over the current state-of-the-arts in LNL.

Optimization/Discrete and Combinatorial Optimization

Jacobus van der Linden, Mathijs de Weerdt, Emir Demirović

Global optimization of decision trees has shown to be promising in terms of accuracy, size, and consequently human comprehensibility. However, many of the methods used rely on general-purpose solvers for which scalability remains an issue.Dynamic programming methods have been shown to scale much better because they exploit the tree structure by solving subtrees as independent subproblems. However, this only works when an objective can be optimized separately for subtrees.We explore this relationship in detail and show the necessary and sufficient conditions for such separability and generalize previous dynamic programming approaches into a framework that can optimize any combination of separable objectives and constraints.Experiments on five application domains show the general applicability of this framework, while outperforming the scalability of general-purpose solvers by a large margin.

Deep Learning/Other Representation Learning

Yiming Lei, Zilong Li, Yangyang Li, Junping Zhang, Hongming Shan

Interpreting the decisions of deep learning models has been actively studied since the explosion of deep neural networks. One of the most convincing interpretation approaches is salience-based visual interpretation, such as Grad-CAM, where the generation of attention maps depends merely on categorical labels. Although existing interpretation methods can provide explainable decision clues, they often yield partial correspondence between image and saliency maps due to the limited discriminative information from one-hot labels. This paper develops a Language-Image COnsistency model for explainable image classification, termed LICO, by correlating learnable linguistic prompts with corresponding visual features in a coarse-to-fine manner. Specifically, we first establish a coarse global manifold structure alignment by minimizing the distance between the distributions of image and language features. We then achieve fine-grained saliency maps by applying optimal transport (OT) theory to assign local feature maps with class-specific prompts. Extensive experimental results on eight benchmark datasets demonstrate that the proposed LICO achieves a significant improvement in generating more explainable attention maps in conjunction with existing interpretation methods such as Grad-CAM. Remarkably, LICO improves the classification performance of existing models without introducing any computational overhead during inference.

Miscellaneous Aspects of Machine Learning/Transfer, Multitask and Meta-learning

Royson Lee, Minyoung Kim, Da Li, Xinchi Qiu, Timothy Hospedales, Ferenc Huszar, Nicholas Lane

Federated learning (FL) research has made progress in developing algorithms for distributed learning of global models, as well as algorithms for local personalization of those common models to the specifics of each client’s local data distribution. However, different FL problems may require different personalization strategies, and it may not even be possible to define an effective one-size-fits-all personalization strategy for all clients: Depending on how similar each client’s optimal predictor is to that of the global model, different personalization strategies may be preferred. In this paper, we consider the federated meta-learning problem of learning personalization strategies. Specifically, we consider meta-nets that induce the batch-norm and learning rate parameters for each client given local data statistics. By learning these meta-nets through FL, we allow the whole FL network to collaborate in learning a customized personalization strategy for each client. Empirical results show that this framework improves on a range of standard hand-crafted personalization baselines in both label and feature shift situations.

Miscellaneous Aspects of Machine Learning/Online Learning, Active Learning and Bandits

James Cheshire, Vincent Laurent, Stephan Clémençon

In this paper, we develop an active learning framework for the bipartite ranking problem.Motivated by numerous applications, ranging from supervised anomaly detection to credit-scoring through the design of medical diagnosis support systems, and usually formulated as the problem of optimizing (a scalar summary of) the ROC curve, bipartite ranking has been the subject of much attention in the passive context. Various dedicated algorithms have been recently proposed and studied by the machine-learning community. In contrast, active bipartite ranking rule is poorly documented in the literature. Due to its global nature, a strategy for labeling sequentially data points that are difficult to rank w.r.t. to the others is required. This learning task is much more complex than binary classification, for which many active algorithms have been designed. It is the goal of this article to provide a rigorous formulation of such a selective sampling approach. We propose a dedicated algorithm, referred to as active-rank, which aims to minimise the distance between the ROC curve of the ranking function built and the optimal one, w.r.t. the sup norm. We show that, for a fixed confidence level $\epsilon$ and probability $\delta$, active-rank is PAC$(\epsilon,\delta)$. In addition, we provide a problem dependent upper bound on the expected sampling time of active-rank and also demonstrate a problem dependent lower bound on the expected sampling time of any PAC$(\epsilon,\delta)$ algorithm. Beyond the theoretical analysis carried out, numerical results are presented, providing strong empirical evidence of the performance of the algorithm proposed, which compares favorably with more naive approaches.