Descent Step Research Articles

Steepest descent method for uncertain multiobjective optimization problems under finite uncertainty set

In this paper, a steepest descent method is developed for the robust counterpart of uncertain multiobjective optimization problems under finite uncertainty. The robust counterpart under consideration is the minimum of objective wise worst case, which is a nonsmooth deterministic multiobjective optimization problem. To solve this robust counterpart with the help of the steepest descent method, a subproblem is constructed and solved to find a descent direction. An Armijo-type inexact line search technique is employed to find a suitable step length. With the help of the descent direction and step length, a sequence is generated. The convergence of the proposed method is justified under some common assumptions. Finally, the algorithm is verified and compared with one existing method with the help of some numerical problems.

Less is more: adjusting convergence of Cooper’s algorithm in the continuous location-allocation problem

This paper proposes a variant of the well-known heuristic by Cooper for continuous location-allocation problems in general, and in particular, the continuous p-median problem. This algorithm alternates between a location phase where the allocations are fixed and an allocation phase where the locations of the p new facilities are fixed. The main idea of the proposed variant is to partially solve the p independent one-median problems in the location phase (instead of solving optimally) in order to induce a larger number of cycles of location-allocation. In effect, we are trying to alter the descent path of the algorithm by slowing the convergence rate using smaller descent steps in the location phase. Computational experiments show that this approach is highly effective in finding alternate descent paths that lead to “deeper” local minima. The best improvements are obtained with random starting locations of the p facilities, and the larger values of p that were tested.

Stabilized distributed online mirror descent for multi-agent optimization

In the domain of multi-agent networks, distributed online mirror descent (DOMD) and distributed online dual averaging (DODA) play pivotal roles as fundamental algorithms for distributed online convex optimization. However, in contrast to DODA, DOMD fails when employed with a dynamic learning rate sequence. To bridge this gap, we introduce two novel variants of DOMD by incorporating a distributed stabilization step in primal space and dual space, respectively. We demonstrate that our stabilized DOMD algorithms achieve a sublinear bound with a sequence of dynamic learning rates. We further evolve our dual-stabilized DOMD by integrating a lazy communicated subgradient descent step, resulting in a re-indexed DODA. This establishes a connection between the two types of distributed algorithms, which enhances our understandings of distributed optimization. Moreover, we extend our proposed algorithms to handle the case of exponentiated gradient, where the iterate is constrained within a simplex probability. Finally, we conduct extensive numerical simulations to validate our theoretical analysis.

Unsupervised Test-Time Adaptation Learning for Effective Hyperspectral Image Super-Resolution With Unknown Degeneration.

Fusing a low-resolution hyperspectral image (HSI) with a high-resolution (HR) multi-spectral image has provided an effective way for HSI super-resolution (SR). The key lies on inferring the posteriori of the latent (i.e., HR) HSI using an appropriate image prior and the likelihood determined by the degeneration between the latent HSI and the observed images. However, in scenarios with complex imaging environments and various imaging scenes, the prior of HSIs can be prohibitively complicated and the degeneration is often unknown, which causes it difficult to accurately infer the posteriori of each latent HSI. To tackle this problem, we present an unsupervised test-time adaptation learning (UTAL) framework for HSI SR under unknown degeneration. Instead of directly modeling the complicated image prior, it first implicitly learns a content-agnostic prior shared across different images through supervisedly pre-training a mutual-guiding fusion module on extensive synthetic data. Then, it adapts the shared prior to those private characteristics in the latent HSI for posteriori inference through unsupervisedly learning a self-guiding adaptation module and a degeneration estimation network on two observed images in the test phase. Such a two-stage learning scheme models the complicated image prior in a divide-and-conquer manner, which eases the modeling difficulty and improves the prior accuracy. Moreover, the unknown degeneration can be estimated properly. Both of these two advantages empower us to accurately infer the posteriori of the latent HSI, thereby increasing the generalization performance in real applications. Additionally, in order to further mitigate the over-fitting in coping with more challenging cases (e.g., degenerations in both spectral and spatial domains are unknown) and speed up, we propose to meta-train UTAL on extensive synthetic SR tasks and solve it using an alternative optimization strategy such that UTAL learns to produce good generalization performance in real challenging cases with a small number of gradient descent steps. To verify the efficacy of UTAL, we evaluate it on HSI SR tasks with different unknown degenerations as well as some other HSI restoration tasks (e.g., compressive sensing), and report strong results superior to that of existing competitors.

Speeding up the Training of Neural Networks with the One-Step Procedure

In the last decade, research and corporate have shown a dramatically growing interest in the field of machine learning, mostly due to the performances of deep neural networks. These increasingly complex architectures solved a wide range of problems. However, training these sophisticated architectures require many computation on advanced hardware. With this paper, we introduce a new approach based on the One-Step procedure that may fasten their training. In this procedure, an initial guess estimator is computed on a subsample that is then improved with only one step of the Newton gradient descent on the whole dataset. To show the efficiency of this framework, we consider regression and classification tasks using simulated and real datasets. We consider classic architectures, namely multi-layer perceptrons and show, on our examples, that the One-Step procedure is often halving the computation time to train the neural networks while preserving the performances.

Topological optimization in 3D-magnetostatics: development of adjoint methods using the equations of magnetic moments

PurposeThe purpose of this paper is mainly to develop the adjoint method within the method of magnetic moment (MMM) and thus, to provide an efficient new way to solve topology optimization problems in magnetostatic to design 3D-magnetic circuits.Design/methodology/approachFirst, the MMM is recalled and the optimization design problem is reformulated as a partial derivative equation-constrained optimization problem where the constraint is the Maxwell equation in magnetostatic. From the Karush–Khun–Tucker optimality conditions, a new problem is derived which depends on a Lagrangian parameter. This problem is called the adjoint problem and the Lagrangian parameter is called the adjoint parameter. Thus, solving the direct and the adjoint problems, the values of the objective function as well as its gradient can be efficiently obtained. To obtain a topology optimization code, a semi isotropic material with penalization (SIMP) relaxed-penalization approach associated with an optimization based on gradient descent steps has been developed and used.FindingsIn this paper, the authors provide theoretical results which make it possible to compute the gradient via the continuous adjoint of the MMMs. A code was developed and it was validated by comparing it with a finite difference method. Thus, a topology optimization code associating this adjoint based gradient computations and SIMP penalization technique was developed and its efficiency was shown by solving a 3D design problem in magnetostatic.Research limitations/implicationsThis research is limited to the design of systems in magnetostatic using the linearity of the materials. The simple examples, the authors provided, are just done to validate our theoretical results and some extensions of our topology optimization code have to be done to solve more interesting design cases.Originality/valueThe problem of design is a 3D magnetic circuit. The 2D optimization problems are well known and several methods of resolution have been introduced, but rare are the problems using the adjoint method in 3D. Moreover, the association with the MMMs has never been treated yet. The authors show in this paper that this association could provide gains in CPU time.

Bilateral neuromuscular control in patients one year after unilateral ACL rupture or reconstruction. A cross-sectional study

ObjectivesTo compare bilateral neuromuscular control in patients one year after anterior cruciate ligament reconstruction (ACL-R) or conservative treatment (ACL-C) to healthy controls (ACL-I). DesignCross-sectional study. SettingElectromyography of vastus medialis (VM) and lateralis (VL), biceps femoris (BF) and semitendinosus (ST) was recorded during stair descent and anterior tibial translation. Each step of stair descent was divided into pre-activity, weight-acceptance and push-off phase. Pre-activation, short, medium (MLR) and long latency responses (LLR) were defined for reflex activity. ParticipantsN = 38 patients one year after ACL reconstruction (ACL-R), N = 26 participants with conservative treatment one year after ACL rupture (ACL-C), N = 38 healthy controls with an intact ACL (ACL-I). Main outcome measuresNormalized root mean squares per muscle and phase (α = 0.05). ResultsDuring stair descent, within-group leg differences were found for the quadriceps in ACL-R during all phases and for the BF in ACL-C during weight-acceptance. Between-group leg differences were found for BF in both patient groups compared to ACL-I during push-off.Between-group differences in pre-activation for VM between ACL-R and ACL-C, and between ACL-C and ACL-I were found, and as LLR between patients and ACL-R versus ACL-I. Pre-activation of BF and MLR of ST differed for each patient group compared to ACL-I. ConclusionsBilateral neuromuscular alterations are still present one year after ACL rupture or reconstruction.

LAFIT: Efficient and Reliable Evaluation of Adversarial Defenses With Latent Features.

Deep convolutional neural networks (CNNs) can be easily tricked to give incorrect outputs by adding tiny perturbations to the input that are imperceptible to humans. This makes them susceptible to adversarial attacks, and poses significant security risks to deep learning systems, and presents a great challenge in making CNNs robust against such attacks. An influx of defense strategies have thus been proposed to improve the robustness of CNNs. Current attack methods, however, may fail to accurately or efficiently evaluate the robustness of defending models. In this paper, we thus propose a unified lp white-box attack strategy, LAFIT, to harness the defender's latent features in its gradient descent steps, and further employ a new loss function to normalize logits to overcome floating-point-based gradient masking. We show that not only is it more efficient, but it is also a stronger adversary than the current state-of-the-art when examined across a wide range of defense mechanisms. This suggests that adversarial attacks/defenses could be contingent on the effective use of the defender's hidden components, and robustness evaluation should no longer view models holistically.

A proximal forward-backward splitting based algorithmic framework for Wasserstein logistic regression using heavy ball strategy

ABSTRACT In this paper, a forward-backward splitting based algorithmic framework incorporating the heavy ball strategy is proposed so as to efficiently solve the Wasserstein logistic regression problem. The proposed algorithmic framework consists two phases: the first phase involves a gradient descent step extension method, whilst the second phase involves a problem of instantaneous optimisation which balances the minimisation of a regularisation term while maintaining close proximity to the interim state given in the first phase. Then, it proves that the proposed algorithmic framework converges to the optimal solution of the Wasserstein logistic regression problem. Finally, numerical experiments are conducted, which illustrate the efficient implementation for high-dimensional sparsity data. The numerical results demonstrate that the proposed algorithmic framework outperforms not only the off-the-shelf solvers, but also some existing first-order algorithms.

Reconstruction of Sparse-View X-ray Computed Tomography Based on Adaptive Total Variation Minimization.

Sparse-view reconstruction has garnered significant interest in X-ray computed tomography (CT) imaging owing to its ability to lower radiation doses and enhance detection efficiency. Among current methods for sparse-view CT reconstruction, an algorithm utilizing iterative reconstruction based on full variational regularization demonstrates good performance. The optimized direction and number of computations for the gradient operator of the regularization term play a crucial role in determining not only the reconstructed image quality but also the convergence speed of the iteration process. The conventional TV approach solely accounts for the vertical and horizontal directions of the two-dimensional plane in the gradient direction. When projection data decrease, the edges of the reconstructed image become blurred. Exploring too many gradient directions for TV terms often comes at the expense of more computational costs. To enhance the balance of computational cost and reconstruction quality, this study suggests a novel TV computation model that is founded on a four-direction gradient operator. In addition, selecting appropriate iteration parameters significantly impacts the quality of the reconstructed image. We propose a nonparametric control method utilizing the improved TV approach as a solution to the tedious manual parameter optimization issue. The relaxation parameters of projection onto convex sets (POCS) are determined according to the scanning number and numerical proportion of the projection data; according to the image error before and after iterations, the gradient descent step of the TV item is adaptively adjusted. Compared with several representative iterative reconstruction algorithms, the experimental results show that the algorithm can effectively preserve edges and suppress noise in sparse-view CT reconstruction.

Superiorization of projection algorithms for linearly constrained inverse radiotherapy treatment planning.

We apply the superiorization methodology to the constrained intensity-modulated radiation therapy (IMRT) treatment planning problem. Superiorization combines a feasibility-seeking projection algorithm with objective function reduction: The underlying projection algorithm is perturbed with gradient descent steps to steer the algorithm towards a solution with a lower objective function value compared to one obtained solely through feasibility-seeking. Within the open-source inverse planning toolkit matRad, we implement a prototypical algorithmic framework for superiorization using the well-established Agmon, Motzkin, and Schoenberg (AMS) feasibility-seeking projection algorithm and common nonlinear dose optimization objective functions. Based on this prototype, we apply superiorization to intensity-modulated radiation therapy treatment planning and compare it with (i) bare feasibility-seeking (i.e., without any objective function) and (ii) nonlinear constrained optimization using first-order derivatives. For these comparisons, we use the TG119 water phantom, the head-and-neck and the prostate patient of the CORT dataset. Bare feasibility-seeking with AMS confirms previous studies, showing it can find solutions that are nearly equivalent to those found by the established piece-wise least-squares optimization approach. The superiorization prototype solved the linearly constrained planning problem with similar dosimetric performance to that of a general-purpose nonlinear constrained optimizer while showing smooth convergence in both constraint proximity and objective function reduction. Superiorization is a useful alternative to constrained optimization in radiotherapy inverse treatment planning. Future extensions with other approaches to feasibility-seeking, e.g., with dose-volume constraints and more sophisticated perturbations, may unlock its full potential for high performant inverse treatment planning.

Amortized Bayesian Meta-Learning with Accelerated Gradient Descent Steps

Recent meta-learning models often learn priors from observed tasks using a network optimized via stochastic gradient descent (SGD), which usually takes more training steps to convergence. In this paper, we propose an accelerated Bayesian meta-learning structure with a stochastic inference network (ABML-SIN). The proposed model aims to solve the training procedure of Bayesian meta-learning to improve the training speed and efficiency. Current approaches of meta-learning hardly converge within a few descent steps, owing to the small number of training samples. Therefore, we introduce an accelerated gradient descent learning network based on teacher–student architecture to learn the meta-latent variable θt for task t. With this amortized fast inference network, the meta-learner is able to learn the task-specific latent θt within a few training steps; thus, it improves the learning speed of the meta-learner. To refine the latent variables generated from the transductive amortization network of the meta-learner, SIN—followed by a conventional SGD-optimized network—is introduced as the student–teacher network to online-update the parameters. SIN extracts the local latent variables and accelerates the convergence of the meta-learning network. Our experiments on simulation data demonstrate that the proposed method provides generalization and scalability on unseen samples, and produces competitive/superior uncertainty estimations on few-shot learning tasks on two widely adopted 2D datasets with fewer training epochs compared to the state-of-the-art meta-learning approaches. Furthermore, the parameters generated by SIN act as perturbations on latent weights, enhancing the probability of accelerating the training efficiency of the meta-learner. Extensive qualitative experiments show that our method performs well across different meta-learning tasks in both simulated and real-world circumstances.

Recursive Control Variates for Inverse Rendering

We present a method for reducing errors---variance and bias---in physically based differentiable rendering (PBDR). Typical applications of PBDR repeatedly render a scene as part of an optimization loop involving gradient descent. The actual change introduced by each gradient descent step is often relatively small, causing a significant degree of redundancy in this computation. We exploit this redundancy by formulating a gradient estimator that employs a recursive control variate , which leverages information from previous optimization steps. The control variate reduces variance in gradients, and, perhaps more importantly, alleviates issues that arise from differentiating loss functions with respect to noisy inputs, a common cause of drift to bad local minima or divergent optimizations. We experimentally evaluate our approach on a variety of path-traced scenes containing surfaces and volumes and observe that primal rendering efficiency improves by a factor of up to 10.

A fast proximal gradient method and convergence analysis for dynamic mean field planning

In this paper, we propose an efficient and flexible algorithm to solve dynamic mean-field planning problems based on an accelerated proximal gradient method. Besides an easy-to-implement gradient descent step in this algorithm, a crucial projection step becomes solving an elliptic equation whose solution can be obtained by conventional methods efficiently. By induction on iterations used in the algorithm, we theoretically show that the proposed discrete solution converges to the underlying continuous solution as the grid becomes finer. Furthermore, we generalize our algorithm to mean-field game problems and accelerate it using multilevel and multigrid strategies. We conduct comprehensive numerical experiments to confirm the convergence analysis of the proposed algorithm, to show its efficiency and mass preservation property by comparing it with state-of-the-art methods, and to illustrate its flexibility for handling various mean-field variational problems.

Network revenue management with online inverse batch gradient descent method

We consider a general class of price‐based network revenue management problems that a firm aims to maximize revenue from multiple products produced with multiple types of resources endowed with limited inventory over a finite selling season. A salient feature of our problem is that the firm does not know the underlying demand function that maps prices to demand rate, which must be learned from sales data. It is well known that for almost all classes of demand functions, such as linear, exponential, multinomial logit, and nested logit models, the revenue rate function is not concave in the products' prices but is concave in products' market shares (or price‐controlled demand rates). This creates challenges in adopting any stochastic gradient descent based methods in the price space. We propose a novel nonparametric learning algorithm termed the online inverse batch gradient descent (IGD) algorithm. This algorithm proceeds in batches. In each batch, the firm implements each product's perturbed prices and then uses the sales information to estimate the market shares. Leveraging these estimates, the firm carries out a stochastic gradient descent step in the market share space that takes into account the relative inventory scarcity for the entire horizon and then inversely maps the updated market shares back to the price space to obtain the prices for the next batch. Moreover, we also propose an inventory‐adjusted algorithm (IGD‐I) that the feasible market share set is dynamically adjusted to capture the real‐time relative inventory scarcity for the remaining season. For the large‐scale systems wherein all resources' inventories and the length of the horizon are proportionally scaled by a parameter k, we establish a dimension‐independent regret bound of [Formula: see text]. This result is independent of the number of products and resources and works for a continuum action‐set prices and the demand functions that are only once differentiable. Our theoretical result guarantees the efficacy of both algorithms in the high‐dimensional systems where the number of products or resources is large and the prices are continuous. Our algorithms also numerically outperform the existing algorithms in the literature.

FedHD: Communication-efficient federated learning from hybrid data

Federated learning (FL) has attracted significant attention in the machine learning community owing to its instinct local privacy awareness. Depending on how the data are distributed over the clients, FL problems can be divided into three categories, namely the horizontal FL (HFL), the vertical FL (VFL) and the hybrid FL (HBFL). Among them, the HBFL problem is the most challenging because each client neither owns the full set of data samples nor knows the complete feature information. While many FL algorithms have been developed for the HFL and VFL problems, unfortunately they cannot handle the HBFL problem. In this paper, we propose a new FL algorithm, termed as FedHD, that can solve the challenging learning task under the HBFL setting. Since the clients cannot perform local optimization on their own under the hybrid data, a tracking variable is introduced to enable the clients to track the global gradient information and update the model based on their local data. FedHD allows the clients to perform multiple steps of local stochastic gradient descent (SGD), and hence has improved communication efficiency. Theoretical analysis is conducted to show that FedHD has a O(1/QT) convergence rate, where Q denotes the local update number and T is the total number of communication rounds. Then we further reveal the insights on how various algorithm parameters impact on the convergence performance. Experiment results show that the proposed FedHD exhibits robust performance on the hybrid data, and is largely superior to the naive local training model.

Fast adaptation of multi-task meta-learning for optical performance monitoring.

An algorithm is proposed for few-shot-learning (FSL) jointing modulation format identification (MFI) and optical signal-to-noise ratio (OSNR) estimation. The constellation diagrams of six widely-used modulation formats over a wide range of OSNR (10-40 dB) are obtained by a dual-polarization (DP) coherent detection system at 32 GBaud. We introduce auxiliary task to model-agnostic meta-learning (MAML) which makes the gradient of meta tasks decline faster in the direction of optimal target. Ablation experiments including multi-task model-agnostic meta-learning (MT-MAML), single-task model-agnostic meta-learning (ST-MAML) and adaptive multi-task learning (AMTL) are executed to train a data set with only 20 examples for each class. First, we discuss the impact from the number of shots and gradient descent steps for support set on the meta-learning based schemes to determine the best hyper parameters and conclude that the proposed method better captures the similarity between new and previous knowledge at 4 shot and 1 step. Withdrawn fine-tuning, the model achieves the lowest error ∼0.37 dB initially. Then, we simulate two other schemes (AMTL and ST-MAML), and the numerical results shows that mean square error (MSE) are ∼0.6 dB, ∼0.3 dB and ∼0.18 dB, respectively, proposed method has faster adaption to main task. For low order modulation formats, the proposed method almost reduces the error to 0. Meanwhile, we reveal the degree of deviation between the prediction and target and find that the deviation is mainly concentrated in the high OSNR range of 25-40 dB. Specifically, we investigate the variation curve of adaptive weights during pretraining and conclude that after 30 epoch, the model's attention was almost entirely focused on estimating OSNR. In addition, we study the generalization ability of the model by varying the transmission distance. Importantly, excellent generalization is also experimentally verified. In this paper, the method proposed will greatly reduce the cost for repetitively collecting data and the training resources required for fine-tuning models when OPM devices need to be deployed at massive nodes in dynamic optical networks.

Gradient-Variation Bound for Online Convex Optimization with Constraints

We study online convex optimization with constraints consisting of multiple functional constraints and a relatively simple constraint set, such as a Euclidean ball. As enforcing the constraints at each time step through projections is computationally challenging in general, we allow decisions to violate the functional constraints but aim to achieve a low regret and cumulative violation of the constraints over a horizon of T time steps. First-order methods achieve an O(sqrt{T}) regret and an O(1) constraint violation, which is the best-known bound under the Slater's condition, but do not take into account the structural information of the problem. Furthermore, the existing algorithms and analysis are limited to Euclidean space. In this paper, we provide an instance-dependent bound for online convex optimization with complex constraints obtained by a novel online primal-dual mirror-prox algorithm. Our instance-dependent regret is quantified by the total gradient variation V_*(T) in the sequence of loss functions. The proposed algorithm works in general normed spaces and simultaneously achieves an O(sqrt{V_*(T)}) regret and an O(1) constraint violation, which is never worse than the best-known (O(sqrt{T}), O(1)) result and improves over previous works that applied mirror-prox-type algorithms for this problem achieving O(T^{2/3}) regret and constraint violation. Finally, our algorithm is computationally efficient, as it only performs mirror descent steps in each iteration instead of solving a general Lagrangian minimization problem.

GASTO: A Fast Adaptive Graph Learning Framework for Edge Computing Empowered Task Offloading

Mobile edge computing (MEC) has become a research trend that solves effectively computationally intensive and latency-sensitive tasks. MEC environments in the real world are dynamic and uncertain and then the changes of the environments bring challenges to the generalization and robustness of offloading algorithms. In order to solve the above problem, we propose a meta-reinforcement learning task offloading algorithm GASTO based on Graph Neural Network and seq2seq network. Meta-learning can learn the optimal initialization parameter through several gradient descent steps and samples to adapt to new environments more quickly. The task generated in the user equipment is composed of multiple subtasks rather than a single task, and there are dependencies between the subtasks. Therefore, the task on the user equipment is modeled as a Directed Acyclic Graph (DAG). The connection relationship between the subtasks in DAG plays an important role. Drawing on the idea of message passing, Graph Neural Network is applied in DAG to extract the intrinsic correlation between subtasks in GASTO. In addition, Seq2Seq network can reduce the dimension of action space effectively, and the scheduling decisions of all subtasks can be generated simultaneously. Besides, in order to enhance the sampling efficiency of tasks and the robustness of GASTO, the priority of sampling tasks is adjusted dynamically during the training process. The experimental results of four algorithms in different environments show that the proposed algorithm GASTO can quickly adapt to the new environment.

A Modified Gradient Method for Distributionally Robust Logistic Regression over the Wasserstein Ball

In this paper, a modified conjugate gradient method under the forward-backward splitting framework is proposed to further improve the numerical efficiency for solving the distributionally robust Logistic regression model over the Wasserstein ball, which comprises two phases: in the first phase, a conjugate gradient descent step is performed, and in the second phase, an instantaneous optimization problem is formulated and solved with a trade-off minimization of the regularization term, while simultaneously staying in close proximity to the interim point obtained in the first phase. The modified conjugate gradient method is proven to attain the optimal solution of the Wasserstein distributionally robust Logistic regression model with nonsummable steplength at a convergence rate of 1/T. Finally, several numerical experiments to validate the effectiveness of theoretical analysis are conducted, which demonstrate that this method outperforms the off-the-shelf solver and the existing first-order algorithmic frameworks.