The sparse dictionary coding framework represents signals as a linear combination of a few predefined dictionary atoms. It has been employed for images, time series, graph signals and recently for 2-way (or 2D) spatio-temporal data employing jointly temporal and spatial dictionaries. Large and over-complete dictionaries enable high-quality models, but also pose scalability challenges which are exacerbated in multi-dictionary settings. Hence, an important problem that we address in this paper is: How to scale multi-dictionary coding for large dictionaries and datasets? We propose a multi-dictionary atom selection technique for low-rank sparse coding named LRMDS. To enable scalability to large dictionaries and datasets, it progressively selects groups of row-column atom pairs based on their alignment with the data and performs convex relaxation coding via the corresponding sub-dictionaries. We demonstrate both theoretically and experimentally that when the data has a low-rank encoding with a sparse subset of the atoms, LRMDS is able to select them with strong guarantees under mild assumptions. Furthermore, we demonstrate the scalability and quality of LRMDS in both synthetic and real-world datasets and for a range of coding dictionaries. It achieves 3 times to 10 times speed-up compared to baselines, while obtaining up to two orders of magnitude improvement in representation quality on some of the real world datasets given a fixed target number of atoms.
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An Efficient Local Search Algorithm for Large GD Advertising Inventory Allocation with Multilinear Constraints
PDF ↗The Guaranteed Delivery (GD) advertising is a crucial component of the online advertising industry, and the allocation of inventory in GD advertising is an important procedure that influences directly the ability of the publisher to fulfill the requirements and increase its revenues. Nowadays, as the requirements of advertisers become more and more diverse and fine-grained, the focus ratio requirement, which states that the portion of allocated impressions of a designated contract on focus media among all possible media should be greater than another contract, often appears in business scenarios. However, taking these requirements into account brings hardness for the GD advertising inventory allocation as the focus ratio requirements involve non-convex multilinear constraints. Existing methods which rely on the convex properties are not suitable for processing this problem, while mathematical programming or constraint-based heuristic solvers are unable to produce high-quality solutions within the time limit. Therefore, we propose a local search framework to address this challenge. It incorporates four new operators designed for handling multilinear constraints and a two-mode algorithmic architecture. Experimental results demonstrate that our algorithm is able to compute high-quality allocations with better business metrics compared to the state-of-the-art mathematical programming or constraint based heuristic solvers. Moreover, our algorithm is able to handle the general multilinear constraints and we hope it could be used to solve other problems in GD advertising with similar requirements.
Minimax problems have achieved success in machine learning such as adversarial training, robust optimization, reinforcement learning. For theoretical analysis, current optimal excess risk bounds, which are composed by generalization error and optimization error, present 1/n-rates in strongly-convex-strongly-concave (SC-SC) settings. Existing studies mainly focus on minimax problems with specific algorithms for optimization error, with only a few studies on generalization performance, which limit better excess risk bounds. In this paper, we study the generalization bounds measured by the gradients of primal functions using uniform localized convergence. We obtain a sharper high probability generalization error bound for nonconvex-strongly-concave (NC-SC) stochastic minimax problems. Furthermore, we provide dimension-independent results under Polyak-Lojasiewicz condition for the outer layer. Based on our generalization error bound, we analyze some popular algorithms such as empirical saddle point (ESP), gradient descent ascent (GDA) and stochastic gradient descent ascent (SGDA). We derive better excess primal risk bounds with further reasonable assumptions, which, to the best of our knowledge, are n times faster than exist results in minimax problems.
Stochastic bilevel optimization (SBO) has been integrated into many machine learning paradigms recently including hyperparameter optimization, meta learning, reinforcement learning, etc. Along with the wide range of applications, there have been abundant studies on concerning the computing behaviors of SBO. However, the generalization guarantees of SBO methods are far less understood from the lens of statistical learning theory. In this paper, we provide a systematical generalization analysis of the first-order gradient-based bilevel optimization methods. Firstly, we establish the quantitative connections between the on-average argument stability and the generalization gap of SBO methods. Then, we derive the upper bounds of on-average argument stability for single timescale stochastic gradient descent (SGD) and two timescale SGD, where three settings (nonconvex-nonconvex (NC-NC), convex-convex (C-C) and strongly-convex-strongly-convex (SC-SC)) are considered respectively. Experimental analysis validates our theoretical findings. Compared with the previous algorithmic stability analysis, our results do not require the re-initialization of the inner-level parameters before each iteration and are suit for more general objective functions.
Nonconvex Multiview Subspace Clustering Framework with Efficient Method Designs and Theoretical Analysis
PDF ↗Multi-view subspace clustering (MvSC) is one of the most effective methods for understanding and processing high-dimensional data. However, existing MvSC methods still have two shortcomings: (1) they adopt the nuclear norm as the low-rank constraint, which makes it impossible to fully exploit the mutually complementary subspace information, and (2) they do not handle disjoint and confounding points carefully, which may degrade the purity and distinctiveness of cross-view fusion. To address these issues, in this paper we propose a novel MvSC model with nonconvex ℓq regularization. Specially, our proposed model can not only effectively capture the intrinsic global low-rank structure, but also accurately cluster disjoint and confounding data samples into corresponding subspaces. Then, an efficient algorithm is developed with convergence guarantee. Furthermore, we prove that the sequence generated by our proposed algorithm converges to the desirable Karush-Kuhn-Tucker (KKT) critical point. Extensive experiments on various datasets verify the superiority of our proposed model. MATLAB code is available at https://github.com/wangzhi-swu/NLRSC-MvSC.
Machine unlearning provides users with the right to remove their privacy data from a well-trained model. Existing approaches of machine unlearning mainly focus on exploring data removing within supervised learning (SL) tasks. However, weakly supervised learning (WSL) is more applicable to real-world scenarios since collecting WSL data is less laborious than collecting fully supervised data. In this paper, we first propose a machine unlearning approach for WSL by updating the model parameters. Motivated by the uniform distributions of untrained model predictions, we derive a formulated target to force the model's predictions of removed data to be indistinguishable. This encourages the model to forget its ability to recognize features of data slated for unlearning. Moreover, we employ formulated targets to transform the classification unlearning into the convex regression, which can significantly reduce computational cost and avoid extra information storage during the training process. Additionally, we discuss how to design a target to ensure the models' predictions of removed data being indistinguishable in different learning scenarios, e.g., SL or WSL. As the flexibility in formulating targets, the proposed approach effectively deals with the WSL problem while still excels in SL models. Empirical studies show the superiority of the proposed approach.
In Multi-View Multi-Label Learning, each instance is described by several heterogeneous features and associated with multiple valid labels simultaneously. Existing methods mainly focus on leveraging feature-level view fusion to capture a common representation for multi-label classifier induction. In this paper, we take a new perspective and propose a new semantic-level fusion model named Common-Individual Semantic Fusion Multi-View Multi-Label Learning Method (CISF). Different from previous feature-level fusion model, our proposed method directly focuses on semantic-level view fusion and simultaneously take both the common semantic across different views and the individual semantic of each specific view into consideration. Specifically, we first assume each view involves some common semantic labels while owns a few exclusive semantic labels. Then, the common and exclusive semantic labels are separately forced to be consensus and diverse to excavate the consistences and complementarities among different views. Afterwards, we introduce the low-rank and sparse constraint to highlight the label co-occurrence relationship of common semantics and the view-specific expression of individual semantics. We provide theoretical guarantee for the strict convexity of our method by properly setting parameters. Extensive experiments on various data sets have verified the superiority of our method.
Provable Acceleration of Nesterov’s Accelerated Gradient Method over Heavy Ball Method in Training Over-Parameterized Neural Networks
PDF ↗Due to its simplicity and efficiency, the first-order gradient method has been extensively employed in training neural networks. Although the optimization problem of the neural network is non-convex, recent research has proved that the first-order method is capable of attaining a global minimum during training over-parameterized neural networks, where the number of parameters is significantly larger than that of training instances. Momentum methods, including the heavy ball (HB) method and Nesterov's accelerated gradient (NAG) method, are the workhorse of first-order gradient methods owning to their accelerated convergence. In practice, NAG often exhibits superior performance than HB. However, current theoretical works fail to distinguish their convergence difference in training neural networks. To fill this gap, we consider the training problem of the two-layer ReLU neural network under over-parameterization and random initialization. Leveraging high-resolution dynamical systems and neural tangent kernel (NTK) theory, our result not only establishes tighter upper bounds of the convergence rate for both HB and NAG, but also provides the first theoretical guarantee for the acceleration of NAG over HB in training neural networks. Finally, we validate our theoretical results on three benchmark datasets.
The heavy ball momentum method is a commonly used technique for accelerating training processes in the machine learning community. However, empirical evidence suggests that the convergence of stochastic gradient descent (SGD) with heavy ball may slow down when the momentum hyperparameter approaches 1. Despite this observation, there are no established theories or solutions to explain and address this issue. In this study, we provide the first theoretical result that elucidates why momentum slows down SGD as it tends to 1. To better understand this inefficiency, we focus on the quadratic convex objective in the analysis. Our findings show that momentum accelerates SGD when the scaling parameter is not very close to 1. Conversely, when the scaling parameter approaches 1, momentum impairs SGD and degrades its stability. Based on the theoretical findings, we propose a descending warmup technique for the heavy ball momentum, which exploits the advantages of the heavy ball method and overcomes the inefficiency problem when the momentum tends to 1. Numerical results demonstrate the effectiveness of the proposed SHB-DW algorithm.
This work studies Stackelberg network interdiction games --- an important class of games in which a defender first allocates (randomized) defense resources to a set of critical nodes on a graph while an adversary chooses its path to attack these nodes accordingly. We consider a boundedly rational adversary in which the adversary's response model is based on a dynamic form of classic logit-based (quantal response) discrete choice models. The resulting optimization is non-convex and additionally, involves complex terms that sum over exponentially many paths. We tackle these computational challenges by presenting new efficient algorithms with solution guarantees. First, we present a near optimal solution method based on path sampling, piece-wise linear approximation and mixed-integer linear programming (MILP) reformulation. Second, we explore a dynamic programming based method, addressing the exponentially-many-path challenge. We then show that the gradient of the non-convex objective can also be computed in polynomial time, which allows us to use a gradient-based method to solve the problem efficiently. Experiments based on instances of different sizes demonstrate the efficiency of our approach in achieving near-optimal solutions.
Knowing that a function is convex ensures that any local minimum is also a global minimum. Here, we implement an approach to certify the convexity of twice-differentiable functions by certifying that their second-order derivative is positive semidefinite. Both the computation of the second-order derivative and the certification of positive semidefiniteness are done symbolically. Previous implementations of this approach assume that the function to be minimized takes scalar or vector inputs, meaning that the second-order derivative is at most a matrix. However, the input of many machine learning problems is naturally given in the form of matrices or higher order tensors, in which case the second-order derivative becomes a tensor of at least fourth order. The familiar linear algebra notations and known rules for determining whether a matrix is positive semidefinite are not sufficient to deal with these higher order expressions. Here, we present a formal language for tensor expressions that allows us to generalize semidefiniteness to higher-order tensors and thereby certify the convexity of a broader set of functions.
Optimization of DR-submodular functions has experienced a notable surge in significance in recent times, marking a pivotal development within the domain of non-convex optimization. Motivated by real-world scenarios, some recent works have delved into the maximization of non-monotone DR-submodular functions over general (not necessarily down-closed) convex set constraints. Up to this point, these works have all used the minimum L-infinity norm of any feasible solution as a parameter. Unfortunately, a recent hardness result due to Mualem and Feldman shows that this approach cannot yield a smooth interpolation between down-closed and non-down-closed constraints. In this work, we suggest novel offline and online algorithms that provably provide such an interpolation based on a natural decomposition of the convex body constraint into two distinct convex bodies: a down-closed convex body and a general convex body. We also empirically demonstrate the superiority of our proposed algorithms across three offline and two online applications.
Vertex Subset Problems (VSPs) are a class of combinatorial optimization problems on graphs where the goal is to find a subset of vertices satisfying a predefined condition. Two prominent approaches for solving VSPs are dynamic programming over tree-like structures, such as tree-decompositions or clique-decompositions, and linear programming. In this work, we establish a sharp connection between both approaches by showing that if a vertex-subset problem Pi admits a solution-preserving dynamic programming algorithm that produces tables of size at most alpha(k,n) when processing a tree decomposition of width at most k of an n-vertex graph G, then the polytope defined as the convex-hull of solutions of Pi in G has extension complexity at most O(alpha(k,n)*n). Additionally, this upper bound is optimal under the exponential time hypothesis (ETH). At the one hand, our results imply that ETH-optimal solution-preserving dynamic programming algorithms for combinatorial problems yield optimal-size parameterized extended formulations for the solution polytopes associated with instances of these problems. At the other hand, unconditional lower bounds obtained in the realm of the theory of extended formulations yield unconditional lower bounds on the table complexity of solution-preserving dynamic programming algorithms.
Advancing Generalized Transfer Attack with Initialization Derived Bilevel Optimization and Dynamic Sequence Truncation
PDF ↗Transfer attacks generate significant interest for real-world black-box applications by crafting transferable adversarial examples through surrogate models. Whereas, existing works essentially directly optimize the single-level objective w.r.t. the surrogate model, which always leads to poor interpretability of attack mechanism and limited generalization performance over unknown victim models. In this work, we propose the BilEvel Transfer AttacK (BETAK) framework by establishing an initialization derived bilevel optimization paradigm, which explicitly reformulates the nested constraint relationship between the Upper-Level (UL) pseudo-victim attacker and the Lower-Level (LL) surrogate attacker. Algorithmically, we introduce the Hyper Gradient Response (HGR) estimation as an effective feedback for the transferability over pseudo-victim attackers, and propose the Dynamic Sequence Truncation (DST) technique to dynamically adjust the back-propagation path for HGR and reduce computational overhead simultaneously. Meanwhile, we conduct detailed algorithmic analysis and provide convergence guarantee to support non-convexity of the LL surrogate attacker. Extensive evaluations demonstrate substantial improvement of BETAK (e.g., 53.41% increase of attack success rates against IncRes-v2_ens victim) against different victims and defense methods in targeted and untargeted attack scenarios.
We consider decentralized learning over a network of workers with heterogeneous datasets, in the presence of Byzantine workers. Byzantine workers may transmit arbitrary or malicious values to neighboring workers, leading to degradation in overall performance. The heterogeneous nature of the training data across various workers complicates the identification and mitigation of Byzantine workers. To address this complex problem, we introduce a resilient decentralized learning approach that combines the gradient descent algorithm with a novel robust aggregator. Specifically, we propose a remove-then-clip aggregator, whereby each benign worker meticulously filters the neighbors' values and subsequently projects the remaining values to a sphere centered at its local value, with an appropriately selected radius. We prove that our proposed method converges to a neighborhood of a stationary point for non-convex objectives under standard assumptions. Furthermore, empirical evaluations are provided to demonstrate the superior performance of our method in comparison to existing algorithms, under various Byzantine attack models.
Reinforcement Learning (RL) has achieved remarkable success in safety-critical areas, but it can be weakened by adversarial attacks. Recent studies have introduced ``smoothed policies" to enhance its robustness. Yet, it is still challenging to establish a provable guarantee to certify the bound of its total reward. Prior methods relied primarily on computing bounds using Lipschitz continuity or calculating the probability of cumulative reward being above specific thresholds. However, these techniques are only suited for continuous perturbations on the RL agent's observations and are restricted to perturbations bounded by the l2-norm. To address these limitations, this paper proposes a general black-box certification method, called ReCePS, which is capable of directly certifying the cumulative reward of the smoothed policy under various lp-norm bounded perturbations. Furthermore, we extend our methodology to certify perturbations on action spaces. Our approach leverages f-divergence to measure the distinction between the original distribution and the perturbed distribution, subsequently determining the certification bound by solving a convex optimisation problem. We provide a comprehensive theoretical analysis and run experiments in multiple environments. Our results show that our method not only improves the tightness of certified lower bound of the mean cumulative reward but also demonstrates better efficiency than state-of-the-art methods.
Randomized smoothing is currently the state-of-the-art method that provides certified robustness for deep neural networks. However, due to its excessively conservative nature, this method of incomplete verification often cannot achieve an adequate certified radius on real-world datasets. One way to obtain a larger certified radius is to use an input-specific algorithm instead of using a fixed Gaussian filter for all data points. Several methods based on this idea have been proposed, but they either suffer from high computational costs or gain marginal improvement in certified radius. In this work, we show that by exploiting the quasiconvex problem structure, we can find the optimal certified radii for most data points with slight computational overhead. This observation leads to an efficient and effective input-specific randomized smoothing algorithm. We conduct extensive experiments and empirical analysis on CIFAR-10 and ImageNet. The results show that the proposed method significantly enhances the certified radii with low computational overhead.
Constrained Meta-Reinforcement Learning for Adaptable Safety Guarantee with Differentiable Convex Programming
PDF ↗Despite remarkable achievements in artificial intelligence, the deployability of learning-enabled systems in high-stakes real-world environments still faces persistent challenges. For example, in safety-critical domains like autonomous driving, robotic manipulation, and healthcare, it is crucial not only to achieve high performance but also to comply with given constraints. Furthermore, adaptability becomes paramount in non-stationary domains, where environmental parameters are subject to change. While safety and adaptability are recognized as key qualities for the new generation of AI, current approaches have not demonstrated effective adaptable performance in constrained settings. Hence, this paper breaks new ground by studying the unique challenges of ensuring safety in nonstationary environments by solving constrained problems through the lens of the meta-learning approach (learning to learn). While unconstrained meta-learning already encounters complexities in end to end differentiation of the loss due to the bi-level nature, its constrained counterpart introduces an additional layer of difficulty, since the constraints imposed on task-level updates complicate the differentiation process. To address the issue, we first employ successive convex-constrained policy updates across multiple tasks with differentiable convex programming, which allows meta-learning in constrained scenarios by enabling end-to-end differentiation. This approach empowers the agent to rapidly adapt to new tasks under nonstationarity while ensuring compliance with safety constraints. We also provide a theoretical analysis demonstrating guaranteed monotonic improvement of our approach, justifying our algorithmic designs. Extensive simulations across diverse environments provide empirical validation with significant improvement over established benchmarks.
Jointly Improving the Sample and Communication Complexities in Decentralized Stochastic Minimax Optimization
PDF ↗We propose a novel single-loop decentralized algorithm, DGDA-VR, for solving the stochastic nonconvex strongly-concave minimax problems over a connected network of agents, which are equipped with stochastic first-order oracles to estimate their local gradients. DGDA-VR, incorporating variance reduction, achieves O(ε^−3) oracle complexity and O(ε^−2) communication complexity without resorting to multi-communication rounds – both are optimal, i.e., matching the lower bounds for this class of problems. Since DGDA-VR does not require multiple communication rounds, it is applicable to a broader range of decentralized computational environments. To the best of our knowledge, this is the first distributed method using a single communication round in each iteration to jointly optimize the oracle and communication complexities for the problem considered here.
Standard approaches for global optimization of non-convex functions, such as branch-and-bound, maintain partition trees to systematically prune the domain. The tree size grows exponentially in the number of dimensions. We propose new sampling-based methods for non-convex optimization that adapts Monte Carlo Tree Search (MCTS) to improve efficiency. Instead of the standard use of visitation count in Upper Confidence Bounds, we utilize numerical overapproximations of the objective as an uncertainty metric, and also take into account of sampled estimates of first-order and second-order information. The Monte Carlo tree in our approach avoids the usual fixed combinatorial patterns in growing the tree, and aggressively zooms into the promising regions, while still balancing exploration and exploitation. We evaluate the proposed algorithms on high-dimensional non-convex optimization benchmarks against competitive baselines and analyze the effects of the hyper parameters.