Noisy Samples Research Articles

Provable bounds for noise-free expectation values computed from noisy samples.

Quantum computing has emerged as a powerful computational paradigm capable of solving problems beyond the reach of classical computers. However, today's quantum computers are noisy, posing challenges to obtaining accurate results. Here, we explore the impact of noise on quantum computing, focusing on the challenges in sampling bit strings from noisy quantum computers and the implications for optimization and machine learning. We formally quantify the sampling overhead to extract good samples from noisy quantum computers and relate it to the layer fidelity, a metric to determine the performance of noisy quantum processors. Further, we show how this allows us to use the conditional value at risk of noisy samples to determine provable bounds on noise-free expectation values. We discuss how to leverage these bounds for different algorithms and demonstrate our findings through experiments on real quantum computers involving up to 127 qubits. The results show strong alignment with theoretical predictions.

Broad Distributed Game Learning for intelligent classification in rolling bearing fault diagnosis

As a new Single Layer Feedforward Network (SLFN) architecture, Broad Learning System (BLS) has been widely used in the field of fault diagnosis because of its fast-training speed and high generalization capability. However, when features in different classes of signals are similar or weak, BLS generates a large number of redundant features that may be difficult to classify accurately. In view of this, a new Broad Distributed Game Learning (BDGL) method is proposed in this paper, which maps data into the game space by constructing two non-parallel game hyperplanes to achieve game and segmentation of different similar features, thereby making the data linearly differentiable in the game space. Meanwhile, a linear distribution constraint term is designed to reduce noise fitting and weak feature learning in training data learning by limiting the complexity of model parameters, thereby making the solution of the objective function simpler and faster. By comparing the Precision, Recall, F-score, Kappa and Accuracy of BDGL and the comparison methods on the two types of rolling bearing experimental data, the results show that BDGL has a high classification accuracy. In addition, the experimental results on small and noisy samples once again demonstrate the effectiveness of BDGL, which provides an efficient solution for rolling bearing fault diagnosis.

Blending is all you need: Data-centric ensemble synthetic data

Deep generative models have gained increasing popularity, particularly in fields such as natural language processing and computer vision. Recently, efforts have been made to extend these advanced algorithms to tabular data. While generative models have shown promising results in creating synthetic data, their high computational demands and the need for careful parameter tuning present significant challenges. This study investigates whether a collective integration of refined synthetic datasets from multiple models can achieve comparable or superior performance to that of a single, large generative model. To this end, we developed a Data-Centric Ensemble Synthetic Data model, leveraging principles of ensemble learning. Our approach involved a data refinement process applied to various synthetic datasets, systematically eliminating noise and ranking, selecting, and combining them to create an augmented, high-quality synthetic dataset. This approach improved both the quantity and quality of the data. Central to this process, we introduced the Ensemble k-Nearest Neighbors with Centroid Displacement (EKCD) algorithm for noise filtering, alongside a density score for ranking and selecting data. Our experiments confirmed the effectiveness of EKCD in removing noisy synthetic samples. Additionally, the ensemble model based on the refined synthetic data substantially enhanced the performance of machine learning models, sometimes even outperforming that of real data.

Adaptive Learning for Dynamic Features and Noisy Labels.

Applying current machine learning algorithms in complex and open environments remains challenging, especially when different changing elements are coupled and the training data is scarce. For example, in the activity recognition task, the motion sensors may change position or fall off due to the intensity of the activity, leading to changes in feature space and finally resulting in label noise. Learning from such a problem where the dynamic features are coupled with noisy labels is crucial but rarely studied, particularly when the noisy samples in new feature space are limited. In this paper, we tackle the above problem by proposing a novel two-stage algorithm, called Adaptive Learning for Dynamic features and Noisy labels (ALDN). Specifically, optimal transport is firstly modified to map the previously learned heterogeneous model to the prior model of the current stage. Then, to fully reuse the mapped prior model, we add a simple yet efficient regularizer as the consistency constraint to assist both the estimation of the noise transition matrix and the model training in the current stage. Finally, two implementations with direct (ALDN-D) and indirect (ALDN-ID) constraints are illustrated for better investigation. More importantly, we provide theoretical guarantees for risk minimization of ALDN-D and ALDN-ID. Extensive experiments validate the effectiveness of the proposed algorithms.

Data Quality-Aware Client Selection in Heterogeneous Federated Learning

Federated Learning (FL) enables decentralized data utilization while maintaining edge user privacy, but it faces challenges due to statistical heterogeneity. Existing approaches address client drift and data heterogeneity issues. However, real-world settings often involve low-quality data with noisy features, such as covariate drift or adversarial samples, which are usually ignored. Noisy samples significantly impact the global model’s accuracy and convergence rate. Assessing data quality and selectively aggregating updates from high-quality clients is crucial, but dynamically perceiving data quality without additional computations or data exchanges is challenging. In this paper, we introduce the FedDQA (Federated learning via Data Quality-Aware) (FedDQA) framework. We discover increased data noise leads to slower loss reduction during local model training. We propose a loss sharpness-based Data-Quality-Awareness (DQA) metric to differentiate between high-quality and low-quality data. Based on the DQA, we design a client selection algorithm that strategically selects participant clients to reduce the negative impact of noisy clients. Experiment results indicate that FedDQA significantly outperforms the baselines. Notably, it achieves up to a 4% increase in global model accuracy and demonstrates faster convergence rates.

Bias Mitigation and Representation Optimization for Noise-Robust Cross-modal Retrieval

The remarkable progress in cross-modal retrieval relies on accurately-annotated multimedia datasets. In practice, most existing datasets used for training cross-modal retrieval models are automatically collected from the Internet to reduce data collection costs. However, it inevitably contains mismatched pairs, i.e. , noisy correspondences, thus degrading the model performance. Recent advances utilize the predicted similarity distribution of individual samples for noise validation and correction, which easily faces two challenging dilemmas: 1) confirmation bias and 2) unstable performance with increasing noise. In light of the above, we propose a generalized B ias M itigation and R epresentation O ptimization framework (BMRO). Specifically, we propose a Bias Estimator (BE) to estimate the unbiased confidence factor of a sample by contrasting it against its nearest neighbors. Unbiased confidence factor can precisely adjust sample contribution and enhance accurate sample division. This facilitates the Adaptive Representation Optimizer (ARO) in providing tailored optimization strategies for clean and noisy samples. ARO performs contrastive learning between clean samples and generated hard samples, thus promoting the generalizability and robustness of the representation. Besides, it utilizes complementary learning to reduce incorrect guidance from noisy samples. Extensive experiments on five visual-text benchmarks verify that our BMRO can significantly improve the matching accuracy and performance stability against noisy correspondences.

Value construction through sequential sampling explains serial dependencies in decision making.

Many decisions are expressed as a preference for one item over another. When these items are familiar, it is often assumed that the decision maker assigns a value to each of the items and chooses the item with the highest value. These values may be imperfectly recalled, but are assumed to be stable over the course of an interview or psychological experiment. Choices that are inconsistent with a stated valuation are thought to occur because of unspecified noise that corrupts the neural representation of value. Assuming that the noise is uncorrelated over time, the pattern of choices and response times in value-based decisions are modeled within the framework of Bounded Evidence Accumulation (BEA), similar to that used in perceptual decision-making. In BEA, noisy evidence samples accumulate over time until the accumulated evidence for one of the options reaches a threshold. Here, we argue that the assumption of temporally uncorrelated noise, while reasonable for perceptual decisions, is not reasonable for value-based decisions. Subjective values depend on the internal state of the decision maker, including their desires, needs, priorities, attentional state, and goals. These internal states may change over time, or undergo revaluation, as will the subjective values. We reasoned that these hypothetical value changes should be detectable in the pattern of choices made over a sequence of decisions. We reanalyzed data from a well-studied task in which participants were presented with pairs of snacks and asked to choose the one they preferred. Using a novel algorithm ( Reval ), we show that the subjective value of the items changes significantly during a short experimental session (about 1 hour). Values derived with Reval explain choice and response time better than explicitly stated values. They also better explain the BOLD signal in the ventromedial prefrontal cortex, known to represent the value of decision alternatives. Revaluation is also observed in a BEA model in which successive evidence samples are not assumed to be independent. We argue that revaluation is a consequence of the process by which values are constructed during deliberation to resolve preference choices.

Improving neural-network-based prediction models for misalignment in off-axis three-mirror anastigmat telescopes

The performance of a telescope system heavily relies on the precise alignment of the mirrors. The off-axis three-mirror anastigmat (TMA) telescope presents unique challenges due to its complex optical design. Each optical element within the off-axis TMA telescope is inherently introduced with theoretical eccentricity and tilt. Furthermore, the incorporation of freeform surfaces and other optical elements with intricate surface features typically leads to low initial alignment accuracy of the optical path. With this low initial alignment accuracy and the noise of the measurements, the prediction of the misalignment of the telescope is getting harder. A fully connected neural network architecture is proposed as a misalignment calculation method for an off-axis TMA telescope system with a freeform surface. Random training data is created by using optical design software. The sensitivity of the mirrors to the wavefront error is quantified and incorporated into the loss function of the neural network to improve prediction accuracy. Adding the noisy measurement samples to the training data creates a noise-immune neural network model. Simulation results show that our model can successfully predict misalignment of the mirrors with noisy measurement values.

A New Joint Training Method for Facial Expression Recognition with Inconsistently Annotated and Imbalanced Data

Facial expression recognition (FER) plays a crucial role in various applications, including human–computer interaction and affective computing. However, the joint training of an FER network with multiple datasets is a promising strategy to enhance its performance. Nevertheless, widespread annotation inconsistencies and class imbalances among FER datasets pose significant challenges to this approach. This paper proposes a new multi-dataset joint training method, Sample Selection and Paired Augmentation Joint Training (SSPA-JT), to address these challenges. SSPA-JT models annotation inconsistency as a label noise problem and selects clean samples from auxiliary datasets to expand the overall dataset size while maintaining consistent annotation standards. Additionally, a dynamic matching algorithm is developed to pair clean samples of the tail class with noisy samples, which enriches the tail classes with diverse background information. Experimental results demonstrate that SSPA-JT achieved superior or comparable performance compared with the existing methods by addressing both annotation inconsistencies and class imbalance during multi-dataset joint training. It achieved state-of-the-art performance on RAF-DB and CAER-S datasets with accuracies of 92.44% and 98.22%, respectively, reflecting improvements of 0.2% and 3.65% over existing methods.

Representing Noisy Image Without Denoising.

A long-standing topic in artificial intelligence is the effective recognition of patterns from noisy images. In this regard, the recent data-driven paradigm considers 1) improving the representation robustness by adding noisy samples in training phase (i.e., data augmentation) or 2) pre-processing the noisy image by learning to solve the inverse problem (i.e., image denoising). However, such methods generally exhibit inefficient process and unstable result, limiting their practical applications. In this paper, we explore a non-learning paradigm that aims to derive robust representation directly from noisy images, without the denoising as pre-processing. Here, the noise-robust representation is designed as Fractional-order Moments in Radon space (FMR), with also beneficial properties of orthogonality and rotation invariance. Unlike earlier integer-order methods, our work is a more generic design taking such classical methods as special cases, and the introduced fractional-order parameter offers time-frequency analysis capability that is not available in classical methods. Formally, both implicit and explicit paths for constructing the FMR are discussed in detail. Extensive simulation experiments and robust visual applications are provided to demonstrate the uniqueness and usefulness of our FMR, especially for noise robustness, rotation invariance, and time-frequency discriminability.

Probability based dynamic soft label assignment for object detection

By defining effective supervision labels for network training, the performance of object detectors can be improved without incurring additional inference costs. Current label assignment strategies generally require two steps: first, constructing a positive sample candidate bag, and then designing labels for these samples. However, the construction of candidate bag of positive samples may result in some noisy samples being introduced into the label assignment process. We explore a single-step label assignment approach: directly generating a probability map as labels for all samples. We design the label assignment approach from the following perspectives: Firstly, it should be able to reduce the impact of noise samples. Secondly, each sample should be treated differently because each one matches the target to a different extent, which assists the network to learn more valuable information from high-quality samples. We propose a probability-based dynamic soft label assignment method. Instead of dividing the samples into positive and negative samples, a probability map, which is calculated based on prediction quality and prior knowledge, is used to supervise all anchor points of the classification branch. The weight of prior knowledge in the labels decreases as the network improves the quality of instance predictions, as a way to reduce noise samples introduced by prior knowledge. By using continuous probability values as labels to supervise the classification branch, the network is able to focus on high-quality samples. As demonstrated in the experiments on the MS COCO benchmark, our label assignment method achieves 40.9% AP in the ResNet-50 under 1x schedule, which improves FCOS performance by approximately 2.0% AP. The code has been available at https://github.com/Liyi4578/PDSLA.

MRCFN: A multi-sensor residual convolutional fusion network for intelligent fault diagnosis of bearings in noisy and small sample scenarios

Bearing fault diagnosis is of great importance to ensure the safe and stable operation of mechanical equipment. The actual collected bearing fault signals are susceptible to strong noise interference and bearing samples for each fault state may be insufficient, which increases the difficulty of capturing effective features. Most of the existing diagnostic methods extract features from a single sensor signal for pattern recognition and fault diagnosis. The fault information provided by a single sensor is limited and incomplete, which is usually very difficult to meet the demand for accurate and reliable fault diagnosis in complex scenarios. To solve these problems, this paper proposes a multi-sensor residual convolutional fusion network (MRCFN) for intelligent fault diagnosis of bearings. Firstly, a convolutional pooling module (CPM) is coupled with the designed double ring residual module (DRRM) to rough feature extraction and deep feature mining, which not only captures the discriminative fault features from multi-sensor signal, but also avoids the performance degradation of network. Secondly, a spatial channel reconstruction module (SCRM) is further introduced to eliminate redundant information in the features and improve the network training efficiency. Finally, the presented global interactive perception fusion module (GIPFM) is connected with a classification block (CB) to globally fuse the features extracted from the acoustic and vibration signals and conduct automatic fault identification, which both can realize the complementarity and calibration of multi-sensor feature information and high precision diagnosis. The experiments and a series of comparisons are implemented on two datasets to efficaciously verify the superiority of the proposed method over five existing representative multi-sensor fusion diagnosis methods (i.e., FAC-CNN, MB-CNN, MsACNN, MRSDF, and IMSFDFL) under strong noise and small samples.

SPORT: A Subgraph Perspective on Graph Classification with Label Noise

Graph neural networks (GNNs) have achieved great success recently on graph classification tasks using supervised end-to-end training. Unfortunately, extensive noisy graph labels could exist in the real world because of the complicated processes of manual graph data annotations, which may significantly degrade the performance of GNNs. Therefore, we investigate the problem of graph classification with label noise, which is demanding because of the complex graph representation learning issue and serious memorization of noisy samples. In this work, we present a novel approach called S ubgra p h Set Netw or k with Sample Selection and Consis t ency Learning (SPORT) for this problem. To release the overfitting of GNNs, SPORT proposes to characterize each graph as a set of subgraphs generated by certain predefined stratagems, which can be viewed as samples from its underlying semantic distribution in graph space. Then we develop an equivariant network to encode the subgraph set with the consideration of the symmetry group. To further release the influences of noisy examples, we leverage the predictions of subgraphs to measure the likelihood of a sample being clean or noisy, followed by effective label updating. In addition, we propose a joint loss to advance the model generalizability by introducing consistency regularization. Comprehensive experiments on a wide range of graph classification datasets demonstrate the effectiveness of our SPORT. Specifically, SPORT outperforms the most competing baseline by up to 6.4%.

On Combining Instance Selection and Discretisation: A Comparative Study of Two Combination Orders

Data discretisation focuses on converting continuous attribute values to discrete ones which are closer to a knowledge-level representation that is easier to understand, use, and explain than continuous values. On the other hand, instance selection aims at filtering out noisy or unrepresentative data samples from a given training dataset before constructing a learning model. In practice, some domain datasets may require processing with both discretisation and instance selection at the same time. In such cases, the order in which discretisation and instance selection are combined will result in differences in the processed datasets. For example, discretisation can be performed first based on the original dataset, after which the instance selection algorithm is used to evaluate the discrete type of data for selection, whereas the alternative is to perform instance selection first based on the continuous type of data, then using the discretiser to transfer the attribute type of values of a reduced dataset. However, this issue has not been investigated before. The aim of this paper is to compare the performance of a classifier trained and tested over datasets processed by these combination orders. Specifically, the minimum description length principle (MDLP) and ChiMerge are used for discretisation, and IB3, DROP3 and GA for instance selection. The experimental results obtained using ten different domain datasets show that executing instance selection first and discretisation second perform the best, which can be used as the guideline for the datasets that require performing both steps. In particular, combining DROP3 and MDLP can provide classification accuracy of 0.85 and AUC of 0.8, which can be regarded as the representative baseline for future related researches.

FQTrack:Object Tracking Method Based on a Feature-Enhanced Memory Network and Memory Quality Selection Mechanism

Visual object tracking technology is widely used in intelligent security, automatic driving and other fields, and also plays an important role in frontier fields such as human–computer interactions and virtual reality. The memory network improves the stability and accuracy of tracking by using historical frame information to assist in the positioning of the current frame in object tracking. However, the memory network is still insufficient in feature mining and the accuracy and robustness of the model may be reduced when using noisy observation samples to update it. In view of the above problems, we propose a new tracking framework, which uses the attention mechanism to establish a feature-enhanced memory network and combines cross-attention to aggregate the spatial and temporal context information of the target. The former introduces spatio-temporal adaptive attention and cross-spatial attention, embeds spatial location information into channels, realizes multi-scale feature fusion, dynamically emphasizes target location information, and obtains richer feature maps. The latter guides the tracker to focus on the area with the largest amount of information in the current frame to better distinguish the foreground and background. In addition, through the memory quality selection mechanism, the accuracy and richness of the feature samples are improved, thereby enhancing the adaptability and discrimination ability of the tracking model. Experiments on benchmark test sets such as OTB2015, TrackingNet, GOT-10k, LaSOT and UAV 123 show that this method achieves comparable performance with advanced trackers.

Time-series China urban land use mapping (2016–2022): An approach for achieving spatial-consistency and semantic-transition rationality in temporal domain

The global urbanization trend is geographically manifested through city expansion and the renewal of internal urban structures and functions. Time-series urban land use (ULU) maps are vital for capturing dynamic land changes in the urbanization process, giving valuable insights into urban development and its environmental consequences. Recent studies have mapped ULU in some cities with a unified model, but ignored the regional differences among cities; and they generated ULU maps year by year, but ignored temporal correlations between years; thus, they could be weak in large-scale and long time-series ULU monitoring. Accordingly, we introduce an temporal-spatial-semantic collaborative (TSS) mapping framework to generating accurate ULU maps with considering regional differences and temporal correlations. Firstly, to support model training, a large-scale ULU sample dataset based on OpenStreetMap (OSM) and Sentinel-2 imagery is automatically constructed, providing a total number of 56,412 samples with a size of 512 × 512 which are divided into six sub-regions in China and used for training different classification models. Then, an urban land use mapping network (ULUNet) is proposed to recognize ULU. This model utilizes a primary and an auxiliary encoder to process noisy OSM samples and can enhance the model's robustness under noisy labels. Finally, taking the temporal correlations of ULU into consideration, the recognized ULU are optimized, whose boundaries are unified by a time-series co-segmentation, and whose categories are modified by a knowledge-data driven method. To verify the effectiveness of the proposed method, we consider all urban areas in China (254,566 km2), and produce a time-series China urban land use dataset (CULU) at a 10-m resolution, spanning from 2016 to 2022, with an overall accuracy of CULU is 82.42%. Through comparison, it can be found that CULU outperforms existing datasets such as EULUC-China and UFZ-31cities in data accuracies, spatial boundaries consistencies and land use transitions logicality. The results indicate that the proposed method and generated dataset can play important roles in land use change monitoring, ecological-environmental evolution analysis, and also sustainable city development.

Semi-supervised adaptive anti-noise meta-learning for few-shot industrial gearbox fault diagnosis

Abstract Real-time and accurate predictive maintenance of industrial equipment is fundamental for ensuring the safety and stability of advanced manufacturing processes. Current fault diagnosis methods based on data mining rely on a large number of labeled samples, and obtaining sufficient labeled data for diagnosing industrial equipment faults is challenging. Meta-learning can achieve the diagnosis of few-shot samples to a certain extent, but the effect is not ideal. Semi-supervision can effectively leverage a large number of unlabeled samples, which is of great practical significance for handling scenarios involving limited labeled samples. However, noise interference can occur when unlabeled samples appear that do not belong to known categories. Therefore, this study proposes adaptive semi-supervised meta-learning networks for noisy few-shot gearbox fault diagnosis. Firstly, a residual network with a Morlet Wavelet layer is used to extract signal features. Next, sample-level attention is defined to select unlabeled samples that are more similar to labeled sample prototypes, thereby reducing the influence of noisy samples. The adaptive metric is used to obtain the relational distance functions of labeled samples and unlabeled samples. Adaptive semi-supervised meta-learning networks uses unlabeled data to refine prototypes for better fault diagnosis. The effectiveness and anti-noise performance of the proposed method are verified by using two gearbox datasets with various few-shot noise scenarios.

Leveraging GANs data augmentation for imbalanced medical image classification

Machine learning algorithms have been widely applied and researched in the field of medical image classification. Most of the current algorithms are designed based on the assumption of data balance. However, in practical applications, medical data often face the problem of imbalanced data. The traditional data augmentation approaches can balance the original data, but they are unable to generate more effective features. Generative Adversarial Networks (GANs) can effectively perform sample augmentation. However, GAN often encounter issues such as intra-class mode collapse and generating noisy samples when faced with imbalanced data. This study proposes a GAN model called IBGAN that focuses on generating intra-class sparse samples and class boundary samples. Specifically, IBGAN addresses the issue of imbalanced data through two stages. In the first stage, the Isolation Forest (iForest) algorithm and boundary sample detection algorithm are designed to identify sparse region samples and boundary samples within the intra-class data. This enables the GAN model to effectively focus on generating such samples during training. The second stage involves the refinement process of the samples. We propose a sample evaluation method based on Support Vector Data Description (SVDD) to filter out noisy data in the generated samples, ensuring the quality of the generated data. We conducted extensive and in-depth experiments on five real-world datasets. The experimental results demonstrate that IBGAN can generate high-quality and diverse augmented samples, which contribute to the improvement of classifier performance. We compared our proposed method with seven baseline methods, including traditional approaches and classic GAN models. The experimental results, including the visualization of the generated data and the evaluation of classification performance, consistently show that our proposed method achieves more competitive results.

A Noisy Sample Selection Framework Based on a Mixup Loss and Recalibration Strategy

Deep neural networks (DNNs) have achieved breakthrough progress in various fields, largely owing to the support of large-scale datasets with manually annotated labels. However, obtaining such datasets is costly and time-consuming, making high-quality annotation a challenging task. In this work, we propose an improved noisy sample selection method, termed “sample selection framework,” based on a mixup loss and recalibration strategy (SMR). This framework enhances the robustness and generalization abilities of models. First, we introduce a robust mixup loss function to pre-train two models with identical structures separately. This approach avoids additional hyperparameter adjustments and reduces the need for prior knowledge of noise types. Additionally, we use a Gaussian Mixture Model (GMM) to divide the entire training set into labeled and unlabeled subsets, followed by robust training using semi-supervised learning (SSL) techniques. Furthermore, we propose a recalibration strategy based on cross-entropy (CE) loss to prevent the models from converging to local optima during the SSL process, thus further improving performance. Ablation experiments on CIFAR-10 with 50% symmetric noise and 40% asymmetric noise demonstrate that the two modules introduced in this paper improve the accuracy of the baseline (i.e., DivideMix) by 1.5% and 0.5%, respectively. Moreover, the experimental results on multiple benchmark datasets demonstrate that our proposed method effectively mitigates the impact of noisy labels and significantly enhances the performance of DNNs on noisy datasets. For instance, on the WebVision dataset, our method improves the top-1 accuracy by 0.7% and 2.4% compared to the baseline method.

Adaptive Bregman–Kaczmarz: an approach to solve linear inverse problems with independent noise exactly

We consider the block Bregman–Kaczmarz method for finite dimensional linear inverse problems. The block Bregman–Kaczmarz method uses blocks of the linear system and performs iterative steps with these blocks only. We assume a noise model that we call independent noise, i.e. each time the method performs a step for some block, one obtains a noisy sample of the respective part of the right-hand side which is contaminated with new noise that is independent of all previous steps of the method. One can view these noise models as making a fresh noisy measurement of the respective block each time it is used. In this framework, we are able to show that a well-chosen adaptive stepsize of the block Bregman–Kaczmarz method is able to converge to the exact solution of the linear inverse problem. The plain form of this adaptive stepsize relies on unknown quantities (like the Bregman distance to the solution), but we show a way how these quantities can be estimated purely from given data. We illustrate the finding in numerical experiments and confirm that these heuristic estimates lead to effective stepsizes.