Non-convex Loss Function Research Articles

Large-scale robust regression with truncated loss via majorization-minimization algorithm

The utilization of regression methods employing truncated loss functions is widely praised for its robustness in handling outliers and representing the solution in the sparse form of the samples. However, due to the non-convexity of the truncated loss, the commonly used algorithms such as difference of convex algorithm (DCA) fail to maintain sparsity when dealing with non-convex loss functions, and adapting DCA for efficient optimization also incurs additional development costs. To address these challenges, we propose a novel approach called truncated loss regression via majorization-minimization algorithm (TLRM). TLRM employs a surrogate function to approximate the original truncated loss regression and offers several desirable properties: (i) Eliminating outliers before the training process and encapsulating general convex loss regression within its structure as iterative subproblems, (ii) Solving the convex loss problem iteratively thereby facilitating the use of a well-established toolbox for convex optimization. (iii) Converging to a truncated loss regression and providing a solution with sample sparsity. Extensive experiments demonstrate that TLRM achieves superior sparsity without sacrificing robustness, and it can be several tens of thousands of times faster than traditional DCA on large-scale problems. Moreover, TLRM is also applicable to datasets with millions of samples, making it a practical choice for real-world scenarios. The codebase for methods with truncated loss functions is accessible at https://i-do-lab.github.io/optimal-group.org/Resources/Code/TLRM.html.

Order Optimal Bounds for One-Shot Federated Learning Over Non-Convex Loss Functions

Order Optimal Bounds for One-Shot Federated Learning Over Non-Convex Loss Functions

Large-Scale Non-convex Stochastic Constrained Distributionally Robust Optimization

Distributionally robust optimization (DRO) is a powerful framework for training robust models against data distribution shifts. This paper focuses on constrained DRO, which has an explicit characterization of the robustness level. Existing studies on constrained DRO mostly focus on convex loss function, and exclude the practical and challenging case with non-convex loss function, e.g., neural network. This paper develops a stochastic algorithm and its performance analysis for non-convex constrained DRO. The computational complexity of our stochastic algorithm at each iteration is independent of the overall dataset size, and thus is suitable for large-scale applications. We focus on the general Cressie-Read family divergence defined uncertainty set which includes chi^2-divergences as a special case. We prove that our algorithm finds an epsilon-stationary point with an improved computational complexity than existing methods. Our method also applies to the smoothed conditional value at risk (CVaR) DRO.

No Internal Regret with Non-convex Loss Functions

Internal regret is a measure of performance of an online learning algorithm, which measures the change in performance by substituting every occurrence of a given action i by an alternative action j. Algorithms for minimizing internal regret are known for the finite experts setting, including a general reduction to the problem of minimizing external regret for this case. The reduction however crucially depends on the finiteness of the action space. In this work we approach the problem of minimizing internal regret for a continuous action space. For the full information setting, we show how to obtain O(sqrt(T)) internal regret for the class of Lipschitz functions, as well as non-Lipschitz dispersed functions, i.e. the non-Lipschitzness may not concentrate in a small region of the action space. We also consider extensions to partial feedback settings, and again obtain sublinear internal regret. Finally we discuss applications of internal regret minimization over continuous spaces to correlated equilibria in pricing problems and auction design, as well as to data-driven hyperparameter tuning.

Near-Optimal Resilient Aggregation Rules for Distributed Learning Using 1-Center and 1-Mean Clustering with Outliers

Byzantine machine learning has garnered considerable attention in light of the unpredictable faults that can occur in large-scale distributed learning systems. The key to secure resilience against Byzantine machines in distributed learning is resilient aggregation mechanisms. Although abundant resilient aggregation rules have been proposed, they are designed in ad-hoc manners, imposing extra barriers on comparing, analyzing, and improving the rules across performance criteria. This paper studies near-optimal aggregation rules using clustering in the presence of outliers. Our outlier-robust clustering approach utilizes geometric properties of the update vectors provided by workers. Our analysis show that constant approximations to the 1-center and 1-mean clustering problems with outliers provide near-optimal resilient aggregators for metric-based criteria, which have been proven to be crucial in the homogeneous and heterogeneous cases respectively. In addition, we discuss two contradicting types of attacks under which no single aggregation rule is guaranteed to improve upon the naive average. Based on the discussion, we propose a two-phase resilient aggregation framework. We run experiments for image classification using a non-convex loss function. The proposed algorithms outperform previously known aggregation rules by a large margin with both homogeneous and heterogeneous data distributions among non-faulty workers. Code and appendix are available at https://github.com/jerry907/AAAI24-RASHB.

Quantization Bits Allocation for Wireless Federated Learning

Federated learning (FL) enables multiple clients to collaborate on a common learning task via only exchanging model updates. With the progressive improvements in deep learning models, communication is becoming a primary bottleneck of FL. Quantization of model updates before transmitting is an effective technique to reduce communication overhead. Most prior literature assumes lossless transmission, but in practice, quantized model updates are distorted by wireless channels due to the variation of client locations. Therefore, this paper focuses on analysis and design of personalized model update quantization with explicitly incorporating channel diversity in wireless FL. We present a novel convergence analysis of quantized FL, which encompasses full and partial client participation, single and multiple local training iterations, and convex and non-convex loss functions. This analysis explicitly embodies the impact of personalized quantization error, channel diversity and model aggregation in FL, and also elucidates their tradeoff on tightening a convergence rate upper bound. An optimization framework, which seeks an optimal allocation scheme given a total budget of quantization bits, is proposed by minimizing an upper bound with respect to channel quality. A nearly optimal solution is derived for this non-convex integer programming problem via analytically solving Karush–Kuhn–Tucker (KKT) optimality conditions and linear search. From a perspective of outlier detection, this channel-aware allocation scheme is also extended to robust model aggregation against client dropouts. Comprehensive numerical evaluation demonstrates the performance enhancement of the proposed scheme over the vanilla allocation scheme with equal quantization bits, particularly in terms of training stability, test accuracy, and robustness.

Deep Neural Networks Training by Stochastic Quasi-Newton Trust-Region Methods

While first-order methods are popular for solving optimization problems arising in deep learning, they come with some acute deficiencies. To overcome these shortcomings, there has been recent interest in introducing second-order information through quasi-Newton methods that are able to construct Hessian approximations using only gradient information. In this work, we study the performance of stochastic quasi-Newton algorithms for training deep neural networks. We consider two well-known quasi-Newton updates, the limited-memory Broyden–Fletcher–Goldfarb–Shanno (BFGS) and the symmetric rank one (SR1). This study fills a gap concerning the real performance of both updates in the minibatch setting and analyzes whether more efficient training can be obtained when using the more robust BFGS update or the cheaper SR1 formula, which—allowing for indefinite Hessian approximations—can potentially help to better navigate the pathological saddle points present in the non-convex loss functions found in deep learning. We present and discuss the results of an extensive experimental study that includes many aspects affecting performance, like batch normalization, the network architecture, the limited memory parameter or the batch size. Our results show that stochastic quasi-Newton algorithms are efficient and, in some instances, able to outperform the well-known first-order Adam optimizer, run with the optimal combination of its numerous hyperparameters, and the stochastic second-order trust-region STORM algorithm.

Gradient complexity and non-stationary views of differentially private empirical risk minimization

In this paper, we study the Differentially Private Empirical Risk Minimization (DP-ERM) problem, considering both convex and non-convex loss functions. For cases where DP-ERM involves smooth (strongly) convex loss functions with or without non-smooth regularization, we propose several novel methods. These methods achieve (near) optimal expected excess risks (i.e., utility bounds) while reducing the gradient complexity compared to existing approaches. When dealing with DP-ERM and smooth convex loss functions in high dimensions, we introduce an algorithm that achieves a superior upper bound with lower gradient complexity than previous solutions.In the second part of the paper, for DP-ERM with non-convex loss functions, we explore both low and high dimensional spaces. In the low dimensional case with a non-smooth regularizer, we extend an existing approach by measuring the utility using the ℓ2 norm of the projected gradient. Furthermore, we introduce a novel error bound measurement, transitioning from empirical risk to population risk by employing the expected ℓ2 norm of the gradient. For the high dimensional case, we demonstrate that by measuring utility with the Frank-Wolfe gap, we can bound the utility using the Gaussian Width of the constraint set, instead of the dimensionality (p) of the underlying space. We also show that the advantages of this approach can be achieved by measuring the ℓ2 norm of the projected gradient. Finally, we reveal that the utility of certain special non-convex loss functions can be reduced to a level (depending only on log⁡p) similar to that of convex loss functions.

Performance Optimization for Noise Interference Privacy Protection in Federated Learning

The data security issue in federated learning is critical. While federated learning allows clients to collaboratively participate in the global model training without sharing private data, external eavesdroppers may intercept the model uploaded by the client to the server, revealing some sensitive information. Noise interference, i.e., adding noise to the client model before transmission, is an effective and efficient privacy-preserving method, but it degrades the learning performance of the system at the same time. In this paper, to address the challenge of system performance degradation caused by noise interference, we propose the FedNoise algorithm, which adopts two separate learning rates at the client and server respectively. By carefully tuning these learning rates, the global model can converge to the optimum. We provide theoretical proofs of the convergence of FedNoise for both strongly convex and non-convex loss functions and conduct simulations on real tasks. Numerical experimental results demonstrate that, under the same privacy protection level, FedNoise significantly outperforms the state-of-art scheme on datasets MNIST, Fashion-MNIST, and CIFAR10.

Fused robust geometric nonparallel hyperplane support vector machine for pattern classification

Recently, introducing nonconvex loss functions in support vector machine (SVM) to improve the robustness against varies noises has been drawing much attention. In this paper, we first construct a new robust capped asymmetric elastic net (CaEN) loss function. Second, we describe a novel robust Huberized kernel-based (HK) loss function and theoretically demonstrate several important properties, such as smoothness, boundness and the trade-off between the standard least squares and the truncated least squares. Finally, we apply the CaEN loss and the HK loss into elastic net nonparallel hyperplane SVM (ENNHSVM) to develop a fused robust geometric nonparallel SVM (FRGNHSVM). The proposed FRGNHSVM not only inherits the advantages of ENNHSVM but also improves the robustness of classification problems. An efficient Pegasos-based DC (difference of convex functions) algorithm is implemented to solve the FRGNHSVM optimization problem. In comparison with four famous SVMs, including Lagrangian SVM, twin SVM, pinball SVM and C-loss twin SVM, experimental results on simulations and twelve UCI datasets show that the proposed FRGNHSVM can often improve more than 5% average prediction accuracy. Moreover, the performance of the high prediction accuracy of FRGHNSVM is more significant along with the ratio of label noise increasing, indicating its superiority in dealing with label-contaminated datasets.

A proximal subgradient algorithm with extrapolation for structured nonconvex nonsmooth problems

In this paper, we consider a class of structured nonconvex nonsmooth optimization problems, in which the objective function is formed by the sum of a possibly nonsmooth nonconvex function and a differentiable function with Lipschitz continuous gradient, subtracted by a weakly convex function. This general framework allows us to tackle problems involving nonconvex loss functions and problems with specific nonconvex constraints, and it has many applications such as signal recovery, compressed sensing, and optimal power flow distribution. We develop a proximal subgradient algorithm with extrapolation for solving these problems with guaranteed subsequential convergence to a stationary point. The convergence of the whole sequence generated by our algorithm is also established under the widely used Kurdyka–Łojasiewicz property. To illustrate the promising numerical performance of the proposed algorithm, we conduct numerical experiments on two important nonconvex models. These include a compressed sensing problem with a nonconvex regularization and an optimal power flow problem with distributed energy resources.

Knowledge-Aided Federated Learning for Energy-Limited Wireless Networks

The conventional model aggregation-based federated learning (FL) approach requires all local models to have the same architecture, which fails to support practical scenarios with heterogeneous local models. Moreover, the frequent model exchange is costly for resource-limited wireless networks since modern deep neural networks usually have over a million parameters. To tackle these challenges, we first propose a novel knowledge-aided FL (KFL) framework, which aggregates light high-level data features, namely knowledge, in the per-round learning process. This framework allows devices to design their machine-learning models independently and reduces the communication overhead in the training process. We then theoretically analyze the convergence bound of the proposed framework under a non-convex loss function setting, revealing that scheduling more data volume in each round helps to improve the learning performance. In addition, large data volume should be scheduled in early rounds if the total scheduled data volume during the entire learning course is fixed. Inspired by this, we define a new objective function, i.e., the weighted scheduled data sample volume, to transform the inexplicit global loss minimization problem into a tractable one for device scheduling, bandwidth allocation, and power control. To deal with unknown time-varying wireless channels, we transform the considered problem into a deterministic problem for each round with the assistance of the Lyapunov optimization framework. Then, we derive the optimal bandwidth allocation and power control solution by convex optimization techniques. We also develop an efficient online device scheduling algorithm to achieve an energy-learning trade-off in the learning process. Experimental results on two typical datasets (i.e., MNIST and CIFAR-10) under highly heterogeneous local data distributions show that the proposed KFL is capable of reducing over 99% communication overhead while achieving better learning performance than the conventional model aggregation-based algorithms. In addition, the proposed device scheduling algorithm converges faster than the benchmark scheduling schemes.

Doubly iteratively reweighted algorithm for constrained compressed sensing models

We propose a new algorithmic framework for constrained compressed sensing models that admit nonconvex sparsity-inducing regularizers including the log-penalty function as objectives, and nonconvex loss functions such as the Cauchy loss function and the Tukey biweight loss function in the constraint. Our framework employs iteratively reweighted $$\ell _1$$ and $$\ell _2$$ schemes to construct subproblems that can be efficiently solved by well-developed solvers for basis pursuit denoising such as SPGL1 by van den Berg and Friedlander (SIAM J Sci Comput 31:890-912, 2008). We propose a new termination criterion for the subproblem solvers that allows them to return an infeasible solution, with a suitably constructed feasible point satisfying a descent condition. The feasible point construction step is the key for establishing the well-definedness of our proposed algorithm, and we also prove that any accumulation point of this sequence of feasible points is a stationary point of the constrained compressed sensing model, under suitable assumptions. Finally, we compare numerically our algorithm (with subproblems solved by SPGL1 or the alternating direction method of multipliers) against the SCP $$_\textrm{ls}$$ in Yu et al. (SIAM J Optim 31: 2024-2054, 2021) on solving constrained compressed sensing models with the log-penalty function as the objective and the Cauchy loss function in the constraint, for badly scaled measurement matrices. Our computational results show that our approaches return solutions with better recovery errors, and are always faster.

On the Convergence of Hybrid Server-Clients Collaborative Training

Modern distributed machine learning (ML) paradigms, such as federated learning (FL), utilize data distributed at different clients to train a global model. In such paradigm, local datasets never leave the clients for better privacy protection, and the parameter server (PS) only performs simple aggregation. In practice, however, there is often some amount of data available at the PS, and its computation capability is strong enough to carry out more demanding tasks than simple model aggregation. The focus of this paper is to analyze the model convergence of a new hybrid learning architecture, which leverages the PS dataset and its computation power for collaborative model training with clients. Different from FL where stochastic gradient descent (SGD) is always computed in parallel across clients, the new architecture has both parallel SGD at clients and sequential SGD at PS. We analyze the convergence rate upper bounds of this <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">aggregate-then-advance</i> design for both strongly convex and non-convex loss functions. We show that when the local SGD has an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {O}(1/t)$ </tex-math></inline-formula> stepsize, the server SGD needs to scale its stepsize to no slower than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {O}(1/t^{2})$ </tex-math></inline-formula> in order to strictly outperform local SGD with strongly convex loss functions. The theoretical findings are corroborated by numerical experiments, where advantages in terms of both accuracy and convergence speed over clients-only (local SGD and FED AVG) and server-only training are demonstrated.

Capped Asymmetric Elastic Net Support Vector Machine for Robust Binary Classification

Recently, there are lots of literature on improving the robustness of SVM by constructing nonconvex functions, but they seldom theoretically study the robust property of the constructed functions. In this paper, based on our recent work, we present a novel capped asymmetric elastic net (CaEN) loss and equip it with the SVM as CaENSVM. We derive the influence function of the estimators of the CaENSVM to theoretically explain the robustness of the proposed method. Our results can be easily extended to other similar nonconvex loss functions. We further show that the influence function of the CaENSVM is bounded, so that the robustness of the CaENSVM can be theoretically explained. Other theoretical analysis demonstrates that the CaENSVM satisfies the Bayes rule and the corresponding generalization error bound based on Rademacher complexity guarantees its good generalization capability. Since CaEN loss is concave, we implement an efficient DC procedure based on the stochastic gradient descent algorithm (Pegasos) to solve the optimization problem. A host of experiments are conducted to verify the effectiveness of our proposed CaENSVM model.

DeceFL: a principled fully decentralized federated learning framework

Traditional machine learning relies on a centralized data pipeline for model training in various applications; however, data are inherently fragmented. Such a decentralized nature of databases presents the serious challenge for collaboration: sending all decentralized datasets to a central server raises serious privacy concerns. Although there has been a joint effort in tackling such a critical issue by proposing privacy-preserving machine learning frameworks, such as federated learning, most state-of-the-art frameworks are built still in a centralized way, in which a central client is needed for collecting and distributing model information (instead of data itself) from every other client, leading to high communication burden and high vulnerability when there exists a failure at or an attack on the central client. Here we propose a principled decentralized federated learning algorithm (DeceFL), which does not require a central client and relies only on local information transmission between clients and their neighbors, representing a fully decentralized learning framework. It has been further proven that every client reaches the global minimum with zero performance gap and achieves the same convergence rate $O(1/T)$ (where $T$ is the number of iterations in gradient descent) as centralized federated learning when the loss function is smooth and strongly convex. Finally, the proposed algorithm has been applied to a number of applications to illustrate its effectiveness for both convex and nonconvex loss functions, time-invariant and time-varying topologies, as well as IID and Non-IID of datasets, demonstrating its applicability to a wide range of real-world medical and industrial applications.

Distributed Learning Over Networks With Graph-Attention-Based Personalization

In conventional distributed learning over a network, multiple agents collaboratively build a common machine learning model. However, due to the underlying non-i.i.d. data distribution among agents, the unified learning model becomes inefficient for each agent to process its locally accessible data. To address this problem, we propose a graph-attention-based personalized training algorithm (GATTA) for distributed deep learning. The GATTA enables each agent to train its local personalized model while exploiting its correlation with neighboring nodes and utilizing their useful information for aggregation. In particular, the personalized model in each agent is composed of a global part and a node-specific part. By treating each agent as one node in a graph and the node-specific parameters as its features, the benefits of the graph attention mechanism can be inherited. Namely, instead of aggregation based on averaging, it learns the specific weights for different neighboring nodes without requiring prior knowledge about the graph structure or the neighboring nodes' data distribution. Furthermore, relying on the weight-learning procedure, we develop a communication-efficient GATTA by skipping the transmission of information with small aggregation weights. Additionally, we theoretically analyze the convergence properties of GATTA for non-convex loss functions. Numerical results validate the excellent performances of the proposed algorithms in terms of convergence and communication cost.

Quantum variational algorithms are swamped with traps

One of the most important properties of classical neural networks is how surprisingly trainable they are, though their training algorithms typically rely on optimizing complicated, nonconvex loss functions. Previous results have shown that unlike the case in classical neural networks, variational quantum models are often not trainable. The most studied phenomenon is the onset of barren plateaus in the training landscape of these quantum models, typically when the models are very deep. This focus on barren plateaus has made the phenomenon almost synonymous with the trainability of quantum models. Here, we show that barren plateaus are only a part of the story. We prove that a wide class of variational quantum models—which are shallow, and exhibit no barren plateaus—have only a superpolynomially small fraction of local minima within any constant energy from the global minimum, rendering these models untrainable if no good initial guess of the optimal parameters is known. We also study the trainability of variational quantum algorithms from a statistical query framework, and show that noisy optimization of a wide variety of quantum models is impossible with a sub-exponential number of queries. Finally, we numerically confirm our results on a variety of problem instances. Though we exclude a wide variety of quantum algorithms here, we give reason for optimism for certain classes of variational algorithms and discuss potential ways forward in showing the practical utility of such algorithms.

Federated Adam-Type Algorithm for Distributed Optimization With Lazy Strategy

For large-scale machine learning tasks, distributing data in multiple clients, and using distributed optimization algorithms with a parameter server can accelerate the training process. The federated average algorithm has been widely used for distributed optimization via training local models in parallel and aggregating local models in a server to obtain the global model. To further improve the performance of the federated average algorithm, a novel federated learning algorithm have been proposed in this article by embedding a lazy strategy in the distributed Adam-type algorithm. In the proposed algorithm, the learning rate is adjusted adaptively in local update and lazy update strategy is applied on the second-order momentum of clients to make the learning rate identical. The convergence of the proposed algorithm is provided for both convex and nonconvex loss functions. Experiments have been conducted on MNIST digit recognition data set and CIFAR-10 data set. Experimental results show that the proposed algorithm can significantly reduce the communication overhead, thereby reduce the training time by 60% for CIFAR-10 data set, and the proposed algorithm achieve better performance than the federated average algorithm and its momentum version.

Clustering multivariate count data via Dirichlet-multinomial network fusion

Clustering of multivariate count data has widespread applications in areas such as text analysis and microbiome studies. The need to account for overdispersion generally results in a nonconvex loss function, which does not fit into the existing convex clustering framework. Moreover, prior knowledge of a network over the samples, often available from citation or similarity relationships, is not taken into account. We introduce Dirichlet-multinomial network fusion (DMNet) for clustering multivariate count data, which models the samples via Dirichlet-multinomial distributions with individual parameters and employs a weighted group L1 fusion penalty to pursue homogeneity over a prespecified network. To circumvent the nonconvexity issue, we present two exponential family approximations to the Dirichlet-multinomial distribution, which are amenable to efficient optimization and theoretical analysis. We derive an ADMM algorithm and establish nonasymptotic error bounds for the proposed methods. Our bounds involve a trade-off between the connectivity of the network and its fidelity to the true parameter. The usefulness of our methods is illustrated through simulation studies and two text clustering applications.